Computational optimal transport for molecular spectra: The fully discrete case

2021 ◽  
Vol 155 (18) ◽  
pp. 184101
Author(s):  
Nathan A. Seifert ◽  
Kirill Prozument ◽  
Michael J. Davis
Keyword(s):  
Optimal Transport ◽  
Discrete Case ◽  
Molecular Spectra ◽  
Fully Discrete

scholarly journals An elementary derivation of Hattendorff’s theorem

2021 ◽  
Author(s):  
Elias S. W. Shiu ◽  
Xiaoyi Xiong
Keyword(s):  
At Risk ◽  
Life Insurance ◽  
Survival Probability ◽  
Random Variable ◽  
Discrete Case ◽  
Insurance Policy ◽  
Summation By Parts ◽  
Fully Discrete ◽  
Insurance Model ◽  
The Moment

AbstractFor a general fully continuous life insurance model, the variance of the loss-at-issue random variable is the expectation of the square of the discounted value of the net amount at risk at the moment of death. In 1964 Jim Hickman gave an elementary and elegant derivation of this result by the method of integration by parts. One might expect that the method of summation by parts could be used to treat the fully discrete case. However, there are two difficulties. The summation-by-parts formula involves shifting an index, making it somewhat unwieldy. In the fully discrete case, the variance of the loss-at-issue random variable is more complicated; it is the expectation of the square of the discounted value of the net amount at risk at the end of the year of death times a survival probability factor. The purpose of this note is to show that one can indeed use the method of summation by parts to find the variance of the loss-at-issue random variable for a fully discrete life insurance policy.

Quantitative stability and error estimates for optimal transport plans

2020 ◽  
Author(s):  
Wenbo Li ◽  
Ricardo H Nochetto
Keyword(s):  
Error Estimates ◽  
Convergence Rates ◽  
Optimal Transport ◽  
Quantitative Stability ◽  
Fully Discrete ◽  
Discrete Algorithms ◽  
Transport Maps ◽  
Transport Problems ◽  
Optimal Transport Map ◽  
Continuous Measures

Abstract Optimal transport maps and plans between two absolutely continuous measures $\mu$ and $\nu$ can be approximated by solving semidiscrete or fully discrete optimal transport problems. These two problems ensue from approximating $\mu$ or both $\mu$ and $\nu$ by Dirac measures. Extending an idea from Gigli (2011, On Hölder continuity-in-time of the optimal transport map towards measures along a curve. Proc. Edinb. Math. Soc. (2), 54, 401–409), we characterize how transport plans change under the perturbation of both $\mu$ and $\nu$. We apply this insight to prove error estimates for semidiscrete and fully discrete algorithms in terms of errors solely arising from approximating measures. We obtain weighted $L^2$ error estimates for both types of algorithms with a convergence rate $O(h^{1/2})$. This coincides with the rate in Theorem 5.4 in Berman (2018, Convergence rates for discretized Monge–Ampère equations and quantitative stability of optimal transport. Preprint available at arXiv:1803.00785) for semidiscrete methods, but the error notion is different.

Reaction of organolithium compounds with 1-substituted 2,4,6-triphenylpyridinium perchlorates

1989 ◽  
Vol 54 (7) ◽  
pp. 1880-1887 ◽  
Author(s):  
Marián Schwarz ◽  
Josef Kuthan
Keyword(s):  
Visible Range ◽  
Molecular Spectra ◽  
Organolithium Compounds ◽  
Phenylmagnesium Bromide

The reaction of organolithium compounds with 1-substituted 2,4,6-triphenylpyridinium perchlorates Ia-Ic produces mixtures of 1,4-dihydropyridines IIa-IIe and 1,2-dihydropyridines IIIa-IIIe. Analogous reactions of phenylmagnesium bromide with compounds Ia-Ic proceed with very low conversions (less than 1%). Photochromism in visible range is observed only with the compounds II which have two aromatic substituents at 4-position, whereas compounds III and IId show no visible photochromism. The molecular spectra of the compounds newly prepared are discussed.

Sign in / Sign up

Export Citation Format

Share Document