Geometric and Physical Interpretation of Fractional Integration and Fractional Differentiation

A solution to the more than 300-years old problem of geometric and physical interpretation of fractional integration and differentiation (i.e., integration and differentiation of an arbitrary real order) is suggested for the Riemann-Liouville fractional integration and differentiation, the Caputo fractional differentiation, the Riesz potential, and the Feller potential. It is also generalized for giving a new geometric and physical interpretation of more general convolution integrals of the Volterra type. Besides this, a new physical interpretation is suggested for the Stieltjes integral.Comment: 18 pages, 7 figures, 1 tabl

A note on comparison of scientific impact expressed by the number of citations in different fields of science

Citation distributions for 1992, 1994, 1996, 1997, 1999, and 2001, which were published in the 2004 report of the National Science Foundation, USA, are analyzed. It is shown that the ratio of the total number of citations of any two broad fields of science remains close to constant over the analyzed years. Based on this observation, normalization of total numbers of citations with respect to the number of citations in mathematics is suggested as a tool for comparing scientific impact expressed by the number of citations in different fields of science.Comment: 5 pages, 1 tabl

Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives

On a series of examples from the field of viscoelasticity we demonstrate that it is possible to attribute physical meaning to initial conditions expressed in terms of Riemann-Liouville fractional derivatives, and that it is possible to obtain initial values for such initial conditions by appropriate measurements or observations.Comment: LaTeX2e, 14 page

Paradoxes of Subdiffusive Infiltration in Disordered Systems

Infiltration of diffusing particles from one material to another where the diffusion mechanism is either normal or anomalous is a widely observed phenomena. When the diffusion is anomalous we find interesting behaviors: diffusion may lead to an averaged net drift from one material to another even if all particles eventually flow in the opposite direction, or may lead to a flow without drift. Starting with an underlying continuous time random walk model we solve diffusion equations describing this problem. Similar drift against flow is found in the quenched trap model. We argue that such a behavior is a general feature of diffusion in disordered systems.Comment: 5 pages, 2 figure

Responsive Graphical User Interface (ReGUI) and its Implementation in MATLAB

In this paper we introduce the responsive graphical user interface (ReGUI) approach to creating applications, and demonstrate how this approach can be implemented in MATLAB. The same general technique can be used in other programming languages.Comment: 8 pages, 3 figure

Transform of Riccati equation of constant coefficients through fractional procedure

We use a particular fractional generalization of the ordinary differential equations that we apply to the Riccati equation of constant coefficients. By this means the latter is transformed into a modified Riccati equation with the free term expressed as a power of the independent variable which is of the same order as the order of the applied fractional derivative. We provide the solutions of the modified equation and employ the results for the case of the cosmological Riccati equation of FRW barotropic cosmologies that has been recently introduced by FaraoniComment: 7 pages, 2 figure

The Little Prince -- The Lost Chapter

A lost chapter from Antoine de Saint-Exupéry\u27s Le Petite Prince about the Little Prince visiting a mathematician, written in French in the style of the original work, is presented along with several translations

State space description of national economies: the V4 countries

We present a new approach to description of national economies. For this we use the state space viewpoint, which is used mostly in the theory of dynamical systems and in the control theory. Gross domestic product, inflation, and unemployment rates are taken as state variables. We demonstrate that for the considered period of time the phase trajectory of each of the V4 countries (Slovak Republic, Czech Republic, Hungary, and Poland) lies approximately in one plane, so that the economic development of each country can be assocated with a corresponding plane in the state space. The suggested approach opens a way to a new set of economic indicators (for example, normal vectors of national economies, various plane slopes, 2D angles between the planes corresponding to different economies, etc.). The tool used for computations is orthogonal regression (alias orthogonal distance regression, alias total least squares method), and we also give general arguments for using orthogonal regression instead of the classical regression based on the least squares method. A MATLAB routine for fitting 3D data to lines and planes in 3D is provided.Comment: 13 pages, 18 figure