Analytic study of continuum spectrum pulse dynamics in optical waveguides

It is demonstrated that for femtosecond supercontinuum generation in optical waveguides with nonresonant electronic nonlinearity the squared root-mean-square pulse duration varies near-parabolically with distance, irrespectively of spectrum ultra-broadening scenarios. The derived formulas for parabolic coefficients are applied to find the areas of spectral and energetic parameters of few-cycle pulses propagating in hollow capillaries filled with noble gases, where temporal pulse broadening, compression, or quasi-stationary propagation occur. Each regime is illustrated by the typical changes in the pulse electrical field and spectrum. PACS 42.65.Re     

Large Deviations for Random Trees

We consider large random trees under Gibbs distributions and prove a Large Deviation Principle (LDP) for the distribution of degrees of vertices of the tree. The LDP rate function is given explicitly. An immediate consequence is a Law of Large Numbers for the distribution of vertex degrees in a large random tree. Our motivation for this study comes from the analysis of RNA secondary structures.     

Decision Making Times in Mean-Field Dynamic Ising Model

We consider a dynamic mean-field ferromagnetic model in the low-temperature regime in the neighborhood of the zero magnetization state. We study the random time it takes for the system to make a decision, i.e., to exit the neighborhood of the unstable equilibrium and approach one of the two stable equilibrium points. We prove a limit theorem for the distribution of this random time in the thermodynamic limit.     

Inequalities for the inner radii of nonoverlapping domains and open sets

We generalize some classical results in the theory of extreme problems for nonoverlapping domains.     

Inequalities for inner radii of disjoint domains and open sets

In geometric function theory there is a well-known field related to estimates for functionals that depend on the inner radii of nonoverlapping domains. Many authors has contributed to the development of this field. Today, great attention is being paid to the study of extremal problems for nonoverlapping domains with free poles. In this paper, one result of this kind is generalized and strengthened.     

Some extremal problems in the theory of nonoverlapping domains with free poles on rays

We obtain new results on the maximization of the product of powers of the interior radii of pairwise disjoint domains with respect to certain systems of points in the extended complex plane. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 12, pp. 1715–1719, December, 2006.     

Generalized (<Emphasis Type="Italic">n</Emphasis>, <Emphasis Type="Italic">d</Emphasis>)-ray systems of points and inequalities for nonoverlapping domains and open sets

We solve the extremal problem of finding the maximum of the functional