iosr-Jm   Volume-7 ~ Issue-3

Abstract: Every count ably normed nuclear space is isomorphic to a subspace of a nuclear Frechet space with basis and a continuous norm. The proof as given in section 2 is a modification of the Komura-Komura inbedding theorem .In this paper, we shall show that a nuclear Frechet space with a continuous norm is isomorphic to a subspace of a nuclear Frechet space with basis and a continuous norm if and only if it is countably normed.The concept of countably normedness is very important in constructing the examples of a nuclear Frechet space. Moreover, the space with basis can be chosen to be a quotient of (s)

Key words and phrases: Nuclear frechet space, Countably normed and Nuclear Kothe space.

[1] Adasch,N.: Topological vector spaces,lecture notes in maths, springer-verlag,1978.
[2] Grothendieck, A: Topological Vector spaces,Goedan and Breach,NewYork,1973.
[3] Holmstrom, L: A note on countably normed nuclear spaces, Pro. Amer. Math. Soc. 89(1983), p. 453.
[4] Litvinov, G.I.: Nuclear Space,Encyclopaedia of Mat Hematics, Kluwer Academic Publishers Spring-Verlag, 2001.
[5] Komura,T. and Y.Komura : Uber die Einbettung der nuclear Raume in (S)
N , Mathe,Ann.162,1966, pp. 284-288.

Abstract: Some popular recreational games have a group- theoretic foundation, and group are useful in modelling games that involve a series of discrete moves, with each move leading to a change in the board state. In this work we present to you our local game called "tsorry" ( meaning they line up straight). Which we carefully examine and found that it has a board and checkers, each possible moves has an element of group, where the effect of performing a sequence of moves corresponds to the product of those elements. The group under consideration here is Klein four- group.

Keyword: Checkerboard, Group theory, Games, Klein four – group, Possible moves and Tsorry Checkerboard,

[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 Publications (2007) http://en.Wikipedia.org/wiki/Group(Mathematics)
[2] Y.K. Algon , "Peg solitaire and Group theory" Retrieved from google search August 2010 ( unpublished) p1 (2006)
[3] Kriloff, " A Group theory explanation for a Rubik's cube". (2005) Retrieved from http://groupprops.subwiki.org/wiki/ Group theory
in Games, August 2010
[4] C. Cecka, "Group Theory and a look at the Slide Puzzle" Retrieved from google search unpublished ( unpublished manuscript )
p1 (2003)
[5] Checker Board Games Retrieved on net August 2010 titled "checker board game review". (unpublished manuscript)
[6] J. Loy's, "The basic rules of Checkers", Jim Loy's Home page www.jimloy.com/checkers/rules2htm unpublished (unpublished
Manuscript)
[7] G.G. Hall , Applied group theory, American Elsevier publishing Co., Inc., New York, (1967) MR0219593, an elementary
introduction.
[8] E. Pavel, "Groups Around Us", Retrieved from Google search unpublished ( Unpublished manuscript) p4 (2009)
[9] Klein four- group, Retrieved from www.jmline.org/math/CourseNotes/GT.pdf unpublished (unpublished manuscript)

Abstract: This paper discusses the concepts and methodology used in the system theory and operations research in respect of planning and control of the sustainable development. The sustainable development of any industrial park is not achieved merely by the introduction of foreign capital and embedding into global value chains system. Any sustainable development problem can be represented mathematically using the concepts of transition of system from the given initial state to the final state. It is established in this paper that sustainable development represents a specific control problem. The aim is to keep the system in the prescribed feasible region of the state space. Further the analysis of planning and control problems of sustainable development has also shown that methods developed in the operations research such as multicriterion optimization, dynamic processes, simulation and goal programming etc. are more than adequate.

Keywords: Sustainability, sustainable development, systems approach, multiple criteria optimization, state transition.

[1] Adams, W.M., "The future of sustainability, rethinking environment and development in the twenty first century", Report IUCN, 2006. [2] Allen, W., "Learning for sustainability", Sustainable Development, 2007.

[3] Donald Brown, 'The role of economics in sustainabledevelopment and environmental protection', in Sustainable Development: Science, Ethics and Public Policy, edited by John Lemons and Donald Brown, Kluwer Academic Publishers, Dordrecht, 1995, pp. 52-63.

[4] ESCAP. 1994. UN Economic and Social Commission for Asia and the Pacific Committee on Environment and Sustainable Development. Note by the Secretariat. E/ESCAP/ESD (2)/4.

[5] Galloppin, G., "A systems approach to sustainability and sustainable development", UN Publication, Santiago, Chile, 2003.

[6] Hanley, N., Shorgen, J., and White. B., Environmental Economics in Theory and Practice,Palgrave, London, 2007.

[7] Harris, J.,"Basic principles of sustainable development", Tufts University, Medford MA,USA, 2000.

[8] Munasinghe, M. (1993) Environmental Economics and Sustainable Development, World Bank Environment Paper Number 3, The World Bank, Washington, D.C. [9] D. Meadows, D. Medows, J. Randers, and W. Berens III, The Limits to Growth, Universe Books, New York, 1972.

[10] Pearce, D., "An intellectual history of environmental economics", Annual Review of Energy and Environment, 27 (2002) 57-81.

Abstract: This paper present valuation models which is define as the expected discounted value of a stream of each flows at a time. Three equivalent forms of this value process is established with each of which has its own merits. Local dynamics of the values process is also considered.

Keywords: Cash Flow, Valuation, Continuous, Deflator, Filtration.

Armerin, F. (2002), 'Valuation of Cash Flows in Continuous Time', Working paper Bias, B., et al (1997), 'Financial Mathematics', Lecture Notes in Mathematics, 1656, Springer – Verlag Delbean, F., et al (1994),

'A General Version of the Fundamental Theorem of Asset Pricing', Math. Annalen, Vol. 300, 463-520 Duffie, D., et al (1995), 'Black's consol rate conjecture', The Annals of Applied Probability, Vol. 5, No. 2, 356-382 Harrison, J.M., et al (1979),

'Martingales and Arbitrage in Multi-period Securities Markets', Journal of Economic

Theory Hubalek F., (2001), 'The Limitations of No-Arbitrage Arguments for Real Options', International Journal of Theoretical and Applied Finance Vol. 4, No. 2