iosr-Jm   Volume-8 ~ Issue-6

Abstract: In this communication, we propose useful fuzzy entropy measure and study its particular cases. Some coding theorems have been proved for decipherable codes using Holder's inequalities. Some known results are the particular cases of our proposed measure. Keywords: Fuzzy set, Fuzzy entropy, Holder's inequalities, Decipherable codes. AMS subject Classification: 94A17, 94A24.

[1] J. Aczel, On Shannon's inequality, optimal coding and characterization of Shannon's and Renyi' entropies, Institute Nazionale di Alta Matematica, Symposia Mathematica XV (1975) 153-179.

[2] Belis and Guiasu, S., Fuzzy sets with inner interdependence, Information Sciences, (1968) 79(3-4), pp. 315-338.

[3] L.L. Campbell, A coding theorem and Renyi's entropy, Inf. Control 8 (1965) 423-429.

[4] A. Feinstin, Foundations of information theory, McGraw-Hill, New York (1958).

[5] S. Guiasu, C.F. Picard, Borne Inferieure de la longueur utile de certain codes, C.R. Acad. Sci. Paris 273 (1971) 248-251

[6] Gurdial, F. Pessoa, On useful information of order JCISS 2 (4) (1977) 150-162.

[7] G.H. Hardy, J.E. Little wood, G. Polya, Inequalities, University Press, Cambridge, (1934).

[8] Kapur, J. N., Measures of fuzzy Information, Mathematical Sciences Trust Society, New Delhi (1997).

[9] Klir,G.J.and B. Parviz, A note on the measure of discord, Proc. Eighth conference on Artificial Intelligence, San Mateo, California, (1992) pp.138-141.

[10] Kraft, D. H., Fuzzy sets and generalized retrieval systems, Intern. J. of Man – Machine studies, (1949) 19(1), pp. 45-56.

Abstract: In this paper, we prove that the Dumbbell graph, Star graph, Cartesian product n P C 2 and K1 + Cn are even-even graceful. The even-even graceful labeling of a graph G with q edges means that there is an injection f: E (G) → {2, 4,…, 2q} so that induced map f *: V (G) →{0, 2,…,(2k-2)} defined by f *(x) ≡ Σf(x, y) (mod 2k) where k = max {p, q} makes all distinct and even. Keywords: Even-even graceful labeling, Dumbbell graph, Star graph and wheel graph.

[1]. A.Rosa, "On certain valuations of the vertices of a graph", Theory of Graphs (Internat. Symposium, Rome, July 1966), Gordon and
Breach, N. Y. and Dunod Paris (1967) 349-355.
[2]. R. L. Graham and N. J. A. Sloane, "On additive bases and harmonious graphs", SIAM
[3]. J. Alg. Discrete Math., 1 (1980) 382-404. [1].
[4]. J. Bondy and U. Murty, Graph Theory with Applications, North- Holland, New York (1979).Christian Barrientos, "Odd-Graceful
Labelings of Trees of Diameter 5", AKCE J.Graphs. Combin., 6, No. 2 (2009) .
[5]. A. Solairaju and K. Chithra, Edge-odd graceful labeling of some graphs, Proceedings of the ICMCS, 1 (2008) 101-107.
[6]. R. Sridevi, S. Navaneethakrishnan, A. Nagarajan and K. Nagarajan," Odd-Even graceful graphs", J. Appl. Math. & Informatics Vol.
30(2012), No. 5 - 6, pp. 913 – 923.
[7]. J.A.Gallian, A dynamic survey of graph labeling, Electronic, J.Comb,
[8]. Sin-Min Lee,Kuo-Jue Chen,Yung-Chin Wang "On the Edge graceful spectra of cycles with one chord and dumbbell graphs",
Congressus Numerantium 170 (2004), pp. 171-183.

Abstract: In this article we introduce and study the notions of ( , ) s v r  -lacunary strongly summable, ( , ) s v r  - Cesàro strongly summable, ( , ) s v r  - statistically convergent and ( , ) s v r  -lacunary statistically convergent sequence of fuzzy numbers. Consequently we construct the spaces   ( , ) F s v r L  ,   1 ( , ) F s v r   ,   ( , ) F s v r S  and   ( , ) F s v r S  respectively and investigate the relationship among these spaces. Further we show that   ( , ) F s v r L  and   1 ( , ) F s v r   are complete metric spaces.

Keywords: Sequence of fuzzy numbers; Difference sequence; lacunary strongly summable; Cesàro strongly summable; statistically convergent; lacunary statistically convergent; Completeness. AMS Classification No: 40A05; 40D25.

[1] P. Diamond and P. Kloeden, Metric spaces of fuzzy sets, Fuzzy Sets and Systems 35 (1990), 241-249.

[2] M. Et and R. Colak, On generalized difference sequence spaces, Soochow Jour. Math. 21 (1995), 377-386.

[3] H. Fast, Sur la convergence statistique, Colloq. Math. (1951), 241-244.

[4] J.A. Fridy and C. Orhan, Statistical limit superior and limit inferior, Proc. Amer. Math. Soc. 125(12) (1997), 3625-3631.

[5] A.R. Freedman, J.J. Sember and M. Raphael, Some Cesàro-type summability spaces, Proc. Lond. Math. Soc. 37(3) (1978), 508-520.

[6] H. Kizmaz, On certain sequence spaces, Canad. Math. Bull. 24(2) (1981), 168-176.

[7] M. Matloka, Sequences of fuzzy numbers, BUSEFAL 28 (1986), 28-37.

[8] I.J. Maddox, A Tauberian condition for statistical convergence, Math. Proc. Camb. Phil. Soc. 106 (1989), 277-280.

[9] S. Nanda, On sequence of fuzzy numbers, Fuzzy Sets and Systems 33 (1989), 28-37.

[10] F. Nuray and E. Savaş, Statistical convergence of sequences of fuzzy numbers, Math. Slovaca 45(3) (1995), 269-273.

Abstract: A closed queuing network that consists of M nodes was studied, and in contrast to an open network, there is a constant population of K customers in the network with no external source or sink. Each node i is a first in first out queue, where the service time is drawn independently. That is, the service time is exponentially distributed with service rates i  i   n and a customer departing from queue i chooses j next with probability i j q , . The Gordon and Newel theorem with the arrival properties of a closed network were used to analyze the Mean values for the number of customers, sojourn time, waiting times and the algorithm for solving the mean value for closed networks were then derived with series of application considered.

[1]. Agboola, S.O. (2007): On the Analysis of M/G/1 Queues. Unpublished M.Sc. (Mathematics) Thesis submitted to Department of
Mathematics, University of Ibadan.
[2]. Agboola, S.O. (2011): The Analysis of Markov Inter-arrival Queues Model with K – Server under Various Service Points.
Unpublished M.Sc. (Statistics) Thesis submitted to Department of Mathematics, Obafemi Awolowo University, Ile-Ife.
[3]. Balbo, G.; Bruell, S. C.; and Schwetman, H. D. (1977): "Customer classes and closed network models – a solution techniques," in
Proc. IFIP Congress 77, North – Holland Publ. Co., Amsterdam, The Netherland.