Approximations of the rate of growth of switched linear systems

The joint spectral radius of a set of matrices is a measure of the maximal asymptotic growth rate that can be obtained by forming long products of matrices taken from the set. This quantity appears in a number of application contexts, in particular it characterizes the growth rate of switched linear systems. The joint spectral radius is notoriously difficult to compute and to approximate. We introduce in this paper the first polynomial time approximations of guaranteed precision. We provide an approximation (p) over cap that is based on ellipsoid norms that can be computed by convex optimization and that is such that the joint spectral radius belongs to the interval [(p) over cap/ rootn (p) over cap] where n is the dimension of the matrices. We also provide a simple approximation for the special case where the entries of all the matrices are non-negative; in this case the approximation is proved to be within a factor at most m (m is the number of matrices) of the exact value

Approximations of the Rate of Growth of Switched Linear Systems

The joint spectral radius of a set of matrices is a measure of the maximal asymptotic growth rate that can be obtained by forming long products of matrices taken from the set. This quantity appears in a number of application contexts, in particular it characterizes the growth rate of switched linear systems. The joint spectral radius is notoriously di#cult to compute and to approximate. We introduce in this paper the first polynomial time approximations of guaranteed precision. We provide an approximation # that is based on ellipsoid norms, that can be computed by convex optimization, and that is such that the joint spectral radius belongs to the interval [ #/ # n, #], where n is the dimension of the matrices. We also provide a simple approximation for the special case where the entries of all the matrices are non-negative; in this case the approximation is proved to be within a factor at most m (m is the number of matrices) of the exact value

Avaliação sensorial da carne de cordeiros machos e fêmeas Texel × Corriedale terminados em diferentes sistemas Sensory evaluation of meat lambs from males and femeles Texel × Corriedale finished in different systems

O objetivo neste trabalho foi avaliar o efeito do sexo e de três sistemas de terminação nas características sensoriais da carne de cordeiros Texel × Corriedale e na aceitação da carne pelo consumidor. Foram utilizados 90 animais, 45 cordeiros machos não-castrados e 45 fêmeas mantidos em pastagem até o desmame (70 dias) e terminados em três sistemas de produção: pastagem; pastagem ao pé da mãe; e pastagem com suplementação (casca de soja em nível correspondente a 1% do peso vivo dos cordeiros). Após o abate, as carcaças foram armazenadas em câmara fria, com ar forçado, a 1ºC, durante 24 horas, para retirada do músculo longissimus dorsi, que foi congelado a -18ºC para análise sensorial. A caracterização sensorial da carne foi realizada por meio da análise descritiva quantitativa: 22 termos descritivos foram desenvolvidos por uma equipe de julgadores selecionados, que geraram também a definição de cada termo e as amostras-referência. Foi realizado um teste de aceitação utilizando escala hedônica híbrida de nove pontos. A carne dos machos e dos animais terminados em pastagem ao pé da mãe caracterizou-se pelo odor e sabor residual mais suaves de carne ovina e gordura, menor maciez e maior mastigabilidade em comparação à das fêmeas e dos animais terminados nos demais sistemas. As carnes dos cordeiros terminados nos sistemas de pastagem e de pastagem com suplementação são semelhantes quanto aos aspectos sensoriais. A carne é igualmente aceita pelos consumidores, independentemente do sexo e do sistema de terminação, apresentando boa aceitação.<br>The objective of this work was to evaluate the effect of sex and of three finishing systems on sensory traits of Texel × Corriedale lamb meat an on the consumer acceptance of the meat. It was used 90 animals, 45 non-castrated male lambs and 45 females kept on pasture until weaning (70 days of age) and finished in three production systems: pasture, pasture with mother, and pasture with supplementation (soybean hull corresponding to 1% of life weight of the lambs). After slaughter, carcasses were stored in cold chamber, with forced air at 1ºC until reaching 24 hours for removal of longissimus dorsi muscle which had been frozen at -18ºC for sensory analysis. Meat sensorial characterization was performed by quantitative descriptive analyses: 22 descriptive terms were developed by a team of selected judges who also created the definition for each term and the reference-samples. Acceptability test was performed by using a nine-point hybrid hedonic scale. Meat of male animals and of animals finished in pasture with the mother was characterized by the flavor and softer residual taste sheep meat and fat, less softness and higher chewiness in comparison to the meat of female and of animals finished in the other systems. Meat of lambs finished in pasture systems and pasture with supplementation were similar regarding to sensory aspects. Meat was equally accepted by the consumers, regardless of sex and finishing system, presenting good acceptability

Stability of linear problems: joint spectral radius of sets of matrices

It is wellknown that the stability analysis of step-by-step numerical methods for differential equations often reduces to the analysis of linear difference equations with variable coefficients. This class of difference equations leads to a family F of matrices depending on some parameters and the behaviour of the solutions depends on the convergence properties of the products of the matrices of F. To date, the techniques mainly used in the literature are confined to the search for a suitable norm and for conditions on the parameters such that the matrices of F are contractive in that norm. In general, the resulting conditions are more restrictive than necessary. An alternative and more effective approach is based on the concept of joint spectral radius of the family F, r(F). It is known that all the products of matrices of F asymptotically vanish if and only if r (F) < 1. The aim of this chapter is that to discuss the main theoretical and computational aspects involved in the analysis of the joint spectral radius and in applying this tool to the stability analysis of the discretizations of differential equations as well as to other stability problems. In particular, in the last section, we present some recent heuristic techniques for the search of optimal products in finite families, which constitute a fundamental step in the algorithms which we discuss. The material we present in the final section is part of an original research which is in progress and is still unpublished