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

Abstract: The tensor product GH of two graphs G and H is well-known graph product and studied in detail in the literature. This concept has been generalized by introducing 2-tensor product 2 G H and it has been discussed for special graphs like n P and n C [5]. In this paper, we discuss 2 G H , where G and H are connected graphs. Mainly, we discuss connectedness of 2 G H and obtained distance between two vertices in it.

Keywords: Bipartite graph, Connected graph, Non-bipartite graph, 2-tensor product of graphs.

[1]. R. Balakrishnan and K. Ranganathan, A Textbook of Graph Theory, Universitext, Springer, New York, 2012.
[2]. I. H. Naga Raja Rao and K. V. S. Sarma, On Tensor Product of Standard Graphs, Int. J. Comp., 8(3), (2010), 99-103.
[3]. Sirous Moradi, A note on Tensor Product of Graphs, Iranian J. of Math. Sci. and Inf., 7(1), (2012), 73-81.
[4]. E. Sampathkumar, On Tensor Product of Graphs, J. Austral. Math. Soc., 20(Series-A), (1975), 268-273.
[5]. U. P. Acharya and H. S. Mehta, 2- Tensor Product of special Graphs, Int. J. of Math. and Scietific Comp., 4(1), (2014), 21-24.
[6]. Biggs. N. L., Algebraic Graph Theory, Cambridge University Press, Cambridge, 1993.
[7]. N. Deo, Graph Theory with Application in Engineering and Computer Science, Prentice-Hall of India, Delhi, 1989.

Abstract: In this note we invoke the Hessian Polyhedron with 3 real and imaginary axes that yield the 27 apices of the graph of E6 shown in Fig.1. In particular we will identify the quarks as belonging to the complex space in the outer ring, which accounts for their tiny masses that can only be estimated.

Keywords: E6 Theta Functions, Quarks, Hessian Polyhedron, Equiharmonic Lattice

[1]. H.S.M. Coxeter, The Polytope 2₂₁ whose 27 vertices correspond to the lines on the general cubic surface, Am. Mathematical Soc. 62(1940) 467-486.
[2]. H.S.M. Coxeter, Regular Complex Polytopes, Camb. Univ. Press (1991)
[3]. J.A.de Wet, An exact derivation of the quark coupling constant without QCD, IOSR-JM,Vol.12(2016),online.
[4]. J.A. de Wet, Icosahedral Symmetry : A Review, International frontier Science Letters,Vol.5(2015) pp1-8.online.
[5]. Bruce Hunt, The Geometry of some special Arithmetic Quotients, Lecture Notes in Mathematics 1637, Springer (1996).
[6]. R.Slansky, Group theory for unified model building, Reprinted in 'Unity of Forces in the Universe' Ed.A.Lee, World Scientific (1992).

Abstract: In this paper a novel technique implementing Bernstein polynomials is introduced for the numerical solution of a Love's integral equations. The Love's integral equation is a class of second kind Fredholm integral equations, and it can be used to describe the capacitance of the parallel plate capacitor (PPC) in the electrostatic field.This numerical technique developed by Huabsomboon et al. bases on using Taylor-series expansion [1], [2], [3] and [4]. We compare the numerical solution using Bernstein technique with the numerical solution obtained by using Chebyshev expansion. It is shown that the numerical results are excellent.

Keywords: Love's integral equation, Fredholm integral equation, Bernstein polynomials method, approximate solution.

[1]. P. Huabsomboon, B. Novaprateep and H. Kaneko, On Taylor-series expansion methods for the second kind Integral equations, J.
Comput. Appl. Math. 234, 2010, pp. 1466–1472.
[2]. P. Huabsomboon, B. Novaprateep and H. Kaneko, Taylor-series expansion methods for nonlinear Hammerstein equations. (to be
appeared in Sci. Math. Jap.)
[3]. P. Huabsomboon, B. Novaprateep and H. Kaneko, Taylor-series expansion method for Volterra Integral Equations of the Second
Kind, Sci. Math. Jap. 73, 2011, pp. 19–29.
[4]. P. Huabsomboon, B. Novaprateep and H. Kaneko, Discrete Taylor-expansion method for Integral Equations, Int. J. Numer. Appl. 1, 2009, pp. 139–153.
[5]. A. T. Bharucha-Reid, Approximation Solution of Random Equations. North Holland, New York (1979).

Abstract: The purpose of this paper is to offer a method for studying the treatment result of gingival recession. A parameter showing the success of the surgical treatment of gingival recessions is the keratinized gingival width. It was measured four times: at baseline, after 1 month, after 3 months and after 6 months. Every patient has data that can be described by an individual trend. Bayesian hierarchical model of the keratinized gingiva width's increase rate is built.

Keywords: Biostatistics, gingival recession, hierarchical modeling, longitudinal data, split-mouth randomized clinical trial

[1] Iv. Chenchev, D. Atanasov, D. Vicheva, The treatment of gingival recessions - Our experience, Romanian Journal of Rhinology, 2016, 6 (22): 85-91.
[2] J. Weinmann, Progress of gingival inflammation into the supporting structures of the teeth, J Periodontol, 1941, 12: 71-82.
[3] C. Ochsenbein, Newer concept of mucogingival surgery, J Periodontol, 1960, 31: 175-185.
[4] U. Hangorsky, NF. Bissada, Clinical assessment of free gingival graft effectiveness on the maintenance of periodontal health, J Periodonol, 1980, 51: 274-278.
[5] NP. Lang, H. Löe, The relationship between the width of keratinized gingiva and gingival health, J Periodontol, 1972, 43(10): 623-627.

Abstract: Let I be a compact interval and f be a continuous function defined from I to I. We study the relationship between tubulent function, erratic function and Devaney Chaos.

Keywords: Devaney Chaos, Erratic Function, Turbulent Function

[1]. H. Roman Flores, Y. Chalco – Cano and G. Nunes Silva, Chaos induced by Turbulent and Erratic functions, IFSA - EUSFLAT.
2009, 231 – 233.
[2]. B. Aulbach and B. Kieninger, On Three Definitions of Chaos, Nonlinear Dynamics and Systems Theory, 1 (1), 2001, 23 – 37.
[3]. T. Y. Li and J. A. Yorke, Period three implies Chaos, Amer. Math. Mon. 82, 1975, 985 – 992.
Books:
[4]. R. L. Devaney, An introduction to Chaotic Dynamical systems, Addison – Wesley, Redwood city, CA 2nd edition (1989).
[5]. L.S. Block and W.A. Coppel, Dynamics in One Dimension, Lecture Notes in Mathematics no. 1513, Springer – Verlag 1992.

Abstract: The Work investigates..............

Keywords: Picone formules

[1]. J.I.Diaz; Nonlinear Differential equations and Free Boundaries. London, Pitman,1985.
[2]. N.Kawano, WM. Ni and S. Yotsutani; A generalized Pohozaev identity and its application, J.Math. Soc. Japan Vol.42,No3,1990
pp.541-564.
[3]. Claus Dohman ; Pohozaev-Type Inequality for a Quasilinear Haraux-Weissler Equation , Funkcialaj Ekvacioj 39 (1996) 389-397.
[4]. Tadi´e; Comparison Results for Quasilinear Elliptic equations via Picone-type Identity: Part II: cases where he coefficient of the
principal part depends on he unknown function J. Nonlinear Anal., T.M.A.,vol. 71(2009)e601-e606 .
[5]. Tadi´e ; Oscillation criteria for damped Quasilinear second-order Elliptic Equations , Electronic J. of Differential Equations vol.
2011(2011), No.151, pp.1-11.

Abstract: This paper deals with the influence of magnetic field on peristaltic flow of an incompressible Williamson fluid in a symmetric channel with heat and mass transfer. Convective conditions of heat and mass transfer are employed. Viscous dissipation and Joule heating are taken into consideration .Channel walls have compliant properties. Analysis has been carried out through long wavelength and low Reynolds number approach. Resulting problems are solved for small Weissenberg number. Impacts of variables reflecting the salient features of wall properties, concentration and heat transfer coefficient are pointed out. Trapping phenomenon is also analyzed.

Keywords: Williamson fluid, compliant walls, convective conditions.

[1]. K. Yazdanpanth-Ardakani and H. Niroomand-Oscii, New approach in modelling peristaltictransport of non-Newtonian fluid, J.
Mech. Med. Biol., 13 (2013) 1350052.
[2]. A.M. Abd-Alla, S. M. Abo-Dahab and H.D.El-Shahrany, Effects of rotation and initialstress on peristaltic transport of fourth grade
fluid with heat transfer and induced
magnetic field, J. Magn. Mater., 349 (2014) 268-280.
[3]. K. Nowar, Peristaltic flow of a nanofluid under the effect of Hall current and porousmedium, Math. Prob. Eng. (2014).
[4]. M. Kothandapani and J. Prakash, Effect of radiation and magnetic field on peristaltictransport of nanofluids through a porous space
in a tapered asymmetric channel, J.Magn. Magn.Mater. 378 (2015) 152-163.29
[5]. G. Radhakrishnamacharya, Ch. Srinivasulu, Influence of wall properties on peristaltictransport with heat transfer, C. R. Mecanique
335 (2007) 369-373.

Abstract: Vogel's Approximation Method (VAM) is one of the conventional methods that gives better Initial Basic Feasible Solution (IBFS) of a Transportation Problem (TP). This method considers the row penalty and column penalty of a Transportation Table (TT) which are the differences between the lowest and next lowest cost of each row and each column of the TT respectively. In a little bit different way, the current method consider the Average Row Penalty (ARP) and Average Column Penalty (ACP) which are the averages of the differences of cell values of each row and each column respectively from the lowest cell value of the corresponding row and column of the TT

Keywords: VAM, IBFS, TP, TT, ARP, ACP

[1] N. Balakrishnan, 'Modified Vogel's Approximation Method for Unbalance Transportation Problem,' Applied Mathematics
Letters 3(2), 9,11,1990.
[2] Serdar Korukoglu and Serkan Balli, 'An Improved Vogel's Approximation Method for the Transportation Problem', Association
for Scientific Research, Mathematical and Computational Application Vol.16 No.2, 370-381, 2011.
[3] H.H. Shore, 'The Transportation Problem and the Vogel's Approximation Method', Decision Science 1(3-4), 441-457, 1970.
[4] D.G. Shimshak, J.A. Kaslik and T.D. Barelay, 'A modification of Vogel's Approximation Method through the use of Heuristics',
Infor 19,259-263, 1981.
[5] Aminur Rahman Khan, 'A Re-solution of the Transportation Problem: An Algorithmic Approach' Jahangirnagar University
Journal of Science, Vol. 34, No. 2, 49-62, 2011.

Abstract: Abacus diagram is a graphical representation for any partition μ of a positive integer t. This study presents the bead positions as a unite square in the graph and de ne a special type of e-abacus called nested chain abacus 𝔑 which is represented by the connected partition. Furthermore, we redefined the polyominoes as a special type of e-abacus diagram. Also, this study reveals new method of enumerating polyominoes special design when e=2.

Keywords: Abacus, Polyominoes, n-connected square, enumeration, Free enumeration, Combinatorial.

[1] E lungo..............

Abstract: The fuzzy set theory has been applied in many fields such as management, engineering, theory of matrices and so on. in this paper, some elementary operations on proposed trapezoidal fuzzy numbers (TrFNS) are defined. We also have been defined some operations on trapezoidal fuzzy matrices(TrFMs). The notion of Hessenberg fuzzy matrices are introduced and discussed. Some of their relevant properties have also been verified.

Keywords:Fuzzy Arithmetic, Fuzzy number, Trapezoidal fuzzy number (TrFN), Trapezoidal fuzzy matrix(TrFM), Hessenberg fuzzy matrix(HFM).

[1]. Dubasis.D and prade.H, (1978), Operations on fuzzy numbers. International journal of systems, vol.9 no.6 pp 613-626
[2]. Dwyer.. (1965), P.S.Fuzzy sets information and control no.8., pp 338 – 353
[3]. Edgar Asplud (1959), Inverse of matrices (𝑎𝑖𝑗 ) which satisfy 𝑎𝑖𝑗 = 0 for 𝑗 > 𝑖 + 𝑝 math S and T, pp 57-60
[4]. Heliporn.S.J(1997), Representation and application of fuzzy numbers, fuzzy sets and systems, vol.91, no.2, pp 259-268.
[5]. Jaisankar.C, Arunvasan.S and Mani.R., On Hessenberg of Triangular fuzzy matrices, ijsret volume 5 issue 12, 586-591.

Abstract: We define an analogous pair of recurrence relations that yield some Fibonacci, Lucas and generalized Fibonacci identities with the coefficients in arithmetic progression. One relation yields same sign identities and the other alternating signs identities. We also show some new results for negative indexed Fibonacci and Lucas Sequences.

Keywords: linear recurrence relation; Fibonacci sequence; Lucas identity; generalized Fibonacci sequence, arithmetic progression.

[1]. Arthur Benjamin and Jennifer Quinn, Proofs That Really Count: The Art of Combinatorial Proof, MAA, 2003, pp1-35.
[2]. Bera Soumendra, Relationships between Ordered Compositions and Fibonacci Numbers, Journal of Mathematics Research, CCSE, Vol. 7, No. 3, 2015. Page 4.

[3]. Knuth Donald, Negafibonacci Numbers and the Hyperbolic Plane, http://www.allacademic.com/meta/p206842_index.html
[4]. Steven Vajda, Fibonacci and Lucas Numbers, and the Golden Section: Theory and Applications, Dover Publications, Inc. New York, 2008.

Abstract: A study of the SIS model of malaria disease with a view to observing the effects of self and cross-diffusion on spatial dynamics is undertaken. Three different cases based on self-diffusion and cross-diffusion are chosen for the investigation. Two cases of cross-diffusion without self-diffusion are also considered in order to see the effects of diffusion on the transmission of malaria. Basic reproductive numbers and bifurcation values are calculated for each case. A series of numerical simulations based on self and cross-diffusion is performed. It is observed that with positive cross-diffusion and self-diffusion in the system........

Keywords: Self and Cross-Diffusion, Disease-free equilibrium state, Endemic equilibrium state, Basic reproduction number, Numerical simulations, Bifurcation Values.

[1]. Center for disease control.............

Abstract: Now a days Surface fitting is applied all engineering and medical fields. Kamron Saniee ,2007 find a simple expression for multivariate LaGrange's Interpolation. We derive a least square plane and least square quadric surface Approximation from a given N+1 tabular points when the function is unique. We used least square method technique. We can apply this method in surface fitting also.

Keywords: Least square, quadric surface , Normal equations

[1]. M.K.JAIN ,S.R.KIYENGAR AND R.K.JAIN .Numerical methods for scientific and Engineering Computation ,Wiley Eastern Limited , 1985.
[2]. M. Gasca and T. Sauer.Polynomial interpolation in several variables. Advances in Computationalv Mathematics, 12(4):377–410, 2000.
[3]. P.J. Olver. On multivariate interpolation. Studies in Applied Mathematics, 116(4):201–240, 2006.
[4]. J.F. Steffensen. Interpolation.Dover Publications, Inc., New York, second edition, 2006. Copyright ©
[5]. Kamron Saniee , A simple expression for multivariate Lagrange Interpolation , 2007.

Abstract: Members of the Camellia genus of plants may be affected by many of the diseases, pathogens and pests that mainly affect the tea plant (Camellia sinensis). One of the most common diseases which are found in tea garden is known as Branch Canker (BC) [1]. Recent outbreaks of these diseases is not only hampering the production of tea, but also stupendously hampering our national economy. In this paper, a simple SEIR model has been used to analyze the dynamics of Branch Canker in the tea garden. The stability of the system is analyzed for the existence of the disease free equilibrium. We established that there exists a disease free equilibrium point that is locally asymptotically stable when the reproduction number Ro< 1 and unstable when Ro> 1. Theoretically, we analyze the BC model. Finally, we numerically tested the theoretical results in MATLAB.

Keywords: Mathematical Model, Tea disease, Stability Analysis, Reproduction Number, Epidemiology, Equilibrium, ODE.

[1]. Ali, M. and Huq, M., 1993, Branch Canker (Macrophoma) In Tea. pp. 5-11.
[2]. Food and Agriculture Organization, 1999.
[3]. PDU (Project Development Unit), 2015. Statistics on Bangladesh Tea Industry-2015. Bangladesh Tea Board, Srimangal, Moulvibazar. pp. 26-66.
[4]. Chen, Z.M. and Chen. X.F., 1990. The diagnosis of tea diseases and their control in Chinese. Shanghai Sci. Tech. publishers, Shanghai, China. pp. 73-88.
[5]. Sana, D. L., 1989, TEA SCIENCE. pp: 1-269.

Abstract: In spatial tasks, the use of cognitive aids reduce mental load and therefore being appealing to trainers and trainees. However, these aids can act as shortcuts and prevents the trainees from active exploration which is necessary to perform the task independently in non-supervised environment. In this paper we used adaptive repetition as control strategy to explore the 3D- Virtual Learning environments. The proposed approach enables the trainee to get the benefits of cognitive support while at the same time he is actively involved in the learning process. Experimental results show the effectiveness of the proposed approach.

Keywords: Virtual Environments, Adaptive 3D-Virtual Learning environments, Cognitive aids, Multimodal feedback, Multimodal Virtual laboratory (MMVL)

[1]. Ali, N., S. Ullah, and A. Alam, The Effect of Multimodalityand 3d Interaction In A Virtual Laboratory on Students'learning In Chemistry Education. Sindh University Research Journal-SURJ (Science Series),. 47(4), 2015.
[2]. Ali, N., et al., 3D Interactive Virtual Chemistry Laboratory for Simulation of High School Experiments. Paper presented at the Proceedings of EURASIA GRAPHICS 2014,Paper 12, 2014.
[3]. Alam, A., Ullah, S., Burqi, M. Z., Ullah, A., & Ali, N. , Evaluating Students Performance in Adaptive 3D-Virtual Learning Environments Using Fuzzy Logic. Sindh University Research Journal-SURJ (Science Series), 48(2): p. 245-250, 2016.
[4]. Troyer, D., F.K. O., and A. Ewais, Enhancing virtual reality learning environments with adaptivity: lessons learned. in Symposium of the Austrian HCI and Usability Engineering Group, Springer., 2010.
[5]. Ewais, A. and O.D. Troyer, Authoring adaptive 3D virtual learning environments. International Journal of Virtual and Personal Learning Environments (IJVPLE). 5(1): p. 1-19, 2014.

Abstract: This paper is concerned with the wxistence of soulation to p-Laplacian dynamic equation.............

Keywords: Time scales, p-Laplacian operator, Leray-Schauder nonlinear alternative threamem.

[1]. B.Aulbach, L.Neidhart, Integration on measure chains, in: Proceedings of the Sixth International Conference on Difference Equations, CRC, Boca Raton, FL, (2004)239-252.
[2]. Chengmin Hou, Yansheng He, Mingzhe Sun, Multiple Positive Solutions of Nonlinear Boundary Value Problem for Finite Fractional Difference, Absract and Applied Analysis, Article ID 147975. (2014)12.
[3]. F.Merdivenci Atici, On Green's function and positive solutions for bound-ary value problems on time scales, Journal of Computational and Applied Mathematics, 144(2012)75-99.
[4]. H.R.Sun, W.T.Li, Existence theory for positive solutions to one-dimensional p-Laplacian boundary value problems on time scales, J. Dif-ferential Equations. 240(2007)217-248.
[5]. I.Yaslan, Multiple positive solutions for nonlinear three-point boundary value problems on time scales, Comput. Math. Appl. 50(2005)1603-1620.

Abstract: This research work investigated the behaviour of a new semiparametric non-linear (SPNL) model on a set of panel data with very small time point (T = 1). The SPNL model incorporates the relationship between individual independent variable and unobserved heterogeneity variable. Five different estimation techniques namely; Least Square (LS), Generalized Method of Moments (GMM), Continuously Updating (CU), Empirical Likelihood (EL) and Exponential Tilting (ET) Estimators were employed for the estimation; for the purpose of modelling the metrical response variable non-linearly on a set of independent variables..........

Keywords: Panel data; Semiparametric Non-Linear; Model; Estimator.

[1]. Akeyede and B.L. Adeleke (2015). Nonlinear time series model. Unpublished Ph. D dissertation, University of Ilorin, Nigeria.
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Abstract: Classification comprises a variety of problems, which are solved in several ways. The need for automatic classification methods arises in a number of areas, from voice recognition, to modern automobiles, to the recognition of tumors through x-rays to assist doctors, by classifying emails as legitimate or spam. Due to the importance and complexity of such problems, there is a need for methods that provide greater accuracy and interpretability of the results. Among them the Boosting methods, which have emerged in the field of computation, work by sequentially applying a classification algorithm to reweighted versions of the training data set, giving greater weight to erroneous observations..........

Keywords: Coronary Heart Disease, Classifiers, AdaBoost, Sensibility, Machine Learning

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