Horizontal accuracy assessment of a novel algorithm for approximate a surface to a DEM

This study evaluates the horizontal positional accuracy of a new algorithm that defines a surface that approximates DEM data by means of a spline function. This algorithm allows evaluating the surface at any point in its definition domain and allows analytically estimating other parameters of interest, such as slopes, orientations, etc. To evaluate the accuracy achieved with the algorithm, we use a reference DEM 2 m × 2 m (DEMref) from which the derived DEMs are obtained at 4 m × 4 m, 8 m × 8 m and 16 m × 16 m (DEMder). For each DEMder its spline approximant is calculated, which is evaluated at the same points occupied by the DEMref cells, getting a resampled DEM 2 × 2 m (DEMrem). The horizontal accuracy is obtained by computing the area amongs the homologous contour lines derived from DEMref and DEMrem, respectively. It has been observed that the planimetric errors of the proposed algorithm are very small, even in flat areas, where you could expect major differences. Therefore, this algorithm could be used when an evaluation of the horizontal positional accuracy of a DEM product at lower resolution (DEMpro) and a different producing source than the higher resolution DEMref is wanted.

i-RheoFT: Fourier transforming sampled functions without artefacts

In this article we present a new open-access code named “i-RheoFT” that implements the analytical method first introduced in [PRE, 80, 012501 (2009)] and then enhanced in [New J Phys 14, 115032 (2012)], which allows to evaluate the Fourier transform of any generic time-dependent function that vanishes for negative times, sampled at a finite set of data points that extend over a finite range, and need not be equally spaced. I-RheoFT has been employed here to investigate three important experimental factors: (i) the ‘density of initial experimental points’ describing the sampled function, (ii) the interpolation function used to perform the “virtual oversampling” procedure introduced in [New J Phys 14, 115032 (2012)], and (iii) the detrimental effect of noises on the expected outcomes. We demonstrate that, at relatively high signal-to-noise ratios and density of initial experimental points, all three built-in MATLAB interpolation functions employed in this work (i.e., Spline, Makima and PCHIP) perform well in recovering the information embedded within the original sampled function; with the Spline function performing best. Whereas, by reducing either the number of initial data points or the signal-to-noise ratio, there exists a threshold below which all three functions perform poorly; with the worst performance given by the Spline function in both the cases and the least worst by the PCHIP function at low density of initial data points and by the Makima function at relatively low signal-to-noise ratios. We envisage that i-RheoFT will be of particular interest and use to all those studies where sampled or time-averaged functions, often defined by a discrete set of data points within a finite time-window, are exploited to gain new insights on the systems’ dynamics.

Shape Preserving Piecewise KNR Fractional Order Biquadratic C 2 Spline

In a recent article, a piecewise cubic fractional spline function is developed which produces C 1 continuity to given data points. In the present paper, an interpolant continuity class C 2 is preserved which gives visually pleasing piecewise curves. The behavior of the resulting representations is analyzed intrinsically with respect to variation of the shape control parameters t and s. The data points are restricted to be strictly monotonic along real line.

C˜2 Continuous Cubic Hermite Interpolation Splines with Second-Order Elliptic Variation

In order to solve the deficiency of Hermite interpolation spline with second-order elliptic variation in shape control and continuity, c-2 continuous cubic Hermite interpolation spline with second-order elliptic variation was designed. A set of cubic Hermite basis functions with two parameters was constructed. According to this set of basis functions, the three-order Hermite interpolation spline curves were defined in segments 02, and the parameter selection scheme was discussed. The corresponding cubic Hermite interpolation spline function was studied, and the method to determine the residual term and the best interpolation function was given. The results of an example show that when the interpolation conditions remain unchanged, the cubic Hermite interpolation spline curves not only reach 02 continuity, but also can use the parameters to control the shape of the curves locally or globally. By determining the best values of the parameters, the cubic Hermite interpolation spline function can get a better interpolation effect, and the smoothness of the interpolation spline curve is the best.

Decentralized issues in bitcoin blockchain and Nakamoto monetary rule

The paper is devoted to studying the bitcoin blockchain as a new global phenomenon in monetary economics, which requires comprehending from the economic theory view - a self-regulating system of decentralized emission without participation of a central monetary authority. Mathematical modelling is the instrument of this studying. The author has analyzed the Bitcoin system parameters that determine dynamics of a self-regulating emission mechanism. This mechanism operates in a peer-to-peer computer network and provides a smooth increasing of the "money supply" with a gradually decreasing rate of growth. The limit of this growth is determined by maximal volume 21 million BTC. Self-regulation is implemented through negative feedback between changes of control parameters (the target interval for the hash function values and the Bitcoin Difficulty level) and the speed of mining process. Control parameters depend on the real speed deviations from the target value. This mechanism provides a stable mining speed and determines annual rate of emission. The author suggests a spline-function for describing the annual rate of the cryptocurrency emission in accordance with the Proof-of-Work protocol in the Bitcoin blockchain algorithm. This spline-function gives possibility to find a monetary rule for annual rate of emission. The author in the paper proposes to call this monetary rule by the name of the Bitcoin system inventor - Nakamoto Monetary Rule. The Nakamoto Monetary Rule could be seen as the first example of a programmable monetary rule of the decentralized emission algorithm on the basis of blockchain technology. Central banks could use a similar approach, with the necessary modifications, to develop their programmable monetary rules for Central Bank Digital Currencies (CBDCs) emission based on DLT or blockchain technology