Skip to main content Accessibility help
×
Home
Hostname: page-component-cf9d5c678-m9wwp Total loading time: 0.175 Render date: 2021-07-27T04:00:08.556Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true, "newUsageEvents": true }

On Tensor-Factorisation Problems,I: The Combinatorial Problem

Published online by Cambridge University Press:  01 February 2010

Peter M. Neumann
Affiliation:
The Queen's College, Oxford 0X1 4AW, United Kingdom, peter.neumann@queens.ox.ac.uk
Cheryl E. Praeger
Affiliation:
School of Mathematics and Statistics, University of Western Australia, Crawley, WA 6009, Australia, praeger@maths.uwa.edu.au, http://www.maths.uwa.edu.au/~praeger

Abstract

A k-multiset is an unordered k-tuple, perhaps with repetitions. If x is an r-multiset {x1, …, xr} and y is an s-multiset {y1, …, ys} with elements from an abelian group A the tensor product x ⊗ y is defined as the rs-multiset {xi yj | 1 ≤ i ≤ r, 1 ≤ j ≤ s}. The main focus of this paper is a polynomial-time algorithm to discover whether a given rs-multiset from A can be factorised. The algorithm is not guaranteed to succeed, but there is an acceptably small upper bound for the probability of failure. The paper also contains a description of the context of this factorisation problem, and the beginnings of an attack on the following division-problem: is a given rs-multiset divisible by a given r-multiset, and if so, how can division be achieved in polynomially bounded time?

Type
Research Article
Copyright
Copyright © London Mathematical Society 2004

References

1. Aho, Alfred V., Hopcroft, John E., Ullman, Jeffrey D., Data structures and algorithms (Addison-Wesley, Reading, MA, 1983).Google Scholar
2. Arratia, Richard, Barbour, A. D. and Tavaré, Simon, ‘On random polynomials over finite fields’, Math. Proc. Cambridge Philos. Soc. 114 (1993) 347368.CrossRefGoogle Scholar
3. Greenhill, Catherine, ‘From multisets to matrix groups: some algorithms related to the exterior square’ DPhil thesis, Oxford, (1996).Google Scholar
4. Greenhill, Catherine, ‘An algorithm for recognising the exterior square of a matrix’, Linear and Multilinear Algebra 46 (1999) 213244.CrossRefGoogle Scholar
5. Greenhill, Catherine, ‘An algorithm for recognising the exterior square of a multiset’, LMS J. Comput. Math. 3 (2000) 96116; http://www.lms.ac.Uk/jcm/3/lmsl999-021.CrossRefGoogle Scholar
6. Huang, Jia Lun, ‘The implementation of a factorisation algorithm for combinatorial tensor products’, MSc thesis, Oxford, August (2003).Google Scholar
7. Jacobson, Nathan, Lectures in abstract algebra, vols I–III (Van Nostrand, New Jersey, 1953).CrossRefGoogle Scholar
8. Knuth, Donald E., The art of computer programming, vol 3: ‘Sorting and searching’ (Addison-Wesley, Reading, MA, 1973).Google Scholar
9. Mignotte, M. and Nicolas, J.-L, ‘Statistique sur Fq [X]’, Ann. Inst. H. Poincaré Sect. B (N.S.) 19 (1983) 113121.Google Scholar
You have Access

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

On Tensor-Factorisation Problems,I: The Combinatorial Problem
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

On Tensor-Factorisation Problems,I: The Combinatorial Problem
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

On Tensor-Factorisation Problems,I: The Combinatorial Problem
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *