Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-10T15:21:49.220Z Has data issue: false hasContentIssue false
  • English
  • Français

A POLYNOMIAL RING CONSTRUCTION FOR THE CLASSIFICATION OF DATA

Part of: Rings and algebras arising under various constructions Theory of error-correcting codes and error-detecting codes

Published online by Cambridge University Press:  13 March 2009

A. V. KELAREV ,
J. L. YEARWOOD  and
P. W. VAMPLEW
A. V. KELAREV*
Affiliation:
School of Information Technology and Mathematical Sciences, University of Ballarat, PO Box 663, Ballarat, Victoria 3353, Australia (email: a.kelarev@ballarat.edu.au)
J. L. YEARWOOD
Affiliation:
School of Information Technology and Mathematical Sciences, University of Ballarat, PO Box 663, Ballarat, Victoria 3353, Australia (email: j.yearwood@ballarat.edu.au)
P. W. VAMPLEW
Affiliation:
School of Information Technology and Mathematical Sciences, University of Ballarat, PO Box 663, Ballarat, Victoria 3353, Australia (email: p.vamplew@ballarat.edu.au)
*
✉For correspondence; e-mail: a.kelarev@ballarat.edu.au
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Drensky and Lakatos (Lecture Notes in Computer Science, 357 (Springer, Berlin, 1989), pp. 181–188) have established a convenient property of certain ideals in polynomial quotient rings, which can now be used to determine error-correcting capabilities of combined multiple classifiers following a standard approach explained in the well-known monograph by Witten and Frank (Data Mining: Practical Machine Learning Tools and Techniques (Elsevier, Amsterdam, 2005)). We strengthen and generalise the result of Drensky and Lakatos by demonstrating that the corresponding nice property remains valid in a much larger variety of constructions and applies to more general types of ideals. Examples show that our theorems do not extend to larger classes of ring constructions and cannot be simplified or generalised.

Keywords

ring constructionsgroup ringsclassification

MSC classification

Secondary: 16S36: Ordinary and skew polynomial rings and semigroup rings 94B60: Other types of codes
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 79 , Issue 2 , April 2009 , pp. 213 - 225
Copyright
Copyright © Australian Mathematical Society 2009

Footnotes

The first author was supported by Discovery grant DP0449469 from the Australian Research Council. The second author was supported by a Queen Elizabeth II Fellowship and Discovery grant DP0211866 from the Australian Research Council. The third author was supported by two research grants of the University of Ballarat.

References

[1] Alfaro, R. and Kelarev, A. V., ‘Recent results on ring constructions for error-correcting codes’, Contemp. Math. 376 (2005), 1–12.CrossRefGoogle Scholar
[2] Alfaro, R. and Kelarev, A. V., ‘On cyclic codes in incidence rings’, Studia Sci. Math. Hungarica 43(1) (2006), 69–77.Google Scholar
[3] Araújo, I. M., Kelarev, A. V. and Solomon, A., ‘An algorithm for commutative semigroup algebras which are principal ideal rings with identity’, Comm. Algebra 32(4) (2004), 1237–1254.CrossRefGoogle Scholar
[4] Bagirov, A. M., Rubinov, A. M. and Yearwood, J., ‘A global optimization approach to classification’, Optim. Eng. 3 (2002), 129–155.CrossRefGoogle Scholar
[5] Cazaran, J. and Kelarev, A. V., ‘Generators and weights of polynomial codes’, Arch. Math. (Basel) 69 (1997), 479–486.CrossRefGoogle Scholar
[6] Cazaran, J., Kelarev, A. V., Quinn, S. J. and Vertigan, D., ‘An algorithm for computing the minimum distances of extensions of BCH codes embedded in semigroup rings’, Semigroup Forum 73 (2006), 317–329.CrossRefGoogle Scholar
[7] Drensky, V. and Lakatos, P., Monomial Ideals, Group Algebras and Error-correcting Codes, Lecture Notes in Computer Science, 357 (Springer, Berlin, 1989), pp. 181–188.Google Scholar
[8] Drensky, V. S., Free Algebras and PI-Algebras (Springer, Singapore, 2000).Google Scholar
[9] Drensky, V. S., Giambruno, A. and Sehgal, S. K., Methods in Ring Theory (Marcel Dekker, New York, 1998).Google Scholar
[10] Easdown, D. and Munn, W. D., ‘Trace functions on inverse semigroup algebras’, Bull. Aust. Math. Soc. 52(3) (1995), 359–372.CrossRefGoogle Scholar
[11] East, J., ‘Cellularity of inverse semigroup algebras’, submitted.Google Scholar
[12] Hall, T. E., ‘The radical of the algebra of any finite semigroup over any field’, J. Aust. Math. Soc. Ser. A 11 (1970), 350–352.CrossRefGoogle Scholar
[13] Kelarev, A. V., ‘On classical Krull dimension of group-graded rings’, Bull. Aust. Math. Soc. 55 (1997), 255–259.CrossRefGoogle Scholar
[14] Kelarev, A. V., Ring Constructions and Applications (World Scientific, River Edge, NJ, 2002).Google Scholar
[15] Kelarev, A. V., Graph Algebras and Automata (Marcel Dekker, New York, 2003).CrossRefGoogle Scholar
[16] Kelarev, A., Kang, B. and Steane, D., Clustering Algorithms for ITS Sequence Data with Alignment Metrics, Lecture Notes in Artificial Intelligence, 4304 (Springer, Berlin, 2006), pp. 1027–1031.Google Scholar
[17] Kelarev, A. V. and Passman, D. S., ‘A description of incidence rings of group automata’, Contemp. Math. 456 (2008), 27–33.CrossRefGoogle Scholar
[18] Kelarev, A. V., Yearwood, J. L. and Mammadov, M. A., ‘A formula for multiple classifiers in data mining based on Brandt semigroups’, Semigroup Forum, to appear, DOI:10.1007/s00233-008-9098-9.CrossRefGoogle Scholar
[19] Kelarev, A. V., Yearwood, J. L. and Watters, P., ‘Rees matrix constructions for data mining’, submitted.Google Scholar
[20] Luger, G., Artificial Intelligence. Structures and Strategies for Complex Problem Solving, 5th edn (Addison-Wesley, Reading, MA, 2005).Google Scholar
[21] Ward, H. N., ‘Visible codes’, Arch. Math. (Basel) 54 (1990), 307–312.CrossRefGoogle Scholar
[22] Witten, I. H. and Frank, E., Data Mining: Practical Machine Learning Tools and Techniques (Elsevier, Amsterdam, 2005).Google Scholar
[23] Yearwood, J. L., Bagirov, A. M. and Kelarev, A. V., ‘Optimization methods and the k-committees algorithm for clustering of sequence data’, submitted.Google Scholar
[24] Yearwood, J. L. and Mammadov, M., Classification Technologies: Optimization Approaches to Short Text Categorization (Idea Group Incorporated, Hershey, PA, 2007).Google Scholar