A theoretical analysis of the growth process of an organism and its dependence on various allometric relations

A new mathematical model regarding the growth process of an organism is proposed, based on the role of surplus power (i.e. power intake minus metabolic cost) and having an allometric dependence on mass. Considering its use in growth, a differential equation has been formed, similar to the von Bertalanffy growth function (VBGF). The time dependence of mass and growth rate, obtained from this equation, has been shown graphically to illustrate the roles played by scaling exponents and other parameters. Concepts of optimum mass, saturation mass and the mass corresponding to the highest growth rate have been discussed under the proposed theoretical framework. Information regarding the dependence of effective growth duration on various parameters has been found graphically. The time of occurrence of the highest growth rate and its dependence on various parameters have been explored graphically. A new parameter (ρ) has been defined, which determines the availability of surplus power at different stages of the growth process of an organism. Depending on its value, there can be three distinctly different modes of growth phenomenon, reflected in the change of surplus power with time. The variations of growth and reproduction efficiencies with time and mass have been shown for different values of the scaling exponent. The limitation regarding the practical measurement of growth rate has been discussed using the present model. Some aspects of length-biomass allometry have been explored theoretically and the results have been depicted graphically.

A Simple Statistical Estimation of One’s Performance in an MCQ Examination, Based Upon Mock Test Results, Using Binomial Distribution of Probability

A simple statistical model is proposed regarding the estimation of one’s overall performance in an MCQ examination along with the calculation of probability of obtaining a certain percentage of marks in the same. Using the data obtained from the results of a sufficiently large number of mock examinations, conducted prior to the main examination, certain parameters quantifying one’s knowledge or preparation for the examination has been calculated. Based on those parameters, the probability of obtaining a certain percentage of marks has been computed using the theory of binomial probability distribution. The dependence of this probability function on various parameters has been depicted graphically. A parameter, called the performance index, has been defined in terms of the expectation value and standard deviation of marks computed from probability distribution. Using this parameter, a new parameter called the relative performance index has been defined. This index estimates one’s performance with respect to the best possible performance. The variation of relative performance index with respect to the preparation index has been shown graphically for different parameter values quantifying various aspects regarding the examination and the examinee.

Modelling and Simulation of Hydro-gel Growth Mechanism by Analysis of Experimental Data

We have experimentally studied the growth process of a water-absorbing polymer commonly known as hydro-gel. We took a number of spherical beads, immersed them in pure water, measured their diameters at regular intervals of time and plotted graphs showing the change in volume with the time of immersion, to analyze their swelling properties. We have formulated a dynamical model on the basis of curve-fitting to our experimental data. Our mathematical modelling is devoted solely towards the explanation of the swelling of spherical hydro-gels and we have interpreted our observations on these basis. And finally evaluating the constants of the process, we have shown that the results obtained from our modelling and numerical simulation exactly match the experimental data. The justification of this new approach lies in its successful application for the complete explanation of our observations through numerical and analytical ways.