8,133 research outputs found
Do Goedel's incompleteness theorems set absolute limits on the ability of the brain to express and communicate mental concepts verifiably?
Classical interpretations of Goedel's formal reasoning imply that the truth
of some arithmetical propositions of any formal mathematical language, under
any interpretation, is essentially unverifiable. However, a language of
general, scientific, discourse cannot allow its mathematical propositions to be
interpreted ambiguously. Such a language must, therefore, define mathematical
truth verifiably. We consider a constructive interpretation of classical,
Tarskian, truth, and of Goedel's reasoning, under which any formal system of
Peano Arithmetic is verifiably complete. We show how some paradoxical concepts
of Quantum mechanics can be expressed, and interpreted, naturally under a
constructive definition of mathematical truth.Comment: 73 pages; this is an updated version of the NQ essay; an HTML version
is available at http://alixcomsi.com/Do_Goedel_incompleteness_theorems.ht
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
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
Oscillation and the mean ergodic theorem for uniformly convex Banach spaces
Let B be a p-uniformly convex Banach space, with p >= 2. Let T be a linear
operator on B, and let A_n x denote the ergodic average (1 / n) sum_{i< n} T^n
x. We prove the following variational inequality in the case where T is power
bounded from above and below: for any increasing sequence (t_k)_{k in N} of
natural numbers we have sum_k || A_{t_{k+1}} x - A_{t_k} x ||^p <= C || x ||^p,
where the constant C depends only on p and the modulus of uniform convexity.
For T a nonexpansive operator, we obtain a weaker bound on the number of
epsilon-fluctuations in the sequence. We clarify the relationship between
bounds on the number of epsilon-fluctuations in a sequence and bounds on the
rate of metastability, and provide lower bounds on the rate of metastability
that show that our main result is sharp
- …