International Journal of Operational Research/Nepal

Golden Ratio: Construction, Geometry, Beauty, and Diversity

Iswar Mani Adhikari — Sun, 31 Dec 2023 00:00:00 +0000

Nature is attractive due to the proper ratios of various components in them and in between others. In nature, leaves, flowers, fruits, Himalayas, hills, valleys, springs, rivers, etc. show a rhythmic balance and seem harmonic and aesthetically pleasing. The golden ratio or the golden (divine) proportion is considered the most pleasing to human visual sensation and is not limited only to aesthetic beauty but also found prominently in the natural world. It is an irrational number denoted by φ and is approximately 1.618034. It is a fascinating topic for mathematicians, artists, natural scientists, philosophers, biologists, artists, architects, musicians, psychologists, designers, etc.

This research work presents a panoramic view of the golden ratio; their construction algorithms, and structures; mathematical presentations, geometry, patterns, and properties from mathematical perspectives and opens the broad horizon for further research. Besides these, the paper also presents their existences, diversities, their relationships with nature, the universe, arts, design, architecture, and engineering in ancient and modern mathematics and sciences.

Numerical Mean-square and Asymptotic Stability Analysis for the Weak Simpson Method

Ram Sharan Adhikari — Sun, 31 Dec 2023 00:00:00 +0000

A weak Simpson method has order of weak convergence one in general and has order of weak convergence three under certain additional assumptions. The proposed method has the potential to overcome some of the numerical instabilities that are often experienced when using explicit Euler methods. This work aims to determine the mean-square stability region of the weak Simpson method for linear stochastic differential equations with multiplicative noises. In this work, a mean-square stability region of the weak Simpson scheme is identified, and step-sizes for the numerical method where errors propagation are under control in a well-defined sense are given. The main results are illustrated with numerical examples.

Contribution of Discrete and Continuum Wave Functions in the Ground State Polarizability of Hydrogen

Chandra M. Adhikari — Sun, 31 Dec 2023 00:00:00 +0000

We analytically derive a discrete radial Green function for a hydrogenic system in terms of associated Leguerre polynomials. We investigate the polarizability of the ground-state hydrogen using the so-called length gauge approximation of electrodynamics. The polarizability is the sum of the polarization matrix elements for the negative and the positive frequency values. We estimate the contribution of discrete wavefunctions to the static ground state polarizability. We found that the contribution of continuum states dominates over that of discrete states to the polarizability by an order of magnitude.