The Effect of the Spacer of Bis(biurea) Ligands on the Structure of A2L3‐type (A=anion) Phosphate Complexes

By tuning the length and rigidity of the spacer of bis(biurea) ligands L, three structural motifs of the A₂L₃ complexes (A represents anion, here orthophosphate PO₄^3?), namely helicate, mesocate, and mono‐bridged motif, have been assembled by coordination of the ligand to phosphate anion. Crystal structure analysis indicated that in the three complexes, each of the phosphate ions is coordinated by twelve hydrogen bonds from six surrounding urea groups. The anion coordination properties in solution have also been studied. The results further demonstrate the coordination behavior of phosphate ion, which shows strong tendency for coordination saturation and geometrical preference, thus allowing for the assembly of novel anion coordination‐based structures as in transition‐metal complexes.     

Sobolev Stability of Plane Wave Solutions to the Nonlinear Schrödinger Equation

This article explores the questions of long time orbital stability in high order Sobolev norms of plane wave solutions to the NLSE in the defocusing case.     

Spectral estimates for saddle point matrices arising in weak constraint four‐dimensional variational data assimilation

We consider the large sparse symmetric linear systems of equations that arise in the solution of weak constraint four‐dimensional variational data assimilation, a method of high interest for numerical weather prediction. These systems can be written as saddle point systems with a 3 × 3 block structure but block eliminations can be performed to reduce them to saddle point systems with a 2 × 2 block structure, or further to symmetric positive definite systems. In this article, we analyse how sensitive the spectra of these matrices are to the number of observations of the underlying dynamical system. We also obtain bounds on the eigenvalues of the matrices. Numerical experiments are used to confirm the theoretical analysis and bounds.     

Heuristic Solution Methods for the Multilevel Generalized Assignment Problem

The multilevel generalized assignment problem is a problem of assigning agents to tasks where the agents can perform tasks at more than one efficiency level. A profit is associated with each assignment and the objective of the problem is profit maximization. Two heuristic solution methods are presented for the problem. The heuristics are developed from solution methods for the generalized assignment problem. One method uses a regret minimization approach whilst the other method uses a repair approach on a relaxation of the problem. The heuristics are able to solve moderately large instances of the problem rapidly and effectively. Procedures for deriving an upper bound on the solution of the problem are also described. On larger and harder instances of the problem one heuristic is particularly effective.     

On the critical exponents of random k‐SAT

There has been much recent interest in the satisfiability of random Boolean formulas. A random k‐SAT formula is the conjunction of m random clauses, each of which is the disjunction of k literals (a variable or its negation). It is known that when the number of variables n is large, there is a sharp transition from satisfiability to unsatisfiability; in the case of 2‐SAT this happens when m/n → 1, for 3‐SAT the critical ratio is thought to be m/n ≈ 4.2. The sharpness of this transition is characterized by a critical exponent, sometimes called ν = ν_k (the smaller the value of ν the sharper the transition). Experiments have suggested that ν₃ = 1.5 ± 0.1. ν₄ = 1.25 ± 0.05, ν₅ = 1.1 ± 0.05, ν₆ = 1.05 ± 0.05, and heuristics have suggested that ν_k → 1 as k → ∞. We give here a simple proof that each of these exponents is at least 2 (provided the exponent is well defined). This result holds for each of the three standard ensembles of random k‐SAT formulas: m clauses selected uniformly at random without replacement, m clauses selected uniformly at random with replacement, and each clause selected with probability p independent of the other clauses. We also obtain similar results for q‐colorability and the appearance of a q‐core in a random graph. © 2002 Wiley Periodicals, Inc. Random Struct. Alg., 21: 182–195, 2002     

Quasi-wavelet calculations of sound scattering behind barriers

Quasi-wavelets (QWs) are a representation of turbulence consisting of self-similar, eddy-like structures with random orientations and positions in space. They are used in this paper to calculate the scattering, due to turbulent velocity fluctuations, of sound behind noise barriers as a function of the size and spatial location of the eddies. The sound scattering cross-section for QWs of an individual size class (eddy size) is derived and shown to reproduce results for the von Kármán spectrum when the scattered energies from a continuous distribution of QW sizes are combined. A Bragg resonance condition is derived for the eddy size that scatters most strongly for a given acoustic wavenumber and scattering angle. Results for scattering over barriers show that, for typical barrier conditions, most of the scattered energy originates from eddies in the size range of approximately one-half to twice the size of the eddies responsible for maximum scattering. The results also suggest that scattering over the barrier due to eddies with a line of sight to both the source and receiver is generally significant only for frequencies above several kilohertz, for sources and receivers no more than a few meters below the top of the barrier, and for very turbulent atmospheric conditions.     

PSL2(59) is a Subgroup of the Monster

A computer construction of the Monster is used to prove thatPSL(2, 59) is a subgroup of the Monster.