4,378 research outputs found
A Universal Ordinary Differential Equation
An astonishing fact was established by Lee A. Rubel (1981): there exists a
fixed non-trivial fourth-order polynomial differential algebraic equation (DAE)
such that for any positive continuous function on the reals, and for
any positive continuous function , it has a
solution with for all . Lee A. Rubel
provided an explicit example of such a polynomial DAE. Other examples of
universal DAE have later been proposed by other authors. However, Rubel's DAE
\emph{never} has a unique solution, even with a finite number of conditions of
the form .
The question whether one can require the solution that approximates
to be the unique solution for a given initial data is a well known open problem
[Rubel 1981, page 2], [Boshernitzan 1986, Conjecture 6.2]. In this article, we
solve it and show that Rubel's statement holds for polynomial ordinary
differential equations (ODEs), and since polynomial ODEs have a unique solution
given an initial data, this positively answers Rubel's open problem. More
precisely, we show that there exists a \textbf{fixed} polynomial ODE such that
for any and there exists some initial condition that
yields a solution that is -close to at all times.
In particular, the solution to the ODE is necessarily analytic, and we show
that the initial condition is computable from the target function and error
function
A Survey on Continuous Time Computations
We provide an overview of theories of continuous time computation. These
theories allow us to understand both the hardness of questions related to
continuous time dynamical systems and the computational power of continuous
time analog models. We survey the existing models, summarizing results, and
point to relevant references in the literature
Exhaustible sets in higher-type computation
We say that a set is exhaustible if it admits algorithmic universal
quantification for continuous predicates in finite time, and searchable if
there is an algorithm that, given any continuous predicate, either selects an
element for which the predicate holds or else tells there is no example. The
Cantor space of infinite sequences of binary digits is known to be searchable.
Searchable sets are exhaustible, and we show that the converse also holds for
sets of hereditarily total elements in the hierarchy of continuous functionals;
moreover, a selection functional can be constructed uniformly from a
quantification functional. We prove that searchable sets are closed under
intersections with decidable sets, and under the formation of computable images
and of finite and countably infinite products. This is related to the fact,
established here, that exhaustible sets are topologically compact. We obtain a
complete description of exhaustible total sets by developing a computational
version of a topological Arzela--Ascoli type characterization of compact
subsets of function spaces. We also show that, in the non-empty case, they are
precisely the computable images of the Cantor space. The emphasis of this paper
is on the theory of exhaustible and searchable sets, but we also briefly sketch
applications
Quantum Algorithm for Hilbert's Tenth Problem
We explore in the framework of Quantum Computation the notion of {\em
Computability}, which holds a central position in Mathematics and Theoretical
Computer Science. A quantum algorithm for Hilbert's tenth problem, which is
equivalent to the Turing halting problem and is known to be mathematically
noncomputable, is proposed where quantum continuous variables and quantum
adiabatic evolution are employed. If this algorithm could be physically
implemented, as much as it is valid in principle--that is, if certain
hamiltonian and its ground state can be physically constructed according to the
proposal--quantum computability would surpass classical computability as
delimited by the Church-Turing thesis. It is thus argued that computability,
and with it the limits of Mathematics, ought to be determined not solely by
Mathematics itself but also by Physical Principles
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
Making big steps in trajectories
We consider the solution of initial value problems within the context of
hybrid systems and emphasise the use of high precision approximations (in
software for exact real arithmetic). We propose a novel algorithm for the
computation of trajectories up to the area where discontinuous jumps appear,
applicable for holomorphic flow functions. Examples with a prototypical
implementation illustrate that the algorithm might provide results with higher
precision than well-known ODE solvers at a similar computation time
The Phillips Machine, The Analogue Computing Traditoin in Economics and Computability
In this paper I try to argue for the desirability of analog computation in economics from a variety of perspectives, using the example of the Phillips Machine. Ultimately, a case is made for the underpinning of both analog and digital computing theory in constructive mathematics. Some conceptual confusion in the meaning of analog computing and its non-reliance on the theory of numerical analysis is also discussed. Digital computing has its mathematical foundations in (classical) recursion theory and constructive mathematics. The implicit, working, assumption of those who practice the noble art of analog computing may well be that the mathematical foundations of their subject is as sound as the foundations of the real analysis. That, in turn, implies a reliance on the soundness of set theory plus the axiom of choice. This is, surely, seriously disturbing from a computation point of view. Therefore, in this paper, I seek to locate a foundation for analog computing in exhibiting some tentative dualities with results that are analogous to those that are standard in computability theory. The main question, from the point of view of economics, is whether the Phillips Machine, as an analog computer, has universal computing properties. The conjectured answer is in the negative.Phillips Machine, Analogue Computation, Digital Computation, Computability, General Purpose Analogue Computer
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
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