iosr-Jm   Volume-13 ~ Issue-1 ~ Jan-Feb 2017

Abstract: In the first part of the paper I willpresentabriefreview on the Hardy-Weinberg equilibrium and it's formulation in projective algebraicgeometry. In the second and last part I willdiscussexamples and generalizations on the topic.

Keywords: Hardy-Weinberg, likelihood function, maximum likelihood estimation.

[1]. D.Agostini, D.Alberelli, F.Grande, P.Lella, The maximum likelihood degree of Fermat hypersurfaces, arXiv:1404.5745.
[2]. E.Arrondo, Introduction to projective varieties, unpublished notes from the website: http://www.mat.ucm.es/~arrondo/projvar.pdf (2007), 9-10.
[3]. F.Catanese, S.Hoşten, A.Khetan, B.Sturmfels, The maximum likelihood degree, Amer. J. Math. 128 (2006), no. 3, 671-697. MR 2230921.
[4]. B.Chor, A.Khetan, S.Snir, Maximum likelihood on four taxa phylogenetic trees: analytic solutions, The 7th Annual Conference on Research in Computational Molecular Biology-RECOMB 2003, Berlin, April 2003, pp. 76-83.
[5]. G.H.Hardy, Mendelian Proportions in a Mixed Population, Science, New Series, Vol.28, 706 (1908), 49-50.

Abstract: The degree equitable connected cototal dominating graph..............

Keywords: Connected dominating graph, Degree equitable connected cototal dominating set, Degree equitable connected cototal dominating graph.

[1] F. Harary, Graph Theory, Addison-Wesley, Reading Mass,(1969).
[2] T. W. Haynes, S.T. Hedetniemi, and P.J.Slater, Fundamentals of domination in graphs, Marcel Dekker, Inc, New York, (1998).
[3] E. J. Cockayne and S. T. Hedetniemi, Towards a theory of domination in Graphs, Networks, (7)(1977),247-26.
[4] B. Basavanagoud and S. M. Hosamani, Degree equitable connected domination in graphs, ADMS, 5(1) (2013), 1-11
[5] V. R. Kulli, B. Janakiram and R. R. Iyer, The cototal domination of graph, Discrete Mathematical Sciences and Cryptography 2 (1999), 179-184.

Abstract: One of the most famous mathematical constants is the Euler's number e. It is a beautiful number and pops up everywhere in nature. It is the natural language of growths and changes. Thus, it has many different graphical representations. However, today I'm going to show you a geometrical representation of e to help you visualize it better.

Keywords: Another third and divide it into a fourth,infinity, remaining half into a third

.................

Abstract: In this paper, we study semi symmetric and pseudo symmetric conditions in S -manifolds, those are RR = 0 , RC = 0 , C R = 0 , C C = 0 , = ( , ) 1 R R LQ g R , = ( , ) 2 RC L Q g C , = ( , ) 3 CR L Q g R , and = ( , ) 4 C C L Q g C , where C is the Concircular curvature tensor and 1 2 3 4 L ,L ,L ,L are the smooth functions on M , further we discuss about Ricci soliton.

Keywords: S -manifold,  -Einstein manifold, Einstein manifold, Ricci soliton.

[1]. C. S. Bagewadi and G. Ingalahalli, Ricci solitons in Lorentzian  -Sasakian manifolds, Acta Mathematica Academiae Paedagogicae
Nyiregyhaziensis, vol. 28, no.1, pp. 59-68,2012.
[2]. C. S. Bagewadi and K.R.Vidyavathi, Ricci soliton of almost C( ) manifolds, Bull. Cal. Math. Soc., 107,(6) 483-494 (2015) .
[3]. D.E. Blair, Geometry of manifolds with structural group U(n)O(s) , J. Differential Geom. 4 (1970), 155–167
[4]. D.E. Blair, G. Ludden and K. Yano., Differential geometric structures on principal toroidal bundles, Trans. Amer. Math. Soc. 181
(1973), 175–184.
[5]. Chenxu He and Meng Zhu Ricci solitons on Sasakian manifolds, Arxiv:1109.4407v2 [math.DG]

Abstract: The aim of this research paper is to establish some linear generating relations involving I-function of two variables.

[1]. Khare, Sandeep: An Advanced Study of I-Function of one Variable with Some Applications, Ph.D. Thesis, A.P.S. University, Rewa (M.P.), 2009.
[2]. Sharma C. K. and Mishra, P. L.: On the I-function of two variables and its certain properties, ACI, 17 (1991), 1-4.
[3]. Shrivastava, Shweta and Shrivastava, B. M. L.: Some new generating relations and identities for H-function, Vijnana Parishad Anusandhan Patrika, Vol. 49, No.1, January, 2006, p. 63-77.

Abstract: Socio-economic status' (SES) is a term used by social scientists and sociologists to describe the position of an individual in a hierarchical social structure which includes both the social and economic status. In looking more closely at why a remarkable number of students may be struggling for improvement in mathematics in comparison to other subjects, it is timely to consider, the SES factor. This study intends to investigate how parents' SES affects their children's performance in that subject. The study adopted a descriptive survey design and data was largely descriptive by nature. Data were collected using questionnaires for the students of 9th standard........

Keywords: Socioeconomic status, mathematical performance.

[1]. Telford, C.W. and Sawrey, J.M. : Educational Psychology, 2nd Edition, New Delhi, Prentice Hall Of India (private) Ltd (1964).
[2]. Jeynes, William H. : Examining The Effects Of Parental Absence On The Academic Achievement Of Adolescents: The Challenge Of Controlling For Family Income, Journal Of Family and Economic Issues (2002), 23(2).
[3]. Hochschild, Jennifer L. : Social Class In Public Schools, Journal Of Social Issues (2003). 59(4), 821-840.
[4]. Grewal, A. : An International Study Of Cognitive And Socioeconomic Correlations Of School Achievement, J. Psy. Res. (1985), 29(1):41-45.
[5]. Velz, E., Schiefelbein, E. & Valenzuella, J. : Factors Affecting Achievement In Primary Education, Washington, DC: Human Resources Development And Operations Policy, The world bank (working paper series) (1993).

Abstract: Idempotent elements are left identity in the Semigroup generated by the left action of each singular element on Symmetric group.

Keywords: Left Identity; Idempotent; Symmetric Group; Singular Semigroup.

[1]. Araújo, Joao. and Cameron, J. Peter (2013) : Permutation Groups and Transformation Semigroups: Results and Problems,
arxiv.org/pdf/1308.3585.pdf, 25, 1-15.
[2]. Gonca Ayik, Hayrullah Ayik, Leyla Bugay and Osman Kelekci (2012): Generating Sets of Finite Singular Transformation
Semigroups, Semigroup Forum, Research Article Dedicated to the Memory of John Mackintosh Howie (1936-2011), DOI
10.1007/s00233 - 012 - 9397 - 1, Springer Science + Business Media, LLC 2012.
[3]. Gomes, G.M.S. and Howie, J.M. (1987): On the Ranks of Certain Finite Semigroups of Transformations, Math. Proc. Camb. Philos. Soc. 101. 395-403.
[4]. Howie, J. M. (1966): The Subsemigroup Generated by the Idempotents of a Full Transformation Semigroup, Journal of London
Math. Soc., (41), 707-716.
[5]. Howie, J. M. (1995): Fundamentals of Semigroup Theory, Oxford University Press New York.

Abstract: The aim of this paper is to introduce pgrw-closed maps and pgrw*-closed maps and to obtain some of their properties. In section 3 pgrw-closed map is defined and compared with other closed maps. In section 4 composition of pgrw-maps is studied. In section 5 pgrw*-closed maps are defined.

Keywords: Pgrw-closed set, pgrw-closed maps, pgrw*-closed maps.

[1]. Malghan S. R. (1982), Generalized Closed Maps, J Karnataka Univ.Sci., 27 : 82-88.
[2]. Nagaveni N. (1999). Studies on Generalizations of Homeomorphisms in Topological Spaces, Ph.D. Thesis, Bharathiar University,Coimbatore,
[3]. Long ,Herington(1978)., Basic Properties of Regular Closed Functions, Rend. Cir. Mat. Palermo, 27, 20-28,
[4]. Gnanambal Y. (1997), On generalized pre regular closed sets in topological spaces, Indian J Pure. Appl. Math., 28(3):351-360.
[5]. S. S. Benchalli, R.S., Wali(2007), On rw-Closed sets is Topological Spaces, Bull, Malays, Math, Sci, Soc., 30, 99-110.

Abstract: In this note we will see how representations of the two-dimensional Unitary Group U(2) used long ago by the Author for the Many Electron Problem, gives rise to an equilateral triangle in a torus that also defines a Theta Function with Equiharmonic frequencies and whose vertices are up and down spins leading to spin-orbit coupling.

Keywords: Equiharmonics, Jacobi Theta Function, Coupling Constant, Lattice.

[1]. J.H. Conway and N.J.A. Sloane, Sphere Packings, Lattices and Groups, Springer NY (1993
[2]. C.T.H. Davies et. al., Update: Accurate determination of q from realistic lattice QCD, arXiv: 0807.1687v [hep-lat] Dec (2008).
[3]. J.A.de Wet , The Many Electron Problem. Proc.Camb. Phil.Soc 72, (1972).
[4]. J.A. de Wet, The Quark Coupling Constant without QCD. IOSR JM Volume 12 Issue 4 Version II (2016).
[5]. Bruce Hunt, The Geometry of some special Arithmetic Quotients, Springer Lecture Notes in Mathematics 1637.

Abstract: The transverse free vibration of electromagnetic nanobeams subjected to an initial axial tension based on nonlocal stress theory is presented. It considers the effects of nonlocal stress field on the natural frequencies and vibration modes. The effects of a small-scale parameter at molecular level unavailable in classical macro-beams are investigated for three different types of boundary conditions: simple supports, clamped supports and elastically constrained supports. Analytical solutions for transverse deformation and vibration modes are derived. Through numerical examples, effects of the dimensionless Hartmann number, nano-scale parameter andpre-tension on natural frequencies are presented and discussed.

Keywords: Nanobeam, Natural frequency, Nonlocal stress, Pre-tensioned, Vibration, mode, Hartmann number

[1]. Mote, C.D.Jr., 1965. A study of band saw vibrations. Journal of the Franklin Institute, 279(6):430-444. [doi:10.1016/0016-0032(65)90273-5]
[2]. Mote, C.D.Jr., Naguleswaran, S., 1966. Theoretical and experimental band saw vibrations. ASME Journal of Engineering Industry, 88(2):151-156.
[3]. Simpson, A., 1973. Transverse modes and frequencies of beams translating between fixed end supports. Journal of Mechanical Engineering Science, 15(3):159-164.[doi:10.1243/JMES_JOUR_1973_015_031_02]
[4]. Oz, H.R., Pakdemirli, M., Boyaci, H., 2001. Non-linear vibrations and stability of an axially moving beam with time-dependent velocity. International Journal of Non-Linear Mechanics, 36(1):107-115. [doi:10.1016/S0020-7462(99)00090-6]
[5]. Liu, Y.Q., Zhang, W., 2007. Transverse nonlinear dynamical characteristic of viscoelastic belt. Journal of Beijing University of Technology, 33(11):10-13.

Abstract: In this paper an introduction of Elliptic curve cryptography explained Then the Diffie- Hellman algorithm was explained with clear examples.

Keywords: Cryptography Elliptic curve cryptography, Diffie-Hellman Key exchange.

[1]. Cole, Eric, Jason Fossen, Stephen Northcutt, Hal Pomeranz. SANS Security Essentials with CISSP CBK, Version 2.1.USA: SANS Press, 2003.
[2]. J.Edge,an introduction to elliptic curve ,cryptography, http://Iwn.net/Articles/174127/.2006.
[3]. N.Koblitz,A course in Number theory and cryptography,2nd ed.,brookes/Cole,1997.
[4]. .J.H.Silverman,The Arithmetic of Elliptic curves, Springer –Verlag,1986.
[5]. RSA‖ Wikipedia.wikipedia,n.d.web.09 feb 2011.Stalings,William. Cryptography and network security.fourth,pearson,2009.print.

Abstract: The performance of a series of Complete and Incomplete Neighbour Balanced Block Designs for Auto Regressive (AR), Moving Average (MA) and Nearest Neighbour (NN) error correlation structure is studied when generalized least squares estimation is used. We have compared the efficiency of AR(2), MA(2) and NN correlation structures. It is observed that efficiency for direct as well as neighbour effects is high, in case of complete block designs for NN correlation structure. In case of incomplete block designs MA(2) correlation...........

Keywords: Neighbour Balanced Block Design; Correlated observations; Generalized least squares; Auto Regressive; Moving Average; Nearest neighbour; Efficiency; Regular Block Design.

[1]. Anderson, A. H., Jensen, E.B., and Schou, G.: Two-way analysis of variance with correlated errors, Int. Statist. Reve. 49, 153-167
(1981).
[2]. Aastveit, A.H.: On the effect of correlation between plots in randomized block experiments. Biometrical J. 25, 129-153 (1983).
[3]. Azais, J.M., Bailey, R. A., and Monod, H.: A catalogue of efficient neighbour designs with border plots, Biometrics, 49, 1252-1261
(1993).
[4]. Box, G.E.P.: Some theorems of quadratic forms applied in the study of analysis of variance problems, II. Effects of inequality of
variance and of correlation between errors in the two-way classification. Ann. Math. Statist. 25, 484-498 (1954).
[5]. Bailey, R. A.: Designs for one-sided neighbour effects, J. Ind. Soc. Agric. Statistics, 56(3), 302-314 (2003).

Abstract: This paper discusses asymptotic stability for autonomous systems by means of the direct method of Liapunov.Lyapunov stability theory of nonlinear systems is addressed .The paper focuses on the conditions needed in order to guarantee asymptotic stability by Lyapunov'ssecond method in nonlinear dynamic autonomous systems of continuous time and illustrated by examples.

Keywords: Lyapunov function, Lyapunov's second method, asymptotic stability, autonomous nonlinear differential system.

[1]. Debeljkovic D. L. J., BuzurovicI. M., (2011), "Lyapunov stability of Linear Continuous Singular", International Journal of
Information and systems sciences, vol. 7,pp. 247-260.
[2]. Luciano M. L., Eduardo N. V., Samuel I. N.,(2014) "On the construction of Regions of Stability", Pure and applied Mathematics,
Journal, vol.3,pp. 87-91.
[3]. Pukdeboon c.,(2011)," A Review of Fundamentals of Lyapunov Theory",The Journal of Applied Science, vol. 10, pp. 55-60.
[4]. Mawhin J.,(2015), "AlexandrMikhailovichLiapunov, The general problem of the stability of motion (1892)", ResearchGate, pp. 1-
17.
[5]. Mitropolskii y. A., Borne P., Martynyuk A.A.,(2007), "Nonlinear Dynamics and systems Theory", vol. 2 pp.113-120.

Abstract: In this paper, some sufficient conditions that ensure second order delay difference inequality have no eventually positive solution are obtained

Keywords: Positive solution, Inequality, Second order and Difference inequality

[1] Agarwal. R.P., Difference Equations and Inequalities, Marcel Decker, New York, 1999.
[2] Erbe. L.H and Zhang. B.G., Oscillation of discrete analogues of delay equations, Diff and Integ. Equations, 2(1989), 300-309.
[3] Hardy G.H., Littlewood J.E and Polya G., Inequalities, Cambridge University Press, 1952.
[4] Ladas G., Explicit conditions for the oscillation of difference equations, J. Math. Anal. Appl., 153(1990), 276-287.
[5] Ladas. G and Stavroulakis. I.P., Oscillation caused by several retarded and advanced arguments, J. Diff. Eqns., 44(1982), 134-152..

Abstract: In this paper, a new algorithm to find an optimal sequence of ' ' jobs that has to be processed through ' ' machines is developed by treating the processing times of jobs on the machines as fuzzy (imprecise) variables. The algorithm is similar to that of Johnson (1954) which considers the processing times as precise in nature. The proposed algorithm makes use of the concepts of 'Credibility Theory' due to Liu (2008). An illustration of the proposed algorithm is given by representing the processing times as triangular fuzzy numbers.

Keywords: Credibility measure, credibility distribution, membership function, fuzzy variable, expectation and variance of fuzzy variable, ranking of fuzzy variable, sequencing

[1]. Johnson, S.M. Optimal Two and Three Stage Production Schedules with Set up Times Included. Nav. Res. Log. Quart. 1954; 1. 61– 68p.
[2]. Palmer, D.S. Sequencing Jobs through a Multi Stage Process in the Minimum Total Time – A Quick Method of Obtaining a Near Optimum. Operational Research Quarterly. 1965; 16(1). 101–107p.
[3]. Smith, R., Dudek, A. A General Algorithm for the Solution of the n Job m Machine Sequencing Problem of the Flow Shop. Operations Research. 1967; 15. 71–82p.
[4]. Campbell, G., Dudek, A., Smith, L. A Heuristic Algorithm for the n Job, m Machine Sequencing Problem. Management Science. 1970; 16(10). B. 630–637p.
[5]. Zadeh, L.A. Fuzzy Sets. Information and Control. 1965; 8(3). 338–353p.

Abstract: A generalized double sampling estimator for the estimation of finite population variance is proposed, its bias and mean squared error are found, and its comparison with the usual estimator of the finite population variance is made. A subclass of estimators depending upon optimum values for which the subclass attains the minimum mean square error is investigated and further a subclass of estimators depending upon estimated optimum value is also searched and its properties are studied.

Keywords: Optimum estimators, Bias and mean square error, Estimated optimum and Efficiency

[1]. COCHRAN, W. G. (1977). Sampling Techniques (3rd Edition), John Wiley & Sons, Inc., New York.
[2]. MURTHY, M. N. (1967). Sampling Theory and Methods, Statistical Publishing Society, India
[3]. RIZVI, S. A. H. and SINGH, R. KARAN (2016). An Alternative Estimator of Finite Population Variance Using Auxiliary Information, Chapter 21, Recent Advances in Applied Statistics and its Applications. pp. 173-176 Lambert Academic Publishing, Republic of MOLDOVA, CHISINAU, EUROPE.
[4]. RIZVI, S. A. H., and RIZVI, S. A. M. (2009). Some Estimators of Population Variance Using Auxiliary Information, Journal of Applied Research, Vol. I, No.1, June-2009, pp. 54-61, INDIA.
[5]. RIZVI, S. A. H., RIZVI, S. A. M., SINGH, R. and SINGH, R. KARAN (2015). Difference and Regression Type Estimators for the Estimation of Finite Population Variance Using Auxiliary information (2015), International Journal of Statistics and Systems (IJSS), vol. 10, No. 1, pp. 111-115, INDIA