iosr-Jm   Volume-7 ~ Issue-6

Abstract: By Andreev's theorem and Choi's theorem, we proved that the degree of each vertex is three and the number of vertices of orderable compact Coxeter polyhedral is at most 10. Therefore a combinatorial polyhedron is a 3-connected planner graph. From the Plantri program, we found that the number of 3-connected planner graphs with at most 10 vertices of degree 3 is 9. We find that only five planner graphs among these 9 graphs satisfy the properties of orderable compact Coxeter polyhedra. Then we verify the polyhedra which are associated with these 5 planner graphs are orderable. Therefore the number of combinatorial polyhedra of orderable and deformable compact hyperbolic Coxeter polyhedra is five up to symmetry.

[1]. Dhrubajit Choudhury. The graphical investigation of orderable and deformable compact Coxeter polyhedral in hyperbolic space.
[2]. E.M. Andreev. On convex polyhedral of finite volume in Lobacevskii space. Math. USSR Sbornik 10, 413-440 (1970)
[3]. E.M. Andreev. On convex polyhedral of finite volume in Lobacevskii space. Math. USSR Sbornik 12, 255-259 (1970)
[4]. S. Choi. Geometric Structures on Orbifolds and Holonomy Representations. Geom. Dedicata 104, 161-199 (2004)
[5]. S. Choi, W.M. Goldman. The deformation spaces of convex
n RP -structures on 2-orbifolds. Amer. J. Math. 127, 1019-1102
(2005)
[6]. S. Choi. The deformation spaces of projective structures on 3-dimensional Coxeter orbifolds. Geom. Dedicata 119, 69-90 (2006)
[7]. D. Cooper, D. Long, M. Thistlethwaite. Computing varieties of representation s of hyperbolic 3-manifolds into SL4,R .
Experiment. Math. 15,291-305 (2006)
[8]. H. Garland. A rigidity theorem for discrete subgroups. Trans. Amer. Math. Soc. 129, 1-25 (1967)
[9]. W.M. Goldman. Convex real projective structures on compact surfaces. J. Differential Geom. 31,791-845 (1990)
[10]. V.G. Kac, E.B Vinberg. Quasi-homogeneous cones. Math. Zamnetki 1, 347-354 (1967)

Abstract: We study ruled surfaces with lightlike ruling in Minkowski 3-space which are said to be null-scrolls. Even that the result is a consequence of some well-known results involving the Gauss map, we give another approach to classify all null-scrolls under the condition where is the Laplace operator with respect to the first fundamental form and the set of 3real matrices.

Key words: Null frame, Lorentzian ruled surface, Gauss map.

[1] T. Takahashi. Minimal immersions of Riemannian manifolds, J. Math. Soc. Jpn. 18(1966),380-385. [2] O. J. Garay. An extension of Takahashi's theorem, Geom. Dedicate 34 (1990) 105-112. [3] C. Baikoussis and D. E. Blair. On the Gauss map of ruled surfaces, Glasgow Math. J. 34 (1992), 355-359. [4] S. M. Choi, On the Gauss map of ruled surfaces in a 3-dimensional Minkowski space,Tsukuba J. Math. 19 (1995), 285-304. [5] L. J. Alias, A. Ferrandez,P. Lucas, and M. A. Merono. On the Gauss map of B-scrolls. Tsukuba J. Math. 22 (1998), 371-377.
[6] B. O'Neill. Semi-Riemannian geometry, Academic Press. Inc., 1983. [7] L. Graves. Codimension one isometric immersions between Lorentz spaces, Trans. Amer. Math. Soc. 252 (1979), 367-392. [8] K. L. Duggal, A. Bejancu. LightlikeSubmanifolds of Semi-Riemannian Manifolds and Applications, Kluwer Academic Publishers, 1996. [9] S. M. Choi,U. H. Ki and Y. J. Suh. On the Gauss map of null scrolls, Tsukuba J. Math. 22 (1998), 273-279. [10] L. K. Graves. Codimension one isometric immersions between Lorentz spaces, Trans. Amer. Math. Soc. 252 (1979), 367-392.

Abstract: Ourmainconcern, in this paper is how group theory quietly involve in decision making , Problem solving and critical thinking in games which could also be helpful to organizational management, policy making, Politics, Business, Science, Health care , Career choice and many other fields, yet unknowingly to many. We show in this work how Group theory can be applied to16- puzzles very effectively. The game we present here is organizational puzzleon management decision making.

Keyword: Decision- making, Group Theory, Games, Organizational Management, 16- Puzzle.

[1] H.Wussing, The Genesis of the abstract group concept: A contribution to the History of the origin of Abstract Group theory, New York: Dover Publication (2007) http://en.wikipedia.org/wiki/Group/(Mathematics).

[2] C. Cecker "Group theory and a look at the slide puzzle" Retrieved from Google search, August 2010 unpublished (unpublished Manuscript) P1 (2003).

[3] Problem Solving and critical thinking Retrieved from Google search July 2013, titled "Mastering soft skills for workplace success" (unpublishedmanuscript) p 98.
[4] Group Theory in games, retrieved from http://groupprops.subwiki.org/wiki/Group_theory_in_games August 2010 titled "Groupprops" (unpublished manuscript) (2008) p1

[5] E. Pavel, "Groups around Us", Retrieved from Google search unpublished (Unpublished manuscript) p1-2

[6] Group theory for puzzles Retrieved from Google search on "Useful Mathematics" page Unpublished (unpublished Manuscript) August, 2008 p11
[7] A. Archer, A Modern Treatment of the 15-puzzle, Retrieved from www.2cs.cmu.edu/afs/cs/academic/class/15859-f01/www/notes/15-puzzle.pdf (1999).

[8] T. Vis, Cycles in groups and graph University of Colorado Denver (2008) pg7

Abstract: This paper deals with a functional equation of an optimal inventory equation having unbounded time period and one period lag in supply .The existence of the solution for this equation is proved through a dynamic programming approach.

Keywords: Inventory, Optimization, Multi objective Allocation.

[1] R.R.Bellman,,Dynamic programming,(Princetion University,Princetion,NJ,1957)

[2] P.C. Bhakta. and S. Mitra, Some existence theorems for functional equations arising in dynamic programming,J.Math.Appl.,98,1984,340-361.

[3] K.D. Senapati, and G. Panda ,Solution of a functional equation arising in continous games:A dynamic programming approach,SIAM J Control and optimization 413 ,2002,820-825.

[4] G. Panda, Some Existence Theorems for functional equations in dynamic programming,Advances in modeling and Analysis(France)Vol.14,No.4, 1993,1-10.

[5] F. Braue,A note on uniqueness and convergence of successive approximations,Canad.Math.Bull.,1959,5-8.

Abstract: The application of kinetic modelling of toxicity, adsorption and biodegradation of phenolic compounds is an active area of research which uses the tool of mathematical modelling to understand the complex interaction of substrate depletion and biomass production. However, it is imperative to test the sensitivity of the model parameters which define these key processes of applied microbiology in order to find out which of these parameters will have either a biggest or smallest cumulative effect on the model output or solution trajectory. Since sensitivity analysis is an integral part of model development and is capable of providing useful insights for a further validation research in applied biodegradation of phenolic compounds, it is a challenging collaborative scientific investigation to attempt to find which model parameters of this microbiological system would require additional research for the purpose of strengthening the continuity of knowledge base thereby reducing output uncertainty. In this numerical study, we have used the technique of sensitivity measures or sensitivity analysis to select the maximum specific growth rate, the experimental time and the starting substrate value as the relatively sensitive parameters while other model parameters can be classified as relatively least sensitive. We will expect these contributions to guide further research in the validation of Monod models of substrate depletion and biomass production.

Key words: Microbial Growth, Substrate Concentration, Kinetics, Monod Model, Sensitivity Measures

[1] Breshears, D.D.: 1987, Uncertainty and sensitivity analyses of simulated concentrations of radionuclides in milk. Fort Collins, CO: Colorado State University, MS Thesis, pp. 1-69.
[2] Crick, M.J., Hill, M.D. and Charles, D.: 1987, 'The Role of Sensitivity Analysis in Assessing Uncertainty. In: Proceedings of an NEA Workshop on Uncertainty Analysis for PerformanceAssessments of Radioactive Waste Disposal Systems, Paris, OECD, pp. 1-258.
[3] Ekaka-a E.N., Computational and Mathematical Modelling of Plant Species Interactions in a Harsh Climate, PhD Thesis, University of Liverpool and University of Chester, 2009.
[4] Ekaka-a, E.N. and Nafo,N.M: 2012, Parameter Ranking of Stock Market Dynamics: A Comparative Study of the Mathematical Models of Competition and Mutualistic Interactions, Scientia African, Vol. 11 (No. 1), June 2012, pp. 36-43.
[5] Gardner, R.H.: Huff, D.D., O'Neill, R.V., Mankin, J.B., Carney, J. and Jones, J.: 1980, 'Application of Error Analysis to a Marsh Hydrology Model', Water Resources Res. 16, 659-664.
[6] O'Neill, R.V., Gardner, R.H., and Mankin, J.B. 1980, 'Analysis of Parameter Error in a Nonlinear Model', Ecol. Modelling.8, 297-311.
[7] Downing, D.J., Gardner, R.H., and Hoffman, EO.: 1985, 'An Examination of Response-Surface Methodologies for Uncertainty Analysis in Assessment Models', Technometrics. 27, 151-163. (See also Letter to the Editor, by R.G. Easterling and a rebuttal, Technometrics28, 91-93, 1986).
[8] Nweke (2010) PhD Thesis
[9] Okpokwasili, G.C and Nweke, C.O 2005, Microbial growth and substrate utilization kinetics, African Journal of Biotechnology, Vol. 5(4), pp. 305-317.
[10] Yu, C., Cheng, J-J., and Zielen, A-J.: 1991, 'Sensitivity Analysis of the RESRAD, a Dose Assessment Code,' Trans. Am. Nuc. Soc. 64; 73-74.

Abstract: In this paper the arrival of calls (new and handoff) in personal communications services (PCS) network are modeled according a Markov Arrival Process (MAP). The purpose of our study is to improve the performance of the cellular mobile system by using directed retry scheme. The cut of priority scheme have also been applied to provide priority to handoff calls. Here we construct tractable model using fixed channel assignment scheme with priority to handoff voice calls. The proposed directed retry scheme with finite buffering for handoff voice calls is useful for improving the handoff blocking probability of handoff voice calls. The integrated traffic model has been constructed for finite population. FIFO fashion has been taken for arrival calls. The balking behavior of the calls has also been taken into consideration. The expressions for various performance indices viz. blocking probabilities for different traffic, overall blocking probability, offered carried load etc. are determined for above model. It is clear from the analytical results that by increasing overlapping area and retrying in overlapping area, a substantial improvement can be achieved.

Keywords: Markov Arrival Process, fixed channel assignment scheme, directed retry scheme, balking, FIFO, Handoff.

[1] Anderson, L. G. [1973]: A simulation study of some dynamic channel assignment algorithms in a high capacity mobile telecommunications system, IEEE Trans. Veh. Technol., Vol. VT-22, pp. 210-217.
[2] Cox, D. C. and Reudink, D. O. [1973]: Increasing channel occupancy in large scale mobile radio systems: Dynamic channel
[3] Eklundh, B. [1986]: Channel utilization and blocking probability in a cellular mobile telephone system with directed retry, IEEE Trans. Commun., Vol. COM-34, No. 4, pp. 329-337.
[4] Zhang, M. and Yum, T. S. [1989]: Comparisons of channel assignment strategies in cellular telephone systems, IEEE Trans. Veh. Technol., Vol. 38, pp.211-215.
[5] Hong, D. and Rappaport, S. S. [1986]: Traffic model and performance analysis for cellular mobile radio telephone systems with prioritized and non-priritized handoff procedures, IEEE Trans. Veh. Technol., Vol. VT-35, No. 3, pp. 77-91.
[6] Srivatava, N. and Rappaport, S. S. [1991]: Models for overlapping coverage area in cellular and micro-cellular communication systems, IEEE GLOBECOM 91, Phoenix, AZ Dec. 2-5, pp. 26.3.1-26.3.5.
[7] Yum, T. S. P. and Yeung, K. L. [1995]: Blocking and handoff performance analysis of directed retry in cellular mobile systems, IEEE Trans. Veh. Technol., Vol. 44, No. 3, pp. 645-650.
[8] Chu, T. P. and Rappaport, S. S. [1997]: Overall coverage with reuse partitioning in cellular communication systems, IEEE Trans. Veh. Technol., Vol. 46, No. 1, pp. 41-54.
[9] Jain, M. [1999]: Finite population cellular radio systems with directed retry, Applied Mathematical Modelling, Vol. 23, pp. 77-86.
[10] Marsan, M. A., Chiasserini C., and Fumagalli, A. [2001]: Performance Models of Handover Protocols and Buffering Policies in Mobile Wireless ATM Networks, IEEE Trans. on Vehicular Technology. Vol. 50, No.4, 925-941.

Abstract: We classified the combinatorial structures of 3-dimensional bounded ODCH (orderable and deformable compact hyperbolic) Coxeter polyhedra up to symmetry. Using Plantri graphs and graph theory, we proved that the number of such combinatorial polyhedra is five. One of such combinatorial polyhedra is tetrahedron. Using graph theory and combinatorics, we find that there are six orderable and deformable compact hyperbolic tetrahedra up to symmetry.

[1] Dhrubajit Choudhury. The graphical investigation of orderable and deformable compact Coxeter polyhedral in hyperbolic space.
[2] Dhrubajit Choudhury, Pranab Kalita. Application of Plantri graph: All Combinatorial structure of Orderable and Deformable
Compact Coxeter Hyperbolic Polyhedra. IOSR Journal of Mathematics (IOSR-JM), e-ISSN: 2278-5728, p-ISSN: 2319-765X,
Volume 7, Issue 3, PP 63-67.
[3] E.M. Andreev. On convex polyhedral of finite volume in Lobacevskii space. Math. USSR Sbornik 10, 413-440 (1970)
[4] E.M. Andreev. On convex polyhedral of finite volume in Lobacevskii space. Math. USSR Sbornik 12, 255-259 (1970)
[5] S. Choi. Geometric Structures on Orbifolds and Holonomy Representations. Geom. Dedicata 104, 161-199 (2004)
[6] S. Choi, W.M. Goldman. The deformation spaces of convex 2 RP -structures on 2-orbifolds. Amer. J. Math. 127, 1019-1102 (2005)
[7] S. Choi. The deformation spaces of projective structures on 3-dimensional Coxeter orbifolds. Geom. Dedicata 119, 69-90 (2006)
[8] D. Cooper, D. Long, M. Thistlethwaite. Computing varieties of representation s of hyperbolic 3-manifolds into SL4,R .
Experiment. Math. 15,291-305 (2006)
[9] H. Garland. A rigidity theorem for discrete subgroups. Trans. Amer. Math. Soc. 129, 1-25 (1967)
[10] W.M. Goldman. Convex real projective structures on compact surfaces. J. Differential Geom. 31,791-845 (1990)

Abstract: The present paper deals with the determination of displacement and thermal stresses in a thick annular disc with internal heat generation. Arbitrary heat 𝑓 𝑟 is applied on the upper surface of disc whereas lower surface dissipates heat by convection and the fixed circular edge are thermally insulated. Here we compute the effects of internal heat generation of a thick annular disc in terms of stresses along radial direction. The governing heat conduction equation has been solved by the method of integral transform technique. The results are obtained in a series form in terms of Bessel's functions. The results for temperature change, displacement and stresses have been computed numerically and illustrated graphically.

Keywords :Thermal stresses, internal heat generation, annular disc, steady state.

[1] Deshmukh K.C., " Generalized transient heat conduction problem in a thin hollow cylinder", Far East Journal of Applied Mathematics, 6(3), 253-264, (2002).

[2] Gogulwar V. S. and Deshmukh K.C., "An inverse quasi-static thermal stresses in an annular disc", Proceeding of ICADS, Narosa Publishing House, New Delhi, (2002).

[3] Kulkarni V. S. and Deshmukh K.C., "Quasi- static steady state thermal stresses in thick annular disc", Journal of thermal stresses, 31, 331-342, (2008).

[4] Naotake Noda, Richard B Hetnarski and Yoshinobu Tanigawa, " Thermal Stresses", 2nd edn. 259-261, Taylor, and Francis, New York, (2003).

[5] Ozisik M. N., " Boundary value problem of heat conduction", International Text Book Company, Scranton, Pennsylvania, 135-148, (1968).

[6] Shang-sheng Wu, "Analysis on transient thermal stresses in an annular fin", Journal of Thermal Stresses, 20, 591-615, (1997).

Abstract: The dynamics of the best-fit mathematical model which can beused to describe the interaction between cowpea and groundnut is an importantscientific problem. In this study, we propose to use the popular Euclideannorm or 2-norm cost function method to select the best-fit model parameters.Having selected the best-fit mathematical model, we intend to determine theimportant model parameters of this interacting system. The application ofstability and stabilization of this proposed mathematical model will studiedand presented. Our novel results which we have not seen elsewhere will bepresented and discussed with the expectation of providing further insightsabout these crop growth time series data.

Key words and phrases: Best-fit parameters, agricultural data, 2-norm, stability, stabilization.

[1] M.A. Ekpo and A.J. Nkanang, Nitrogen fixing capacity of legumes and their Rhizosphereal microflora in diesel oil polluted soil in the tropics, Journal of Petroleum and Gas Engineering 1(4), (2010), pp. 76-83.

[2] E.N. Ekaka-a, Computational and Mathematical Modelling of Plant Species Interactions in a Harsh Climate, PhD Thesis, Department of Mathematics, The University of Liverpool and The University of Chester, United Kingdom, 2009.

[3] C. Damgaard, Evolutionary Ecology of Plant-Plant Interactions-An Empirical ModellingAp-proach, Aarthus University Press, 2004.

[4] E. Renshaw, Modelling Biological Populations in Space and Time, Cambridge University Press, 1991.

[5] M. Kot, Elements of Mathematical Ecology, Cambridge University Press, 2001.

[6] J.D. Murray, Mathematical Biology, 2nd Edition Springer Berlin, 1993.

[7] D. Tilman, Dynamics and Structure of Plant Communities, Princeton University Press, 1988.

[8] T.O. Ibia, M.A. Ekpoand L.D. Inyang, Soil Characterisation, Plant Diseases and Microbial Survey in Gas Flaring Community in Nigeria, World J. Biotechnol. 3, (2002), pp. 443-453.

[9] R. Hunt, Studies in Biology, no. 96: Plant Growth Analysis, London: Edward Arnold (Pub-lishers) Limited , 1981.

[10] J. Goudriaan, J.L Monteith, A mathematical function for crop growth based on light inter-ception and leaf area expansion, Annals of Botany 66, (1990), pp. 695-701.

Abstract: In this study, we discuss alternative methods of stabilizing theunique positive stable steady-state solutions of interacting biogas solids undersome simplifying assumptions of higher carrying capacities and initial conditions. Numerical results are presented and discussed.

Key words and phrases: Stabilization, Biogas Solids, Higher Carrying Capacities

[1] Yubin Yan, Enu-Obari N. Ekaka-a, Stabilizing a mathematical model of population system,Journal of the Franklin Institute, 348, pp. 2744-2758, 2011.

[2] X. Zhou, J. Cui, Stability and Hopf bifurcation analysis of an eco-epidemiological model withdelay, Journal of the Franklin Institute, 347, pp. 1654-1680, 2010.

[3] Y. Yan, D. Coca, V. Barbu, Feedback control for Navier-Stokes equation, Nonlinear FunctionalAnalysis and Optimization, 29, pp. 225-242, 2008.

[4 Y. Yan, D. Coca, V. Barbu, Finite-dimensional controller design for semilinearparabolicsystems, Nonlinear Analysis: Theory and Applications, 70, pp. 4451-4475, 2009.

Abstract: This paper studied the 2–point improved block backward differentiation formula for solving stiff initial value problems proposed by Musa et al (2013) and further established the necessary conditions for the convergence of the method. It is shown that the method is both zero stable and consistent. The order of the method is also derived.

Keywords: Improved block BDF, Zero stability, consistency conditions for block method, convergence, order.

[1] C. Curtiss, J.O. Hirschfelder, Integration of stiff equations, Proceedings of the National Academy of Sciences of the United States
of America. 38 (1952), pp. 235-243.
[2] D. Voss, S. Abbas, Block predictor-corrector schemes for the parallel solution of ODEs, Computers and Mathematics with
Applications. 33 (1997),
65-72.
[3] G. Hall, J. M. Watt Modern Numerical Methods for Ordinary Differential Equations, Clarendon Press, 1976.
[4] H. Musa, M. B. Suleiman, F. Ismail, N. Senu, Z. B. Ibrahim The convergence and order of the 3-point block extended backward
differentiation formula, ARPN Journal of Engineering and Applied Sciences. 7(12) (2012) pp. 1539-1545.
[5] H. Musa, M. B. Suleiman, F. Ismail, N. Senu, Z. B. Ibrahim An improved 2-point block backward differentiation formula for
solving stiff initial value problems, AIP Conference Proceedings. 1522 (2013), pp. 211-220.
[6] Henrici, P. Discrete variable methods in ordinary differential equations, New York: Wiley, 1 1962.
[7] J. D. Lambert, Computational Methods in Ordinary Differential Equations, Chichester, New York, 1973.
[8] J. R. Cash, On the integration of stiff systems of ODEs using extended backward differentiation formulae, Numerische Mathematik.
34 (1980), 235-246.
[9] L. F. Shampine, S. Thompson, Stiff systems, Scholarpedia. 2(3) (2007), 28-55.
[10] L. Brugnano, F. Mazzia, D. Trigiante, Fifty years of stiffness, Recent Advances in Computational and Applied Mathematics. 2011,
pp. 1-21.

Abstract: This paper explores a Mathematical model for obtaining the minimum total cost of Multi-items with Mult-objectives in a Fuzzy Environment. The paper incorporates four constraints such as warehouse space and the constraint, investment amount constraint, percentage of utilization of volume of the ware house space and the shortage level constraint by allowing shortages. The previous researchers had analyzed different models on this relevant topic without shortages. But this paper mainly focuses on model with shortages and above mentioned constraints. Ware house maintenance is one of the essential parts of service operation. The ware house space in selling stores plays an important role in stocking the goods. In the proposed model, the ware house space is considered in volume. The demand is taken as function of unit cost price which is taken as triangular fuzzy number. This paper also includes a detailed numerical example to elucidate our proposed strategy.

Keywords: Inventory model, volume of the ware house, triangular fuzzy number, shortages, capital investment.

[1]. Inventory systems by Eliezer Naddor
[2]. Fuzzy sets and logics by Zadeh
[3]. E.A Silver, R. Peterson, Decision systems for Inventory Management and production planning, New York: John Wiley, 1985
[4]. R.E. Bellman, L.A Zadeh, "Decision- making in a fuzzy environment", Management science, 17(4), B141-B164, 1970.
[5]. H.J. Zimmermann, "Description and optimization of fuzzy systems", Internet. J. General systems 2(4), 209-215,1976
[6]. Debdul Panda, Samarjit Kar, "Multi – item stochastic and fuzzy – stochastic inventory models under imprecise goal and cjhance constraints" Advanced modeling in optimization, (7), 155-167, 2005.
[7]. G.M Arun Prasath, C.V Seshaiah, "Multi-item multi objective fuzzy inventory model with possible constraints"ARPN Journal of Science and Technology VOL 2 NO 7 ,August 2012.
[8]. K.S Park, "Fuzzy set theoretic interpretation of economic order quantity", IEEEE Trans. Systems Man cybernet, 17(6), 1082-1084 ,1987.
[9]. Nirmal Kumar Mandal, ET. Al., "Multi-objective fuzzy inventory model with three constraints: a geometric programming approach". Fuzzy sets and Systems, (150), 87-106,2005
[10]. H. Tanaka, et al., "On fuzzy mathematical programming", J. Cybernet,3(4), 37-46,1974

Abstract: In this present paper we introduce a subclass of analytic functions with negative coefficients. We study coefficient bounds, distortion properties, covering theorem, extreme points, radius of close to convexity, star likeness, and convexity and integral transformations for the functions in this class. The results of this paper generalize many earlier results in this direction. Mathematics subject classification: Primary 30C45

Keywords: Analytic functions, star like and convex functions, k-uniformly convex functions, negative coefficients.

[1] Goodman A.W.:- "On uniformly convex functions", Ann. Pal. Math. 56 (1991), 87-92.
[2] Goodman A.W.: "On uniformly starlike functions" J.of Math. Anal And Appl. 155 (1991), 364-370.
[3] Kanas.S and Srivastava.H.M.: Linear operators associated with k-uniformly convex functions, Integral transforms Spec. Funct., 9,
(2000), 121 – 132.
[4] Wisniowska. A. and Kanas. S: Conic regions and K-uniform convexity, J. Comput, Appl. Math., 105 (1999), 327-336.
[5] Ma.W and Minda.D: Uniformly convex functions, Ann, Polon, Math, 57 (1992), 165 – 175.
[6] Ronning. F.:- "Uniformly convex functions and a corresponding class of starlike functions, proc. Amer. Math. Soc. 118 (1993), 189 -
196.
[7] Silverman H.:- Univalent functions with negative coefficients, Proc. Amer. Math. Soc. 51(1975), 109-116.

Abstract: In 1990 M. Matsumoto has considered the projectively flatness of Finsler spaces with (, )-metric [5]. In particular, the Randers space, the Kropina space and the special generalized Kropina space are considered in detail. Here  is Riemannian metric and  is a differential one-form. In the present chapter we consider the projective flatness of Finsler space with special (, )-metrics. In particular, Matsumoto metric 2/(  ), special generalized Matsumoto metric 2/(  ) and the metric  + (2/) are considered in detail.

[1] M. Hashiguchi and Y. Ichijyo: Rep. Fac. Sci. Kagoshima Univ. (Math., Phys. & Chem), 13 (1980), 33
[2] L. Berwald: Ann. of Math., 48 (1947), 755.
[3] M. Matsumoto: Tensor, N. S., 24 (1972), 29.
[4] M. Matsumoto: Foundations of Finsler Geometry and Special Finsler Spaces, Kaiseisha Press, Saikawa, Otsu 520, Japan, 1986.
[5] M. Matsumoto: Projectively flat Finsler space with an ( ) metric, Rep. Math. Phys. 30 (1991), 15.
[6] M. Matsumoto: Proc. Fifth Natl. Sem. Finsler & Lagrange spaces, Univ. Brasov, Romania, 1988, 233.
[7] C. Shibata: Rep. Math. Phys. 13 (1978), 117.

Abstract: The present manuscript deals with the heat transfer and thermal stress analysis of thick annular cylinder under steady temperature conditions. A annular cylinder is subjected to arbitrary heat flux applied on the upper surface with lower surface is thermally insulated. The fixed circular edges are at zero temperature. The integral transform methods are used for heat transfer analysis to determine temperature change. The theory of linearized thermoelasticity based on solution of Naviers equation in terms of Goodiers thermoelastic displacement potential, Michell's function, and the Boussinesq's function for cylindrical co-ordinate system have been used for thermal stress anaylsis. The results for temperature change, displacement and stresses have been computed numerically and illustrated graphically.

Key Words: Heat transfer analysis, Steady state, Thermal Stress analysis.

[1]. Kulkarni V.S. and Deshmukh K.C., "Quasi-static thermal stresses in a thick circular plate‟ Applied Mathematical Modelling, Vol-31, issue 8, pp 1479-1488, 2007.
[2]. Kulkarni V.S. and Deshmukh K.C., "Thermal stresses in a thick annular disc‟ Journal of Thermal stresses, Vol-31, issue 4, pp 1-8, 2008.
[3]. Lee C.W., "Thermal stresses in thick plate", Journal of Solids and Structures, Vol.6, issue 5, pp 605-615, 1970.
[4]. Lee Z.Y. Chen C.K. and Hung C.L., "Transient Thermal stress analysis of multilayered hollow cylinder", Acta Mechanica, Vol.151, pp 75-88,2001
[5]. Naotake Noda, Richard B Hetnarski and Yoshinobu Tanigawa. "Thermal Stresses‟ 2nd edn. pp 259-261, 2003. Taylor and Francis New York.
[6]. Ozisik,M.N., "Boundary value problem of heat conduction‟ ,International Text book Company,Scranton,Pennsylvania,pp84-87,1968.
[7]. Singh D.V., "Thermal stresses in an infinite thick with circular cylindrical hole" Vol.33, No.1 and 2, 1966, Sneddon I.N., "The Use of Intergral Transform‟, McGraw Hill,New York,pp 235-238, 1972.
[8]. Toshiaki Hata, "Thermal stresses in a nonhomogeneous thick plateunder steady distribution of temperature", Journal of Thermal Stresses, Vol.5,v issue 1, pp 1-11, 1982.

Abstract: The time series data of interacting legumes grown in oil uncontam- inated utisol have been collected. However, the dynamics of a best-fit math- ematical model that can be used to describe the interaction between cowpea and groundnut poses a challenging interdisciplinary approach. We propose to use the 1-norm penalty function method to select the best-fit model parame- ters from a list of other candidate logistic models. A mathematical analysis of this best-fit interspecific interaction model will be conducted. The novel results which we have achieved in this study will be presented and discussed.

Key words: and phrases. Best-fit parameters, agricultural data, 1-norm.

[1] M.A. Ekpo and A.J. Nkanang, Nitrogen fixing capacity of legumes and their Rhizosphereal microflora in diesel oil polluted soil in the tropics, Journal of Petroleum and Gas Engineering 1(4), (2010), pp. 76-83.

[2] E.N. Ekaka-a, Computational and Mathematical Modelling of Plant Species Interactions in a Harsh Climate, PhD Thesis, Department of Mathematics, The University of Liverpool and The University of Chester, United Kingdom, 2009.

[3] C. Damgaard, Evolutionary Ecology of Plant-Plant Interactions-An Empirical Modelling Ap- proach, Aarthus University Press, 2004.

[4] E. Renshaw, Modelling Biological Populations in Space and Time, Cambridge University Press,1991.

[5] M. Kot, Elements of Mathematical Ecology, Cambridge University Press, 2001.

[6] J.D. Murray, Mathematical Biology, 2nd Edition Springer Berlin, 1993.

[7] D. Tilman, Dynamics and Structure of Plant Communities, Princeton University Press, 1988.

[8] T.O. Ibia, M.A. Ekpoand L.D. Inyang, Soil Characterisation, Plant Diseases and Microbial Survey in Gas Flaring Community in Nigeria, World J. Biotechnol. 3, (2002), pp. 443-453.

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Abstract: In the classical Harris Wilson inventory model al the cost associated with the formula was taken to be constant and which does not dependent on any quantity. In this paper we have taken Ordering cost, holding cost and order quantity all are triangular fuzzy numbers. Graded mean integration representation method is used for defuzzification. In this paper we consider an inventory model where the holding cost depends on order quantity. An algorithm is developed to find the economic order quantity along with numerical examples.

Keywords: Fuzzy inventory model, Triangular fuzzy numbers, Defuzzification

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Abstract: Examine the theoretical foundations of Galois algebra and its contemporary applications. It is generally acknowledged that one of mathematics' most complex subfields is the theory of Galois. In A Classical Introduction to Galois Theory, the topic is treated further from a historical perspective, with the main emphasis being on the radical solvability of polynomials. Through the usage of the book, the computational techniques that are typical of early writing on the subject are progressively changed into the more abstract approach that is typical of the bulk of current expositions. Fundamental principles are presented by the author in a clear and understandable manner......

Key word: mathematics, Galois theory

[1]. Artin, Emil (2013) Galois Theory. Dover. ISBN 0-486-62342-4.
[2]. Bewersdorff, Jörg (2013). Galois Theory for Beginners: A Historical Perspective. The Student Mathematical Library. Vol. 35. American Mathematical Society. doi:10.1090/stml/035. ISBN 0-8218-3817-2. S2CID 118256821.
[3]. Brantner, Lukas; Waldron, Joe (2013), Purely Inseparable Galois theory I: The Fundamental Theorem, arXiv:2010.15707
[4]. Cardano, Gerolamo (2013). Artis Magnæ (PDF) (in Latin).
[5]. Edwards, Harold M. (2013). Galois Theory. Springer-Verlag. ISBN 0-387-90980-X. (Galois' original paper, with extensive background and commentary.)