Reducible Dupin submanifolds

Pinkall's standard constructions for obtaining a Dupin hypersurface W in ^N from a Dupin hypersurface M in ⁿ , N>n, are studied in the context of Lie sphere geometry. It is shown that a compact Dupin hypersurface W in ^N with g distinct principal curvatures at each point is reducible to a compact Dupin hypersurface M in ⁿ if and only if g=2.This research was supported by NSF Grant No. DMS 87-06015.     

Relaxation methods in control theory

Two different relaxed problems associated with a problem of optimal control theory, governed by an ODE, are considered: the first is obtained by Young's methods and the second by semicontinuity arguments. A formula which relates the two relaxed functionals to each other is given.     

Poisson structure of a periodic classical XYZ -chain

Classical variables ( -variables) are constructed for a periodic classical XYZ -chain (of the discrete Landau-Lifshitz equation). The connection of the quantum method of the inverse problem and the theory of finite-zone integration is discussed.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 150, pp. 154–180, 1986.     

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.     

On 2-spannedness for the adjunction mapping

LetL be a line bundle on a smooth connected projective manifold X of dimension n. We extend to any dimension the definition of k-spannedness forL; this is a notion of k-th order embedding which was recently given in the case of curves and surfaces. Then, by a reduction to the surfaces case, we prove that the adjoint bundle K_x+(n–1)L is 2-spanned ifL is (at least) 3-spanned.     

The Cauchy Problem for a Nonlinear Diffusion Equation with Absorption and Convection

In this paper we construct the large-time solution of the equation which models the effectof nonlinear absorption on thin film flow of a viscous fluidon a sloping bed. For initial data with finite support we showhow the asymptotic structure of the Cauchy problem depends onthe absorption index . An important addition motivation forthe study is to provide a basis for extending the results tothe more general equation which has applications in plasma physics.     

A remark on partial linear spaces of girth 5 with an application to strongly regular graphs

We derive a lower bound on the number of points of a partial linear space of girth 5. As an application, certain strongly regular graphs with=2 are ruled out by observing that the first subconstituents are partial linear spaces.     

Effects of speaking rate on tongue position and velocity of movement in vowel production

This study used glossometry to examine the position of the tongue and the velocity of its movements in vowels spoken normally and at a self-selected fast rate. The subject in experiment 1 showed lingual undershoot for stressed vowels in "a big again" and "a bob again." The tongue was lower for /I/ and higher for /a/ at the fast rate than at the normal rate. The stressed vowels exerted an affect on unstressed vowels: The tongue was lower in the schwas that preceded and followed /a/ than /I/. Only one of the three subjects in experiment 2 showed no lingual undershoot for fast-rate /I/. The tongue was higher at the fast rate than at the normal rate in the schwas flanking /I/ so that the displacement was less at the fast rate than at the normal rate. Another talker increased the peak velocity of tongue movements at the fast rate and showed no undershoot for /a/. Multiple regression analyses showed that the timing of movements for successive phonetic segments accounted well for undershoot in only one of the three subjects. The results suggest that in order to model the effects of speaking rate on the tongue movements used in forming stressed vowels, it will be necessary to take into account: (1) how much vowels are shortened at a fast rate: (2) how much the peak velocity of tongue movements is increased, if at all; and (3) the position of the tongue before and after the stressed vowels. All three factors are likely to be influenced by how clearly the talker wishes to speak.     

Progressing waves on spherical spacetimes

A family of spherically symmetrical spacetimes that are transparent to minimally coupled scalar multipole waves is investigated. The concept of transparency is reformulated in terms of the established concept of progressing waves. The set of spherically symmetric spacetimes and the family of wave equations for which the transparency condition can be shown to hold is generalized.     

Quasi-genericity of bifurcations to high dimensional invariant tori for maps

We consider a family of maps in a Banach spaceE near the situation when the derivative at the fixed point has two pairs of complex eigenvalues lying on the unit circle, the other part of the spectrum being strictly inside the unit disc. We focus our attention on the region of the parameter space where the truncated normal form of the maps shows a bifurcation of a family of invariantT ¹-circles into a family of invariantT ²-tori. We show that this problem needs a 3 dimensional parameter unfolding and that, for the complete maps, bifurcation occurs at points _,, where is the rotation number on the non-normally hyperbolicT ¹-circle, ande ^±2i are the eigenvalues of the constant matrix conjugated to the non-contracting part of the linearization on the normal fiber bundle overT ¹. Making some non-resonance and diophantine assumptions on (, ) leading to a positive measure Cantor set inT ², we show that in paraboloïdal regions of the 3 dim. parameter space we have clean bifurcations as for the truncated normal form. The complement of these regions forms a set of bubbles such as the ones obtained by Chenciner in [Chen] for a codimension 2 problem for maps in ². The main tool here is a generalization for a matrix function onT ¹, close to a constant, of the quasi-conjugacy to a constant, modulo a minimum of additional parameters (moved quasi-conjugacy). For the infinite dimensional case we use aC decoupling result on the angular dependent linear parts into a contraction, still angular dependent, and another part quasi-conjugated to a constant matrix. This type of analysis applies for a wide range of problems, where truncated normal forms of the maps give bifurcations fromT ⁿ toT ⁿ⁺¹ tori, and this needs a (n+1)-dimensional parameter unfolding.We gratefully acknowledge the DRET (contrat 86/1445) who supported one of the authors (J.L.) during this work. This research has been also supported by the E.E.C. contract No. ST 2J-0316-C (EDB) on Mathematical problems in nonlinear Mechanics