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Internet security applications of the Munn rings. (English) Zbl 1203.94142
Effective multiple clustering systems, or clusterers, have important applications in information security. The aim of the present article is to introduce a new method of designing multiple clusterers based on the Munn rings (A {\it Munn ring} over ring $R$ with sandwich matrix $P$ is the set $M(R;I,\Lambda;P)$, consisting of all $I\times\Lambda$ matrices with a finite number of nonzero entries over $R$, equipped with the usual addition and multiplication $\cdot$ defined by $A\cdot B=APB$) and describe a class of optimal clusterers which can be obtained in this construction. Theorems proven in the paper characterize an optimal class of multiple clusterers of large weight, which can be used by other researchers in the design of experiments.

94B60Other types of codes
16W50Graded associative rings and modules
68U01Computing methodologies (general aspects)
Full Text: DOI
[1] Antony, N., Coleman, C., Easdown, D., Group presentations for a class of radical rings of matrices. In: Semigroups, Algorithms, Automata and Languages, Coimbra, 2001, pp. 293--311. World Sci. Publ., River Edge (2002) · Zbl 1030.16011
[2] Bagirov, A.M., Yearwood, J.L.: A new nonsmooth optimization algorithm for minimum sum-of-squares clustering problems. Eur. J. Oper. Res. 170, 578--596 (2006) · Zbl 1085.90045 · doi:10.1016/j.ejor.2004.06.014
[3] Bagirov, A.M., Rubinov, A.M., Yearwood, J.L.: A global optimization approach to classification. Optim. Eng. 3, 129--155 (2002) · Zbl 1035.90060 · doi:10.1023/A:1020911318981
[4] Bagirov, A.M., Rubinov, A.M., Soukhoroukova, N.V., Yearwood, J.L.: Unsupervised and supervised data classification via nonsmooth and global optimization. Top 11, 1--93 (2003) · Zbl 1048.65059 · doi:10.1007/BF02578945
[5] Boslaugh, S., Watters, P.: Statistics in a Nutshell: A Desktop Quick Reference. O’Reilly & Associates, Sebastopol (2008)
[6] Cárdenas, A.A., Baras, J.S.: Evaluation of classifiers: practical considerations for security applications. In: Evaluation Methods for Machine Learning, pp. 30--35. AAAI Press, Menlo Park (2006)
[7] Cazaran, J., Kelarev, A.V.: Generators and weights of polynomial codes. Arch. Math. (Basel) 69, 479--486 (1997) · Zbl 0898.94010
[8] Cazaran, J., Kelarev, A.V., Quinn, S.J., Vertigan, D.: An algorithm for computing the minimum distances of extensions of BCH codes embedded in semigroup rings. Semigroup Forum 73, 317--329 (2006) · Zbl 1145.94023 · doi:10.1007/s00233-006-0647-9
[9] Crabb, M.J., McGregor, C.M., Munn, W.D.: A property of the complex semigroup algebra of a free monoid. J. Aust. Math. Soc. 81(1), 97--103 (2006) · Zbl 1104.16021 · doi:10.1017/S1446788700014658
[10] Easdown, D., Munn, W.D.: Trace functions on inverse semigroup algebras. Bull. Aust. Math. Soc. 52(3), 359--372 (1995) · Zbl 0845.20051 · doi:10.1017/S0004972700014854
[11] Giacinto, G., Roli, F., Didaci, L.: Fusion of multiple classifiers for intrusion detection in computer networks. Pattern Recogn. Lett. 24(12), 1795--1803 (2003) · Zbl 01977699 · doi:10.1016/S0167-8655(03)00004-7
[12] Hall, T.E.: The radical of the algebra of any finite semigroup over any field. J. Aust. Math. Soc. Ser. A 11, 350--352 (1970) · Zbl 0241.20055 · doi:10.1017/S1446788700006753
[13] Howie, J.M.: Fundamentals of Semigroup Theory. Clarendon, Oxford (1995) · Zbl 0835.20077
[14] Jackson, M., Volkov, M.V.: Undecidable problems for completely 0-simple semigroups. J. Pure Appl. Algebra 213(10), 1961--1978 (2009) · Zbl 1179.20052 · doi:10.1016/j.jpaa.2009.02.011
[15] Kang, B.H., Kelarev, A.V., Sale, A.H.J., Williams, R.N.: A new model for classifying DNA code inspired by neural networks and FSA. Lect. Notes Comput. Sci. 4303, 187--198 (2006) · doi:10.1007/11961239_17
[16] Kelarev, A.V.: Ring Constructions and Applications. World Scientific, River Edge (2002) · Zbl 0999.16036
[17] Kelarev, A.V.: Graph Algebras and Automata. Marcel Dekker, New York (2003) · Zbl 1070.68097
[18] Kelarev, A.V., Göbel, R., Rangaswamy, K.M., Schultz, P., Vinsonhaler, C.: Abelian Groups Rings and Modules. Contemporary Mathematics, vol. 273. American Mathematical Society, New York (2001) · Zbl 0960.00043
[19] Kelarev, A., Kang, B., Steane, D.: Clustering algorithms for ITS sequence data with alignment metrics. Lect. Notes Artif. Intell. 4304, 1027--1031 (2006)
[20] Kelarev, A.V., Passman, D.S.: A description of incidence rings of group automata. Contemp. Math. 456, 27--33 (2008) · Zbl 1154.16015
[21] Kelarev, A.V., Praeger, C.E.: On transitive Cayley graphs of groups and semigroups. Eur. J. Comb. 24(1), 59--72 (2003) · Zbl 1011.05027 · doi:10.1016/S0195-6698(02)00120-8
[22] Kelarev, A., Ryan, J., Yearwood, J.: Cayley graphs as classifiers for data mining: The influence of asymmetries. Discrete Math. 309(17), 5360--5369 (2009) · Zbl 1206.05050 · doi:10.1016/j.disc.2008.11.030
[23] Kelarev, A.V., Sokratova, O.V.: On congruences of automata defined by directed graphs. Theor. Comput. Sci. 301, 31--43 (2003) · Zbl 1022.68068 · doi:10.1016/S0304-3975(02)00544-3
[24] Kelarev, A.V., Yearwood, J.L., Mammadov, M.A.: A formula for multiple classifiers in data mining based on Brandt semigroups. Semigroup Forum 78(2), 293--309 (2009) · Zbl 1168.94007 · doi:10.1007/s00233-008-9098-9
[25] Kelarev, A.V., Yearwood, J.L., Vamplew, P.W.: A polynomial ring construction for classification of data. Bull. Aust. Math. Soc. 79, 213--225 (2009) · Zbl 1181.16024 · doi:10.1017/S0004972708001111
[26] Kelarev, A.V., Yearwood, J.L., Watters, P.: Rees matrix construction for clustering of data. J. Aust. Math. Soc. 87, 377--393 (2009) · Zbl 1233.16021 · doi:10.1017/S1446788709000299
[27] López-Permouth, S.R., Shum, K.P., Sanh, N.V.: Kasch modules and pV-rings. Algebra Colloq. 12(2), 219--227 (2005) · Zbl 1078.16004
[28] McCombie, S., Watters, P., Ng, A., Watson, B.: Forensic characteristics of phishing--petty theft or organized crime? In: Proc. 4th Internat. Conf. on Web Information Systems and Technologies, WEBIST, Madeira, Portugal (2008)
[29] Munn, W.D.: Some results on semigroup-graded rings. In: Semigroups, Algorithms, Automata and Languages, Coimbra, 2001, pp. 215--234. World Sci. Publ., River Edge (2002) · Zbl 1031.16024
[30] Munn, W.D.: $\mathcal{D}$ -faithful semigroup-graded rings. Proc. Edinb. Math. Soc. 45(3), 549--556 (2002) · Zbl 1062.16048 · doi:10.1017/S001309150100030X
[31] Munn, W.D.: Rings graded by inverse semigroups. In: Semigroups, Braga, 1999, pp. 136--145. World Sci. Publ., River Edge (2000) · Zbl 0982.16032
[32] Munn, W.D.: Rings graded by bisimple inverse semigroups. Proc. R. Soc. Edinb. Sect. A 130(3), 603--609 (2000) · Zbl 0961.16029
[33] Munn, W.D.: A class of band-graded rings. J. Lond. Math. Soc. 45(1), 1--16 (1992) · Zbl 0786.16023 · doi:10.1112/jlms/s2-45.1.1
[34] Munn, W.D.: Nil right ideals in inverse semigroup algebras. Semigroup Forum 44(1), 93--95 (1992) · Zbl 0748.16012 · doi:10.1007/BF02574326
[35] Munn, W.D.: The Jacobson radical of a band ring. Math. Proc. Cambr. Philos. Soc. 105(2), 277--283 (1989) · Zbl 0677.20053 · doi:10.1017/S0305004100067761
[36] Munn, W.D.: Nil ideals in inverse semigroup algebras. J. Lond. Math. Soc. 35(3), 433--438 (1987) · Zbl 0621.20039 · doi:10.1112/jlms/s2-35.3.433
[37] Munn, W.D.: The algebra of a combinatorial inverse semigroup. J. Lond. Math. Soc. 27(1), 35--38 (1983) · Zbl 0506.20036 · doi:10.1112/jlms/s2-27.1.35
[38] Pellikaan, R., Wu, X.-W., Bulygin, S.: Codes and Cryptology as Applications of Algebra, Combinatorics and Geometry. Cambridge University Press, Cambridge (2010)
[39] Reilly, N.R.: Varieties generated by completely 0-simple semigroups. J. Aust. Math. Soc. 84(3), 375--403 (2008) · Zbl 1161.20051 · doi:10.1017/S1446788708000311
[40] Silva, P.V.: A note on primeness of semigroup rings. Proc. R. Soc. Edinb. Sect. A 120(3--4), 191--197 (1992) · Zbl 0780.20043
[41] Witten, I.H., Frank, E.: Data Mining: Practical Machine Learning Tools and Techniques. Elsevier/Morgan Kaufman, Amsterdam (2005) · Zbl 1076.68555
[42] Wolpert, D.H.: The lack of a priori distinctions between learning algorithms. Neural Computation 8(7), 1341--1390 (1996) · Zbl 05475233 · doi:10.1162/neco.1996.8.7.1341
[43] Yearwood, J.L., Bagirov, A.M., Kelarev, A.V.: Optimization methods and the k-committees algorithm for clustering of sequence data. J. Appl. Comput. Math. 8(1), 92--101 (2009) · Zbl 1184.49027
[44] Yearwood, J.L., Kang, B.H., Kelarev, A.V.: Experimental investigation of classification algorithms for ITS dataset. In: PKAW 2008, Pacific Rim Knowledge Acquisition Workshop, part of PRICAI-08, Tenth Pacific Rim Internat. Conf. Artificial Intelligence, pp. 262--272
[45] Yearwood, J.L., Mammadov, M.A.: Classification Technologies: Optimization Approaches to Short Text Categorization. Idea Group Inc., Hershey (2007)