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]     

Multinuclear magnetic resonance study of some mesoionic oxatriazoles containing nitrogenous exocyclic groups

¹H,¹³C,¹⁴N and¹⁵N NMR measurements are reported for four mesoionic 1-oxa-2, 3, 4-triazoles containing exocyclic nitrogenous groups. The NMR signal assignments are discussed and compared with those previously published for some corresponding oxatriazoles. The results obtained support the proposed cyclic mesoionic structures for the compounds studied. The questions of possible charge delocalization and valence tautomerism are addressed. Compound with N^– H as a exocyclic group (Fig. 1) is found to be relatively unstable, this is attributed to proton migration in the corresponding non-cyclic form of this molecule.Published in Khimiya Geterotsikiicheskikh Soedinenii, No. 9, pp. 1260–1263, September, 1995.     

Application of Microwave Energy to Cleavage of Sulfur Bonds in the S8Ring in the Simplest Amide Solvents

Microwave energy has been employed for cleavage of sulfur bonds in the S₈ring. Cleavage of S–S bonds was carried out in extraction of elemental sulfur in simple amide solvents. Due to the use of microwave energy and the polarity of amides, up to, 100 times lower concentration of a nucleophilic reducer, such as azide or nitrite, may be used to cleave the sulfur bonds. The time needed for cleavage of sulfur bonds at sulfur concentration higher than 0.1 mg S/100 mg solvent was reduced from hours to minutes. This cleavage was quantitative when carried out in a Teflon container in a microwave oven with continuous or pulsed heating. UV spectra show from which value of absorption there will be a sulfur sol at different sulfur concentrations. This allows us to adjust the microwave exposure power and time to avoid sulfur sol formation and not destroy the sample.     

New insights into the reaction of zinc alkyls with dioxygen

Studies on the reaction of zinc alkyls with O2 are reported which demonstrate that the selective oxygenation of organozinc compound is viable. The reaction of [EtZn(azol)]n (azol = deprotonated 1-aziridineethanol) with an excess of dry O2 in toluene affords the zinc ethylperoxide [EtOOZn(azol)]2[EtZn(azol)]2, while the analogous reaction between Me2Zn and O2 results in the isolation of the Me6Zn7(OMe)8 cluster in high yield.     

Formal Mathematical Analysis of the Existence of the Common Intersection Point in Relation to Determining the Parameters Describing Ion Adsorption at the Oxide/Electrolyte Interface: Comparison of the Triple and Four-Layer Models

For most oxide/electrolyte systems potentiometric titration curves measured for different ionic strengths have a Common Intersection Point (CIP) which corresponds to the Point of Zero Charge (PZC). However, there are systems where a CIP exists but the surface charge at this point does not equal zero (PZC CIP). In this paper theoretical analysis of the systems in which the PZC and CIP do not coincide is presented. It is based on the well-known 2-pK surface charging approach and Triple Layer Model (TLM) as well as the Four Layer Model (FLM) of the electric double layer. The appropriate mathematical criterion for CIP existence was applied with detailed derivations, both for TLM and FLM. Having determined in this manner the parameter values, one can draw proper conclusions about the features of oxide/electrolyte adsorption systems, in which PZC and CIP do not coincide. The values of adsorption parameters are found by fitting simultaneously the obtained theoretical expressions to both of the experimental titration isotherms, and to the individual isotherms of electrolyte cation adsorption measured using radiometric methods.     

Secondary deuterium kinetic isotope effects in irreversible additions of allyl reagents to benzaldehyde

The competitive kinetics of additions of allyl to benzaldehyde-h and -d from allyltributyl tin, from diisopropyltartrylallyl boronate, and from allyl bromide and zinc dust in aqueous tetrahydrofuran have inverse secondary deuterium kinetic isotope effects, SDKIEs. These inverse SDKIEs are in contrast to the normal SDKIEs that were obtained with allyl lithium and allyl Grignard, suggesting rate-determining single-electron transfer in these cases. By various MO calculations the transition state for addition of allyl boronate occurs with substantial B-O bond formation and little C-C bond formation. The magnitudes of the SDKIEs with the other two allylating reagents, when compared with reasonable equilibrium isotope effects for the addition, suggest transition states with substantial C-C bond formation.     

The synthesis of 8‐hydroxyquinazoline derivatives and their acid‐base interactions

The reaction of N‐(2‐R¹‐oxyphenyl)benzimidoyl chlorides with cyanamide derivatives in the presence of titanium tetrachloride as a catalyst has yielded some new 4‐amino‐8‐R¹‐oxy‐2‐phenylquinazolines. pK_a values have been determined for these compounds and the influence of substituents at the basicity of the parent system has been discussed. Some investigations on the methyl‐quinazolinyl ether cleavage in different medium have been done yielding the appropriate hydroxyquinazoline derivatives. In those cases when the deprotection of 4‐amino‐8‐methoxy‐2‐phenylquinazoline was carried in aqueous acidic solutions, the formation of the hydrolysis products 3,4‐dihydro‐2‐phenyl‐4‐quinazolone derivatives was observed as well.