9,693 research outputs found
Level sets of multiple ergodic averages
We propose to study multiple ergodic averages from multifractal analysis
point of view. In some special cases in the symbolic dynamics, Hausdorff
dimensions of the level sets of multiple ergodic average limit are determined
by using Riesz products.Comment: This note was refused by Proceedings of AMS although the referee said
"In my opinion this is a nice application of the Riesz product technique to
solve, in principle, a hard problem when considered in its full generality.
Nevertheless, I think it needs some extra work to see how this example seats
in a more general context and explore how far this technique can go." We
should say that Riesz product works perfectly in the situation described in
this note, but Riesz product has its limit--we don't think that Riesz product
technique can solve the problem in its generalit
AutoEncoder by Forest
Auto-encoding is an important task which is typically realized by deep neural
networks (DNNs) such as convolutional neural networks (CNN). In this paper, we
propose EncoderForest (abbrv. eForest), the first tree ensemble based
auto-encoder. We present a procedure for enabling forests to do backward
reconstruction by utilizing the equivalent classes defined by decision paths of
the trees, and demonstrate its usage in both supervised and unsupervised
setting. Experiments show that, compared with DNN autoencoders, eForest is able
to obtain lower reconstruction error with fast training speed, while the model
itself is reusable and damage-tolerable
Novel Two-dimensional Carbon Allotrope with Strong Electronic Anisotropy
Two novel two-dimensional carbon allotropes comprised of octagons and
pentagons are proposed based on the first-principles calculations. The two
carbon allotropes, named OPG-L and OPG-Z, are found to have distinct
properties. OPG-L is metallic, while OPG-Z is a gapless semimetal. Remarkably,
OPG-Z exhibits pronounced electronic anisotropy with highly anisotropic Dirac
points at the Fermi level. A tight-binding model is suggested to describe the
low-energy quasiparticles, which clarifies the origin of the anisotropic Dirac
points. Such an anisotropic electronic characteristic of OPG-Z is expected to
have wide implications in nano-electronics.Comment: 6 pages, 5 figures (accepted by Physical Review B
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