iosr-Jm   Volume-12 ~ Issue-5 ~ Sep-Oct 2016

Abstract: The article aims at knowing the type of factors by which a multiplication of integers is produced. By putting forward the concept of composite seed of an odd composite number, the article constructs a sieve to select and produce odd composite numbers of the form 6n1 and then demonstrates several theorems related with factorization of an odd composite number in terms of its seed. It shows that factorization of a big odd number can be converted into that of small numbers incorporated with the big number's seed. The article also makes an investigation on new characteristics of odd numbers............

Keywords: Sieve, Odd numbers, Integer Factorization, Information security

[1] L E Dickson. History of the Theory of Numbers, Chelesea publishing Company, New York.1971
[2] S Sarnaik，D Gadekar，U Gaikwad. An Overview to Integer Factorization and RSA in Cryptography, International Journal for Advance Research in Engineering and Technology (IJARET),2014,2(IX):21-27
[3] XX Liu，XX Zou and JL Tan. Survey of large integer factorization algorithms, Application Research of Computers,2014,13(11):3201-3207
[4] WANG Xingbo. Valuated Binary Tree: A New Approach in Study of Integers, International Journal of Scientific and Innovative Mathematical Research (IJSIMR), Volume 4, Issue 3, March 2016, PP 63-67
[5] CD Pan and CB Pan, Elementary Number Theory(3rd Edition), Press of Peking University, 2013,pp-22

Abstract: Among the many families of wavelets available in the literature, Shannon wavelets offer some more specific advantages which are usually missing in the others such as infinite differentiability, analytical expressions, shapely boundedness in the frequency domain, enjoying a generalization of the Shannon sampling theorem, giving rise to the connection coefficients which can be analytically defined for any order derivatives and since it is a known fact that wavelets have been finding enormous applications in science and technology since 1980, so it becomes very fruitful..............

Keywords: Connection coefficients, 𝐶2− functions, Integro – differential equations, Multiresolution Analysis, Shannon scaling function, Shannon wavelet

[1]. Bachmana, N.L. and Beckenstein , E., (2000), Fourier and wavelet Analysis (springer - verlay New york , Inc.)
[2]. Cooley , J., Tukey , J . (1965) An algorithm for the machine calculation of complex Fourier series , Math . Com . 19 , 297-301 .
[3]. Debnath, L., (2002), Wavelet Transforms and their Applications (Birkhauser.)
[4]. Hong, D. Wang , J. and Gardner , R . (2005), (Real Analyisis with an introduction to wavelets and Applications (Academic Press.)
[5]. Cattani, C.,(2006), Connection coefficients of Shannon wavelets, Mathematical Modelling and Analysis, vol. 11, no. 2, pp. 117–132.

Abstract: In this paper we study some properties of the modified new Bernstein type operators introduced by M. A. Siddiqui et. al. in 2014. We show that some properties of the original function such as shape preserving properties, smoothness properties, etc. are preserved by these modified operators. We obtain an estimate on rate of convergence of these operators in terms of modulus of continuity. We also study pointwise approximation by these operators with use of Peetre's K-functional and Ditzian-Totik moduli of smoothness and obtain a direct theorem for these operators. 2010 AMS Subject Classification: Primary 41A36

Keywords: Linear positive operators, Bernstien operators, Rate of convergence, Modulus of continuity.

[1] B. M. Brown, D. Elliot and D. F. Paget , The Lipschitz constant for the Bernstein polynomials of a Lipschitz continuous function, J. Approx. Theory , 49 (1987), 196–199
[2] N. Deo, M. A. Noor and M. A. Siddiqui, On approximation by a class of new Bernstein type operators, Appl. Math. Comput., 201 (2008), 604–612.
[3] Z. Ditzian, Direct estimate for Bernstein polynomials, J. Approx. Theory, 97 (1994)(1), 165–166.
[4] Z. Ditzian, V. Totik, Moduli of smoothness, Springer Ser. Comput. Math., 9 (1987), Springer-Verlag, New York.
[5] G. G. Lorentz, Bernstein Polynomials, Chelsea Publishing Company, New York, 1986.

Abstract: The combined effect of thermal radiation and viscous dissipation on steady Magnetohydrodynamic heat and mass transfer flow of viscous incompressible micropolar fluid past a continuous moving plate with suction and injection is studied. A set of similarity parameters is employed to convert the governing partial differential equations into ordinary differential equations. Numerical results for the dimensionless velocity, microrotation, temperature and concentration profiles as well as the local skin-friction coefficient, the couple
stress coefficient, the local Nusselt number and the local Sherwood number are displayed graphically for various physical parameters. Results show that the skin friction coefficient as well as the local Sherwood number is increases as radiation parameter R increases for both the cases section and injection. It is also observed that the local Nusselt number decreases as radiation parameter R increases...........

Keywords: Radiation; Viscous Dissipation; MHD; Micropolar Fluid; Suction/Injection

[1]. A.C. Eringen, Theory of micropolar fluids, J. Math. Mech., 16, (1966), pp. 1-18.
[2]. A.C. Eringen , Theory of thermomicrofluids, J. Math. Anal. Appl., 38, (1972), pp. 480 - 496.
[3]. B.C. Sakiadis, Boundary layer behavior on continuous solid surface; the boundary layer on a continuous flat surface, A.I.Ch.E Journal, 7 (1961), pp. 221-231.
[4]. T Ariman, M.A Turk and N.D Sylvester: "Boundary-layer theory for a micropolar fluid". Recent Adv. Engng Sci., 5, (1974) pp. 405-426.
[5]. J. Peddieson and R.P. McNitt, Boundary layer theory for micropolar fluid, Recent Adv. Eng. Sci., 5 (1970) pp. 405–426.

Abstract: Suppose K is a nonempty closed convex nonexpansive retract of real uniformly convex Banach space E with P as a nonexpansive retraction. Let T: K → E be a nonexpansive non-self map with F(T)  {xK :Tx  x} . suppose { } n x is generated iteratively by , ((1 ) [(1 ) ]), 1 n 1 n n n n n n n x K x  P  x  TP  x  Tx  n  1 Where { } n  and { } n  are real sequences [ ,1 ]for some (0,1) . (1) If the dual E* of E has the Kadec-Klee property, then weak convergence of { } n x to some ( ) * x F T is proved. (2) If T satisfies condition (A), then strong convergence of { } n x to some ( ) * x F T is obtained.............

Keywords: Nonexpansive non-self map, Demiclosed map, Kadec-Klee property, Strong convergence.

[1]. F. E. Browder, Semicontractive and semiaccretive nonlinear mappings in Banach spaces, Bull. Amer. Math.soc 74(1968)660-665.
[2]. W. J. Davis, P. Enflo, Contractive projections on lp- spaces, Analysis at Urbana 1, Cambrtdge Universtiy Press, New York, 1989, pp. 151-161.
[3]. J. Diestel, Geoery of Banach spaces- Selected Topics, Lecture Notes in Mathematics, vol. 485, Springer, New York 1975.
[4]. J. G. Falset, W. Kaczor, T. Duczumow, S. Reich, weak convergence theorem forasymptotically nonexpansive mapping and semigroups, Nonlinear Aanl. 43(2001) 377-401.
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Abstract: We find necessary and sufficient conditions under which a regular shifted sampling expansion hold on 𝑉 (𝜑(𝑡𝑑)) 𝑚𝑑=1 and obtain truncation error estimates of the sampling series. We also find a sufficient condition for a function in 𝐿2(ℝ) that belongs to a sampling subspace of 𝐿2(ℝ). We use Fourier duality between 𝑉 (𝜑(𝑡𝑑)) 𝑚𝑑=1 and 𝐿2[0,2𝜋] to find conditions under which there is a stable asymmetric multi-channel sampling formula on 𝑉 (𝜑(𝑡𝑑)) 𝑚𝑑=1....

Keywords: Shift invariant space, sampling expansion, Multi-channel sampling , Frame Riesz basis.

[1]. Adam zakria , Ahmed Abdallatif , Yousif Abdeltuif [17] sampling expansion in a series of shift invariant spaces ISSN 2321 3361 © 2016 IJESC
[2]. O. Christensen, An Introduction to Frames and Riesz Bases (Birkh¨auser, 2003).
[3]. J. R. Higgins, Sampling Theory in Fourier and Signal Analysis: Foundations (Oxford University Press, 1996).
[4]. G. G.Walter, A sampling theorem for wavelet subspaces, IEEE Trans. Inform. Theory 38 (1992) 881–884.
[5]. O. Christensen, An Introduction to Frames and Riesz Bases, Birkhäuser, Boston, 2001.

Abstract: A proper vertex coloring of a graph G is equitable if the size of color classes differ by at most 1. The equitable chromatic number, denoted by ( ) e  G , is the smallest integer k for which the graph G is equitably colorable. We investigate the equitable chromatic number of some path related graphs..............

Keywords: Proper Coloring, Chromatic Number, Equitable

[1] R. Balakrishnan and K. Ranganathan, A Text book of Graph Theory (2/e, Springer, 2012).
[2] W. Meyer, Equitable Coloring, The American Mathematical Monthly, 80(8), 1973, 920-922.
[3] K. W. Lih and P. L. Wu, On Equitable coloring of Bipartite Graphs, Discrete Mathematics, 151 , 1996, 155-160.
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Sciences, 13(2) , 2000, 190-194.

Abstract: All successive natural numbers less than 10n, for any positive integer n, are extensively analyzed of successive occurrence of digit 0. The formula for the number of successive occurrences of 0's is given. Formulae for the very first instance of successive 0's and their last occurrence are also provided.

Keywords: Digit 0, natural numbers, successive occurrences.

[1] Neeraj Anant Pande, Numeral Systems of Great Ancient Human Civilizations, Journal of Science and Arts, Year 10, No. 2 (13),
2010, 209-222.
[2] Neeraj Anant Pande, Analysis of Occurrence of Digit 1 in Natural Numbers Less Than 10n, Advances in Theoretical and Applied
Mathematics, Volume 11, Number 2, 2016, 99-104.
[3] Neeraj Anant Pande, Analysis of Successive Occurrence of Digit 1 in Natural Numbers Less Than 10n, American International
Journal of Research in Science, Technology, Engineering & Mathematics, 16(1), 2016, 37-41.
[4] Neeraj Anant Pande, Analysis of Non-successive Occurrence of Digit 1 in Natural Numbers Less Than 10n, International Journal of
Advances in Mathematics and Statistics, Communicated, 2016.
[5] Neeraj Anant Pande, Analysis of Occurrence of Digit 0 in Natural Numbers Less Than 10n, American International Journal of
Research in Formal, Applied and Natural Sciences, Communicated, 2016