Linearization of a class of n coupled Bernoulli equations

Analytic linearization maps are found for autonomous systems of n coupled Bernoulli equations of a certain class. The class of equations is determined by the requirement that the nonlinear system linearize to its linear part. These equations have applications as models for interacting biological systems.     

A strategy for characterizing the mixing state of immiscible aerosol components and the formation of multiphase aerosol particles through coagulation

We demonstrate that the coagulation of two aerosol droplets of different chemical composition can be studied directly through the unique combination of optical tweezers and Raman spectroscopy. Multiple optical traps can be established, allowing the manipulation of multiple aerosol droplets. Spontaneous Raman scattering allows the characterization of droplet composition and mixing state, permitting the phase segregation of immiscible components in multiphase aerosol to be investigated with spatial resolution. Stimulated Raman scattering allows the integrity of the droplet and uniformity of refractive index to be probed. The combination of these spectroscopic probes with optical tweezers is shown to yield unprecedented detail in studies of the coagulation of decane and water droplets.     

Electronic spectroscopy of the A1A' <-- X1A' system of CDF

To further investigate the Renner-Teller (RT) effect and barriers to linearity and dissociation in the simplest singlet carbene, we recorded fluorescence excitation spectra of bands involving the pure bending levels 2(n)(0) with n = 0-9 and the combination states 1(1)(0)2(n)(0) with n = 1-8 and 2(n)(0)3(1)(0) with n = 0-5 in the A(1)A'<-- X(1)A' system of CDF, in addition to some weak hot bands. The spectra were measured under jet-cooled conditions using a pulsed discharge source, and rotationally analyzed to yield precise values for the band origins and rotational constants; fluorescence lifetimes were also measured to probe for lifetime lengthening effects due to the RT interaction. The derived A state parameters are compared with previous results for CHF and with predictions of ab initio electronic structure theory. The approach to linearity in the A state is evidenced in a sharp increase in the A rotational constant with bending excitation, and a minimum in the vibrational intervals near 2(9). A fit of the vibrational intervals for the pure bending levels yields an A state barrier to linearity in good agreement both with that previously derived for CHF and ab initio predictions. From the spectra and lifetime measurements, the onset of extensive RT perturbations is found to occur at a higher energy than in CHF, consistent with the smaller A constant.     

Packing measure of super separated iterated function systems

Let J be the limit set of an iterated function system in \(\mathbb {R}^d\) satisfying the open set condition. It is well known that the h-dimensional packing measure of J is positive and finite when h is given by Hutchinson’s formula. However, it may be hard to find a formula for the h-dimensional packing measure of J. We introduce the super separation condition and use it to reduce the problem of computing the packing measure to checking densities of a finite number of balls around each point in the limit set. We then use this fact to find formulas for the packing measure of a class of Cantor sets in \(\mathbb {R}\), a class of fractals based on regular convex polygons in \(\mathbb {R}^2\), and a class of fractals based on regular simplexes in \(\mathbb {R}^d\) for \(d \ge 3\).     

Statistical Thermodynamics Unveils How Ions Influence an Aqueous Diels-Alder Reaction

The kinetics of Diels-Alder (DA) reactions in water has been known to be altered by salts for a long time. Yet the question how salts influence the reaction rate, either as rate-enhancing or rate-reducing additives, has so far remained unresolved. Conflicting hypotheses involve (i) indirect salt contributions through the modulation of internal pressure and (ii) making (or breaking) of the so-called “water-structure” by salts that strengthen (or weaken) the hydrophobic effect. In contrast to the qualitative nature of these hypotheses, here we answer this question quantitatively through a combination of transition state theory and fluctuation adsorption-solvation theory (FAST) using the DA reaction between anthracene-9-carbinol and N-ethylmaleimide as an example. We show that rate enhancement is driven by the salting out of the hydrophobic reactant, while rate-enhancing salts exhibit stronger affinity to the transition state.