iosr-Jm   Volume-16 ~ Issue-3 ~ may – june 2020

Abstract: We present here a result on congruent numbers elliptic curves. We construct an isomorphism class of elliptic curves associated to congruent numbers. We show that, two elliptic curves defined overQ and associated
to congruent numbers which are the areas of two congruent right-angled triangles are Q-isomorphic. We prove a relation on the discriminants of congruent numbers elliptic curves, and we pose a conjecture on the
conductors of congruent numbers elliptic curves.

Key Word:Congruent number, congruent elliptic curves, isomorphism, rank of an elliptic curve, BSDconjecture.

[1]. R. Alter, T. B. Curtz and K. K. Kubota. Remarks and results on congruent numbers. In Proc. 3rd South Eastern Conf. Combin.,
Graph theory and Comput., (1972), p. 27-35.
[2]. B.J. Birch and H.P.F. Swinnerton-Dyer. Notes on elliptic curves II, J. reineangew. Math., 218,(1965), p. 79-108.
[3]. P. Colmez, Le problème des nombres congruents. Séminaire des élèves de l'Ecole Polytechnique, (2005), available at
https://webusers.imj-prg.fr/~pierre.colmez/congruents.pdf.
[4]. W. A. Coppel, Number theory: An introduction to Mathematics. Part B, Springer-Verlag, New York,(2006).
[5]. G. Cornnel, J. Silverman and G. Stevens, Modular forms and Fermat's last theorem, Springer, (1997).

Abstract: Cooperative game theory made a great impression on several aspects of life sciences and the impression of scientific theory had started fifty years ago. This research paper illustrates a top-level view of the implementation of skills gained by students in Mathematics for Business and Economics to unravel problems of the game theory. Economically speaking, people have choices, demands, wishes, and desires. While it's expected that individuals will act in a very rational way to comply with social norms, the method is complicated thanks to external factors like prices, production, gains, and expenses. The act of constructing a rational decision regarding choices involves what's said as a social exchange economy.

Key Word: Game Theory, Cooperative Game, Strategy, Player, Zero Sum Game,Penalty, Equilibrium, Payoff Matrix, Probability.Key Word: Game Theory, Cooperative Game, Strategy, Player, Zero Sum Game,Penalty, Equilibrium, Payoff Matrix, Probability.

[1]. Cano-Berlanga (2017). Enjoying cooperative games: The R package GameTheory.
[2]. Baskov O (2017) Bounded computational capacity equilibrium in repeated two-player zero-sum games.
[3]. Elena Yevsyeyeva, Olena Skafa (2016) Game Theory in Economics Education.
[4]. Varian Hal R. (2014). Intermediate Microeconomics: A Modern Approach. Eighth Edition. W. Norton & Company.
[5]. Albert Michael H, Nowakowski Richard J, Wolfe David (2007) Lessons in play: an introduction to combinatorial game theory. A. K. Peters, Wellesley.

Abstract: Price stability is one of the main objectives of every government as it is an important economic indicator that governments, politicians, economists and other stakeholders use as basis of argument when debating on the state of the economy (Suleman and Sarpong, 2012). In recent years, rising inflation has become one of the major economic challenges facing most countries in the world, especially developing countries such as Nigeria. David (2001) described inflation as a major focus of economic policy worldwide. This is rightly so as inflation is the frequently used economic indicator of the performance of a country's economy due to the fact that it has a direct effect on the state of the economy.Inflation is one of the major economic challenges facing most countries in the world especially those in Africa including Nigeria.

[1]. Asogu, J. O. (1991): An Econometric Analysis of the Nature and Causes of Inflation in Nigeria.CBN Economic and Financial Review, Vol.29, No.3, pp.239-254.
[2]. David F.H. (2001): Modeling UK inflation, 1875-1991.J. Appl. Economics., 16(3): 255-275.
[3]. Fakiyesi, O.M. (1996): Further Empirical analysis of Inflation in Nigeria. CBN Economic and Financial Review, 34(1), 489-500.
[4]. Hall, R., (1982): Inflation, Causes and Effects. Chicago University Press, Chicago.
[5]. Webster, D. (2000): Webster's New Universal Unabridged Dictionary. Barnes and Noble Books, New York.

Abstract: The performances of five traditional methods for unconstrained nonlinear optimization problems are evaluated using a test problem. Efficiency index is based on convergence rate, application to a wider class of
functions and ease of manual application. It is seen that optimization techniques that make use of the gradient vector are better-off than those that do not involve it in their operations. The former is observed to converge
quadratically..

Keywords: Optimization, nonlinear, unconstrained, minimize, objective function

[1]. Kreyzig, E. Advanced Engineering Mathematics, John Wiley & Sons, 2010.
[2]. Rao, S. S. Optimization Theory and Applications, Wiley Eastern Ltd, 1979.
[3]. Hillier, F. S. and Lieberman, G. J. Introduction to Operations Research, McGraw-Hill, New York, 2009.
[4]. Taha, H. A., Operation Research: An Introduction, Pearson Prentice Hall Inc, New Jersey, U.S.A . 2017.
[5]. Powell, M. J. D. An Efficient Method of Finding the Minimum of a Function of Several Variables Without Calculating Derivatives,
The Computer Journal, Vol 7 (2), 1964, pp. 155–162.

Abstract: The purpose of this review study focused on identifying and verifying an effective and reliable mathematical growth model for prediction of future Population size of United Republic of Tanzania. To specify the best model out of the three models, two criteria's, that have been used using numerical and graphical techniques, were in order to compare and contracts the models. The Exponential, Logistic growth model and Method of Least square (MLS) used to estimate the model parameters with previous census data from 1980 to 2016 inclusive. During estimation, different software used such as MATLAB (R2015a) and R version 3.6.3 software. The review proposed exponential model via the estimation of method of least square is most effective and reliable model with the least SEE (241,806) and MAD (188,413) as well as the highest R2(99.95%) relative to the other models.

Keyword: least square, estimation, Diagnosis, parameters, prediction, statistical models, Exponential

[1]. A.J.Patel and M.B.Prajapati, 2016, Estimation for Future Population Growth of china by using logistic model, International Journal of scientific development and research (IJSDR), vol.1, no.9, pp.52-56
[2]. Abdelrahim.M. et al, 2017, A mathematical and statistical approach for predicting the population Growth, WWJMRD;3(7):50-59
[3]. Augustus Wali, 2011, Mathematical modeling of Rwanda's Population growth, Applied Mathematics sciences. vol.5, No.53, 2617-2628
[4]. Blossfeld, H. P., et al, 2014, Event history analysis: Statistical theory and application in the social sciences, Psychology Press.
[5]. C.Darwin,1859, On the origin of Species by Means of Natural Selection of the Preservation of Favored Races in the struggle for life, reprinted by Random House, New York, (1993)

Abstract: We introduce the exact odd composites counting function and the exact prime counting function.

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Abstract: In this study, we define a type of generalized quaternions whose coefficients are a generalization of Fibonacci and Lucas quaternions. We give Binet – like formulas and generating functions for these kind of quaternions. By using Binet – like formulas, we obtain generalizations of some well – known identities such as, Vajda's, Catalan's, Cassini's and d'Ocagne's identities.

Key Word: Generalized Fibonacci numbers; Generalized Lucas numbers; Generating function; Binet – like formula

[1]. Akyigit, M., Kosal, H.H. and Tosun, M. Fibonacci generalized quaternions. Advances in Applied Clifford Algebras. 2014: 24; 631-641.
[2]. Akyigit, M., Kosal, H.H. and Tosun, M. Split Fibonacci quaternions. Advances in Applied Clifford Algebras. 2013: 23; 535-545.
[3]. Bilgici, G. New generalizations of Fibonacci and Lucas sequences. Applied Mathematical Sciences. 2014; 8(29): 1429-1437.
[4]. Cimen, C. B. and Ipek, A. On Pell quaternions and Pell-Lucas quaternions. Advances in Applied Clifford Algebras. 2016: 26(1); 39-51.
[5]. Halici, S. On Fibonacci quaternions. Advances in Applied Clifford Algebras. 2012: 22; 321-327.

Abstract: The mathematical proof of Goldbach's conjecture in number theory is drawn in this paper by applying a specific bounding condition from Bertrand's postulate or Chebyshev's theorem

Keywords: Bertrand's postulate & Chebyshev's theorem, Goldbach's conjecture, prime number, even & odd number, natural numbers series

[1]. A. E. Ingham (1990), The Distribution of Prime Numbers, Cambridge University Press. pp. 2-5. ISBN 978-0-521-39789-6.
[2]. N. Costa Pereira, (1985), A Short Proof of Chebyshev's Theorem, American Mathematical Monthly (Aug.-Sep. 1985). 92(7): 494-495. Doi: 10.2307/2322510. JSTOR 2322510.
[3]. M. Nair(1982), On Chebyshev-Type Inequalities for Primes, American Mathematical Monthly (Feb. 1982). 89(2): 126-129.doi: 10.2307/2320934. JSTOR 2320934.
[4]. P. Hoffman (1998), The Man Who Loved Only Numbers, New York: Hyperion Books. P. 227. ISBN 978-0-7868-8406—3. MR !666054.
[5]. H. M. Edwards (2001), Riemann's zeta function, Courier Dover Publications, ISBN 978-0-486-41740-0.

Abstract: In this article, some facts about the beauty of Ramanujan number 1729 are mentioned. Ramanujan number 1729 is the smallest positive integer which can be expressed as the sum of cubes of two numbers in two different ways. The Diophantine equation N = X3+Y3= W3+ Z3 for 1729 is solved and the relations among the terms of this equation are derived. Different types of representations of 1729 and magic squares containing 1729 are given in this article. 1729 can be expressed in terms of Fibonacci numbers, but it is not a Fibonacci number. 1729 is a Harshad number.

Key words: Ramanujan number, Diophantine equation, Fibonacci number, Harshad number, Magic square.

[1]. Debajit D., Mystery of Ramunujan number a3 + b3 = c3 + d3, International Journal of Scientific &Engineering Research. 2014;5(4):461-466.
[2]. Inder J.T., Hardy – Ramanujan number 1729, Research Report Collection.(2017);20:1-50.
[3]. Kapil Hari P., Ramanujan-type numbers, Resonance, Journal of Science Education, Indian Academy of Sciences.(2020);25(1):133-139.
[4]. Ramaswamy A.M.S., Contributions of Srinivasa Ramanujan to number theory, Research Gate.(2016):1-6.
[5]. Sagar B., Extending the beauty of Ramanujan's number, The Mathematics Student, Indian Mathematical Society. (2017);86(3-4):97-11O..

Abstract: The group test strategy, also known as pooling samples test /screening test / group sample test, has been used since WWII to test cases of HIV, chlamydia, malaria and influenza. It not only can increase test capacity, but also decrease test cost. Given the current shortage of COVID-19 tests, the group test strategy can be an appropriate and necessary tool for increasing testing and enabling safe returns to normal activity during the COVID-19 pandemic. Outside of China, the majority of COVID-19 tests are currently individual tests, which are less cost efficient than group testing. The selection of ideal group size is based on many factors including prevalence levels and operational challenges. The objective of this brief report is to recommend guide group size selection at various prevalence levels for a simple group test strategy, and discuss other pertinent issues governing group size selection.

[1]. Bilder CR, Iwen PC Abdalhamid A, Tebbs JM and McMahan CS (2020) Increasing testing capacity for SARS-CoV-2 by pooling specimens, Significance,https://www.significancemagazine.com/science/651-increasing-testing-capacity-for-sars-cov-2-by-pooling-specimens
[2]. Dorfman R. (1943) The detection of defective members of large populations. The Annals of Mathematical Statistics14, 436-440..