Maximal Lyapunov exponent and rotation numbers for two coupled oscillators driven by real noise

Asymptotic expansions for the exponential growth rate, known as the Lyapunov exponent, and rotation numbers for two coupled oscillators driven by real noise are constructed. Such systems arise naturally in the investigation of the stability of steady-state motions of nonlinear dynamical systems and in parametrically excited linear mechanical systems. Almost-sure stability or instability of dynamical systems depends on the sign of the maximal Lyapunov exponent. Stability conditions are obtained under various assumptions on the infinitesimal generator associated with real noise provided that the natural frequencies are noncommensurable. The results presented here for the case of the infinitesimal generator having a simple zero eigenvalue agree with recent results obtained by stochastic averaging, where approximate ItÔ equations in amplitudes and phases are obtained in the sense of weak convergence.Dedicated to Thomas K. Caughey on the occasion of his 65th birthday.     

Finding numerical solution of linear partial differential equations of first order with real coefficients

Applying Bittner's operational calculus we present a method to give approximate solutions of linear partial differential equations of first order

The effectivity of solvents as electron pair donors

The effectivity of solvents as electron pair donors, their donicity as expressed by their donor numbers DN, is reexamined. The linear dependence of the enthalpy and the Gibbs free energy for the reaction of donor solvents with antimony pentachloride is affirmed. Extension of the DN scale by other measured quantities, via their linear correlations for sets of solvents where both kinds of data are known is applied to Drago's E-C scale, Kamlet's scale, and Koppel's B scale. This extension, added to previous extensions employing Selbin's D_{I, II} and Popov's ²³Na, produces a DN scale for 170 solvents. The relation of DN measured for isolated solvent molecules in 1,2-dichloroethane to DN values measured for bulk solvents is examined by means of the scaled particle theory and solute-solvent interaction terms. A scale of normalized donor numbers DN ^N is presented. These values are practically the same as the scale, and for those solvents for which DN values are presented here and values are lacking, they can serve in their place for those preferring to use the scale for donor solvents.     

KINETIC ANALYSIS OF CO-CONDENSATION POLYMERIZATION OF AB2 AND AB MONOMERS

This work deals with the kinetics of co-condensation polymerization of AB2 and AB monomers, giving expressions of the two-dimensional molecular weight distribution function and the number/weight average molecular weights of the resulting copolymers. The two-dimensional molecular weight distribution depends on two indices, n and l, which are the respective numbers of AB2 and AB units in a copolymer species. The evolution of the two-dimensional weight and z distributions during the co-condensation polymerization has been evaluated systematically. Finally, the two-dimensional distribution was transformed into a one-dimensional molecular weight distribution with only one variable (the molecular weight of the products instead of the degree of polymerization). The calculated results show that the highly branched copolymer has a very broad molecular weight distribution when the co-condensation polymerization approaches completion.     

Doubly excited3Se,3De and3Ge states of two-electron atomic systems

Summary Time-dependent perturbation theory has been applied to calculate the doubly excited triplet statesNsns:³S^e,Npnp:³D^e andNdnd:³G^e (N=2, 3, 4,n=N+1, ... ,5) for He, Li⁺, Be²⁺ and B³⁺. A time-dependent harmonic perturbation causes simulataneous excitation of both the electrons with a change of spin state. The doubly excited energy levels have been identified as the poles of an appropriately constructed linearized variational functional with respect to the driving frequency. In addition to the transition energies, effective quantum numbers of these doubly excited states have been calculated and analytic representations of their wave functions are obtained. These are utilized to estimate the Coulomb repulsion term for these states which checks the consistency of the wave functions. These wave functions may also be used for calculating other physical properties of the systems.     

Clustering of nucleosides in the presence of alkali metals: Biologically relevant quartets of guanosine,deoxyguanosine and uridine observed by ESI-MS/MS

Electrospray ionization (ESI) mass spectra of nucleosides, recorded in the presence of alkali metals, display alkali metal ion-bound quartets and other clusters that may have implications for understanding non-covalent interactions in DNA and RNA. The tetramers of guanosine and deoxyguanosine and also their metaclusters (clusters of clusters), cationized by alkali metals, were observed as unusually abundant magic number clusters. The observation of these species in the gas phase parallels previous condensed-phase studies, which show that guanine derivatives can form quartets and metaclusters of quartets in solution in the presence of metal cations. This parallel behavior and also internal evidence suggest that bonding in the guanosine tetramers involves the bases rather than the sugar units. The nucleobases thymine and uracil are known to form magic number pentameric adducts with K+, Cs+ and NH4+ in the gas phase. In sharp contrast, we now show that the nucleosides uridine and deoxythymidine do not form the pentameric clusters characteristic of the corresponding bases. More subtle effects of the sugars are evident in the fact that adenosine and cytidine form numerous higher order clusters with alkali metals, whereas deoxyadenosine and deoxycytidine show no clustering. It is suggested that hydrogen bonding between the bases in the tetramers of dG and rG are the dominant interactions in the clusters, hence changing the ribose group to deoxyribose (and vice versa) generally has little effect. However, the additional hydroxyl group of RNA nucleosides enhances the non-selective formation of higher-order aggregates for adenosine and cytidine and results in the lack of highly stable magic number clusters. Some clusters are the result of aggregation in the course of ionization (ESI) whereas others appear to be intrinsic to the solution being examined.     

Simple correction for perturbations of peak amplitudes in statistical models of overlap

Summary The application of statistical models of overlap (SMOs) to saturated separations is made possible by theory that addresses variable peak amplitudes. These amplitudes cause peak widths to vary, and this variation can be modeled by a random variable whose effect on the probability of overlap is expressed by a convolution integral. Modified probabilities of overlap are derived for both homogeneous and nonhomogeneous one-dimensional separations, and the new probabilities are compared to results determined from published computer simulations. The new theory can describe overlap at saturations that are 3 to 4 times larger than before. Previously reported experimental chromatograms are reinterpreted to show the capabilities of theory. The theoretical extension is an important step towards making SMOs into practical tools for screening analytical separations.     

Range-Renewal Processes: SLLNs and Power Laws*

Given n samples(viewed as an n-tuple) of a γ-regular discrete distribution π,in this article the authors concern with the weighted and unweighted graphs induced by the n samples. They first prove a series of SLLN results(of Dvoretzky-Erd¨os’ type). Then they show that the vertex weights of the graphs under investigation obey asymptotically power law distributions with exponent 1 + γ. They also give a conjecture that the degrees of unweighted graphs would exhibit asymptotically power law distribu...     

The peak and descent statistics over ballot permutations

A ballot permutation is a permutation π such that in any prefix of π the descent number is not more than the ascent number. By using a reversal-concatenation map, we (i) give a formula for the joint distribution (pk, des) of the peak and descent statistics over ballot permutations, (ii) connect this distribution and the joint distribution (pk, des) over ordinary permutations in terms of generating functions, and (iii) confirm Spiro's conjecture which finds the equidistribution of the descent statistic for ballot permutations and an analogue of the descent statistic for odd order permutations.     

Yet another way of calculating moments of the Kesten's distribution and its consequences for Catalan numbers and Catalan triangles

We calculate moments of the so-called Kesten distribution by means of the expansion of the denominator of the density of this distribution and then integrate all summands with respect to the semicircle distribution. By comparing this expression with the formulae for the moments of Kesten's distribution obtained by other means, we find identities involving polynomials whose power coefficients are closely related to Catalan numbers, Catalan triangles, binomial coefficients. Finally, as applications of these identities we obtain various interesting relations between the aforementioned numbers, also concerning Lucas, Fibonacci and Fine numbers.