On parametric equivalent, isomorphic and unique sets
ODAM-Vol. 4 (2021), Issue 1, pp. 19 – 24 Open Access Full-Text PDF
J. Kok, J. Shiny
Abstract: This short paper introduces the notions of parametric equivalence, isomorphism and uniqueness in graphs. Results for paths, cycles and certain categories (or types) of trees with regards to minimum confluence sets are presented.
On the exponential Diophantine equation \((2^{2m+1}-1)+(13)^n=z^2\)
EASL-Vol. 4 (2021), Issue 1, pp. 77 – 79 Open Access Full-Text PDF
Sudhanshu Aggarwal
Abstract: Nowadays, scholars are very interested to determine the solution of different Diophantine equations because these equations have numerous applications in the field of coordinate geometry, cryptography, trigonometry and applied algebra. These equations help us for finding the integer solution of famous Pythagoras theorem and Pell’s equation. Finding the solution of Diophantine equations have many challenges for scholars due to absence of generalize methods. In the present paper, author studied the exponential Diophantine equation \((2^{2m+1}-1)+(13)^n=z^2\), where \(m,n\) are whole numbers, for its solution in whole numbers. Results show that the exponential Diophantine equation \((2^{2m+1}-1)+(13)^n=z^2\), where \(m\), \(n\) are whole numbers, has no solution in whole number.
The effect of time-varying delay damping on the stability of porous elastic system
OMS-Vol. 5 (2021), Issue 1, pp. 147 – 161 Open Access Full-Text PDF
Soh Edwin Mukiawa
Abstract: In the present work, we study the effect of time varying delay damping on the stability of a one-dimensional porous-viscoelastic system. We also illustrate our findings with some examples. The present work improve and generalize existing results in the literature.
Mathematical analysis of a delayed HIV/AIDS model with treatment and vertical transmission
OMS-Vol. 5 (2021), Issue 1, pp. 128 – 146 Open Access Full-Text PDF
Gratien Twagirumukiza, Edouard Singirankabo
Abstract: None can underestimate the importance of mathematical modelling for their role in clarifying dynamics of epidemic diseases. They can project the progress of the disease and demonstrate the result of the epidemic to public health in order to take precautions. HIV attracts global attention due to rising death rates and economic burdens and many other consequences that it leaves behind. Up to date, there is no medicine and vaccine of HIV/AIDS but still many researches are conducted in order to see how to mitigate this epidemic and reduce the death rate or increase the life expectancy of those who are infected. A delayed HIV/AIDS treatment and vertical transmission model has been investigated. The model took into account both infected people from the symptomatics group and asymptomatic group to join AIDS group. We considered that a child can be infected from the mother to an embryo, fetus or childbirth. Those who are infected, it will take them some time to get mature and spread the disease. By using mathematical model, reproduction number, positivity, boundedness, and stability analysis were determined. The results showed that the model is much productive if time delay is considered.
Atomic localization via superposition of three standing wave fields in a four level tripod atomic configuration
EASL-Vol. 4 (2021), Issue 1, pp. 69 – 76 Open Access Full-Text PDF
Anwar Hussain, Muhammad Izhar, Mian Azhar Uddin, Muhammad Wahab, Anwar Ali Khan, Masood Rauf Khan
Abstract: We theoretically present the physical realization of one dimensional (1D) atom localization by superposition of three standing wave fields in a four-level tripod atomic configuration. The most interesting result that we observe is the variation of the bandwidth of the localization peaks with the intensity of the space independent Rabi frequency. A sharp single and double localized peaks are observed at different direction of the wave numbers. The bandwidth of a localized peak is reduced as the intensity of the space independent Rabi frequency goes on increasing, which corresponds to the reduction in the uncertainty. These results will hopefully contribute to the development of current high tech-applications.
A note on Jeśmanowicz’ conjecture for non-primitive Pythagorean triples
OMS-Vol. 5 (2021), Issue 1, pp. 115 – 127 Open Access Full-Text PDF
Van Thien Nguyen, Viet Kh. Nguyen, Pham Hung Quy
Abstract: Let \((a, b, c)\) be a primitive Pythagorean triple parameterized as \(a=u^2-v^2,\ b=2uv,\ c=u^2+v^2\), where \(u>v>0\) are co-prime and not of the same parity. In 1956, L. Jeśmanowicz conjectured that for any positive integer \(n\), the Diophantine equation \((an)^x+(bn)^y=(cn)^z\) has only the positive integer solution \((x,y,z)=(2,2,2)\). In this connection we call a positive integer solution \((x,y,z)\ne (2,2,2)\) with \(n>1\) exceptional. In 1999 M.-H. Le gave necessary conditions for the existence of exceptional solutions which were refined recently by H. Yang and R.-Q. Fu. In this paper we give a unified simple proof of the theorem of Le-Yang-Fu. Next we give necessary conditions for the existence of exceptional solutions in the case \(v=2,\ u\) is an odd prime. As an application we show the truth of the Jeśmanowicz conjecture for all prime values \(u < 100\).
