5,164 research outputs found
Small gaps between products of two primes
Let denote the number that is a product of exactly two
distinct primes. We prove that
This sharpens an earlier result of the authors (arXivMath NT/0506067), which
had 26 in place of 6. More generally, we prove that if is any positive
integer, then
We also prove several other results on the representation of numbers with
exactly two prime factors by linear forms.Comment: 11N25 (primary) 11N36 (secondary
Unusual structural tuning of magnetism in cuprate perovskites
Understanding the structural underpinnings of magnetism is of great
fundamental and practical interest. Se_{1-x}Te_{x}CuO_{3} alloys are model
systems for the study of this question, as composition-induced structural
changes control their magnetic interactions. Our work reveals that this
structural tuning is associated with the position of the supposedly dummy atoms
Se and Te relative to the super-exchange (SE) Cu--O--Cu paths, and not with the
SE angles as previously thought. We use density functional theory,
tight-binding, and exact diagonalization methods to unveil the cause of this
surprising effect and hint at new ways of engineering magnetic interactions in
solids.Comment: 4 pages, with 4 postscript figures embedded. Uses REVTEX4 and
graphicx macro
Self-Supervised Intrinsic Image Decomposition
Intrinsic decomposition from a single image is a highly challenging task, due
to its inherent ambiguity and the scarcity of training data. In contrast to
traditional fully supervised learning approaches, in this paper we propose
learning intrinsic image decomposition by explaining the input image. Our
model, the Rendered Intrinsics Network (RIN), joins together an image
decomposition pipeline, which predicts reflectance, shape, and lighting
conditions given a single image, with a recombination function, a learned
shading model used to recompose the original input based off of intrinsic image
predictions. Our network can then use unsupervised reconstruction error as an
additional signal to improve its intermediate representations. This allows
large-scale unlabeled data to be useful during training, and also enables
transferring learned knowledge to images of unseen object categories, lighting
conditions, and shapes. Extensive experiments demonstrate that our method
performs well on both intrinsic image decomposition and knowledge transfer.Comment: NIPS 2017 camera-ready version, project page:
http://rin.csail.mit.edu
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