Unfolding the torus: Oscillator geometry from time delays

Summary We present a simple method of plotting the trajectories of systems of weakly coupled oscillators. Our algorithm uses the time delays between the “firings” of the oscillators. For any system ofn weakly coupled oscillators there is an attracting invariantn-dimensional torus, and the attractor is a subset of this invariant torus. The invariant torus intersects a suitable codimension-1 surface of section at an (n−1)-dimensional torus. The dynamics ofn coupled oscillators can thus be reduced,in principle, to the study of Poincaré maps of the (n−1)-dimensional torus. This paper gives apractical algorithm for measuring then−1 angles on the torus. Since visualization of 3 (or higher) dimensional data is difficult we concentrate onn=3 oscillators. For three oscillators, a standard projection of the Poincaré map onto the plane yields a projection of the 2-torus which is 4-to-1 over most of the torus, making it difficult to observe the structure of the attractor. Our algorithm allows a direct measurement of the 2 angles on the torus, so we can plot a 1-to-1 map from the invariant torus to the “unfolded torus” where opposite edges of a square are identified. In the cases where the attractor is a torus knot, the knot type of the attractor is obvious in our projection.     

Anionic oxycope rearrangements in acetylenic spiro systems

The anionic oxy-Cope rearrangement of the acetylenic carbinol 3 furnishes the tricyclic diketone 5; substitution at the key position furnishes only the bicyclic 1,3-bridged dione 6.     

Crystal structure of N-m-tolyl phthalimide

N-m-tolyl phthalimide, C₁₅NO₂H₁₁ crystallizes in the space group Cc with unit cell dimensions,a=8·54(1),b=19·89(2),c = 7·59(1)A, β=114·53(1)° andZ=4.V=1173(2)A³,D _m =1·35(1),D _c = 1·344 mg.m⁻³,M _r =237 λCoK_a=1·7903A. The structure was solved byMULTAN and refined to an R-factor of 0·116 for 632 counter reflections. The molecules are held together by van der Waal’s forces. The angle between the tolyl plane and the plane through the phthalimide group is 53·4(4)°. Contribution No. 607.     

Catalytic conjugate additions of carbonyl anions under neutral aqueous conditions

The conjugate addition of carbonyl anions catalyzed by thiazolium salts that is fully operative under neutral aqueous conditions has been accomplished. The combination of alpha-keto carboxylates and thiazolium-derived zwitterions produces reactive carbonyl anions in a buffered protic environment that readily undergo conjugate additions to substituted alpha,beta-unsaturated 2-acyl imidazoles. The scope of the reaction has been examined and found to accommodate various alpha-keto carboxylates and beta-aryl substituted unsaturated 2-acyl imidazoles. The optimal precatalyst for this process is the commercially available thiazolium salt 5, a simple analogue of thiamin diphosphate. In this process, no benzoin products from carbonyl anion dimerization are observed. The corresponding 1,4-dicarbonyl compounds can be efficiently converted into esters and amides by way of activation of the N-methylimidazole ring via alkylation.     

Influence of magnetic quantization on the debye screening length in degenerate semiconductors having parabolic energy bands

The Debye screening length in degenerate semiconductors having parabolic energy bands is shown for the first time to have an oscillatory dependence on magnetic field under the influence of Landau quantization when the higher lying sub-bands are occupied.The authors are indebted to Professor J. N.Bhar for his keen interest in the work.     

A note on the Valiron method of summability

Acta Mathematica Hungarica -     

Flow visualization of supersonic transverse jet over Mach 2 freestream of a sharp cone

Graphical abstract