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81.
Very often, in the course of uncertainty quantification tasks or data analysis, one has to deal with high-dimensional random variables. Here the interest is mainly to compute characterizations like the entropy, the Kullback–Leibler divergence, more general f $$ f $$ -divergences, or other such characteristics based on the probability density. The density is often not available directly, and it is a computational challenge to just represent it in a numerically feasible fashion in case the dimension is even moderately large. It is an even stronger numerical challenge to then actually compute said characteristics in the high-dimensional case. In this regard it is proposed to approximate the discretized density in a compressed form, in particular by a low-rank tensor. This can alternatively be obtained from the corresponding probability characteristic function, or more general representations of the underlying random variable. The mentioned characterizations need point-wise functions like the logarithm. This normally rather trivial task becomes computationally difficult when the density is approximated in a compressed resp. low-rank tensor format, as the point values are not directly accessible. The computations become possible by considering the compressed data as an element of an associative, commutative algebra with an inner product, and using matrix algorithms to accomplish the mentioned tasks. The representation as a low-rank element of a high order tensor space allows to reduce the computational complexity and storage cost from exponential in the dimension to almost linear.  相似文献   
82.
Nowadays, Green Analytical Chemistry is widely applied to provide various analytical methods with eco-friendly procedures employing the least toxic, harmful reagents on humans and the environment without affecting the efficacy of the determination. Accordingly, two eco-friendly, accurate, and reliable high-performance thin-layer chromatography-densitometry and high-performance liquid chromatographic methods were established for the determination and separation of two antispasmodic drugs, namely phloroglucinol and trimethylphloroglucinol in their pure, combined dosage form along with phloroglucinol toxic impurity, 3,5-dichloroaniline. For high-performance thin-layer chromatography-densitometry, efficient separation was developed via utilizing the stationary phase of high-performance thin-layer chromatography silica gel 60 F254 plates and developing a system comprising of ethyl acetate-butanol-ammonia in the ratio of 8.0:2.0:0.2, by volume and scanning of the developed bands at 210.0 nm. The subsequent method is isocratic high-performance liquid chromatography with diode array detection in which separation was successively attained using XTerra RP-C18 (250 × 4.6 mm, 5 μm) column as stationary phase and methanol-10.0 mM phosphate buffer, pH 3.7 ± 0.1 as mobile phase in the ratio of 75.0:25.0, v/v at flow rate 1.0 ml/min and scanning at 220.0 nm. The developed liquid chromatography methods were validated according to the International Council for Harmonization guidelines, and all results acknowledged their efficacy. Additionally, the proposed methods worked well for assessing the cited drugs in binary combined commercially available pharmaceutical formulation. The greenness profile of the present methods was assessed and estimated using various assessment tools, namely; Green Analytical Procedure Index, analytical eco-scale method, National Environmental Method Index in addition to Analytical GREEnness tool to evaluate the greenness of the provided methods with a statistical comparison between them to assess the more green ones.  相似文献   
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