43,730 research outputs found
Groups generated by two elliptic elements in PU(2,1)
Let and be two elliptic elements in of order
and respectively, where . We prove that if the distance
between the complex lines or points fixed by and is large
than a certain number, then the group is discrete nonelementary and
isomorphic to the free product .Comment: 9 page
Wishart Mechanism for Differentially Private Principal Components Analysis
We propose a new input perturbation mechanism for publishing a covariance
matrix to achieve -differential privacy. Our mechanism uses a
Wishart distribution to generate matrix noise. In particular, We apply this
mechanism to principal component analysis. Our mechanism is able to keep the
positive semi-definiteness of the published covariance matrix. Thus, our
approach gives rise to a general publishing framework for input perturbation of
a symmetric positive semidefinite matrix. Moreover, compared with the classic
Laplace mechanism, our method has better utility guarantee. To the best of our
knowledge, Wishart mechanism is the best input perturbation approach for
-differentially private PCA. We also compare our work with
previous exponential mechanism algorithms in the literature and provide near
optimal bound while having more flexibility and less computational
intractability.Comment: A full version with technical proofs. Accepted to AAAI-1
Topological Imbert-Fedorov shift in Weyl semimetals
The Goos-H\"anchen (GH) shift and the Imbert-Fedorov (IF) shift are optical
phenomena which describe the longitudinal and transverse lateral shifts at the
reflection interface, respectively. Here, we report the GH and IF shifts in
Weyl semimetals (WSMs) - a promising material harboring low energy Weyl
fermions, a massless fermionic cousin of photons. Our results show that GH
shift in WSMs is valley-independent which is analogous to that discovered in a
2D relativistic material - graphene. However, the IF shift has never been
explored in non-optical systems, and here we show that it is valley-dependent.
Furthermore, we find that the IF shift actually originates from the topological
effect of the system. Experimentally, the topological IF shift can be utilized
to characterize the Weyl semimetals, design valleytronic devices of high
efficiency, and measure the Berry curvature
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