Representation of continuous change with discrete time

By careful control of the electron beam it is possible to simultaneously induce and observe the phase transformation from monomer to polymer in certain solid-state polymcrizable diacetylenes. The continuous change in the crystal structure from DCHD diacetylene monomer (a=1.76 nm, b=1.36 nm, c=0.455 nm, γ=94 degrees, P2l/c) to polymer (a=1.74 nm, b=1.29 nm, c=0.49 nm, γ=108 degrees, P2l/c) occurs at a characteristic dose (10−4C/cm2) which is five orders of magnitude smaller than the critical end point dose (20 C/cm2). Previously we discussed the progress of this phase transition primarily as observed down the [001] zone (the chain axis direction). Here we report on the associated changes of the dark field (DF) images and selected area electron diffraction (SAED) patterns of the crystals as observed from the side (i.e., in the [hk0] zones).High resolution electron micrographs (HREM), DF images, and SAED patterns were obtained on a JEOL 4000 EX HREM operating at 400 kV.

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Orthogonal functions approach to the analysis of discrete-time systems is unnecessary

Journal of the Franklin Institute ◽

10.1016/s0016-0032(98)90003-5 ◽

1998 ◽

Vol 355 (3) ◽

pp. 441-449

Author(s):

B Mohan

Keyword(s):

Discrete Time ◽

Orthogonal Functions ◽

Discrete Time Systems ◽

Time Systems

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The Effect of Partly Missing Covariates on Statistical Power in Randomized Controlled Trials With Discrete-Time Survival Endpoints

Methodology ◽

10.1027/1614-2241/a000121 ◽

2017 ◽

Vol 13 (2) ◽

pp. 41-60

Author(s):

Shahab Jolani ◽

Maryam Safarkhani

Keyword(s):

Randomized Controlled Trials ◽

Discrete Time ◽

Treatment Effect ◽

Survival Data ◽

Controlled Trials ◽

Missing Covariates ◽

Indicator Method ◽

Imputation Methods ◽

Randomized Controlled ◽

Baseline Covariates

Abstract. In randomized controlled trials (RCTs), a common strategy to increase power to detect a treatment effect is adjustment for baseline covariates. However, adjustment with partly missing covariates, where complete cases are only used, is inefficient. We consider different alternatives in trials with discrete-time survival data, where subjects are measured in discrete-time intervals while they may experience an event at any point in time. The results of a Monte Carlo simulation study, as well as a case study of randomized trials in smokers with attention deficit hyperactivity disorder (ADHD), indicated that single and multiple imputation methods outperform the other methods and increase precision in estimating the treatment effect. Missing indicator method, which uses a dummy variable in the statistical model to indicate whether the value for that variable is missing and sets the same value to all missing values, is comparable to imputation methods. Nevertheless, the power level to detect the treatment effect based on missing indicator method is marginally lower than the imputation methods, particularly when the missingness depends on the outcome. In conclusion, it appears that imputation of partly missing (baseline) covariates should be preferred in the analysis of discrete-time survival data.

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