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With the rapid development of urbanization in China, urban spatial form has increasingly gained research attention. In this study, the DEA (data envelopment analysis)-Malmquist index model and a panel data model are used to examine the relationship between the urban spatial form and total factor productivity (TFP) of 30 provincial cities in China. Our method of measuring urban spatial form is different from the current entropy method, but we use remote sensing GIS (Geographic Information System) technology to measure the relevant data on urban compactness and urban elongation. The average values of urban compactness and urban elongation first rise, then fall, and then rise again, and there are alternate situations of urban compact development and urban sprawl and expansion. Furthermore, there is a significant positive correlation between urban compactness and TFP. Therefore, cities with high urban compactness can promote TFP. In addition, there is a significant negative correlation between urban extension rate and TFP, indicating that an increase in urban elongation has a restraining effect on TFP. Finally, the average TFP of each city shows a fluctuating trend of rising first and then declining, which is determined mainly by technological change and efficiency change. These results are expected to provide a scientific basis for the development of urban agglomerations in China.

Let k ⩾ 1 be an integer. In this study, we derive an asymptotic formula for the average number of representations of integers n = p 1 k + p 2 3 + p 3 3 + p 4 3 + p 5 3 in short intervals, where p 1 , p 2 , p 3 , p 4 , p 5 are prime numbers.

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<p style='text-indent:20px;'>In this work, we investigate a class of fractional Schrödinger - Poisson systems</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \left\{\begin{array}{ll}(-\triangle)^s u +V(x)u+\lambda\phi u = \mu u+|u|^{p-1}u, &amp; x\in\ \mathbb{R}^3, \\(-\triangle)^s \phi = u^2, &amp; x\in\ \mathbb{R}^3, \end{array}\right. \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ s\in(\frac{3}{4}, 1) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ p\in(3, 5) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula> is a positive parameter. By the variational method, we show that there exists <inline-formula><tex-math id="M4">\begin{document}$ \delta(\lambda)&gt;0 $\end{document}</tex-math></inline-formula> such that for all <inline-formula><tex-math id="M5">\begin{document}$ \mu\in[\mu_1, \mu_1+\delta(\lambda)) $\end{document}</tex-math></inline-formula>, the above fractional Schrödinger -Poisson systems possess a nonnegative bound state solutions with positive energy. Here <inline-formula><tex-math id="M6">\begin{document}$ \mu_1 $\end{document}</tex-math></inline-formula> is the first eigenvalue of <inline-formula><tex-math id="M7">\begin{document}$ (-\triangle)^s +V(x) $\end{document}</tex-math></inline-formula>.</p>