## Fachbereich Mathematik

### Refine

#### Year of publication

#### Document Type

- Report (106) (remove)

#### Language

- English (106) (remove)

#### Keywords

- Elastoplastizität (4)
- Elastoplasticity (3)
- Hysterese (3)
- Mathematikunterricht (3)
- Modellierung (3)
- modelling (3)
- praxisorientiert (3)
- Elastic BVP (2)
- Elastisches RWP (2)
- Elastoplastisches RWP (2)
- Inverses Problem (2)
- Jiang's model (2)
- Jiang-Modell (2)
- Lineare Algebra (2)
- Ratenunabhängigkeit (2)
- Regularisierung (2)
- Variationsungleichungen (2)
- Wavelet (2)
- algorithmic game theory (2)
- hysteresis (2)
- linear algebra (2)
- mathematical education (2)
- polynomial algorithms (2)
- praxis orientated (2)
- variational inequalities (2)
- (dynamic) network flows (1)
- Abstract linear systems theory (1)
- Automatische Differentiation (1)
- Bell Number (1)
- Berechnungskomplexität (1)
- Betriebsfestigkeit (1)
- Biot-Savart Operator (1)
- Biot-Savart operator (1)
- CHAMP <Satellitenmission> (1)
- Competitive Analysis (1)
- Continuum mechanics (1)
- Convex sets (1)
- Core (1)
- Differentialinklusionen (1)
- Dynamic Network Flows (1)
- Education (1)
- Elastoplastic BVP (1)
- FPTAS (1)
- Filippov theory (1)
- Filippov-Theorie (1)
- FlowLoc (1)
- Geomagnetic Field Modelling (1)
- Geomagnetismus (1)
- Geomathematik (1)
- Geostrophic flow (1)
- Gravimetrie (1)
- Greedy Heuristic (1)
- Green’s function (1)
- Helmholtz-Decomposition (1)
- Helmholtz-Zerlegung (1)
- Homotopie (1)
- Homotopiehochhebungen (1)
- Homotopy (1)
- Homotopy lifting (1)
- Hysteresis (1)
- Injectivity of mappings (1)
- Injektivität von Abbildungen (1)
- Inkorrekt gestelltes Problem (1)
- Jiang's constitutive model (1)
- Jiangsches konstitutives Gesetz (1)
- Kaktusgraph (1)
- Kontinuumsmechanik (1)
- Konvexe Mengen (1)
- Linear kinematic hardening (1)
- Linear kinematische Verfestigung (1)
- Lineare Optimierung (1)
- Locational Planning (1)
- Methode der Fundamentallösungen (1)
- Mie-Darstellung (1)
- Mie-Representation (1)
- Multi-dimensional systems (1)
- Nash equilibria (1)
- Nichtlineare/große Verformungen (1)
- Nonlinear/large deformations (1)
- One-dimensional systems (1)
- Online Algorithms (1)
- Parameter identification (1)
- Parameteridentifikation (1)
- Poroelastizität (1)
- Pseudopolynomial-Time Algorithm (1)
- Rate-independency (1)
- Restricted Shortest Path (1)
- SAW filters (1)
- SGG (1)
- SST (1)
- Satellitengradiometrie (1)
- Sensitivitäten (1)
- Shapley value (1)
- Shapleywert (1)
- Simplex (1)
- Spieltheorie (1)
- Standortplanung (1)
- Stop- and Play-Operators (1)
- Stop- und Play-Operator (1)
- Stop-und Play-Operator (1)
- Stücklisten (1)
- Theorie schwacher Lösungen (1)
- Train Rearrangement (1)
- Variational inequalities (1)
- Variationsungleichugen (1)
- Vectorial Wavelets (1)
- Vektor-Wavelets (1)
- Vektorkugelfunktionen (1)
- Vektorwavelets (1)
- automatic differentiation (1)
- autoregressive process (1)
- basic systems theoretic properties (1)
- bills of materials (1)
- bin coloring (1)
- cactus graph (1)
- change analysis (1)
- competitive analysis (1)
- complexity (1)
- cooperative game (1)
- core (1)
- delay management (1)
- differential inclusions (1)
- discrete time setting (1)
- durability (1)
- dynamic network flows (1)
- earliest arrival flows (1)
- eigenvalue problems (1)
- elastoplastic BVP (1)
- elastoplasticity (1)
- extreme equilibria (1)
- fptas (1)
- geomathematics (1)
- harmonic density (1)
- harmonische Dichte (1)
- heuristic (1)
- kooperative Spieltheorie (1)
- linear optimization (1)
- local approximation of sea surface topography (1)
- 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)

In this paper we continue the study of p - groups G of square order \(p^{2n}\) and investigate the existence of partial congruence partitions (sets of mutually disjoint subgroups of order \(p^n\)) in G. Partial congruence partitions are used to construct translation nets and partial difference sets, two objects studied extensively in finite geometries and combinatorics. We prove that the maximal number of mutually disjoint subgroups of order \(p^n\) in a group G of order \(p^{2n}\) cannot be more than \((p^{n-1}-1)(p-1)^{-1}\) provided that \(n\ge4\)and that G is not elementary abelian. This improves a result in [6] and as we do not distinguish the cases p=2 and p odd in the present paper, we also have a generalization of D. FROHARDT' s theorem on 2 - groups in [4]. Furthermore we study groups of order \(p^6\). We can show that for each odd prime number, there exist exactly four nonisomorphic groups which contain at least p+2 mutually disjoint subgroups of order \(p^3\). Again, as we do not distinguish between the even and the odd case in advance, we in particular obtain
D. GLUCK' s and A. P. SPRAGUE' s classification of groups of order 64 which contain at least 4 mutually disjoint subgroups of order 8 in [5] and [13] respectively.

In this paper the existence of translation transversal designs which is equivalent to the existence of certain particular partitions in finite groups is studied. All considerations are based on the fact that the particular component of such a partition (the component representing the point classes of the corresponding design) is a normal subgroup of the translation group. With regard to groups admitting an (s,k,\(\lambda\))-partiton, on one hand the already known families of such groups are determined without using R. BAER's, 0.H.KEGEL's and M. SUZUKI' s classification of finite groups with partition and on the other hand some new results on the special structure of p - groups are proved. Furthermore, the existence of a series of nonabelian p - groups of odd order which can be represented as translation groups of certain (s,k,1) - translation transversal designs is shown; moreover, the translation groups are normal subgroups of collineation groups acting regularly on the set of flags of the same designs.

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.

Limits of instantons
(1991)

Moduli for singularities
(1991)

The aim of this article is to give a survey on recent results about moduli spaces for curve singularities and for modules over the local ring of a fixed curve singularity. We emphasize especially the general concept which lies behind these constructions.
Therefore, the article might be useful to the reader who wishes to have the leading ideas and the main steps of the proofs explained without going into all the details. We also calculate explicit examples (for singularities and for modules) which illustrate
the general theorems.

We show that the different module structures of GF(\(q^m\)) arising from the intermediate fields of GF(\(q^m\))and GF(q) can be studied simultaneously with the help of some basic properties of cyclotomic polynomials. We use this ideas to give a detailed and constructive proof of the most difficult part of a Theorem of D. Blessenohl and K. Johnsen (1986), i.e., the existence of elements v in GF(\(q^m\)) over GF(q) which generate normal bases over any intermediate field of GF(\(q^m\)) and GF(q), provided that m is a prime power. Such elements are called completely free in GF(\(q^m\)) over GF(q). We develop a recursive formula for the number of completely free elements in GF(\(q^m\)) over GF(q) in the case where m is a prime power. Some of the results can be generalized to finite cyclic Galois extensions
over arbitrary fields.

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.

We are concerned with a parameter choice strategy for the Tikhonov regularization \((\tilde{A}+\alpha I)\tilde{x}\) = T* \(\tilde{y}\)+ w where \(\tilde{A}\) is a (not necessarily selfadjoint) approximation of T*T and T*\(\tilde y\)+ w is a perturbed form of the (not exactly computed) term T*y. We give conditions for convergence and optimal convergence rates.

Let \(a_i i:= 1,\dots,m.\) 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 \(P\), integral representations of their first two moments are given which lead to asymptotic estimations of variances for special "additive variables" known from stochastic approximation theory in case of rotationally symmetric distributions.

Weighted k-cardinality trees
(1992)

We consider the k -CARD TREE problem, i.e., the problem of finding in a given undirected graph G a subtree with k edges, having minimum weight. Applications of this problem arise in oil-field leasing and facility layout. While the general problem is shown to be strongly NP hard, it can be solved in polynomial time if G is itself a tree. We give an integer programming formulation of k-CARD TREE, and an efficient exact separation routine for a set of generalized subtour elimination constraints. The polyhedral structure of the convex huLl of the integer solutions is studied.

Facility location problems in the plane are among the most widely used tools of Mathematical Programming in modeling real-world problems. In many of these problems restrictions have to be considered which correspond to regions in which a placement of new locations is forbidden. We consider center and median problems where the forbidden set is
a union of pairwise disjoint convex sets. As applications we discuss the assembly of printed circuit boards, obnoxious facility location and the location of emergency facilities.

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.

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.

Max ordering (MO) optimization is introduced as tool for modelling production
planning with unknown lot sizes and in scenario modelling. In MO optimization a feasible solution set \(X\) and, for each \(x\in X, Q\) individual objective functions \(f_1(x),\dots,f_Q(x)\) are given. The max ordering objective
\(g(x):=max\) {\(f_1(x),\dots,f_Q(x)\)} is then minimized over all \(x\in X\).
The paper discusses complexity results and describes exact and approximative
algorithms for the case where \(X\) is the solution set of combinatorial
optimization problems and network flow problems, respectively.

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.

The article provides an asymptotic probabilistic analysis of the variance of the number of pivot steps required by phase II of the "shadow vertex algorithm" - a parametric variant of the simplex algorithm, which has been proposed by Borgwardt [1] . The analysis is done for data which satisfy a rotationally
invariant distribution law in the \(n\)-dimensional unit ball.

Despite their very good empirical performance most of the simplex algorithm's variants require exponentially many pivot steps in terms of the problem dimensions of the given linear programming problem (LPP) in worst-case situtation. The first to explain the large gap between practical experience and the disappointing worst-case was Borgwardt (1982a,b), who could prove polynomiality on tbe average for a certain variant of the algorithm-the " Schatteneckenalgorithmus (shadow vertex algorithm)" - using a stochastic problem simulation.

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.

Given Q different objective functions, three types of single-facility problems
are considered: Lexicographic, pareto and max ordering problems. After discussing the interrelation between the problem types, a complete characterization of lexicographic locations and some instances of pareto and max ordering locations is given. The characterizations result in efficient solution algorithms for finding these locations. The paper relies heavily on the theory of restricted locations developed by the same authors, and can be further extended, for instance, to multifacility problems with several objectives. The proposed approach is more general than previously published results on multicriteria planar location problems and is particulary suited for modelling real-world problems.

We investigate two versions of multiple objective minimum spanning tree
problems defined on a network with vectorial weights. First, we want to minimize the maximum of Q linear objective functions taken over the set of all spanning trees (max linear spanning tree problem ML-ST). Secondly, we look for efficient spanning trees (multi criteria spanning tree problem MC-ST). Problem ML-ST is shown to be NP-complete. An exact algorithm which is based on ranking is presented. The procedure can also be used as an approximation scheme. For solving the bicriterion MC-ST, which in the worst case may have an exponential number of efficient trees, a two-phase procedure is presented. Based on the computation of extremal efficient spanning trees we use neighbourhood search to determine a sequence of solutions with the property that the distance
between two consecutive solutions is less than a given accuracy.

Order-semi-primal lattices
(1994)

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.

Let (\(a_i)_{i\in \bf{N}}\) be a sequence of identically and independently distributed random vectors drawn from the \(d\)-dimensional unit ball \(B^d\)and let \(X_n\):= convhull \((a_1,\dots,a_n\)) be the random polytope generated by \((a_1,\dots\,a_n)\). Furthermore, let \(\Delta (X_n)\) : = (Vol \(B^d\) \ \(X_n\)) be the deviation of the polytope's volume from the volume of the ball. For uniformly distributed \(a_i\) and \(d\ge2\), we prove that tbe limiting distribution of \(\frac{\Delta (X_n)} {E(\Delta (X_n))}\) for \(n\to\infty\) satisfies a 0-1-law. Especially, we provide precise information about the asymptotic behaviour of the variance of \(\Delta (X_n\)). We deliver analogous results for spherically symmetric distributions in \(B^d\) with regularly varying tail.