Dynamics of heart rate

Heart rate oscillates on several different time scales and has long-term variability in the form of 1/f noise. The physiological control of heart rate is briefly reviewed, and several typical patterns of heart rate variability, in health and sickness, are described. Considered briefly are some possible dynamical mechanisms for heart rate variability.     

Upper bounds for the diameter and height of graphs of convex polyhedra

Let Δ(d, n) be the maximum diameter of the graph of ad-dimensional polyhedronP withn-facets. It was conjectured by Hirsch in 1957 that Δ(d, n) depends linearly onn andd. However, all known upper bounds for Δ(d, n) were exponential ind. We prove a quasi-polynomial bound Δ(d, n)≤n ^{2 logd+3}. LetP be ad-dimensional polyhedron withn facets, let ϕ be a linear objective function which is bounded onP and letv be a vertex ofP. We prove that in the graph ofP there exists a monotone path leading fromv to a vertex with maximal ϕ-value whose length is at most . This research was supported in part by a BSF grant, by a GIF grant, and by ONR-N00014-91-C0026.     

Cycloaddition of methyl 2-(2,6-dichorophenyl)-2H-azirine-3-carboxylate to electron-rich 2-azadienes

tert-Butyldimethylsililoxy-2-aza-1,3-butadienes react with 2H-azirine 3 leading to Diels-Alder cycloadducts in moderate yields. The reactions are endo- and regioselective with the azirine being added by its less hindered face. There is only one product in the case of 1b, 4b. There are two isomers (4 and 5) from 1a, 1c and 1d. A different result was obtained with the diene 1e. Diene 1e formed products 4e and 8. Some of compounds 4 and 5 have been hydrolysed leading to functionalised aziridines 7. Compound 8 gave aziridine 9.