## Fachbereich Mathematik

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- Elastoplastizität (4)
- Elastoplasticity (3)
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- Betriebsfestigkeit (1)
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- Biot-Savart operator (1)
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- elastoplastic BVP (1)
- elastoplasticity (1)
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- linear optimization (1)
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- locally supported (Green’s) vector wavelets (1)
- locally supported wavelets (1)
- location theory (1)
- magnetic field (1)
- mechanism design (1)
- method of fundamental solutions (1)
- minimaler Schnittbaum (1)
- minimum cut tree (1)
- multiple objective optimization (1)
- network congestion game (1)
- neural network (1)
- nonparametric regression (1)
- online optimization (1)
- piezoelectric periodic surface acoustic wave filters (1)
- poroelasticity (1)
- price of anarchy (1)
- price of stability (1)
- public transportation (1)
- quickest path (1)
- rate-independency (1)
- robust network flows (1)
- selfish routing (1)
- sensitivities (1)
- sequential test (1)
- series-parallel graphs (1)
- simplex (1)
- single layer kernel (1)
- spherical decomposition (1)
- stop- and play-operators (1)
- strong equilibria (1)
- total latency (1)
- vector spherical harmonics (1)
- vectorial wavelets (1)
- wave propagation (1)
- weak solution theory (1)
- weakly/ strictly pareto optima (1)

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.

In this article we give a sufficient condition that a simply connected flexible body does not penetrate itself, if it is subjected to a continuous deformation. It is shown that the deformation map is automatically injective, if it is just locally injective and injective on the boundary of the body. Thereby, it is very remarkable that no higher regularity assumption than continuity for the deformation map is required. The proof exclusively relies on homotopy methods and the Jordan-Brouwer separation theorem.

The notion of Q-Gorenstein smoothings has been introduced by Kollar. ([KoJ], 6.2.3). This notion is essential for formulating Kollar's conjectures on smoothing components for rational surface singularities. He conjectures, loosely speaking, that every smoothing of a rational surface singularity can be obtained by blowing down a deformation of a partial resolution, this partial resolution having the property (among others) that the singularities occuring on it all have qG-smoothings. (For more details and precise statements see [Ko], ch. 6.). It is therefore of interest to construct singularities having qG-smoothings.

We present a generalization of Proth's theorem for testing certain large integers for primality. The use of Gauß sums leads to a much simpler approach to these primality criteria as compared to the earlier tests. The running time of the algorithms is bounded by a polynomial in the length of the input string. The applicability of our algorithms is linked to certain diophantine approximations of \(l\)-adic roots of unity.

In this article, we give an explicit homotopy between the solutions (i.e. stress, strain, displacement) of the quasistatic linear elastic and nonlinear elastoplastic boundary value problem, where we assume a linear kinematic hardening material law. We give error estimates with respect to the homotopy parameter.

A method to correct the elastic stress tensor at a fixed point of an elastoplastic body, which is subject to exterior loads, is presented and analysed. In contrast to uniaxial corrections (Neuber or ESED), our method takes multiaxial phenomena like ratchetting or cyclic hardening/softening into account by use of Jiang's model. Our numerical algorithm is designed for the case that the scalar load functions are piecewise linear and can be used in connection with critical plane/multiaxial rainflow methods in high cycle fatigue analysis. In addition, a local existence and uniqueness result of Jiang's equations is given.

In this article, we present an analytic solution for Jiang's constitutive model of elastoplasticity. It is considered in its stress controlled form for proportional stress loading under the assumptions that the one-to-one coupling of the yield surface radius and the memory surface radius is switched off, that the transient hardening is neglected and that the ratchetting exponents are constant.

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.

Let \(a_1, i:=1,\dots,m\), be an i.i.d. sequence taking values in \(\mathbb{R}^n\), whose convex hull is interpreted as a stochastic polyhedron \(P\). For a special class of random variables, which decompose additively relative to their boundary simplices, eg. the volume of \(P\), simple integral representations of its first two moments are given in case of rotationally symmetric distributions in order to facilitate estimations of variances or to quantify large deviations from the mean.

Let \(A\):= {\(a_i\mid i= 1,\dots,m\)} be an i.i.d. random sample in (\mathbb{R}^n\), which we consider a random polyhedron, either as the convex hull of the \(a_i\) or as the intersection of halfspaces {\(x \mid a ^T_i x\leq 1\)}. We introduce a class of polyhedral functionals we will call "additive-type functionals", which covers a number of polyhedral functionals discussed in different mathematical fields, where the emphasis in our contribution will be on those, which arise in linear optimization theory. The class of additive-type functionals is a suitable setting in order to unify and to simplify the asymptotic probabilistic analysis of first and second moments of polyhedral functionals. We provide examples of asymptotic results on expectations and on variances.

Algebraic Systems Theory
(2004)

Control systems are usually described by differential equations, but their properties of interest are most naturally expressed in terms of the system trajectories, i.e., the set of all solutions to the equations. This is the central idea behind the so-called "behavioral approach" to systems and control theory. On the other hand, the manipulation of linear systems of differential equations can be formalized using algebra, more precisely, module theory and homological methods ("algebraic analysis"). The relationship between modules and systems is very rich, in fact, it is a categorical duality in many cases of practical interest. This leads to algebraic characterizations of structural systems properties such as autonomy, controllability, and observability. The aim of these lecture notes is to investigate this module-system correspondence. Particular emphasis is put on the application areas of one-dimensional rational systems (linear ODE with rational coefficients), and multi-dimensional constant systems (linear PDE with constant coefficients).

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.

In this paper we develop monitoring schemes for detecting structural changes
in nonlinear autoregressive models. We approximate the regression function by a
single layer feedforward neural network. We show that CUSUM-type tests based
on cumulative sums of estimated residuals, that have been intensively studied
for linear regression in both an offline as well as online setting, can be extended
to this model. The proposed monitoring schemes reject (asymptotically) the null
hypothesis only with a given probability but will detect a large class of alternatives
with probability one. In order to construct these sequential size tests the limit
distribution under the null hypothesis is obtained.

Efficient algorithms and structural results are presented for median
problems with 2 new facilities including the classical 2-Median problem,
the 2-Median problem with forbidden regions and bicriterial 2-Median
problems. This is the first paper dealing with multi-facility multiobjective location problems. The time complexity of all presented algorithms is O(MlogM), where M is the number of existing facilities.

This publication tries to develop mathematical subjects for school from realistic problems. The center of this report are business planning and decision problems which occur in almost all companies. The main topics are: Calculation of raw material demand for given orders, consumption of existing stock and the lot sizing.

The purpose of this paper is the canonical connection of classical global gravity field determination following the concept of Stokes (1849), Bruns (1878), and Neumann (1887) on the one hand and modern locally oriented multiscale computation by use of adaptive locally supported wavelets on the other hand. Essential tools are regularization methods of the Green, Neumann, and Stokes integral representations. The multiscale approximation is guaranteed simply as linear difference scheme by use of Green, Neumann, and Stokes wavelets, respectively. As an application, gravity anomalies caused by plumes are investigated for the Hawaiian and Iceland areas.

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.

We compare different notions of differentiability of a measure along a vector field on a locally convex space. We consider in the \(L^2\)-space of a differentiable measure the analoga of the classical concepts of gradient, divergence and Laplacian (which coincides with the Ornstein-Uhlenbeck
operator in the Gaussian case). We use these operators for the extension of the basic results of Malliavin and Stroock on the smoothness of finite dimensional image measures under certain nonsmooth mappings to the case of non-Gaussian measures. The proof of this extension is quite direct and does not use any Chaos-decomposition. Finally, the role of this Laplacian in the
procedure of quantization of anharmonic oscillators is discussed.

A nonequilibrium situation governed by kinetic equations with strongly contrasted Knudsen numbers in different subdomains is discussed. We consider a domain decomposition problem for Boltzmann- and Euler equations, establish the correct coupling conditions and prove the validity of the obtained coupled solution . Moreover numerical examples comparing different types of coupling conditions are presented.

In the Dynamic Multi-Period Routing Problem, one is given a new set of requests at the beginning of each time period. The aim is to assign requests to dates such that all requests are fulfilled by their deadline and such that the total cost for fulling the requests is minimized. We consider a generalization of the problem which allows two classes of requests: The 1st class requests can only be fulfilled by the 1st class server, whereas the 2nd class requests can be fulfilled by either the 1st or 2nd class server. For each tour, the 1st class server incurs a cost that is alpha times the cost of the 2nd class server, and in each period, only one server can be used. At the beginning of each period, the new requests need to be assigned to service dates. The aim is to make these assignments such that the sum of the costs for all tours over the planning horizon is minimized. We study the problem with requests located on the nonnegative real line and prove that there cannot be a deterministic online algorithm with a competitive ratio better than alpha. However, if we require the difference between release and deadline date to be equal for all requests, we can show that there is a min{2*alpha, 2 + 2/alpha}-competitive algorithm.

Error estimates for quasistatic global elastic correction and linear kinematic hardening material
(2006)

We consider in this paper the quasistatic boundary value problems of linear elasticity and nonlinear elastoplasticity with linear kinematic hardening material. We derive expressions and estimates for the difference of solutions (i.e. stress, strain and displacement) of both models. Further, we study the error between the elastoplastic solution and the solution of a postprocessing method, that corrects the solution of the linear elastic problem in order to approximate the elastoplastic model.

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.

Piezoelectric filters are used in telecommunication to filter electrical signals. This report deals with the problem of calculating passing and damped frequency intervals for a filter with given geometrical configurations and materials. Only periodic filters, which are widely used in practice, were considered. These filters consist of periodically arranged cells. For a small amount of cells a numerical procedure to visualise the wave propagation in the filter was developed. For a big number of cells another model of the filter was obtained. In this model it is assumed that the filter occupies an infinite domain. This leads to a differential equation, with periodic coefficients, that describes propagation of the wave with a given frequency in the filter. To analyse this equation the Spectral Theory for Periodic Operators had to be employed. Different ways -- analytical and numerical -- to apply the theory were proposed and analysed.

In the generalized max flow problem, the aim is to find a maximum flow in a generalized network, i.e., a network with multipliers on the arcs that specify which portion of the flow entering an arc at its tail node reaches its head node. We consider this problem for the class of series-parallel graphs. First, we study the continuous case of the problem and prove that it can be solved using a greedy approach. Based on this result, we present a combinatorial algorithm that runs in O(m*m) time and a dynamic programming algorithm with running time O(m*log(m)) that only computes the maximum flow value but not the flow itself. For the integral version of the problem, which is known to be NP-complete, we present a pseudo-polynomial algorithm.

We study the complexity of finding extreme pure Nash equilibria in symmetric network congestion games and analyse how it depends on the graph topology and the number of users. In our context best and worst equilibria are those with minimum respectively maximum total latency. We establish that both problems can be solved by a Greedy algorithm with a suitable tie breaking rule on parallel links. On series-parallel graphs finding a worst Nash equilibrium is NP-hard for two or more users while finding a best one is solvable in polynomial time for two users and NP-hard for three or more. Additionally we establish NP-hardness in the strong sense for the problem of finding a worst Nash equilibrium on a general acyclic graph.

Hyperidentities
(1992)

The concept of a free algebra plays an essential role in universal algebra and in computer science. Manipulation of terms, calculations and the derivation of identities are performed in free algebras. Word problems, normal forms, system of reductions, unification and finite bases of identities are topics in algebra and logic as well as in computer science. A very fruitful point of view is to consider structural properties of free algebras. A.I. Malcev initiated a thorough research of the congruences of free algebras. Henceforth congruence permutable, congruence distributive and congruence modular varieties are
intensively studied. A lot of Malcev type theorems are connected to the congruence lattice of free algebras. Here we consider free algebras as semigroups of compositions of terms and more specific as clones of terms. The properties of these semigroups and clones are adequately described by hyperidentities. Naturally a lot of theorems of "semigroup" or "clone" type can be derived. This topic of research is still in its beginning and therefore a lot öf concepts and results cannot be presented in a final and polished form. Furthermore a lot of problems and questions are open which are of importance for the further development of the theory of hyperidentities.

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.

Limits of instantons
(1991)

Linear Optimization is an important area from applied mathematics. A lot of practical problems can be modelled and solved with this technique. This publication shall help to introduce this topic to pupils. The process of modelling, the reduction of problems to their significant attributes shall be described. The linear programms will be solved by using the simplex method. Many examples illustrate the topic.

In this article, we give some generalisations of existing Lipschitz estimates for the stop and the play operator with respect to an arbitrary convex and closed characteristic a separable Hilbert space. We are especially concerned with the dependency of their outputs with respect to different scalar products.

This paper deals with the problem of determining the sea surface topography from geostrophic flow of ocean currents on local domains of the spherical Earth. In mathematical context the problem amounts to the solution of a spherical differential equation relating the surface curl gradient of a scalar field (sea surface topography) to a surface divergence-free vector field(geostrophic ocean flow). At first, a continuous solution theory is presented in the framework of an integral formula involving Green’s function of the spherical Beltrami operator. Different criteria derived from spherical vector analysis are given to investigate uniqueness. Second, for practical applications Green’s function is replaced by a regularized counterpart. The solution is obtained by a convolution of the flow field with a scaled version of the regularized Green function. Calculating locally without boundary correction would lead to errors near the boundary. To avoid these Gibbs phenomenona we additionally consider the boundary integral of the corresponding region on the sphere which occurs in the integral formula of the solution. For reasons of simplicity we discuss a spherical cap first, that means we consider a continuously differentiable (regular) boundary curve. In a second step we concentrate on a more complicated domain with a non continuously differentiable boundary curve, namely a rectangular region. It will turn out that the boundary integral provides a major part for stabilizing and reconstructing the approximation of the solution in our multiscale procedure.

We provide a space domain oriented separation of magnetic fields into parts generated by sources in the exterior and sources in the interior of a given sphere. The separation itself is well-known in geomagnetic modeling, usually in terms of a spherical harmonic analysis or a wavelet analysis that is spherical harmonic based. However, it can also be regarded as a modification of the Helmholtz decomposition for which we derive integral representations with explicitly known convolution kernels. Regularizing these singular kernels allows a multiscale representation of the magnetic field with locally supported wavelets. This representation is applied to a set of CHAMP data for crustal field modeling.

In this paper we consider the location of stops along the edges of an already existing public transportation network, as introduced in [SHLW02]. This can be the introduction of bus stops along some given bus routes, or of railway stations along the tracks in a railway network. The goal is to achieve a maximal covering of given demand points with a minimal number of stops. This bicriterial problem is in general NP-hard. We present a nite dominating set yielding an IP-formulation as a bicriterial set covering problem. We use this formulation to observe that along one single straight line the bicriterial stop location problem can be solved in polynomial time and present an e cient solution approach for this case. It can be used as the basis of an algorithm tackling real-world instances.

Dealing with problems from locational planning in schools can enrich the mathematical education. In this report we describe planar locational problems which can be used in mathematical lessons. The problems production of a semiconductor plate, design of a fire brigade building and the warehouse problem are from real-world. The problems are worked out detailed so that the usage for school lessons is possible.

This report gives an insight into basics of stress field simulations for geothermal reservoirs.
The quasistatic equations of poroelasticity are deduced from constitutive equations, balance
of mass and balance of momentum. Existence and uniqueness of a weak solution is shown.
In order of to find an approximate solution numerically, usage of the so–called method of
fundamental solutions is a promising way. The idea of this method as well as a sketch of
how convergence may be proven are given.