The Greek letter φ (Phi) represents one of the most mysterious numbers (1.618…) known to humankind. Historical approbation for φhas led to the monikers “The Golden Number” or “The Divine Proportion”. This simple, but inscrutable number, is inseparably linked to the recursive mathematical sequence which produces Fibonacci numbers. The study of the Fibonacci sequence exists in most aspects of life starting from the leaves of a non-flowering plant, design, paintings, animals, and even human body. Despite its wide-spread prevalence and existence, the Fibonacci series and also the Rule of Golden Proportions have not been widely documented within the human body. The main objective of this study is to prove that the length of the human hand bone is in step with the Fibonacci series to spot the degree of movement and variation for every finger. Victimization of the sample z test with 95% confidence interval, this analysis shows that just one of the four bone length ratios contained the ratio phi φ within the 95% confidence interval and follow the Fibonacci series, that of the little finger metacarpal and proximal phalanx in both hands. The largest variability was seen within the little finger phalangeal relationships and other fingers will follow mathematical relative series. Due to the relationship with the golden number, it will facilitate in monitoring the individual with an injured hand, especially if injured in small fingers throughout a medical aid, or to identify the cause of the problem of physical functioning of the hands or individual fingers. Hence, it should be helpful for the length of the clenched fist to perform in reconstruction or placement of the prosthesis.
In this work we use an analytical technique to analyse the effect of a vertical uniform magnetic field on the onset of steady Benard-Marangoni convection in a horizontal layer of electrically conducting fluid subject to a uniform vertical temperature gradient in the asymptotic limit short waves. We found that in the limit of short waves, the leading order expression for the marginal curve is not affected by the magnetic field.
Dalam makalah ini kesan medan magnet menegak seragam ke atas lengkung sut permulaan olakan mantap Benard-Marangoni dalam lapisan bendalir mengufuk berpengalir elektrik dikaji tertakluk kepada kecerunan suhu yang seragam dalam had asimptot gelombang pendek. Kami dapati medan magnet tidak memberi kesan kepada sebutan utama lengkung sut dalam had gelombang pendek.
Ordinary Differential Equations (ODEs) are usually used in numerous fields especially in solving the modelling problem. Numerical methods are one of the vital mathematical tools to solve the ODEs that appear in various modelling problems by determining the approximation solution close to the in exact solution if it exists. Runge-Kutta methods (RK) are the numerical methods used to integrate the ODEs by applying multistage methods at the midpoint of an interval which can efficiently produce a more accurate result or small magnitude of error. We proposed Runge-Kutta methods (RK) to solve the 1st_ order nonlinear stiff ODEs. The RK methods used in this research are known as the RK-2, RK-4, and RK-5 methods. We proved the existence and uniqueness of the ODEs before we solved it numerically. We also proved the absolute-stability of the RK methods to determine the overall stability of these methods. We found two suitable test cases which are the standard test problem and manufactured solution. We proved that by combining the adaptive step size with RK methods can result in more efficient computation. We implemented the 2nd_, 4th_ and 5th_ order of RK methods with step size adaptively algorithm to solve the test problem and manufactured solution via Octave programming language. The resulting numerical error and the stability of each method can be studied. We compared our results using several error plots versus the Central Processing Unit (CPU) time required to compute a given nonlinear 1st_ order stiff ODE problem. In a conclusion, RK methods which combine with the adaptive step size can result in more efficient computation and accuracy compare with the fixed step size RK methods.