Improved bounds for the expected behaviour of AVL trees

In this paper we improve previous bounds on expected measures of AVL trees by using fringe analysis. A new way of handling larger tree collections that are not closed is presented. An inherent difficulty posed by the transformations necessary to keep the AVL tree balanced makes its analysis difficult when using fringe analysis methods. We derive a technique to cope with this difficulty obtaining the exact solution for fringe parameters even when unknown probabilities are involved. We show that the probability of a rotation in an insertion is between 0.37 and 0.73 (and seems to be less than 0.56), that the fraction of balanced nodes is between 0.56 and 0.78, and that the expected number of comparisons in a search seems to be at most 12% more than in the complete balanced tree.The work of the first author was also supported by the Institute for Computer Research of the University of Waterloo, the second author by a Natural Sciences and Engineering Research Council of Canada Grant No. A-3353, and the third by a Brazilian Coordenação do Aperfeiçoamento de Pessoal de Nível Superior Contract No. 4799/77 and by the University of Waterloo.     

Calcul approché des contributions dipolaires et orbitales au couplage J cc à l'aide de fonctions d'onde LCAO-SCF non empiriques

The contributions of the spin-dipolar and orbital terms to the nuclear spin coupling constants in acetylene, ethylene and ethane have been evaluated by systematically using the Mulliken-Ruedenberg approximation for the matrix elements of the corresponding operators. Both contributions are found to be negligeable as compared to the usual contact term.     

Etudes theoriques de molecules interstellaires soufrees

The potential value of quantum chemical methods in the study of interstellar molecules is illustrated by ab initio calculations on sulphur compounds of astrophysical interest. This work includes: (1) assignments of rotation lines from “non-terrestrial” molecules (i.e., those for which laboratory spectra were lacking, such as the HCS⁺ ion and its metastable isomer HSC⁺), through structure computations; (2) energy balances for the formation and dissociation reactions of protonated species from simpler sulphur-containing molecules, and (3) a determination of the energy curves for the first Σ and Π states of the (H---S)⁺ system, to be used later in a collision treatment of the charge-exchange reaction between the sulphur and hydrogen atoms. ab]Lea possibilités deg méthodes de la chimie quantique dans le domaine des molécules interstellaires sont illustrées par des calculs relatifs á des composés soufrés d'intérêt astrophysique. Les recherches effectuées comprennent: (1) des attributions de raies de rotation provenant de molécules “non-terrestres” (c'est à dire de composés pour lesquels on manquait de données de laboratoire, tels l'ion HCS⁺ et son isomzère métastable (HSC⁺), à l'aide de calculs de structure; (2) des évaluations de bilans énerg étiques pour lea réactions de formation et de destruction des espcés protonés dans l'espace à partir de molécules soufrées plus simples, et (3) une détermination des courbes d'énergle des premiers états Σ et Π du système (H---S)⁺ pour une étude collisionnelle ult erieure de la réaction d'échange de charge entre les atomes de soufre et d'hydrogène.     

NMR kinetic investigations of the photochemical and thermal reactions of a photochromic chromene

The photochromic behavior of 2,2-di(4-fluorophenyl)-6-methoxy-2H-1-chromene has been investigated by 19F NMR spectroscopy. Photocoloration under UV irradiation at low temperature led to the formation of three interconverting photoisomers including two merocyanines and an unprecedented allenyl-phenol isomer. Photobleaching with visible light, which was known to lead to reversion to the initial closed chromene, was shown to increase allenyl-phenol concentration. Thermal relaxation of the preirradiated system was also studied at various temperatures. In each case (UV and visible irradiations, thermal isomerization), the kinetics of each of the four species was monitored. Numerical analysis of concentration vs time profiles enabled us to unequivocally establish the global mechanism occurring in each of the experimental conditions and to interpret the specific reactivity of each photoisomer. It has been shown that, among the 12 possible isomerization processes, only some paths were active. For the first time, it has been possible to determine their corresponding thermal activation parameters and photochemical quantum yield ratios.     

The photochemical behaviour of bis aryl-1,3 triazenes

The photolysis of bis aryl-1,3 triazenes carried out in non-aromatic solvents gives products whose structures are consistent with a cage recombination process of homolytically formed radicals and the subsequent abstraction of hydrogen from the solvent molecules by these arylamino radicals.In aromatic solvents, a free-radical chain process leads to the formation of products resulting from the homolytic substitution on the solvent.     

Microwave-assisted organic synthesis: scale-up of palladium-catalyzed aminations using single-mode and multi-mode microwave equipment

Batch wise scale-up of Buchwald-Hartwig aminations under microwave irradiation has been investigated for the first time. Multi-mode (microSYNTH and MARS) (several vessels irradiated in parallel per batch) as well as single-mode (Discover) (one vessel irradiated per batch) platforms can be successfully used for this purpose with trifluoromethylbenzene (benzotrifluoride: BTF) as amination solvent. The obtained yields indicate a direct scalability in BTF for all the studied aminations. The Voyager equipment (based on a Discover platform) is the most convenient system since it allows an automatic continuous batch wise production without the necessity to manually load and unload reaction vessels.     

Corrélation électronique et effets de base dans l'étude de la liaison hydrogène: le dimère mixte ammoniac-eau

Résumé L'énergie de la liaison hydrogène N...H-O dans le dimère ammoniac — eau a été calculée par la méthode LCAO-MO, à l'aide d'une base minimale d'orbitales de Slater dont les exposants sont déterminés variationnellement dans les monomères isolés. Les orbitales moléculaires occupées et virtuelles ont été déterminées par un calcul SCF classique; la corrélation électronique a été introduite par un calcul de perturbation du second ordre utilisant différentes bases d'orbitales moléculaires équivalentes. L'énergie de liaison du dimère est de 7,66 kcal/mole à l'étape SCF et de 9,65 kcal/mole après correction du second ordre. Ce dernier résultat est donné par un jeu d'orbitales moléculaires équivalentes obtenues par projection des orbitales canoniques des monomères dans l'espace des orbitales du dimère. Les valeurs précédentes sont réduites à 3,96 kcal/mole à l'étape SCF et à 4,63 kcal/mole au second ordre, si l'on tient compte de l'élargissement de base provenant de la proximité des monomères dans le dimère.

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Electron correlation and basis effects in the theory of hydrogen bonds: The mixed dimer ammonia-water

The energy of the hydrogen bond N...H-O in the dimer (NH₃, H₂O) has been computed by the LCAO-MO method, using a minimal set of Slater-type orbitals optimized for the isolated monomers. The doubly occupied and virtual orbitals have been determined by the standard SCF technique, and electron correlation has been introduced by a complete second-order perturbation calculation, using different sets of equivalent MO's. The bonding energy is found to be equal to 7.66 kcal/mol at the SCF step and to 9.65 kcal/mol after second order corrections. The latter value is given by a set of equivalent MO's obtained by projecting the canonical MO's of the monomers into the space of the dimer MO's. The preceding values are reduced to 3.96 kcal/mol at the SCF step and to 4.63 kcal/mol at the second-order, if the basis extension arising from the vicinity of the two monomers inside the dimer is taken into account.

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Zusammenfassung Mit Hilfe der LCAO-MO-Methode wurde die Energie der N...H-O-Wasserstoffbrücke im Dimeren (NH₃, H₂O) berechnet. Es wurde dazu ein minimaler, für die isolierten Monomeren optimierter Basissatz von Slater-Orbitalen benutzt. Die doppelt besetzten und die virtuellen Orbitale sind mit Standard-SCF-Techniken bestimmt worden, die Elektronenkorrelation wurde durch eine Störungsrechnung zweiter Ordnung unter Benutzung von anderen SÄtzen Äquivalenter Orbitale eingeführt. Mit der SCF-Rechnung erhielt man eine Bindungsenergie von 7,66 kcal/mol und nach Berücksichtigung der Störung zweiter Ordnung von 9,65 kcal/mol. Der letztgenannte Wert wird aus einem Satz Äquivalenter Orbitale erhalten, die durch Projektion der kanonischen Orbitale der Monomeren in den Raum der MO's der Dimeren entstehen. Diese beiden Werte werden zu 3,96 kcal/mol im SCF-Schritt und zu 4,63 kcal/mol in der zweiten Ordnung reduziert, wenn die Basiserweiterung, die durch die AnnÄherung der beiden Monomeren im Dimeren zustande kommt, in Betracht gezogen wird.

     

On the asymptotic behavior of the solutions of semilinear nonautonomous equations

We consider nonautonomous semilinear evolution equations of the form $$\frac{dx}{dt}= A(t)x+f(t,x) . $$ Here A(t) is a (possibly unbounded) linear operator acting on a real or complex Banach space $\mathbb{X}$ and $f: \mathbb{R}\times\mathbb {X}\to\mathbb{X}$ is a (possibly nonlinear) continuous function. We assume that the linear equation (1) is well-posed (i.e. there exists a continuous linear evolution family {U(t,s)}_(t,s)∈Δ such that for every s∈?₊ and x∈D(A(s)), the function x(t)=U(t,s)x is the uniquely determined solution of Eq. (1) satisfying x(s)=x). Then we can consider the mild solution of the semilinear equation (2) (defined on some interval [s,s+δ),δ>0) as being the solution of the integral equation $$x(t) = U(t, s)x + \int_s^t U(t, \tau)f\bigl(\tau, x(\tau)\bigr) d\tau,\quad t\geq s . $$ Furthermore, if we assume also that the nonlinear function f(t,x) is jointly continuous with respect to t and x and Lipschitz continuous with respect to x (uniformly in t∈?₊, and f(t,0)=0 for all t∈?₊) we can generate a (nonlinear) evolution family {X(t,s)}_(t,s)∈Δ , in the sense that the map $t\mapsto X(t,s)x:[s,\infty)\to\mathbb{X}$ is the unique solution of Eq. (4), for every $x\in\mathbb{X}$ and s∈?₊. Considering the Green’s operator $(\mathbb{G}{f})(t)=\int_{0}^{t} X(t,s)f(s)ds$ we prove that if the following conditions hold

• the map $\mathbb{G}{f}$ lies in $L^{q}(\mathbb{R}_{+},\mathbb{X})$ for all $f\in L^{p}(\mathbb{R}_{+},\mathbb{X})$ , and
• $\mathbb{G}:L^{p}(\mathbb{R}_{+},\mathbb{X})\to L^{q}(\mathbb {R}_{+},\mathbb{X})$ is Lipschitz continuous, i.e. there exists K>0 such that $$\|\mathbb{G} {f}-\mathbb{G} {g}\|_{q} \leq K\|f-g\|_{p} , \quad\mbox{for all}\ f,g\in L^p(\mathbb{R}_+,\mathbb{X}) , $$

then the above mild solution will have an exponential decay.     

Circular spectrum and bounded solutions of periodic evolution equations

We consider the existence and uniqueness of bounded solutions of periodic evolution equations of the form u^′=A(t)u+?H(t,u)+f(t), where A(t) is, in general, an unbounded operator depending 1-periodically on t, H is 1-periodic in t, ? is small, and f is a bounded and continuous function that is not necessarily uniformly continuous. We propose a new approach to the spectral theory of functions via the concept of “circular spectrum” and then apply it to study the linear equations u^′=A(t)u+f(t) with general conditions on f. For small ? we show that the perturbed equation inherits some properties of the linear unperturbed one. The main results extend recent results in the direction, saying that if the unitary spectrum of the monodromy operator does not intersect the circular spectrum of f, then the evolution equation has a unique mild solution with its circular spectrum contained in the circular spectrum of f.     

Almost automorphic solutions of nonautonomous evolution equations

In this paper, we establish some new theorems about the existence of almost automorphic solutions to nonautonomous evolution equations u^′(t)=A(t)u(t)+f(t) and u^′(t)=A(t)u(t)+f(t,u(t)) in Banach spaces. As we will see, our results allow for a more general A(t) to some extent. An example is also given to illustrate our results. In addition, by means of an example, we show that one cannot ensure the existence of almost automorphic solutions to u^′(t)=A(t)u(t)+f(t) even if the evolution family U(t,s) generated by A(t) is exponentially stable and fAA(X).