On the number of full levels in tries

We study the asymptotic distribution of the fill‐up level in a binary trie built over n independent strings generated by a biased memoryless source. The fill‐up level is the number of full levels in a tree. A level is full if it contains the maximum allowable number of nodes (e.g., in a binary tree level k can have up to 2^k nodes). The fill‐up level finds many interesting applications, e.g., in the internet IP lookup problem and in the analysis of level compressed tries (LC tries). In this paper, we present a complete asymptotic characterization of the fill‐up distribution. In particular, we prove that this distribution concentrates on one or two points around the most probably value k = ?log_1/qn ? log log log n + 1 + log log(p/q)?, where p > q = 1 ? p is the probability of generating the more likely symbol (while q = 1 ? p is the probability of the less likely symbol). We derive our results by analytic methods such as generating functions, Mellin transform, the saddle point method, and analytic depoissonization. We also present some numerical verification of our results. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 2004     

Chern numbers for two families of noncommutative Hopf fibrations

We consider noncommutative line bundles associated with the Hopf fibrations of SU_q(2) over all Podle? spheres and with a locally trivial Hopf fibration of S³_pq. These bundles are given as finitely generated projective modules associated via 1-dimensional representations of U(1) with Galois-type extensions encoding the principal fibrations of SU_q(2) and S³_pq. We show that the Chern numbers of these modules coincide with the winding numbers of representations defining them. To cite this article: P.M. Hajac et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Chaı̂nes de Markov TripletTriplet Markov Chains

The Hidden Markov Chains (HMC) are widely applied in various problems. This succes is mainly due to the fact that the hidden process can be recovered even in the case of very large set of data. These models have been recetly generalized to ‘Pairwise Markov Chains’ (PMC) model, which admit the same processing power and a better modeling one. The aim of this note is to propose further generalization called Triplet Markov Chains (TMC), in which the distribution of the couple (hidden process, observed process) is the marginal distribution of a Markov chain. Similarly to HMC, we show that posterior marginals are still calculable in Triplets Markov Chains. We provide a necessary and sufficient condition that a TMC is a PMC, which shows that the new model is strictly more general. Furthermore, a link with the Dempster–Shafer fusion is specified. To cite this article: W. Pieczynski, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 275–278.     

Chiroptical properties of steroid 1,3-diaxial diols in the presence of [Mo2(OAc)4]

Summary Bidentate complexes of dimolybdenum tetraacetate with 1,3-diaxial diols show Cotton effects around 490–540 nm (CD band I), 400–440 nm (II), 350–380 nm (III), 320 nm (IV) and 280 nm (V). The sign of the Cotton effect II can be predicted by a proposed sector rule. It seems that from the Cotton effects for cyclic 1,3-diols the absolute configurations or conformations can be obtained.A part of this material was presented at III-rd International Conference on CD Spectroscopy [1]     

Spatial dependence of meson correlation functions at high temperature

The spatial dependence of meson correlation functions at high temperature is studied in perturbative QCD, keeping only the lowest order term. We obtain analytic results for the static correlation function in this approximation. Problems connected with the regularization of the divergent expressions are discussed in detail. The meson screening mass is determined from the form of the correlation function at large distances. We obtain , which agrees with the results of Eletskii and Ioffe, for massless quarks (M=0). Finally, the correlation function in the temporal direction is briefly discussed. For massless quarks, we find the corresponding screening massm _scr=2T.Research supported in part by the Polish State Committee for Scientific Research under Grant No. 2.0204.91.01     

Ultra trace analysis of refractory metals by solid state mass spectrometry — A comparison of GDMS,SSMS and SIMS

The analytical performance of a glow discharge quadrupole mass spectrometer is demonstrated using sintered tungsten as an example. The inherent problem of molecular interferences in glow discharge mass Spectrometry has been considerably reduced using neon as a complementing discharge gas. Particular attention has been paid to time dependences. Analytical figures of merit are presented.     

The use of bathoferroine for extraction-spectrophotometric determinations of microamounts of perchlorate and perrhenate

Bathoferroine can be used for anion determinations by a method based on selective ion-pair extraction if the selectivity of extraction has been enhanced by addition of nonelectrolyte to the water phase. The method of perchlorate and perrhenate determinations characterizes parameters similar to those obtained by methods in which other cation complexes have been used. The advantage of the proposed method is the possibility of considerable concentration of ion association compound extracts by nitromethane reextraction.     

An easy preparation of per- and poly-fluoroalkyl propenyl sulfones

Sodium tridecafluorohexanesulfinate (1a) and sodium 1-chloro-2,2,2-trifluoroethanesulfinate (1b) were prepared by the treatment of 1-iodo-tridecafluorohexane and 1-bromo-1-chloro-2,2,2-trifluoroethane with sodium dithionite in a water-acetonitrile solution. Prolonged reaction of 1a with allyl bromide in DMF afforded tridecafluorohexane 1-propenyl sulfone 2 as the only product in good yield. A similar treatment of 1b gave exclusively 1-chloro-2,2,2-trifluoroethane 3-propenyl sulfone 4. Bromination of 4 followed by dehydrobromination with Et₃N resulted in a mixture of 1-chloro-2,2,2-trifluoroethane 3-bromo-1-propenyl sulfone 6 and 1-chloro-2,2,2-trifluoroethane 2-bromo-3-propenyl sulfone 7, while dehydrobromination with pyridine gave sulfone 6 practically as the only product. α,β-Unsaturated sulfones 2 and 6 were shown to be active dienophiles.     

An indirect flame emission spectrometry method for the determination of traces of sulphur. I

Summary An accurate and simple indirect method of flame emission spectrometric determination of small quantities of elemental sulphur is described. The method is based on the reduction of elemental sulphur by means of sodium borohydride and oxidation of the resulting boron-sulphur compounds to sulphate by means of hydrogen peroxide. The determinations can be carried out in the presence of methanol, acetone, dimethylformamide, toluene and benzene. The detection limit of the reported method is 1 ppm of sulphur.

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Indirekte etnissionsspektroskopische Bestimmungsmethode für Spuren elementaren Schwefels. I.

Zusammenfassung Eine genaue und einfache, indirekte, emissionsspektroskopische Methode zur Bestimmung geringer Mengen elementaren Schwefels wurde ausgearbeitet. Sie besteht in der Reduktion des elementaren Schwefels mit Hilfe von Natriumborhydrid und in der Oxydation der entstehenden Bor-Schwefel-Verbindung zu Sulfat mit Hilfe von Wasserstoffperoxid. Die Analyse wurde in Methanol, Aceton, Dimethylformamid, Toluol und Benzol durchgeführt. Die Erfassungsgrenze beträgt 1 ppm Schwefel.

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Presented at the 8th International Microchemical Symposium, Graz, August 25–30, 1980.     

Syntheses with organoboranes. IX. Vinyl- and 1-alkenyldichloroboranes as ethylene and 1-alkene equivalents for the Diels–Alder reaction

Vinyl- and 1-alkenyldichloroboranes were used as dienophiles for the Diels–Alder reaction with representative aliphatic and cyclic 1,3-dienes. The organoborane adducts were transformed into the corresponding olefins either by protonolysis or by oxidation–mesylation–reduction. Direct protonolysis of the adducts gave in most cases mixtures of olefins whereas the reduction of mesylates with lithium triethylborohydride produced pure olefins in good yields.