Pseudo-Exponential-Type Solutions of Wave Equations Depending on Several Variables

Using matrix identities, we construct explicit pseudo-exponential-type solutions of linear Dirac, Loewner and Schr\"odinger equations depending on two variables and of nonlinear wave equations depending on three variables

Variable Metric Random Pursuit

We consider unconstrained randomized optimization of smooth convex objective functions in the gradient-free setting. We analyze Random Pursuit (RP) algorithms with fixed (F-RP) and variable metric (V-RP). The algorithms only use zeroth-order information about the objective function and compute an approximate solution by repeated optimization over randomly chosen one-dimensional subspaces. The distribution of search directions is dictated by the chosen metric. Variable Metric RP uses novel variants of a randomized zeroth-order Hessian approximation scheme recently introduced by Leventhal and Lewis (D. Leventhal and A. S. Lewis., Optimization 60(3), 329--245, 2011). We here present (i) a refined analysis of the expected single step progress of RP algorithms and their global convergence on (strictly) convex functions and (ii) novel convergence bounds for V-RP on strongly convex functions. We also quantify how well the employed metric needs to match the local geometry of the function in order for the RP algorithms to converge with the best possible rate. Our theoretical results are accompanied by numerical experiments, comparing V-RP with the derivative-free schemes CMA-ES, Implicit Filtering, Nelder-Mead, NEWUOA, Pattern-Search and Nesterov's gradient-free algorithms.Comment: 42 pages, 6 figures, 15 tables, submitted to journal, Version 3: majorly revised second part, i.e. Section 5 and Appendi

Two New Bounds on the Random-Edge Simplex Algorithm

We prove that the Random-Edge simplex algorithm requires an expected number of at most 13n/sqrt(d) pivot steps on any simple d-polytope with n vertices. This is the first nontrivial upper bound for general polytopes. We also describe a refined analysis that potentially yields much better bounds for specific classes of polytopes. As one application, we show that for combinatorial d-cubes, the trivial upper bound of 2^d on the performance of Random-Edge can asymptotically be improved by any desired polynomial factor in d.Comment: 10 page

Larval condition and growth of Sardinella brasiliensis (Steindachner, 1879): preliminary results from laboratory studies

Brazilian sardine, the most important resource along the southeastern Brazilian coast, presented great variations and declines in its stocks. The main factors contributing to this are: oceanographic structure changes; recruitment failures; excessive catches of juveniles and increase in fishery effort. In spite of this, no alterations in the density-dependent parameters were detected. Consequently, methods analysing the condition of the larvae coupled with methods determining growth using sagittae otolith increment width were applied to evaluate growth under experimental conditions. The results of the readings on the sagittae were compared with the age of the laboratory-reared sardine larvae and confirmed that increments are formed on a daily basis. Under poor feeding conditions, sardine larvae showed a low growth expressed by dry weight, RNA/DNA ratio and tryptic enzyme activity and by the narrow and low contrast increments in the otoliths. The results of the biochemical indices showed an unexpected decline in the feeding group coupled with a decrease in width of increment numbers 8 and 10. Other factors than food availability were affecting the condition of the larvae and might be indicative of physiological processes and ontogenetic changes occurring in sardine larvae