iosr-Jm   Volume-18 ~ Issue-2 ~ Mar. – Apr. 2022

Abstract: 3x+1 sequences of numbers are the special cases of px+q sequences of numbers. Px+q sequences of numbers are the mapping recurrent sequences of numbers. The general term an in the classical sequences of numbers is a function of n, while the general term an of the px+q sequences of numbers is not a function of n. Usually, the classical sequences of numbers are not circular, while circularity is the most important property of the px+q sequences of numbers. The classical sequences of numbers only extend from left to right, while the px+q sequences of numbers can be extended from right to left.

Key words: Mapping recurrent sequences of numbers; px+q sequences of numbers; equiratio residual sequences of numbers; leftward-extendable sequences of numbers; px+q infinite trees

[1]. Ming Xian, Xunwei Zhou, Zi Xian, The proof of 3x+1 problem, IOSR Journal of Mathematics, Volume 17, Issue 2, Series 3, 05-12, Mar.-Apr. 2021
[2]. Kenneth H. Rosen, Elementary Number Theory and Its Applications, Fifth Edition, Beijing: Pearson Education Asia Limited and China Machine Press, 2005

Abstract:In this paper,we establish the Pohozaev-type identity for a kind of fourth order elliptic problem,which has the biharmonic operator.We discuss the problem in a class of domains that are more general than star-shaped ones.

Key words: Pohozaev-type identity;biharmonic operator.

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[2]. T.F. Feng, Pohoˇ zaev Identities of Elliptic Equations and Systems with Variable Exponents and Some Applications. Math. Appl. (Wuhan) 32 (2019) 581-589.
[3]. T.Q. An, Non-existence of positive solutions of some elliptic equations in positive-type domains, Appl. Math. Lett. 20 (2007) 681- 685.
[4]. Y. Li, R. An, K.T. Li, New Pohozaev identity and application to fourth order quasilinear elliptic equation, Journal of Xian J iaotong University 41(10) (2007) 1245-1247.
[5]. M. Otani,Existence and nonexistence of nontrivial solution of some nonlinear degenerate elliptic equations,J. Funct. Anal. 76(1)(1988) 140-159.

Abstract: In this present paper , we defined complex valued functions that are univalent of the form f = h+ where h and g are analytic in the open unit disk . we obtain several sufficient coefficient conditions for normalized harmonic functions that are starlike of order.....

Key words: Harmonic function ,univalent function sense-preserving; starlike.convex combination

[1]. Y. Avci and E. Zlotkiewicz, On harmonic univalent mappings, Ann. Uni¨. Mariae CurieSklodowska Sect. A
[2]. J. Clunie and T. Sheil-Small, Harmonic univalent functions, Ann. Acad. Aci. Fenn. Ser. A I Math.
[3]. P. L. Duren, A survey of harmonic mappings in the plane, Texas Tech. Uni¨. Math. Ser. 18 Ž . 1992
[4]. T. Sheil-Small, Constants for planar harmonic mappings, J. London Math. Soc.
[5]. H. Silverman, Univalent functions with negative coefficients, Proc. Amer. Math. Soc. 51
[6]. H. Silverman, Harmonic univalent functions with negative coefficients, J. Math. Anal.

Abstract: The concept of a bipolar intuitionistic fuzzy α-ideal and bipolar intuitionistic anti fuzzy α-ideal are a new algebraic structure of BP-algebra and to use special operators. The purpose of this study is to implement the fuzzy set theory and ideal theory of a BP-algebra.........

Key words: BP-algebra, fuzzy ideal, bipolar fuzzy ideal, bipolar intuitionistic fuzzy........

[1]. S.Abdullah and M.M.M. Aslam, Bipolar fuzzy ideals in LA-semigroups, World Appl. Sci. J., 17.12 (2012), 1769-1782.
[2]. S.S.Ahn and J.S.Han, On BP-algebra, Hacettepe Journal of Mathematics and Statistics, 42 (2013), 551-557.
[3]. K.Chakrabarthy, Biswas R.Nanda, A note on union and intersection of intuitionistic fuzzy sets, Notes on intuitionistic fuzzy sets, 3(4), 1997.
[4]. K.J.Lee, Bipolar-valued fuzzy sets and their basic operations, Proc. Int. Conf., Bangkok, Thailand, 2007, 307-317.
[5]. Osama Rashad EI-Gendy, Bipolar fuzzy -ideal of BP –algebra, American Journal of Mathematics and Statistics 2020, 10(2): 33-37.

Abstract: A cantilever is a thin unvaried cross-sectional beam unchanging horizontally at one end and loaded at the additional end. In this paper, we talk about the theory of the cantilever and the beam supported at its ends and loaded in the centervia the application of residue theorem to findtotal depression at a given point on them and it comes out to be very effective tool for analyzing the theory of the cantilever and the beam supported at its ends and loaded in the middle for finding thetotal depression at a given point on their length.

Key words: Residue Approach, Cantilever, Beam, Mahgoub Transformation

[1]. Mahgoub, M.A.M. (2016) The new integral transform "Mahgoub Transform", Advances in theoretical and applied mathematics, 11(4), 391- 398. [2]. Fadhil, R.A. (2017) Convolution for Kamal and Mahgoub transforms, Bulletin of mathematics and statistics research, 5(4), 11-16. [3]. M.M. AbdelrahimMahgoub, "The NewIntegral Transform ''Mahgoub Transform'',Advances in Theoretical and Applied Mathematics,vol. 11, pp. 391-398, 2016. [4]. Mohamed and A. E. Elsayed, "Elzaki Transformation for Linear Fractional and Differential Equations",Journal of Computational Theoretical Nanoscience,vol. 12, pp. 2303-2305, 2015. application of new transform "Mahgoub Transform" to partial differential equations,
[5]. Rohit Gupta , Rahul Gupta, Residue approach to mathematical analysis of the moving coil galvanometer, International Journal of Advanced Trends in Engineering and Technology, Volume 4, Issue 1, 2019.

Abstract: Most of the answers so far have been along the general lines of 'Why hard problems are important', rather than 'Why the Collatz conjecture is important'; I will try to address the latter.
The Collatz conjecture is the simplest open problem in mathematics. You can explain it to all your non-mathematical friends, and even to small children who have just learned to divide by 2. It doesn't require understanding divisibility, just evenness. The lack of connections between this conjecture and existing mathematical theories (as complained of in some other answers) is not an inadequacy of this conjecture, but of our theories. This problem has led directly to theoretical work by Conway showing that very similar questions are formally undecidable, certainly a surprising result.

[1]. Boolos, G.S., J.P. Burgess, and R.C. Jeffrey, Computability and Logic. Fourth edition. Cambridge University Press, Cambridge, 2002.
[2]. Conway, J.H., "Unpredictable Iterations". In: Proceedings 1972 Number Theory Conference. University of Colorado, Boulder, CO, 1972, pp. 49–52.
[3]. Edwards, H. M., Riemann's Zeta Function. Academic Press, New York, 1974 (Dover Publications, New York, 2001).
[4]. Hayes, B., "Computer Recreations. On the ups and downs of hailstone numbers." Scientific American, vol. 250, nr. 1, 1984, 13–17.
[5]. Lagarias, J. C., "The 3x+1 problem and its generalizations". American Mathemat- ical Monthly, 92, 1985, 3–23.