Circles through two points that always enclose many points

This paper proves that any set of n points in the plane contains two points such that any circle through those two points encloses at least points of the set. The main ingredients used in the proof of this result are edge counting formulas for k-order Voronoi diagrams and a lower bound on the minimum number of semispaces of size at most k.Work on this paper by the first author has been supported by Amoco Fnd. Fac. Dev. Comput. Sci. 1-6-44862 and by the National Science Foundation under Grant CCR-8714565, by the second author has been partially supported by the Digital Equipment Corporation, by the fourth author has been partially supported by the Office of Naval Research under Grant N00014-86K-0416.     

Algorithms for flows with parametric capacities

We consider the problem of finding maximal flows with respect to capacities which are linear functions of a parametert [0,T]. Since this problem is a special case of a parametric linear program the classichorizontal approach can be applied in which optimal solutions are computed for successive subintervals of [0,T]. We discuss an alternative algorithm which approximates in each iteration the optimal solution for allt [0,T]. Thisvertical algorithm is a labeling type algorithm where the flow variables are piecewise linear functions. Flow augmentations are done alongconditional flow augmenting paths which can be found by modified path algorithms. The vertical algorithm can be used to solve the parametric flow problem optimally as well as to compute a good approximation for allt if the computation of the optimal solution turns out to be too time consuming.Partially supported by NSF Grants ECS-8412926 and INT-8521433, and NATO Grant RG 85/0240.     

Disorder arising from the location of enantiomers in a twist-boat structure: 2,6-trans-2,4,6-tris(trichloromethyl)-1-oxa-3,5-dithian

Crystals of the title compound are orthorhombic, Pca2₁;a=27.586(4),b=10.509(3),c=11.080(2) Å,V=3212(1) ų,Z=8. The structure was solved by direct methods, from data collected at room temperature on an Enraf-Nonius CAD4 diffractometer, and refined by full-matrix least squares to a finalR value of 0.055 using 2043 reflections. One of the two independent molecules in the asymmetric unit is disordered. The disorder is manifested mainly between the oxygen atom and one sulfur atom and is caused by opposite enantiomers occupying corresponding positions in different unit cells. The endocyclic torsion angles in both independent molecules are very similar but the disposition of the heteroatoms on the twist-boat framework is different in the two molecules. Site occupancy of the two models on this framework in the disordered molecule is in the ratio 2179 with the smaller proportion being very similar to the ordered molecule and the remainder similar to that found in the normal molecule oftrans-4-chloro-2,4,6-tris(trichloromethyl)-1-oxa-3,5-dithian (Irving and Irving, 1988).     

Crystal and molecular structure of 1R,3S,5S,7R,10R-3,6,6,10,14-Pentamethyltricyclo[10.3.0.05,7]pentadeca-11,14-diene-1, 10-dihydroxy-2,13-dione (Japodagrol), C20H28O4

A new cytotoxic macrocyclic diterpenoid named Japodagrol, C₂₀H₂₈O₄, has been isolated from the ornamental plantJatropha podagrica, family Euphorbiaceae, grown in Nigeria. The structure of the title compound, C₂₀H₂₈O₄, was solved by X-ray analysis.M _r =332.44, monoclinic space groupC2,a=23.285(4),b=6.5105(12),c=12.505(3) Å,=98.504(17),V=1874.8 ų,Z=4,D _c =1.178 Mg/m³. CuK radiation (graphite crystal monochromator, =1.54184 A), (CuK)=6.13 cm^–1,F(000)=720,T=290 K. Final conventionalR factor=0.035,R _w =0.038 for 3349 observed reflections. The structure was solved with the programDirdif. The 5-membered ring is closed to a half-chair form. The compound contains inter- and intramolecular hydrogen bonds.     

Eine direkt formulierte lineare Theorie für viskoelastische Platten und Schalen

Übersicht Eine Möglichkeit der Entwicklung einer allgemeinen Theorie für Platten und Schalen besteht in der Modellierung des Flächentragwerks durch ein zweidimensionales deformierbares Kontimmm und der direkten Ableitung der kinematischen Beziehungen, der Bewegungsgleichungen und der konstitutiveu Gleichungen. Eine so erhaltene Theorie ist mathematisch und physikalisch widerspruchsfrei, jedoch ist es für ihre Anwendung notwendig, die sogenannten Ersatzeigenschaften zu ermitteln. Im Beitrag wird eine geometrisch und physikalisch lineare Theorie behandelt. Jeder Punkt des Kontinuums ist ein infinitesimal kleiner Starrkörper mit nur 5 Freiheitsgraden (3 Translationen, 2 Rotationen). Bei Annahme dieser Einschränkungen gelingt es, eine Theorie direkt abzuleiten, wobei alle Ersatzeigenschaften des Flächentragwerks bestimmt werden können. Im Beitrag werden die Möglichkeiten der allgemeinen Theorie am Beispiel isotropen viskoelastischen Materials mit über die Dicke veränderlichen Eigenschaften gezeigt. Die Theorie schließt die Betrachtung mehrschichtiger Flächentragwerke ein.

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A directly formulated linear theory of viscoelastic plates and shells

Summary One kind to develop a general theory for plates and shells is the modelling of the structure as a two-dimensional deformable continuum and the direct approach to the kinematical relations, equations of motion and constitutive equations. Such a theory is physically and mathematically correct. For the application of such directly formulated theories it is nessecery to identify the so-called effective properties. In the paper the theory is formulated for geometrical and physical linearity. Each point of the continuum is an infinitesimal small rigid body with only 5 degrees of freedom (3 translations, 2 rotations). For such a kinematical assumption it is possible to develop a direct theory and to determinate all effective properties of the structure. The paper demonstrates the possibilities of the general theory for an isotropic viscoelastic plate with material properties varying over the thickness. The theory includes also the analysis of multilayered plates and shells.

     

A fourth order method for a singular two-point boundary value problem

Recently, Chawla et al. described a second order finite difference method for the class of singular two-point boundary value problems:

Forces of a fluid in a rotating straight pipe

The fluid flow through a rotating straight pipe is considered, the axis of rotation being perpendicular to the pipe axis. The flow of the fluid is taken as fully developed, i.e. the velocity field is assumed to be the same in all transverse cross sections of the pipe. The derivation presented applies to viscous and nonviscous incompressible fluids. For constant angular pipe velocity a simple and exact (Coriolis type) relationF=2Q(t) is derived between the forceF by which the fluid acts on the (unit length of the) pipe in the direction perpendicular to the two axes, the fluid mass flow rateQ(t) through the pipe, and the angular velocity. Variable angular velocities, i.e. , introduce an additional term into the expression for the inertial forceF, which depends only on and on known (constant) parameters; this term is known for given angular velocity(t). The flow configuration investigated here is an idealization of those appearing (over short space and time intervals) in the devices measuring mass flow rateQ(t) through the (Coriolis) forceF. Therefore the exact results derived here cast some light on the degree of precision one expects in these devices, where more complicated flow configurations are present than those looked at in this paper.     

Optimization of scrapping decisions: Theory and applications

Contrary to the variety of models and procedures for optimizing the ordering decision in inventory control theory, no comparable tools exist for the scrapping decision. Thus, the current paper constitutes a step in this direction. Using a simple deterministic model, the basic economics and structure of the problem are introduced first. Then we determine explicitly scrap-optimal turn-around-times as well as the corresponding scrap- and rest-quantities. Accounting for the stochastic nature of demand, the model is generalized and algorithms for determination of optimal scrapping decision are discussed. Finally we compare the results and discuss practical applications.     

Oscillatory Perturbations of Linear Problems at Resonance

This paper is concerned with a study of bounded perturbations of resonant linear problems. It follows from our results that for certain types of bounded domains Ω ? Rⁿ, n ≥ 2, the Dirichlet problem $\matrix{\Delta u+\lambda_{1}u+g(u)=h(x),\ \ \ x\in\Omega\cr \quad\quad\quad\quad\quad\quad u=0,\ \ \ x\in\partial\Omega,}$ has infinitely many positive solutions, in case λ₁ is the principal eigenvalue of ?Δ subject to trivial Dirichlet boundary conditions, g is a nontrivial periodic nonlinearity of zero mean and ∫_03A9h(x)?(x)dx = 0, where ? is an eigenfunction corresponding to λ₁.     

Automata accepting primitive words

LeA be an automaton whose set of inputs equalsX (|X|≧2) and whose cardinality of the set of states equalsn (n≧2), and letQ be the set of all primitive words overX. ByT(A) we denote the language accepted byA. In this paper, we give the following results: