12,568 research outputs found
On the resolution power of Fourier extensions for oscillatory functions
Functions that are smooth but non-periodic on a certain interval possess
Fourier series that lack uniform convergence and suffer from the Gibbs
phenomenon. However, they can be represented accurately by a Fourier series
that is periodic on a larger interval. This is commonly called a Fourier
extension. When constructed in a particular manner, Fourier extensions share
many of the same features of a standard Fourier series. In particular, one can
compute Fourier extensions which converge spectrally fast whenever the function
is smooth, and exponentially fast if the function is analytic, much the same as
the Fourier series of a smooth/analytic and periodic function.
With this in mind, the purpose of this paper is to describe, analyze and
explain the observation that Fourier extensions, much like classical Fourier
series, also have excellent resolution properties for representing oscillatory
functions. The resolution power, or required number of degrees of freedom per
wavelength, depends on a user-controlled parameter and, as we show, it varies
between 2 and \pi. The former value is optimal and is achieved by classical
Fourier series for periodic functions, for example. The latter value is the
resolution power of algebraic polynomial approximations. Thus, Fourier
extensions with an appropriate choice of parameter are eminently suitable for
problems with moderate to high degrees of oscillation.Comment: Revised versio
Fast Algorithms for the computation of Fourier Extensions of arbitrary length
Fourier series of smooth, non-periodic functions on are known to
exhibit the Gibbs phenomenon, and exhibit overall slow convergence. One way of
overcoming these problems is by using a Fourier series on a larger domain, say
with , a technique called Fourier extension or Fourier
continuation. When constructed as the discrete least squares minimizer in
equidistant points, the Fourier extension has been shown shown to converge
geometrically in the truncation parameter . A fast algorithm has been described to compute Fourier extensions for the case
where , compared to for solving the dense discrete
least squares problem. We present two algorithms for
the computation of these approximations for the case of general , made
possible by exploiting the connection between Fourier extensions and Prolate
Spheroidal Wave theory. The first algorithm is based on the explicit
computation of so-called periodic discrete prolate spheroidal sequences, while
the second algorithm is purely algebraic and only implicitly based on the
theory
The Role of Correlated Noise in Quantum Computing
This paper aims to give an overview of the current state of fault-tolerant
quantum computing, by surveying a number of results in the field. We show that
thresholds can be obtained for a simple noise model as first proved in [AB97,
Kit97, KLZ98], by presenting a proof for statistically independent noise,
following the presentation of Aliferis, Gottesman and Preskill [AGP06]. We also
present a result by Terhal and Burkard [TB05] and later improved upon by
Aliferis, Gottesman and Preskill [AGP06] that shows a threshold can still be
obtained for local non-Markovian noise, where we allow the noise to be weakly
correlated in space and time. We then turn to negative results, presenting work
by Ben-Aroya and Ta-Shma [BT11] who showed conditional errors cannot be
perfectly corrected. We end our survey by briefly mentioning some more
speculative objections, as put forth by Kalai [Kal08, Kal09, Kal11]
Expressivism and Arguing about Art
Peter Kivy claims that expressivists in aesthetics cannot explain why we argue about art. The situation would be different in the case of morals. Moral attitudes lead to action, and since actions affect people, we have a strong incentive to change people’s moral attitudes. This can explain why we argue about morals, even if moral language is expressive of our feelings. However, judgements about what is beautiful and elegant need not significantly affect our lives. So why be concerned with other people’s feelings about art? Kivy thinks the best explanation of our tendency to argue about art is that we implicitly believe in objective facts about aesthetics. This would count against expressivism. I argue two things: that there is no good reason to think that we don’t care about preferences and emotions unless they have significant practical consequences and that the truth of expressivism about aesthetic language is compatible with beliefs about objective aesthetic facts
Koka: Programming with Row Polymorphic Effect Types
We propose a programming model where effects are treated in a disciplined
way, and where the potential side-effects of a function are apparent in its
type signature. The type and effect of expressions can also be inferred
automatically, and we describe a polymorphic type inference system based on
Hindley-Milner style inference. A novel feature is that we support polymorphic
effects through row-polymorphism using duplicate labels. Moreover, we show that
our effects are not just syntactic labels but have a deep semantic connection
to the program. For example, if an expression can be typed without an exn
effect, then it will never throw an unhandled exception. Similar to Haskell's
`runST` we show how we can safely encapsulate stateful operations. Through the
state effect, we can also safely combine state with let-polymorphism without
needing either imperative type variables or a syntactic value restriction.
Finally, our system is implemented fully in a new language called Koka and has
been used successfully on various small to medium-sized sample programs ranging
from a Markdown processor to a tier-splitted chat application. You can try out
Koka live at www.rise4fun.com/koka/tutorial.Comment: In Proceedings MSFP 2014, arXiv:1406.153
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