Discrete Chebyshev Research Articles

Discrete Chebyshev polynomials for nonsingular variable‐order fractional KdV Burgers' equation

In this article, nonlinear variable‐order (VO) fractional Korteweg‐de Vries (KdV) Burgers' equation with nonsingular VO time fractional derivative is introduced and discussed. The approximate solution of the expressed problem is obtained in the form of a series expansion in terms of the shifted discrete Chebyshev polynomials (CPs) with great accuracy. The method is a computational procedure based on the collocation technique and the shifted discrete CPs together with their operational matrices (ordinary and VO fractional derivatives). The main advantage of the designed approach is that it provides a global solution for the problem. In order to examine the efficiency of the designed algorithm, some numerical problems have been provided. The obtained solutions confirm that the present method is computationally effective and sufficiently accurate in solving this equation.

Discrete Manhattan and Chebyshev pair correlation functions in k dimensions.

Pair correlation functions provide a summary statistic which quantifies the amount of spatial correlation between objects in a spatial domain. While pair correlation functions are commonly used to quantify continuous-space point processes, the on-lattice discrete case is less studied. Recent work has brought attention to the discrete case, wherein on-lattice pair correlation functions are formed by normalizing empirical pair distances against the probability distribution of random pair distances in a lattice with Manhattan and Chebyshev metrics. These distance distributions are typically derived on an ad hoc basis as required for specific applications. Here we present a generalized approach to deriving the probability distributions of pair distances in a lattice with discrete Manhattan and Chebyshev metrics, extending the Manhattan and Chebyshev pair correlation functions to lattices in k dimensions. We also quantify the variability of the Manhattan and Chebyshev pair correlation functions, which is important to understanding the reliability and confidence of the statistic.

Measurement and Chebyshev spectral analysis of Brownian coordinate of electrochemical noise

For the first time, the electrochemical Brownian motion is considered in the context of discrete Chebyshev spectroscopy. The computer processing of output signal of analog-to-digital converter operating in the first-order sinc filter mode enables measuring the discrete Chebyshev spectrum of electrochemical analog of Brownian particle coordinate. It is shown that there is the Chebyshev power spectral density with respect to the normalized Chebyshev polynomial degree. The Chebyshev power spectral density is determined for the Brownian coordinate of corrosion noise and for the Brownian coordinate of intrinsic noise of EVAL-AD7176-2SDZ evaluation board with a shorted input. For the first time, it is shown that the Chebyshev power spectral density for the Brownian coordinate of electrochemical noise depends on the first-order sinc filter output data rate. The trend-resistant method of Chebyshev power spectral density of Brownian coordinate can be used for the noise diagnostics of electrochemical devices and processes.

Design of secure cryptosystem based on chaotic components and AES S-Box

In this paper, we design, realize and evaluate a new secure cryptosystem based on a Pseudo-Chaotic Number Generator(PCNG), a global diffusion and a block cipher in Cipher Block Chaining(CBC) mode. The proposed PCNG, defined over a finite field, is based on a discrete Chebyshev map coupled with a Pseudo-Random Number Generator(PRNG) and a discrete skew tent map. Our PCNG can remove a hidden danger of deteriorated security caused by the dynamical degradation encountered in numerical implementation of chaotic maps defined on real numbers. The global diffusion uses a Horizontal Addition Diffusion(HAD) and a Vertical Addition Diffusion(VAD) followed by a modified 2-D cat map. The block cipher in CBC mode is based on a confusion layer using the AES S-Box, followed by a diffusion layer built on the modified 2-D cat map and a key addition operation. The global diffusion enhances the diffusion process of the block cipher. Conducted security analysis and the obtained results have proven that the proposed cryptosystem design is robust with high security and that it can be applied to practical implementations.

A new collocation method using near-minimal Chebyshev quadrature nodes on a square

A new collocation method using near-minimal Chebyshev quadrature nodes is proposed on the square [−1,1]2. For a (total) degree of precision 2n−1, the number of nodes of this near-minimal quadrature rule amounts only to n(n+1)2+⌊n2⌋+1, which is one more than the Möller's lower bound, n(n+1)2+⌊n2⌋, i.e., the minimal number of nodes in a quadrature rule of degree 2n−1 in two dimensions. Firstly, a new Chebyshev interpolation based on the near-minimal Chebyshev quadrature rule is constructed. An optimal error estimate on the new interpolation is obtained, and fast algorithms for the corresponding discrete Chebyshev transformation (DCT) between the function values and the discrete Chebyshev coefficients are then devised. Next, spectral differentiation schemes are developed both in the physical space and in the frequency space. Finally, a new Chebyshev collocation method, which uses nearly half nodes of the tensorial Chebyshev collocation method, is proposed to solve second order partial differential equations on the square. Numerical experiments illustrate that our new Chebyshev collocation method also possesses an exponential order of convergence for smooth problems. In comparison to the tensorial collocation method, it can offer better accuracy for most problems with the same degrees of freedom.

The Chebyshev Difference Equation

We define and investigate a new class of difference equations related to the classical Chebyshev differential equations of the first and second kind. The resulting “discrete Chebyshev polynomials” of the first and second kind have qualitatively similar properties to their continuous counterparts, including a representation by hypergeometric series, recurrence relations, and derivative relations.

Discrete Chebyshev Wavelet Transformation with Image Processing

This article introduces novel image processing techniques due to improve the images under examine based on Chebyshev wavelet filter. The technique enabling to obtain Chebyshev wavelet transform using four kinds of Chebyshev wavelet operation matrix of integration, which conduct to enhance the images quality under examine. After comparing the results with standard waves that have a large role in the image processing, it was concluded that the second type of proposed wavelets is the most efficient of the three types, all this work was done using the MatLab program. The obtained results indicate decreasing of MSE= 1.777910-29, increasing both PSNR=340.2918, BPP=0.0078125 and compression ratio CR=0.03% reveal the effectiveness and accuracy of the suggested technique. The results can be utilized in various areas such as science treatment, medicine, noise removal and compression images.

Chebyshev type inequalities for interval-valued functions

We introduce the interval double integral for interval-valued functions, and give some fundamental properties. Further, we establish some continuous and discrete Chebyshev type inequalities, and some interesting equalities for interval-valued functions. Also, some examples are presented to illustrate our results.

The discrete Chebyshev algorithm for nonparametric estimation of autocorrelation function of electrochemical random time series

The discrete Chebyshev algorithm for nonparametric estimation of autocorrelation function of electrochemical random time series is presented. The algorithm is resistant to a trend of electrochemical noise. The discrete Chebyshev algorithm is tested using the model electrochemical noise corresponding to the equilibrium two-element Voigt circuit. The algorithm is used for nonparametric estimation of autocorrelation function of corrosion noise and electronic noise of measuring instrument.

A discrete orthogonal polynomials approach for coupled systems of nonlinear fractional order integro-differential equations

This paper develops a numerical approach for solving coupled systems of nonlinear fractional order integro-differential equations(NFIDE). Shifted discrete Chebyshev polynomials (SDCPs) have been introduced and their attributes have been checked. Fractional operational matrices for the orthogonal polynomials are also acquired. A numerical algorithm supported by the discrete orthogonal polynomials and operational matrices are used to approximate solution of coupled systems of NFIDE. The operational matrices of fractional integration and product are applied for approximate the unknown functions directly. These approximations were put in the coupled systems of NFIDE. A comparison has been made between the absolute error of approximate solutions of SDCPs method with previous published. The gained numerical conclusions disclose that utilizing discrete Chebyshev polynomials are more efficient in comparison to the other methods.

A New Approach for American Option Pricing: The Dynamic Chebyshev Method

We introduce a new method to price American options based on Chebyshev interpolation. In each step of a dynamic programming time-stepping we approximate the value function with Chebyshev polynomials. The key advantage of this approach is that it allows us to shift the model-dependent computations into an offline phase prior to the time-stepping. In the offline part a family of generalized (conditional) moments is computed by an appropriate numerical technique such as a Monte Carlo, PDE, or Fourier transform based method. Thanks to this methodological flexibility the approach applies to a large variety of models. Online, the backward induction is solved on a discrete Chebyshev grid, and no (conditional) expectations need to be computed. For each time step the method delivers a closed form approximation of the price function along with the options' delta and gamma. Moreover, the same family of (conditional) moments yield multiple outputs including the option prices for different strikes, maturities, and different payoff profiles. We provide a theoretical error analysis and find conditions that imply explicit error bounds for a variety of stock price models. Numerical experiments confirm the fast convergence of prices and sensitivities. An empirical investigation of accuracy and runtime also shows an efficiency gain compared with the least-squares Monte Carlo method introduced by Longstaff and Schwartz [Rev. Financ. Stud., 14 (2001), pp. 113--147]. Moreover, we show that the proposed algorithm is flexible enough to price barrier and multivariate barrier options.

Chebyshev Spectra Resistance to Trend of Random Noise

Spectral analysis of random noise in the space of discrete Chebyshev polynomials is an alternative to spectral Fourier analysis. The importance of Chebyshev spectral approach is associated with the fact that the discrete Chebyshev transformation of the [Formula: see text]-th order eliminates automatically the polynomial trend of the ([Formula: see text]−1) order. Using the method of artificial trend, it was found that, under the real experimental conditions, the intensity of Chebyshev spectral lines with numbers higher than 1 is resistant to a strong trend of random process. This effect is observed when we use both the arithmetic averaging and the median. The Chebyshev spectral approach is a powerful tool for spectral analysis of random time series with a strong trend.

A Comparative Approach for Time‐Delay Fractional Optimal Control Problems: Discrete Versus Continuous Chebyshev Polynomials

AbstractThis paper aims to demonstrate the superiority of the discrete Chebyshev polynomials over the classical Chebyshev polynomials for solving time‐delay fractional optimal control problems (TDFOCPs). The discrete Chebyshev polynomials have been introduced and their properties are investigated thoroughly. Then, the fractional derivative of the state function in the dynamic constraint of TDFOCPs is approximated by these polynomials with unknown coefficients. The operational matrix of fractional integration together with the dynamical constraints is used to approximate the control function directly as a function of the state function. Finally, these approximations were put in the performance index and necessary conditions for optimality transform the under consideration TDFOCPs into an algabric system. A comparison has been made between the required CPU time and accuracy of the discrete and continuous Chebyshev polynomials methods. The obtained numerical results reveal that utilizing discrete Chebyshev polynomials is more efficient and less time‐consuming in comparison to the continuous Chebyshev polynomials.

Hot spot method for pedestrian detection using saliency maps, discrete Chebyshev moments and support vector machine

The increasing risks of border intrusions or attacks on sensitive facilities and the growing availability of surveillance cameras lead to extensive research efforts for robust detection of pedestrians using images. However, the surveillance of borders or sensitive facilities poses many challenges including the need to set up many cameras to cover the whole area of interest, the high bandwidth requirements for data streaming and the high-processing requirements. Driven by day and night capabilities of the thermal sensors and the distinguished thermal signature of humans, the authors propose a novel and robust method for the detection of pedestrians using thermal images. The method is composed of three steps: a detection which is based on a saliency map in conjunction with a contrast-enhancement technique, a shape description based on discrete Chebyshev moments and a classification step using a support vector machine classifier. The performance of the method is tested using two different thermal datasets and is compared with the conventional maximally stable extremal regions detector. The obtained results prove the robustness and the superiority of the proposed framework in terms of true and false positives rates and computational costs which make it suitable for low-performance processing platforms and real-time applications.

Lossy Data Compression Effects on Wall-bounded Turbulence: Bounds on Data Reduction

Postprocessing and storage of large data sets represent one of the main computational bottlenecks in computational fluid dynamics. We assume that the accuracy necessary for computation is higher than needed for postprocessing. Therefore, in the current work we assess thresholds for data reduction as required by the most common data analysis tools used in the study of fluid flow phenomena, specifically wall-bounded turbulence. These thresholds are imposed a priori by the user in L2-norm, and we assess a set of parameters to identify the minimum accuracy requirements. The method considered in the present work is the discrete Legendre transform (DLT), which we evaluate in the computation of turbulence statistics, spectral analysis and resilience for cases highly-sensitive to the initial conditions. Maximum acceptable compression ratios of the original data have been found to be around 97%, depending on the application purpose. The new method outperforms downsampling, as well as the previously explored data truncation method based on discrete Chebyshev transform (DCT).

Chebyshev Rational Approximations for the Rosenau-KdV-RLW Equation on the Whole Line

In this paper, we consider the use of a modified Chebyshev rational approximations for the Rosenau-KdV-RLW equation on the whole line with initial-boundary values. It is shown that the proposed scheme leads to optimal error estimates. Furthermore, the stability and convergence of the proposed schemes are proved. The fully discrete Chebyshev pseudo-spectral scheme is constructed. Numerical results confirm well with the theoretical results. The idea and techniques presented in this paper will be useful to solve many other problems.

Discrete version of Wiener-Khinchin theorem for Chebyshev’s spectrum of electrochemical noise

A discrete version of Wiener-Khinchin theorem for Chebyshev’s spectrum of electrochemical noise is developed. Based on the discrete version of Wiener-Khinchin theorem, the theoretical discrete Chebyshev spectrum for the Markov random process is calculated. It is characterized by two parameters: the dispersion and the relaxation frequency (or relaxation time). The noise of corrosion process and the noise of recording equipment are measured. Using the theoretical Chebyshev spectrum, the Markov parameters were found both for the noise of the corrosion process and for the noise of the measuring equipment.

Generalizations of Szőkefalvi Nagy and Chebyshev inequalities with applications in spectral graph theory

Two weighted inequalities for real non-negative sequences are proven. The first one represents a generalization of the Szőkefalvi Nagy inequality for the variance, and the second a generalization of the discrete Chebyshev inequality for two real sequences. Then, the obtained inequalities are used to determine lower bounds for some degree-based topological indices of graphs.

New discrete orthogonal moments for signal analysis

The paper addresses some computational aspects and applications of the polynomial unbiased finite impulse response functions originally derived by Shmaliy as a new class of a one-parameter family of discrete orthogonal polynomials. We present two new explicit formulas to compute these polynomials directly. They are based on the discrete Rodrigues’ representation and hypergeometric form, respectively. These straightforward calculations lead us to propose another discrete signal representation in moment domain. Experimental results show a comparison between the proposed moments and discrete Chebyshev moments in terms of noise-free/noisy signal feature extraction and reconstruction error.

On the Generating Function of Discrete Chebyshev Polynomials

We give a closed form for the generating function of the discrete Chebyshev polynomials. It is the MacWilliams transform of Jacobi polynomials together with a binomial multiplicative factor. It turns out that the desired closed form is a solution to a special case of the Heun differential equation, and that it implies combinatorial identities that appear quite challenging to prove directly.