82,834 research outputs found
Computability and analysis: the legacy of Alan Turing
We discuss the legacy of Alan Turing and his impact on computability and
analysis.Comment: 49 page
Sampling algebraic sets in local intrinsic coordinates
Numerical data structures for positive dimensional solution sets of
polynomial systems are sets of generic points cut out by random planes of
complimentary dimension. We may represent the linear spaces defined by those
planes either by explicit linear equations or in parametric form. These
descriptions are respectively called extrinsic and intrinsic representations.
While intrinsic representations lower the cost of the linear algebra
operations, we observe worse condition numbers. In this paper we describe the
local adaptation of intrinsic coordinates to improve the numerical conditioning
of sampling algebraic sets. Local intrinsic coordinates also lead to a better
stepsize control. We illustrate our results with Maple experiments and
computations with PHCpack on some benchmark polynomial systems.Comment: 13 pages, 2 figures, 2 algorithms, 2 table
Can a computer be "pushed" to perform faster-than-light?
We propose to "boost" the speed of communication and computation by immersing
the computing environment into a medium whose index of refraction is smaller
than one, thereby trespassing the speed-of-light barrier.Comment: 7 pages, 1 figure, presented at the UC10 Hypercomputation Workshop
"HyperNet 10" at The University of Tokyo on June 22, 201
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