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

Abstract: This paper is concerned with calculating weekday of any given date. There are two methods, namely, Direct method and Months code method. Direct method is described briefly with an example. Months code method involves : Whether the Year in the date is leap year or non-leap year, finding the weekday of January 1 of the year of the given date and using the month code corresponding to the month in the date from the months code sequence with January 1 of the year in the date and the relation month day + month code = weekday of the given date and obtaining the remainder when the sum is divided by 7 and the remainder gives the weekday for the given date

[1]. Atul Saxena . Mathematical codes to help predict exact day of any date, Hindustan times January 3, 2020 , pp. 1, reported by SURJIT DAS.

Abstract: By studying the ^ S function whose integer zeros are the prime numbers, and being inspired by the article [2], I give a new proof of the Riemann hypothesis.

[1]. Roshdi Rashed, Entre arithmétique et algèbre : Recherches sur l'histoire des mathématiques arabes, journal Paris, 1984,
[2]. M. Sghiar. The Mertens function and the proof of the Riemann's hypothesis, International Journal of Engineering and Advanced
Technology (IJEAT), ISNN:2249-8958, Volume- 7 Issue-2, December 2017
[3]. https://en.wikipedia.org/wiki/Gamma_function.
[4]. https://en.wikipedia.org/wiki/Riemann_zeta_function.

Abstract: The paper proposes a method for proving the Goldbach binary conjecture, based on the properties of representations of even numbers and the rules of formal logic. Numerical evaluations confirming the
correctness of the method are carried out. The connection between the Goldbach conjecture and the Legendre hypothesis is considered.

Keywords: number theory, prime numbers, Goldbach's binary hypothesis. Classification code: 11A41 – Primes; 11D85 – Representation problems

[1]. Deshouillers J.–M., Riele H.J.J.te., Sauoter Y. New experimental results concerning the Goldbach conjecture. Amsterdam: Stichting math. centrum, 1998.
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[3]. Jutila M. On the least Golbach's number in an arithmetical progression with a prime difference. Turku, 1968.
[4]. Romanov V.N. Representation of integer positive number as a sum of natural summands. IJERA, V.7, Issue 6 (Part 4), June 2017, pp. 01 – 08.
[5]. Romanov V.N. Study of fundamental problems of number theory. Saint-Petersburg, Publishing house "Asterion", 2015 (in Russian).

Abstract: A double sequence A = (𝑎𝑖𝑗𝑘𝑙) is said to belong to the class (X, Y), where X and Y are two sequence spaces, if any sequence x = 𝑥𝑚𝑛 in X is transformed to a sequence 𝑦={𝑦𝑚𝑛} in Y by the matrix transformation 𝑦𝑚𝑛= 𝑎𝑚𝑛𝑗𝑘 𝑥𝑗𝑘∞𝑗=𝑜∞𝑘=0such that the sequence 𝑦𝑚𝑛 exists and converges in the Pringsheim sense.A sequence 𝑥={𝑥𝑚𝑛} of reals is said to be [𝜆,𝜇]-almost convergent (briefly,ℱ[𝜆,𝜇]−𝑐𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑛𝑡) to some number 𝑙 if 𝑥 𝜖 ℱ[𝜆,𝜇], whereumber 𝑙 if 𝑥 𝜖 ℱ[𝜆,𝜇], where

[1]. Lorentz, G. G., A contribution to the theory of divergent sequences, Acta Mathematica, 1948, 80, 167–190.
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[3]. Muhiuddine, S. A., An application of almost convergence in approximation theoems, Applied Mathematics Letters, 2011, 24(11), 1856 – 1860.
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[5]. Mohiuddine, S. A. and A. Alotaibi, Almost conservative four-dimensional matrices through de la Vallèe-Poussin mean,, Abstract and Applied Analysis, 2014, Article ID 412974, 6pages

Abstract: The extended 𝑓-divergence between two functions of probability distributions is defined for a given convex function 𝑓and an increasing function 𝑔. A universal portfolio is generated from the zero gradient set of an objective function involving the estimated daily rate of wealth increase and the extended 𝑓-divergence. For specific convex functions f and increasing functions 𝑔 the form of the universal portfolio is derived. There exists a convex function such that the Bregman universal portfolio generated by this convex function is similar to the universal portfolio generated by the extended 𝑓-divergence

Keywords-universal portfolio, extended f-divergence,Bregman divergence.

[1]. T. M. Cover and E. Ordentlich, Universal portfolios with side information, IEEE Transactions on Information Theory,vol.42, no.2, pp.348-363, Mar. 1996.
[2]. D. P. Helmbold, R. E. Shapire, Y. Singer and M. K. Warmuth, On-line portfolio selection using multiplicative updates,Mathematical Finance, vol.8, no.4, pp.325-347, Oct. 1998.
[3]. C. P. Tan, Performance bounds for the distribution-generated universal portfolios, Proc. 59thISI World Statistics Congress, Hong Kong, 5327-5332, 2013.
[4]. C. P. Tan and K. S. Kuang, Universal portfolios generated by the 𝑓 and Bregman divergences, IOSR Journal of Mathematics, vol. 14, 19-25, May – June 2018.
[5]. C. P. Tan and K. S. Kuang, Universal portfolios generated by 𝑓-disparity differences, AIP Conference Proceedings, vol. 2184, 050024, 2019.

Abstract: Monitoring the presence or absence of a trait in a large population one – at – a – time is tedious, uneconomical and bound to errors. The remedy is to group the population into homogeneous groups and test each group for the presence of a trait. Multi – stage group testing procedure involves testing groups for the presence or absence of trait in a population and sequentially subdividing the positive groups into sub – groups. The sub – groups to be tested at a particular stage are based on the information obtained from the previous stage. This paper proposed an M – stage hierarchical design for testing the presence of multiple traits in a finite population. The design improves the efficiency of the estimators as evident via the computation of asymptotic variance.

Keywords: M – stage, ith group, hth stage.

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[2]. Brookmeyer, R. (1999). Analysis of multistage pooling studies of biological specimens for estimating disease incidence and prevalence. Biometrics 55, 608 – 612.
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[5]. Hughes-Oliver and Shallow. W.H (1994). A two-stage adaptive group design for group testing of only one trait. American statistical association, 89, 982 – 993.

Abstract: In this paper, a simple but efficient method of solving harmonic Dirichlet problems of heat flow in solids which can be reduced to two dimensional problems using conformal mapping is presented. The method employs an appropriate mapping function to transform the domain and boundary of the given problem in the 𝑤 plane onto onein the upper half of the 𝑧 plane and the appropriate portions of the 𝑥 axis where its solution for steady state temperature is easily identified as the imaginary part of some branch of the logarithmic function. The required solution in the 𝑤 plane was the obtained from the mapping function by simply substituting 𝑢 𝑎𝑛𝑑 𝑣for 𝑥 𝑎𝑛𝑑 𝑦, respectively. Furthermore, the isothermal lines or level curves of the solution were also obtained to show the lines/surfaces of constant temperature within the given solid. This method gave exact analytical solutions for the steady state temperature within the given solid and can therefore be a suitable alternative method for solving such problems in two dimensions..

Keywords and Phrases: Harmonic function, Conformal Map,Analytic Function, Schwarz-Christoffel Map,Joukowski map, Steady State Temperature, Isothermal lines.

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[5]. Gonzalo, R., Hernan, C., and Ruben, P. (2008). The Schwarz-Christoffel Conformal Mapping for "Polygons" with Infinitely many Sides. International Journal of Mathematics and Mathematical Sciences. 2008: pp.1-20.

Abstract: Background: Annually, avian influenza causes high morbidity and mortality rate predominately among the immunodeficiency persons worldwide. Treatment and vaccination remain the optimal strategies in curbing the spread of avian influenza infection. Methods: In this paper, a mathematical model of the dynamics of influenza infection is formulated and both qualitative and quantitative analyses are carried out extensively.
Results: The qualitative analysis of the model is given in terms of the basic reproduction number, equilibria points and their stability analyses. The disease dies out whenever the basic reproduction number is less than a unit. The disease free equilibrium (DFE) is locally asymptotically stable provided R0 <1 and unstable if otherwise. The endemic equilibrium only occurs whenever the disease threshold.....

Key word: Avian Influenza, modeling, basic reproduction number, equilibrium, numerical solution.

[1]. Asquith, B. and Bangham, C.R. (2003). An introduction to lymphocyte and viral dynamics: the power and limitations of mathematical analysis, prove Biol. Sci., 270:1651-1657.
[2]. Caroline, W.K., Kimathi, M. and Livingstone, L. (2018). Mathematical analysis of influenza A dynamics in the emergence of drug resistance. Computational and Mathematical Methods in Medicine, 1-14.
[3]. Centres for Disease Control and Prevention (CDC). Types of influenza virus, http://www.cdc.gov/?u/about/viruses/types.html.
[4]. Cox, N.J. and Subbarao, K. (1999). Influenza. Lancet. 354:1277-1282. http://dx.dio.org/50140-6736(99)012410-6.
[5]. Gumel, A.B. (2009). Global dynamics of a two strain avian influenza model. International Journal of Computer Mathematics, 86: 85-108..

Abstract: Influenza virus infection represents a global threat causing seasonal outbreaks and pandemics, and as a result, it is associated with considerable morbidity and mortality worldwide despite the availability of vaccine and antiviral drugs. However, medical experts and published data has shown that deaths are largely caused by respiratory complications resulting from secondary bacterial infections, the most common being bacterial
pneumonia. In order to understand the transmission and control dynamics of this infection in the presence of secondary bacterial infections, we formulated an eight compartmental mathematical model, which incorporated
vaccination and treatment parameters into the deterministic model that studies the infection behaviour in the presence of secondary bacterial infections..........

Keywords: Influenza, Bacterial infection, Complication, Vaccination, Critical points, Basic reproduction number, Stability

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