40,356 research outputs found
On the alleged simplicity of impure proof
Roughly, a proof of a theorem, is “pure” if it draws only on what is “close” or “intrinsic” to that theorem. Mathematicians employ a variety of terms to identify pure proofs, saying that a pure proof is one that avoids what is “extrinsic,” “extraneous,” “distant,” “remote,” “alien,” or “foreign” to the problem or theorem under investigation. In the background of these attributions is the view that there is a distance measure (or a variety of such measures) between mathematical statements and proofs. Mathematicians have paid little attention to specifying such distance measures precisely because in practice certain methods of proof have seemed self- evidently impure by design: think for instance of analytic geometry and analytic number theory. By contrast, mathematicians have paid considerable attention to whether such impurities are a good thing or to be avoided, and some have claimed that they are valuable because generally impure proofs are simpler than pure proofs. This article is an investigation of this claim, formulated more precisely by proof- theoretic means. After assembling evidence from proof theory that may be thought to support this claim, we will argue that on the contrary this evidence does not support the claim
Type classes for efficient exact real arithmetic in Coq
Floating point operations are fast, but require continuous effort on the part
of the user in order to ensure that the results are correct. This burden can be
shifted away from the user by providing a library of exact analysis in which
the computer handles the error estimates. Previously, we [Krebbers/Spitters
2011] provided a fast implementation of the exact real numbers in the Coq proof
assistant. Our implementation improved on an earlier implementation by O'Connor
by using type classes to describe an abstract specification of the underlying
dense set from which the real numbers are built. In particular, we used dyadic
rationals built from Coq's machine integers to obtain a 100 times speed up of
the basic operations already. This article is a substantially expanded version
of [Krebbers/Spitters 2011] in which the implementation is extended in the
various ways. First, we implement and verify the sine and cosine function.
Secondly, we create an additional implementation of the dense set based on
Coq's fast rational numbers. Thirdly, we extend the hierarchy to capture order
on undecidable structures, while it was limited to decidable structures before.
This hierarchy, based on type classes, allows us to share theory on the
naturals, integers, rationals, dyadics, and reals in a convenient way. Finally,
we obtain another dramatic speed-up by avoiding evaluation of termination
proofs at runtime.Comment: arXiv admin note: text overlap with arXiv:1105.275
Uniqueness of reconstruction and an inversion procedure for thermoacoustic and photoacoustic tomography
The paper contains a simple approach to reconstruction in Thermoacoustic and
Photoacoustic Tomography. The technique works for any geometry of point
detectors placement and for variable sound speed satisfying a non-trapping
condition. A uniqueness of reconstruction result is also obtained
Some generalizations of Calabi compactness theorem
In this paper we obtain generalized Calabi-type compactness criteria for
complete Riemannian manifolds that allow the presence of negative amounts of
Ricci curvature. These, in turn, can be rephrased as new conditions for the
positivity, for the existence of a first zero and for the
nonoscillatory-oscillatory behaviour of a solution of ,
subjected to the initial condition , . A unified approach for
this ODE, based on the notion of critical curve, is presented. With the aid of
suitable examples, we show that our new criteria are sharp and, even for , in borderline cases they improve on previous works of Calabi, Hille-Nehari
and Moore.Comment: 20 pages, submitte
On the role of entanglement in quantum computational speed-up
For any quantum algorithm operating on pure states we prove that the presence
of multi-partite entanglement, with a number of parties that increases
unboundedly with input size, is necessary if the quantum algorithm is to offer
an exponential speed-up over classical computation. Furthermore we prove that
the algorithm can be classically efficiently simulated to within a prescribed
tolerance \eta even if a suitably small amount of global entanglement
(depending on \eta) is present. We explicitly identify the occurrence of
increasing multi-partite entanglement in Shor's algorithm. Our results do not
apply to quantum algorithms operating on mixed states in general and we discuss
the suggestion that an exponential computational speed-up might be possible
with mixed states in the total absence of entanglement. Finally, despite the
essential role of entanglement for pure state algorithms, we argue that it is
nevertheless misleading to view entanglement as a key resource for quantum
computational power.Comment: Main proofs simplified. A few further explanatory remarks added. 22
pages, plain late
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