Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Computing with Polynomial Ordinary Differential Equations   In 1941, Claude Shannon introduced the General Purpose Analog Computer(GPAC)
as a mathematical model of Differential Analysers, that is to say as a model of
continuous-time analog (mechanical, and later one electronic) machines of that
time.
  Following Shannon's arguments, functions generated by GPACs must be
differentially algebraic. As it is known that some computable functions like
Euler's $\Gamma(x)=\int_{0}^{\infty}t^{x-1}e^{-t}dt$ or Riemann's Zeta function
$\zeta(x)=\sum_{k=0}^\infty \frac1{k^x}$ are not differentially algebraic, this
argument has been often used to demonstrate in the past that the GPAC is less
powerful than digital computation.
  It was proved in JOC2007, that if a more modern notion of computation is
considered, i.e. in particular if computability is not restricted to real-time
generation of functions, the GPAC is actually equivalent to Turing machines.
  Our purpose is first to discuss the robustness of the notion of computation
involved in JOC2007, by establishing that natural variants of the notion of
computation from this paper leads to the same computability result.
  Second, to go considerations about (time) complexity, we explore several
natural variants for measuring time/space complexity of a computation.
  Rather surprisingly, whereas defining a robust time complexity for general
continuous time systems is a well known open problem, we prove that all
variants are actually equivalent even at the complexity level. As a
consequence, it seems that a robust and well defined notion of time complexity
exists for the GPAC, or equivalently for computations by polynomial ordinary
differential equations.
  Another side effect of our proof is also that we show in some way that
polynomial ordinary differential equations can be used as a kind of programming
model, and that there is a rather nice and robust notion of ODE programming.

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The categories of this paper are: hep-ph
28
The categories of this paper are: alg-geom
30
The categories of this paper are: q-alg
20
The categories of this paper are: acc-phys
37
The categories of this paper are: supr-con
5
The categories of this paper are: gr-qc
11
The categories of this paper are: hep-lat
36
The categories of this paper are: cmp-lg
16
The categories of this paper are: stat
32
The categories of this paper are: atom-ph
17
The categories of this paper are: q-fin
25
The categories of this paper are: patt-sol
27
The categories of this paper are: comp-gas
9
The categories of this paper are: nlin
21
The categories of this paper are: adap-org
35
The categories of this paper are: mtrl-th