Linear dependence of bivariate Minimal Support and Locally Refined B-splines over LR-meshes

The focus on locally refined spline spaces has grown rapidly in recent years due to the need in Isogeoemtric analysis (IgA) of spline spaces with local adaptivity: a property not offered by the strict regular structure of tensor product B-spline spaces. However, this flexibility sometimes results in collections of B-splines spanning the space that are not linearly independent. In this paper we address the minimal number of B-splines that can form a linear dependence relation for Minimal Support B-splines (MS B-splines) and for Locally Refinable B-splines (LR B-splines) on LR-meshes. We show that the minimal number is six for MS B-splines, and eight for LR B-splines. The risk of linear dependency is consequently significantly higher for MS B-splines than for LR B-splines. Further results are established to help detecting collections of B-splines that are linearly independent

1993 Earth Observing System reference handbook

Mission to Planet Earth (MTPE) is a NASA-sponsored concept that uses space- and ground-based measurement systems to provide the scientific basis for understanding global change. The space-based components of MTPE will provide a constellation of satellites to monitor the Earth from space. Sustained observations will allow researchers to monitor climate variables overtime to determine trends; however, space-based monitoring alone is not sufficient. A comprehensive data and information system, a community of scientists performing research with the data acquired, and extensive ground campaigns are all important components. Brief descriptions of the various elements that comprise the overall mission are provided. The Earth Observing System (EOS) - a series of polar-orbiting and low-inclination satellites for long-term global observations of the land surface, biosphere, solid Earth, atmosphere, and oceans - is the centerpiece of MTPE. The elements comprising the EOS mission are described in detail

The Extraction of Titanium from Titaniferous Magnetite

The titanium in the titaniferous magnetite under study, as well as most other titaniferous iron ores, is so intimately associated with the iron that liberation cannot be achieved by mechanical means. It is obvious from the preliminary investigations that some chemical means must be utilized from efficient extraction of the titanium. Two possible pyrometallurgical methods for this extraction are either soda sintering or direct smelting

The Effects of Mixing Variables on Settling Rates and Particle Size Distribution of Dicalcium Phosphate Made by the Hydrolysis of Monocalcium Phosphate

Summary: A process is under investigation for the manufacture of dicalcium phosphate by the hydrolosis of concentrated superphosphate containing recycled monocalcium phosphate. The hydrolysis also results in the formation of an aqueous solution of monocalcium phosphate and free phosphoric acid. The phases are separated, followed by washing and drying of the solid dicalcium phosphate. The wash water is used in the hydrolyzer. The solution is returned to the superphosphate production step, where phosphate rock and additional phosphoric acid are added, and where water is evaporated to form the solid superphosphate. Pilot plant results have indicated that filtration rates vary widely under almost identical mixing conditions, presumably due to variations in particle size ranges. It was thought worthwhile, therefore, to study the effects of different mixing variables on the relative particle sizes as indicated by the settling rates of the mixture

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Scattered Data Approximation by LR B-Spline Surfaces: A Study on Refinement Strategies for Efficient Approximation

Locally refined B-spline (LRB) surfaces provide a representation that is well suited to scattered data approximation. When a data set has local details in some areas and is largely smooth elsewhere, LR B-splines allow the spatial distribution of degrees of freedom to follow the variations of the data set. An LRB surface approximating a data set is refined in areas where the accuracy does not meet a required tolerance. In this paper we address, in a systematic study, different LRB refinement strategies and polynomial degrees for surface approximation. We study their influence on the resulting data volume and accuracy when applied to geospatial data sets with different structural behaviour. The relative performance of the refinement strategies is reasonably coherent for the different data sets and this paper concludes with some recommendations. An overall evaluation indicates that bi-quadratic LRB are preferable for the use cases tested, and that the strategies we denote as “full span" have the overall best performance.publishedVersio