Set-valued orthogonal additivity

We study the set-valued Cauchy equation postulated for orthogonal vectors. We give its general solution as well as we look for selections of functions satisfying the equation

Scientific, Technical and Economic Committee for Fisheries. Review of scientific advice for 2012 - Part 3 (STECF-11-15)

The STECF review of scientific advice for 2012 Part 3 was drafted by the STECF-EWG 11-17 Working Group held in Ancona, Italy from 17-21 October 2011. The Report was reviewed and adopted by the STECF at its 38th plenary session held in Brussels from 7-11 November 2011

A composite functional equation from algebraic aspect

In this paper we discuss the composite functional equation f(x+2f(y))=f(x)+y+f(y) on an Abelian group. This equation originates from Problem 10854 of the American Mathematical Monthly. We give an algebraic description of the solutions on uniquely 3-divisible Abelian groups, and then we construct all solutions f of this equation on finite Abelian groups without elements of order 3 and on divisible Abelian groups without elements of order 3 including the additive group of real numbers

Orthogonalities and functional equations

In this survey we show how various notions of orthogonality appear in the theory of functional equations. After introducing some orthogonality relations, we give examples of functional equations postulated for orthogonal vectors only. We show their solutions as well as some applications. Then we discuss the problem of stability of some of them considering various aspects of the problem. In the sequel, we mention the orthogonality equation and the problem of preserving orthogonality. Last, but not least, in addition to presenting results, we state some open problems concerning these topics. Taking into account the big amount of results concerning functional equations postulated for orthogonal vectors which have appeared in the literature during the last decades, we restrict ourselves to the most classical equations

Discrete exterior calculus (DEC) for the surface Navier-Stokes equation

We consider a numerical approach for the incompressible surface Navier-Stokes equation. The approach is based on the covariant form and uses discrete exterior calculus (DEC) in space and a semi-implicit discretization in time. The discretization is described in detail and related to finite difference schemes on staggered grids in flat space for which we demonstrate second order convergence. We compare computational results with a vorticity-stream function approach for surfaces with genus 0 and demonstrate the interplay between topology, geometry and flow properties. Our discretization also allows to handle harmonic vector fields, which we demonstrate on a torus.Comment: 21 pages, 9 figure

Using model selection to choose a size-based condition index that is consistent with operational welfare indicators

Quantitative and qualitative measures of fish health and welfare are essential for management of both wild capture and aquaculture species. These measures include morphometric body condition indices, energetic condition, and aquaculture operational welfare indicators (OWI). Measures vary in ease of measurement (and may require destructive sampling), and it is critical to know how well they correlate with fish health and welfare so appropriate management decisions can be based on them. Lumpfish (Cyclopterus lumpus) is a new farming species that needs non-destructive OWIs to be developed and validated. In this study, we developed a C. lumpus fin damage score. Four different body condition indexes based on individual weight relative to either length-weight relationships, or relative to other fish in its local environment were tested (using model selection) as predictors of individual fin damage. Results showed severity of fin damage was predicted by small size relative to the other individuals in the tank or cage. Body condition based on length-weight relationship was not found to predict fin damage, indicating that using established indices from fisheries or from other species would not predict welfare risks from fin damage. Implications are that especially in hatchery conditions grading will improve the condition index, and is expected to mitigate fin damage, but that low weight at length was not of use in predicting fin damage. Model selection to choose between a suite of possible indices proved powerful, and should be considered in other applications where an easily measured index is needed to correlate with other health measures

37^th plenary meeting report of the scientific, technical and economic committee for fisheries (PLEN-11-02)

The Scientific, Technical and Economic Committee for Fisheries hold its 37th plenary on 11-15 July 2011 in Copenhagen (Denmark). The terms of reference included both issues assessments of STECF Expert Working Group reports and additional requests submitted to the STECF by the Commission. Topics dealt with ranged from fisheries economics to management plan evaluation issues