iosr-Jm   Volume-14 ~ Issue-1 ~ Jan-Feb 2018

Abstract: In this paper we extend to infinite triangular arrays the concept of triangular domino systems and triangular tiling systems which recognize infinite triangular arrays and show that the class of -triangular array languages recognized by triangular domino systems is same as the class of -triangular array languages recognized by triangular tiling systems. Also we introduce triangular Wang systems and prove that the class of -triangular languages obtained by triangular Wang systems is the same as the class of recognizable -triangular languages.

Keywords: Infinite triangular domino systems,Wang recognizable infinite triangular array languages, labelled
triangular Wang tile.

[1]. Ahmed Saoudi,Takashi Yokomoni,Learning "Local and Recognizable Omega languages and Monadic Logic Programs", proceedings of the first European conference on computational learning theory,pages 157-169,October 1994, Royal Holloay Univ of Londan,United Kingdom.
[2]. Dare V.R., K.G.Subramanian, D.G.Thomas and R.Siromoney, Infinite arrays and recognizability, Int.J.Pattern Recognition and Artificial intelligence 14(2000), 525-536.
[3]. V.Devi Rajaselvi, T.Kalyani, D.G.Thomas "HRL-Local infinite triangular picture languages" Published in Brazillian Archives of Biology and Technology" vol. 59, n.spe. 2 e16161076,January-December 2016,ISSN 1678-4324.
[4]. V.Devi Rajaselvi, T.Kalyani, D.G.Thomas, "Domino Recognizability of triangular picture languages", International Journal of Computer Applications Vol.57, No.15, (ISSN 0975-8887), Nov-2012.
[5]. Prophetis L.De and S.Varricchio.Recognizability of rectangular pictures by Wang systems.Journal of Automata,Languages and Combinatorics,4:269-288,1997.

Abstract: Cholera is an infection of the small intestine of humans caused by a gram negative bacterium called Vibrio cholerae. It is spread through eating food or drinking water contaminated with faeces from an infected person. It causes rapid dehydration and general body imbalance, and can lead to death since untreated individuals suffer severely from diarrhoea and vomiting. In this paper we formulate a mathematical model to assess the role of rehydration and antibiotic treatment on reduction of cholera mortality. All solutions in our model are positive and bounded hence well posed. The stability analysis of the model has been done. Numerical simulation shows that rehydration and administration of antibiotics play a major role in reducing cholera deaths.

Keywords: Cholera Disease, Role of Rehydration and Antibiotic treatment.

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Abstract: An analysis of complex functions and mapping that are holomorphic in nature are studied and discussed in this paper through Riemann surfaces, which involves Riemann Mapping theorem and Caratheodory's theorem. Furthermore Montel's theorem, Runge's theorem and Mergelyan's theorem over the holomorphic nature is studied with its basic properties and developed in this paper. To enhance the reliability over the nature of holomorphism the metrics of Riemann surface and conformal maps of plane to disk is analyzed and studied in this paper.

Keywords: Holomorphic functions, Conformal mapping, Riemann mappings, Riemann surface and Holomorphic open sets.

[1]. Carl Friedrich Gauss (1831) "The geometric representation of complex numbers since 1796, submitted his ideas to the Royal Society of Gottingen.

[2]. Carathéodory, C. (1954), Theory of functions of a complex variable, Vol. 2, translated by F. Steinhardt, Chelsea

[3]. John B. Conway (1978). Functions of One Complex Variable I. Springer-Verlag. ISBN 0-387-90328-3.

[4]. Carathéodory, C. (1998), Conformal representation (reprint of the 1952 second edition), Dover, ISBN 0-486-40028-X.

[5]. Pommerenke, C. (1992), Boundary behaviour of conformal maps, Grundlehren der Mathematischen Wissenschaften, 299, Springer, ISBN 3-540-54751-7.

Abstract: To study problems in geometry the technique known as Differential geometry is used. Through which in calculus, linear algebra and multi linear algebra are studied from theory of plane and space curves and of surfaces in the three-dimensional Euclidean space formed the basis for development of differential geometry during the 18th century and the 19th century. Since the late 19th century, differential geometry has grown into a field concerned more generally with the geometric structures on differentiable manifolds. The differential geometry of surfaces captures many of the key ideas and techniques characteristic of this field..

Keywords: Curvature Manifolds, Riemannian geometry and surface of revolutions.

[1]. Hestenes, David (2011). "The Shape of Differential Geometry in Geometric Calculus" (PDF). In Dorst, L.; Lasenby, J. Guide to Geometric Algebra in Practice. Springer Verlag. pp. 393–410. There is also a pdf available of a scientific talk on the subject.

[2]. Calavi and Roselicht ―complex analytic manifolds without countable base, Proc. Amer. Math. Soc (1953)pg335-340.

[3]. Manton, Jonathan H. (2005). "On the role of differential geometry in signal processing". doi:10.1109/ICASSP.2005.1416480.

[4]. Love, David J.; Heath, Robert W., Jr. (October 2003). "Grassmannian Beamforming for Multiple-Input Multiple-Output Wireless Systems" (PDF). IEEE Transactions on Information Theory. 49 (10): 2735–2747. doi:10.1109/TIT.2003.817466.

[5]. Micheli, Mario (May 2008). The Differential Geometry of Landmark Shape Manifolds: Metrics, Geodesics, and Curvature(PDF) (Ph.D.)..