Is it necessary to measure hearing protector attenuation at 3.15 and 6.3 kHz?

The real-ear attenuation data for 81 different hearing protectors were analyzed with respect to the errors that would arise if, instead of averaging the 1/3-octave-band results at 3.15 and 4 kHz and 6.3 and 8 kHz, respectively, the octave-band attenuation at 4 and 8 kHz was estimated from only the 1/3-octave-band data at those two frequencies. Errors as large as 3-4 dB were found to occur in rare instances, but more typically were in the range of 0.5-1.5 dB. However, in terms of computation of an overall noise reduction rating such as the NRR, the effect of excluding the 3.15- and 6.3-kHz data led to errors that averaged only 0.1 dB and never exceeded 0.3 dB, except in one instance, where the error was 0.6 dB. It was concluded that there is little value in measuring real-ear attenuation in a diffuse sound field at the frequencies of 3.15 and 6.3 kHz for applications in which hearing protector attenuation data are normally utilized.     

Evaluation of mutual information and genetic programming for feature selection in QSAR

Feature selection is a key step in Quantitative Structure Activity Relationship (QSAR) analysis. Chance correlations and multicollinearity are two major problems often encountered when attempting to find generalized QSAR models for use in drug design. Optimal QSAR models require an objective variable relevance analysis step for producing robust classifiers with low complexity and good predictive accuracy. Genetic algorithms coupled with information theoretic approaches such as mutual information have been used to find near-optimal solutions to such multicriteria optimization problems. In this paper, we describe a novel approach for analyzing QSAR data based on these methods. Our experiments with the Thrombin dataset, previously studied as part of the KDD (Knowledge Discovery and Data Mining) Cup 2001 demonstrate the feasibility of this approach. It has been found that it is important to take into account the data distribution, the rule "interestingness", and the need to look at more invariant and monotonic measures of feature selection.     

Stable helical solitons in optical media

We present a review of new results which suggest the existence of fully stable spinning solitons (self-supporting localised objects with an internal vorticity) in optical fibres with self-focusing Kerr (cubic) nonlinearity, and in bulk media featuring a combination of the cubic self-defocusing and quadratic nonlinearities. Their distinctive difference from other optical solitons with an internal vorticity, which were recently studied in various optical media, theoretically and also experimentally, is that all the spinning solitons considered thus far have been found to be unstable against azimuthal perturbations. In the first part of the paper, we consider solitons in a nonlinear optical fibre in a region of parameters where the fibre carries exactly two distinct modes, viz., the fundamental one and the first-order helical mode. From the viewpoint of application to communication systems, this opens the way to doubling the number of channels carried by a fibre. Besides that, these solitons are objects of fundamental interest. To fully examine their stability, it is crucially important to consider collisions between them, and their collisions with fundamental solitons, in (ordinary or hollow) optical fibres. We introduce a system of coupled nonlinear Schrödinger equations for the fundamental and helical modes with nonstandard values of the cross-phase-modulation coupling constants, and show, in analytical and numerical forms, results of collisions between solitons carried by the two modes. In the second part of the paper, we demonstrate that the interaction of the fundamental beam with its second harmonic in bulk media, in the presence of self-defocusing Kerr nonlinearity, gives rise to the first ever example of completely stable spatial ring-shaped solitons with intrinsic vorticity. The stability is demonstrated both by direct simulations and by analysis of linearized equations.     

Leone Burton: an appreciation

Some mathematical statements can be validated by a supportive example or refuted by a counterexample. Our study investigated secondary school teachers' knowledge of such proofs. Fifty practising secondary school teachers were first asked to validate/refute six elementary number theory statements, then to suggest justifications that students might give for the same statements, and finally to judge eighteen numerical justifications for the same statements. The findings indicated that teachers are well acquainted with numerical examples and counterexamples as proofs. We also found that teachers' considerations for accepting given justifications involve mathematical aspects as well as didactical ones. Teachers are less familiar with students' tendencies to bring more than one example or counterexample in such proofs.     

Toward a language theoretic proof of the four color theorem

This paper considers the problem of showing that every pair of binary trees with the same number of leaves parses a common word under a certain simple grammar. We enumerate the common parse words for several infinite families of tree pairs and discuss several ways to reduce the problem of finding a parse word for a pair of trees to that for a smaller pair. The statement that every pair of trees has a common parse word is equivalent to the statement that every planar graph is four-colorable, so the results are a step toward a language theoretic proof of the four color theorem.