648 research outputs found
The prospects for mathematical logic in the twenty-first century
The four authors present their speculations about the future developments of
mathematical logic in the twenty-first century. The areas of recursion theory,
proof theory and logic for computer science, model theory, and set theory are
discussed independently.Comment: Association for Symbolic Logi
Dimension in the realm of transseries
Let be the differential field of transseries. We establish some
basic properties of the dimension of a definable subset of ,
also in relation to its codimension in the ambient space . The
case of dimension is of special interest, and can be characterized both in
topological terms (discreteness) and in terms of the
Herwig-Hrushovski-Macpherson notion of co-analyzability. The proofs use results
by the authors from "Asymptotic Differential Algebra and Model Theory of
Transseries", the axiomatic framework for "dimension" in [L. van den Dries,
"Dimension of definable sets, algebraic boundedness and Henselian fields", Ann.
Pure Appl. Logic 45 (1989), no. 2, 189-209], and facts about co-analyzability
from [B. Herwig, E. Hrushovski, D. Macpherson, "Interpretable groups, stably
embedded sets, and Vaughtian pairs", J. London Math. Soc. (2003) 68, no. 1,
1-11].Comment: 16 pp; version 2, taking into account comments by the refere
Analog computers and recursive functions over the reals
This paper revisits one of the rst models of analog computation, the
General Purpose Analog Computer (GPAC). In particular, we restrict our
attention to the improved model presented in [11] and we show that it
can be further re ned. With this we prove the following: (i) the previous
model can be simpli ed; (ii) it admits extensions having close connec-
tions with the class of smooth continuous time dynamical systems. As a
consequence, we conclude that some of these extensions achieve Turing
universality. Finally, it is shown that if we introduce a new notion of
computability for the GPAC, based on ideas from computable analysis,
then one can compute transcendentally transcendental functions such as
the Gamma function or Riemann's Zeta function
Selected non-holonomic functions in lattice statistical mechanics and enumerative combinatorics
We recall that the full susceptibility series of the Ising model, modulo
powers of the prime 2, reduce to algebraic functions. We also recall the
non-linear polynomial differential equation obtained by Tutte for the
generating function of the q-coloured rooted triangulations by vertices, which
is known to have algebraic solutions for all the numbers of the form , the holonomic status of the q= 4 being unclear. We focus on the
analysis of the q= 4 case, showing that the corresponding series is quite
certainly non-holonomic. Along the line of a previous work on the
susceptibility of the Ising model, we consider this q=4 series modulo the first
eight primes 2, 3, ... 19, and show that this (probably non-holonomic) function
reduces, modulo these primes, to algebraic functions. We conjecture that this
probably non-holonomic function reduces to algebraic functions modulo (almost)
every prime, or power of prime numbers. This raises the question to see whether
such remarkable non-holonomic functions can be seen as ratio of diagonals of
rational functions, or algebraic, functions of diagonals of rational functions.Comment: 27 page
The extended analog computer and functions computable in a digital sense
In this paper we compare the computational power of the Extended Analog Computer (EAC) with partial recursive functions. We first give a survey of some part of computational theory in discrete and in real space. In the last section we show that the EAC can generate any partial recursive function defined over N. Moreover we conclude that the classical halting problem for partial recursive functions is an equivalent of testing by EAC if sets are empty or not
Analogue of Newton-Puiseux series for non-holonomic D-modules and factoring
We introduce a concept of a fractional-derivatives series and prove that any
linear partial differential equation in two independent variables has a
fractional-derivatives series solution with coefficients from a differentially
closed field of zero characteristic. The obtained results are extended from a
single equation to -modules having infinite-dimensional space of solutions
(i. e. non-holonomic -modules). As applications we design algorithms for
treating first-order factors of a linear partial differential operator, in
particular for finding all (right or left) first-order factors
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