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Wed, 08 Nov 2017 16:03:14 +0100Wed, 08 Nov 2017 16:03:14 +0100An improved asymptotic analysis of the expected number of pivot steps required by the simplex algorithm
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5049
Let \(a_1,\dots,a_m\) be i.i .d. vectors uniform on the unit sphere in \(\mathbb{R}^n\), \(m\ge n\ge3\) and let \(X\):= {\(x \in \mathbb{R}^n \mid a ^T_i x\leq 1\)} be the random polyhedron generated by. Furthermore, for linearly independent vectors \(u\), \(\bar u\) in \(\mathbb{R}^n\), let \(S_{u, \bar u}(X)\) be the number of shadow vertices of \(X\) in \(span (u, \bar u\)). The paper provides an asymptotic expansion of the expectation value \(E (S_{u, \bar u})\) for fixed \(n\) and \(m\to\infty\). The first terms of the expansion are given explicitly. Our investigation of \(E (S_{u, \bar u})\) is closely connected to Borgwardt's probabilistic analysis of the shadow vertex algorithm - a parametric variant of the simplex algorithm. We obtain an improved asymptotic upper bound for the number of pivot steps required by the shadow vertex algorithm for uniformly on the sphere distributed data.Karl-Heinz Küferreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5049Wed, 08 Nov 2017 16:03:14 +0100A Note on Approximation Algorithms for the Multicriteria \(\Delta\)-TSP
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5031
The Tree and Christofides heuristic are weil known 1- and \(\frac{1} {2}\)- approximate algorithms for the \(\Delta\)-TSP. In this note their performance for the multicriteria case is described, depending on the norm in \(\mathbb{R}^Q\) in case of \(Q\) criteria.Matthias Ehrgott; Alexander Feldmannreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5031Mon, 06 Nov 2017 11:45:28 +0100Mathematik als Vehikel der Philosophie und Weltanschauung
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4887
Knut Radbruchreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4887Thu, 19 Oct 2017 10:05:34 +0200On Matroids with Multiple Objectives
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4850
In this paper we investigate two optimization problems for matroids with multiple objective functions, namely finding the pareto set and the max-ordering problem which conists in finding a basis such that the largest objective value is minimal. We prove that the decision versions of both problems are NP-complete. A solution procedure for the max-ordering problem is presented and a result on the relation of the solution sets of the two problems is given. The main results are a characterization of pareto bases by a basis exchange property and finally a connectivity result for proper pareto solutions.Matthias Ehrgottreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4850Mon, 16 Oct 2017 09:14:56 +0200Lexicographic Max-Ordering - A Solution Concept for Multicriteria Combinatorial Optimization
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4849
In this paper we will introduce the concept of lexicographic max-ordering solutions for multicriteria combinatorial optimization problems. Section 1 provides the basic notions of
multicriteria combinatorial optimization and the definition of lexicographic max-ordering solutions. In Section 2 we will show that lexicographic max-ordering solutions are pareto optimal as well as max-ordering optimal solutions. Furthermore lexicographic max-ordering solutions can be used to characterize the set of pareto solutions. Further properties of lexicographic max-ordering solutions are given. Section 3 will be devoted to algorithms. We give a polynomial time algorithm for the two criteria case where one criterion is a sum and one is a bottleneck objective function, provided that the one criterion sum problem is solvable in polynomial time. For bottleneck functions an algorithm for the general case of Q criteria is presented.Matthias Ehrgottreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4849Mon, 16 Oct 2017 08:57:51 +0200Connectedness of Efficient Solutions in Multiple Criteria Combinatorial Optimization
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4844
In multiple criteria optimization an important research topic is the topological structure of the set \( X_e \) of efficient solutions. Of major interest is the connectedness of \( X_e \), since it would allow the determination of \( X_e \) without considering non-efficient solutions in the
process. We review general results on the subject,including the connectedness result for efficient solutions in multiple criteria linear programming. This result can be used to derive a definition of connectedness for discrete optimization problems. We present a counterexample to a previously stated result in this area, namely that the set of efficient solutions of the shortest path problem is connected. We will also show that connectedness does not hold for another important problem in discrete multiple criteria optimization: the spanning tree problem.Matthias Ehrgott; Kathrin Klamrothreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4844Mon, 16 Oct 2017 08:41:05 +0200An Overview of the Method of Smoothed Particle Hydrodynamics
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/585
This report is intended to provide an introduction to the method of SmoothedParticle Hydrodynamics or SPH. SPH is a very versatile, fully Lagrangian, particle based code for solving fluid dynamical problems. Many technical aspects of the method are explained which can then be employed to extend the application of SPH to new problems.J.P. Morrispreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/585Fri, 23 Jun 2000 00:00:00 +0200Direct Coupling of Fluid and Kinetic Equations: I
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/556
J. Schneiderpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/556Wed, 07 Jun 2000 00:00:00 +0200A Model for the Cloudiness of Fabrics
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/562
Cloudy inhomogenities in artificial fabrics are graded by a fast method which is based on a Laplacian pyramid decomposition of the fabric image. This band-pass representation takes into account the scale character of the cloudiness. A quality measure of the entire cloudiness is obtained as a weighted mean over the variances of all scales.Joachim Weickertarticlehttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/562Wed, 07 Jun 2000 00:00:00 +0200Locally Supported Kernels for Spherical Spline Interpolation
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/572
By the use of locally supported basis functions for spherical spline interpolation the applicability of this approximation method is spread out since the resulting interpolation matrix is sparse and thus efficient solvers can be used. In this paper we study locally supported kernels in detail. Investigations on the Legendre coefficients allow a characterization of the underlying Hilbert space structure. We show now spherical spline interpolation with polynomial precision can be managed with locally supported kernels, thus giving the possibility to combine approximation techniques based on spherical harmonic expansions with those based on locally supported kernels.Michael Schreinerpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/572Wed, 07 Jun 2000 00:00:00 +0200On a New Condition for Strictly Positive Definite Functions on Spheres
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/573
Recently, Xu and Cheney (1992) have proved that if all the Legendre coefficients of a zonal function defined on a sphere are positive then the function is strictly positive definite. It will be shown in this paper, that even if finitely many of the Legendre coefficients are zero, the strict positive definiteness can be assured. The results are based on approximation properties of singular integrals, and provide also a completely different proof of the results ofXu and Cheney.Michael Schreinerpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/573Wed, 07 Jun 2000 00:00:00 +0200Learning and Replication of Periodic Signals in Neural-Like Networks
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/575
The paper describes the concepts and background theory for the analysis of a neural-like network for learning and replication of periodic signals containing a finite number of distinct frequency components. The approach is based on the combination of ideas from dynamic neural networks and systems and control theory where concepts of dynamics, adaptive control and tracking of specified time signals are fundamental. The proposed procedure is a two stage process consisting of a learning phase when the network is driven by the required signal followed by a replication phase where the network operates in an autonomous feedback mode whilst continuing to generate the required signal to a desired acccuracy for a specified time. The analysis draws on currently available control theory and, in particular, on concepts from model reference adaptive control.R. Reinke; D.H. Owens; D. Prätzel-Wolterspreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/575Wed, 07 Jun 2000 00:00:00 +0200Lectures on the Problem of Space and Time in Einstein" s Theory of Gravitation
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/578
C. Müllerpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/578Wed, 07 Jun 2000 00:00:00 +0200The Solution of Linear Inverse Problems in Satellite Geodesy by Means of Spherical Spline Approximation
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/581
In this paper we consider a certain class of geodetic linear inverse problems LambdaF=G in a reproducing kernel Hilbert space setting to obtain a bounded generalized inverse operator Lambda. For a numerical realization we assume G to be given at a finite number of discrete points to which we employ a spherical spline interpolation method adapted to the Hilbertspaces. By applying Lambda to the obtained spline interpolant we get an approximation of the solution F. Finally our main task is to show some properties of the approximated solution and to prove convergence results if the data set increases.F. Schneiderpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/581Wed, 07 Jun 2000 00:00:00 +0200The Mathematical Simulation of an Electrolytic Cell for Aluminium Production
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/583
A. Buikis; H. Kalispreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/583Wed, 07 Jun 2000 00:00:00 +0200Periodic Signals in Neural-Like Networks - an Averaging Analysis
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/587
The paper describes the concepts and background theory of the analysis of a neural-like network for the learning and replication of periodic signals containing a finite number of distinct frequency components. The approach is based on a two stage process consisting of a learning phase when the network is driven by the required signal followed by a replication phase where the network operates in an autonomous feedback mode whilst continuing to generate the required signal to a desired accuracy for a specified time. The analysis focusses on stability properties of a model reference adaptive control based learning scheme via the averaging method. The averaging analysis provides fast adaptive algorithms with proven convergence properties.R. Reinke; D. Prätzel-Wolterspreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/587Wed, 07 Jun 2000 00:00:00 +0200Holomorphie und Laplace Transformation banachraumwertiger Funktionen
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/480
Peter Vietendoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/480Mon, 03 Apr 2000 00:00:00 +0200A: New Wavelet Methods for Approximating Harmonic Functions; B: Satellite Gradiometry - from Mathematical and Numerical Point of View
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/537
Some new approximation methods are described for harmonic functions corresponding to boundary values on the (unit) sphere. Starting from the usual Fourier (orthogonal) series approach, we propose here nonorthogonal expansions, i.e. series expansions in terms of overcomplete systems consisting of localizing functions. In detail, we are concerned with the so-called Gabor, Toeplitz, and wavelet expansions. Essential tools are modulations, rotations, and dilations of a mother wavelet. The Abel-Poisson kernel turns out to be the appropriate mother wavelet in approximation of harmonic functions from potential values on a spherical boundary.Willi Freeden; Michael Schreinerpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/537Mon, 03 Apr 2000 00:00:00 +0200Second Order Scheme for the Spatially Homogeneous Boltzmann Equation with Maxwellian Molecules
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/557
In the standard approach, particle methods for the Boltzmann equation are obtained using an explicit time discretization of the spatially homogeneous Boltzmann equation. This kind of discretization leads to a restriction of the discretization parameter as well as on the differential cross section in the case of the general Boltzmann equation. Recently, it was shown, how to construct an implicit particle scheme for the Boltzmann equation with Maxwellian molecules. The present paper combines both approaches using a linear combination of explicit and implicit discretizations. It is shown that the new method leads to a second order particle method, when using an equiweighting of explicit and implicit discretization.Jens Struckmeier; Konrad Steinerpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/557Mon, 03 Apr 2000 00:00:00 +0200Numerical Simulation of the Stationary One-Dimensional Boltzmann Equation by Particle Methods
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/558
The paper presents a numerical simulation technique - based on the well-known particle methods - for the stationary, one-dimensional Boltzmann equation for Maxwellian molecules. In contrast to the standard splitting methods, where one works with the instationary equation, the current approach simulates the direct solution of the stationary problem. The model problem investigated is the heat transfer between two parallel plates in the rarefied gas regime. An iteration process is introduced which leads to the stationary solution of the exact - space discretized - Boltzmann equation, in the sense of weak convergence.Jens Struckmeier; A.V. Bobylevpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/558Mon, 03 Apr 2000 00:00:00 +0200Normalized Coprime Factorizations in Continuous and Discrete Time - A Joint State-Space Approach
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/559
Based on state-space formulas for coprime factorizations over ... and an algebraic characterization of J-inner functions, normalized doubly-coprime factorizations for different classes of continuous- and discrete-time transfer functions are derived by using a single general construction method. The parametrization of the factors is in terms of the stabilizing solutions of general degenerate continuous- respectively discrete-time Riccati equations, which are obtained by examining state-space representations of J-normalized factor matrices.Jörg Hoffmannpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/559Mon, 03 Apr 2000 00:00:00 +0200Wavelet Thresholding: Beyond the Gaussian I.I.D. Situation
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/564
With this article we first like to give a brief review on wavelet thresholding methods in non-Gaussian and non-i.i.d. situations, respectively. Many of these applications are based on Gaussian approximations of the empirical coefficients. For regression and density estimation with independent observations, we establish joint asymptotic normality of the empirical coefficients by means of strong approximations. Then we describe how one can prove asymptotic normality under mixing conditions on the observations by cumulant techniques.; In the second part, we apply these non-linear adaptive shrinking schemes to spectral estimation problems for both a stationary and a non-stationary time series setup. For the latter one, in a model of Dahlhaus on the evolutionary spectrum of a locally stationary time series, we present two different approaches. Moreover, we show that in classes of anisotropic function spaces an appropriately chosen wavelet basis automatically adapts to possibly different degrees of regularity for the different directions. The resulting fully-adaptive spectral estimator attains the rate that is optimal in the idealized Gaussian white noise model up to a logarithmic factor.Michael H. Neumann; Rainer von Sachsarticlehttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/564Mon, 03 Apr 2000 00:00:00 +0200Stochastic Reconstruction of Loading Histories from a Rainflow Matrix
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/565
This paper is devoted to the mathematica l description of the solution of the so-called rainflow reconstruction problem, i.e. the problem of constructing a time series with an a priori given rainflow m atrix. The algorithm we present is mathematically exact in the sense that no app roximations or heuristics are involved. Furthermore it generates a uniform distr ibution of all possible reconstructions and thus an optimal randomization of the reconstructed series. The algorithm is a genuine on-line scheme. It is easy adj ustable to all variants of rainflow such as sysmmetric and asymmetric versions a nd different residue techniques.Klaus Dreßler; Michael Hack; Wilhelm Krügerarticlehttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/565Mon, 03 Apr 2000 00:00:00 +0200Fatigue Lifetime Estimation Based on Rainflow Counted Data Using the Local Strain Approach
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/566
In the automotive industry both the loca l strain approach and rainflow counting are well known and approved tools in the numerical estimation of the lifetime of a new developed part especially in the automotive industry. This paper is devoted to the combination of both tools and a new algorithm is given that takes advantage of the inner structure of the most used damage parameters.Klaus Dreßler; Michael Hackarticlehttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/566Mon, 03 Apr 2000 00:00:00 +0200Asymptotic-Induced Domain Decomposition Methods for Kinetic and Drift Diffusion Semiconductor Equations
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/568
This paper deals with domain decomposition methods for kinetic and drift diffusion semiconductor equations. In particular accurate coupling conditions at the interface between the kinetic and drift diffusion domain are given. The cases of slight and strong nonequilibrium situations at the interface are considered and some numerical examples are shown.Axel Klarpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/568Mon, 03 Apr 2000 00:00:00 +0200