Linear extension of the Erdos-Heilbronn conjecture

The famous Erdos-Heilbronn conjecture plays an important role in the development of additive combinatorics. In 2007 Z. W. Sun made the following further conjecture (which is the linear extension of the Erdos-Heilbronn conjecture): For any finite subset A of a field F and nonzero elements $a_1,...,a_n$ of F, the set {a_1x_1+...+a_nx_n: x_1,....,x_n are distinct elements of A} has cardinality at least min{p(F)-delta, n(|A|-n)+1}, where the additive order p(F) of the multiplicative identity of F is different from n+1, and delta=0,1 takes the value 1 if and only if n=2 and $a_1+a_2=0$. In this paper we prove this conjecture of Sun when $p(F)\geq n(3n-5)/2$. We also obtain a sharp lower bound for the cardinality of the restricted sumset {x_1+...+x_n: x_1\in A_1,...,x_n\in A_n, and P(x_1,...,x_n)\not=0}, where $A_1,...,A_n$ are finite subsets of a field F and $P(x_1,...,x_n)$ is a general polynomial over F

Semi-Supervised Text Simplification with Back-Translation and Asymmetric Denoising Autoencoders

Text simplification (TS) rephrases long sentences into simplified variants while preserving inherent semantics. Traditional sequence-to-sequence models heavily rely on the quantity and quality of parallel sentences, which limits their applicability in different languages and domains. This work investigates how to leverage large amounts of unpaired corpora in TS task. We adopt the back-translation architecture in unsupervised machine translation (NMT), including denoising autoencoders for language modeling and automatic generation of parallel data by iterative back-translation. However, it is non-trivial to generate appropriate complex-simple pair if we directly treat the set of simple and complex corpora as two different languages, since the two types of sentences are quite similar and it is hard for the model to capture the characteristics in different types of sentences. To tackle this problem, we propose asymmetric denoising methods for sentences with separate complexity. When modeling simple and complex sentences with autoencoders, we introduce different types of noise into the training process. Such a method can significantly improve the simplification performance. Our model can be trained in both unsupervised and semi-supervised manner. Automatic and human evaluations show that our unsupervised model outperforms the previous systems, and with limited supervision, our model can perform competitively with multiple state-of-the-art simplification systems

Carbon Nanostructures Production by AC Arc Discharge Plasma Process at Atmospheric Pressure

Carbon nanostructures have received much attention for a wide range of applications. In this paper, we produced carbon nanostructures by decomposition of benzene using AC arc discharge plasma process at atmospheric pressure. Discharge was carried out at a voltage of 380 V, with a current of 6 A–20 A. The products were characterized by scanning electron microscopy (SEM), high-resolution transmission electron microscopy (HRTEM), powder X-ray diffraction (XRD), and Raman spectra. The results show that the products on the inner wall of the reactor and the sand core are nanoparticles with 20–60 nm diameter, and the products on the electrode ends are nanoparticles, agglomerate carbon particles, and multiwalled carbon nanotubes (MWCNTs). The maximum yield content of carbon nanotubes occurs when the arc discharge current is 8 A. Finally, the reaction mechanism was discussed