Preprints (rote Reihe) des Fachbereich Mathematik
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Fachbereich / Organisatorische Einheit
208
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.
277
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.
243
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.
330
In this paper we study linear ill-posed problems Ax = y in a Hilbert space setting where instead of exact data y noisy data y^delta are given satisfying |y - y^delta| <= delta with known noise level delta. Regularized approximations are obtained by a general regularization scheme where the regularization parameter is chosen from Morozov's discrepancy principle. Assuming the unknown solution belongs to some general source set M we prove that the regularized approximation provides order optimal error bounds on the set M. Our results cover the special case of finitely smoothing operators A and extends recent results for infinitely smoothing operators.
301
We extend the methods of geometric invariant theory to actions of non reductive groups in the case of homomorphisms between decomposable sheaves whose automorphism groups are non recutive. Given a linearization of the natural actionof the group Aut(E)xAut(F) on Hom(E,F), a homomorphism iscalled stable if its orbit with respect to the unipotentradical is contained in the stable locus with respect to thenatural reductive subgroup of the automorphism group. Weencounter effective numerical conditions for a linearizationsuch that the corresponding open set of semi-stable homomorphismsadmits a good and projective quotient in the sense of geometricinvariant theory, and that this quotient is in additiona geometric quotient on the set of stable homomorphisms.
207
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.
307
Seinen Versuch, den Begriff der negativen Größen in die Weltweisheit einzuführen beginnt der neununddreißigjährige Immanuel Kant mit einer grundsätzlichen Erörterung über einen etwaigen Gebrauch, den man in der Weltweisheit von der Mathematik ma-chen kann. Dabei stellt er die These auf, daß Mathematik grundsätzlich nur auf zweierlei Art in die Philosophie eingreifen könne. Eine erste Möglichkeit sieht Kant in der Nachahmung mathematischer Methoden bei der Darstellung von Philosophie, die andere Möglichkeit besteht für ihn in der konkreten Anwendung mathematischer Theorien in der Naturlehre. Die zuerst genannte Möglichkeit beurteilt Kant ausgesprochen negativ; seine Kritik an dem von Comenius zunächst ganz allgemein formulierten und dann von Christian Wolff insbesondere für die Philosophie favorisierten Programm einer Präsentation der Philosophie nach mathematischem Vorbild einer Darstellung more geometrico demonstrata ist hinlänglich bekannt. Die Verwendung von Mathematik in der Naturlehre sieht Kant zwar durchaus positiv; in den Metaphysischen Anfangsgründen der Naturwissenschaft wird er gut zwei Jahrzehnte später sogar jene berühmte Behauptung hinzufügen, daß in jeder besonderen Naturlehre nur so viel eigentliche Wissenschaft angetroffen werden könne, als darin Mathematik anzutreffen ist. Dennoch weist Kant mit aller Deutlichkeit auf die engen Grenzen des Wirkungsbereichs solcher Anwendungen von Mathematik hin, denn seiner Meinung nach würden aber auch nur die zur Naturlehre gehörigen Einsichten von derartigem mathematischem Zugriff profitieren.
254
268
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.
331
Strict order relations are defined as strict asymmetric and transitive binary relations. For classes of so-called levelled strict orders it is analyzed, under which conditions the endomorphism monoids of two relations coincide; in particular the case of direct sums of strict antichains is studied. Further, it is shown that these orders differ in their sets of binary order preserving functions.
336
Hyperquasivarieties
(2003)
220
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.
321
334
We define a class of topological spaces (LCNT-spaces) which come together with a nuclear Frechet algebra. Like the algebra of smooth functions on a manifold, this algebra carries the differential structure of the object. We compute the Hochschild homology of this object and show that it is isomorphic to the space of differential forms. This is a generalization of a result obtained by Alain Connes in the framework of smooth manifolds.
332
In recent years a considerable attention was paid to an investigation of finite orders relative to different properties of their isotone functions [2,3]. Strict order relations are defined as strict asymmetric and transitive binary relations. Some algebraic properties of strict orders were already studied in [6]. For the class K of so-called 2-series strict orders we describe the partially ordered set EndK of endomorphism monoids, ordered by inclusion. It is obtained that EndK possesses a least element and in most cases defines a Boolean algebra. Moreover, every 2-series strict order is determined by its n-ary isotone functions for some natural number n.
297
In the Banach space co there exists a continuous function of bounded semivariation which does not correspond to a countably additive vector measure. This result is in contrast to the scalar case, and it has consequences for the characterization of scalar-type operators. Besides this negative result we introduce the notion of functions of unconditionally bounded variation which are exactly the generators of countably additive vector measures.
279
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.
308
In this paper we discuss a special class of regularization methods for solving the satellite gravity gradiometry problem in a spherical framework based on band-limited spherical regularization wavelets. Considering such wavelets as a reesult of a combination of some regularization methods with Galerkin discretization based on the spherical harmonic system we obtain the error estimates of regularized solutions as well as the estimates for regularization parameters and parameters of band-limitation.
219
A Remark on Primes of the Form \(2^{3n}a + 2^{2n}b+2^nc+1\). Necessary and sufficient conditions for the numbers in the title to be prime are given. The tests are well suited for practical purposes.
289
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.
286
An analogue of the classical Riemann-Siegel integral formula for Dirichlet series associated to cusp forms is developed. As an application of the formula, we give a comparatively simple proof of the approximate functional equation for this type of Dirichlet series.
275
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