Incompatible sets of gradients and metastability

We give a mathematical analysis of a concept of metastability induced by incompatibility. The physical setting is a single parent phase, just about to undergo transformation to a product phase of lower energy density. Under certain conditions of incompatibility of the energy wells of this energy density, we show that the parent phase is metastable in a strong sense, namely it is a local minimizer of the free energy in an $L^1$ neighbourhood of its deformation. The reason behind this result is that, due to the incompatibility of the energy wells, a small nucleus of the product phase is necessarily accompanied by a stressed transition layer whose energetic cost exceeds the energy lowering capacity of the nucleus. We define and characterize incompatible sets of matrices, in terms of which the transition layer estimate at the heart of the proof of metastability is expressed. Finally we discuss connections with experiment and place this concept of metastability in the wider context of recent theoretical and experimental research on metastability and hysteresis.Comment: Archive for Rational Mechanics and Analysis, to appea

Spectral variables, growth analysis and yield of sugarcane Variáveis espectrais e indicadores de desenvolvimento e produtividade da cana-de-açúcar

Spectral information is well related with agronomic variables and can be used in crop monitoring and yield forecasting. This paper describes a multitemporal research with the sugarcane variety SP80-1842, studying its spectral behavior using field spectroscopy and its relationship with agronomic parameters such as leaf area index (LAI), number of stalks per meter (NPM), yield (TSS) and total biomass (BMT). A commercial sugarcane field in Araras/SP/Brazil was monitored for two seasons. Radiometric data and agronomic characterization were gathered in 9 field campaigns. Spectral vegetation indices had similar patterns in both seasons and adjusted to agronomic parameters. Band 4 (B4), Simple Ratio (SR), Normalized Difference Vegetation Index (NDVI), and Soil Adjusted Vegetation Index (SAVI) increased their values until the end of the vegetative stage, around 240 days after harvest (DAC). After that stage, B4 reflectance and NDVI values began to stabilize and decrease because the crop reached ripening and senescence stages. Band 3 (B3) and RVI presented decreased values since the beginning of the cycle, followed by a stabilization stage. Later these values had a slight increase caused by the lower amount of green vegetation. Spectral variables B3, RVI, NDVI, and SAVI were highly correlated (above 0.79) with LAI, TSS, and BMT, and about 0.50 with NPM. The best regression models were verified for RVI, LAI, and NPM, which explained 0.97 of TSS variation and 0.99 of BMT variation.<br>A informação espectral tem boa relação com variáveis agronômicas e pode contribuir com informações para o monitoramento, acompanhamento e previsão de safras. O presente trabalho descreve a análise multitemporal do comportamento espectral da variedade de cana-de-açúcar SP80-1842 e a relação com variáveis agronômicas como índice de área foliar (IAF), número de perfilhos por metro (NPM), produtividade (TCH) e biomassa total (BMT). Nas safras 2000/2001 e 2001/2002, um talhão comercial, localizada no município de Araras/SP foi monitorado em nove campanhas de coleta de dados radiométricos e agronômicos. O comportamento temporal das variáveis espectrais acompanhou o comportamento das variáveis agronômicas. A banda 4 (B4), o índice de vegetação da razão simples (SR), o índice de vegetação por diferença normalizada (NDVI) e o índice de vegetação ajustado ao solo (SAVI) aumentaram seus valores até o fim da fase de crescimento vegetativo, aproximadamente até os 240 dias após o corte, a partir do qual os valores se estabilizaram e diminuíram em função da entrada da cultura na fase de maturação. A banda 3 (B3) e o índice de vegetação da razão (RVI) tiveram queda em seus valores desde o início do ciclo, com posterior estabilização e aumento em seus valores devido ao aumento da quantidade de palha e da queda da biomassa foliar. As variáveis espectrais B3, RVI, NDVI e SAVI tiveram correlações maiores que 0,79 com as variáveis IAF e BMT e de aproximadamente 0,50 com o NPM. Os melhores modelos de regressão linear múltipla foram os com RVI, IAF e NPM e explicaram 0,97 da variação da TCH e 0,99 da BMT