Uniform lower bound for the least common multiple of a polynomial sequence

Let $n$ be a positive integer and $f(x)$ be a polynomial with nonnegative integer coefficients. We prove that ${\rm lcm}_{\lceil n/2\rceil \le i\le n} \{f(i)\}\ge 2^n$ except that $f(x)=x$ and $n=1, 2, 3, 4, 6$ and that $f(x)=x^s$ with $s\ge 2$ being an integer and $n=1$, where $\lceil n/2\rceil$ denotes the smallest integer which is not less than $n/2$. This improves and extends the lower bounds obtained by Nair in 1982, Farhi in 2007 and Oon in 2013.Comment: 6 pages. To appear in Comptes Rendus Mathematiqu

The elementary symmetric functions of a reciprocal polynomial sequence

Erd\"{o}s and Niven proved in 1946 that for any positive integers $m$ and $d$, there are at most finitely many integers $n$ for which at least one of the elementary symmetric functions of $1/m, 1/(m+d), ..., 1/(m+(n-1)d)$ are integers. Recently, Wang and Hong refined this result by showing that if $n\geq 4$, then none of the elementary symmetric functions of $1/m, 1/(m+d), ..., 1/(m+(n-1)d)$ is an integer for any positive integers $m$ and $d$. Let $f$ be a polynomial of degree at least $2$ and of nonnegative integer coefficients. In this paper, we show that none of the elementary symmetric functions of $1/f(1), 1/f(2), ..., 1/f(n)$ is an integer except for $f(x)=x^{m}$ with $m\geq2$ being an integer and $n=1$.Comment: 4 pages. To appear in Comptes Rendus Mathematiqu

Recent progress on molecular breeding of rice in China

Molecular breeding of rice for high yield, superior grain quality, and strong environmental adaptability is crucial for feeding the world’s rapidly growing population. The increasingly cloned quantitative trait loci and genes, genome variations, and haplotype blocks related to agronomically important traits in rice have provided a solid foundation for direct selection and molecular breeding, and a number of genes have been successfully introgressed into mega varieties of rice. Here we summarize China’s great achievements in molecular breeding of rice in the following five traits: high yield, biotic stress resistance, abiotic stress resistance, quality and physiology. Further, the prospect of rice breeding by molecular design is discussed

Towards SAR Tomographic Inversion via Sparse Bayesian Learning

Existing SAR tomography (TomoSAR) algorithms are mostly based on an inversion of the SAR imaging model, which are often computationally expensive. Previous study showed perspective of using data-driven methods like KPCA to decompose the signal and reduce the computational complexity. This paper gives a preliminary demonstration of a new data-driven method based on sparse Bayesian learning. Experiments on simulated data show that the proposed method significantly outperforms KPCA methods in estimating the steering vectors of the scatterers. This gives a perspective of data-drive approach or combining it with model-driven approach for high precision tomographic inversion of large areas.Comment: accepted in preliminary version for EUSAR2020 conferenc

Impact of Females on the Top Management Team on Firm Performance: Evidence from Chinese Public Firms

Gender diversity has become a popular issue within corporate government. This paper focus on gender diversity in top management team to explore the influence of gender diversity on firm performance. This research takes a sample of A-share listed companies in Shanghai and Shenzhen stock exchanges spanning 5 years from 2015 to 2019. Despite that fact that the status of Chinese female have largely improved, female still have to face a lot of barriers in their workplace. The finding of this research indicates that female on top management team have a negative effect on firm performance (Tobin’s Q), which is contrary to many other researches

Bayesian imaging inverse problem with SA-Roundtrip prior via HMC-pCN sampler

Bayesian inference with deep generative prior has received considerable interest for solving imaging inverse problems in many scientific and engineering fields. The selection of the prior distribution is learned from, and therefore an important representation learning of, available prior measurements. The SA-Roundtrip, a novel deep generative prior, is introduced to enable controlled sampling generation and identify the data's intrinsic dimension. This prior incorporates a self-attention structure within a bidirectional generative adversarial network. Subsequently, Bayesian inference is applied to the posterior distribution in the low-dimensional latent space using the Hamiltonian Monte Carlo with preconditioned Crank-Nicolson (HMC-pCN) algorithm, which is proven to be ergodic under specific conditions. Experiments conducted on computed tomography (CT) reconstruction with the MNIST and TomoPhantom datasets reveal that the proposed method outperforms state-of-the-art comparisons, consistently yielding a robust and superior point estimator along with precise uncertainty quantification

γ\boldsymbol{\gamma}-Net: Superresolving SAR Tomographic Inversion via Deep Learning

Synthetic aperture radar tomography (TomoSAR) has been extensively employed in 3-D reconstruction in dense urban areas using high-resolution SAR acquisitions. Compressive sensing (CS)-based algorithms are generally considered as the state of the art in super-resolving TomoSAR, in particular in the single look case. This superior performance comes at the cost of extra computational burdens, because of the sparse reconstruction, which cannot be solved analytically and we need to employ computationally expensive iterative solvers. In this paper, we propose a novel deep learning-based super-resolving TomoSAR inversion approach, $\boldsymbol{\gamma}$-Net, to tackle this challenge. $\boldsymbol{\gamma}$-Net adopts advanced complex-valued learned iterative shrinkage thresholding algorithm (CV-LISTA) to mimic the iterative optimization step in sparse reconstruction. Simulations show the height estimate from a well-trained $\boldsymbol{\gamma}$-Net approaches the Cram\'er-Rao lower bound while improving the computational efficiency by 1 to 2 orders of magnitude comparing to the first-order CS-based methods. It also shows no degradation in the super-resolution power comparing to the state-of-the-art second-order TomoSAR solvers, which are much more computationally expensive than the first-order methods. Specifically, $\boldsymbol{\gamma}$-Net reaches more than $90\%$ detection rate in moderate super-resolving cases at 25 measurements at 6dB SNR. Moreover, simulation at limited baselines demonstrates that the proposed algorithm outperforms the second-order CS-based method by a fair margin. Test on real TerraSAR-X data with just 6 interferograms also shows high-quality 3-D reconstruction with high-density detected double scatterers