In this paper, we construct some unirational Calabi–Yau threefolds in characteristic 3. We adopt the method by Schoen, but we use quasi-elliptic surfaces instead of elliptic surfaces. We find new examples which do not admit a lifting to characteristic zero.

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In its simplest form, convolution neural networks (CNNs) consist of a fully connected layer g composed with a sequence of convolution layers T . Although g is known to have the universal approximation property, it is not known if CNNs, which has the form g ◦ T inherits this property, especially when the kernel size in T is small. In this paper, we show that under suitable conditions, CNNs does inherit the universal approximation property and its sample complexity can be characterized. In addition, we discuss concretely how the nonlinearity of T can improve the approximation power. Finally, we show that when the target function class has a certain compositional form, convolutional networks are far more advantageous compared with fully connected networks, in terms of number of parameters needed to achieve a desired accuracy.

Let$K$be the class of trigonometric series of power type, i.e. Taylor series $$\sum\nolimits_{n = 0}^\infty {c_n z^n } $$ for$z$=$e$^{$ix$}, whose partial sums for all$x$in$E$, where$E$is a nondenumerable subset of [0, 2π), lie on a$finite$number of circles (a priori depending on$x$) in the complex plane. The main result of this paper is that for every member of the class$K$, there exist a complex number ω, |ω|=1, and two positive integers $$\nu , \kappa , \nu < \kappa $$ , such that for the coefficients$c$_{$n$}we have: $$c_{\mu + \lambda \left( {\kappa - \nu } \right)} = c_\mu \omega ^\lambda , \mu = \nu ,\nu + 1, \ldots , \kappa - 1, \lambda = 1,2,3, \ldots $$ . Thus, every member of the class$K$has (with minor modifications) a representation of the form: $$P(x)\sum\nolimits_{n = 0}^\infty {e^{iknx} ,} $$ where P($x$) is a suitable trigonometric polynomial and$k$a positive integer. The proof is elementary but rather long. This result is closely related to a theorem of Marcinkiewicz and Zygmund on the circular structure of the set of limit points of the sequence of partial sums of ($C$, 1) summable Taylor series.

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.