Two-dimensional choked transonic flow

Summary Choked transonic flow about a thin two-dimensional airfoil in a wide wind tunnel is analyzed. Asymptotic expansions are constructed in terms of the transonic similarity variableK _T as K_T0 (and the tunnel height), in an inner region, near the body, and in an outer region, near the walls, and are matched. The inner expansion consists of sonic free flow plus a correction. The outer expansion accounts for the meeting of the sonic line and the limit characteristic at the wall. The magnitude of the correction to sonic free flight at the body as well as the dependence of the choking Mach number on the wind tunnel height are determined. A boundary value problem is formulated to determine the correction to the free flight sonic flow.

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Zusammenfassung Es wird die schallnahe Sperrströmung um einen zweidimensionalen Flügel in einem weiten Windkanal analysiert. Asymptotische Reihenentwicklungen werden in der transsonischen Ähnlich-KeitsvariablenK _T im inneren Bereich in der Nähe des Körpers und im äußeren Bereich nahe der Wandung konstruiert und verglichen. Die innere Reihenentwicklung besteht aus freier Schallströmung und einem Korrektionsglied. Die äußere Reihenentwicklung steht für das Zusammentreffen der Schallinie und der Grenzcharacteristik an der Wandung. Es werden die Größe des Korrektionsgliedes zur freien Schallströmung am Körper, als auch die Abhängigkeit der Machschen Zahl der freien Strömung von der Höhe des Windkanals bestimmt. Ein Randwertproblem wird formuliert um die Korrektur zur freien Schallströmung zu bestimmen.

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Research partly sponsored by AFOSR under Grant No. 82-0155 and by NSF under Grant DMS-8401738. The U.S. Government is authorized to reproduce and distribute reprints notwithstanding any copyright notation thereon.     

Simplifying assumptions for the order of partitioned multivalue methods

In solving a system of ordinary differential equations by multivalue methods, the concept of partitioning such a method into two parts corresponding to internal and external computations has recently been introduced by Burrage and Butcher [3]. In this paper the order conditions for such methods will be given and simplifying assumptions for them derived. Various examples are presented and, in particular, a new class of methods is introduced and its order properties are studied.     

Isotropic random walks in a tree

Summary Let T be an infinite homogeneous tree of order a+1. We study Markov chains {X _n} in T whose transition functions p(x, y)=A[d(x,y)] depend only on the shortest distance between x and y in the graph. The graph T can be represented as a symmetric space of a p-adic matrix group; we prove a series of results using essentially the spherical functions of this symmetric space. Theorem 1. d(X _n,x) n a.s., where >0 if A(0) 1, X ₀=x. Assuming {X _n} is strongly aperiodic, Theorem 2. p ₂(x, y)CRⁿ/n^3/2 for fixed x, y where R=(d) A(d)<1, and if E[d(X₁, X₀)²]<, Theorem 3. R(1–u, x, y) = (1–u)ⁿp_n(x, y)=Ca^–d[exp(–du/)+o_d(1)] as d=d(x,y) uniformly for 0u2. Using Theorem 3, we calculate the Martin boundary Dirichlet kernel of p(x, y) on T, which turns out to be independent of {itA(d)}. We also consider a stepping-stone model of a randomly-mating-and-migrating population on the nodes of T. If initially all individuals are distinct, then in generation n approximately half of the individuals of a given type are within n of a typical one and essentially all are within 2n.This work was partially supported by the National Science Foundation under grant number MCS 75-08098-A01For the academic year 1977–78: Department of Mathematics GN-50, University of Washington, Seattle, Washington 98195 USA     

Some Asymptotic Results on <Emphasis Type="Italic">q</Emphasis>-Binomial Coefficients

We look at the asymptotic behavior of the coefficients of the q-binomial coefficients (or Gaussian polynomials) \({\left(\begin{array}{c}{a+k}\\{k}\end{array}\right)_{q}}\), when k is fixed. We give a number of results in this direction, some of which involve Eulerian polynomials and their generalizations.     

Comparison of "Polarization inversion with spin exchange at magic angle" and "geometric analysis of labeled alanines" methods for transmembrane helix alignment

Using the model alpha-helical peptide acetyl-GGALW5LALALALALALALW19LAGA-ethanolamide ("GWALP23"), we have compared the polarization inversion with spin exchange at magic angle method and geometric analysis of labeled alanines method for estimating the transmembrane helix orientation. For GWALP23 in bilayers of a short lipid, dilauroylphosphatidylcholine, we find general agreement between the two methods, with a static helix tilt of about 11degrees-13degrees with respect to the bilayer normal.     

Synthesis of new polymetallic iron(III) clusters containing polycarboxylate ligands based on cavity-blocked cyclen

The synthesis and magnetic properties of enneametallic and octametallic Fe(III) cage complexes from tetraazamacrocycle ligands (1,7-H2DO2A) and (H3DO3A) respectively, are reported.     

An efficient asymmetric synthesis of manzacidin C

A brief synthesis of manzacidin C based on a chiral silane-promoted diastereo- and enatioselective acylhydrazone-alkene [3 + 2] cycloaddition reaction has been achieved. This synthesis is the first synthesis of any of the manzacidins wherein the C(4) and C(6) stereocenters are established in a single highly stereoselective step.     

Synthesis of the unnatural amino acid N-N-(ferrocene-1-acetyl)-l-lysine: A novel organometallic nuclease

The synthesis of a new unnatural amino acid, N^α-N^ε-(ferrocene-1-acetyl)-l-lysine, was achieved by coupling a ferroceneacetic acid molecule onto the side chain amine of a lysine. The structure of the compound provides options for incorporation of the molecule into peptides or large proteins. In addition, N^α-N^ε-(ferrocene-1-acetyl)-l-lysine exhibits nuclease activity. It is expected that incorporation of this ferrocenyl amino acid into any nucleic acid-binding protein will endow the protein with nuclease capability.