| EPSRC Reference: |
GR/M85906/01 |
| Title: |
ROUND-OFF ERRORS AND P-ADIC NUMBERS |
| Principal Investigator: |
Vivaldi, Professor F |
| Other Investigators: |
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| Department: |
Sch of Mathematical Sciences |
| Organisation: |
Queen Mary University of London |
| Scheme: |
Standard Research (Pre-FEC) |
| Starts: |
23 May 2000 |
Ends: |
22 September 2000 |
Value (£): |
3,600
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| EPSRC Research Topic Classifications: |
| Algebra & Geometry |
Logic & Combinatorics |
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| EPSRC Industrial Sector Classifications: |
| No relevance to Underpinning Sectors |
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Summary on Grant Application Form |
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Igor Vladimirov and I wish to collaborate on an investigation on the relation between round-off errors in dynamical systems an algebraic number theory, involving both arithmetical and probabilistic aspects. The study of round-off errors in spatial discretisations of dynamical systems has attracted considerable interest in recent years, due to its significance in theory and computations. A much-studied problem is that of uniform spatial discretisations of linear planar rotations (harmonic oscillator), where the exact dynamics is regular and well-understood, and where round-off errors can be studied in isolation from other dynamical phenomena [2, 6, 14, 15, 8, 11]. Among the most intriguing and complex objects associated to a discrete dynamical system are the so-called period functions describing the period of orbits (typically maximal, or average), as a function of the system 'size' N [13, 7, 1, 14]. The latter is invariably a large parameter, which for round-off systems may be taken to be the inverse of the discretisation length. These functions of a class of strongly chaotic system (toral automorphisms), have been related to some classical number-theoretic problems, centred around the so-called Artin's conjecture [12, 7, 5]. In particular, their average order may be inferred assuming the validity of the so-called generalised Riemann hypothesis. Recently [4], a connection has been found between round-off errors in planar rotations and algebraic number theory. Specifically, the uniform spatial discretisation of the linear mapping F: (x,y) (r)( ax-y,x) a=q/pn (1)where p is a prime number, can be represented as a mapping on the p-adic image of the algebraic number ring generated by the eigenvalues of F. Such mapping can be extended to a chaotic mapping of the p-adic integers, consisting of multiplication by a unit, followed by a Bernoulli shift. The existence of an embedding with complete symbolic dynamics affords an explicit construction and classification of all periodic orbits, and opens the possibility of a number-theoretic interpretation of the associated period functions. This is the problem Vladimirov and I would like to address in a joint research project, of which his visit to London would constitute the first step. We intend to proceed as follows:* Work towards a proof that all orbits of F are periodic, whence bounded. A probabilistic proof (valid for a set of full density) is the first target: this is equivalent to a probabilistic upper bound for the period function.* Work towards a formula for the average order of the period function. The first target is to isolate a minimal set of number-theoretic assumptions necessary to derive it.Igor Vladimirov is a young Russian researchers, who has recently produced remarkable work on the probabilistic aspects of round-off errors in dynamical systems, including the first central limit theorem for round-off errors in linear systems [15].
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