iosr-Jm   Volume-5 ~ Issue-4

Abstract: The British colonial masters in Nigeria did not encourage the teaching of science in the schools. At best science was taught as Nature Study and General Science. There was also shortage of middle-level manpower besides the gross shortage of qualified science teachers. To combat the acute shortage of qualified science teachers in Nigeria the Special Entry Preparatory programme (SEPP) was instituted in the University of Lagos to absorb and upgrade the unqualified science teachers to the level of graduate science teachers with education background. The SEPP students were brought up to the advance level of General Certificate in Education (GCE A/L) which was the required qualification for direct admission into the university. Mathematics was one of the science-education courses taught through distance learning for a B.Sc degree. Using the Mann-Whitney test the performance of successful SEPP students was compared with those of "direct" students in the six courses offered. The two groups performed at the same level in four courses while outdoing each other in each of the remaining two courses.

Key Words: Distance Education, Special Entry Preparatory Programme (SEPP), Direct Entry.

1] AFFENDY, L.S., Paris, I.H.M., Mustapha, N., Nasir Sulaiman, Md. And Muda, Z. 2010. Ranking of influence Factors in predicting
students' performance. Information Technology Journal, 9:832-837.
[2] ALDOUS, C., Leeder, S., Price, J., Sefton, A. and Tuebner, J. 1997. A selection of tests for Australian graduate-entry medical
schools. Medical Journal of Australia. 166:247-250.
[3] BLACKMAN, Ian and Darmawan, I. Gusti Ngurah. 2004. Graduate-Entry Medical Student variables that Predict Academic and
Clinical Achievement: International Education Journal, Vol.4 (No.4) 2004. Educational Research Conference 2003. Special Issue
http://iej.cjb; http://iej.net.
[4] CHANSARKAR, B.A., and Michaeloudis, A. 2004. Student profiles and factors affecting performance. International Journal of
Mathematics, Education Science and Technology, Volume 32(1) 97-104.
[5] CRAIG, P.L., Gordon, J.J., Clark, R.M. and Langendy, K.V. 2004. Medical Education, Volume 38. Issue 11, pp. 1164-1168.
www.mededuc.com
[6] GOLDING, P. and Donalson, O. 2006. Predicting academic performance. Proceedings of the 36th ASEE/IEEE Frontiers in
Education Conference TID-21, October 28-31, San Diego, CA, pp. 1-6.
[7] GRACIA, L. and Jenkins, E. 2003. A quantitative exploration of students' performance on an undergraduate accounting programme
of study. Accounting Education, 12(1) 15-32.
[8] HUFF, K. and Fang, D. 1999. When are students most at risk of encountering academic difficulty? A study of the 1992 matriculants
to US Medical Schools. Academic Medicine 74, 454-460.
[9] JAMES, D. and Chilvers, C. 2001. Academic and non-academic predictors of success on the Nottingham undergraduate medical
course. 1970-1995. Medical Education 35:1056-1064.
[10] JOSSEY, Bass. 1977. Selective admissions in higher education. Carnegie Foundation for the Advancement of Teaching.

Abstract: We know that the smallest positive integer f such that a f  1 mod m is called the exponent of 'a' modulo m and is denoted by expma. We say that 'a' is a semi-primitive root mod m if expma = 2  (m) . In this paper we discuss the properties of the semi primitive roots and examine for which prime 2 is a semi-primitive root. If S is the sum of semi-primitive roots less than p then we proved that p p S )mod 2 1 (    .Also we proved that if 'a' is a semi primitive root then 'a' is a quadratic residue, converse need not be true. It was established that whenever a is a semi-primitive root mod p where p is of the form 4n+3 then –a is a semi primitive root and if p = 4n+1 then expm (–a) = 4 p 1 .We establish that 2 is semi-primitive root for mod p whenever 'p' is of the form 2q+1where 'q' is an odd prime of the form 4n+3 and if 4n+1,8n+3 are primes then –2 is a semi-primitive root mod 8n+3 by using Gauss Lemma [1].

[1] Apostol Tom M: Introduction to Analytic Number Theory, Springer International Student Edition.
[2] Leveque W.J.: Topics in Number Theory, Vol. 1 Reading mass (1956).
[3] Vegh E: - primitive roots. Portugaliae Mathematica 40(1981) 297-303.
[4] An Introduction to Theory of Numbers by Ivan Niven, Herbert S. Zuckerman – Wiley Eastern Limited.
[5] Elementary Number theory by David M .Burtone

Abstract: The aim of this paper is to develop the implicit finite difference scheme for space fractional soil moisture diffusion equation (SFSMDE) with initial and boundary conditions. We prove that the scheme is unconditionally stable and convergent. Also, as an application of this scheme numerical solution for space fractional soil moisture diffusion equation is obtained by Mathematica software. Keywords: Space fractional, Soil moisture diffusion equation, Finite difference scheme, Fractional derivatives, Mathematica.

[1] Florica MATEL and BUDIU , Differential equations and their applications to the SoilMoisture Study, Bulletin UASVM,
Horticulture 65(2)(2008) .
[2] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore (2000).
[3] Daniel Hillel, Introduction to Soil Physics, Academic Press (1982).
[4] Don Kirkham and W.L. Powers, Advanced Soil Physics, Wiley-Interscience (1971).
[5] F.Liu, P. Zhuang, V. Anh, I, Turner, A Fractional Order Implicit Difference Approximation for the Space-Time Fractional
Diffusion equation , ANZIAM J.47 (EMAC2005), pp. C48-C68:(2006).
[6] Pater A.C., Raats and Martinus TH. Ven Genuchten, Milestones in Soil Physics, J. Soil Science 171,1(2006). 10
[7] I. Podlubny, Fractional Differential equations, Academic Press, San Diago (1999).
[8] B.Ross (Ed), Fractional Calculus and Its Applications, Lecture Notes in Mathematics, Vol.457, Springer-Verlag, New York ( 1975).
[9] S. Shen, F.Liu, Error Analysis of an explicit Finite Difference Approximation for the SpaceFractional Diffusion equation with
insulated ends, ANZIAM J.46 (E), pp. C871-C887:(2005).

Abstract: In this paper, three distance measures and their corresponding similarity measures of Intuitionistic Fuzzy Multi sets (IFMS) are introduced and compared. The measures are based on Hausdroff distance measure, Geometric distance measure and the Normalized distance measure.

Key Words - Multi set, Intuitionistic fuzzy set, Intuitionistic Fuzzy Multi sets, Distance Measure, Similarity measure

[1] Zadeh L. A., Fuzzy sets, Information and Control 8 (1965) 338-353.
[2] Atanassov K., Intuitionistic fuzzy sets, Fuzzy Sets and System 20 (1986) 87-96.
[3] Atanassov K., More on Intuitionistic fuzzy sets, Fuzzy Sets and Systems 33 (1989) 37-46.
[4] Yager R. R., On the theory of bags,(Multi sets), Int. Jou. Of General System, 13 (1986) 23-37.
[5] Blizard W. D., Multi set Theory, Notre Dame Journal of Formal Logic, Vol. 30, No. 1 36-66, (1989).
[6] Shinoj T.K., Sunil Jacob John , Intuitionistic Fuzzy Multi sets and its Application in Medical Diagnosis, World Academy of
Science, Engineering and Technology, Vol. 61 (2012).
[7] Szmidt E., Kacprzyk J., On measuring distances between Intuitionistic fuzzy sets, Notes on IFS, Vol. 3 (1997) 1- 13.
[8] Szmidt E., Kacprzyk J., Distances between Intuitionistic fuzzy sets. Fuzzy Sets System, 114 (2000) 505-518.
[9] Yanhong L., Olson D., Qin Z., Similarity Measures between Intuitionistic Fuzzy sets : A Comparative Analysis, Pattern
Recognition Letters, 28(2) (2007) 278 - 285.
[10] Hung W.L., Yang M. S. , Similarity Measures of Intuitionistic Fuzzy Sets Based on Hausdorff Distance, Pattern Recognition
Letters, 25 (2004) 1603-1611.

Abstract: This paper deal with the inverse version of capacitated transportation problem, in which the unit transportation cost of some cells in the original problem are adjusted as little as possible so that the given feasible solution become an optimal one. In our proposed method, first we reduce the capacitated transportation problem into linear programming problem and then obtained its inverse problem using dual and the optimality conditions. The method has been illustrated by a numerical example also.

Keywords: Inverse optimization, Capacitated transportation problem, Complementary slackness, Optimal Solution.

[1] Burton, D., Toint, Ph. L. (1992). On an instance of shortest paths problem. Mathematical Programming, 53, 45-61.
[2] Zhang, J., & Liu, Z. (1996). Calculating some inverse linear programming problems. Journal of Computational and Applied
Mathematics. 72, 261-273.
[3] Zhang, J., & Liu, Z. (1999). A further study on inverse linear programming problems. Journal of Computational and Applied
Mathematics. 106, 345-359.
[4] Ahuja, R. K. & Orlin, J. B. (2001). Inverse Optimization. Operation Research, 49,771-783.
[5] Huang, S. & Liu, Z. (1999). On the inverse problem of linear programming and its application to minimum weight perfect kmatching.
European Journal of Operational Research, 112, 421-426.
[6] Amin, G.R., Emrouznejad, A. (2007). Inverse Forecasting: A New approach for predictive modeling, Computers & Industrial
Engineering. DOI:10.1016/j.cie. 2007.05.007.
[7] Yibing, L., Tiesong, H., Zhongping, W. (2008). A penalty function method for solving inverse optimal value problem. J. Comp.
Appl. Math., 220(1-2):175-180.
[8] Zhang, J., Zhang, Li., Xiao, X. (2010). A perturbation approach for an inverse quadratic programming problem. doi: 10.1007/s00186- 010-0323-4.
[9] Zhang, J., Zhang, Li. (2010). An Augmented Lagrangian Method for a Class of Inverse Quadratic Programming Problems. Applied
Math. & Opt. 61(1), 57-83.
[10] Zhang, J., Zhang, Li. (2009). Solving a class of inverse QP problems by a smoothing Newton method. Journal of Computational
Mathematics, 27:6, 787-801.

Abstract: In this paper two types of duals are considered for a class of variational problems involving 3rd order derivatives. The duality results are derived without any use of optimality conditions. One set of results is based on Mond weir type dual that has the same objective functional as the primal problem but different constraints. The second set of results is based on a dual of an auxiliary primal with single objective function. Under various convexity and generalized convexity assumptions, duality relationships between primal and its various duals are established. Problems with natural boundary values are considered and the analogs of our results in nonlinear programmings are also indicated. Our results are generalizations of those presented by I.Husain, B.Ahmad and Z.Jaben.

Keywords: Variational Problems, Multiobjective duality, Concavity, Generalized convexity

[1] Mond B, Hanson M.A: Duality for variational problems Math Anal Appl 1967; 18:355-64.
[2] Mond B, Weir T, Generalized concavity and duality. In Schiable S, Ziemba WT, Eds. Generalized concavity in optimization and
economics. New York: Academic press 1981.
[3] Wolfe P.A duality theorem for nonlinear programming .Q Appl Math 1961; 19:133-239.
[4] Mangasarian OL. nonlinear programming, New York., McGraw Hill 1969.
[5] I. Husain, B Ahmad, Z. Jabeen; on multiobjective Duality for Variational problems; Open Operational Research Journal, 2012, 6, 1-8.
[6] C.R.Bector, I.Husain, Duality in multiobjective variational problems J.Math.Anal.Appl.166 (1992) 214-229.

Abstract: The Purpose of this paper is to prove common fixed point theorems in metric space by using occasionally weakly compatible six self mappings satisfying the implicit relation which unify and generalize most of the existing relevant fixed point theorems. 2000 Mathematics Subject Classification : 54H25, 47H10.

Keywords: Compatible, Weakly compatible, Occasionally Weakly Compatible and implicit relations.

[1] Y. J. Cho, M. S. Khan and S. L. Singh, Common fixed points of weakly commuting mappings, Univ. u NovomSaduZb. Rad.
Prirod.-Mat. Fak. Ser. Mat. 18 (1988), no. 1, 129-142.
[2] J. Danes, Two fixed point theorems in topological and metric spaces, Bull. Austral.Math. Soc. 14 (1976), no. 2, 259-265.
[3] S. A. Husain and V. M. Sehgal, On common fixed points for a family of mappings, Bull. Austral.Math. Soc. 13 (1975), no. 2, 261-
267.
[4] G. Jungck, Compatible mappings and common fixed points, Internat. J. Math and Math Sci., 9(4)(1986), 771-779.
[5] G.Jungck, P.P. Murthy, Y.J.Cho., Compatible mappings of type (A) and common fixed point theorems, Math. Japonica, 38( 2)
(1993), 381-390.
[6] G.Jungck, and B.E. Rhoades, Fixed point for set valued functions without continuity, Indian J.Pure Appl. Math, 29(3)(1998) , 227-
238.
[7] G. Jungck and B. E. Rhoades, Fixed Point Theorems for occasionally weakly compatible Mappings, Fixed Point Theory, Vol
7,(2)(2006), No. 287-296.
[8] R. Kannan, Some results on fixed points, Bull. Cal. Math. Soc., 60(1968), 71-76.
[9] M. S. Khan and M. Imdad, A common fixed point theorem for a class of mappings, Indian J. Pure Appl. Math. 14 (1983), no. 10,
1220-1227.
[10] S. V. R. Naidu and J. Rajendra Prasad, Common fixed points for four self-maps on a metric space, Indian J. Pure Appl. Math. 16
(1985), no. 10, 1089-1103.

Abstract: We constructed a hash function by using the idea of cayley graph, hash function based on computing a suitable matrix product in groups of the form SL2(F2 n ). We found collision between palindrome bit strings of length 2n+2 for the new construction. Here we reinforce the hash function by adding key to it and we claim that it will resists palindrome collision.

Keywords - Cayley graph, Group, Hash function, Irreducible polynomial, Message Authentication Code, palindrome collision.

[1] Bart Van Rom pay, Analysis and Design of Cryptographic Hash Functions, MAC algorithms and Block Ciphers, doctoral
dissertation, KU Leuven2004D/2004/7515 ISBN 90-5682-527-5
[2] B. Praneel, Analysis and Design of Cryptographic Hash Functions, doctoral dissertation K.U .Leuven Jan. 1993
[3] J.P. Tillich and G. Zemor, Hashing with SL2, Advances in Cryptology Lecture Notes in Computer Science, vol. 839(1994),
Springer-Verlag, pp. 40-49.
[4] K.T .Joju and P.L. Lilly , Keyed Tillich-Zemor Hash Function. Research Journal of Pure Algebra-vol 3(1),2013,www .rjpa .info
ISSN 2248-9037,page: 24-32.
[5] K.T .Joju and P.L. Lilly , Alternate form of Hashing with Polynomials. Proceedings of the International workshop in Cyber
Security, St. Joseph‟s College, Irinjalakuda pp. 43-45, 2011
[6] K.T .Joju and P.L. Lilly, Tillich-Zemor Hash Function with New Generators and Analysis. Research Journal of Pure Algebra-
2(11),2012 ,www.rjpa.info ISSN 2248-9037,page:338-343.
[7] John R Durbin, Modern Algebra, (John Wiley & Sons 2005).
[8] K.T .Joju and P.L. Lilly, Alternate form of Tillich - Zemor Hash Function which Resist Second Preimage(accepted) International
J. of Math. Sci. & Engg. Appls. (IJMSEA) ISSN 0973-9424, Vol. 7 No. II (March, 2013), pp.
[9] Markus Grassl, Ivana Ilic, Spyros Magliveras, and Rainer Steinwandt, Cryptanalysis of the Tillich-Zemor hash function, Journal
of Cryptology, Volume 24 Number-1.pp 148-156.
[10] Jill P. Mesirov and Melvin M. Sweet. Continued Fraction Expansions of Rational Expressions with Irreducible Denominators in
Characteristic 2. Journal of Number Theory, 27 (1987), pp.144-148.

Abstract: In the context of information theory, Shannon's entropy 16 plays an important role. In case, one has information about the current age of the component which can be taken into account for measuring its uncertainty, Shannon's entropy is not suitable as such. Consequently, Ebrahimi 3 proposed an alternative approach for characterization of distribution functions in terms of conditional Shannon's measure of uncertainty. In this paper, we propose generalized residual entropy function for characterization of some life time models. Also, upper and lower bound of hazard rate function in terms of generalized residual entropy have been obtained. Based on the proposed measure, we derive the generalized residual entropy function for some continuous lifetime models.

Key Words: Shannon;s entropy, Mathai-Haubold entropy, Residual entropy, Generalized residual entropy, Life time distribution. AMS Subject Classification: 60E15 , 62N05, 90B25 , 94A17 , 94A24

1 Asaid, M and Ebrahimi, N (2000), " Residual entropy and its characterizations in terms of hazard function and mean residual life
function". Statist. Prob. Lett. 49: 263-269.
2 Belzunce, F., Navarror, J., Ruiz, J.M and Aguila, Y (2004), "Some results on residual entropy function".Metrika 59:147-161.
3 Ebrahimi, N (1996), "How to measure uncertainty in the life time distributions".Sankhya. vol. 58 , Ser.A , 48-57.
4 Ebrahimi, N (1997), "Testing whether life time distribution is decreasing uncertainty".J.Statist. Plann.Infer. 64:9-19
5 Hooda, D. S and Saxena, S (2011), " Generalized measures of discrimination between past life time distributions", Pak.J.Stat. Oper.
Res. Vol. VII, No. II,PP. 233-243.
6 Hooda, D.S and Kumar, P (2007), " Generalized residual entropies in survival analysis", Journal of Applied Probability and
Statistics, Vol. 2, No. 2, PP. 241-249.
7 Mathai, A.M and Haubold, H. J (2006), " Pathway models, Tsallis statistics, superstatistics and a generalized measure of entropy",
Physics A, Vol. 375, PP. 110-122.
8 Mathai, A.M and Haubold, H. J (2006a), " On generalized distributions and pathways". arXiv:cond-mat/0609526v2.
9 Mukherjee, S.P and Islam, A (1983), " A finite range distribution of failure times" , Navel Res. Logist. Quat. Vol. 30, PP. 487-491.
10 Nanda, A. K. and Gupta, R. D (2001), " Some properties of reversed hazard rate function." Statistical Methods, Vol. 3, PP. 108-124.

Abstract: In this paper the notion of interval-valued fuzzy KUS-ideals (briefly i-v fuzzy KUS-ideal) in KUSalgebras is introduced. Several theorems are stated and proved. The image and inverse image of i-v fuzzy KUSideals are defined and how the homomorphic images and inverse images of i-v fuzzy KUS-ideals become i-v fuzzy KUS-ideals in KUS-algebras is studied as well.

Keywords: KUS-algebras, fuzzy KUS-ideals, interval-valued fuzzy KUS-sub-algebras, interval-valued fuzzy KUS-ideals in KUS-algebras. 2000 Mathematics Subject Classification: 06F35, 03G25, 03B52, 94D05.

[1] Biswas R., Rosenfeld's fuzzy subgroups with interval valued membership, function, Fuzzy Sets and ystems , vol.63, no.1 (1994),87- 90.
[2] Dudek W. A. and Zhang X., On ideal and congruences in BCC-algebras, Czechoslovak Math. Journal, vol.48 , no. 123 (1998), 21-29.
[3] Dudek W. A., On proper BCC-algebras, Bull. Ins. Math. Academic Science, vol. 20 (1992),137-150.
[4] Mostafa S. M., Abdel Naby M. A. , Abdel-Halim F. and Hameed A. T., Fuzzy KUS-ideals in KUS-algebras. To appear .
[5] Prabpayak C. and Leerawat U. , On ideals and congurences in KU-algebras , scientia magna journal , vol.5, no .1 (2009) , 54-57 .
[6] Prabpayak C. and Leerawat U. , On isomorphisms of KU-algebras , scientia magna journal , vol.5, no .3 (2009), 25-31 .
[7] Xi O. G., Fuzzy BCK-algebra, Math. Japon. , vol.36 (1991) 935-942.
[8] Zadeh L. A., Fuzzy sets, Inform. And Control, vol. 8 (1965) 338-353.
[9] Zadeh L. A., The concept of a linguistic variable and its application to approximate I, Information Sci. And Control , vol.8 (1975) ,
199-249.