## Preprints (rote Reihe) des Fachbereich Mathematik

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#### Keywords

- average density (3)
- tangent measure distributions (3)
- Brownian motion (2)
- Palm distributions (2)
- average densities (2)
- density distribution (2)
- lacunarity distribution (2)
- occupation measure (2)
- order-two densities (2)
- Algebraic Geometry (1)

228

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.

312

Vigenere-Verschlüsselung
(1999)

322

Integral equations on the half of line are commonly approximated by the finite-section approximation, in which the infinite upper limit is replaced by apositie number called finite-section parameter. In this paper we consider the finite-section approximation for first kind intgral equations which are typically ill-posed and call for regularization. For some classes of such equations corresponding to inverse problems from optics and astronomy we indicate the finite-section parameters that allows to apply standard regularization techniques. Two discretization schemes for the finite-section equations ar also proposed and their efficiency is studied.

296

We show that the occupation measure on the path of a planar Brownian motion run for an arbitrary finite time intervalhas an average density of order three with respect to thegauge function t^2 log(1/t). This is a surprising resultas it seems to be the first instance where gauge functions other than t^s and average densities of order higher than two appear naturally. We also show that the average densityof order two fails to exist and prove that the density distributions, or lacunarity distributions, of order threeof the occupation measure of a planar Brownian motion are gamma distributions with parameter 2.

293

Tangent measure distributions were introduced by Bandt and Graf as a means to describe the local geometry of self-similar sets generated by iteration of contractive similitudes. In this paper we study the tangent measure distributions of hyperbolic Cantor sets generated by contractive mappings, which are not similitudes. We show that the tangent measure distributions of these sets equipped with either Hausdorff or Gibbs measure are unique almost everywhere and give an explicit formula describing them as probability distributions on the set of limit models of Bedford and Fisher.

295

Tangent measure distributions are a natural tool to describe the local geometry of arbitrary measures of any dimension. We show that for every measure on a Euclidean space and every s, at almost every point, all s-dimensional tangent measure distributions define statistically self-similar random measures. Consequently, the local geometry of general measures is not different from the local geometry of self-similar sets. We illustrate the strength of this result by showing how it can be used to improve recently proved relations between ordinary and average densities.

292

Symmetry properties of average densities and tangent measure distributions of measures on the line
(1995)

Answering a question by Bedford and Fisher we show that for every Radon measure on the line with positive and finite lower and upper densities the one-sided average densities always agree with one half of the circular average densities at almost every point. We infer this result from a more general formula, which involves the notion of a tangent measure distribution introduced by Bandt and Graf. This formula shows that the tangent measure distributions are Palm distributions and define self-similar random measures in the sense of U. Zähle.

274

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.

227

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.

206

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.

280

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.

231

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.

200

Das sind die Texte der Vorlesungen, die ich im Dezember 1988 - März 1989 an der Universität Kaiserslautern hielt. Die Sektionen 1-4 enthalten Materialien, die in Russisch im Buch [33] und in früheren Arbeiten [27,28] [30-33] publiziert sind.
Sektion 5 enthält neue Ergebnisse, die wir während meines Aufenthaltes in Kaiserslautern in Zusammenarbeit mit Herrn Robert Plato
(TU Berlin) ausarbeiteten (siehe [21,22]). Sektion 6 ist eine Erweiterung der Arbeit [31].

246

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.

338

320

Power-ordered sets are not always lattices. In the case of distributive lattices we give a description by disjoint of chains. Finite power-ordered sets have a polarity. We introduct the leveled lattices and show examples with trivial tolerance. Finally we give a list of Hasse diagrams of power-ordered sets.

284

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.

306

In this paper we study the space-time asymptotic behavior of the solutions and derivatives to th incompressible Navier-Stokes equations. Using moment estimates we obtain that strong solutions to the Navier-Stokes equations which decay in \(L^2\) at the rate of \(||u(t)||_2 \leq C(t+1)^{-\mu}\) will have the following pointwise space-time decay \[|D^{\alpha}u(x,t)| \leq C_{k,m} \frac{1}{(t+1)^{ \rho_o}(1+|x|^2)^{k/2}} \]
where \( \rho_o = (1-2k/n)( m/2 + \mu) + 3/4(1-2k/n)\), and \(|a |= m\). The dimension n is \(2 \leq n \leq 5\) and \(0\leq k\leq n\) and \(\mu \geq n/4\)

224

Jede Wissenschaft entfaltet sich in einem Spannungsverhältnis zu ihren Nachbardisziplinen. In diesem Beitrag wird insbesondere das Disziplinenpaar Mathematik-Philosophie in den Blick genommen. Dies geschieht entlang der Leitfrage, ob und gegebenenfalls wie Philosophie auf die Entwicklung und Ausformung der Mathematik Einfluß genommen hat. Dazu wird nach philosophischen Spuren in der Mathematik gefragt, wobei jene historischen Konstellationen bevorzugt betrachtet werden, die eine grundlegende Änderung im Mathematikverständnis erbracht haben. Deshalb gilt das Hauptinteresse dieser Untersuchung dem Verhältnis von Philosophie und Mathematik in der klassischen Antike, bei Kant und in der Gegenwart.

319

The Kallianpur-Robbins law describes the long term asymptotic behaviour of the distribution of the occupation measure of a Brownian motion in the plane. In this paper we show that this behaviour can be seen at every typical Brownian path by choosing either a random time or a random scale according to the logarithmic laws of order three. We also prove a ratio ergodic theorem for small scales outside an exceptional set of vanishing logarithmic density of order three.

236

Es wird anhand von Beispielen, an denen der Autor in der Vergangenheit gearbeitet hat, gezeigt, wie man Modelle der exakten Naturwissenschaften auf wirtschaftliche Probleme
anwenden kann. Insbesondere wird diskutiert, wo Grenzen dieser Übertragbarkeit liegen. Die Arbeit ist eine Zusammenfassung eines Vortrags, der im SS 1992 im Rahmen des Studium Generale an der Universität Kaiserslautern gehalten wurde.

313

A class of regularization methods using unbounded regularizing operators is considered for obtaining stable approximate solutions for ill-posed operator equations. With an a posteriori as well as an priori parameter choice strategy, it is shown that the method yields optimal order. Error estimates have also been obtained under stronger assumptions on the the generalized solution. The results of the paper unify and simplify many of the results available in the literature. For example, the optimal results of the paper includes, as particular cases for Tikhonov regularization, the main result of Mair (1994) with an a priori parameter choice and a result of Nair (1999) with an a posteriori parameter choice. Thus the observations of Mair (1994) on Tikhonov regularization of ill-posed problems involving finitely and infinitely smoothing operators is applicable to various other regularization procedures as well. Subsequent results on error estimates include, as special cases, an optimal result of Vainikko (1987) and also recent results of Tautenhahn (1996) in the setting Hilbert scales.

248

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.

238

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.

233

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.

271

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.

282

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 \(L_k\).

299

We propose a new discretization scheme for solving ill-posed integral equations of the third kind. Combining this scheme with Morozov's discrepancy principle for Landweber iteration we show that for some classes of equations in such method a number of arithmetic operations of smaller order than in collocation method is required to appoximately solve an equation with the same accuracy.

316

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250

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.

239

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.

328

In this short note we prove some general results on semi-stable sheaves on P_2 and P_3 with arbitrary linear Hilbert polynomial. Using Beilinson's spectral sequence, we compute free resolutions for this class of semi-stable sheaves and deduce that the smooth moduli spaces M_{r m + s}(P_2) and M_{r m + r - s}(P_2) are birationally equivalent if r and s are coprime.

267

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.

215

285

On derived varieties
(1996)

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.

232

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.

201

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.

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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.

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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.