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.     

Self-Interactions of Strands and Sheets

Physical strands or sheets that can be modelled as curves or surfaces embedded in three dimensions are ubiquitous in nature, and are of fundamental importance in mathematics, physics, biology, and engineering. Often the physical interpretation dictates that self-avoidance should be enforced in the continuum model, i.e., finite energy configurations should not self-intersect. Current continuum models with self-avoidance frequently employ pairwise repulsive potentials, which are of necessity singular. Moreover the potentials do not have an intrinsic length scale appropriate for modelling the finite thickness of the physical systems. Here we develop a framework for modelling self-avoiding strands and sheets which avoids singularities, and which provides a way to introduce a thickness length scale. In our approach pairwise interaction potentials are replaced by many-body potentials involving three or more points, and the radii of certain associated circles or spheres. Self-interaction energies based on these many-body potentials can be used to describe the statistical mechanics of self-interacting strands and sheets of finite thickness.     

Some rules forS(-2k) dipole sums?

Stieltjes conditions and the ratio test provide necessary but not sufficient conditions onS(-2k) dipole sums. If the dipole sums are accurate the associated [n, n –1] Padé approximant provides a better representation of (), the frequency-dependent dipole polarizability, than a truncated series expression and, in addition, should bound () below. It is shown how constraints on the dipole sums effect the form of the [2,1] Padé approximant and an additional constraint is derived that ensures the analyticity of the approximant on 0 < ₁. There then follows a discussion of the reliability of available literature dipole sum values for small molecules containing H, C, N and O.     

Design, synthesis, and biological evaluation of simplified analogues of (+)-discodermolide. Additional insights on the importance of the diene, the C7 hydroxyl, and the lactone

[structure: see text] The design, synthesis, and biological evaluation of seven totally synthetic analogues of the antitumor agent (+)-discodermolide are reported. Saturation of the terminal diene system, alteration of the substituents on the lactone, and alkylation of the C7-hydroxyl group reveal significant structure-activity relationships.     

Design, synthesis, and binding affinities of pyrrolinone-based somatostatin mimetics

[structure: see text] Tetrapyrrolinone somatostatin (SRIF) mimetics (cf. 1), based on a heterochiral (D,L-mixed) pyrrolinone scaffold, were designed, synthesized, and evaluated for biological activity. The iterative synthetic sequence, incorporating the requisite functionalized coded and noncoded amino acid side chains, comprised a longest linear synthetic sequence of 23 steps. Binding affinities at two somatostatin receptor subtypes (hsst 4 and 5) reveal micromolar activity, demonstrating that the d,l-mixed pyrrolinone scaffold can be employed to generate functional mimetics of peptide beta-turns.