Blocking is commonly used in randomized experiments to increase efficiency of estimation. A generalization of blocking removes allocations with imbalance in covariate distributions between treated and control units, and then randomizes within the remaining set of allocations with balance. This idea of rerandomization was formalized by Morgan and Rubin (Annals of Statistics, 2012, 40, 1263–1282), who suggested using Mahalanobis distance between treated and control covariate means as the criterion for removing unbalanced allocations. Kallus (Journal of the Royal Statistical Society, Series B: Statistical Methodology, 2018, 80, 85–112) proposed reducing the set of balanced allocations to the minimum. Here we discuss the implication of such an ‘optimal’ rerandomization design for inferences to the units in the sample and to the population from which the units in the sample were randomly drawn. We argue that, in general, it is a bad idea to seek the optimal design for an inference because that inference typically only reflects uncertainty from the random sampling of units, which is usually hypothetical, and not the randomization of units to treatment versus control.

We prove that there are no stable intersections of regular Cantor sets in the$C$^{1}topology: given any pair ($K$,$K$′) of regular Cantor sets, we can find, arbitrarily close to it in the$C$^{1}topology, pairs $ \left( {\tilde{K},\tilde{K}'} \right) $ of regular Cantor sets with $ \tilde{K} \cap \tilde{K}' = \emptyset $ .

Surface-related-multiple wavefields constitute redundant information in conventional migration and can often be difficult to attenuate. However, when used for migration, multiple wavefields can improve subsurface illumination. Unfortunately, the process of imaging using multiples involves the management of crosstalk, which largely restricts its application. Crosstalk causes phantom images formed by spurious correlation of unrelated events in a migration process. These events can be unrelated orders of multiples in the source and receiver wavefields; they can also be one event associated with a reflector in the source wavefield and another event generated by a different reflector in the receiver wavefield. We have first examined crosstalk by explicitly investigating its generation mechanisms in a migration process and classifying it into different categories based on causality. Following this analysis, crosstalk can be predicted in a migration process and subtracted in the image domain; however, this method is usually difficult to apply due to the complexity of wavefield separation and adaptive subtraction. Furthermore, we develop different algorithms to attenuate the crosstalk, including a deconvolution imaging condition, a least-squares migration (LSM) method, and an advanced algorithm combining LSM with a deconvolution imaging condition. We evaluate these different strategies with synthetic examples. A deconvolution imaging condition can attenuate some crosstalk, but it is less effective at suppressing strong coherent crosstalk events. However, the LSM method can fundamentally address the crosstalk issue, and this approach is further optimized when combined with a deconvolution imaging condition.

This paper gives a classification of first order polynomial differential operators of form $\mathscr{X} = X_1(x_1,x_2)\delta_1 + X_2(x_1,x_2)\delta_2$, $(\delta_i = \partial/\partial x_i)$. The classification is given through the order of an operator that is defined in this paper. Let $X=\mathscr{X}y$ to be the differential polynomial associated with $\mathscr{X}$, the order of $\mathscr{X}$, $\mathrm{ord}(\mathscr{X})$, is defined as the order of a differential ideal $\Lambda$ of differential polynomials that is a nontrivial expansion of the ideal $\{X\}$ and with the lowest order. In this paper, we prove that there are only four possible values for the order of a differential operator, $0$, $1$, $2$, $3$, or $\infty$. Furthermore, when the order is finite, the expansion $\Lambda$ is generated by $X$ and a differential polynomial $A$, which can be obtained through a rational solution of a partial differential equation that is given explicitly in this paper. When the order is infinite, the expansion $\Lambda$ is just the unit ideal. In additional, if, and only if, the order of $\mathscr{X}$ is $0$, $1$, or $2$, the polynomial differential equation associating with $\mathscr{X}$ has Liouvillian first integrals. Examples and connections with Godbilon-Vey sequences are also discussed.

No abstract uploaded!