Scaling behavior of laser population dynamics with time-delayed coupling: theory and experiment

We study the influence of asymmetric coupling strengths on the onset of light intensity oscillations in an experimental system consisting of two semiconductor lasers cross coupled optoelectronically with a time delay. We discover a scaling law that relates the amplitudes of oscillations and the coupling strengths. These observations are in agreement with a theoretical model. These results could be applicable to the population dynamics of other systems, such as the spread of disease in human populations coupled by migration.     

Classical scattering with long range forces

We discuss the classical two-body scattering problem for potentials which decrease at infinity liker ^?α, 1≧α>0. We prove existence and uniqueness theorems for scattering orbits parametrized by their asymptotic data. Wave operators are constructed and their properties discussed. We also discuss and prove cluster properties of theS-operator.     

TRANES Spectra of Fluorescence Probes in Lipid Bilayer Membranes: An Assessment of Population Heterogeneity and Dynamics

Time-resolved fluorescence of eight fluorescence probes were studied in EggPC bilayer membrane vesicles. Emission wavelength dependent fluorescence decays were analyzed in a model-independent way to obtain time resolved area normalized emission spectra (TRANES). The TRANES spectra of the probes studied were classified into four types: (1) spectra that are identical at all time (one emissive species), (2) spectra that show an isoemissive point (two emissive species), (3) spectra that shift continuously with time (slow solvation dynamics or multiple species), and (4) spectra that shift for a short time and thereafter one or two emissive species are indicated. The TRANES spectra of these eight probes, except RH421, belong to the type 1, 2, or 4. The continuous shift of the TRANES spectra that was observed for the probe RH421 is attributed to multiple ground state species and not due to slow solvation dynamics.     

Stirling polynomials

An investigation is made of the polynomials f_k(n) = S(n + k, n) and g_k(n) = (?1)^ks(n, n ? k), where S and s denote the Stirling numbers of the second and first kind, respectively. The main result gives a combinatorial interpretation of the coefficients of the polynomial (1 ? x)^2k+1Σ_n=0^∞f_k(n)xⁿ analogous to the well-known combinatorial interpretation of the Eulerian numbers in terms of descents of permutations.     

Tournaments and Vandermond's determinant

We prove that det |x_i^i–1|_{n × n} = Π_1≤i<i≤n (X_j – X_i) by associating a tournament to each term in the expansion of the product. All terms cancel except those corresponding to transitive tournaments, and their sum of the determinant.     

Set-based corral control in stochastic dynamical systems: making almost invariant sets more invariant

We consider the problem of stochastic prediction and control in a time-dependent stochastic environment, such as the ocean, where escape from an almost invariant region occurs due to random fluctuations. We determine high-probability control-actuation sets by computing regions of uncertainty, almost invariant sets, and Lagrangian coherent structures. The combination of geometric and probabilistic methods allows us to design regions of control, which provide an increase in loitering time while minimizing the amount of control actuation. We show how the loitering time in almost invariant sets scales exponentially with respect to the control actuation, causing an exponential increase in loitering times with only small changes in actuation force. The result is that the control actuation makes almost invariant sets more invariant.     

Free channel fourier transform in the long-rangen-body problem

Research supported by NSF grant DMS-8807816.     

Hypergraphs and a functional equation of Bouwkamp and de Bruijn

Let . Bouwkamp and de Bruijn found that there exists a power series Ψ(u,v) satisfying the equation . We show that this result can be interpreted combinatorially using hypergraphs. We also explain some facts about Φ(u,0) and Ψ(u,0), shown by Bouwkamp and de Bruijn, by using hypertrees, and we use Lagrange inversion to count hypertrees by number of vertices and number of edges of a specified size.     

On solutions of nonlinear Schrödinger equations with Cantor-type spectrum

The solvability of the Cauchy problem for the Nonlinear Nonfocusing Schrödinger equation (NNSE) with almost periodic initial data satisfying certain conditions is studied. It is shown that solutions are uniform almost periodic functions with respect to each variable. An example of initial data with Cantor-type spectrum is given. The Cauchy problem for NNSE is solved in the class of limit periodic functions which are well approximated by periodic ones.