Efficient Algorithms for Large Scale Global Optimization: Lennard-Jones Clusters

A stochastic global optimization method is applied to the challenging problem of finding the minimum energy conformation of a cluster of identical atoms interacting through the Lennard-Jones potential. The method proposed incorporates within an already existing and quite successful method, monotonic basin hopping, a two-phase local search procedure which is capable of significantly enlarging the basin of attraction of the global optimum. The experiments reported confirm the considerable advantages of this approach, in particular for all those cases which are considered in the literature as the most challenging ones, namely 75, 98, 102 atoms. While being capable of discovering all putative global optima in the range considered, the method proposed improves by more than two orders of magnitude the speed and the percentage of success in finding the global optima of clusters of 75, 98, 102 atoms.     

The distribution of dense Sidon subsets of $ \mathbb{Z}_m $

Let S \subseteqq \mathbbZ_m S \subseteqq \mathbb{Z}_m be a Sidon set of cardinality | S | = m^1/₂ + O(1) \mid S \mid = m^{1 \over 2} + O(1) . It is proved, in particular, that for any interval á = {a, a + 1, ?, a + l- 1} {\cal I} = \{a, a + 1, \ldots, a + \ell - 1\} in \mathbbZ_m \mathbb{Z}_m , 0 \leqq l 0 \leqq \ell < m, we have | | S ?á | - | S | l/m | = O( | S | ^1/₂ln m) \big| {\mid S \cap {\cal I} \mid - \mid S \mid \ell/m} \big| = O(\mid S \mid^{1 \over 2}\textrm{ln}\, m) .     

The first Steklov eigenvalue, conformal geometry, and minimal surfaces

We consider the relationship of the geometry of compact Riemannian manifolds with boundary to the first nonzero eigenvalue σ₁ of the Dirichlet-to-Neumann map (Steklov eigenvalue). For surfaces Σ with genus γ and k boundary components we obtain the upper bound σ₁L(∂Σ)?2(γ+k)π. For γ=0 and k=1 this result was obtained by Weinstock in 1954, and is sharp. We attempt to find the best constant in this inequality for annular surfaces (γ=0 and k=2). For rotationally symmetric metrics we show that the best constant is achieved by the induced metric on the portion of the catenoid centered at the origin which meets a sphere orthogonally and hence is a solution of the free boundary problem for the area functional in the ball. For a general class of (not necessarily rotationally symmetric) metrics on the annulus, which we call supercritical, we prove that σ₁(Σ)L(∂Σ) is dominated by that of the critical catenoid with equality if and only if the annulus is conformally equivalent to the critical catenoid by a conformal transformation which is an isometry on the boundary. Motivated by the annulus case, we show that a proper submanifold of the ball is immersed by Steklov eigenfunctions if and only if it is a free boundary solution. We then prove general upper bounds for conformal metrics on manifolds of any dimension which can be properly conformally immersed into the unit ball in terms of certain conformal volume quantities. We show that these bounds are only achieved when the manifold is minimally immersed by first Steklov eigenfunctions. We also use these ideas to show that any free boundary solution in two dimensions has area at least π, and we observe that this implies the sharp isoperimetric inequality for free boundary solutions in the two-dimensional case.     

The properties of liquid nitrogen

The light-scattering ‘anisotropy’ spectrum of liquid nitrogen has been measured for the liquid along the liquid-vapour coexistence line from 69·4 K (almost the triple point) to near the critical point at 125 K and for the gas under pressure at 128 K. The spectrum is very broad (HWHH～50 cm^-1) due to rapid molecular reorientation. The molecular anisotropy spectrum is approximately gaussian, corresponding to a correlation time for molecular reorientation of order 2 × 10^-13 s at 80 K which is comparable with that obtained from nuclear magnetic resonance. A gaussian rather than a lorentzian form arises because molecular reorientation is not a ‘slow’ variable.

Above the critical temperature the molecular anisotropy spectrum can be roughly described as a collision-broadened rotational line spectrum.

The spectrum and its time Fourier Transform are analysed in terms of the dynamical correlation of orientation of the molecules.

A component of the spectrum due to induced polarization is separated from the anisotropy contribution by a study of the far wing of the spectrum and is observed on the Stokes side out to about 250 cm^-1. This depends exponentially on frequency shift and is interpreted in terms of molecular collision dynamics. The correlation time for this motion is about a factor three shorter than that for molecular reorientation.

The spectra are markedly asymmetric after all experimental corrections for asymmetry have been made. The asymmetry is shown to correspond to the detailed balance factor. It is pointed out that this factor should be allowed for in the case of induced scattering, in particular.

A theoretical analysis is given of the effect of correlation of molecular orientation on the light-scattering spectrum for centrosymmetric linear molecules. In particular it is shown that the normalized second moment of the spectrum is unaffected by correlation of orientation.     

A survey on the global optimization problem: General theory and computational approaches

Several different approaches have been suggested for the numerical solution of the global optimization problem: space covering methods, trajectory methods, random sampling, random search and methods based on a stochastic model of the objective function are considered in this paper and their relative computational effectiveness is discussed. A closer analysis is performed of random sampling methods along with cluster analysis of sampled data and of Bayesian nonparametric stopping rules.     

Spectroscopic studies of polymerized surfactants: 1,2-bis(10,12-tricosadiynoyl)-sn-glycero-3-phosphocholine

Spectroscopic studies have been performed on aqueous dispersions of the surfactant 1,2-bis(10,12-tricosadiynoyl)-sn-glycero-3-phosphocholine before and after polymerization with ul-traviolet light. Monomers of this lipid can, under certain conditions, convert from the expected spherical liposomal form to a unique phase consisting of hollow tubules. To determine the molecular conformation of these structures we have used Raman and infrared spectroscopies to probe the structure of the hydrocarbon chains and head groups of the lipids, and used absorption spectroscopy and resonance enhanced Raman scattering of the colored polymer to monitor the length and structure of the diacetylenic polymer backbone. Unusual C? H stretch-ing Raman bands imply that the hydrocarbon chain packing in the monomeric bilayers is different from that observed in other phosphatidylcholines, and that a distrubance in alkyl chain packing occurs on polymerization. Depending on irradiation conditions and the dispersal state of the lipid the polymer chains may be of at least three different colors, from which distinct resonance Raman spectra are obtained. The effective bond conjugation lengths range from quite short in the yellow polymer produced in sonicated vesicles to extremely long in a blue component seen in polymerized tubules.