Consistency of V = HOD with the wholeness axiom

The Wholeness Axiom (WA) is an axiom schema that can be added to the axioms of ZFC in an extended language , and that asserts the existence of a nontrivial elementary embedding . The well-known inconsistency proofs are avoided by omitting from the schema all instances of Replacement for j-formulas. We show that the theory ZFC + V = HOD + WA is consistent relative to the existence of an embedding. This answers a question about the existence of Laver sequences for regular classes of set embeddings: Assuming there is an -embedding, there is a transitive model of ZFC +WA + “there is a regular class of embeddings that admits no Laver sequence.” Received: 7 July 1998 / Revised version: 5 November 1998     

On Bayes procedures

Summary A result of Doob regarding consistency of Bayes estimators is extended to a large class of Bayes decision procedures in which the loss functions are not necessarily convex. Rather weak conditions are given under which the Bayes procedures are consistent. One set involves restrictions on the a priori distribution and follows an example in which the choice of a priori distribution determines whether the Bayes estimators are consistent. Another example shows that the maximum likelihood estimators may be consistent when the Bayes estimators are not. However, the conditions given are of an essentially weaker nature than those established for consistency of maximum likelihood estimators.This paper was prepared with the partial support of the Office of Ordnance Research, U. S. Army, under Contract DA-04-200-ORD-171. While the paper was in press, its author Dr. Lorraine Schwartz suddenly died. Sit ei terra levis! (L. LC., J. N.)I am deeply indebted to Professor L. LeCam, under whose guidance I obtained the results established in my thesis and given in this paper.     

Algebraic Coding of Expansive Group Automorphisms and Two-sided Beta-Shifts

 Let α be an expansive automorphisms of compact connected abelian group X whose dual group is cyclic w.r.t. α (i.e. is generated by for some ). Then there exists a canonical group homomorphism ξ from the space of all bounded two-sided sequences of integers onto X such that , where σ is the shift on . We prove that there exists a sofic subshift such that the restriction of ξ to V is surjective and almost one-to-one. In the special case where α is a hyperbolic toral automorphism with a single eigenvalue and all other eigenvalues of absolute value we show that, under certain technical and possibly unnecessary conditions, the restriction of ξ to the two-sided beta-shift is surjective and almost one-to-one. The proofs are based on the study of homoclinic points and an associated algebraic construction of symbolic representations in [13] and [7]. Earlier results in this direction were obtained by Vershik ([24]–[26]), Kenyon and Vershik ([10]), and Sidorov and Vershik ([20]–[21]). (Received 27 October 1998; in revised form 17 May 1999)     

Extending the Limits of Supertree Methods

Recently, two exact polynomial-time supertree methods have been developed in which the traditional input of rooted leaf-labelled trees has been extended in two separate ways. The first method, called RankedTree, allows for the inclusion of relative divergence dates and the second method, called AncestralBuild, allows for the inclusion of rooted trees in which some of the interior vertices as well as the leaves are labelled. The latter is particularly useful for when one has information that includes nested taxa. In this paper, we present two supertree methods that unite and generalise RankedTree and AncestralBuild. The first method is polynomial time and combines the allowable inputs of RankedTree and AncestralBuild. It determines if the original input is compatible, in which case it outputs an appropriate ‘ranked semi-labelled tree’. The second method lists all ‘ranked semi-labelled trees’ that are consistent with the original input. While there may be an exponential number of such trees, the second method outputs the next such tree in the list in polynomial time. Supported by the New Zealand Institute of Mathematics and its Applications funded programme Phylogenetic Genomics and the work was conducted at the University of Canterbury. Supported by the New Zealand Marsden Fund (UOC301). Received June 14, 2004     

Analytic Semigroups on C [0,1] Generated by Some Classes of Second Order Differential Operators

C [0,1], α > 0 in (0,1) and α(1), we consider the second order differential operator on C[0,1] defined by Au: = αu″ + βu′, where D(A) may include Wentzell boundary conditions. Under integrability conditions involving √α and β/√α, we prove the analyticity of the semigroup generated by (A,D(A)) on Co[0,1], Cπ[0,1] and on C[0,1], where Co[0,1]: {u∈ C[0,1]|u (1)} and Cπ[0,1]: = {u∈ C[0,1]| u (0) = u (1)}. We also prove different characterizations of D(A) related to some results in [1], where β≡ 0, exhibiting peculiarities of Wentzell boundary conditions. Applications can be derived for the case αx = x ^k (1 - x )^kγ(x )(k≥j/2, x∈ [0,1], γ∈ C{0,1}).     

Maximization of the transmission rate of a discrete,constant channel

Summary The paper is concerned with the assignment of input probabilities to achieve the maximum rate of transmission when the transition matrix is prespecified. Unlike the standard reference [Muroga, 1953], in whichLagrange multipliers are used in conventional form, mathematical programming approach has been proposed.

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Zusammenfassung Die Zuordnung von Eingabe-Wahrscheinlichkeiten zur Erzielung einer maximalen Übertragungsrate bei vorgegebenen Übergangswahrscheinlichkeiten wird untersucht. Entgegen der Behandlung unter Verwendung vonLagrangeschen Multiplikatoren [Muroga, 1953] wird hier der Zugang mit Hilfe konvexer Programmierung betrachtet.

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Vorgel. v.:J. Nitsche     

Non-group Ranks in Finite Full Transformation Semigroups

T _n be the full transformation semigroup on a finite set. Both rank and idempotent rank of the semigroup K(n,r) = {α∈T _n : | im α | ≤r, 2 ≤ r ≤ n - 1. In this paper we prove that the non-group rank, defined as the cardinality of a minimal generating set of non-group elements, of K(n,r) is S(n,r) , the Stirling number of the second kind.     

A theorem on supports of a convex functional in a realBanach space

Summary A theorem on supports of a convex functional in a realBanach space is proved which provides us with the principal tool to establish certain duality results for a non-linear programming problem in a realBanach space.

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Zusammenfassung Es wird ein Satz über die Stützhyperebenen eines konvexen Funktionals in einem reellenBanach-Raum bewiesen. Dieser ist ein wesentliches Hilfsmittel zur Herleitung von Dualitätssätzen für nichtlineare Programme inBanach-Räumen.

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Vorgel. v.:J. Nitsche.     

Die HOWARDsche Berechnung optimalen Anschaffungs- und Betriebsalters für Kraftwagen: Ein ergänzender Befund mit deutschen Daten

Zusammenfassung Mit Hilfe des Iterationsverfahrens vonHoward wird das optimale Anschaffungs-und Betriebsalter für ein deutsches PKW-Fabrikat bestimmt. Diese Zeiten liegen unterhalb der entsprechenden Werte, dieHoward auf Grund amerikanischer Daten erhalten hat.

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Summary By aid of theHoward-Algorithm the optimal purchase- and operating age for a German automobile is determined. These time-spans are lying below the corresponding values which were received byHoward on account of American data.

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Vorgel. v.: H.P.Künzi.     

The special theory of relativity as a case study of the importance of the philosophy of science for the history of science

Summary 1. The philosophical justification of Einstein's conception of distant simultaneity as conventional depends on two cardinal physical assumptions which are stated. Awareness of these two assumptions poses the following historical problem: On what grounds didEinstein feel entitled to make them in 1905? In an endeavor to answer this question, the contribution of experimental results to Einstein's postulational achievement in the Special Theory of Relativity (?RT?) is examined. As a consequence, the author rejects (i)M. Polanyi's recent citation of the history of RT as evidence against an empiricist account of scientific knowledge, and (ii)G. Holton's assessment of the relevance of knowledge of the history of RT to the philosophical mastery of its logical foundations. 2. An analysis of theKennedy-Thorndike experiment is used to provide a refutation of the widespread belief that the aether-theoreticLorentz-Fitzgerald contraction hypothesis was ad hoc in the logical sense. A distinction is drawn between a logical and a psychological sense in which an auxiliary hypothesis can be ad hoc. 3. E. T. Whittaker's disparaging estimate ofEinstein's contributions to RT vis -à -vis those ofLorentz andPoincare is shown to rest on fundamental philosophical misunderstandings ofEinstein's conception of theLorentz transformations.Holton's maxim for the study of the history of RT is then tested in the light of his evaluation ofWhittaker's belittlement ofEinstein's role. To Enrico Bompiani on his scientific Jubilee     

On the Least Property of the Semilattice Congruences on PO-Semigroups

n on po-semigroups. We study the least property of (ordered) semilattice congruences, and prove: 1. N is the least ordered semilattice congruence on pr-semigroups (cf.[1]). 2. n is the least semilattice congruence on po-semigroups. 3. N is not the least semilattice congruence on po-semigroups in general. Thus, we give a complete solution to the problem posed by N. Kehayopulu in [1].     

Sur l’ordre de grandeur des quotients partiels du développement en fractions continues régulières

 For any irrational , let denote the regular continued fraction expansion of x and define f, for all z > 0 by and by J. GALAMBOS proved that (μ the Gauss measure)

A simple proof of a general duality theorem of convex programming

Summary The first part of this note presents concisely and partially proves in logical terms the relations betweenUzawa's andKuhn andTucker's equivalence theorems of nonlinear programming. In the second part we give simple mathematical proofs of two lemmata linking theKuhn-Tucker conditions and dual solutions and use them to establish a nonlinear duality theorem of considerable generality.

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Zusammenfassung Im ersten Abschnitt dieses Beitrags demonstrieren wir in kurzer, symbolischlogischer Form die Relationen zwischenUzawas undKuhn undTuckers Äquivalenzsätzen des nichtlinearen Programmierens. Im zweiten Teil geben wir mathematische Beweise zweier Lemmata, die dieKuhn-Tucker-Bedingungen und duale Lösungen in Zusammenhang bringen, und benützen sie und die logische Struktur des ersten Teils, um ein nichtlineares Dualitätstheorem von besonderer Allgemeinheit zu beweisen.

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The author is indebted toP. Van Moeseke for teaching him programming theory some time ago, and toD. Bent for helpful comments and criticism.

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Vorgel. v.:G. Tintner.     

Duality for non-linear programming in aBanach space

Summary A constrained maximization problem in a realBanach space is considered, where the objective function is a non-linear pseudo concave functional and constraints are given bym non linear quasi convex functionals. The optimality conditions and converse duality theorem, given byRitter, have been extended to this class of programming problems.

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Zusammenfassung Die Maximierungsaufgabe in einem reellenBanachraum mit nichtlinearerpseudo konkaver Zielfunktion und endlich vielen nichtlinearen quasi konvexen Nebenbedingungen wird untersucht. Die vonRitter aufgestellten Optimalitätskriterien sowie der Dualitätssatz werden für diese Klasse von Programmen als gültig erkannt.

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Vorgel. v.:J. Nitsche.     

Extension ofUV n -valued mappings

A new method for extending upper semicontinuousUV ⁿ -valued mappings is introduced. Any upper semicontinuousUV ⁿ -valued mapping Ψ:A→Y of a closed subsetA of a separable metric spaceX into ann-connected, locallyn-connected complete metric spaceY satisfying the property of disjoint (n+1)-disks is proved to be extendable to an upper semicontinuousUV ⁿ -valued mapping Ψ′:X→Y such that Ψ′|a=Ψ. As an application, some results aboutn-soft mappings are obtained. Translated fromMatematicheskie Zametki, Vol. 66, No. 3, pp. 351–363, September, 1999.     

Zwei Algorithmen zur Lösung eines komplexen Reihenfolgeproblems

Zusammenfassung Bei der Ordnung von Input-Output-Matrizen treten Kriterien auf, die wesentlich komplexer sind als die Ordnungsbedingungen bei klassischen Reihenfolgeproblemen (z. B.Traveling-Salesman-Problem). Ausgehend von den Arbeiten vonHelmstädter werden im folgenden die mathematische Formulierung eines solchen Reihenfolgeproblems sowie zwei Algorithmen zu dessen Lösung angegeben. Ein Verfahren ist ein lexikographischer Suchalgorithmus, das andere eine Modifikation des bekanntenJacobi-Verfahrens zur Berechnung von EigenwertenHermitescher Matrizen.

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Summary On the ordering of input-output matrices we use some criteria which are more complex than the order conditions of classical sequencing problems (e. g.Traveling-Salesman-Problem). Basing on the papers ofHelmstädter we will give a mathematical formulation of this sequencing problem and two algorithms solving them. One procedure is a lexicographic searching algorithm, the other one a modification of the well knownJacobi-method which calculates the eigenvalues ofHermiteian matrices.

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Ein Teil dieser Arbeit wurde als Referat auf der Jahrestagung der Deutschen Gesellschaft für Unternehmensforschung (DGU) am 23. September 1968 in Dortmund vorgetragen.

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Vorgel. v.:W. Krelle.     

Primal- und Dual-Algorithmen zur Optimierung von Markov-Prozessen

Zusammenfassung In dem folgenden Aufsatz sollen zwei Algorithmen zur Optimierung vonMarkov-Prozessen, die vonDantzig-Wolfe undHoward entwickelt wurden, diskutiert werden. Es kann gezeigt werden, daß der Algorithmus vonHoward ein Dualproblem behandelt, während beim Verfahren vonDantzig-Wolfe ein Primal-Problem zugrunde gelegt wurde. Beide Algorithmen weisen eine Schar von sich entsprechenden Vor- und Nachteilen auf, die miteinander verglichen werden.

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Summary In the following paper two algorithms byDantzig-Wolfe andHoward dealing with the optimization ofMarkov processes are discussed. It can be shown thatHoward's algorithm is based on the dual formulation of the problem, whereasDantzig-Wolfe are treating the primal problem. Both algorithms have a host of corresponding advantages and disadvantages, which are compared.

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Résumé Dans le traité suivant deux algorithmes pour l'optimisation des procès deMarkov développés parDantzig-Wolfe etHoward, sont discutés. Il est exposé, que l'algorithme deHoward est un problème dual pendant que l'algorithme deDantzig-Wolfe est un problème primal. Les deux algorithmes ont certaines avantages et désavantages correspondants, qui sont comparés.

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Vorgel. v.: H. v.Falkenhausen     

Prüfung endlicher homogener Markoffketten auf Ergodizität

Zusammenfassung Die zur Prüfung endlicher homogener Markoffketten auf Ergodizität bekannten Kriterien werden zusammengestellt und in bezug auf ihre Anwendbarkeit diskutiert: Das (am geeignetsten erscheinende)Gordonsche Kriterium wird modifiziert und durchBoolesche Methoden ergänzt.

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Summary There are several criteria to decide whether a finite homogeneous Markoff chain is ergodic or not. These criteria are discussed with respect to their applicability.Gordon's criterion (which seems to be the most appropriate) is modified and combined withboolean methods.

     

Expected utility maximization and capital budgeting subgoals

Summary It will be shown that the capital budgeting subgoals which generally are used to determine the optimal investment and financing program of a corporation are not necessarily consistent with the objective of maximizing the expected utility of the terminal wealth of its shareholders. This holds true also if there is no conflict of interest between the shareholders. The analysis is based on a portfolio model and a stock market equilibrium model.

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Zusammenfassung Es wird gezeigt, daß die Zielfunktionen, mit denen in der Regel bei der Bestimmung des optimalen Investitionsprogramms einer Kapitalgesellschaft gearbeitet wird, nicht zwingend im Einklang stehen mit dem Oberziel der Maximierung des erwarteten Nutzens der Anteilseigner. Dies gilt auch dann, wenn kein Interessenkonflikt zwischen den Anteilseignern besteht. Die Analyse basiert auf einem Portfolio Modell und einem darauf aufbauenden Modell zur Bestimmung der Gleichgewichtskurse von Aktien.

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Vorgel. v.:H. Schneeweiss