8,138 research outputs found
Casimir effect of two conducting parallel plates in a general weak gravitational field
We calculate the finite vacuum energy density of the scalar and
electromagnetic fields inside a Casimir apparatus made up of two conducting
parallel plates in a general weak gravitational field. The metric of the weak
gravitational field has a small deviation from flat spacetime inside the
apparatus and we find it by expanding the metric in terms of small parameters
of the weak background. We show that the found metric can be transformed via a
gauge transformation to the Fermi metric. We solve the Klein-Gordon equation
exactly and find mode frequencies in Fermi spacetime. Using the fact that the
electromagnetic field can be represented by two scalar fields in the Fermi
spacetime, we find general formulas for the energy density and mode frequencies
of the electromagnetic field. Some well-known weak backgrounds are examined and
consistency of the results with the literature is shown.Comment: 25 pages, 1 figur
Sparse Hopsets in Congested Clique
We give the first Congested Clique algorithm that computes a sparse hopset
with polylogarithmic hopbound in polylogarithmic time. Given a graph ,
a -hopset with "hopbound" , is a set of edges
added to such that for any pair of nodes and in there is a path
with at most hops in with length within of
the shortest path between and in .
Our hopsets are significantly sparser than the recent construction of
Censor-Hillel et al. [6], that constructs a hopset of size
, but with a smaller polylogarithmic hopbound. On the other
hand, the previously known constructions of sparse hopsets with polylogarithmic
hopbound in the Congested Clique model, proposed by Elkin and Neiman
[10],[11],[12], all require polynomial rounds.
One tool that we use is an efficient algorithm that constructs an
-limited neighborhood cover, that may be of independent interest.
Finally, as a side result, we also give a hopset construction in a variant of
the low-memory Massively Parallel Computation model, with improved running time
over existing algorithms
Massively Parallel Approximate Distance Sketches
Data structures that allow efficient distance estimation (distance oracles, distance sketches, etc.) have been extensively studied, and are particularly well studied in centralized models and classical distributed models such as CONGEST. We initiate their study in newer (and arguably more realistic) models of distributed computation: the Congested Clique model and the Massively Parallel Computation (MPC) model. We provide efficient constructions in both of these models, but our core results are for MPC. In MPC we give two main results: an algorithm that constructs stretch/space optimal distance sketches but takes a (small) polynomial number of rounds, and an algorithm that constructs distance sketches with worse stretch but that only takes polylogarithmic rounds.
Along the way, we show that other useful combinatorial structures can also be computed in MPC. In particular, one key component we use to construct distance sketches are an MPC construction of the hopsets of [Elkin and Neiman, 2016]. This result has additional applications such as the first polylogarithmic time algorithm for constant approximate single-source shortest paths for weighted graphs in the low memory MPC setting
Brief Announcement: Massively Parallel Approximate Distance Sketches
Data structures that allow efficient distance estimation have been extensively studied both in centralized models and classical distributed models. We initiate their study in newer (and arguably more realistic) models of distributed computation: the Congested Clique model and the Massively Parallel Computation (MPC) model. In MPC we give two main results: an algorithm that constructs stretch/space optimal distance sketches but takes a (small) polynomial number of rounds, and an algorithm that constructs distance sketches with worse stretch but that only takes polylogarithmic rounds. Along the way, we show that other useful combinatorial structures can also be computed in MPC. In particular, one key component we use is an MPC construction of the hopsets of Elkin and Neiman (2016). This result has additional applications such as the first polylogarithmic time algorithm for constant approximate single-source shortest paths for weighted graphs in the low memory MPC setting
Shear waves in prestrained poroelastic media
Shear wave elastography measures shear wave speed in soft tissues for diagnostic purposes. In [1], shear wave speed was shown to depend on prestrain, but not necessarily prestress, in a perfused canine liver. We model this phenomenon by examining incremental waves in a pressurized poroelastic medium with incompressible phases. The analysis suggests novel restrictions on the strain energy functions for soft tissues
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