Fourth kind Chebyshev Wavelet Method for the solution of multi-term variable order fractional differential equations

PurposeMulti-term variable-order fractional differential equations (VO-FDEs) are powerful tools in accurate modeling of transient-regime real-life problems such as diffusion phenomena and nonlinear viscoelasticity. In this paper the Chebyshev polynomials of the fourth kind is employed to obtain a numerical solution for those multi-term VO-FDEs.Design/methodology/approachTo this end, operational matrices for the approximation of the VO-FDEs are obtained using the Fourth kind Chebyshev Wavelets (FKCW). Thus, the VO-FDE is condensed into an algebraic equation system. The solution of the system of those equations yields a coefficient vector, the coefficient vector in turn yields the approximate solution.FindingsSeveral examples that we present at the end of the paper emphasize the efficacy and preciseness of the proposed method.Originality/valueThe value of the paper stems from the exploitation of FKCWs for the numerical solution of multi-term VO-FDEs. The method produces accurate results even for relatively small collocation points. What is more, FKCW method provides a compact mapping between multi-term VO-FDEs and a system of algebraic equations given in vector-matrix form.

Enumeration of Dumont permutations avoiding certain four-letter patterns

In this paper, we enumerate Dumont permutations of the fourth kind avoiding or containing certain permutations of length 4. We also conjecture a Wilf-equivalence of two 4-letter patterns on Dumont permutations of the first kind.

MATHEMATICAL MODELING OF CONCRETE STRUCTURE CORROSION IN BIOLOGICALLY AGGRESSIVE ENVIRONMENTS

In the aquatic environment, biocorrosion is an important factor affecting the reliability and durability of concrete structures. The destruction of cement concretes during biological corrosion is determined by the processes of mass transfer. The article presents the development of a calculated mathematical model of liquid corrosion in cement concrete, taking into account the biogenic factor. For the first time, a model of mass transfer in an unbounded two-layer plate is considered in the form of differential equations of parabolic type in partial derivatives with boundary conditions of the second kind at the interface between concrete and liquid and of the fourth kind at the interface between concrete and biofilm. The results of a numerical experiment are presented to study the influence of the coefficients of mass conductivity and mass transfer on the kinetics and dynamics of the process.

Coefficient quantization effects on new filters based on Chebyshev fourth-kind polynomials

The aim of this paper is to construct non-recursive filters, extensively used type of digital filters in digital signal processing applications, based on Chebyshev orthogonal polynomials. The paper proposes the use of the fourth-kind Chebyshev polynomials as functions in generating new filters. In this kind, low-pass filters with linear phase responses are obtained. Comprenhansive study of the frequency response characteristics of the generated filter functions is presented. The effects of coefficient quantization as one type of quantization that influences a filter characteristic are investigated here also. The quantized-coefficient errors are considered based on the number of bits and the implementation algorithms.

Racism, the Highest Stage of Anti-Communism

There are many and different types of racism in contemporary Russia: institutional racism, far-right racism, everyday (bytovoi) racism, and a fourth kind to which this essay will be devoted, the racism of the liberal intelligentsia. Russian liberal media's reaction to the BLM protests of 2020 has offered abundant material for the study of its social base, main tropes, and underlying logic. This article attempts to historicize it, locating its origins in the anti-Soviet pro-western dissidence of the stagnation era and illustrating its workings through some statements made by Joseph Brodsky and his milieu. Furthermore, the article identifies the intersection of two main ideas from which this racism emerges. In the first place, this is Cold-War rejection of real or perceived Soviet alliances with newly decolonized countries of Africa and Asia or with African Americans during the Civil Rights era. In the second place, this is dissident civilizational hierarchies that placed the west at the top and saw the east or the south as a backward space best avoided.

Gegenbauer and Other Planar Orthogonal Polynomials on an Ellipse in the Complex Plane

We show that several families of classical orthogonal polynomials on the real line are also orthogonal on the interior of an ellipse in the complex plane, subject to a weighted planar Lebesgue measure. In particular these include Gegenbauer polynomials $$C_n^{(1+\alpha )}(z)$$ C n ( 1 + α ) ( z ) for $$\alpha >-1$$ α > - 1 containing the Legendre polynomials $$P_n(z)$$ P n ( z ) and the subset $$P_n^{(\alpha +\frac{1}{2},\pm \frac{1}{2})}(z)$$ P n ( α + 1 2 , ± 1 2 ) ( z ) of the Jacobi polynomials. These polynomials provide an orthonormal basis and the corresponding weighted Bergman space forms a complete metric space. This leads to a certain family of Selberg integrals in the complex plane. We recover the known orthogonality of Chebyshev polynomials of the first up to fourth kind. The limit $$\alpha \rightarrow \infty $$ α → ∞ leads back to the known Hermite polynomials orthogonal in the entire complex plane. When the ellipse degenerates to a circle we obtain the weight function and monomials known from the determinantal point process of the ensemble of truncated unitary random matrices.