iosr-Jm   Volume-12 ~ Issue-4 ~ Jul-Aug 2016

Abstract: In this paper, we analyses a stochastic model of tumorigenesis which takes into account the transient defects of mitotic spindle geometry and chromosome segregation errors in cell division process. The model is essentially a Markov branching process evolving in a random environment. Using the model, we predict that the two defects may specifically contribute to the tumorigenesis initiation and progression

[1]. Abramowitz, M. and Stegun, I. J., 1965. "Handbook of Mathematical Functions," Dover.
[2]. Andersson, D. I. and Hughes, D., 2009. "Gene Amplification and Adaptive Evolution inBacteria," Annual Review of Genetics, Vol. 43, pp. 167-195.
[3]. Gusev, Y., Kagansky, V. and Dooley, W. C., 2000. "A stochastic model of chromosomesegregation errors with reference to cancer cells," Mathematical and Computer Modelling,Vol. 32, pp. 97-111.
[4]. Kimmel, M. and Stivers, D. N., 1994. "Time-continuous branching walk models of unstablegene amplification," Bulletin of Mathematical Biology, Vol. 56, pp. 337-357.
[5]. Pease, J. C., and Tirnauer, J. S., 2011. "Mitotic spindle misorientation in cancer out ofalignment and into the fire," Journal of Cell Science, Vol. 124, pp. 1007-1016.

Abstract: In this paper, invariant submanifolds in a indefinite trans-Sasakian manifold are studied. Necessary and sufficient condition are given on submanifold of a indefinite trans-Sasakian manifold to be invariant submanifold.Here we shown that an invariant submanifold of a indefinite trans-Sasakian manifold is totally geodesic. AMS Subject Classification: 53C15,53C17, 53C20, 53C25.
Keywords: Indefinite trans-Sasakian manifold, Invariant submanifold, Covariant differentiation, Bianchi identity,totally geodesic

[1]. B.Y Chen , Geometry of submanifolds, M.Dekker,Newyork,1973

[2]. B.Y.Chen and K.Ogive , on totally real submanifolds, Transaction of the Amer.Math.Society, 193(1974), 257-266

[3]. D.E. Blair, Contact manifold in Riemannian Geometry, Lecture Notes in Mathematics, 509, Springer-Verlag, Berlin, 1976.

[4]. K.L.Duggal and B.Sahin, Lightlike submanifolds of Indefinite sasakian manifolds, International Journal of mathematics and Matematical Sciences, Hindawi Publishing Corporation (2007),1-22

[5]. T.H.Kang,S.D.Jung,B.H.Kim,H.K.pak and J.S.Pak, Light like hypersurfaces of indefinite sasakian manifolds, Indian Journal of Pure and Applied mathematics, Vol.34,no.9(2003),1369-1380

Abstract: This paper explores the potential of the Compound Weibull Distribution (CW) in theory and practice. Statistical lifetime distributions are commonly used in various fields for modeling data sets where the CW distribution is the most applicable. This distribution is presented with its properties and graphical representations. Some special cases and related distributions to the CW distribution are derived. Moments (M), maximum likelihood (ML) and Bayesian - based on informative prior - methods are used to estimate the unknown parameters of the distribution...............
Keywords: Compound Weibull Distribution, maximum likelihood, Bayesian estimation, MCMC, error loss functions

[1] Dubey, S.D., A compound Weibull distribution, Naval Research Logistics Quarterly, 15,1968, 179-188.
[2] Gottschalk, L., Tallaksen, L.M., Perzyna, G., Derivation of low flow distribution functions using recession curves, Journal of
hydrology, 194, 1997, 239-262.
[3] Qutb, N.S., Estimation of the Parameters of Compound Distributions, Ph.D.dissertation, Al-Azhar University, Egypt,2002.
[4] Mahmoud, M.A., Moshref, M., Yhiea, N.M., Mohamed, N.M., Progressively Censored Data from The Weibull Gamma Distribution Moments and Estimation, Journal of Statistics Applications & Probability, 3,2014, 45-60.
[5] Fisz, M.,Probability theory and mathematical statistics (New York: John Wiley and sons,1963).

Abstract: In this chapter we study about finite lattices with isoform congruences. We prove that every finite distributive lattice D can be represented as the congruence lattice of a finite isoform lattice. Infact, we prove that every finite distributive lattice D can be represented as the congruence lattice of a finite lattice L with the following properties:
(i) L is isoform
(ii) For every congruence  of L, the congruence classes of  are projective intervals.
(iii) L is a finite pruned Boolean lattice.
(iv) L is discrete –transitive........
Keywords:Interactions, Siri', Pacce

[1]. G.Gratzer, H.Lakser, and E.T.Schmidt, Congruence lattices of small planar lattices. Proc. Amer. Math. Soc. 123(1995), 2619-2623.
[2]. G.Gratzer, General Lattice Theory, Second edition, Birkhauser Verlag, Basel, 2010.
[3]. G.Gratzer and E.T.Schmidt, Congruence-preserving extensions of finite lattices into sectionally complemented lattice, Proc. Amer.Math.Soc.127(1999), 1903-1915.
[4]. G.Gratzer and F.Wehrung, Proper congruence-preserving extensions of lattices, Acta Math. Hungar. 85(1999),175-185.
[5]. G.Gratzer and E.T.Schmidt, Regular congruence-preserving extensions. Algebra Universalis 46(2008), 119-130.

Abstract:In this paper, we consider a mixed integral equation (MIE) of the second kind. Under certain conditions, the existence of a unique solution of is discussed and proved. The kernel of position takes asingular form, while the kernel of time is continuous. Using a quadratic numerical method, the MIE leads us to a linear system of Fredholm integral equations (SFIEs). Then,SFIEsafter using Toeplitz matrix method (TMM),tends to a linear algebraic system (LAS). The existence of a unique solution of LAS is proved. Finally, numerical
examples are considered, and the error, in each case, is calculated..
Keywords: Mixed integral equation, Linear system of Fredholm integral equations, Linear algebraic system, Carleman function, logarithmickernel, Toeplitz matrix method.

[1]. T. Allahviranloo, Z. Gouyandeh, A. Armand, Numerical solutions for fractional differential equations by Tau-Collocation method,
Appl. Math.and Compute.271(2015)979-990.
[2]. F. Ghoreishi, M. Hadizadeh, Numerical computation of the Tau approximation for the Volterra-Hammerstein integral equations,
Numer.Algor.(2009) 52:541559.
[3]. H.R. Marzban, H.R. Tabrizidooz, M. Razzaghi, A composite collocation method for the nonlinear mixed Volterra-Fredholm-
Hammerstein integral equations, Commun. Non. Sci. Numer. Simul.16 (2011) 1186-1194.
[4]. ] H.L. Dastjerdia, F.M.M. Ghainia, M. Hadizadeh, A meshless approximate solution of mixedVolterra-Fredholm integral equations,
Inter. J. Comput. Math., 90 (2013) 527-538.
[5]. C.D.Green,Integral Equation Methods,Nelson, New York, 1969

Abstract: The present paper aims at studying a Ricci solitons on para-Sasakian manifolds. We give the results on Ricci solitons in para-Sasakian manifold satisfying the conditions S( , X) R = 0, R( , X ) P = 0 and R( , X) H = 0 ,where P ,H are pseudo-projective and quasi-conharmonic curvature tensors, respectively. MSC(2000): 53C21; 53C44; 53C25.ee.
Keywords: Para-Sasakian manifold; Ricci Solitons; Pseudo-projective; Quasi-conharmonic curvature tensor..

[1] R. S. Hamilton, The Ricci flow on surfaces, Mathematics and general relativity (Santa Cruz, CA, 1986), Contemp. Math., vol. 71,
Amer. Math. Soc., Providence, RI, 1988, pp. 237–262.
[2] H. D. Cao, Recent progress on Ricci solitons, arXiv:0908:2006v1, Adv. Lect. Math. (ALM), 11 (2009), 1-38.
[3] C. L. Bejan and M. Crasmareanu, Second order parallel tensors and Ricci solitons in 3-dimensional normal paracontact geometry,
Ann. Glob. Anal. Geom., DOI 10.1007/s10455-014-9414-4.
[4] C. S. Bagewadi, Gurupadavva Ingalahalli, and S. R. Ashoka, A study on Ricci solitons in Kenmotsu manifolds, ISRN Geometry,
Hindawi Publishing Corporation., Volume 2013, Artcle ID 412593, 6 pages.
[5] G. Calvaruso and D. Perrone, Geometry of H-paracontact metric manifolds, arXiv:1307.7662v1.2013.

Abstract:In this paper, we consider the bivariate Vieta-Fibonacci and bivariate Vieta-Lucas polynomials which are generalized of Vieta-Fibonacci, Vieta-Lucas, Vieta-Pell, Vieta-Pell-Lucas polynomials. Also, we give the some properties. Afterwards, we obtain the some identities for the bivariate Vieta-Fibonacci and bivariate Vieta-Lucas polynomials by using the known properties of bivariate Vieta-Fibonacci and bivariate Vieta-Lucas polynomials.
Keywords: Binet's Formula, Fibonacci Polynomials, Lucas Polynomials, Vieata Polynomials.

[1] A. F. Horadam, Vieta Polynomials, The Fibonacci Quarterly, 40(3), 2002, 223-232.
[2] M. Catalini, Some Formulae for Bivariate Fibonacci and Lucas Polynomials, arXiv: math.CO/0406323 v1, 16 Jun 2004.
[3] M. Catalini, Identities for Fibonacci and Lucas Polynomials derived from a book Gould, arXiv: math.CO/0407105 v1, 7 Jul 2004.
[4] M. Catalini, Generalized Bivariate Fibonacci Polynomials, arXiv: math.CO/0211366 v2, 4 Jun 2004.
[5] M. N. S. Swammy, Generalized Fibonacci and Lucas Polynomials and Their Associated Diagonal Polynomials , The Fibonacci
Quarterly, 37(3), 1999, 213-222.

Abstract: In this paper we discuss theGeneralized Inverses, Group Inverses And Reverse Order Law For Range Quaternion Hermitian Matrices (q-EP).se...........
Keywords: Moore-Penrose inverse , q-EP matrix, Generalized Inverses for q-EP, Group Inverses for q-EP, Reverse Order Law for q-EP.

[1]. Ben Isreal . A and Greville . TNE : Generalized Inverses , Theory andapplications ; Wiley and Sons , New York( 1974).
[2]. KatzT.J and Pearl M.H: on Epr and Normal Epr matriceJ.res.Nat.Bur.Stds. 70B, 47-77(1966).
[3]. Gunasekaran.K and Sridevi.S: On Range Quaternion Hermittian MatricesInter;J;Math.,Archieve-618, 159-163
[4]. Gunasekaran.G and Sridevi.S: On Sums of Range Quaternion Hermittian Matrices; Inter;J.Modern Engineering Res-.5, ISS.11. 44 –
49(2015)
[5]. Marsagila.GandStyanG.P.H :Equalities and Inequalities for rank of Matrices;lin.Alg.Appl.,2,269-292(1974)

Abstract: The problem of generating bivariate life distributions from univariate once is drawing the attention of the reliability analyst. Amongst those approaches, the characterization approach and the modeling approach are very appealing. In fact characterization approach is of interest to both theoreticians and applied workers. Here we have used a bivariate distribution for application by extending the univariate distribution through characterization approach. In our application we have considered cancer patients with chemotherapy treatment and taking GnRHa, FSH, and Estradiol as women stress effects
Keywords: GnRHa , FSH, Estradiol

[1] Woodruff T K, Preserving fertility during cancer treatment, NAT MED 15,2009,1124-1125.
[2] Peccatori F, Demeestere I, GnRH analogue for chemotherapy-induced ovarian damage; Too early to say?Fertil Steril,92;2009,e33.
[3] Beck-Fruchter R, Weiss A, Shalev E,GnRH agonist therapy as a ovarian protectants in female patients undergoing chemotherapy; a review of clinical data. Hum Reprod update 14,2008, 553-561.
[4] Thomas P Curran, Miguel Ángel Prieto Lage, Yvonne Anders, José Antonio Vázquez Álvarez, Miguel Anxo Murado García ,Development of bivariate mathematical model to characterize simultaneously the dose-time responses of pro-oxidant agents, Paper number 131620326, 2013.
[5] Pereyra Pacheco B, Mendez Ribas JM, Milone J, Use of GnRH analogs for functional protection of the ovaries and the preservation of fertility during cancer treatment in adolesents: a preliminary report, Gynecol Oncol 81,2001,391-397.

Abstract: This paper examined the didactic conceptual structures of preservice teachers in the absolute and quadratic inequalities to deduce mistakes and errors. Quasi-experimental and mixed exploratory sequential designs were adopted on the participants who worked in 37 groups of 10 members in the Department of Basic Education, University of Education, Winneba in Ghana. The data collection instruments consisted of 15 open-ended items on the basic ideas of the absolute and quadratic inequalities to identify the didactic content knowledge in solving the problems..........
Keywords: Absolute and quadratic inequalities; didactic conceptual structures; didactic facets; pre-service teachers

[1]. Appleby, A., Letal, R. & Ranieri, G. (2006). Pure Math Grade 11 Workbook. Alberta: Absolute Value Publications.

[2]. Bäckman, K. & Attorps, I. (2012). Teaching Mathematics in the Pre-school Context. US- China Education Review B 1 (2012) 1-16. [3]. Ball, D. L., Thames, M. H. & Phelps, G. (2008). Content knowledge for teaching. What makes it special? Journal of Teacher Education, 59(5), 389-407.

[4]. Clandinin, D.J. & Connelly, M.F. (1987). Teachers' personal knowledge: What counts as personal in studies of the personal. Journal of Curriculum Studies, 19, 6, pp. 487-500.

[5]. Cook, C.E. & Tx, R. (2009). Alg II, Unit 1 Solving linear equations and inequalities. Teacher Version 1.01. New York: Blue Pelican Alg II.

Abstract: In this study, we model the occurrence and length of wet, medium wet and dry spells by Markov chain that best describes the rainfall pattern of Bungoma County (Western Kenya).This is achieved by Markov chain theory and estimation of probabilities of the chain by MLE. Also computed is the distribution of the length of each spells; wet, medium wet and dry from which the central moments of the rainfall pattern are computed. The model developed is applied to rainfall data from Bungoma meteorological station.............
Keywords: Markov chain, Wet spell, Medium wet spell, Dry spell, Prediction, Stationary distribution

[1]. Barkotulla M.A.B, Rahmann M.S. (2012). Multi-state Markov chain modeling system for Environmental Impact of climate change. International Journal of statistical sciences Vol. 12: 29 - 46.
[2]. Biamah E. K, Gichuki N, Kaumbutho P.G. (1993). Tillage methods and soil and water conservation in East Africa. Soil and Tillage Research, Vol. 27: 105-123.
[3]. Biamah E. K, Sterk G, Shamah T.C. (2004). Analysis of agricultural drought in Liuni eastern Kenya: Application of Markov model. Journal of Hydrology Processes Vol.19: 1307-1322.12
[4]. Bogardi J, Duckstein L, Ruhambo O.H. (1988). Practical generation of synthetic rainfall event series in a semi-arid climate zone. Journal of hydrology Vol. 103: 357-313.
[5]. Fischer B. M. C, Mul M.L and Savinije H.H.G. (2012). Determining spatial variability of dry Spells – a Markov based method, applied to the Makanya catchment. Tanzania Hydrology And Earth Sci system. 11707 - 11731.

Abstract: The current microprocessors are concentrating on the multiprocessor or multi - core system architecture. The parallel algorithms are recently focusing on multi - core system to take full utilization of multiple processors available in the system. The design of parallel algorithm and performance measurement is the major issue on today's multi - core environment. Numerical problems arise in almost every branch of science which requires fast solution.............
Keywords: Approximate roots, Börsch – Supan method, Durand-Kerner method, Parallelization, Simultaneous root finding methods

[1]. M.S. Petkovicª,*, Ð. Herceg, Point estimation of simultaneous methods for solving polynomial equations: a survey, J. Computational and Applied Mathematics 136 (2001) 283–307
[2]. P.Batra, Improvement of a convergence condition for the Durand–Kerner iteration, J.Comput.Appl.Math. 96 (1998) 117–125.
[3]. M.S. Petkovic, On initial conditions for the convergence of simultaneous root - finding methods, Computing 57 (1996) 163–177.
[4]. M.S. Petkovic, C.Carstensen, M.Trajkovic, Weierstrass' formula and zero - finding methods, Numer.Math.69 (1995) 353–372.
[5]. M.S. Petkovic ".Herceg, S.Ilic, Safe convergence of simultaneous methods for polynomial zeros, Numer. Algorithms 17 (1998) 313–331.

Abstract: A near ring N is called weak commutative if xyz = xzy for every x,y,z ϵ N[14]. N is called pseudo commutative if xyz = zyx for every x,y,z ϵ N[15]. N is called quasi weak commutative if xyz = yxz for every xyz = yxz forevery x,y,z ϵ N [11]. N is called pseudo m – power commutative if xmyz =zymx for every x,y,z ϵ N [10]. We obtain more results, generalising the results of [15].

[1]. H.E.Bell, Quasi centres, Quasi Commutators, and Ring Commutativity, Acta Maths, Hungary 4 (1 – 2) (1983),127-136
[2]. L.o.Chung and Jiang Luh, Scalar central elements in an algebra over a Principal ideal domain, Acta Sci. Maths 41, (1979), 289-293
[3]. Dheena .P; On Strongly regular near –rings, Journal of the Indian Maths.Soc, 49 (1985), 201 – 208
[4]. Dheena.P. A note on a paper of Lee, Journal of the Indian Maths.Soc, 53(1988), 227 - 229
[5]. Fou'n, Some structure Theorem for near – rings, Doctoral dissertation, University of lahama, 1968

Abstract: A right near – ring N is called weak commutative if xyz = xzy for every x,y,z ϵ N (Definition 9.4 [10]). A right near – ring N is called pseudo commutative (Definition 2.1 [11] ) if xyz = zyx for all x,y,z ϵ N.A right near – ring N is called quasi – weak commutative (Definition 2.1 [7] ) if xyz = yxz for all x,y,z ϵ N. We call a right near – ring N to be quasi – weak m – power commutative if xm y z = ym x z for all x,y,z ϵ N. N is said to be Quasi – weak (m,n) power commutative near – ring if xm yn z = ym xn z for all x,y,z ϵ N.In this paper we study and establish various results of Quasi – weak m – power commutative near – ring and Quasi – weak (m,n) power commutative near -- ring.

[1]. H.E.Bell,Quasi Centres,Quasi Commutators, and Ring Commutativity,Acta Maths,Hungary 4( 1 – 2 )(1983), 127 – 136.
[2]. L.O.Chung and Jiang Luh,Scalar Central elements in an algebra over a Principal ideal domain,Acta Sci. Maths 41,(1979),289 – 293.
[3]. .G.Gopalakrishnamoorthy and R.Veega, On Quasi – Periodic,Generalised Quasi-Periodic Algebras,Jour.of Inst. of Mathematics
and Computer Sciences,Vol 23,No 2 (2010).
[4]. G.Gopalakrishnamoorthy and S.Geetha,On (m,n) – Power Commutativity of rings and Scalar (m,n) – Power Commutativity of
Algebras,Jour. of Mathermatical Sciences ,Vol 24(3),2013, 97-110.
[5]. G.Gopalakrishnamoorthy and R.Veega,On Scalar Power Central Elements in an Algrbra over a Principal ideal domain,Jour. of
Mathematical Sciences,Vol 24(3),2013,111-128.

Abstract: This paper presents some contexts relative to a matroid, which satisfy: all the closure operators derived from their Galois connections which to these contexts are the closure operator of the matroid. Using these results and ready-made algorithms for building up a concept lattice or an extent lattice, we believe that we may search out the class of all closed sets of a matroid, and meanwhile, create the construction of the dual of a matroid
Keywords: Context; closed set; matroid; Galois connection; closure operator

[1]. Whitney H. On the abstract properties of linear dependence. American Journal of Math- ematics, 1935, 57(3), 509-533.
[2]. Welsh D.J.A. Matroid Theory. London: Academic Press, 1976.
[3]. Hlinˇeny´ P. A parametrized algorithm for matorid branch-width. SIAM Journal on Com- puting, 2005, 35(2), 259-277.
[4]. Gabow H.N., Westermann H.H., Forests, frames, and games: algorithms for matroid sums and applications. Algorithmica, 1992, 7, 465-497.
[5]. Mao H. On closure axioms for a matroid using Galois connections. Mathematical Com- munications, 14(2)(2009), 425-432.

Abstract: In this paper we prove a common fixed point theorem in fuzzy metric space using an implicit relation. 2010 AMS Classification: 47H10, 54H25
Keywords: Fuzzy metric space, Fixed point, Fixed point Theorem, Compatible mapping, Reciprocal continuity

[1] P. Balasubramaniam, S. Murlisankar and R.P. Pant, Common fixed points of four mappings in a fuzzy metric space, J. Fuzzy Math.
10(2) (2002) 379{384.
[2] M. Grabiec, Fixed points in fuzzy metric space, Fuzzy Sets and System 27 (1988) 385{389.
[3] A. George and P. Veermani, On some results in fuzzy metric spaces, Fuzzy Sets and System 64 (1994) 395{399.
[4] O. Kramosil and J. Michelak, Fuzzy metric and statistical metric spaces, Kybernetika 11 (1975) 336{344.
[5] S. N. Mishra, N. Sharma and S. L. Singh, Common ¯xed points of maps on fuzzy metric spaces, Int. J. Math. Math. Sci. 17 (1994)

Abstract: Peristaltic pumping is a form of the fluid transport in a flexible tube caused by a progressive wave of contraction or expansion from a region of lower pressure to higher pressure. Peristalsis is one of the major mechanisms for fluid in many biological systems. It is an automatic and vital process that moves food through the digestive tract, urine from the kidneys through the ureters into the bladder, and bile from the gall bladder into the duodenum and transport of blood through the artery with mild stenosis. In addition, the mechanism of peristalsis is applied in the design of biomechanical instruments in chas heart-lung machine.........
Keywords: Pulsatile Flow, Newtonian Fluid, Reynolds Number.

[1]. Fung and Yih, Maximum flow rate versus Reynolds number for small amplitude ratios. J. Applied Mech, 35, 669 – 675, (1968).

[2]. A.H. Shapiro, M. Y. Jaffrin and S. L. Weinberg, Peristaltic pumping with long wavelengths at low Reynolds number. J. fluid Mech, 37, 799 – 825, (1969).

[3]. Li and Brasseur, Non-Study Peristaltic transport in finite length tubes. J.fluid Mech, 248, 129-151, (1993). [4]. Eytan and D.Elad, Analysis of intra – uterine fluid motion induced by uterine contractions. J. Bulletin of Mathematical Biology, 61, 221- 238,(1999).

[5]. Srivastava, M. Interaction of peristaltic flow with pulsatile flow in a circular cylindrical tube. Biomechanics,, 18(4), 247 – 253, (1985).

[6]. Afifi, N. A. S., and N. S. Gad, Interaction of peristaltic flow with pul¬satile fluid throngh a porous medium, Applied Mathematics and Compu-tation,142, 167 – 176, (2003).

Abstract: In this paper we used Weighted Exponential distribution (WE) using the exponentiated class family of distributions. The proposed model named as Exponentiated Weighted Exponential distribution and it serves as an alternative to both the Weighted Exponential distribution and the Exponential distribution The exact role of ghrelin in the control of growth hormone (GH) secretion has not been completely clarified as yet. Eighteen prepubertal children with short stature were included in the study. Using theresults of two GH-provocative tests (glucagon and clonidine), theparticipants were divided into two groups:...........
Keywords: Clonidine, Ghrelin, Growth hormone, Exponential Distribution, Reliability Analysis

[1] Kojima M, Hosoda H, Date Y, Nakazato M, Matsuo H, Kangawa K, 1999 Ghrelin is a growth hormone-releasingacylated peptide
from stomach. Nature 402: 656-660
[2] Tschöp M, Smiley DL, Heiman ML, 2000 Ghrelin induces adiposity in rodents. Nature; 407: 908-913.
[3] Wren AM, Seal LJ, Cohen MA, et al, 2001 Ghrelin enhances appetite and increases food intake in humans.J Clin Endocrinol Metab
86: 5992-5995.
[4] AlexandraC, Cordeiro GM, sarabia JM,2012 Generallized Beta –Generalized Distributions Computational Statistics&Data
Analysis,56(6),1880-1897
[5] Andjelic G,Milosev 1,Djakovic v2010, Extreme Value Theory in Engineering Markets Economic Annals,LV(185),63-105

Abstract: Background:This research work deals with the travelling salesman problem, the aim of this research is to obtain the cheapest possible route to be taken during the distribution of product (water) in some districts in Obio-Apkor Port Harcourt by Junac Table Water Company in other to maximize the company's profit. MaterialsandMethods: The metric method is adopted in solving the problem analytically, and the result was also confirmed and compared with aid of a statistical software (TSP solver and Generator v0.1). Results: The route with the minimum cost from both the matrix method and the statistical software is N1010 (one thousand and ten naira only).
Keywords: Transportation, Travelling salesman problem, routing, genetic algorithm, greedy algorithm, metric method.

[1] Akandwanaho, S. M., Adewumi, A. O., &Adebiyi, A. A. (2014). Solving Dynamic Traveling Salesman Problem Using Dynamic Gaussian Process Regression. Journal of Applied Mathematics, 2014.

[2] Alhamdy, S., Ahmad, S., Noudehi, A. N., &Majdara, M. (2012). Solving traveling salesman problem (TSP) using ants colony (ACO) algorithm and comparing with tabu search, simulated annealing and genetic algorithm. J. Appl. Sci. Res, 8(1), 434-440.

[3] Fereidouni, S. (2011). Solving traveling salesman problem by using a fuzzy multi-objective linear programming. African Journal of mathematics and computer science research, 4(11), 339-249.

[4] Filip, E., &Otakar, M. (2011). The travelling salesman problem and its application in logistic practice. WSEAS Transactions on Business and Economics, 8(4), 163-173.

[5] Knox, J. (1994). Tabu search performance on the symmetric traveling salesman problem. Computers operations research, 21(8), 867-876.

Abstract: In this paper we resolve the question of whether or not P and BPP are unequal.We do this by showing the existence of a set that is a subset of P and not in P,then using a property of the definition of this set to show it is within BPP.
Keywords: ..........

[1]. Navarro, Gonzalo "A guided tour to approximate string matching".p.7 (2001)ACM Computing Surveys
[2]. Stephen Cook" http://www.claymath.org/sites/default/_les/pvsnp.pdf" (2000)Clay mathematics institute 3.Rogers, H. "Theory of Recursive Functions and Ef-fective Computability." Cambridge, MA: MIT Press, 1987.
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[4]. HenkBarendregt, Wil Dekkers, Richard Statman. "Lambda Calculus withTypes", Association of Symbolic Logic (2013)
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Abstract: This paper presents some contexts relative to a matroid, which satisfy: all the closure operators derived from their Galois connections which to these contexts are the closure operator of the matroid. Using these results and ready-made algorithms for building up a concept lattice or an extent lattice, we believe that we may search out the class of all closed sets of a matroid, and meanwhile, create the construction of the dual of a matroid.
Keywords: context; closed set; matroid; Galois connection; closure operator

[1]. Whitney H. On the abstract properties of linear dependence. American Journal of Mathematics, 1935, 57(3), 509-533.
[2]. Welsh D.J.A. Matroid Theory. London: Academic Press, 1976.
[3]. Hlinˇeny´ P. A parametrized algorithm for matorid branch-width. SIAM Journal on Computing, 2005, 35(2), 259-277.
[4]. Gabow H.N., Westermann H.H., Forests, frames, and games: algorithms for matroid sums and applications. Algorithmica, 1992, 7, 465-497.
[5]. Mao H. On closure axioms for a matroid using Galois connections. Mathematical Com- munications, 14(2)(2009), 425-432.

Abstract: This work deals with exact solutions of (2+1)-dimensional Calogero-Bogoyavlenskii-Schiff (CBS) equation. Solutions so obtained are derived by using similarity transformations method via Lie-group theory. Exploiting Lie-symmetries and using invariance property, the number of independent variables is reduced by one by using the method, which turns the CBS equation into a new ordinary differential equation...........
Keywords: CBS equation. Similarity transformations method. Similarity variables. Exact solutions

[1]. M. Kumar, and R. Kumar, Some more solutions of Burgers' equation. IC-MSQUARE 2014. J. Phys.: Conference Series 574, 2015, 012038. DOI:10.1088/1742-6596/574/1/012038
[2]. M. Kumar, A. Kumar, and R. Kumar, Similarity solutions of the Konopelchenko-Dubrovsky system using Lie group theory, Computer & Mathematics with Application, 71(10), 2016, 2051-2059.
[3]. M. Kumar, and R. Kumar, On some new exact solutions of incompressible steady state Navier-Stokes equations, Meccanica 49, 2014, 335-345.
[4]. M. Kumar, and R. Kumar, on new similarity solutions of the Boiti-Leon-Pempinelli system, Communication of Theoretical Physics (Beijing) 61, 2014, 121-126.
[5]. Z. Jiang, and R. K. Bullough, Combined  and Riemann-Hilbert inverse methods for integrable nonlinear evolution equations in (2 + 1)-dimensions, Journal of Physics A: Mathematical and General, 20, 1987, L429-L435.

Abstract: The purpose of this paper is to study fuzzy sets and their real applications. Also, we study some properties of fuzzy sets. As an application of fuzzy sets, we solve some test problems and their solutions are represented graphically using Mathematica
Keywords: Fuzzy set, Membership function, Fuzzy Norm, α-Cuts, Support

[1]. Intrduction to Fuzzy Sets, Fuzzy Logic , and Fuzzy Control Systems By Guanrong Chen Trung Tat Pham .
[2]. A Course in Fuzzy Systems and Control -by Li-Xin Wang.
[3]. Fuzzy Sets : L.A. Zadeh .
[4]. Fuzzy Logic a pratical approach by F, Martin Mcneill, Ellen Thro.
[5]. Fuzzy Logic Toolbox by L. A. Zadeh .