iosr-Jm   Volume-18 ~ Issue-6 ~ Nov. – Dec. 2022

Abstract : In the course of the iteration of a function of a complex variable, the resulting family of iterates provide a surprising occurrence. This paper carefully investigates this occurrence and provides a general characterization for obtaining the Fatou sets for a class of polynomials. The contrast in the nature of these sets for different classes is then illustrated using computer programming techniques and graphics and the results are consistent with what obtains in the literature.

[1]. Braverman, M. & Yampolski, M. (2008). Computability of Julia sets, Vol 23, Algorithm and Computation in Mathematics, Springer, 5, 18 – 19
[2]. Brolin, H. (1965). Invariant sets under iteration of rational functions. Archive for Mathematika, 103-144
[3]. Jian Hua, Z., Sheng, W., & Zhi Gang, H., (2002). Some properties of Fatou and Julia sets of transcendental meromorphic functions. Bulletin. American Mathematical Society, 66(1), 1-8
[4]. Fatou, P., (1919). Sur l'iteration des fonctions trancendentes entieres, Acta Mathmatica 47, 161 -271
[5]. Fatou, P., (1920). Sur les, equationelles fonctionelles. Bulletin, Societe Mathematique de France, 48, 208-314

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Abstract In this paper, we investigate the relationship between some classes of operators and their spectral and numerical gaps. We characterize these gaps for equivalent operators. 2020 Mathematics Subject Classification: Primary 47A10, 47A12; Secondary 47B20, 47N20

Keywords and phrases: spectral gap, spectral radius, numerical gap, numerical radius, similar, metrically equivalent

[1]. R. Bhatia, Matrix analysis, Springer Verlag, New York, 1997.
[2]. P. Bourdon and T. Harbilas, Numerical range of a square matrix, www.maplesoft.com/applications/view.aspx?sid=4128 view =html (2001).
[3]. E. O. Çevik and Z. I. Ismailov, Spectral radius and operator norm of infinite upper triangular double band block operator matrices, J. Ana. Num, Theor. 9, No.2 (2021), 19-22.
[4]. M. Demuth, Mathematical aspects of Physics with non-self-adjoint operators: List of problem, American Institute of Mathematics Workshop (San Jose, California), 8-12, June 2015.
[5]. S. S. Dragomir, Reverse inequalities for the numerical radius of linear operators in Hilbert spaces, Bull. Austral. Math. Soc. 73 (2006), 255-262.

Abstract :This article addresses an assignment problem that employs a branching approach and a lower bound. To address the assignment problem, we provide a novel approach. The approach uses branch and bound decisional problem versions to solve the problem using column creation.First, a generalized assignment problem that is currently solved by a Hungarian approach using row generation is solved in this study by using column generation, and the best solution is produced, which is quite comparable to the original Hungarian solution.For the purpose of determining the lower bound, bound tree column generation is carried out at each node of the branch.

[1]. Basirzadeh, H. (2012). Ones assignment method for solving assignment problems. Applied Mathematical Sciences, 6 (47), 2345-2355.
[2]. Wulan, E. R., Pratiwi, A., & Zaqiah, Q. Y. (2020, September). The Analysis of Unbalanced Assignment Problems Using the Kotwal-Dhope Method to Develop A Massive Open Online Course. In 2020 6th International Conference on Wireless and Telematics (ICWT) (pp. 1-5). IEEE.
[3]. Burkard, R., Dell'Amico, M., & Martello, S. (2012). Assignment problems: revised reprint. Society for Industrial and Applied Mathematics.
[4]. Geetha, S., & Nair, K. P. K. (1993). A variation of the assignment problem. European Journal of Operational Research, 68 (3), 422-426.
[5]. Caramia, M., Dell'Olmo, P., & Italiano, G. F. (2008). Novel local-search-based approaches to university examination timetabling. INFORMS Journal on Computing, 20 (1), 86-99..

Abstract :Random numbers are extremely important in cryptography (both quantum and classical), Statistics, probabilistic computation (Monte Carlo methods), numerical simulations, industrial testing and labeling, hazard games, scientific research, gaming, gambling and other fields. Random assignment in randomized controlled trials enables scientists to test hypotheses, and random numbers or pseudo random numbers help e-banking operations by generating 'One Time Passwords' or OTPs. Random numbers are generated through random number generators (RNGs). Random number generation is a process which, with the help of a device, creates a string of numbers......

Key Word: Random Numbers, Random Number Generators, Minimum Effort, Binary Digits, Decimal Digits.

[1]. Bahl, R.K. and Sabharwal, Y.P. Puzzle 257.The Actuary, March 2001, p.38, 2001.
[2]. Trkulja, V. and Hrabač, P..The Role of t-test in Beer Brewing. Croatian Medical Journal, 61(1), p.69, 2020

Abstract :In this paper, we study the Black-Scholes' option pricing model and the Heston option pricing model for calculating theoretical call value in European options. We consider the real market data and compare the call premium value for the two models. Sensitivity analysis is done for both the model at different moneyness options. We compare the Greeks for Black-Scholes' and Heston option pricing models and elucidate them graphically. We use Matlab software for all mathematical computations and graphs.

Key Word: Black-Scholes' model, Heston model, European call option, Underlying price, Volatility, Interest rate, Greeks and, Sensitivity.

[1]. Ayres, H. F. (1963). Risk aversion in the warrant markets.
[2]. Baumol, W. J., Malkiel, B. G., & Quandt, R. E. (1966). The valuation of convertible securities. The Quarterly Journal of Economics, 80(1), 48-59.
[3]. Black F. and Scholes M. (1973), The Pricing of Options and Corporate Liabilities, Journal of Political Economy,81(3):637-644.
[4]. Boness, A. J. (1964). Elements of a theory of stock-option value. Journal of Political Economy, 72(2), 163-175.
[5]. Chao Chen, Jamie T. and Jonathan M. (2017), A new accuracy measure based on bounded relative error for time series forecasting, Tianjin University, China, PLoS ONE 12(3): e0174202, 6-7
[6]. Chauhan, A., & Gor, R. (2020). Study of India VIX Options pricing using Black-Scholes model. NMIMS- Management Review, 1-15.

Abstract :In this paper, we introduce the solution of systems of nonlinear partial differential equations subject to the general initial conditions by using Adomian decomposition method (ADM) and Modified decomposition method(MDM).Four illustrated examples has been introduced, and the method has shown a high-precision, fast approach to solve nonlinear system of partial differential equations(PDEs) with initial conditions. The steps of the method are easy implemented and high accuracy.

Key Word: Systems of nonlinear partial differential equations,Adomian decomposition method, Modified decomposition method..

[1]. Batiha B, Noorani MSM, Hashim I. Numerical simulations of systems of PDEs by Variational iteration method. Phys. Lett. A. 2008;372:822-829.
[2]. Wazwaz AM. The Variatioal iteration method for solving Linear and nonlinear systems of PDEs, Comput. Math. Appl.2007;54:895-902.
[3]. Tawfiq LNM and Altaie H. 2002. Recent Modification of Homotopy Perturbation Method for Solving System of Third Order PDEs. Journal of Physics: Conference Series.1530 (012073):1-8. IOP Publishing.
[4]. Somjate Duangpithak. Variational iteration method for special nonlinear partial differential equations. Int. JoUrnal of Math.Analysis. 2012;6(22):1071-1077.
[5]. Enadi, M. O., and Tawfiq, L.N.M, 2019,New Technique for Solving Autonomous Equations, Ibn Al-Haitham Journal for Pure and Applied Science, 32(2),p:123-130, Doi:10.30526/32.2.2150..

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Abstract : In the era of internet, transmitting data through network communication via mail, social group, online banking,and many more are essential. Hence the requirement of securing the information occurs. Moreover, cryptographic techniques like symmetric key, asymmetric key and hashing techniques are employed. In this paper, plaintext is encrypted and decrypted using SEE transformation.

KEYWORDS: Encryption, Decryption, SEE transformation

[1]. Dhingra, S., Savalgi, A. A., & Jain, S. (2016). Laplace transformation based cryptographic technique in network security. International Journal of Computer Applications, 136(7), 0975-8887.
[2]. Kumar, P. S., & Vasuki, S. (2018). An Application of MAHGOUB Transform in Cryptography. Advances in Theoretical and Applied Mathematics, 13(2), 91-99.
[3]. Mansour, E. A., Kuffi, E. A., & Mehdi, S. A. (2021). The new integral transform "SEE transform" and its applications. Periodicals of Engineering and Natural sciences (PEN), 9(2), 1016-1029.
[4]. Priya, D. S., Amirtha, M. A., & Kumar, S. P. (2019). Application of N-transform in cryptography. International Journal of Scientific Research and Reviews, 8(1), 2143-2147.
[5]. Srushti, G., and Gor, R. (2022). Digital Image Encryption using RSA and LFSR. International Journal of Engineering Science Technologies, 6(4), 1-16. doi: 10.29121/IJOEST.v6.i4.2022.351

Abstract : Dirichlet average is average given by Dirichlet. The Dirichlet average of elementary function like power function, exponential function etc is given by many notable mathematician, Actually, We have convert the elementary function into the summation form after that taking Dirichlet average of those function, using fractional integral and get new results. These results will be used in future by mathematician and scientist. Thus we have find a connection Dirichlet average of a function and fractional integral.

[1]. Carlson, B.C. Special Function of Applied Mathematics. Academic Press, New York. 1977.
[2]. Carlson, B.C. Appell's function F4 as a double average.SIAM J. Math. Anal.6. 1975; 960–965p.
[3]. Carlson, B.C. Hidden symmetries of special functions. SIAM Rev. 1970; 12: 332–345p.
[4]. Carlson, B.C. Dirichlet averages of x t log x. SIAM J. Math. Anal. 1987; 18(2): 550–565p.
[5]. Carlson, B.C. A connection between elementary functions and higher transcendental functions. SIAM J. Appl. Math. 1969; 17: 116–140p.

Abstract : The impact of lines on how long it takes for clients to receive banking services is becoming a major concern due to the possibility that making consumers wait too long may cost them money. In Delhi, queues are a common sight at numerous banks, leading to lengthy suffering and financial costs. This study aims to determine the trade-off between minimizing the total economic cost and providing a satisfactory and reasonably short time of service to customers. Data was gathered over a month at the commercial banks' Main Branch in Delhi and organized into a multi-server single line queuing model. Management should adopt a five teller model to reduce economic costs and customer satisfaction, as it is better than a four or six teller system in terms of average waiting time and total economic cost.

[1]. Sharma, J. K. Operations Research: Theory and Application, 3rd Ed. (Macmillan Ltd., India 2017)
[2]. Taha, A. H., Operations Research: An Introduction, 7th Ed. (Prentice Hall, India, 2018)
[3]. Hiray, J, Waiting Lines and Queuing System, Article of Business Management, 2018
[4]. Agbadudu, A.B. Elementary Operation Research,Vol. I, 2017 (Benin City, A.B. Mudiaga Limited).
[5]. Elegalam (2015) "Customer Retention Versus Cost Reduction technique" A Paper Presented at the Bankers Forum held at Lagos, pg.9-10.1978..