3,772 research outputs found
Alternative mechanism of avoiding the big rip or little rip for a scalar phantom field
Depending on the choice of its potential, the scalar phantom field
(the equation of state parameter ) leads to various catastrophic fates of
the universe including big rip, little rip and other future singularity. For
example, big rip results from the evolution of the phantom field with an
exponential potential and little rip stems from a quadratic potential in
general relativity (GR). By choosing the same potential as in GR, we suggest a
new mechanism to avoid these unexpected fates (big and little rip) in the
inverse-\textit{R} gravity. As a pedagogical illustration, we give an exact
solution where phantom field leads to a power-law evolution of the scale factor
in an exponential type potential. We also find the sufficient condition for a
universe in which the equation of state parameter crosses divide. The
phantom field with different potentials, including quadratic, cubic, quantic,
exponential and logarithmic potentials are studied via numerical calculation in
the inverse-\textit{R} gravity with correction. The singularity is
avoidable under all these potentials. Hence, we conclude that the avoidance of
big or little rip is hardly dependent on special potential.Comment: 9 pages,6 figure
Linear and Range Counting under Metric-based Local Differential Privacy
Local differential privacy (LDP) enables private data sharing and analytics
without the need for a trusted data collector. Error-optimal primitives (for,
e.g., estimating means and item frequencies) under LDP have been well studied.
For analytical tasks such as range queries, however, the best known error bound
is dependent on the domain size of private data, which is potentially
prohibitive. This deficiency is inherent as LDP protects the same level of
indistinguishability between any pair of private data values for each data
downer.
In this paper, we utilize an extension of -LDP called Metric-LDP or
-LDP, where a metric defines heterogeneous privacy guarantees for
different pairs of private data values and thus provides a more flexible knob
than does to relax LDP and tune utility-privacy trade-offs. We show
that, under such privacy relaxations, for analytical workloads such as linear
counting, multi-dimensional range counting queries, and quantile queries, we
can achieve significant gains in utility. In particular, for range queries
under -LDP where the metric is the -distance function scaled by
, we design mechanisms with errors independent on the domain sizes;
instead, their errors depend on the metric , which specifies in what
granularity the private data is protected. We believe that the primitives we
design for -LDP will be useful in developing mechanisms for other analytical
tasks, and encourage the adoption of LDP in practice
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