Preprints (rote Reihe) des Fachbereich Mathematik
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- tangent measure distributions (3)
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Fachbereich / Organisatorische Einheit
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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.
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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].
339
Caloric Restriction (CR) is the only intervention proven to retard aging and extend maximum lifespan in mammalians. A possible mechanism for the beneficial effects of CR is that the mild metabolic stress associated with CR induces cells to express stress proteins that increase their resistance to disease processes. In this article we therefore model the retardation of aging by dietary restriction within a mathematical framework. The resulting model comprises food intake, stress proteins, body growth and survival. We successfully applied our model to growth and survival data of mice exposed to different food levels.
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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.
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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.
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.
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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.
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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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In einem Beitrag zu Platons Philosophie des Abstiegs schreibt C.F. v. Weizsäcker, er sei "überzeugt, daß die griechische Philosophie, dieses in allen Weltkulturen einzigartige Kunstwerk, ohne das mathematische Paradigma undenkbar gewesen wäre" . Und in seiner berühmten Kant-Vorlesung im WS 1935/36 erklärte M. Heidegger, es sei "kein Zufall, daß die Kritik der reinen Vernunft... ständig von einer Besinnung auf das Wesen des Mathematischen und der Mathematik begleitet sei" . Was hier über Platon und Kant gesagt wird, trifft auf fast alle abendländischen Philosophen von Rang zu: Explizit oder implizit spielt die Mathematik eine entscheidende Rolle für die neue philosophische Konzeption. Welche Gründe sind es, die der Mathematik einen so hohen Stellenwert im Denken der maßgebenden Philosophen sichern? Mit welchen Intentionen und Zielvorstellungen montieren Philosophen seit Platon bis Heidegger, seit Aristoteles bis Bloch immer wieder Aussagen über Mathematik in ihre Philosophie? Weshalb war in den vergangenen zweieinhalb Jahrtausenden keine andere Wissenschaft für die Philosophie so >>frag-würdig<< wie die Mathematik? Die Philosophie hat - dies ist offensichtlich - den Dialog mit der Mathematik immer wieder gesucht. Und wie steht es um das Interesse der Mathematik an einem Dialog mit der Philosophie? In einem äußerst gehaltvollen und auch heute noch sehr lesenswerten Aufsatz Mathematik und Antike stellt der Mathematiker O. Toeplitz 1925 die Frage, "ob einmal im Dasein der Mathematik die Philosophie bestimmend in sie eingegriffen hat, ihre eigentliche definitive Gestalt gebildet hat" ? Eine derartige Initiative aus der Mathematik heraus zum Dialog mit der Philosophie ist kein Einzelfall. Cantor, Hilbert, Weyl, Gödel und Robinson - um nur einige Repräsentanten der neueren Mathematik in Erinnerung zu rufen - haben sich immer wieder um Kontakte mit der Philosophie bemüht.
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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.
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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.
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An a posteriori stopping rule connected with monitoringthe norm of second residual is introduced forBrakhage's implicit nonstationary iteration method, applied to ill-posed problems involving linear operatorswith closed range. It is also shown that for someclasses of equations with such operators the algorithmconsisting in combination of Brakhage's method withsome new discretization scheme is order optimal in the sense of Information Complexity.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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