Geometrically Convex Solutions of Certain Difference Equations and Generalized Bohr-Mollerup Type Theorems

Let G: (0, ∞) → (0, ∞) be logarithmically concave on a neighbourhood of ∞ and suppose lim_x→∞ G(x + δ)/G(x) = 1 for some δ > 0. Then, the functional equation $$g(x+1)=G(x)\cdot g(x),\ \ \ x\in (0,\infty),$$ admits, up to a multiplicative constant, at most one solution g: (0, ∞) → (0, ∞), geometrically convex on a neighbourhood of ∞. Sufficient conditions on G are given, for which also such a unique geometrically convex solution of (D) exists. This result improves the classical theorems of Bohr-Mollerup type and gives a new characterization of the gamma function and the q-gamma function for q ∈ (0, 1).     

An Asymptotic Formula for the Iterates of a Function

Let IK be either IR or ? and D an open set of IK containing 0 and starlike with respect to 0 (i.e. an open interval containig 0 in the case IK = IR). If f: D » IK is a continuous function with fixed point 0, then under certain conditions stated below we can prove for the kn- th iterates of f the following asymptotic formula: 1 $$f^{(kn)}\bigg({x \over n}\bigg )=\sum_{i-1}^r{1\over (nk)^i}\ f_i(kx)+o \bigg({1\over n^r}\bigg),$$ for n » ∞, k, n and r beeing positive integers and x close enough to 0. The functions f _i are continuous and uniquely determined by f. In particular (1) holds for any function holomorphic on a neighbourhood of zero, having a convergent power series expansion of the form $$f(z)=z+a_2z^2+\cdots=\sum_{j=1}^\infty\ a_jz^j,\ a_j\in {\cal C},a_1=1,$$ and for any integers k, r with r > 0.     

Theory-enforced Re-investigation of the Origin of the Large Metal–Carbon Bond Strength in PdCH2I+ and its reactions with unsaturated hydrocarbons

State-of-the-art ab initio studies demonstrate that the reaction Pd⁺ + CH₃I → PdCH₂I⁺ + H^. is endothermic by ca. 20 kcal/mol, which translates into a bond dissociation energy (BDE) of ca. 83 kcal/mol for the Pd⁺? CH₂I bond. This figure is in agreement with an experimental bracket of 68 kcal/mol < BDE(Pd⁺? CH₂I) < 92 kcal/mol. Based on these findings, the previously studied Pd⁺/CH₃I system was re-investigated, and double-resonance experiments demonstrate that the formation of PdCH₂I⁺ occurs stepwise via PdCH as a reactive intermediate. Further, ion/molecule reactions of PdCH₂I⁺ with unsaturated hydrocarbons are studied, which reveal the formation of carbon–carbon bonds in the gas phase.     

A global equilibrium as the foundation of quantum randomness

We analyze the origin of quantum randomness within the framework of a completely deterministic theory of particle motion—Bohmian mechanics. We show that a universe governed by this mechanics evolves in such a way as to give rise to the appearance of randomness, with empirical distributions in agreement with the predictions of the quantum formalism. Crucial ingredients in our analysis are the concept of the effective wave function of a subsystem and that of a random system. The latter is a notion of interest in its own right and is relevant to any discussion of the role of probability in a deterministic universe.Research supported in part by NSF Grant DMS-9105661.     

Drag reduction in bubbly Taylor-Couette turbulence

In Taylor-Couette flow the total energy dissipation rate and therefore the drag can be determined by measuring the torque on the system. We do so for Reynolds numbers between Re=7 x 10(4) and Re=10(6) after having injected (i) small bubbles (R=1 mm) up to a volume concentration of alpha=5% and (ii) buoyant particles (rhop/rhol=0.14) of comparable volume concentration. In case (i) we observe a crossover from little drag reduction at smaller Re to strong drag reduction up to 20% at Re=10(6). In case (ii) we observe at most little drag reduction throughout. Several theoretical models for bubbly drag reduction are discussed in view of our findings.     

Imaging delocalized electron clouds: photoionization of C(60) in Fourier reciprocal space

The dynamics of the photoionization of the two outermost orbitals of C(60) has been studied in the oscillatory regime from threshold to the carbon K edge. We show that geometrical properties of the fullerene electronic hull, such as its diameter and thickness, are contained in the partial photoionization cross sections by examining ratios of partial cross sections as a function of the photon wave number in the Fourier conjugated space. Evaluated in this unconventional manner photoemission data reveal directly the desired spatial information.     

The Smoluchowski limit for a simple mechanical model

We consider a vertical stick constantly accelerated along thex-axis by a forceF and which elastically collides with point particles of the same mass (atoms). The atoms are initially Poisson distributed and are allowed to have four velocities only. It is shown that under suitable scaling of the system the displacementQ(t) of the stick satisfies a nontrivial CLT:Q(t)=vFt+D ^1/2 W(t) (Smoluchowski equation), where the values ofv andD depend on the fact that one atom may collide several times.     

"Compression-only" behavior: a second-order nonlinear response of ultrasound contrast agent microbubbles

Oscillating phospholipid-coated ultrasound contrast agent microbubbles display a so-called "compression-only" behavior, where it is observed that the bubbles compress efficiently while their expansion is suppressed. Here, a theoretical understanding of the source of this nonlinear behavior is provided through a weakly nonlinear analysis of the shell buckling model proposed by Marmottant et al. [J. Acoust. Soc. Am. 118, 3499-3505 (2005)]. It is shown that the radial dynamics of the bubble can be considered as a superposition of a linear response at the fundamental driving frequency and a second-order nonlinear low-frequency response that describes the negative offset of the mean bubble radius. The analytical solution deduced from the weakly nonlinear analysis shows that the compression-only behavior results from a rapid change of the shell elasticity with bubble radius. In addition, the radial dynamics of single phospholipid-coated microbubbles was recorded as a function of both the amplitude and the frequency of the driving pressure pulse. The comparison between the experimental data and the theory shows that the magnitude of compression-only behavior is mainly determined by the initial phospholipids concentration on the bubble surface, which slightly varies from bubble to bubble.     

Knudsen gas provides nanobubble stability

We provide a model for the remarkable stability of surface nanobubbles to bulk dissolution. The key to the solution is that the gas in a nanobubble is of Knudsen type. This leads to the generation of a bulk liquid flow which effectively forces the diffusive gas to remain local. Our model predicts the presence of a vertical water jet immediately above a nanobubble, with an estimated speed of ～3.3 m/s, in good agreement with our experimental atomic force microscopy measurement of ～2.7 m/s. In addition, our model also predicts an upper bound for the size of nanobubbles, which is consistent with the available experimental data.