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.