Comment on 'Classification scheme for phenomenological universalities in growth problems in physics and other sciences'

In this communication, the incorrectness of phenomenological approach to the logistic growth equation, proposed by Castorina et al. is presented in detail. The correct phenomenological approach to logistic growth equation is also proposed here. It is also shown that the same leads to different types of biological growths also.Comment: 3 pages, 1 figur

Existence and uniqueness theorem for ODE: an overview

The study of existence and uniqueness of solutions became important due to the lack of general formula for solving nonlinear ordinary differential equations (ODEs). Compact form of existence and uniqueness theory appeared nearly 200 years after the development of the theory of differential equation. In the article, we shall discuss briefly the differences between linear and nonlinear first order ODE in context of existence and uniqueness of solutions. Special emphasis is given on the Lipschitz continuous functions in the discussion.Comment: 10 page

Role of specific growth rate in the development of different growth processes

Effort has been given for the development of an analytical approach that helps to address several sigmoidal and non-sigmoidal growth processes found in literature. In the proposed approach, the role of specific growth rate in different growth processes has been considered in an unified manner. It is found that the different growth equations can be derived from the same functional form of rate equation of specific growth rate. A common functional form of growth and growth velocity have been derived analytically and it has been shown that different values of the parameters involved in the description lead to different growth function. The theta logistic growth can be explained in this proposed framework. It is found that the competitive environment may increase the saturation level of population size.Comment: 16 Pages, 6 Figures, 1 Tabl

Under what kind of parametric fluctuations is spatiotemporal regularity the most robust?

It was observed that the spatiotemporal chaos in lattices of coupled chaotic maps was suppressed to a spatiotemporal fixed point when some fraction of the regular coupling connections were replaced by random links. Here we investigate the effects of different kinds of parametric fluctuations on the robustness of this spatiotemporal fixed point regime. In particular we study the spatiotemporal dynamics of the network with noisy interaction parameters, namely fluctuating fraction of random links and fluctuating coupling strengths. We consider three types of fluctuations: (i) noisy in time, but homogeneous in space; (ii) noisy in space, but fixed in time; (iii) noisy in both space and time. We find that the effect of different kinds of parameteric noise on the dy- namics is quite distinct: quenched spatial fluctuations are the most detrimental to spatiotemporal regularity; spatiotemporal fluctuations yield phenomena similar to that observed when parameters are held constant at the mean-value; and interestingly, spatiotemporal regularity is most robust under spatially uniform temporal fluctuations, which in fact yields a larger fixed point range than that obtained under constant mean-value parameters.Comment: 12 pages, 5 figure

Oscillatory Growth: A Phenomenological View

In this communication, the approach of phenomenological universalities of growth are considered to describe the behaviour of a system showing oscillatory growth. Two phenomenological classes are proposed to consider the behaviour of a system in which oscillation of a property may be observed. One of them is showing oscillatory nature with constant amplitude and the other represents oscillatory nature with a change in amplitude. The term responsible for damping in the proposed class is also been identified. The variations in the nature of oscillation with dependent parameters are studied in detail. In this connection, the variation of a specific growth rate is also been considered. The significance of presence and absence of each term involved in phenomenological description are also taken into consideration in the present communication. These proposed classes might be useful for the experimentalists to extract characteristic features from the dataset and to develop a suitable model consistent with their data set.Comment: 15 pages, 8 figure

Method of variation of parameters revisited

The method of variation of parameter (VOP) for solving linear ordinary differential equation is revisited in this article. Historically, Lagrange and Euler explained the method of variation of parameter in the context of perturbation method. In this article, we explain the construction of particular solutions of a linear ordinary differential equation in the light of linearly independent functions in a more systematic way. In addition, we have shown that if the time variation of the constants contribute substantially to the velocity then also the solution remains invariant. VOP method for system of n linear ODE is discussed. Duhamels principle has also been studied in reference to a system of n linear ODE for completeness of this review. Finally, applications of VOP method for constructing Green's function is reported.Comment: 12 pages, 1 figur

Circular hydraulic jump in generalized-Newtonian fluids

We carry out an analytical study of laminar circular hydraulic jumps, in generalized-Newtonian fluids obeying the two-parametric power-law model of Ostwald-de Waele. Under the boundary-layer approximation we obtained exact expressions determining the flow, an implicit relation for the jump radius is derived. Corresponding results for Newtonian fluids can be retrieved as a limiting case for the flow behavior index n=1, predictions are made for fluids deviating from Newtonian behavior.Comment: 4 pages, 3 figures, added references, corrected typo

Phenomenological approach for describing environment dependent growths

Different classes of phenomenological universalities of environment dependent growths have been proposed. The logistic as well as environment dependent West-type allometry based biological growth can be explained in this proposed framework of phenomenological description. It is shown that logistic and environment dependent West-type growths are phenomenologically identical in nature. However there is a difference between them in terms of coefficients involved in the phenomenological descriptions. It is also established that environment independent and enviornment dependent biological growth processes lead to the same West-type biological growth equation. Involuted Gompertz function, used to describe biological growth processes undergoing atrophy or a demographic and economic system undergoing involution or regression, can be addressed in this proposed environment dependent description. In addition, some other phenomenological descriptions have been examined in this proposed framework and graphical representations of variation of different parameters involved in the description are executed.Comment: 16 pages, 11 figure

Constant rotation of two-qubit equally entangled pure states by local quantum operations

We look for local unitary operators $W_1 \otimes W_2$ which would rotate all equally entangled two-qubit pure states by the same but arbitrary amount. It is shown that all two-qubit maximally entangled states can be rotated through the same but arbitrary amount by local unitary operators. But there is no local unitary operator which can rotate all equally entangled non-maximally entangled states by the same amount, unless it is unity. We have found the optimal sets of equally entangled non-maximally entangled states which can be rotated by the same but arbitrary amount via local unitary operators $W_1 \otimes W_2$, where at most one these two operators can be identity. In particular, when $W_1 = W_2 = (i/\sqrt{2})({\sigma}_x + {\sigma}_y)$, we get the local quantum NOT operation. Interestingly, when we apply the one-sided local depolarizing map, we can rotate all equally entangled two-qubit pure states through the same amount. We extend our result for the case of three-qubit maximally entangled state.Comment: 11 pages, Latex fil

Nonlinear Dynamics of a position-dependent mass driven Duffing-type oscillator

We examine some nontrivial consequences that emerge from interpreting a position-dependent mass (PDM) driven Duffing oscillator in the presence of a quartic potential. The propagation dynamics is studied numerically and sensi- tivity to the PDM-index is noted. Remarkable transitions from a limit cycle to chaos through period doubling and from a chaotic to a regular motion through intermediate periodic and chaotic routes are demonstrated