Roots under convolution of sequences

The convolution a * b of the sequences a = a₀, a₁, a₂, and b is the sequence with elements ∑₀ⁿ a_kb_{n − k}. One sets 1, 1, 1, equal to σ. Given that a * a with a ≥ 0 is close to σ * σ, how close is a to σ? More generally, one asks how close a is to σ if the p-th convolution power, a*P with a ≥ 0, is close to σ*P. Power series and complex analysis form a natural tool to estimate the ‘summed deviation’ ρ = σ * (a — σ) in terms of b = a * a — σ * σ or b = a*P − σ*P. Optimal estimates are found under the condition ∑_k=0ⁿ b_k² = %plane1D;512;(n^{2β + 1}) whenever −½ < β < p − 1. It is not known what the optimal estimates are for the special case b_n = %plane1D;512;(n^β).     

Characterization of algebraic curves by Chebyshev quadrature

Complex potential theory is used to show that Chebyshev-type quadrature works particularly well on algebraic Jordan curves Γ in ℝ ^d , supplied with normalized arc length or a similar probability measure μ. Evaluating the integral ∫_Γ fdμ by the arithmetic mean of the value off on any cycle ofN equally spaced nodes on Γ (relative to μ), the quadrature error will, be bounded byAe ^−bN sup_Γ|f| for allN and all polynomialsf(x) of degree ≤cN. It is plausible that small shifts of the nodes would give quadrature error zero for such polynomials. There are related results for algebraic Jordan arcs and certain algebraic surfaces. The situation is completely different for nonalgebraic curves and surfaces, where corresponding quadrature remainders are at least of order 1/N.     

Discrete π‐Stacks from Self‐Assembled Perylenediimide Analogues

The formation of well‐defined finite‐sized aggregates represents an attractive goal in supramolecular chemistry. In particular, construction of discrete π‐stacked dye assemblies remains a challenge. Reported here is the design and synthesis of a novel type of discrete π‐stacked aggregate from two comparable perylenediimide (PDI) dyads ( PEP and PBP ). The criss‐cross PEP ‐ PBP dimers in solution and ( PBP ‐ PEP )‐( PEP ‐ PBP ) tetramers in the solid state are well elucidated using single‐crystal X‐ray diffraction, dynamic light scattering, and diffusion‐ordered NMR spectroscopy. Extensive π–π stacking between the PDI units of PEP and PBP as well as repulsive interactions of swallow‐tailed alkyl substituents are responsible for the selective formation of discrete dimer and tetramer stacks. Our results reveal a new approach to preparing discrete π stacks that are appealing for making assemblies with well‐defined optoelectronic properties.     

Controlling chemical self-assembly by solvent-dependent dynamics

The influence of the ratio between poor and good solvent on the stability and dynamics of supramolecular polymers is studied via a combination of experiments and simulations. Step-wise addition of good solvent to supramolecular polymers assembled via a cooperative (nucleated) growth mechanism results in complete disassembly at a critical good/poor solvent ratio. In contrast, gradual disassembly profiles upon addition of good solvent are observed for isodesmic (non-nucleated) systems. Due to the weak association of good solvent molecules to monomers, the solvent-dependent aggregate stability can be described by a linear free-energy relationship. With respect to dynamics, the depolymerization of π-conjugated oligo(p-phenylene vinylene) (OPV) assemblies in methylcyclohexane (MCH) upon addition of chloroform as a good solvent is shown to proceed with a minimum rate around a critical chloroform/MCH solvent ratio. This minimum disassembly rate bears an intriguing resemblance to phenomena observed in protein unfolding, where minimum rates are observed at the thermodynamic midpoint of a protein denaturation experiment. A kinetic nucleation-elongation model in which the rate constants explicitly depend on the good solvent fraction is developed to rationalize the kinetic traces and further extend the insights by simulation. It is shown that cooperativity, i.e., the nucleation of new aggregates, plays a key role in the minimum polymerization and depolymerization rate at the critical solvent composition. Importantly, this shows that the mixing protocol by which one-dimensional aggregates are prepared via solution-based processing using good/poor solvent mixtures is of major influence on self-assembly dynamics.     

On the nondegeneracy of constant mean curvature surfaces

We prove that many complete, noncompact, constant mean curvature (CMC) surfaces are nondegenerate; that is, the Jacobi operator Δ_f + | A_f |² has no L² kernel. In fact, if ∑ has genus zero with k ends, and if f (∑) is embedded (or Alexandrov immersed) in a half-space, then we find an explicit upper bound for the dimension of the L² kernel in terms of the number of non-cylindrical ends. Our main tool is a conjugation operation on Jacobi fields which linearizes the conjugate cousin construction. Consequences include partial regularity for CMC moduli space, a larger class of CMC surfaces to use in gluing constructions, and a surprising characterization of CMC surfaces via spinning spheres. R.K. partially supported by NSF grants DMS-0076085 at GANG/UMass and DMS-9810361 at MSRI, and by a FUNCAP grant in Fortaleza, Brazil. J.R. partially supported by an NSF VIGRE grant at Utah. Received: January 2005; Accepted: June 2005     

Logarithmic Convexity for Supremum Norms of Harmonic Functions

We prove the following convexity property for supremum normsof harmonic functions. Let be a domain in Rⁿ, ₀ and E a subdomainand a compact sebset of ,respectively. Then there exists a constant = (E, ₀, ) (0, 1) such that for all harmonic functions u on, the inequality is valid.The case of concentric balls E plays a key role in the proof.For positive harmonic funcitons ono osuch balls, we determinethe sharp constant in the inequlity.