iosr-Jm   Volume-18 ~ Issue-1 ~ Jan. – Feb. 2022

Abstract:We construct and analyse the orbits of the 3·n+1i.e. the (3·n+1)/2 problem in a finite set of the integer n, and we observe the presence of a "saturation point" for the 3·n+1 at n=118 (notice l(97)=118) and for the (3·n+1)/2 formulation at l(73)=73. The point is a value n0 for which l(n) ≤ n, n≥ n0 where l(n) is the length of the orbit of the integer n to reach the unit i.e. 1, in the cycle 421 or 21.
Alternatively, we then pose the conjecture that, above the saturation point, for the tree of the inverse orbits starting at 1 and of depth k, the number of integers not exceeding k present on the tree is equal to k for k≥k0where k0. is the depth of the chalice at the saturation point, i.e. k0=118 respectively k0 =73 in the second formulation........

Key words: Collatz problem in the two formulation (3n+1) and (3n+1/2), inverse orbits, total stopping time, saturation point, conjecture, stochastic like Fibonacci Sequences, numerical experiment

[1]. Lagarias, J. C. (2003). The 3x+ 1 problem: An annotated bibliography (1963–1999). The ultimate challenge: the 3x, 1, 267-341.
[2]. Lagarias, J. C. (2006). The 3x+ 1 problem: An annotated bibliography, II (2000-2009). arXiv preprint math/0608208.
[3]. Merlini, D., and Sala, N. (1999). Mappe discrete e numeri naturali: alcune considerazioni sul problema del 3n+1, in Bollettino della Società Ticinese delle Scienze Naturali, Vol. 87, n. 1-2, 33 – 37.
[4]. Johannesen, S., and Merlini, D. (1982). On "critical points" in the 2-d classical Jellium, Physics Letters, Vol 91A, number 1, 21-23.
[5]. Merlini, D., and Sala N. (1999). On the Fibonacci's Attractor and the Long Orbits in the 3n+1 Problem" in International Journal of Chaos Theory and Applications, Special Issue Volume 4, Number 2-3, pp. 75 – 85.

Abstract: Corona Virus Disease is highly contagious respiratory disease that lead to the dangerous complications in the respiratory system. The virus weakens the body immunity making a way to some opportunistic diseases, pneumonia being highly common. The co-infection of COVID 19 and bacterial pneumonia is more dangerous because both attack the lungs blocking the respiratory system of the human body. The model was developed under the assumptions that human population is not constant, no vertical transmission of diseases and no vaccination of the human population. The basic reproduction numbers
were calculated and........

Key words: Co-infection, COVID-19, Pneumonia, Reproduction number, Stability

[1]. F. Bettenay, J. De Campo, and D. McCrossin. Differentiating bacterial from viral pneumonias in children. Pediatric radiology, 18(6):453–454, 1988.
[2]. X. Chen, B. Liao, L. Cheng, X. Peng, X. Xu, Y. Li, T. Hu, J. Li, X. Zhou, and B. Ren. The microbial coinfection in covid-19. Applied microbiology and biotechnology, pages 1–9, 2020.
[3]. K. Dietz. The estimation of the basic reproduction number for infectious diseases. Statistical methods in medical research, 2(1):23–41, 1993.
[4]. Q. Ding, P. Lu, Y. Fan, Y. Xia, and M. Liu. The clinical characteristics of pneumonia patients co-infected with 2019 novel coronavirus and influenza virus in wuhan, china. Journal of medical virology, 2020.
[5]. R. Elie, E. Hubert, and G. Turinici. Contact rate epidemic control of covid-19: an equilibrium view. Mathematical Modelling of Natural Phenomena, 15:35, 2020..

Abstract: Let........

Key words: Domination, Secure domination, Secure dominating set, Secure domination number.

[1]. S. Alikhani and Y-H. Peng, Dominating Sets and Domination Polynomials of Paths, International Journal of Mathematics and Mathematical Sciences, Vol.2009, 2009, Article ID 542040.
[2]. G. Chartrand and L. Lesniak (2005), Graphs & Digraphs, Fourth Edition, Champman \& Hall / CRC.
[3]. E.J. Cockayne, P.J.P. Grobler, W.R. Grundlingh J. Munganga and J.H. Van Vuuren, Protection of a graph, Util., Math., 67(2005),19-32.
[4]. S.V. Divya Rashmi, S. Arumugam and Ibrahim Venkat, Secure Domination in Graphs, Int. J. Advance. Soft Comput. Appl., 8(2) (2016), 79-83.
[5]. Frank Harary, Graph Theory, Addison-Wesley Publishing Company, Inc.

Abstract: To date, HIV/AIDS has remained the most widespread viral STD with no cure and the leading cause of death. All what is available are viral load control regimen and treatment of opportunistic diseases. The highly advocated control measure remains as Abstinence, Being faithful and use of Condom, popularly denoted as (ABC) method. The ABC control measures relate directly to individual behavior, and the success depends on individual commitment and self-discipline. However, environmental and cultural practices have for a long time been blamed for observed high levels of prevalence, especially among the fisher folk community. In this paper, we consider the interaction dynamics.....

Key words: Metapopulation, Disease Free Equilibrium, Endemic Equilibrium, Optimal Control, Stability, Boundedness, Positivity, Elasticity, Reproductive Ratio

[1]. Murray, J.D., Mathematical biology: I. An introduction. Vol. 17. 2007: Springer Science & Business Media.
[2]. Keeling, M.J. and P. Rohani, Modeling infectious diseases in humans and animals. 2011: Princeton University Press.
[3]. Tanzarn, N. and C. Bishop-Sambrook, The dynamics of HIV/AIDS in small-scale fishing communities in Uganda. Rome, Italy: HIV/AIDS Programme, Food and Agriculture Organization (FAO) of the United Nations, 2003.
[4]. Olowosegun, T., et al., Sexuality and HIV/AIDS among Fisher folks in Kainji Lake Basin. Global Journal of Medical Research Diseases, 2013. 13(2): p. 4-18.
[5]. WHO, W.H.O., 2008 Report on the Global AIDS Epidemic. 2008: World Health Organization

Abstract: Background: All most all the function spaces over real or complex domains and spaces of sequences, which arise in practice as examples of normed complete linear spaces (Banach spaces), are reflexive. These Banach spaces are dual to their respactive spaces of continuous linear functionals over the corresponding Banach spaces, meaning that the double dual space coincides with the Banach space. For each of these Banach spaces, a countable vector space basis exists, which is responsible for their reflexivity. Materials and Methods: A vector space.....

Key words: Normed Linear Spaces; Banach Spaces; Vector Space Basis; Weak Topology

[1]. Johm B. Conway. A Course in Functional Analysis. Springer-Verlag Inc. New York, 1990
[2]. Serge Lang. Real and Functional Analysis. Springer-Verlag Inc. New York, 1993
[3]. Robert E. Megginson. An Introduction to Banach Space Theory. Springer-Verlag Inc. New York, 1998
[4]. George F. Simmons. Introduction to Topology and Modern Analysis. McGraw-Hill Inc. Printed by Robert E. Krieger Publishing Company Inc. 1963
[5]. Walter Rudin. Principles of Mathematical Analysis. McGraw-Hill Inc. 1976

Abstract:In this paper, we present the sum formula of generalized Tribonacci numbers via generating functions. 2020 Mathematics Subject Classification. 11B37, 11B39, 11B83.

Key words:generalized Tribonacci numbers, generalized Tribonacci sequence, sum, Tribonacci numbers, Tribonacci-Lucas numbers

xxx

Abstract: This note is consequent upon the important paper of Bede that was published in Fuzzy Sets and Systems, 2006, 986-989. His paper demonstrated by a counterexample that the equivalence between the twopoint boundary value problem for a fuzzy differential equation and integral equation proposed by Lakshmikantham et al. (2001) and O'Regan et al. (2003) does not hold. The purpose of this note is twofold.
First, we show that the derivative of fuzzy numbers can be derived component-wise. Second, according to our findings, we provide an easy way to simplify Bede's derivation for derivatives. Our results may be useful to help researchers to understand the significant contribution of Bede's paper......

Key words: Fuzzy differential equations; Two-point boundary value problems; Green's function

[1]. B. Bede, A note on "two-point boundary value problems associated with non-linear fuzzy differential equations", Fuzzy Sets and
Systems, (2006) 986989.
[2]. V. Lakshmikantham, K.N. Murty, J. Turner, Two-point boundary value problems associated with non-linear fuzzy differential
equations, Mathematical Inequalities and Application (2001) 527533.
[3]. D. O'Regan, V. Lakshmikantham, J. Nieto, Initial and boundary value problems for fuzzy differential equations, Nonlinear Anal. (2003) 405415.
[4]. Ahn, J.Y., Mun, K.S., Kim, Y.H., Oh, S.Y., Han, B.S., 2008. A fuzzy method for medical diagnosis of headache. IEICE Transactions on Information and Systems E91-D (4), 1215-1217.
[5]. Atanassov, K.T., 1984. Intuitionistic fuzzy sets. in: V. Sgurev (Ed), VII ITKR's Session. Sofia (June 1983 Central Sci and Techn Library Bulg Academy of Sciences).

Abstract: human is the product of highly stable evolution of nature, and the optimal result of managing and controlling its function and structure in the earth's energy field. This paper gives the results and reasons of the division of human nervous system function. It has reference value for DNA interpretation.

Key words: nervous system; DNA； neuron

.........

Abstract: In this study, we have applied a numerical method to calculate the effect of the principle of parsimony assumption on the uncertainty analysis index by a 1-norms error value provided a multiplicative random perturbation of 0.01 is applied on the initial condition values of x(0) = 1, y(0) = 1. We will expect this alternative method of calculating the effect of the principle of parsimony on the basis of a probabilistic
assumption on the initial data to complement and move the frontier of knowledge in numerical simulation. The novel result that we have obtained that we have not seen elsewhere are presented and discussed quantitatively.

Key words: Principle of parsimony, uncertainty analysis, 1-norms error analysis, random perturbation, initial data, numerical simulation.

[1]. Ekaka-a, E.N and Nafo, N.M (2012). Theoretical Bifurcation analysis of a biological interspecific competition interaction. Scientia Africana, Vol II, Number 2, Pp 30 – 34.
[2]. Ekaka-a, E.N., Chukwuocha, E.O. and Nafo, N.M. (2012). Sensitivity Analysis of a Physiochemical Interaction method undergoing changes in the initial condition and duration of experimental time. Journal of Applied Science and Environmental Management, Vol 16(2), 180-184.
[3]. Ekaka-a, E.N. (2009). Computational and Mathematical Modelling of Plant Species Interactions in a Harsh Climate, PhD Thesis, Department of Mathematics, The University of Liverpool and the University of Chester, United Kingdom, 2009.
[4]. Nafo, N.M, Enu Ekaka-a, Olowu, B.U, Agwu, I.A. and Nwachukwu, C.E. (2013). Determining the importance of carrying capacities of prey-predator interaction with harvesting. Journal of the Nigerian Association of Mathematical Physics, Vol. 23, Pp 183-186
[5]. Nafo, N.M., Ekaka-a, E.N. and Weli, A. (2014). Toward a reduction of uncertainty of a model parameter value: Induced random Noise characterization on resource Biomass and Implication for Biodiversity Policy control. Journal of the Nigerian Association of mathematical Physics, Vol. 28, Pp 437-444.

Abstract: In this study, the harvest scheduling problem of a group of cane growers in Maharashtra is addressed. Each member in a group is required to consistently supply sugar cane to a mill for the entire harvest season. However, the current scheduling does not account for the time-variant cane production of each cane field, which leads to unequal opportunities for growers to harvest. A portion of growers could have the opportunity to harvest in periods that provide higher sugar cane yields, while others in the same group do not. This inefficient harvest scheduling causes conflicts between growers and unnecessary loss of sugar cane and sugar yields.

Key words: Sugarcane industry, Harvesting, optimization problem , Mathematical model

[1] Ahumada, O., Villalobos, J.R., 2009. Application of planning models in the agrifood supply chain: a review. European Journal of Operational Research 169, 1–20.
[2] Bezuidenhout, C.N., Baier, T.J.A., 2009. A global review and synthesis of literature pertaining to integrated sugarcane production systems. In: 82nd Annual Congress of the South African Sugar Technologists' Association, pp. 93–101
[3] Singh S. P., Parashar Anil K. & Singh H. P. (1977), Econometrics and Mathematical Economics, S. Chand and Company Limited, New Delhi.
[4] Higgins, A.J., 1999. Optimizing cane supply decisions within a sugar mill region. Journal of Scheduling 2, 229–244. Higgins, A.J., 2006. Scheduling of road vehicles in sugarcane transport: a case study at an Australian sugar mill. European Journal of Operational Research 170, 987– 1000..

Abstract: In this article, we introduce a new concept of Neutrosophic continuous functions called totally N continuous functions and study their properties in Neutrosophic topological spaces

Key words: Ng#-closed set, Ng#-continuous function, totally Ng#-continuous function

[1]. K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, (1986), 20, 87-96.
[2]. C. L. Chang, Fuzzy topological spaces, J.Math.Anal.Appl., (1968), 24, 182- 190.
[3]. R. Dhavaseelan and S. Jafari, Generalized Neutrosophic closed sets, New trends in Neutrosophic theory and applications (2018), 2, 261-273.
[4]. Floretin Smarandache, Neutrosophic Set:- A Generalization of Intuitionistiic Fuzzy set, Journal of Defense Resourses Management , (2010), 1, 107–116.
[5]. D. Jayanthi, On Generalized closed sets in Neutrosophic topological spaces, International Conference on Recent Trends in Mathematics and Information Technology, (2018), pecial Issue March, 88-91.