Sudoku’s french ancestors

This column is a place for those bits of contagious mathematics that travel from person to person in the community, because they are so elegant, suprising, or appealing that one has an urge to pass them on. Contributions are most welcome.     

Ant colony optimization: Introduction and recent trends

Ant colony optimization is a technique for optimization that was introduced in the early 1990's. The inspiring source of ant colony optimization is the foraging behavior of real ant colonies. This behavior is exploited in artificial ant colonies for the search of approximate solutions to discrete optimization problems, to continuous optimization problems, and to important problems in telecommunications, such as routing and load balancing. First, we deal with the biological inspiration of ant colony optimization algorithms. We show how this biological inspiration can be transfered into an algorithm for discrete optimization. Then, we outline ant colony optimization in more general terms in the context of discrete optimization, and present some of the nowadays best-performing ant colony optimization variants. After summarizing some important theoretical results, we demonstrate how ant colony optimization can be applied to continuous optimization problems. Finally, we provide examples of an interesting recent research direction: The hybridization with more classical techniques from artificial intelligence and operations research.     

Discrete and Embedded Eigenvalues for One-Dimensional Schrödinger Operators

I present an example of a discrete Schrödinger operator that shows that it is possible to have embedded singular spectrum and, at the same time, discrete eigenvalues that approach the edges of the essential spectrum (much) faster than exponentially. This settles a conjecture of Simon (in the negative). The potential is of von Neumann-Wigner type, with careful navigation around a previously identified borderline situation.     

On Gittins’ index theorem in continuous time

We give a new and comparably short proof of Gittins’ index theorem for dynamic allocation problems of the multi-armed bandit type in continuous time under minimal assumptions. This proof gives a complete characterization of optimal allocation strategies as those policies which follow the current leader among the Gittins indices while ensuring that a Gittins index is at an all-time low whenever the associated project is not worked on exclusively. The main tool is a representation property of Gittins index processes which allows us to show that these processes can be chosen to be pathwise lower semi-continuous from the right and quasi-lower semi-continuous from the left. Both regularity properties turn out to be crucial for our characterization and the construction of optimal allocation policies.     

Beyond nanosizing: an approach to shape analysis of fluorescent nanostructures by SMI-microscopy

Using spatially modulated illumination (SMI) light microscopy it is possible to measure the sizes of fluorescent structures that have an extension far below the conventional optical resolution limit (“subresolution size”). Presently, the sizes are determined as the object extension along the optical axis of the SMI microscope. For this, however, “a priori” assumptions on the fluorochrome distribution (“shape”) within the examined fluorescent structure have to be made. Usually it is assumed that the fluorochrome follows a Gauss-distribution or a spherical distribution. In this report we overcome the necessity to make an assumption on the shape of the fluorochrome distribution. We introduce two new experimentally obtained parameters which allow the determination of a shape measure to describe the spatial distribution of the fluorescent dye. This becomes possible by independent measurements with different excitation wavelengths. As an example, we present shape parameter measurements on individual fluorescent microspheres with a nominal geometrical diameter (“size”) of 190 nm. In the case investigated, the experimental shape correlated well with a homogeneous fluorochrome distribution (“spherical shape”) but not with a variety of other “shapes”.     

On Kemnitz’ conjecture concerning lattice-points in the plane

In 1961, Erdős, Ginzburg and Ziv proved a remarkable theorem stating that each set of 2n−1 integers contains a subset of size n, the sum of whose elements is divisible by n. We will prove a similar result for pairs of integers, i.e. planar lattice-points, usually referred to as Kemnitz’ conjecture. Dedicated to Richard Askey on the occasion of his 70th birthday. 2000 Mathematics Subject Classification Primary—11B50.     

Surface polarity of cellulose derivates observed by coumarin 151 and 153 as solvatochromic and fluorochromic probes

Solvent‐dependent ultraviolet–visible (UV–vis) absorption and Stokes shifts including strong hydrogen‐bond‐donating (HBD) solvents such as 2,2,2‐trifluoroethanol and 1,1,1,3,3,3‐hexafluoro‐2‐propanol of two coumarine dyes (Co 151 and Co 153) were analyzed with multiple‐square analyses of linear solvation energy relationships and the Kamlet–Taft solvent parameter set to α (HBD capacity), β (hydrogen‐bond‐accepting capacity), and π* (dipolarity/polarizability). The UV–vis absorption and emission spectra of Co 151 and Co 153 were measured when adsorbed on various polysaccharides such as different cellulose batches, carboxymethylcelluloses with different degrees of substitution, and chitine. As a result of this evaluation, Co 153 is recommended as an alternative UV–vis probe for evaluating the dipolarity/polarizability of cellulose and cellulose derivates. Multiple adsorption of Co 153 on Linters cellulose took place indicating a wide‐surface polarity distribution, which makes the determination of a rigid polarity parameter questionable. Thus, fluorescence measurements of adsorbed Co 153 are suitable to detect inhomogenities on a surface but not for the determination of empirical polarity parameters. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 41: 1210–1218, 2003     

Latex particles bearing hydrophilic grafted hairs with controlled chain length and functionality synthesized by reversible addition–fragmentation chain transfer

A method is described for synthesizing latex particles with anchored hairs by the grafting of hydrophilic chains, synthesized by reversible addition–fragmentation chain transfer, onto functionalized latex particles. These have the potential to bind biologically active species. © 2003 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 41: 1188–1195, 2003