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Fri, 10 Nov 2017 09:47:24 +0100Fri, 10 Nov 2017 09:47:24 +0100Polynomial functions of modular lattices
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5060
A polynomial function \(f : L \to L\) of a lattice \(\mathcal{L}\) = \((L; \land, \lor)\) is generated by the identity function id \(id(x)=x\) and the constant functions \(c_a (x) = a\) (for every \(x \in L\)), \(a \in L\) by applying the operations \(\land, \lor\) finitely often. Every polynomial function in one or also in several variables is a monotone function of \(\mathcal{L}\).
If every monotone function of \(\mathcal{L}\)is a polynomial function then \(\mathcal{L}\) is called orderpolynomially complete. In this paper we give a new characterization of finite order-polynomially lattices. We consider doubly irreducible monotone functions and point out their relation to tolerances, especially to central relations. We introduce chain-compatible lattices
and show that they have a non-trivial congruence if they contain a finite interval and an infinite chain. The consequences are two new results. A modular lattice \(\mathcal{L}\) with a finite interval is order-polynomially complete if and only if \(\mathcal{L}\) is finite projective geometry. If \(\mathcal{L}\) is simple modular lattice of infinite length then every nontrivial interval is of infinite length and has the same cardinality as any other nontrivial interval of \(\mathcal{L}\). In the last sections we show the descriptive power of polynomial functions of
lattices and present several applications in geometry.Dietmar Schweigertreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5060Fri, 10 Nov 2017 09:47:24 +0100On derived varieties
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5059
Derived varieties play an essential role in the theory of hyperidentities. In [11] we have shown that derivation diagrams are a useful tool in the analysis of derived algebras and varieties. In this paper this tool is developed further in order to use it for algebraic constructions of derived algebras. Especially the operator \(S\) of subalgebras, \(H\) of homomorphic irnages and \(P\) of direct products are studied. Derived groupoids from the groupoid \(N or (x,y)\) = \(x'\wedge y'\) and from abelian groups are considered. The latter class serves as an example for fluid algebras and varieties. A fluid variety \(V\) has no derived variety as a subvariety and is introduced as a counterpart for solid varieties. Finally we use a property of the commutator of derived algebras in order to show that solvability and nilpotency are preserved under derivation.Dietmar Schweigertreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5059Fri, 10 Nov 2017 09:22:23 +0100Error estimates for Tikhonov regularization with unbounded regularizing operators
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5056
It is shown that Tikhonov regularization for ill- posed operator equation
\(Kx = y\) using a possibly unbounded regularizing operator \(L\) yields an orderoptimal algorithm with respect to certain stability set when the regularization parameter is chosen according to the Morozov's discrepancy principle. A more realistic error estimate is derived when the operators \(K\) and \(L\) are related to a Hilbert scale in a suitable manner. The result includes known error estimates for ordininary Tikhonov regularization and also the estimates available under the Hilbert scale approach.M. Thamban Nairreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5056Thu, 09 Nov 2017 12:01:16 +0100On the expected number of shadow vertices of the convex hull of random points
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5051
Let \(a_1,\dots,a_m\) be independent random points in \(\mathbb{R}^n\) that are independent and identically distributed spherically symmetrical in \(\mathbb{R}^n\). Moreover, let \(X\) be the random polytope generated as the convex hull of \(a_1,\dots,a_m\) and let \(L_k\) be an arbitrary \(k\)-dimensional
subspace of \(\mathbb{R}^n\) with \(2\le k\le n-1\). Let \(X_k\) be the orthogonal projection image of \(X\) in \(L_k\). We call those vertices of \(X\), whose projection images in \(L_k\) are vertices of \(X_k\)as well shadow vertices of \(X\) with respect to the subspace \(L_k\) . We derive a distribution independent sharp upper bound for the expected number of shadow vertices of \(X\) in \(L_k\).Karl-Heinz Küferreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5051Thu, 09 Nov 2017 10:49:33 +0100A comparison method for expectations of a class of continuous polytope functionals
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5047
Let \(a_1,\dots,a_n\) be independent random points in \(\mathbb{R}^d\) spherically symmetrically but not necessarily identically distributed. Let \(X\) be the random polytope generated as the convex hull of \(a_1,\dots,a_n\) and for any \(k\)-dimensional subspace \(L\subseteq \mathbb{R}^d\) let \(Vol_L(X) :=\lambda_k(L\cap X)\) be the volume of \(X\cap L\) with respect to the \(k\)-dimensional Lebesgue measure \(\lambda_k, k=1,\dots,d\). Furthermore, let \(F^{(i)}\)(t):= \(\bf{Pr}\) \(\)(\(\Vert a_i \|_2\leq t\)),
\(t \in \mathbb{R}^+_0\) , be the radial distribution function of \(a_i\). We prove that the expectation
functional \(\Phi_L\)(\(F^{(1)}, F^{(2)},\dots, F^{(n)})\) := \(E(Vol_L(X)\)) is strictly decreasing in
each argument, i.e. if \(F^{(i)}(t) \le G^{(i)}(t)t\), \(t \in {R}^+_0\), but \(F^{(i)} \not\equiv G^{(i)}\), we show \(\Phi\) \((\dots, F^{(i)}, \dots\)) > \(\Phi(\dots,G^{(i)},\dots\)). The proof is clone in the more general framework
of continuous and \(f\)- additive polytope functionals.Karl-Heinz Küferreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5047Wed, 08 Nov 2017 13:33:06 +0100New Integrals for \(\zeta(s)\zeta(s+1)\)
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5038
Andreas Guthmannreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5038Tue, 07 Nov 2017 11:46:13 +0100Derived Varieties of Semigroups and Groupoids
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4889
Dietmar Schweigert; S.L. Wismathreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4889Thu, 19 Oct 2017 11:35:28 +0200Determinantal Rational Surface Singularities
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4866
In this paper we give explicit equations for determinantal rational surface singularities and prove dimension formulas for the \(T^1\) and \(T^2\) for those singularities.Theo de Jongreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4866Tue, 17 Oct 2017 11:38:01 +0200Superlinear convergence rates for the Lanczos method applied to elliptic operators
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4864
This paper investigates the convergence of the Lanczos method for computing the smallest eigenpair of a selfadjoint elliptic differential operator via inverse iteration (without shifts).
Superlinear convergence rates are established, and their sharpness is investigated for a simple model problem. These results are illustrated numerically for a more difficult problem.Martin Hankereporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4864Tue, 17 Oct 2017 09:10:44 +0200Regularizing properties of a truncated Newton-CG algorithm for nonlinear inverse problems
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4863
This paper develops truncated Newton methods as an appropriate tool for nonlinear inverse problems which are ill-posed in the sense of Hadamard. In each Newton step an approximate solution for the linearized problem is computed with the conjugate gradient method as an inner iteration. The conjugate gradient iteration is terminated when the residual has been reduced to a prescribed percentage. Under certain assumptions on the nonlinear operator it is shown that the algorithm converges and is stable if the discrepancy principle is used to terminate the outer iteration.
These assumptions are fulfilled , e.g., for the inverse problem of identifying the diffusion coefficient in a parabolic differential equation from distributed data.Martin Hankereporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4863Tue, 17 Oct 2017 08:51:54 +0200Nonstationary lterated Tikhonov Regularization
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4862
A convergence rate is established for nonstationary iterated Tikhonov regularization, applied to ill-posed problems involving closed, densely defined linear operators, under general conditions on the iteration parameters. lt is also shown that an order-optimal accuracy is attained when a certain a posteriori stopping rule is used to determine the iteration number.Martin Hanke; C.W. Groetschreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4862Mon, 16 Oct 2017 16:20:02 +0200A regularization Levenberg-Marquardt scheme, with applications to inverse groundwater filtration problems
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4861
The first part of this paper studies a Levenberg-Marquardt scheme for nonlinear inverse problems where the corresponding Lagrange (or regularization) parameter is chosen from an inexact Newton strategy. While the convergence analysis of standard implementations based on trust region strategies always requires the invertibility of the Fréchet derivative of the nonlinear operator at the exact solution, the new Levenberg-Marquardt scheme is suitable for ill-posed problems as long as the Taylor remainder is of second order in the interpolating metric between the range and dornain
topologies. Estimates of this type are established in the second part of the paper for ill-posed parameter identification problems arising in inverse groundwater hydrology. Both, transient and steady state data are investigated. Finally, the numerical performance of the new Levenberg-Marquardt scheme is
studied and compared to a usual implementation on a realistic but synthetic 2D model problem from the engineering literature.Martin Hankereporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4861Mon, 16 Oct 2017 15:27:12 +0200Infinitesimal module deformations in the Thom-Sebastiani Problem
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4851
Florian Enescu; Gerhard Pfister; Dorin Popescureporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4851Mon, 16 Oct 2017 09:24:50 +0200On the Solution Region for Certain Scheduling Problems with Preemption
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4839
The paper deals with parallel-machine and open-shop scheduling problems with preemptions and arbitrary nondecreasing objective function. An approach to describe
the solution region for these problems and to reduce them to minimization problems on polytopes is proposed. Properties of the solution regions for certain problems are investigated. lt is proved that open-shop problems with unit processing times are equivalent to certain parallel-machine problems, where preemption is allowed at arbitrary time. A polynomial algorithm is presented transforming a schedule of one type into a schedule of the other type.Heidemarie Bräsel; Natalia Shakhlevichreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4839Fri, 13 Oct 2017 14:01:28 +0200Das dynamische Travelling-Salesman Problem
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1144
Das TSP wird auf zeitabhängige Kosten und Wegelängen verallgemeinert, der Komplexitätstatus untersucht, verschiedene Formulierungen verglichen, Spezialfälle untersucht und ein auf Lagrange-Relaxation und Branch&Bound beruhendes exaktes Lösungsverfahren von Lucena erweitert, implementiert und getestet. Für das TDTSP wird die Dimension des ganzzahligen Polyeders bestimmt.Martin C. Müllerdiplomhttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1144Tue, 05 Dec 2000 00:00:00 +0100Singular Optimal Control - The State of the Art
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/603
The purpose of this paper is to present the state of the art in singular optimal control. If the Hamiltonian in an interval \([t_1,t_2]\) is independent of the control we call the control in this interval singular. Singular optimal controls appear in many applications so that research has been motivated since the 1950s. Often optimal controls consist of nonsingular and singular parts where the junctions between these parts are mostly very difficult to find. One section of this work shows the actual knowledge about the location of the junctions and the behaviour of the control at the junctions. The definition and the properties of the orders (problem order and arc order), which are important in this context, are given, too. Another chapter considers multidimensional controls and how they can be treated. An alternate definition of the orders in the multidimensional case is proposed and a counterexample, which confirms a remark given in the 1960s, is given. A voluminous list of optimality conditions, which can be found in several publications, is added. A strategy for solving optimal control problems numerically is given, and the existing algorithms are compared with each other. Finally conclusions and an outlook on the future research is given.Volker Michelpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/603Fri, 23 Jun 2000 00:00:00 +0200The C Programmes for "Numerical Methods (Programmes and Implementation)"
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/594
Michael Schreinerpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/594Fri, 23 Jun 2000 00:00:00 +0200Multidimensionale Systemtheorie
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/598
Die Theorie der mehrdimensionalen Systeme ist ein relativ junges Forschungsgebiet innerhalb der Systemtheorie, erste Arbeiten stammen aus den 70er Jahren. Hauptmotiv für das Studium multidimensionaler Systeme war die Notwendigkeit einer Erweiterung der Theorie der digitalen Filter, die in der klassischen, eindimensionalen Signalverarbeitung (zeitabhängige Signale) Anwendung finden, auf den Bereich der Bildverarbeitung, also auf zweidimensionale Signale.; Die Vorlesung beschäftigt sich daher in ihrem ersten Teil mit skalaren zweidimensionalen Systemen und beschränkt sich im wesentlichen auf den linearen Fall. Untersucht werden zweidimensionale Filter, ihre wichtigsten Eigenschaften, Kausalität und Stabilität, sowie ihre Zustandsraum- realisierungen, etwa die Modelle von Roesser und Fornasini-Marchesini. Parallelen und Unterschiede zur eindimensionalen Systemtheorie werden betont.; Im zweiten Teil der Vorlesung werden allgemeine höherdimensionale und multivariable Systeme behandelt. Für diese Systeme erweist sich der von Jan Willems begründete Zugang zur Systemtheorie, der sogenannte behavioral approach, als zweckmäßig. Grundlegende Ideen dieses Ansatzes sowie eine der wichtigsten Methoden zum Rechnen mit Polynomen in mehreren Variablen, die Theorie der Gröbnerbasen, werden vorgestellt.E. Zerzpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/598Fri, 23 Jun 2000 00:00:00 +0200Asymptotic Behaviour of Self-Organizing Maps with Non-Uniform Stimuli Distribution
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/600
Here the almost sure convergence of one dimensional Kohonen" s algorithm in its general form, namely, 2k point neightbour setting with a non-uniform stimuli distribution is proved. We show that the asymptotic behaviour of the algorithm is governed by a cooperative system of differential equations which in general is irreducible. The system of differential equation has an asymptotically stable fixed point which a compact subset of its domain of attraction will be visited by the state variable Xn infinitely often.Ali A. Sadeghipreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/600Fri, 23 Jun 2000 00:00:00 +0200Heuristics for the K-Cardinality Tree and Subgraph Problems
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/524
In this paper we consider the problem of finding in a given graph a minimal weight subtree of connected subgraph, which has a given number of edges. These NP-hard combinatorial optimization problems have various applications in the oil industry, in facility layout and graph partitioning. We will present different heuristic approaches based on spanning tree and shortest path methods and on an exact algorithm solving the problem in polynomial time if the underlying graph is a tree. Both the edge- and node weighted case are investigated and extensive numerical results on the behaviour of the heuristics compared to optimal solutions are presented. The best heuristic yielded results within an error margin of less than one percent from optimality for most cases. In a large percentage of tests even optimal solutions have been found.Matthias Ehrgott; Horst. W. Hamacher; J. Freitag; F. Maffiolipreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/524Mon, 03 Apr 2000 00:00:00 +0200Deformation Analysis Using Navier Spline Interpolation
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/567
The static deformation of the surface of the earth caused by surface pressure like the water load of an ocean or an artificial lake is discussed. First a brief mention is made on the solution of the Boussenesq problem for an infinite halfspace with the elastic medium to be assumed as homogeneous and isotropic. Then the elastic response for realistic earth models is determinied by spline interpolation using Navier splines. Major emphasis is on the derteminination of the elastic field caused by water loads from surface tractions on the (real) earth" s surface. Finally the elastic deflection of an artificial lake assuming a homogeneous isotropic crust is compared for both evaluation methods.Willi Freeden; E. Groten; Michael Schreiner; W. Söhhne; M. Tücckspreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/567Mon, 03 Apr 2000 00:00:00 +0200Particle Methods for Evolution Equations
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/602
Michael Junk; Axel Klar; Jens Struckmeier; Sudarshan Tiwaripreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/602Mon, 03 Apr 2000 00:00:00 +0200Spherical Wavelet Transform and its Discretization
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/555
A continuous version of spherical multiresolution is described, starting from continuous wavelet transform on the sphere. Scale discretization enables us to construct spherical counterparts to Daubechies wavelets and wavelet packets (known from Euclidean theory). Essential tool is the theory of singular integrals on the sphere. It is shown that singular integral operators forming a semigroup of contraction operators of class (Co) (like Abel-Poisson or Gauß-Weierstraß operators) lead in canonical way to (pyramidal) algorithms.Willi Freeden; U. Windheuserpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/555Mon, 03 Apr 2000 00:00:00 +0200An Adaptive Hierarchical Approximation Method on the Sphere Using Axisymmetric Locally Supported Basis Functions
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/561
The paper discusses the approximation of scattered data on the sphere which is one of the major tasks in geomathematics. Starting from the discretization of singular integrals on the sphere the authors devise a simple approximation method that employs locally supported spherical polynomials and does not require equidistributed grids. It is the basis for a hierarchical approximation algorithm using differently scaled basis functions, adaptivity and error control. The method is applied to two examples one of which is a digital terrain model of Australia.Willi Freeden; J. Fröhhlich; R. Brandpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/561Mon, 03 Apr 2000 00:00:00 +0200Gradiometry - an Inverse Problem in Modern Satellite Geodesy
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/595
Satellite gradiometry and its instrumentation is an ultra-sensitive detection technique of the space gravitational gradient (i.e. the Hesse tensor of the gravitational potential). Gradeometry will be of great significance in inertial navigation, gravity survey, geodynamics and earthquake prediction research. In this paper, satellite gradiometry formulated as an inverse problem of satellite geodesy is discussed from two mathematical aspects: Firstly, satellite gradiometry is considered as a continuous problem of harmonic downward continuation. The space-borne gravity gradients are assumed to be known continuously over the satellite (orbit) surface. Our purpose is to specify sufficient conditions under which uniqueness and existence can be guaranteed. It is shown that, in a spherical context, uniqueness results are obtainable by decomposition of the Hesse matrix in terms of tensor spherical harmonics. In particular, the gravitational potential is proved to be uniquely determined if second order radial derivatives are prescribed at satellite height. This information leads us to a reformulation of satellite gradiometry as a (Fredholm) pseudodifferential equation of first kind. Secondly, for a numerical realization, we assume the gravitational gradients to be known for a finite number of discrete points. The discrete problem is dealt with classical regularization methods, based on filtering techniques by means of spherical wavelets. A spherical singular integral-like approach to regularization methods is established, regularization wavelets are developed which allow the regularization in form of a multiresolution analysis. Moreover, a combined spherical harmonic and spherical regularization wavelet solution is derived as an appropriate tool in future (global and local) high-presision resolution of the earth" s gravitational potential.Willi Freeden; F. Schneider; Michael Schreinerpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/595Mon, 03 Apr 2000 00:00:00 +0200