Näherungstheorien für die Berechnung von Strömungsgrenzschichten

Summary Calculations of turbulent boundary layers in a compressible flow with heat transfer at the wall require the knowledge of empirical laws for the wall-friction and dissipation valid under these general conditions. An approximate theory is developed which involves a generalization of the empirical wall-friction and dissipation laws well known for the incompressible case to be valid up to about Mach number 10 and arbitrary heat transfer conditions. Universal functions occurring in this theory have been evaluated so that practical calculations may be performed in a short time. Comparisons of theoretical results with experiments show good agreement. The fundamental idea of deriving averaged boundary layer conditions may also be applied to the complete equations ofNavier andStokes, thus creating a fairly simple method of checking or improving the results ofPrandtl's simplified theory, e.g., in cases, where the conditionsδp/δy=0 and high Reynolds numbers are no longer fulfilled. This approximate theory, however, has so far been advanced to a preliminary state only.      

Eine Ungleichung von van der Corput und Kemperman

The tool of van der Corput's difference theorem in the theory of uniform distribution is his so-called fundamental inequality.Kemperman showed that even the non-constructive proofs of the difference theorem byBass, Bertrandias andCigler implicitly use a more general form of van der Corput's fundamental inequality. In this article, the inequality which constitutes the basis of the difference theorem will be proved under a very general setting, applications will be demonstrated in connection with the uniform distribution of products of linear forms and a quantitative version of the difference theorem, i. e. an estimation of discrepancies, will be derived.     

Numerical calculation of elliptic integrals and elliptic functions. III

Summary This paper is a continuation of [2, 3]. It contains anALGOL program for the incomplete elliptic integral of the third kind based on a theory described in [4]. This program replaces the inadequate one based on the Gauß-transformation which was published in [2]. In addition, anAlgol program for a general complete elliptic integral is presented. Editor's note. In this fascicle, prepublication of algorithms from the Special Functions Series of the Handbook for Automatic Computation is continued. Algorithms are published inAlgol 60 reference language as approved by the IFIP. Contributions in this series should be styled after the most recently published ones.This research was supported by the Advanced Research Projects Agency of the Department of Defense and was monitored by the U. S. Army Research Office —Durham under Contract DA-31-124-ARO-D-257.     

The Wholeness Axioms and V=HOD

If the Wholeness Axiom wa is itself consistent, then it is consistent with v=hod. A consequence of the proof is that the various Wholeness Axioms are not all equivalent. Additionally, the theory zfc+wa is finitely axiomatizable. Received: 16 February 1999 / Revised version: 1 June 1999     

On the carleman's model for the boltzmann equation and its generalizations

Summary T. Carleman [1] showed that a certain 2 x 2 hyperbolic, semilinear system of partial differential equations has many features in common with the integro-differential equation ofBoltzmann. We produce here a global existence theory for theCauchy problem for this system and its generalizations. The main result is derived by exploiting the fact that the solution of a certainRiccati equation depends isotonically (with respect to a suitable partial order) on the forcing term and the initial data. Research supported by the National Science Foundation, Grant NSF G-21053.     

Über die Zulässigkeit des Verfahrens von Galerkin bei linearen,nichtselbstadjungierten Eigenwertproblemen

Summary In this report it is proved that the method ofGalerkin remains valid for a certain class of linear, non-self-adjoint eigenvalue problems. This class contains all eigenvalue problems which arise from originally self-adjoint problems by addition of a linear differential expression which destroys the former self-adjointness.Helge von Koch's theory of infinite determinants is used in deriving the above result.     

Rohrströmung mit schwachem Drall

Summary The steady laminar flow at high Reynolds' numbers in a straight tube of circular cross-section is investigated under the condition that swirl is present. Given any axially symmetrical distribution of the rotational velocities, the flow downstream is calculated in the special case that the rotational velocities are small compared to the axial velocity. The problem is reduced to an eigen-value problem of an ordinary linear differential equation of second order, the theory of which is developed in detail. Numerical values for the five lowest eigen-values and the corresponding eigen-functions are given. As an example, the decay of an initially rigid rotation is considered.

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HerrnFr. A. Willers zum siebzigsten Geburtstag     

Geometries involving affine connections and general linear connections

Summary Part I of this paper is concerned with the theory of differential invariants of a symmetric affine connection and a general linear connection of theK?nig type. Part II deals with a geometry in which the components of the affine connection areChristoffel symbols, and the general linear connection is of a special sort. This section can be considered in part as an extension of theEinstein-Mayer (5) geometry,     

Theory of oscillation type viscometers V: Disk oscillating between fixed plates

A disk oscillating between two fixed plates is considered as an instrument for absolute measurements of viscosity. The existing theories relating the viscosity to the decrement of oscillation are improved by calculating the effects of the edge. This is done by assuming that the separation between the plates is small compared with both the radius of the disk and the boundary layer thickness. A comparison is made with the experimental data ofKestin andPilarczyk for which the present theory is estimated to be correct to 0·1%.

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Zusammenfassung Es wird untersucht, ob eine zwischen zwei feststehenden Platten schwingende Scheibe als Gerät für die Absolutmessung der Zähigkeit von Flüssigkeiten benutzt werden kann. Die dafür bestehenden Theorien, bei denen die Viskosität aus dem logarithmischen Dekrement der Schwingung berechnet wird, sind hier durch die Berücksichtigung der Randeffekte erweitert worden, unter der Voraussetzung, dass der Abstand zwischen den beiden feststehenden Platten klein ist im Verhältnis zum Radius der Scheibe und zur Grenzschichtdicke. Die Theorie stimmt mit den vonKestin undPilarczyk experimentell gefundenen Werten auf 0,1% genau überein.

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This research was supported by the United States Air Force through the Air Force Office of Scientific Research of the Air Research and Development Command, under contract No. AF 18 (600) 1548. Reproduction in whole or in part is permitted for any purpose of the United States Govermnent. Part of the work was done while the author was being supported by a grant from the Alfred P. Sloan Foundation.     

Duale Optimierungsaufgaben und Sattelpunktsätze

Zusammenfassung Es wird der formale Zusammenhang zwischen der Existenz eines Sattelpunktes und einem Paar dualer Optimierungsaufgaben angegeben. Dabei werden Voraussetzungen wie konvex oder konkav nicht benötigt. Die Theorie ist so aufgezogen, daß dieFenchelsche Theorie der konjugierten Funktionen gut in diesen Rahmen eingebaut werden kann. An vielen Beispielen wird der Zusammenhang mit bekannten Dualitätssätzen dargestellt.

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Summary We treat the formal relations between the existence of a saddle-point and a pair of dual programming problems without using assumptions like convex or concave. The theory is built in such a way, thatFenchels theory of conjugate functions fits very well in its frame. The relationship between the theory and other well known duality theorems are explained by many examples.

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Vorgel. v.:W. Krelle.     

Eine Feldtheorie der Versetzungen imCosserat-Kontinuum

Summary There is a mathematical analogy between the continuum theory of moving dislocations and electrodynamics. This remarkable analogy points out that the theory of dislocations is incomplete with regard to missing constitutive equations. The generalized electrodynamic theory byMie shows a way to complete the theory of moving dislocations, leading finally to the formulation of a Lagrangian density.     

Operatorenkalkül über freien Monoiden III: Lagrangeinversion und Sheffersysteme

In this part the relations between the theory of binomial systems for noncommuting variables (of part II) and the results ofRota, Cigler et al. for the commutative case are studied in some detail. Afterwards for binomial systems of diagonal type there are given generalizations of the Rodrigues formula and the theorem ofLagrange-Good. A short development of Sheffer sequences follows. Finally the results are extended to binomial systems the structure of which is determined by any partial order of linear type.     

Aerodynamische Kräfte auf schlanke Rotationskörper

Summary Comparison between measurements of the normal force of fiveslender bodies of revolution with different base areas in supersonic flow and the simple airshiphull theory ofMunk or the more accurate linearised theory ofpotential flow show considerable discrepancy not only at large, but also at small angles of attack.To get better agreement with the experimental results, it is shown to be more important to estimate theeffects of viscosity than to consider higher orders of potential flow theory.

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Vorgetragen am Astronautischen Kongress 1953 in Zürich, zufolge Abwesenheit des Verfassers durch HerrnB. Chaix, Institut für Aerodynamik der ETH.     

On a regular cauchy problem for the Euler-Poisson-Darboux equation

Summary The explicit solution of a particularCauchy problem for the n-dimensionalEuler-Poisson-Darboux equation is found. To obtain the solution the method ofM. Riesz is extended to include non self-adjoint equations. Existence and uniqueness are shown. This research was supported in part by the United States Air Force under Contract No. AF18(600)-573 — monitored by the Office of Scientific Research, Air Research and Development Command.     

Globalization strategies for Mesh Adaptive Direct Search

The class of Mesh Adaptive Direct Search (Mads) algorithms is designed for the optimization of constrained black-box problems. The purpose of this paper is to compare instantiations of Mads under different strategies to handle constraints. Intensive numerical tests are conducted from feasible and/or infeasible starting points on three real engineering applications.     

A generalization of Riemann's method for partial differential equations

Summary A bilinear divergence identity is obtained, which differs from the usualLagrange divergence identity employed byRiemann. In the case of two independent variables, this new identity is used to unify the treatment ofCauchy's problem for hyperbolic equations, the initial value problem for parabolic equations, and theDirichlet problem for elliptic equations. This research was supported in whole or in part by the United States Air Force under Contract N^o. AF18(600)-573 monitored by the Office of Scientific Research, Air Research and Development Command.     

Some contributions to the asymptotic theory of Bayes solutions

Summary This paper deals with the asymptotic theory of Bayes solutions in (i) Estimation (ii) Testing when hypothesis and alternative are separated at least by an indifference region, under the assumption that the observations are independent and indentically distributed. The estimation results which are partial generalizations of results of LeCam begin with a proof of the convergence of the normalized posterior density to the appropriate normal density in a strong sense. From this result we derive the asymptotic efficiency of Bayes estimates obtained from smooth loss functions and in particular of the posterior mean. The last two theorems of this section deal with asymptotic expansions for the posterior risk in such estimation problems. The section on testing contains a limit theorem for the n-th root of the posterior risk under weak conditions on the prior and the loss function. Finally we discuss generalizations and some open problems.Part of this research was done while P. J. Bickel was on leave at Imperial College, London. — This research was partially supported by National Science Foundation Grant GP-5059.This research was partially supported by National Science Foundation Grant GP-5705. Part of this research was done while J. A. Yahav was visiting the department of Statistics at Stanford University.     

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.     

Applications of Nagumo-Hukuhara theory on the boundary value problems for nonlinear ordinary differential equations to abrikosov problem and Falkner-Skan problem

Summary This paper studies the two boundary value problems (A) and (B) as applications of Nagumo-Hukuhara theory on the boundary value problems for the second order nonlinear ordinary differential equations. Dedicated to Profs.Masuo Hukuhara andMitio Nagumo on their Seventieth Birthdays Entrata in Redazione il 4 giugno 1976.     

Bimeasures and their integral extensions

Summary The object of this paper is to motivate three papers byM. Morse andW. Transue, which will presently appear, where the Authors will lay the foundations of a general theory of C-bimeasures and their integral extensions. To Mauro Picone on his 70^th birth day.