3,861 research outputs found
A reformulation of Hilbert's tenth problem through Quantum Mechanics
Inspired by Quantum Mechanics, we reformulate Hilbert's tenth problem in the
domain of integer arithmetics into either a problem involving a set of
infinitely coupled differential equations or a problem involving a Shr\"odinger
propagator with some appropriate kernel. Either way, Mathematics and Physics
could be combined for Hilbert's tenth problem and for the notion of effective
computability
Adaptive Galerkin approximation algorithms for partial differential equations in infinite dimensions
Space-time variational formulations of infinite-dimensional Fokker-Planck (FP) and Ornstein-Uhlenbeck (OU) equations for functions on a separable Hilbert space are developed. The well-posedness of these equations in the Hilbert space of functions on , which are square-integrable with respect to a Gaussian measure on , is proved. Specifically, for the infinite-dimensional FP equation, adaptive space-time Galerkin discretizations, based on a tensorized Riesz basis, built from biorthogonal piecewise polynomial wavelet bases in time and the Hermite polynomial chaos in the Wiener-Itô decomposition of , are introduced and are shown to converge quasioptimally with respect to the nonlinear, best -term approximation benchmark. As a consequence, the proposed adaptive Galerkin solution algorithms perform quasioptimally with respect to the best -term approximation in the finite-dimensional case, in particular. All constants in our error and complexity bounds are shown to be independent of the number of "active" coordinates identified by the proposed adaptive Galerkin approximation algorithms
Compressive Space-Time Galerkin Discretizations of Parabolic Partial Differential Equations
We study linear parabolic initial-value problems in a space-time variational
formulation based on fractional calculus. This formulation uses "time
derivatives of order one half" on the bi-infinite time axis. We show that for
linear, parabolic initial-boundary value problems on , the
corresponding bilinear form admits an inf-sup condition with sparse tensor
product trial and test function spaces. We deduce optimality of compressive,
space-time Galerkin discretizations, where stability of Galerkin approximations
is implied by the well-posedness of the parabolic operator equation. The
variational setting adopted here admits more general Riesz bases than previous
work; in particular, no stability in negative order Sobolev spaces on the
spatial or temporal domains is required of the Riesz bases accommodated by the
present formulation. The trial and test spaces are based on Sobolev spaces of
equal order with respect to the temporal variable. Sparse tensor products
of multi-level decompositions of the spatial and temporal spaces in Galerkin
discretizations lead to large, non-symmetric linear systems of equations. We
prove that their condition numbers are uniformly bounded with respect to the
discretization level. In terms of the total number of degrees of freedom, the
convergence orders equal, up to logarithmic terms, those of best -term
approximations of solutions of the corresponding elliptic problems.Comment: 26 page
Computing Solution Operators of Boundary-value Problems for Some Linear Hyperbolic Systems of PDEs
We discuss possibilities of application of Numerical Analysis methods to
proving computability, in the sense of the TTE approach, of solution operators
of boundary-value problems for systems of PDEs. We prove computability of the
solution operator for a symmetric hyperbolic system with computable real
coefficients and dissipative boundary conditions, and of the Cauchy problem for
the same system (we also prove computable dependence on the coefficients) in a
cube . Such systems describe a wide variety of physical
processes (e.g. elasticity, acoustics, Maxwell equations). Moreover, many
boundary-value problems for the wave equation also can be reduced to this case,
thus we partially answer a question raised in Weihrauch and Zhong (2002).
Compared with most of other existing methods of proving computability for PDEs,
this method does not require existence of explicit solution formulas and is
thus applicable to a broader class of (systems of) equations.Comment: 31 page
Computational Complexity of Smooth Differential Equations
The computational complexity of the solutions to the ordinary
differential equation , under various assumptions
on the function has been investigated. Kawamura showed in 2010 that the
solution can be PSPACE-hard even if is assumed to be Lipschitz
continuous and polynomial-time computable. We place further requirements on the
smoothness of and obtain the following results: the solution can still
be PSPACE-hard if is assumed to be of class ; for each , the
solution can be hard for the counting hierarchy even if is of class
.Comment: 15 pages, 3 figure
Instruction sequence processing operators
Instruction sequence is a key concept in practice, but it has as yet not come
prominently into the picture in theoretical circles. This paper concerns
instruction sequences, the behaviours produced by them under execution, the
interaction between these behaviours and components of the execution
environment, and two issues relating to computability theory. Positioning
Turing's result regarding the undecidability of the halting problem as a result
about programs rather than machines, and taking instruction sequences as
programs, we analyse the autosolvability requirement that a program of a
certain kind must solve the halting problem for all programs of that kind. We
present novel results concerning this autosolvability requirement. The analysis
is streamlined by using the notion of a functional unit, which is an abstract
state-based model of a machine. In the case where the behaviours exhibited by a
component of an execution environment can be viewed as the behaviours of a
machine in its different states, the behaviours concerned are completely
determined by a functional unit. The above-mentioned analysis involves
functional units whose possible states represent the possible contents of the
tapes of Turing machines with a particular tape alphabet. We also investigate
functional units whose possible states are the natural numbers. This
investigation yields a novel computability result, viz. the existence of a
universal computable functional unit for natural numbers.Comment: 37 pages; missing equations in table 3 added; combined with
arXiv:0911.1851 [cs.PL] and arXiv:0911.5018 [cs.LO]; introduction and
concluding remarks rewritten; remarks and examples added; minor error in
proof of theorem 4 correcte
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