Convergence of Alternating Least Squares Optimisation for Rank-One Approximation to High Order Tensors

The approximation of tensors has important applications in various disciplines, but it remains an extremely challenging task. It is well known that tensors of higher order can fail to have best low-rank approximations, but with an important exception that best rank-one approximations always exists. The most popular approach to low-rank approximation is the alternating least squares (ALS) method. The convergence of the alternating least squares algorithm for the rank-one approximation problem is analysed in this paper. In our analysis we are focusing on the global convergence and the rate of convergence of the ALS algorithm. It is shown that the ALS method can converge sublinearly, Q-linearly, and even Q-superlinearly. Our theoretical results are illustrated on explicit examples.Comment: tensor format, tensor representation, alternating least squares optimisation, orthogonal projection metho

On the Convergence of Alternating Least Squares Optimisation in Tensor Format Representations

The approximation of tensors is important for the efficient numerical treatment of high dimensional problems, but it remains an extremely challenging task. One of the most popular approach to tensor approximation is the alternating least squares method. In our study, the convergence of the alternating least squares algorithm is considered. The analysis is done for arbitrary tensor format representations and based on the multiliearity of the tensor format. In tensor format representation techniques, tensors are approximated by multilinear combinations of objects lower dimensionality. The resulting reduction of dimensionality not only reduces the amount of required storage but also the computational effort.Comment: arXiv admin note: text overlap with arXiv:1503.0543

Analyse des Einflusses von Biolandbau-Verordnungen und Richtlinien auf die Sensorik von Produkten

In the EU funded research project Ecropolis (www.ecropolis.eu), a comparative analysis of relevant sensory related requirements in regulations and standards for mainly processed organic products was made. A potential impact matrix of standards requirements on sensory properties of organic food has been made. The empirical verification of product qualities through consumers and sensory laboratory testing showed significant impacts for oil and salami, but no impact for apple, cookies, and tomato sauce

Re-determination of the Pseudobinary System Li2O-MoO3

The quasi-binary phase diagram lithium oxide - molybdenum(VI) oxide was investigated by differential scanning calorimetry and X-ray diffraction. The four intermediate phases Li4MoO5, Li2MoO4, Li4Mo5O17, and Li2Mo4O13 show incongruent melting. The system has one eutectic point at 50.5 mol% MoO3 and 49.5 mol% LiO0.5 with a eutectic temperature of 524.6 deg C. At this point the melt is in equilibrium with Li2MoO4 and Li4Mo5O17.Comment: excerpt from a published article, with 3 figures and 1 tabl

Atributos químicos do solo e potencial contaminante de fósforo em áreas cultivadas com olerícolas por longo prazo

TCC (graduação) - Universidade Federal de Santa Catarina, Campus Curitibanos, Agronomia.A olericultura é uma atividade caracterizada principalmente pelo cultivo de plantas folhosas, tubérculos e alguns frutos, onde as principais espécies são a couve-folha, cebolinha verde, salsa e alface. São consideradas espécies exigentes em nutrientes prontamente disponíveis e plantas esgotantes do solo. A cama de aviário pode ser utilizada como fonte de nutrientes para olerícolas, porém o uso sem critério pode levar ao excesso de nutrientes no solo e perda para ambientes aquáticos. O fósforo (P) é um elemento essencial para as plantas, mas em excesso pode saturar o solo e ficar disponível para ambientes aquáticos, onde pode se tornar um poluente e causar eutrofização. O objetivo desse trabalho foi de avaliar atributos químicos do solo, fósforo disponível e fósforo em cloreto de Cálcio (CaCl2) em diferentes profundidades de solos cultivados com olerícolas por longo prazo. O estudo foi realizado em uma propriedade de Ilhota – SC, onde foram selecionadas áreas cultivadas durante 6, 20, 30 e 50 anos, que utiliza principalmente a cama de aviário como fonte de nutrientes e uma área de mata nativa como testemunha, das quais foram coletadas amostras de solo nas camadas de 0 – 5, 5 – 10, 10 – 15, 15 – 20, 20 – 30 e 0 – 20 cm, as quais foram avaliados granulometria, pH em água, acidez potencial (H+Al), Carbono orgânico total (COT), cálcio (Ca2+), magnésio (Mg2+) e alumínio (Al3+) trocáveis, P disponível, P solúvel em CaCl2. Também foram avaliados massa fresca (Mf), massa seca (Ms) e P em tecido, de salsa, cebolinha verde e couve-folha cultivadas na propriedade, para estimar a exportação de P pelas culturas. Os resultados mostram que os solos cultivados, adubados com cama de aves por longo prazo, apresentam valores de pH próximo a neutro e menores teores de H + Al e Al3+ em comparação com a mata nativa, bem como os teores de Ca2+ foram superiores nos solos cultivados. Além disso os solos cultivados apresentaram teores elevados de P disponível e P em CaCl2, superando o teor crítico para as culturas e o limite crítico ambiental de P no solo. Isso indica a possibilidade de perda de P para recursos hídricos. Já os teores de P no tecido das plantas cultivadas estão dentro dos teores considerados adequados. Para solos cultivados a mais de 50 anos, poderiam ser realizados 98 colheitas de couve-folha, aproximadamente 4 anos, sem aplicação de fertilizantes fosfatados até os valores atingirem níveis adequados de P no solo, sem expectativa de alteração na produtividade.Olericulture is an activity characterized mainly by the cultivation of leafy plants, tubers and some fruits, where the main species are cabbage, green onions, parsley and lettuce. They are considered demanding species in readily available nutrients and soil-depleting plants. Poultry litter can be used as a source of nutrients for vegetable crops, but improper use can lead to excess nutrients in the soil and loss to aquatic environments. Phosphorus (P) is an essential element for plants, but in excess it can saturate the soil and become available for aquatic environments, where it can become a pollutant and cause eutrophication. The objective of this work was to evaluate soil chemical attributes, available phosphorus and phosphorus in Calcium chloride (CaCl2) at different depths in soils cultivated with vegetable crops for long term. The study was carried out on a property in Ilhota - SC, where areas cultivated for 6, 20, 30 and 50 years were selected, which mainly use poultry litter as a source of nutrients and an area of native forest as a witness, of which Soil samples were collected in layers of 0 – 5, 5 – 10, 10 – 15, 15 – 20, 20 – 30 and 0 – 20 cm, which were evaluated for granulometry, pH in water, potential acidity (H+Al), Total organic carbon (TOC), calcium (Ca2+), magnesium (Mg2+) and aluminum (Al3+) exchangeable, available P, soluble P in CaCl2. Fresh mass (Mf), dry mass (Ms) and P in tissue of parsley, green onions and kale grown on the property were also evaluated to estimate P exportation by crops. The results show that cultivated soils, fertilized with poultry litter for a long time, have pH values close to neutral and lower levels of H + Al and Al3+ compared to native forest, as well as higher levels of Ca2+ in soils cultivated. In addition, the cultivated soils showed high levels of available P and P in CaCl2, exceeding the critical level for crops and the environmental critical limit of P in the soil. This indicates the possibility of loss of P to water resources. The levels of P in the tissue of cultivated plants are within levels considered adequate. For soils cultivated for more than 50 years, 98 cabbage harvests could be carried out, approximately 4 years, without application of phosphate fertilizers until the values reach adequate levels of P in the soil, with no expectation of change in productivity