Nummer Datum Autor Titel Abstraact/MSC
2624 01.12.2010 Domschke, Pia
Kolb, Oliver
Lang, Jens
Adjoint-Based Control of Model and Discretization Errors for Gas and Water Supply Networks
2623 03.11.2010 Lindner, Marko
Roch, Steffen
Finite sections of random Jacobi operators MSC: 65J10; 47B36; 47B80
2622 10.10.2010 Domschke, Pia
Kolb, Oliver
Lang, Jens
Adjoint-Based Control of Model and Discretization Errors for Gas Flow in Networks
2621 19.10.2010 Farwig,Reinhard
Galdi, Giovanni P.
Kyed, Mads
Asymptotic Structure of a Leray Solution to the Navier-Stokes Flow Around a Rotating Body MSC: 35Q30; 76D05; 35B40
2620 09.08.2010 Sawada, Okihiro
Takada, Ryo
On the analyticity and the almost periodicity of the solution to the Euler equations with non-decaying initial velocity MSC: 35Q31; 76B03; 35B15
2619 14.06.2010 Farwig, Reinhard
Kozono, Hideo
Sohr, Hermann
Global weak solutions of the Navier-Stokes equations with nonhomogeneous boundary data and divergence MSC: 35Q30; 35J65; 76D05
2618 10.06.2010 Lindner, Marko
Roch, Steffen
On the integer points in a lattice polytope: n-fold Minkowski sum and boundary MSC: 52B20; 52C07; 65J10
2617 21.05.2010 Nesenenko, Sergiy Homogenization of Viscoplastic Models of Monotone Type with Positive Semi-Definite Free Energy
2616 14.05.2010 Kyed, Mads
Galdi,Giovanni P.
Steady-State Navier-Stokes Flows Past a Rotating Body: Leray Solutions are Physically Reasonable MSC: 35Q30; 76D05; 76U05
2615 14.05.2010 Kyed, Mads
Galdi,Giovanni P.
Asymptotic Behavior of a Leray Solution around a Rotating Obstacle MSC: 35Q30; 76D05; 76U05
2584 21.05.2010 Nesenenko, Sergiy
Neff, Patrizio
Well-posedness for dislocation based gradient visco-plasticity I: subdifferential case
2614 26.04.2010 Farwig, Reinhard
Sohr, Hermann
Varnhorn, Werner
Extensions of Serrin's uniqueness and regularity conditions for the Navier-Stokes equations MSC: 35Q30; 67D05
2613 13.04.2010 Farwig, Reinhard
Taniuchi, Yasushi
On the energy equality of Navier-Stokes equations in general unbounded domains MSC: 35Q35; 76D05
We present a sufficient condition for the energy equality of Leray-Hopf's weak solutions to the Navier-Stokes equations in general unbounded 3--dimensional domains.
2612 12.04.2010 Farwig, Reinhard
Morimoto, Hiroko
Leray's inequality for fluid flow in symmetric multi-connected two-dimensional domains MSC: 35Q30; 76D03; 76D05
2611 11.04.2010 Farwig, Reinhard
Kozono, Hideo
Yanagisawa, Taku
Leray's inequality in general multi-connected domains in $R^n$ MSC: 35Q35; 76D05
2610 06.04.2010 Ullmann, Sebastian
Lang, Jens
A POD-Galerkin Reduced Model with Updated Coefficients for Smagorinsky LES MSC: 76F65; 76D05; 76M25
2609 31.03.2010 Gottermeier, Bettina
Lang, Jens
Adaptive Two-Step Peer Methods for Thermally Coupled Incompressible Flow MSC: 65M99; 76D05; 76M10
2608 24.03.2010 Schieche, Bettina
Lang, Jens
Stochastic Analysis of Nusselt Numbers for Natural Convection with Uncertain Boundary Conditions MSC: 35R60; 65N35; 76D05
2607 04.03.2010 Kolb, Oliver
Domschke, Pia
Lang, Jens
Modified QR Decomposition to Avoid Non-Uniqueness in Water Supply Networks with Extension to Adjoint Calculus
2606 04.03.2010 Domschke, Pia
Kolb, Oliver
Lang, Jens
An Adaptive Model Switching and Discretization Algorithm for Gas Flow on Networks
2605 23.02.2010 Alber, Hans-Dieter
Zhu, Peicheng
Solutions to a model with Neumann boundary conditions for phase transitions driven by configurational forces MSC: 74N20; 35Q72
2604 22.02.2010 Rabinovich, Vladimir S.
Roch, Steffen
Finite sections of band-dominated operators on discrete groups
2603 10.02.2010 Roch, Steffen
Rabinovich, Vladimir S.
Exponential estimates of solutions of pseudodifferential equations with operator-valued symbols. Applications to Schrödinger operators with operator-valued potentials MSC: 35xx; 58Jxx; 81Q10
2602 04.02.2010 Farwig, Reinhard
Taniuchi, Yasushi
Uniqueness of almost periodic-in-time solutions to Navier-Stokes equations in unbounded domains MSC: 35Q30; 35Q35; 76D05
2601 03.02.2010 Okabe, Takahiro Periodic solutions of the Navier-Stokes equations with inhomogeneous boundary conditions MSC: 35Q30; 76D05
Nummer Datum Autor Titel Abstract/MSC
2600 21.12.2009 Otto, Martin Avoiding Incidental Homomorphisms Into Guarded Covers
2599 21.12.2009 Otto, Martin Acyclicity in Hypergraph Covers
2598 15.12.2009 Farwig, Reinhard
Necasova, Sarka
Neustupa, Jiri
Spectral Analysis of a Stokes-Type Operator Arising from Flow around a Rotating Body MSC: 35Q35; 35P99; 47A10; 76D07
2597 05.01.2010 Nesenenko, Sergiy L^q-almost Solvability of Viscoplastic Models of Monotone Type
2596 04.12.2009 Roch, Steffen Spatial discretization of restricted group algebras
2595 20.11.2009 Clever, Debora
Lang, Jens
Optimal Control of Radiative Heat Transfer in Glass Cooling with Restrictions on the Temperature Gradient MSC: 35K10; 35K58; 35R15; 65M99; 35Q80; 35Q93; 65Z05
2594 18.11.2009 Gruber, Peter
Knees, Dorothee
Nesenenko, Sergiy
Thomas, Marita
Analytical and numerical aspects of time-dependent models with internal variables MSC: 74C05; 74C10; 49N60; 65M60
2593 06.10.2009 Debrabant, Kristian
Jakobsen, Espen R.
Semi-Lagrangian schemes for linear and fully non-linear diffusion equations MSC: 65M12; 65M15; 65M06; 35K10; 35K55; 35K65; 49L25; 49L20
2592 01.10.2009 Gottermeier, Bettina
Lang, Jens
Adaptive Two-Step Peer Methods for Incompressible Navier-Stokes Equations MSC: 65M99; 76D05; 76M10
2591 26.08.2009 Farwig, Reinhard
Hishida, Toshiaki
Leading term at infinity of steady Navier-Stokes flow around a rotating obstacle MSC: 35B40; 35Q30; 76D05
2590 23.06.2009 Ali Mehmeti, Felix
Haller-Dintelmann, Robert
Régnier, Virginie
The Klein-Gordon equation with multiple tunnel effect on a star-shaped network: expansions in generalized eigenfunctions MSC: 34B45(Primary); 42A38; 47A10; 47A60; 47A70(Secondary)
2589 16.06.2009 Debrabant, Kristian Runge-Kutta methods for third order weak approximation of SDEs with multidimensional additive noise MSC: 65C30; 60H35; 65C20; 68U20
2588 15.06.2009 Spatial discretization of $C^*$-algebras Regularity of weak solutions to the Navier-Stokes equations in exterior domains MSC: 76D05; 35Q30; 35B65
2587 28.05.2009 Roch, Steffen Spatial discretization of $C^*$-algebras
2586 28.05.2009 Ebobisse, Francois
Neff, Patrizio
Existence and uniqueness for rate-independent infinitesimal gradient plasticity with isotropic hardening and plastic spin MSC: 74C05; 49J40; 49J52; 35J25; 35Q72
2585 20.05.2009 Farwig, Reinhard
Neustupa, Jiri
Spectral Properties in $L^q$ of an Oseen Operator Modelling Fluid Flow past a Rotating Body MSC: 35Q35; 35P99; 76 D 07 35\,P\,99, 76\,D\,07
2583 17.04.2009 Mößner, Bernhard
Reif, Ulrich
Stability of Tensor Product B-Splines on Domains
2582 17.04.2009 Hechler, Jochen
Mößner, Bernhard
Reif, Ulrich
$C^1$-Continuity of the Generalized Four-Point Scheme
2581 17.04.2009 Ulrich Reif Polynomial Approximation on Domains Bounded by Diffeomorphic Images of Graphs
2580 17.04.2009 Mößner, Bernhard
Reif, Ulrich
Error Bounds for Polynomial Tensor Product Interpolation We provide estimates for the maximum error of polynomial tensor product interpolation on regular grids in $\mathbb{R}$^d. The set of partial derivatives required to form these bounds depends on the clustering of interpolation nodes. Also bounds on the partial derivatives of the error are derived.
2579 17.04.200 App, Andreas
Reif, Ulrich
Piecewise Linear Orthogonal Approximation We derive Sobolev-type inner products with respect to which hat functions on arbitrary triangulations of domains in $\mathbb{R}^d$ are orthogonal. Compared with linear interpolation, the resulting approximation schemes yield superior accuracy at little extra cost.
2578 17.04.2009 Farwig, Reinhard
Hishida, Toshiaki
Asymptotic profile of steady Stokes flow around a rotating obstacle MSC: 35Q30; 35Q35; 35B40; 76D07
2577 09.03.2009 Steffen Roch
Pedro A. Santos
Bernd Silbermann
A sequence algebra of finite sections, convolution and multiplication operators on $L^p(R)$ MSC: 65R20
2576 25.02.2009 Hofmann, Karl H.
Morris, Sidnay A.
The Structure of Almost Connected Pro-Lie Groups
2575 16.02.2009 Riechwald, Paul Felix
Schumacher, Katrin
A Large Class of Solutions for the Instationary Navier-Stokes System MSC: 35Q30; 76D05; 76D07
2574 05.02.2009 Neff, Patrizio
Jeong, Jena
Fischle, Andreas
Stableidentification of linear isotropic Cosserat parameters: bounded stiffness in bending and torsion implies conformal invariance of curvature
2573 21.01.2009 Farwig, Reinhard
Kozono, Hideo
Sohr, Hermann
Global Weak Solutions of the Navier-Stokes System with Nonzero Boundary Conditions MSC: 76D05; 35Q30; 35J65
2572 19.01.2009 Rabinovich, Vladimir
Roch, Steffen
Essential spectra and exponential estimates of eigenfunctions of lattice operators of quantum mechanics MSC: 39A47; 47B39; 81Q10
Nummer Datum Autor Titel Abstract/MSC
2571 22.12.2008 Ferreira, Carlos
Günther, Ute
Martin, Alexander
Mathematical Models and Polyhedral Studies for Integral Sheet Metal Design
2570 05.12.2008 Huang, Weizhang
Kamenski, Lennard
Lang, Jens
A New Anisotropic Mesh Adaptation Method Based upon Hierarchical A Posteriori Error Estimates MSC: 65N50; 65N30; 65N15
2569 21.11.2008 Alber, Hans-Dieter
Nesenenko, Sergiy
Local and global regularity in time dependent viscoplasticity
2568 31.10.2008 Neff, Patrizio
Jeong, Jena
Ramezani, Hamid
Subgrid interaction and micro-randomness – novel invariance requirements in infinitesimal gradient elasticity
2567 31.10.2008 Alber, Hans-Dieter
Nesenenko, Sergiy
Justification of homogenization in viscoplasticity: From convergence on two scales to an asymptotic solution in ${L^2(\Omega)}$
2566 04.01.2010 Nesenenko, Sergiy A Note on Existence Result for Viscoplastic Models with Nonlinear Hardening
2565 14.10.2008 Ehrhardt, Torsten
Roch, Steffen
Silbermann, Bernd
The Strong Szegö-Widom Limit Theorem for operators with almost periodic diagonal MSC: 47B35; 47B37; 47N40
2564 13.10.2008 Debrabant, Kristian
Kværnø, Anne
Stochastic Taylor Expansions: Weight functions of B-series expressed as multiple integrals MSC: 65C30; 60H10
The exact solution of stochastic differential equations can be expressed as stochaastic B-series. In this paper, we present an algorithm using rooted trees for expanding the weight functions occurring in this representation in terms of multiple integrals using multi-indices.
2563 13.10.2008 Wille, Rudolf
Wille-Henning, Renate
The Mathematical in Music Thinking 00A05
2562 26.10.2008 Kolb, Oliver
Domschke, Pia
Lang, Jens
Moving Penalty Functions for Optimal Control with PDEs on Networks An adaptive penalty technique to find feasible solutions of mixed integer nonlinear optimal control problems on networks is introduced. This new approach is applied to problems arising in the operation of gas and water supply networks.
2561 30.09.2008 Scheffold, Egon Interessante Kongruenzen im Zusammenhang mit den Formeln von Abel und Barlow We derive special congruences from the formulas of Abel and Barlow and we remember of König's theory.
2560 12.09.2008 Farwig, Reinhard
Sohr, Hermann
Varnhorn, Werner
On optimal initial value conditions for local strong solutions of the Navier-Stokes equations MSC: 35Q30; 76D05
2559 07.09.2008 Neff, Patrizio
Jeong, Jena
A new paradigm: the linear isotropic Cosserat model with conformally invariant curvature energy.
2558 27.08.2008 Jeong, Jena
Ramezani, Hamid
Münch, Ingo
Neff, Patrizio
Simulation of linear isotropic Cosserat elasticity with conformally invariant curvature
2557 05.08.2008 Neff, Patrizio
Chelminski, Krzysztof
$H^1_{loc}$-stress and strain regularity in Cosserat plasticity
2556 05.08.2008 Neff, Patrizio
Jeong, Jena
Münch, Ingo
Ramezani, Hamid
Mean field modeling of isotropic random Cauchy elasticity versus microstretch elasticity.
2555 05.08.2008 Neff, Patrizio
Hong, Kwon-Il
Jeong, Jena
The Reissner-Mindlin plate is the $\Gamma$-limit of Cosserat elasticity.
2554 12.069.2008 Bales, Pia
Kolb, Oliver
Lang, Jens
Hierarchical modelling and model adaptivity for gas flow on networks MSC: 76N25; 65K99; 65M99
2553 26.06.2008 Kolb, Oliver
Lang, Jens
Bales, Pia
Adaptive linearization for the optimal control problem of gas flow in pipeline networks MSC: 90C35; 65K99; 65M12
2552 09.03.2008 Bales, Pia
Geißler, Björn
Kolb, Oliver
Lang, Jens
Martin, Alexander
Morsi, Antonio
Combination of Nonlinear and Linear Optimization of Transient Gas Networks MSC: 76N25; 90C11; 90C30; 90C90
2551 11.09.2008 Delia Teleaga
Jens Lang
Numerically Solving Maxwell’s Equations. Implementation Issues for Magnetoquasistatics Having experienced of couple of tricky issues during the implementation of edge elements within the fully space-time adaptive PDE solver KARDOS to solve magnetoquasistatic problems we found it useful to share our exciting learning process with interested readers and beginners.
2549 01.04.2008 Wille, Rudolf Formal Concept Analysis and Contextual Logic MSC: 03B; 03B
2548 26.05.2008 Reinhard Farwig
Jiri Neustupa
Patrick Penel
Vorticity, Rotation and Symmetry -- Stabilizing and Destabilizing Fluid Motion Vorticity, Rotation and Symmetry – Stabilizing and Destabilizing Fluid Motion
2547 26.05.2008 Neeb, Karl-Hermann Semi-bounded unitary representations of infinite-dimensional Lie groups Semi-bounded unitary representations of infinite-dimensional of Lie groups
2550 05.08.2008 Jeong, Jena
Neff, Patrizio
Existence, uniqueness and stability in linear Cosserat elasticity for weakest curvature conditions
2546 28.04.2008 Farwig, Reinhard
Sohr, Hermann
The Largest Possible Initial Value Space for Local Strong Solutions of the Navier-Stokes Equations in General Domains The Largest Possible Initial Vaue Space for Local Strong Solutions of the Navier-Stokes Equations in General Domains
2545 24.04.2008 Roch, Steffen Spatial discretization of Cuntz algebras Spatial discretization of Cuntz algebras
2544 28.04.2008 Alber, Hans-Dieter
Zhu, Peicheng
Interface motion by interface diffusion driven by bulk energy: justification of a diffusive interface model Interface motion by interface diffusion driven by bulk energy: justification of a diffusive interface model
2543 28.04.2008 Rabinovich, Vladimir
Roch, Steffen
Agmon's type estimates of exponential behavior of solutions of systems of partial differential equations. Applications to Schrödinger, Moisil-Theodorescu and Dirac operators. Agmon's type estimates of exponential behaviour of solutionsof systems of elliptic partial differential equations. Applications to Schrödinger, Moisil-Theodorescu and Dirac operators.
2542 18.03.2008 Neeb, Karl-Hermann
Vizman, Cornelia
An abstract setting for hamiltonian actions MSC: 17B56; 35Q53
In this paper we develop an abstract setup for hamiltonian group actions as follows: Starting with a continuous $2$-cochain $\omega$ on a Lie algebra $h$ with values in an $h$-module $V$, we associate subalgebras $sp(h,\omega) \supseteq ham(h,\omega)$ of symplectic, resp., hamiltonian elements. Then $ham(h,\omega)$ has a natural central extension which in turn is contained in a larger abelian extension of $sp(h,\omega)$. In this setting, we study linear actions of a Lie group $G$ on $V$ which are compatible with a homomorphism $g \rightarrow ham(h,\omega)$, i.e. abstract hamiltonian actions, corresponding central and abelian extensions of $G$ and momentum maps $J : g \rightarrow V$.
2541 18.03.2008 Karl-Hermann Neeb Lie group extensions associated to projective modules of continuous inverse algebras MSC: 22E65; 58B34
We call a unital locally convex algebra $A$ a continuous inverse algebra if its unit group $A^\times$ is open and inversion is a continuous map. For any smooth action of a, possibly infinite-dimensional, connected Lie group $G$ on a continuous inverse algebra $A$ by automorphisms and any finitely generated projective right $A$-module $E$, we construct a Lie group extension $\hat G$ of $G$ by the group $GL_A(E)$ of automorphisms of the $A$-module $E$. This Lie group extension is a ``non-commutative'' version of the group $Aut(V)$ of automorphism of a vector bundle over a compact manifold $M$, which arises for $G = Diff(M)$, $A = C^\infty(M,C)$ and $E = \Gamma V$. We also identify the Lie algebra $\hat g$ of $\hat G$ and explain how it is related to connections of the $A$-module $E$.
2540 18.03.2008 Rudolf Wille Concept Graphs as Semantic Structures for Contextual Judgment Logic MSC: 03B
This paper presents a mathematization of the philosophical dcotrine of judgments as an extension of the mathem atization of the philosophical doctrine of concepts delveloped in Formal Concept Analysis. The chosen approach was strongly sti mulated by J.F. Sowa's theory of conceptual graphs. The mathematized conceptual graphs, called concept graphs, are mathematical semantic structures based on formal contexts and their formal concepts; those semantic structures are viewed as formal judgmen ts in the underlying Contextual Judgment Logic. In this papper concept graphs are systematically built up with simple concepts graphs in section 2 and continuing with existential graphs in section 3, with implicational and clausal concept graphs in secti on 4, and finally with generalizations of concept graphs in section 5. Examples are illustrating the different types of concept graphs.
2539 18.03.2008 Rudolf Wille An Algebraization of Linear Continuum Structures MSC: 06F
This paper continues the approach of developing an order-theoretic structure theory of one-dimensional continu um structures as elaborated in [Wi07] (see also [Wi83],[Wi03]. The aim is to extend the order-theoretic structure theory by a m eaningful algebraization; for this, we concentrate on the real linear continuum structure with its derived concept lattice whic h gives rise to the so-called „real half-numbers“. The algebraization approaches an ordered algebraic structure on the set of a ll real half-numbers to make the continuum structure ot the reals more transparent and tractable.
2538 18.03.2008 Alber, H.-D.
Ramm, A. G.
Asymptotics of the solution to Robin problem MSC: 35J10; 35J15
Convergence of the solution to the exterior Robin problem to the solution of the Dirichlet problem, as the impedance tends to infinity, is proved. The rate of convergence is established. A method for deriving higher order terms of the asymptotics of the solution is given.
2537 18.03.2008 Alber, Hans-Dieter
Nesenenko, Sergiy
Local $H^1$--regularity and $H^{1/3-\delta}$--regularity up to the boundary in time dependent viscoplasticity MSC: 35B65; 35D10; 74C10; 74D10; 35J25; 34G20; 34G25; 47H04; 47H05
Local and boundary regularity for quasistatic initial-boundary value problems from viscoplasticity is studied. The problems considered belong to a general class with monotone constitutive equations modelling materials showing kinematic hardening. A standard example is the Melan-Prager model. It is shown that the strain/stress/internal variable fields have $H^{1+1/3-\delta}/H^{1/3-\delta}/H^{1/3-\delta}$ regularity up to the boundary. The proof uses perturbation estimates for monotone operator equations.
2536 01.03.2008 Hofmann, Karl Heinrich
Neeb, Karl-Hermann
Solvable Subgroups of Locally Compact Groups MSC: 22A05; 22D05; 22E15
It is shown that a closed solvable subgroup of a connected Lie groupis compactly generated. In particular, every discrete solvable subgroupof a connected Lie group is finitely generated.Generalizations to locally compact groupsare discussed as far as they carry.
Nummer Datum Autor Titel Abstract/MSC
2535 14.12.2007 Wille, Rudolf Communicative Rationality, Logic, and Mathematics MSC: 00-99
In this article the following thesis is explained and substantiated: Sense and meaning of mathematics finally lie in the fact that mathematics is able to report the rational communication of humans. The essence of the argumentation is that the effective support becomes possible by the close connection between mathematics and logic (in the sense of Peirce's latest philosophy) by which, in his turn, the communicative rationality (in the sense of Habermas' theory of communicative action) can be activated. How such a support may be concretely performed shall be illustrated by the development of a retrieval system for the library of the Center of Interdisciplinary Technology Research at Darmstadt University of Technology.
2534 29.11.2007 Clemens, Markus
Lang, Jens
Teleaga, Delia
Wimmer, Georg
Adaptivity in Space and Time for Magnetoquasistatics MSC: 65M60; 65L06; 78M10
This paper addresses fully space-time adaptive magnetic field computations. We describe an adaptive Whitney finite element method for solving the magnetoquasistatic formulation of Maxwell's equations on unstructured 3D tetrahedral grids. Spatial mesh refinement and coarsening are based on hierarchical error estimators especially designed for combining tetrahedral H(curl)-conforming edge elements in space with linearly implicit Rosenbrock methods in time. An embedding technique is applied to get efficiency in time through variable time steps. Finally, we present numerical results for the magnetic recording write head benchmark problem proposed by the Storage Research Consortium in Japan.
2533 10.12.2007 Farwig, Reinhard
Sohr, Hermann
Optimal Initial Value Conditions for the Existence of Local Strong Solutions of the Navier-Stokes Equations MSC: 35Q30; 76D05; 35B65
Consider the instationary Navier-Stokes system in a smooth bounded domain $\Omega\subset R^3$ with vanishing force and initial value $u_0\in L^2_\sigma(\Omega)$. Since the work of Kiselev-Ladyzhenskaya in 1963 there have been found several conditions on $u_0$ to prove the existence of a unique strong solution $u\in L^s(0,T; L^q(\Omega))$ with $u(0) = u_0$ in some time interval $[0,T)$, $0 < T \leq \infty$, where the exponents $2 < s < \infty$, $3 < q < \infty$ satisfy $\frac{2}{s} + \frac{3}{q} = 1$. Indeed, such conditions could be weakened step by step, thus enlarging the corresponding solution classes. Our aim is to prove the following optimal result with the weakest possible initial value condition and the largest possible solution class: Given $u_0,¸q,¸s$ as above and the Stokes operator $A_q$, we prove that the condition $\int_0^\infty \| e^{-tA_q}u_0\|_q^s¸ dt < \infty$ is necessary and sufficient for the existence of such a strong solution $u$. The proof rests on arguments from the recently developed theory of very weak solutions.
2532 03.11.2007 Debrabant, Kristian
Rö{ß}ler, Andreas
Diagonally Drift--Implicit Runge--Kutta Methods of Weak Order One and Two for It{ô} SDEs and Stability Analysis MSC: 65C30; 60H35; 65C20; 68U20
Families of first and second order diagonally drift--implicit SRK (DDISRK) methods for the weak approximation of SDEs contained in the class of SRK methods proposed by R{ö}{ß}ler are calculated. Their asymptotic stability as well as mean--square stability (MS--stability) properties are studied for a linear stochastic test equation with multiplicative noise. The stability functions for the DDISRK methods are determined and their domains of stability are compared to the corresponding domain of stability of the considered test equation. Stability regions are presented for various coefficients of the families of DDISRK methods in order to determine step size restrictions such that the numerical approximation reproduces the characteristics of the solution process.
2531 27.11.2007 Wille, Rudolf Generalistic Mathematics as Mathematics for the General Public MSC: 0099
What mathematics could and should mean for humans in general may only be clarified in a broader process of communication and understanding. This process of understanding needs a general culture of discourse which should not only be restricted to the discourse between mathematicians, but as a matter of principle should include all humans whether they are actively concerned with mathematics or only been affected by consequences of mathematical developments. Such a culture of discourse is dependent on a 'generalistic mathematics' which makes understandable the conception of mathematics, its connection to the world, and sense, meaning, and connection of mathematical acitivities; moreover, generalistic mathematics is guided by the idea of an open, meaningful, communicative and critical mathematics.
2530 27.11.2007 Wille, Rudolf Logisch denken lernen im Mathematikunterricht MSC: 97
Logisch denken lernen im Mathematikunterricht wird jeweils getragen von konkret-realer, philsosophisch-logischer und mathematischer Semantik.
2529 27.11.2007 Wille, Rudolf Formal Concept Analysis as Applied Lattice Theory MSC: 06A
Formal Concept Analysis is a mathematical theory of concept hierarchies which is based on Lattice Theory. It has been developed to support humans in their thought and knowledge. The aim of this paper is to show how successful the lattice-theoretic foundation can be in applying Formal Concept Analysis in a wide range. This is demonstraded in three sections dealing with representation, processing and measurement of conceptual knowledge. Finally, further relationships between abstract Lattice Theory and Formal Concept Analysis are briefly discussed.
2528 15.11.2007 Debrabant, Kristian
Rö{ß}ler, Andreas
Families of efficient second order Runge-Kutta methods for the weak approximation of Itô stochastic differential equations MSC: 65C30; 60H35; 65C20; 68U20
Recently, a new class of second order Runge-Kutta methods for Itô stochastic differential equations with a multidimensional Wiener process was introduced by Rö{ß}ler. In contrast to second order methods earlier proposed by other authors, this class has the advantage that the number of function evaluations depends only linearly on the number of Wiener processes and not quadratically. In this paper, we give a full classification of the coefficients of all explicit methods with minimal stage number. Based on this classification, we calculate the coefficients of an extension with minimized error constant of the well-known RK32 method to the stochastic case. For three examples, this method is compared numerically with known order two methods and yields very promising results.
2527 14.11.2007 Farwig, Reinhard
Kozono, Hideo
Sohr, Hermann
Very weak, weak and strong solutions to the instationary Navier-Stokes system MSC: 35Q30; 35B65; 76D05; 76D07
In this survey paper we discuss the theory of very weak solutions to the stationary and instationary (Navier-)Stokes system in a bounded domain of $R^3$ and show how this new notion of solutions may be used to prove regularity locally or globally in space and time of a given weak solution.
2526 19.09.2007 Schumacher, Katrin The Instationary Navier-Stokes Equations in Weighted Bessel-Potential Spaces MSC: 35Q30; 35D05
We investigate the solvability of the instationary Navier-Stokes equations with fully inhomogeneous data in a bounded domain. The class of solutions is contained in the space variable in a Bessel-Potential space weighted with a Muckenhoupt weight. In this context we derive solvability for small data, where this smallness can be realized by the restriction on a short time interval. Depending on the order of this Bessel-Potential space we are dealing with strong solutions, weak solutions, or with very weak solutions.
2525 11.09.2007 Schumacher, Katrin The Instationary Stokes Equations in Weighted Bessel-Potential Spaces MSC: 35Q30; 35D05
We investigate the solvability of the instationary Stokes equations with fully inhomogeneous data in a weighted Bessel-Potential space. Depending on the order of this Bessel-Potential space we are dealing with strong solutions or with very weak solutions. Whereas in the context of lowest regularity one obtains solvability with respect to inhomogeneous data by dualization, this is more delicate in the case of higher regularity, where one has to introduce some additional time regularity. As a preparation, we introduce a generalization of the Stokes operator that is appropriate to the context of very weak solutions in weighted Bessel-Potential spaces. \end{abstract} \emph{Key Words and Phrases:} Instationary Stokes equations, Muckenhoupt weights, very weak solutions, Bessel-Potential spaces, nonhomgeneous data
2524 23.08.2007 Schumacher, Katrin The Stationary Navier-Stokes Equations in Weighted Bessel-Potential Spaces MSC: 35Q30; 35D05; 76D05; 35J65
We investigate the stationary Navier-Stokes equations in Bessel-potential spaces with Muckenhoupt weights. Since in this setting it is possible that the solutions do not posses any weak derivatives, we use the notation of very weak solutions introduced by Amann [1]. The basic tool is complex interpolation, thus we give a characterization of the interpolation spaces of the spaces of data and solutions. Then we establish a theory of solutions to the Stokes equations in weighted Bessel-potential spaces and use this to prove solvability of the Navier-Stokes equations for small data by means of Banach's Fixed Point Theorem.
2523 23.07.2007 Schumacher, Katrin Very Weak Solutions to the Stationary Stokes and Stokes Resolvent Problem in Weighted Function Spaces MSC: 35Q30; 35D05; 76D07; 35J25
We investigate very weak solutions to the stationary Stokes and Stokes resolvent problem in function spaces with Muckenhoupt weights. The notion used here is similar but even more general than the one used in [2] or [14]. Consequently the class of solutions is enlarged. To describe boundary conditions we restrict ourselves to more regular data. We introduce a Banach space that admits a restriction operator and that contains the solutions according to such data.
2522 16.07.2007 Lang, Jens
Teleaga, Delia
Towards a Fully Space-Time Adaptive FEM for Magnetoquasistatics MSC: 65M60; 78M10
This paper is concerned with fully space-time adaptive magnetic field computations. We describe a Whitney finite element method for solving the magnetoquasistatic formulation of Maxwell's equations on unstructured 3D tetrahedral grids. Spatial discretization is done by employing hierarchical tetrahedral H(curl)-conforming elements proposed by Ainsworth and Coyle. For the time discretization, we use a newly constructed one-step Rosenbrock method ROS3PL with 3rd order accuracy in time. Adaptive mesh refinement and coarsening are based on hierarchical error estimators especially designed for Rosenbrock methods. An embedding technique is applied to get efficiency in time through variable time steps. Finally, we present numerical results for the benchmark problem TEAM 7.
2521 18.07.2007 Farwig, Reinhard
Kozono, Hideo
Sohr, Hermann
Energy-Based Regularity Criteria for the Navier-Stokes Equations MSC: 35Q30; 76D05; 35B65
We present several new regularity criteria for weak solutions $u$ of the instationary Navier-Stokes system which additionally satisfy the strong energy inequality. (i) If the kinetic energy $1/2 \| u(t) \|_2^2$ is Hölder continuous as a function of time $t$ with Hölder exponent $\alpha \in (1/2,1)$, then $u$ is regular. (ii) If the dissipation energy satisfies the left-side condition $\liminf_{\delta \to 0} \delta^{-\alpha} \int_{t-\delta}^t \| \na u\|_2^2 ¸ d\tau < \infty$, $\alpha \in (1/2,1)$, for all $t$ of the given time interval, then $u$ is regular. The proofs use local regularity results which are based on the theory of very weak solutions and on uniqueness arguments for weak solutions. Finally, in the last section, we mention a local space-time regularity condition.
2520 29.06.2007 Neff, Patrizio
Knees, Dorothee
Regularity up to the boundary for nonlinear elliptic systems arising in time-incremental infinitesimal elasto-plasticity MSC: 74C05; 35B65; 49N60; 74A35; 74G40
In this note we investigate the question of higher regularity up to the boundary for quasilinear elliptic systems which origin from the time-discretization of models from infinitesimal elasto-plasticity. Our main focus lies on an elasto-plastic Cosserat model. More specifically we show that the time discretization renders $H^2$-regularity of the displacement and $H^1$-regularity for the symmetric plastic strain $\varepsilon_p$ up to the boundary provided the plastic strain of the previous time step is in $H^1$, as well. This result contrasts with classical Hencky and Prandtl-Reuss formulations where it is known not to hold due to the occurrence of slip lines and shear bands. Similar regularity statements are obtained for other regularizations of ideal plasticity like viscosity or isotropic hardening. In the first part we recall the time continuous Cosserat elasto-plasticity problem, provide the update functional for one time step and show various preliminary results for the update functional (Legendre-Hadamard/monotonicity). Using non standard difference quotient techniques we are able to show the higher global regularity. Higher regularity is crucial for qualitative statements of finite element convergence. As a result we may obtain estimates linear in the mesh-width $h$ in error estimates.
2519 10.06.2007 Abels, Helmut
Krbec, Miroslav
Schumacher, Katrin
On the Trace Space of a Sobolev Space with a Radial Weight MSC: 46E35; 46E30
Our concern in this paper lies with trace spaces for weighted Sobolev spaces, when the weight is a power of the distance to a point at the boundary. For a large range of powers we give a full description of the trace space.
2518 04.06.2007 Neff, Patrizio
Fischle, Andreas
Muench, Ingo
Symmetric Cauchy stresses do not imply symmetric Biot strains in weak formulations of isotropic hyperelasticity with rotational degrees of freedom. MSC: 74A35; 74B20
We show that symmetric Cauchy stresses do not imply symmetric Biot strains in weak formulations of finite isotropic hyperelasticity with exact rotational degrees of freedom. This is contrary to claims in the literature which are valid, however, in the linear isotropic case.
2517 22.05.2007 Billig, Yuly
Neeb, Karl-Hermann
On the cohomology of vector fields on parallelizable manifolds MSC: 17B56; 17B65; 17B68
In the present paper we determine for each parallelizable smooth compact manifold ${\sst M}$ the cohomology spaces ${\sst H^2({\cal V}_M,\oline\Omega^p_M)}$ of the Lie algebra ${\sst {\cal V}_M}$ of smooth vector fields on ${\sst M}$ with values in the module ${\sst \oline\Omega^p_M = \Omega^p_M/d\Omega^{p-1}_M}$. The case of ${\sst p=1}$ is of particular interest since the gauge algebra ${\sst C^\infty (M,\k)}$ has the universal central extension with center ${\sst \oline\Omega^1_M}$, generalizing affine Kac-Moody algebras. The second cohomology ${\sst H^2(\V_M, \oline\Omega^1_M)}$ classifies twists of the semidirect product of ${\sst \V_M}$ with the universal central extension ${\sst C^\infty (M,\k) \oplus \oline\Omega^1_M}$.
2516 21.05.2007 Rabinovich, Vladimir S.
Roch, Steffen
Fredholm properties of band-dominated operators on periodic discrete structures MSC: 47B36; 47B39; 47A53
Let $(X, \sim)$ be a combinatorial graph the vertex set $X$ of which is a discrete metric space. We suppose that a discrete group $G$ acts freely on $(X, \sim)$ and that the fundamental domain with respect to the action of $G$ contains only a finite set of points. A graph with these properties is called periodic with respect to the group $G$. We examine the Fredholm property and the essential spectrum of band-dominated operators acting on the spaces $l^p(X)$ or $c_0(X)$, where $(X, \sim)$ is a periodic graph. Our approach is based on the thorough use of band-dominated operators. It generalizes the results obtained by the authors and B. Silbermann in the special case $X = G = \sZ^n$ and by J. Roe in case $X = G$ is a general finitely generated discrete group.
2515 01.05.2007 Scheffold, Egon Kongruenzen im Zusammenhang mit den Formeln von Barlow und Abel
2514 05.05.2007 Mößner, Bernhard
Reif, Ulrich
Stability of B-Splines on Bounded Domains We construct a uniformly stable family of bases for tensor product spline approximation on bounded domains in $\R^d$. These bases are derived from the standard B-spline basis by normalization with respect to the $L^p$-norm and a selection process relying on refined estimates for the de Boor-Fix functionals.
2513 01.05.2007 Debrabant, Kristian
Rößler, Andreas
Continuous Runge--Kutta methods for Stratonovich stochastic differential equations MSC: 65C30; 60H35; 65C20; 68U20
In this article we give order conditions for continuous stochastic Runge--Kutta methods of second order for the weak approximation of Stratonovich stochastic differential equations. As an example, by using these order conditions, two time discrete order two SRK schemes are extended to continuous schemes. Finally, numerical examples confirm our theoretical results.
2512 09.05.2007 Debrabant, Kristian
Lang, Jens
On Global Error Estimation and Control for Parabolic Equations MSC: 65M15; 65M06; 65M20; 65M60
The aim of this paper is to extend the global error estimation and control addressed in Lang and Verver [SIAM J. Sci. Comput., 2007] for initial value problems to parabolic partial differential equations. The classical ODE approach based on the first variational equation is combined with an estimation for the PDE spatial truncation error to estimate the overall error in the computed solution. Control is achieved through tolerance proportionality and uniform mesh refinement. Numerical examples are used to illustrate the reliability of the estimation and control strategies.
2511 01.05.2007 Schumacher, Katrin Solutions to the Equation $\div u=f$ in Weighted Sobolev Spaces MSC: 35F15
We consider the problem $\div u=f$ in a bounded Lipschitz domain $\Omega$, where $f$ with $\int_\Omega f=0$ is given. It is shown that the solution $u$, that is constructed as in Bogowski's approach in [1] fulfills estimates in the weighted Sobolev spaces $W^{k,q}_{w}(\Omega)$, where the weight function $w$ is contained in the class of Muckenhoupt weights $A_q$.
2510 02.05.2007 Schumacher, Katrin A Chart Preserving the Normal Vector and Extensions of Normal Derivatives in Weighted Function Spaces MSC: 47A20; 35A99; 46E35
Given a domain $\Omega$ of class $C^{k,1}$, $k\in \N$ we construct a chart that maps normals to the boundary of the half space to normals to the boundary of $\Omega$ in the sense that $\frac\pa{\pa x_n}\alpha(x',0)= – N(x')$ and that still is of class $C^{k,1}$. As an application we prove the existence of a continuous extension operator for all normal derivatives of order 0 to $k$ on domains of class $C^{k,1}$. The construction of this operator is performed in weighted function spaces where the weight function is taken from the class of Muckenhoupt weights.
2509 14.03.2007 Heck, Horst Stability Estimates for the inverse conductivity problem for less regular conductivities MSC: 35R30; 35J25
We prove a $\log$-type stability estimate for the inverse conductivity problem in space dimension $n\geq 3$, if the conductivity has $C^{3/2+\varepsilon}$ regularity.
2508 19.03.2007 Teleaga, Ioan
Lang, Jens
Higher-order linearly implicit one-step methods for three-dimensional incompressible Navier-Stokes equations MSC: 76D05; 76M10
In this work higher-order methods for integrating the three-dimensional incompressible Navier-Stokes equations are proposed. The numerical solution is achieved by using linearly implicit one-step methods up to third order in time coupled with up to third order stable finite element discretizations in space. These orders of convergence are demonstrated by comparing the numerical solution with exact Navier-Stokes solutions. Finally, we present benchmark computations for flow around a cylinder.
2507 18.04.2007 Rabinovich, V. S.
Roch, S.
Essential spectra of difference operators on $\sZ^n$-periodic graphs MSC: 81Q10; 46N50; 47B36
Let $(\cX, ¸ \rho)$ be a discrete metric space. We suppose that the group $\sZ^n$ acts freely on $X$ and that the number of orbits of $X$ with respect to this action is finite. Then we call $X$ a $\sZ^n$-periodic discrete metric space. We examine the Fredholm property and essential spectra of band-dominated operators on $l^p(X)$ where $X$ is a $\sZ^n$-periodic discrete metric space. Our approach is based on the theory of band-dominated operators on $\sZ^n$ and their limit operators. In case $X$ is the set of vertices of a combinatorial graph, the graph structure defines a Schrödinger operator on $l^p(X)$ in a natural way. We illustrate our approach by determining the essential spectra of Schrödinger operators with slowly oscillating potential both on zig-zag and on hexagonal graphs, the latter being related to nano-structures.
2506 01.04.2007 Zahn, Peter Eine pragmatische Rechtfertigung des klassischen Argumentierens MSC: 03A05
We introduce a `meaningful' (i.e. not only formal) language L, the use and the semantic of their sentences are determined by `external facts' on the one hand and rules of assertion on the other. To reduce the problem of beginning reasoning, we stipulate certain rules to restrict assertions, and we also agree that assertions of sentences of L must not be restricted besides. By liberalizing the resulting use of assertions we establish a `classical game' of assertion which permits to apply classical logic an serves essential purposes of reasoning most favorably. At the end of this paper we analyze the meaning of general conditionals that may be applied like rules of inference. To this end, we consider temporal details of assertion. By the way, we obtain a rule-logical approach to intuitionistic logic, and an approach to deontic logic, too.
2505 03.04.2007 An, Jinpeng
Neeb, Karl-Hermann
An implicit function theorem for Banach spaces and some applications MSC: 22E65; 57N2
We prove a generalized implicit function theorem for Banach spaces, without the usual assumption that the subspaces involved being complemented. Then we apply it to the problem of parametrization of fibers of differentiable maps, the Lie subgroup problem for Banach--Lie groups, as well as Weil's local rigidity for homomorphisms from finitely generated groups to Banach--Lie groups.
2504 12.04.2007 Wille, Rudolf Towards a Semantology of Music The aim of this paper is to approach a Semantology of Music which is understood as the theory and methodology of musical semantic structures. The analysis of music structures is based on a threefold semantics which is performed on the musical level, the abstract philosophic-logical level, and the hypothetical mathematical level. Basic music structures are discussed by examples, in particular: tone systems, chords, harmonies, scales, modulations, musical time flow, and music forms. A specific concern of this paper is to clarify how a Semantology of Music may support the understanding of music.
2503 14.03.2007 Neeb, Karl-Hermann
Friedrich Wagemann
Lie group structures on groups of smooth and holomorphic maps on non-compact manifolds MSC: 22E65; 22E67; 22E15; 22E30
We study Lie group structures on groups of the form ${\sst C^\infty(M,K)}$, where ${\sst M}$ is a non-compact smooth manifold and ${\sst K}$ is a, possibly infinite-dimensional, Lie group. First we prove that there is at most one Lie group structure with Lie algebra ${\sst C^\infty(M,\k)}$ for which the evaluation map is smooth. We then prove the existence of such a structure if the universal cover of ${\sst K}$ is diffeomorphic to a locally convex spa ce and if the image of the left logarithmic derivative in ${\sst \Omega^1(M,\k)}$ is a smooth submanifold, the latter being the case in particular if ${\sst M}$ is one-dimensional. We also obtain analogs of these results for the group ${\sst {\cal O}(M,K)}$ of holomorphic maps on a complex manifold with values in a complex Lie group ${\sst K}$. We further show that there exists a natural Lie group structure on ${\sst {\cal O}(M,K)}$ if ${\sst K}$ is Banach and ${\sst M}$ is a non-compact complex curve with finitely generated fundamental group.
2502 19.03.2007 Neff, Patrizio
Chelminski, Krzysztof
Alber, Hans-Dieter
Notes on strain gradient plasticity: Finite strain covariant modelling and global existence in the infinitesimal rate-independent case. MSC: 74A35; 74A30; 74C05; 74C10
We propose a model of finite strain gradient plasticity including phenomenological Prager-Ziegler type linear kinematical hardening and nonlocal kinematical hardening due to dislocation interaction. Based on the multiplicative decomposition a thermodynamically admissible flow rule for $F_p$ is described involving as plastic gradient $\Curl F_p$. The formulation is covariant w.r.t. superposed rigid rotations of the reference, intermediate and spatial configuration but the model is not spin-free due to the nonlocal dislocation interaction and cannot be reduced to a dependence on $C_p$. The linearization leads to a thermodynamically admissible model of infinitesimal plasticity involving only the $\Curl$ of the non-symmetric plastic variable $p$. Linearized spatial and material covariance under constant infinitesimal rotations is satisfied. Uniqueness of strong solutions of the infinitesimal model is obtained if two non-classical boundary conditions on the non-symmetric small strain plastic variable $p$ are introduced: $\dot{p}.\tau=0$ on the microscopically hard boundary $\Gamma_D\subset\partial\Omega$ and $[\Curl p].\tau=0$ on the microscopically free boundary $\partial\Omega\setminus\Gamma_D$, where $\tau$ are the tangential vectors at the boundary $\partial\Omega$. Moreover, we show that a weak reformulation of the infinitesimal model allows for a global in-time solution of the corresponding rate-independent initial boundary value problem. The method of choice are a formulation as a variational inequality with symmetric and coercive bilinear form. Use is made of new Hilbert-space suitable for dislocation density dependent plasticity.
2501 06.03.2007 Eklund, Peter
Wille, Rudolf
Semantology as Basis for Conceptual Knowledge Processing MSC: 03B42
Semantology has been introduced as the theory of semantic structures and their connections which, in particular, covers the methodology of activating semantic structures for representing conceptual knowledge. It it the main aim of this paper to explain and demonstrate that semantic structures are in fact basic for conceptual knowledge processing which comprises activities such as representing, infering, acquiring and communicating conceptual knowledge.
2500 26.02.2007 Weinberg, Kerstin
Neff, Patrizio
A geometrically exact thin membrane model -investigation of large deformations and wrinkling. MSC: 74K15; 74K20; 74G65
We investigate a geometrically exact membrane model with respect to its capabilities in describing buckling and wrinkling. Contrary to more classical tension-field or relaxed approaches, our model is able to capture the detailed geometry of wrinkling while the balance of force equations remains elliptic throughout. This is achieved by introducing artficial viscosity related to the movement of an adjusted orthonormal frame (rotations) given by a local evolution equation. We discuss the consistent linearization of the model and investigate the efficiency of the local update of rotations. Numerical examples are presented that demonstrate the effectiveness of the new model for predicting wrinkles in membranes undergoing large deformation.
2499 23.02.2007 Farwig, Reinhard
Kozono, Hideo
Sohr, Hermann
Local space-time regularity criteria for weak solutions of the Navier-Stokes equations beyond Serrin's condition MSC: 35Q30; 76D05; 35D05
Consider a weak solution $u$ of the Navier-Stokes equations for a general domain $\Omega \subset R^3$ on the time interval $[0,\infty)$ and a parabolic cylinder $Q_r =Q_r(t_0,x_0) \subset (0,\infty) \times \Omega$ with $r>0$, $t_0 \in (0,\infty)$, $x_0 \in \Omega$. Then we show that there exists an absolute constant $\varepsilon_*>0$ such that the local condition $\|u \|_{L^q(Q_r)} \leq \varepsilon_* ¸ r^{\frac{2}{q} + \frac{3}{q}-1}, \frac{2}{q} + \frac{3}{q} \leq 1 + \frac{1}{4}$, implies the regularity of $u$ in the smaller cylinder $Q_{r/2}$. The special case $\frac{2}{q} + \frac{3}{q}=1$ yields the well-known local Serrin condition $\| u \|_{L^q(Q_r)} \leq varepsilon_*$. Thus our criterion extends Serrin's condition admitting smaller exponents $q$ and replacing the barrier $1$ by $1+\frac{1}{4}$.
2498 20.02.2007 Farwig, Reinhard
Kozono, Hideo
Sohr, Hermann
On the Stokes Operator in General Unbounded Domains MSC: 76D05; 35Q30
It is known that the Stokes operator is not well-defined in $L^q$-spaces for certain unbounded smooth domains unless $q=2$. In this paper, we generalize a new approach to the Stokes resolvent problem and to maximal regularity in general unbounded smooth domains from the three-dimensional case, see R. Farwig, H. Kozono, H. Sohr, {\it An $L^q$--approach to Stokes and Navier-Stokes equations in general domains,} Acta Math. 195, 21--53 (2005), to the $n$-dimensional one, $n \geq 2,$ replacing the space $L^q, 1 < q < \infty,$ by $\tilde L^q$ where $\tilde L^q = L^q \cap L^2$ for $q \geq 2$ and $\tilde L^q = L^q + L^2$ for $1 < q < 2.$ In particular, we show that the Stokes operator is well-defined in $\tilde L^q$ for every unbounded domain of uniform $C^{1,1}$-type in $R^n, n \geq 2$, satisfies the classical resolvent estimate, generates an analytic semigroup and has maximal regularity.
2497 16.02.2007 Reif, Ulrich An Appropriate Geometric Invariant for the $C^2$-Analysis of Subdivision Surfaces MSC: 46G05; 46B22; 28B05
We introduce the embedded Weingarten map as a geometric invariant of piecewise smooth surfaces. It is given by a $(3\times 3)$-matrix and provides complete curvature information in a continuous way. Thus, it is the appropriate tool for the $C^2$-analysis of subdivision surfaces near extraordinary points. We derive asymptotic expansions and show that the convergence of the sequence of embedded Weingarten maps to a constant limit is necessary and sufficient for curvature continuity.
2496 13.02.2007 Bergner, Matthias On the Dirichlet problem for the prescribed mean curvature equation over nonconvex domains We study and solve the Dirichlet problem for graphs of prescribed mean curvature $H$ in $\mathbb R^{n+1}$ over general domains $\Omega$ without requiring a mean convexity assumption. By using pieces of nodoids as barriers we first give sufficient conditions for the solvability in case of zero boundary values. Applying a result by Schulz and Williams we can then also solve the Dirichlet problem for boundary values satisfying a Lipschitz condition.
2495 08.02.2007 Bergner, Matthias A simple proof for Brouwer's fixed point theorem Using only basic tools from calculus, we give a relatively simple proof for Brouwer's fixed point theorem.
2494 19.01.2007 Vladimir S. Rabinovich, Steffen Roch Essential spectra of pseudodifferential operators and exponential decay of their solutions. Applications to Schrödinger operators MSC: 47G30; 35J10; 35P05; 35S05; 47A53; 47N20
The main aim of this paper is to study the relations between the location of the essential spectrum and the exponential decay of eigenfunctions of pseudodifferential operators on $L^p(\sR^n)$ perturbed by singular potentials. For a solution of this problem we apply the limit operators method. This method associates with each band-dominated operator $A$ a family $op (A)$ of so-called limit operators which reflect the properties of $A$ at infinity. Consider the compactification of $\sR^n$ by the „infinitely distant“ sphere $S^{n-1}$. Then the set $op (A)$ can be written as the union of its components $op_{\eta_\omega} (A)$ where $\omega$ runs through the points of $S^{n-1}$ and where $op_{\eta_\omega} (A)$ collects all limit operators of $A$ which reflect the properties of $A$ if one tends to infinity „in the direction of $\omega$. Set $sp_{\eta_\omega} A := \cup_{A_h \in op_{\eta_\omega} (A)} sp ¸ A_{h}$. We show that “the distance" of an eigenvalue $\lambda \notin sp_{ess} A$ to $sp_{\eta_\omega} A$ determines the exponential decay of the $\lambda$-eigenfunctions of $A$ in the direction of $\omega$. We apply these results to estimate the exponential decay of eigenfunctions of electromagnetic Schrödinger operators for a large class of electric potentials, in particular, for multiparticle Schrödinger operators and periodic Schrödinger operators perturbed by slowly oscillating at infinity potentials.
2493 17.01.2007 Wille, Rudolf The Basic Theorem on Labelled Line Diagrams of Finite Concept Lattices MSC: 06A; 06B
This paper offers a mathematical analysis of labelled line diagrams of finite concept lattices to gain a better understanding of those diagrams. The main result is the Basic Theorem on Labelled Line Diagrams of Finite Concept Lattices. This Theorem can be applied to justify, for instance, the training tool 'CAPESSISMUS – A Game of Conceiving Concepts' which has been created to support the understanding and the drawing of appropriate line diagrams of finite concept lattices.
2492 14.01.2007 Glöckner, Helge Applications of hypocontinuous bilinear maps in infinite-dimensional differential calculus MSC: 26E15; 26E20; 17B63; 22E65; 46A32; 46G20; 46T25
Paradigms of bilinear maps $\beta$ between locally convex spaces (like evaluation or composition) are not continuous, but merely hypocontinuous. We describe situations where, nonetheless, compositions of $\beta$ with Keller $C^n_c$-maps (on suitable domains) are $C^n_c$. Our main applications concern holomorphic families of operators, and the foundations of locally convex Poisson vector spaces.
2491 11.01.2007 Alber, Hans-Dieter
Zhu, Peicheng
Solutions to a Model for Interface Motion by Solutions to a Model for Interface Motion by Interface Diffusion MSC: 35Q72; 35M20; 35Q72
Existence of weak solutions is proved for a phase field model describing an interface in an elastically deformable solid, which moves by diffusion of atoms along the interface. The volume of the different regions separated by the interface is conserved, since no exchange of atoms across the interface occurs. The diffusion is only driven by reduction of the bulk free energy. The evolution of the order parameter in this model is governed by a degenerate parabolic fourth order equation. If a regularizing parameter in this equation tends to zero, then solutions tend to solutions of a sharp interface model for interface diffusion. The existence proof is valid only for a $1\frac{1}{2}$--dimensional situation.
2490 09.01.2007 Farwig, Reinhard
Krbec, Miroslav
Necasova, Sarka
A Weighted $L^q$-Approach to Oseen Flow Around a Rotating Body MSC: 76D05; 35Q30
We study time-periodic Oseen flows past a rotating body in $R^3$ proving weighted \it{a priori} estimates in $L^q$-spaces using Muckenhoupt weights. After a time-dependent change of coordinates the problem is reduced to a stationary Oseen equation with the additional terms $(\omega\times x)\cdot\nabla u$ and $-\omega \wedge u$ in the equation of momentum where $\omega$ denotes the angular velocity. Due to the asymmetry of Oseen flow and to describe its wake we use anisotropic Muckenhoupt weights, a weighted theory of Littlewood-Paley decomposition and of maximal operators as well as one-sided univariate weights, one-sided maximal operators and a new version of Jones' factorization theorem.
2489 07.01.2007 Glöckner, Helge Instructive examples of smooth, complex differentiable and complex analytic mappings into locally convex spaces MSC: 46G20; 26E05; 26E15; 26E20; 46T25
For each positive integer $k$, we describe a map $f$ from the complex plane to a suitable non-complete complex locally convex space such that $f$ is $k$ times continuously complex differentiable but not $k+1$ times, and hence not complex analytic. As a preliminary, we prove that the sequences $(n^kz^n)_n$ are linearly independent in the space of complex sequences, for $k$ ranging through the integers and $z$ through the set of non-zero complex numbers. We also describe a complex analytic map from $l^1$ to a suitable complete complex locally convex space which is unbounded on each non-empty open subset of $l^1$. Furthermore, we present a smooth map from the real line to a non-complete locally convex space which is not real analytic although it is given locally by its Taylor series around each point. As a byproduct, we find that free locally convex spaces over subsets of the complex plane with non-empty interior are not Mackey complete.
Nummer Datum Autor Titel Abstract/MSC
2488 20.12.2006 V. S. Rabinovich, S. Roch, B. Silbermann The finite sections approach to the index formula for band-dominated operators MSC: 47A53; 46N40; 47B36; 65J10
In a previous paper, two of the authors together with J. Roe derived an index formula which expresses the Fredholm index of a band-dominated operator on $l^2(\sZ)$ in terms of local indices of its limit operators. The proof makes thoroughly use of $K$-theory for $C^*$-algebras (which, of course, appears as a natural approach to index problems). The purpose of this short note is to develop a completely different approach to the index formula for band-dominated operators which is exclusively based on ideas and results from asymptotic numerical analysis.
2487 14.12.2006 Matthias Bergner The Dirichlet problem for graph of prescribed anisotropic mean curvature in $\mathbb R^{n+1}$ MSC: 53A10
We consider the Dirichlet problem for graphs of prescribed mean curvature in $\mathbb R^{n+1}$ where the prescribed mean curvature function $H=H(X,N)$ may depend on the point $X$ in space and on the normal $N$ of the graph as well. In some special cases this Dirichlet problem arises as the Euler equation of a generalised nonparametric area funcional.
2486 13.12.2006 Rabinovich, Vladimir
Roch, Steffen
Silbermann, Bernd
On finite sections of band-dominated operators MSC: 47N40; 47L40; 65J10
In an earlier paper we showed that the sequence of the finite sections $P_nAP_n$ of a band-dominated operator $A$ on $l^p(\sZ)$ is stable if and only if the operator $A$ is invertible, every limit operator of the sequence $(P_n A P_n)$ is invertible, and if the norms of the inverses of the limit operators are uniformly bounded. The purpose of this short note is to show that the uniform boundedness condition is redundant.
2485 15.11.2006 Floater, Michael
Rasmussen, Atgeirr
Reif, Ulrich
Extrapolation Methods for Approximating Arc Length and Surface Area MSC: 65D30; 65B05
A well-known method of estimating the length of a parametric curve in $R^d$ is to sample some points from it and compute the length of the polygon passing through them. In this paper we show that for uniform sampling of regular smooth curves Richardson extrapolation can be applied repeatedly giving a sequence of derivative-free length estimates of arbitrarily high orders of accuracy. Further, a similar result is derived for the approximation of the area of parametric surfaces.
2484 30.11.2006 Farwig, Reinhard
Neustupa, Jiri
On the Spectrum of an Oseen--Type Operator Arising from Flow past a Rotating Body MSC: 35Q35; 35P99; 76D07
We present the description of the spectrum of a linear perturbed Oseen--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 an infinite set of overlapping parabolic regions in the left half--plane of 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.
2483 14.11.2006 Hofmann, Karl H.
Morris, Sidney A.
Iwasawa's Local Splitting Theorem for Pro-Lie Groups MSC: 22E67; 22D05
If the nilradical of the Lie algebra of a pro-Lie group $G$ is finite dimensional modulo the center, then every identity neighborhood $U$ of $G$ contains a closed normal subgroup $N$ such that $G/N$ is a Lie group and $G$ and $N\times G/N$ are locally isomorphic.
2482 14.11.2006 A. Neuenkirch
I. Nourdin
A. Rößler
S. Tindel
Trees and asymptotic developments for fractional stochastic differential equations MSC: 60H05; 60H07; 60G15
In this paper we consider a $n$-dimensional stochastic differential equation driven by a fractional Brownian motion with Hurst parameter $H>1/3$. After solving this equation in a rather elementary way, following the approach of M. Gubinelli (2004), we show how to obtain an expansion for $E[f(X_t)]$ in terms of $t$, where $X$ denotes the solution to the SDE and $f:\R^n\to\R$ is a regular function. With respect to F. Baudoin and L. Coutin (2006), where the same kind of problem is considered, we try an improvement in three different directions: we are able to take a drift into account in the equation, we parametrize our expansion with trees (which makes it easier to use), and we obtain a sharp control of the remainder.
2481 05.11.2006 Glöckner, Helge
Willis, George A.
Directions of automorphisms of Lie groups over local fields compared to the directions of Lie algebra automorphisms MSC: 22D05; 20G25; 22D45; 22E15; 22E35
To each totally disconnected, locally compact topological group $G$ and each group $A$ of automorphisms of $G$, a pseudo-metric space $\partial A$ of «directions» has been associated by U. Baumgartner and the second author. Given a Lie group $G$ over a local field, it is a natural idea to try to define a map $\Phi\colon \partial Aut_{C^\omega}(G)\to \partial Aut(L(G))$, $\partial \alpha\mapsto \partial L(\alpha)$ which takes the direction of an analytic automorphism of $G$ to the direction of the associated Lie algebra automorphism. We show that, in general, $\Phi$ is not well defined. Also, it may happen that $\partial L(\alpha)=\partial L(\beta)$ although $\partial \alpha\not=\partial\beta$. However, such pathologies are absent for a large class of groups: we show that $\Phi\colon \partial Inn(G)\to \partial Aut(L(G))$ is a well-defined isometric embedding for each generalized Cayley group $G$. Some counterexamples concerning the existence of small joint tidy subgroups for flat groups of automorphisms are also provided.
2480 01.11.2006 Hofmann, K.H.
Morris, S. A.
Open Mapping Theorem for Topological Groups MSC: 22A05; 22E65, 46A30
We survey sufficient conditions that force a surjective continuous homomorphism between topological groups to be open. We present the shortest proof yet of an open mapping theorem between projective limits of finite dimensional Lie groups.
2479 24.10.2006 Rößler, Andreas Second Order Runge--Kutta Methods for Itô Stochastic Differential Equations MSC: 65C30; 65L06; 60H35; 60H10
A new class of stochastic Runge--Kutta methods for the weak approximation of the solution of Itô stochastic differential equation systems with a multi--dimensional Wiener process is introduced. As the main innovation, the number of stages of the methods does not depend on the dimension of the driving Wiener process and the number of the necessary random variables is reduced considerably. This reduces the computational effort significantly. Order conditions for the stochastic Runge--Kutta methods assuring weak convergence with order two are calculated by applying the colored rooted tree analysis due to the author. Further, some coefficients for explicit second order stochastic Runge--Kutta schemes are presented.
2478 24.10.2006 Rößler, Andreas Second Order Runge--Kutta Methods for Stratonovich Stochastic Differential Equations MSC: 65C30; 65L06; 60H35; 60H10
The weak approximation of the solution of a system of Stratonovich stochastic differential equations with a $m$--dimensional Wiener process is studied. Therefore, a new class of stochastic Runge--Kutta methods is introduced. As the main novelty, the number of stages does not depend on the dimension $m$ of the driving Wiener process which reduces the computational effort significantly. The colored rooted tree analysis due to the author is applied to determine order conditions for the new stochastic Runge--Kutta methods assuring convergence with order two in the weak sense. Further, some coefficients for second order stochastic Runge--Kutta schemes are calculated explicitly.
2477 18.10.2006 Bergner, Matthias A mixed boundary value problem for the prescribed mean curvature equation MSC: 53A10
We solve a mixed boundary value problem for the nonparametric prescribed mean curvature equation, prescribing continuous Dirichlet boundary values at some strictly convex boundary part and Neumann zero boundary values at the remaining part of the boundary. We assume that Dirichlet and Neumann boundary parts are some positive distance away from each other.
2476 25.09.2006 Hofmann, K. H.
Neeb, K.-H.
Pro-Lie groups which are infinite-dimensional Lie groups MSC: 22E65; 17B65; 22D05
A pro-Lie group is a projective limit of a family of finite-dimensional Lie groups. In this note we show that a pro-Lie group $G$ is a Lie group in the sense that its topology is compatible with a smooth manifold structure for which the group operations are smooth if and only if $G$ is locally contractible. We also characterize the corresponding pro-Lie algebras in various ways. Furthermore, we characterize those pro-Lie groups which are locally exponential, that is, they are Lie groups with a smooth exponential function which maps a zero neighborhood in the Lie algebra diffeomorphically onto an open identity neighborhood of the group.
2475 07.09.2006 Magata, Frederick An Integration Formula for Polar Actions MSC: 57S25; 53C20
We prove an analogue of Weyl's Integration Formula for compact Lie groups in the context of polar actions. We also show how certain classical examples from the literature can be viewed as special cases of our result.
2474 12.09.2006 Rudolf Wille Formal Concept Analysis of One-Dimensional Continuum Structures This paper offers an approach of developing an order-theoretic structure theory of one-dimensional continuum structures. The chosen approach is based on continua and their subcontinua as primitive notions. In a first step linear and circular continuum structures are defined as ordered sets and concretized by a real number model. In a second step 'points' are deduced as limits of continua by methos of Formal Concept Analysis. The continuum structures extended by those points are analysed and represented by an enlarged real number model. Further research is planned to extend the approach of this paper to higher dimensional continuum structures.
2473 01.09.2006 Glöckner, Helge Comparison of some notions of $C^k$-maps in multi-variable non-archimedian analysis MSC: 26E30; 26E20; 46A16; 46G05; 46S10
Various definitions of $C^k$-maps on open subsets of finite-dimensional vector spaces over a complete valued field have been proposed in the literature. We show that the $C^k$-maps considered by Schikhof and De Smedt coincide with those of Bertram, Glöckner and Neeb. By contrast, Ludkovsky's $C^k$-maps need not be $C^k$ in the former sense, at least in positive characteristic. We also compare various types of Hölder differentiable maps on finite-dimensional and metrizable spaces.
2472 01.09.2006 Glöckner, Helge
Ludkovsky, Sergey V.
Ultrametric and non-locally convex analogues of the general curve lemma of convenient differential calculus MSC: 26E30; 26E15; 26E20; 45T20; 46A16; 46S10
The General Curve Lemma is a tool of Infinite-Dimensional Analysis, which enables refined studies of differentiability properties of mappings between real locally convex spaces. In this article, we generalize the General Curve Lemma in two ways: First, we remove the condition of local convexity in the real case. Second, we adapt the lemma to the case of curves in topological vector spaces over ultrametric fields.
2471 09.08.2006 Dintelmann, Eva
Geissert, Matthias
Hieber, Matthias
Strong $L^p$-Solutions to the Navier-Stokes Flow past Moving Obstacles: The Case of Several Obstacles and Time Dependent Velocity MSC: 76D03; 35Q30; 35B30
Consider the Navier-Stokes flow past several moving obstacles. It is shown that there exists a unique strong local solution in the $L^p$-setting, $1 < p < \infty$. Moreover, it is proved that the strong solution coincides with the known mild solution in the very weak sense.
2470 01.08.2006 Neff, Patrizio
Chelminski, Krzysztof
Müller, Wolfgang
Wieners, Christian
A numerical solution method for an infinitesimal elasto-plastic Cosserat model We present a finite element implementation of a Cosserat elasto-plastic model allowing for non-symmetric stresses and we provide a numerical analysis of the introduced time-incremental algorithm. The model allows the use of standard tools from convex analysis as known from classical Prandtl-Reuss plasticity. We derive the dual stress formulation and show that for vanishing Cosserat couple modulus $\mu_c\to 0$ the classical problem with symmetric stresses is approximated. Our numerical results testify to the robustness of the approximation. Notably, for positive couple modulus $\mu_c>0$ there is no need for a safe-load assumption. For small $\mu_c$ the response is numerically indistinguishable from the classical response.
2469 16.07.2006 Bergner, Matthias The Dirichlet problem for graphs of prescribed anisotropic mean curvature MSC: 53A10; 49Q05
We consider the Dirichlet problem for two-dimensional graphs of prescribed mean curvature in $\mathbb R^3$ where the prescribed mean curvature function $H=H(X,N)$ may depend on the point $X$ in space and on the normal $N$ of the graph as well. In special situations this Dirichlet problem arises as the Euler equation of a generalised nonparametric area funcional. Under certain smallness conditions we will solve the Dirichlet problem and construct minimizers of the generalized area functional.
2468 17.07.2006 Neff, Patrizio
Chelminski, Krzysztof
Approximation of Prandtl-Reuss Plasticity through Cosserat-Plasticity MSC: 35Q72; 74A35; 74A30; 74C05; 74C10
In this article we investigate the regularizing properties of Cosserat elasto-plastic models in a geometrically linear setting. The models feature an independent microrotation field which allow the Cauchy-stress to become non-symmetric while the contribution of the microrotations itself remains linear elastic. Extending previous work we show that for the large class of all quasistatic models of monotone type, solutions to the problem with microrotations are $\mathbb{H}^1$ well-posed. A similar result does not hold for the classical case without microrotations. For vanishing Cosserat effects we show also that the model with microrotations approximates the classical Prandtl-Reuss solution in an appropriate measure valued sense.
2467 10.07.2006 Zahn, Peter Approximative Computation and Generalizations of Metric Spaces MSC: 2000; 68Q99; 54E25
We introduce certain 'computation spaces', neighbourhood spaces, and generalizations of metric spaces (with generalizations of +), and we investigate the relationalship between those spaces. We also present calculus-like methodes to obtain programs to compute functions on computation spaces and, for such functions, computable moduli of continuity, which are suitable for individual applications.
2466 10.07.2006 Zahn, Peter Eine pragmatische Rechtfertigung des klassischen Argumentierens MSC: 2000; 03A05
We introduce a «meaningful» (i.e. not only formal) language L, the use and the semantic of their sentences are determined by «external facts» on the one hand and rules of assertion on the other. To reduce the problem of beginning reasoning, we stipulate certain rules to restrict assertions, and we also agree that assertions of sentences of L must not be restricted besides. By liberalizing the resulting use of assertions we establish a «classical game» of assertion which permits to apply classical logic an serves essential purposes of reasoning most favorably.
2465 19.06.2006 Farwig, Reinhard
Kozono, Hideo
Sohr, Hermann
Local in time regularity properties of the Navier-Stokes equations beyond Serrin's condition MSC: 76D05; 35Q30; 35B65
Let $u$ be a weak solution of the Navier-Stokes equations in a domain $\Omega \subset R^3$ and a time interval $[0,T)$, $0< T \leq \infty$, with initial value $u_0$, and vanishing external force. As is well known, global regularity of $u$ for general $u_0$ is an unsolved problem unless we pose additional assumptions on $u_0$ or on the solution $u$ itself such as Serrin's condition $\| u \|_{L^s(0,T;L^q(\Omega))} < \infty$ where $\frac{2}{s} + \frac{3}{q} =1$. In the present paper we prove several new local and global regularity properties by using assumptions beyond Serrin's condition e.g. as follows: If the norm $\|u\|_{L^r(0,T;L^q(\Omega))}$, with Serrin's number $\frac{2}{r} + \frac{3}{q} =1+\alpha$ $(\alpha>0)$ strictly larger than $1$, is sufficiently small, or if $u$ satisfies a {\it local leftward} $L^s(L^q(\Omega))$--condition for every $t\in(0,T)$, where $\frac{2}{s} + \frac{3}{q} =1$, then $u$ is regular in $(0,T)$. Further results deal with similar regularity conditions based on energy quantities only.
2464 01.06.2006 Bergner, Matthias
Froehlich, Steffen
On two-dimensional immersions of prescribed mean curvature in $\mathbb R^n$ MSC: 35J60; 53A07; 53A10
We consider two-dimensional immersions of disc-type in $\mathbb R^n.$ We focus on 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 graphs with prescribed mean curvature fields and derive a theorem of Bernstein type for minimal graphs.
2463 06.06.2006 Glöckner, Helge Direct limits of infinite-dimensional Lie groups compared to direct limits in related categories MSC: 22E65; 22E67; 46A13; 46F05; 46T20; 54B30; 54H11; 58B10; 58D05
Let $G$ be a Lie group which is the union of an ascending sequence $G_1 \subseteq G_2 \subseteq \cdots$ of Lie groups (each of which may be infinite-dimensional). We study the question when $G$ is the direct limit of the $G_n$'s in the category of Lie groups, topological groups, smooth manifolds, resp., topological spaces. Full answers are obtained for $G$ the group $Diff_c(M)$ of compactly supported $C^\infty$-diffeomorphisms of a $\sigma$-compact smooth manifold $M$; and for test function groups $C^\infty_c(M,H)$ of compactly supported smooth maps with values in a finite-dimensional Lie group $H$. We also discuss the cases where $G$ is a direct limit of unit groups of Banach algebras, a Lie group of germs of Lie group-valued analytic maps, or a weak direct product of Lie groups.
2462 04.06.2006 Glöckner, Helge Direct limit groups do not have small subgroups MSC: 22E65
We show that countable direct limits of finite-dimensional Lie groups do not have small subgroups. The same conclusion is obtained for suitable direct limits of infinite-dimensional Lie groups.
2461 01.06.2006 Petra Gehring
Rudolf Wille
Semantology: Basic Methods for Knowledge Respresentations MSC: 030
In this paper, we introduce the term 'Semantology' for naming the theory of semantic structures and their connections. Semantic structures are fundamental for representing knowledge which we demonstrate by discussing basic methods of knowledge representation. In this context we discuss why, in the field of knowledge representation, the term 'Semantology' should be given perference to the term 'Ontology'.
2460 23.05.2006 Debrabant, Kristian
Rößler, Andreas
Classification of Stochastic Runge--Kutta Methods for the Weak Approximation of Stochastic Differential Equations MSC: 65C30; 60H35; 65C20; 68U20
In the present paper, a class of stochastic Runge--Kutta methods for weak approxi­ma­tion of Itô stochastic differential equation systems with a multi--dimensional Wiener process is considered. Order one and order two conditions for the coefficients of explicit stochastic Runge--Kutta methods are solved and the solution space of all possible coefficients is analyzed. A full classification of the coefficients for such stochastic Runge--Kutta schemes of order one and two as well as coefficients for optimal schemes are presented.
2459 23.05.2006 Neeb, Karl-Hermann Towards a Lie theory of locally convex groups MSC: 22E65; 22E15
In this survey, we report on the state of the art of some of the fundamental problems in the Lie theory of Lie groups modeled on locally convex spaces, such as integrability of Lie algebras, integrability of Lie subalgebras to Lie subgroups, and integrability of Lie algebra extensions to Lie group extensions. We further describe how regularity or local exponentiality of a Lie group can be used to obtain quite satisfying answers to some of the fundamental problems. These results are illustrated by specialization to some specific classes of Lie groups, such as direct limit groups, linear Lie groups, groups of smooth maps and groups of diffeomorphisms.
2458 13.04.2006 Nesenenko, Sergiy Homogenization in viscoplasticity MSC: 74Q15; 64C05; 74D10; 35J25; 34G20; 47H04; 47H05
In this work we present the justification of the formally derived homogenized problem for the quasistatic initial boundary value problem with internal variables, which models the deformation behavior of viscoplastic materials with a periodic microstructure.
2457 16.06.2006 Grundling, Hendrik
Neeb, Karl-Hermann
Abelian topological groups with host algebras MSC: 46L05; 43A10; 43A65; 46L60; 22E65
The concept of a host algebra generalises that of a group $C^*$-algebra to groups which are not locally compact in the sense that its non-degenerate representations are in one-to-one correspondence with representations of the group under consideration. Here we consider the question of the existence of host algebras for abelian topological groups and also for multiplier representations. Our main negative result is essentially that a topological abelian group has a full host algebra (covering all its continuous unitary representations) if and only if it embeds into a locally compact group. On the positive side, we show that the canonical symplectic form on a countably dimensional complex vector space leads to an abelian group with multiplier for which a full host algebra exists. This provides a host algebra for the set of regular representations of the CCR algebra.
2456 11.04.2006 Froehlich, Steffen
Winklmann, Sven
Curvature estimates for graphs with prescribed mean curvature and flat normal bundle MSC: 53J60; 53A10; 49Q05
We consider graphs $\Sigma^n \subset \R^m$ with prescribed mean curvature and flat normal bundle. Using techniques of Schoen, Simon and Yau, and Ecker-Huisken, we derive the interior curvature estimate $$\sup_{\Sigma \cap B_R} |A|^2 \leq \frac{C}{R^2}$$ up to dimension $n \leq 5$, where $C$ is a constant depending on natural geometric data of $\Sigma$ only. This generalizes previous results of Smoczyk, Wang and Xin, and Wang for minimal graphs with flat normal bundle.
2455 01.04.2006 Neff, Patrizio
Muench, Ingo
Curl bounds Grad on ${\rm SO(3)}$ MSC: 74A35; 74E15; 74G65; 74N15
We show that the operator Curl is isomorphic to the operator Grad on ${\rm SO(3)}$
2454 04.04.2006 Glöckner, Helge
Willis, George A.
Classification of the simple factors appearing in composition series of totally disconnected contraction groups MSC: 22D05; 20E15; 20E36
Let $G$ be a totally disconnected, locally compact group admitting a contractive automorphism $\alpha$. We prove a Jordan-Hölder theorem for series of $\alpha$-stable closed subgroups of $G$, classify all possible composition factors and deduce consequences for the structure of $G$.
2453 03.04.2006 Niese, Birgit A generalized order statistic property MSC: 60G55; 62G30; 60Jxx
In this article we generalize the uniform order statistic property of mixed Poisson processes by the use of a wider model of ordered random variables. The corresponding point processes will be characterized. We deduce alternative representations of their distribution.
2452 30.03.2006 Krohne, Katrin Very Weak Solutions to the Stokes and Stokes-Resolvent Problem in Weighted Function Spaces MSC: 35Q30; 35D05; 76D07; 35J25
We investigate very weak solutions the stationary Stokes- and Stokes resolvent problem in function spaces with Muckenhoupt weights. The notion used here is similar but even more general than the one used in \cite{ama1} or \cite{gss}. Consequently the class of solutions is enlarged. To describe boundary conditions we restrict ourselves to more regular data. We introduce a Banach space admitting a restriction operator and containing the solutions according to such data. As a preparation we prove a weighted analogue to Bogowski's Theorem and extension theorems for functions defined on the boundary.
2451 30.03.2006 Krohne, Katrin Stationary Stokes- and Navier-Stokes Equations with Low Regularity Data in Weighted Bessel-Potential Spaces MSC: 35Q30; 35D05; 76D05; 35J65
We investigate the stationary Navier-Stokes equations in Spaces with Muckenhoupt weights. The aim is to find a class of solutions as large as possible. We join the notation of very weak solutions in [1] and [10]. When estimating the nonlinear term the weighted context causes difficulties. For this reason we consider solutions in weighted Bessel-potential spaces. Thus using complex interpolation we establish a theory of solutions to the Stokes equations in weighted Bessel-potential spaces.
2450 01.03.2006 Gramlich, Ralf
Horn, Max
Nickel, Werner
Odd-dimensional orthogonal groups as amalgams of unitary groups. \\ Part 2: machine computations In the first part, a characterization of central quotients of the group $\Spin(2n+1,q)$ is given for $n \geq 3$ and all odd prime powers $q$, with the exception of the cases $n=3$, $q\in{3,5,7,9}$. The present article treats these cases computationally, thus completing the Phan-type theorem for the group $\Spin(2n+1,q)$.
2449 01.03.2006 Bennett, Curt
Gramlich, Ralf
Hoffman, Corneliu
Shpectorov
Sergey
Odd-dimensional orthogonal groups as amalgams of unitary groups. \\ Part 1: general simple connectedness We extend the Phan theory described in previous articles to the last remaining infinite series of classical Chevalley groups over finite fields. Namely, we prove that the twin buildings for the group $\Spin(2n+1,q^2)$, $q$ odd, admit a unique unitary flip and that the corresponding flipflop geometry is simply connected for almost all finite fields $\Fqsq$. Applying standard methods from amalgam theory, this results in a characterization of central quotients of the group $\Spin(2n+1,q)$ by a Phan system of rank one and rank two subgroups. In the present first part of a series of two articles we present simple connectedness results for sufficiently large fields or sufficiently large rank. To be precise, the result stated in the present paper is proved for all cases but $n=3$ and $q \in {3, 5, 7, 9}$, the remaining cases are dealt with in the sequel \cite{Part2} computationally.
2448 01.03.2006 Gramlich, Ralf
Horn, Max
Nickel, Werner
The complete Phan-type theorem for $\mathrm{Sp}(2n,q)$ Previous articles give a characterization of central quotients of the group $\mathrm{Sp}(2n,q)$ for $n \geq 3$ and all prime powers $q$ up to some small cases that are left open. The present article fills in this gap, thus providing the definitive version of the Phan-type theorem for $\Sp(2n,q)$.
2447 16.03.2006 Ri Myong-Hwan
Farwig, Reinhard
Existence and Exponential Stability in $L^r$-spaces of Stationary Navier-Stokes Flows with Prescribed Flux in Infinite Cylindrical Domains MSC: 35Q30; 76D05; 76D07; 76E99; 35B35
We prove existence, uniqueness and exponential stability of stationary Navier-Stokes flows with prescribed flux in an unbounded cylinder of $R^n, n\geq 3,$ with several exits to infinity provided the total flux and external force are sufficiently small. The proofs are based on analytic semigroup theory, perturbation theory and $L^r-L^q$-estimates of a perturbation of the Stokes operator in $L^q$-spaces.
2446 01.03.2006 Roch, Steffen
Silbermann, Bernd
Szegö limit theorems for operators with almost periodic diagonals MSC: 47B36; 47A75; 47B35
The classical Szegö theorems study the asymptotic behaviour of the determinants of the finite sections $P_n T(a) P_n$ of Toeplitz operators, i.e., of operators which have constant entries along each diagonal. We generalize these results to operators which have almost periodic functions on their diagonals.
2445 02.03.2006 Farwig, Reinhard
Hishida, Toshiaki
Stationary Navier-Stokes flow around a rotating obstacle MSC: 35Q30; 76D05
Consider a viscous incompressible fluid filling the whole 3-dimensional space exterior to a rotating body with constant angular velocity $\omega$. By using a coordinate system attached to the body, the problem is reduced to an equivalent one in a fixed exterior domain. The reduced equation involves the crucial drift operator $(\omega\wedge x)\cdot\nabla$, which is not subordinate to the usual Stokes operator. This paper addresses stationary flows to the reduced problem with an external force $f=\mbox{div $F$}$, that is, time-periodic flows to the original one. Generalizing previous results of G. P. Galdi we show the existence of a unique solution $(\nabla u,p)$ in the class $L_{3/2,\infty}$ when both $F\in L_{3/2,\infty}$ and $\omega$ are small enough; here $L_{3/2,\infty}$ is the weak-$L_{3/2}$ space.
2444 25.02.2006 Kristian Debrabant and Andreas Rößler Continuous Extension of Stochastic Runge--Kutta Methods for the Weak Approximation of SDEs MSC: 65C30; 60H35; 65C20; 68U20
A continuous extension of the class of stochastic Runge--Kutta methods for weak approximation is introduced. Order conditions for continuous Runge--Kutta schemes of weak order one and two for the approximation of Itô stochastic differential equations with respect to a multi--dimensional Wiener process are stated. Further, a full classification of the coefficients for continuous stochastic Runge--Kutta schemes of order one and two is calculated and coefficients for optimal schemes are presented.
2443 27.02.2006 Vladimir S. Rabinovich, Steffen Roch The essential spectrum of Schrödinger operators on lattices MSC: 81Q10; 47B36; 46N50
The paper is devoted to the study of the essential spectrum of discrete Schrödinger operators on the lattice $\mathbb{Z}^{N}$ by means of the limit operators method. This method has been applied by one of the authors to describe the essential spectrum of (continuous) electromagnetic Schrödinger operators, square-root Klein-Gordon operators, and Dirac operators under quite weak assumptions on the behavior of the magnetic and electric potential at infinity. The present paper is aimed to illustrate the applicability and effectivity of the limit operators method to discrete problems as well. We consider the following classes of the discrete Schrödinger operators: 1) operators with slowly oscillating at infinity potentials, 2) operators with periodic and semi-periodic potentials; 3) Schrödinger operators which are discrete quantum analogs of the acoustic propagators for waveguides; 4) operators with potentials having an infinite set of discontinuities; and 5) three-particle Schrödinger operators which describe the motion of two particles around a heavy nuclei on the lattice $\mathbb{Z}^3$.
2442 22.02.2006 Müller, Christoph
Wockel, Christoph
Equivalences of Smooth and Continuous Principal Bundles with Infinite-Dimensional Structure Group MSC: 22E65; 55R10; 57R10
This paper is on the equivalence of continuous and smooth principal bundles. Throughout the text, let K be a a Lie group, modeled on a locally convex space, and M be a finite-dimensional paracompact manifold with corners. We show that each continuous principal K-bundle over M is continuously equivalent to a smooth one and that two smooth principal K-bundles over M which are continuously equivalent are also smoothly equivalent. In the concluding section, we relate our results to neighboring topics.
2441 19.02.2006 Glöckner, Helge Aspects of $p$-Adic Non-Linear Functional Analysis MSC: 26E30; 22E65; 26E15; 37D10; 46S10; 47H10; 58C15; 58C20; 58D05
The article provides an introduction to infinite-dimensional differential calculus over topological fields and surveys some of its applications, notably in the areas of infinite-dimensional Lie groups and dynamical systems.
2440 29.02.2006 Glöckner, Helge Finite order differentiability properties, fixed points and implicit functions over valued fields MSC: 58C15; 26E15; 26E30; 46S10; 47H10; 58C20
We prove an implicit function theorem for $C^k$-maps from arbitrary topological vector spaces over valued fields to Banach spaces (for $k \geq 2$). As a tool, we show the $C^k$-dependence of fixed points on parameters for suitable families of contractions of a Banach space. Similar results are obtained for $k$ times strictly differentiable maps, and for $k$ times Lipschitz differentiable maps. In the real case, our results subsume an implicit function theorem for Keller $C^k_c$-maps from arbitrary topological vector spaces to Banach spaces.
2439 19.02.2006 Glöckner, Helge Fundamental Problems in the Theory of Infinite-Dimensional Lie Groups MSC: 22E65
In a preprint from 1982, John Milnor formulated various fundamental questions concerning infinite-dimensional Lie groups. In this note, we describe some of the answers (and partial answers) obtained in the preceding years.
2438 19.02.2006 Glöckner, Helge
Lucht, Lutz G.
Porubský, Stefan
Solutions to Arithmetic Convolution Equations MSC: 11A25; 46H30
In the complex algebra $A$ of arithmetic functions $g: N \to C$, endowed with the usual pointwise linear operations and the Dirichlet convolution, let $g^{*k}$ denote the convolution power $g*\cdots*g$ with $k$ factors $g \in A$. We investigate the solvability of polynomial equations of the form $a_d*g^{*d}+a_{d-1}*g^{*(d-1)}+\cdots+a_1*g+a_0 = 0$ with fixed coefficients $a_d,a_{d-1},\ldots,a_1,a_0 \in A$. In some cases the solutions have specific properties and can be determined explicitly. We show that the property of the coefficients to belong to convergent Dirichlet series transfers to those solutions $g \in A$, whose values $g(1)$ are simple zeros of the polynomial $a_d(1)z^d+a_{d-1}(1)z^{d-1}+\cdots+a_1(1)z+a_0(1)$. We extend this to systems of convolution equations, which need not be of polynomial type.
2437 25.02.2006 Reinhard Farwig
Ri Myong-Hwan
The Resolvent Problem and $H^\infty$-calculus of the Stokes Operator in Unbounded Cylinders MSC: 35Q30; 76D07
It is proved that the Stokes operator in $L^q$-space on an infinite cylindrical domain of $R^n,¸n\geq 3,$ with several exits to infinity generates a bounded and exponentially decaying analytic semigroup and admits a bounded $H^\infty$-calculus. For the resolvent estimates, the Stokes resolvent system with a prescribed divergence in an infinite straight cylinder with bounded cross-section $\Sigma$ is studied in $L^q(R;L^r_\omega(\Sigma))$ where $1< q , r<\infty$ and $\omega\in A_r(R^{n-1})$ is an arbitrary Muckenhoupt weight. The proofs use cut-off techniques and the theory of Schauder decomposition of {\em UMD} spaces based on ${\cal R}$-boundedness of operator families and on square function estimates involving Muckenhoupt weights.
2436 01.02.2006 Christoph Wockel The Samelson Product and Rational Homotopy of Gauge Groups MSC: 57T20; 57S05; 81R10; 55P62
This paper is on the connecting homomorphism in the long exact homotopy sequence of the evaluation fibration $\tx{ev}_{p_{0}}:C(P,K)^{K}\to K$, where $C(P,K)^{K}\cong\Gau(\cP)$ is the gauge group of a continuous principal $K$-bundle $P$ over a closed orientable surface or a sphere. We show that in this cases the connecting homomorphism in the corresponding long exact homotopy sequence is given in terms of the Samelson product. As applications, we exploit this correspondence to get an explicit formula for $\pi_{2}(\Gau(\cP_{k}))$, where $\cP_{k}$ denotes the principal \mbox{$\bS^{3}$-bundle} over $\bS^{4}$ of Chern number $k$ and derive explicit formulae for the rational homotopy groups $\pi_{n}(\Gau(\cP))\otimes \Q$.
2435 01.01.2006 Lahti, Pekka
Maczynski, Maciej J.
Scheffold, Egon
Ylinen, Kari
Noise sequences of infinite matrices and their applications to the characterization of the canonical phase and box localization observables Noise sequences of infinite matrices associated with covariant phase and box localization observables are defined and determined. The canonical observables are characterized within the relevant classes of observables as those with asymptotically minimal of minimal noise, i.e., the noise tending to $0$ or having the value $0$.
2434 17.01.2006 Wille, Rudolf Methods of Conceptual Knowledge Processing MSC: 03B
The offered Methods of Conceptual Knowledge Processing are procedures which are well-planed to mean and purpose and therewith lead to skills for solving practical tasks. The used means and skills have been mainly created as translations of mathematical means and skills of Formal Concept Analysis. Those transdisciplinary translations may be understood as transformations from mathematical thinking, dealing with potential realities, to logical thinking, dealing with actual realities. Each of the 38 presented methods is discussed in a general language of logical nature, while citations give links to the underlying mathematical background. Applications of the methods are demonstrated by concrete examples mostly taken from the literature to which explicit references are given.
2433 09.01.2006 Neeb, Karl-Hermann Monastir Summer School: Infinite-Dimensional Lie Groups MSC: 22E65
These are lecture notes of a course given at a summer school in Monastir in July 2005. The main purpose of this course is to present some of the main ideas of infinite-dimensional Lie theory and to explain how it differs from the finite-dimensional theory. In the introductory section, we present some of the main types of infinite-dimensional Lie groups: linear Lie groups, groups of smooth maps and groups of diffeomorphisms. We then turn in some more detail to manifolds modeled on locally convex spaces and the corresponding calculus (Section II). In Section III, we present some basic Lie theory for locally convex Lie groups. The Fundamental Theorem for Lie group-valued-functions on manifolds and some of its immediate applications are discussed in Section IV. For many infinite-dimensional groups, the exponential function behaves worse than for finite-dimensional ones or Banach--Lie groups. Section V is devoted to the class of locally exponential Lie groups, i.e., those for which the exponential function is a local diffeomorphism in 0. We conclude these notes with a brief discussion of the integrability problem for locally convex Lie algebras: When is a locally convex Lie algebra the Lie algebra of a global Lie group?
2432 04.01.2006 Burgmann, Christian
Wille, Rudolf
The Basic Theorem on Preconcept Lattices MSC: 03B; 06D10
Preconcept Lattices are identified to be (up to isomorphism) the completely distributive complete lattices in which the supremum of all atoms is equal or greater than the infimum of all coatoms. This is a consequence of the Basic Theorem on Preconcept Lattices, which also offers means for checking line diagrams of preconcept lattices.