29,843 research outputs found
Computable Cyclic Functions
This dissertation concerns computable analysis where the idea of a representation of a set is of central importance. The key ideas introduced are those commenting on the computable relationship between two newly constructed representations, a representation of integrable cyclic functions, and the continuous cyclic function representation. Also, the computable relationship of an absolutely convergent Fourier series representation is considered. It is observed that the representation of integrable cyclic functions gives rise to a much larger set of computable functions than obtained by the continuous cyclic function representation and that integration remains a computable operation, but that basic evaluation of the function is not computable. Many other representations are acknowledged enhancing the picture of the partial order structure on the space of representations of cyclic functions. The paper can also be seen as a foundation for the study of Fourier analysis in a computable universe and concludes with an investigation into the computability of the Fourier transform
Parameterized Uniform Complexity in Numerics: from Smooth to Analytic, from NP-hard to Polytime
The synthesis of classical Computational Complexity Theory with Recursive
Analysis provides a quantitative foundation to reliable numerics. Here the
operators of maximization, integration, and solving ordinary differential
equations are known to map (even high-order differentiable) polynomial-time
computable functions to instances which are `hard' for classical complexity
classes NP, #P, and CH; but, restricted to analytic functions, map
polynomial-time computable ones to polynomial-time computable ones --
non-uniformly!
We investigate the uniform parameterized complexity of the above operators in
the setting of Weihrauch's TTE and its second-order extension due to
Kawamura&Cook (2010). That is, we explore which (both continuous and discrete,
first and second order) information and parameters on some given f is
sufficient to obtain similar data on Max(f) and int(f); and within what running
time, in terms of these parameters and the guaranteed output precision 2^(-n).
It turns out that Gevrey's hierarchy of functions climbing from analytic to
smooth corresponds to the computational complexity of maximization growing from
polytime to NP-hard. Proof techniques involve mainly the Theory of (discrete)
Computation, Hard Analysis, and Information-Based Complexity
Borders, Trade and Welfare
International economic integration yields large potential welfare effects, even in a static constant returns competitive world economy. Our method is novel. The effect of border barriers on trade flows is often inferred from gravity models. But their rather atheoretic structure precludes welfare analysis. Computable general equilibrium models are designed for tight welfare analysis, but lack econometric foundation. Our method combines these approaches. Gravity models based on Anderson's (1979) interpretation are full general equilibrium models of a special simple sort. In Anderson and van Wincoop (NBER WP 8079, 2001) we develop and estimate this structure, then calculate the comparative static effects on trade flows of border barriers. In this paper we further deploy the model to explore the comparative statics of welfare with respect to borders, to currency unions and to NAFTA. Our NAFTA exercise does a much better job of replicating the actual trade flow changes than do computable general equilibrium models. An interesting implication is that terms of trade changes are very important, even for small' countries such as Mexico.
The isometry degree of a computable copy of
When is a computable real so that , the isometry degree of a
computable copy of is defined to be the least powerful
Turing degree that computes a linear isometry of onto .
We show that this degree always exists and that when these degrees
are precisely the c.e. degrees
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