There are currently intense efforts being directed towards extending the range and energy of long distance nonlinear pulse propagation in the atmosphere by moving to longer infrared wavelengths, with the purpose of mitigating the effects of turbulence. In addition, picosecond and longer pulse durations are being used to increase the pulse energy. While both of these tacks promise improvements in applications, such as remote sensing and directed energy, they open up fundamental issues regarding the standard model used to calculate the nonlinear optical properties of dilute gases. Amongst these issues is that for longer wavelengths and longer pulse durations, exponential growth of the laser-generated electron density, the so-called avalanche ionization, can limit the propagation range via nonlinear absorption and plasma defocusing. It is therefore important for the continued development of the field to assess the theory and role of avalanche ionization in gases for longer wavelengths. Here, after an overview of the standard model, we present a microscopically motivated approach for the analysis of avalanche ionization in gases that extends beyond the standard model and we contend is key for deepening our understanding of long distance propagation at long infrared wavelengths. Our new approach involves the mean electron kinetic energy, the plasma temperature, and the free electron density as dynamic variables. The rate of avalanche ionization is shown to depend on the full time history of the pulsed excitation, as opposed to the standard model in which the rate is proportional to the instantaneous intensity.

The effect of pretreatment with fentanyl on the pharmacokinetics of a single bolus of propofol was studied in 17 female patients (mean age 35 yr), ASA grade I. Eight patients received fentanyl 1.5 micrograms kg-1 5 min before induction of anaesthesia. In all patients anaesthesia was induced with propofol 2.5 mg kg-1 and maintained with halothane and nitrous oxide in oxygen. Pretreatment with fentanyl resulted in prolonged apnoea in all eight patients compared with three of nine patients in the control group. The pharmacokinetic values for propofol were described by a three-compartment mammillary model with rapid distribution phases (T1/2 alpha mean (SEM) 3.1 (2.0) min and T1/2 beta 44 (9.1) min) and a slower final phase of T1/2 gamma 520 (96) min. The clearance of propofol was rapid (mean 1.6 (0.24) litre min-1). Propofol was distributed initially into a relatively large central compartment (mean 23.7 (6.6) litre) and was extensively redistributed (mean Vss 593 (157) litre). There was no difference in the pharmacokinetic profile of propofol between the two groups.

[ Proc. Roy. Soc. Edinburgh Sect. A 91 (1982), 205–212]

An (n, q) graph is a graph on n labelled points and q lines, no loops and no multiple lines. We write N = ½n(n – 1), B(a, b) = a!/{b!(a – b)!} and B(a, 0) = 1, so that there are just B(N, q)different (n, q) graphs. Again h(n, q) is the number of Hamiltonian (n, q) graphs. Much attention has been devoted to the problem of determining for which q = q(n) “almost all” (n, q) graphs are Hamiltonian, i.e. for which q we haveas n → ∞. I proved [8, Theorem 4] that qn–3/2; → ∞ is a sufficient condition by showing that, for such q, almost all (n, q) graphs have about the average number of Hamiltonian circuits (H.c.s).

AbstractThe number of nonseparable graphs on n labeled points and q lines is u(n, q). In the second paper of this series an exact formula for u(n, n + k) was found for general n and successive (small) k. The method would give an asymptotic approximation for fixed k as n → ∞. Here an asymptotic approximation to u(n, n + k) is found when k = O(n1/2) and an approximation to logu(n, n + k) when k < (1 ‐ ϵ)(1/3 n)1/2. The problem of finding an approximation to u(n, q) when (q ‐ n)/n1/2 → + → and q/n ‐ 1/2 logn ‐ 1/2 log logn → ‐ ∞ is open.

We consider bipartite graphs on m red points and n blue points, where m ⩽ n, and prove that, for any fixed k, almost all such graphs (labelled or unlabelled) are k-connected as n → ∞, provided m > C log n, where C depends on k. If Tmn is the number of such unlabelled graphs, we show that Tmn ∼ 2mn/(m!n!). If T′mn is the number of such unlabelled graphs with the colours removed, then T′mn ∼ Tmn if m < n and T′mn ∼ ½Tnn. We deduce that almost all bipartite graphs on p points in all, whether labelled or unlabelled, are k-connected and so prove a conjecture of Harary and Robinson.

SynopsisAn (n,q) graph is a graph on n labelled points andqlines without loops or multiple lines. We write ν(n,q) for the number of smooth (n,q) graphs, i.e. connected graphs without end points, and ν =V(Z,Y) = ∑n,qν(n,q)ZnYq/n! for the exponential generating function of ν(n,q). We use the Riddell “core and mantle” method to find an explicit form forV(not, as usual with this method, only a functional equation). From this we deduce a partial differential equation satisfied byV.We interpret this equation in purely combinatorial terms. We writeVk= ∑nν(n,n+k)Xn/n! and find a recurrence formula forVkfor successivek.We use these and other results to find an asymptotic expansion for ν(n,q) asn→∞ when (q/n) −logn−log logn→ + ∞ and an asymptotic approximation to ν(n,n+k) when 0

Journal of the London Mathematical SocietyVolume s2-24, Issue 3 p. 397-404 Notes and papers The Proportion of Labelled Bipartite Graphs which are Connected V. L. Klee, V. L. Klee Department of Mathematics, University of Washington, Seattle, Washington 98195Search for more papers by this authorD. G. Larman, D. G. Larman Department of Mathematics, University College, London WC1E 6BTSearch for more papers by this authorE. M. Wright, E. M. Wright Department of Mathematics, University of Aberdeen, Aberdeen AB9 2TYSearch for more papers by this author V. L. Klee, V. L. Klee Department of Mathematics, University of Washington, Seattle, Washington 98195Search for more papers by this authorD. G. Larman, D. G. Larman Department of Mathematics, University College, London WC1E 6BTSearch for more papers by this authorE. M. Wright, E. M. Wright Department of Mathematics, University of Aberdeen, Aberdeen AB9 2TYSearch for more papers by this author First published: December 1981 https://doi.org/10.1112/jlms/s2-24.3.397Citations: 5AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Citing Literature Volumes2-24, Issue3December 1981Pages 397-404 RelatedInformation

Burnside himself correctly ascribed the lemma now given his name to Frobenius. We explain how the subsequent error seems to have arisen.

We write and x1 = 1. In a recent paper (3), Stein and Everett consider the sequence defined byand investigate whethertends to a limit as n → ∞. The case b = 0 has a combinatorial interpretation (see (1)) and, in (2), they use this to prove that xn → e−1 in this case. Even for positive integral b, the number Sn has no known combinatorial interpretation, but they prove (3) that xn → x under the hypothesis that Sn+1Sn−1 ≥ Sn i.e. that Sn is convex. By computation and induction, they prove this hypothesis for b = k/5, where k = 1, 2,…, 50.

AbstractThe number of connected graphs on n labeled points and q lines (no loops, no multiple lines) is f(n,q). In the first paper of this series I showed how to find an (increasingly complicated) exact formula for f(n,n+k) for general n and successive k. The method would give an asymptotic approximation to f(n,n+k) for any fixed k as n → ∞. Here I find this approximation when k = o(n1/3), a much more difficult matter. The problem of finding an approximation to f(n,q) when q > n + Cn1/3 and (2 q/n) ‐ log n → ‐ ∞ is open.

Wille found an asymptotic approximation to r ( n , q ), the number of unlabeled oriented graphs on n points and q directed lines, for a wide interval of q and conjectured that, for given n , the maximum of r ( n , q ) occurs at q = [ 2(N + 1) 3 ] . We find the (different) asymptotic approximation to r ( n , q ) valid for the remaining interval of q and prove Wille's conjecture for all large n .

Journal of the London Mathematical SocietyVolume s2-18, Issue 3 p. 397-402 Notes and papers The k-Connectedness of Unlabelled Graphs T. R. S. Walsh, T. R. S. Walsh Computing Centre, U.S.S.R. Academy of Sciences, MoscowSearch for more papers by this authorE. M. Wright, E. M. Wright Aberdeen University, Aberdeen, ScotlandSearch for more papers by this author T. R. S. Walsh, T. R. S. Walsh Computing Centre, U.S.S.R. Academy of Sciences, MoscowSearch for more papers by this authorE. M. Wright, E. M. Wright Aberdeen University, Aberdeen, ScotlandSearch for more papers by this author First published: December 1978 https://doi.org/10.1112/jlms/s2-18.3.397Citations: 3AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Citing Literature Volumes2-18, Issue3December 1978Pages 397-402 RelatedInformation

Journal Article FORMULAE FOR THE NUMBER OF SPARSELY-EDGED STRONG LABELLED DIGRAPHS Get access E. M. WRIGHT E. M. WRIGHT University of AberdeenScotland Search for other works by this author on: Oxford Academic Google Scholar The Quarterly Journal of Mathematics, Volume 28, Issue 3, September 1977, Pages 363–367, https://doi.org/10.1093/qmath/28.3.363 Published: 01 September 1977 Article history Received: 01 December 1976 Published: 01 September 1977