3,905 research outputs found
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
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 . In this
paper we prove this conjecture of Sun when . 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 are
finite subsets of a field F and 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 380βV, with a current of 6βAβ20βA. 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 20β60βnm 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 8βA. Finally, the reaction mechanism was discussed
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