Generalizing trigonometric functions from different points of view

Contributors

The starting point is the autocorrelation function of the signal. We assume the latter to be the sum of a random component (noise) plus d trigonometric functions with incommensurate frequencies. The time is descretized and the signal has a finite length N. The autocorrelation is thus itself a set of N random numbers Ci, i = 0,…(N - 1); - periodic boundary conditions are assumed. Considered as Fourier coefficients, the Ci define a random measure dμ with support on the unit. One can then build by a recurrence relation the family of polynomials orthogonal with respect to this measure. These Szego polynomials are simply related to the transfer functions of a family of Wierner-Levinson filters. The n complex random zeros of the random orthogonal polynomial of degree n are the focus point of the method, (i) all the zeros are within the unit circle [1]. When n and N increase, they all have their mean distance D to the unit circle going to zero [2]. (ii) d pairs of complex conjugate zeros can be associated to the d trigonometric components of the signal [3]. In the mean their distance (D) to the unit circle goes to zeros like n –2 [4]. Asymptotically, the modulus of their phases are equal to the frequencies of the d trigonometric components [2]. (iii) the (n – 2d) remaining zeros can be associated to the noise. The bulk of these “noisy zeros” is further away from the unit circle [3]. In the mean, the distance to the latter is decreasing like n – 1 [4]. (iv) if only noise is present the mean positions of the n zeros are regularly spaced on an inner circle [5]. This exactly regular spacing or crystallization is the signature of the noise. It is universal with respect to the probability law of the noise and to the length of the signal, provided lnN.

What is a complex number? The question may be well solved today, but about 400 years ago it would have baffled many mathematicians. For example, how was it possible to define the square root of a negative number, which was an "impossible" quantity at the time, but they were all real quantities in mathematics. Complex analysis is now recognized as a fundamental part of mathematics because of its many applications in physics and engineering as well as its connections to other branches of mathematics. By solving some equations of more than one degree, complex numbers were produced. Most mathematicians focused on real analysis and physics applications until the nineteenth century. Cauchy was the first figure to make a considerable effort in a challenging examination. He calculated a number of complex integrals using his integral formula, establishing the foundation for defining the operations and characteristics of complex analysis. Gauss furthered the development of the complex number geometrical theory. The first part of this essay focuses on the history discovery of mathematical and geometrical aspects of complex numbers. The topological features of the complex plane are then carefully understood after that. The beautiful characteristics of complex functions and series are finally compiled based on the topology of the complex field.

Spaces of entire functions of several complex variables occupy an important position in view of their vast applications in various branches of mathematics, for instance, the classical analysis, theory of approximation, theory of
 topological bases etc. With an idea of correlating entire functions with certain aspects in the theory of basis in locally convex spaces, we have investigated in this paper the bornological aspects of the space $X$ of integral functions of several complex variables. By $Y$ we denote the space of all power series with positive radius of convergence at the origin. We introduce bornologies on $X$
 and $Y$ and prove that $Y$ is a convex bornological vector space which is the completion of the convex bornological vector space $X$.

The paper is devoted to the study of the function E γ ρ,μ(z) defined for complex ρ, μ, γ (Re(ρ) > 0) by which is a generalization of the classical Mittag-Leffler function E ρ,μ(z) and the Kummer confluent hypergeometric function Φ(γ, μ; z). The properties of E γ ρ,μ(z) including usual differentiation and integration, and fractional ones are proved. Further the integral operator with such a function kernel is studied in the space L(a, b). Compositions of the Riemann–Liouville fractional integration and differentiation operators with E γ ρ,μ,ω;a+ are established. An analogy of the semigroup property for the composition of two such operators with different indices is proved, and the results obtained are applied to construct the left inversion operator to the operator E γ ρ,μ,ω;a+. Since, for γ = 0, E 0 ρ,μ,ω;a+ coincides with the Riemann–Liouville fractional integral of order μ, the above operator and its inversion can be considered as generalized fractional calculus operators involving the generalized Mittag-Leffler function E γ ρ,μ(z) in the kernels. Similar assertions are presented for the integral operators containing the Mittag-Leffler and Kummer functions, E ρ,μ(z) and Φ(γ, μ; z), in the kernels, and applications are given to obtain solutions in closed form of the integral equations of the first kind.

This qualitative study aimed to diagnose the challenges facing teachers of the (A2) program for the Arabic language in developing 21st century learning skills among students of the Arabic language in Comoros high schools. The 21st century imposes an interest in acquiring students the skills that can help them solve their learning problems, prepare for work and live in this 21st century, the ability to innovate, innovate and collaborate in their practices rather than compete, and to be able to adapt to the rapid technological change and development of a modern society. This research dealt with semi-organized personal interviews from six teachers for this program in one of the islands, and the researcher interviewed them one by one. The content of the interviews was analyzed, and the topics were extracted. This research is divided into two sections. The first section deals with the effects that impede the application of 21st century learning skills among Arabic language students from the teachers' point of view. Administratively, there is a problem in the curriculum and teaching methods, weak response of students towards learning the Arabic language, while the second section deals with the suggestions of teachers on how to develop the application of skills among students of the Arabic language from the perspective of teachers which are: Raising interest in Arabic (2) Providing modern teaching aids (3) Raising students' desire towards learning Arabic. Keywords: Challenges, Secondary Teachers, (A2) Program, Learning Skills, Comoros. Keywords Challenges, Secondary Teachers, (A2) Program, Learning Skills, Comoros

The paper gives some insight on the use of Levinson's recursion on non positive Toeplitz matrices. The critical case, where a principal minor is zero, leads to define a new lossless lattice structure with zeros on the unit circle, also useful in the standard case. The stability of the AR models built on the eigen vectors of a Toeplitz is also examined, most of the corresponding zeros are shown to lie on the unit circle. The results apply to noise cancellation, spectral line analysis and computation of eigenvectors of correlation matrices.

Chapter 15 - Algebraic Methods for Computer Aided Geometric Design

This research will provide an implementation of the study about the area in terms of numerical point of view. Regular n-shaped constructions that load a unit circle and load in unit circles can be made for at least 3 points on the unit circle. The irregular n-shaped construction contained in the unit circle can also be made without conditions with a minimum of 3 points on the unit circle. The problem arises in the irregular n-shaped construction that contains the unit circle, apparently it cannot be made by taking 3 points freely. The purpose of this research is to conduct an analysis related to what conditions are needed so that the irregular n terms can be made so that it contains a unit circle. The method used is a literature review, carried out by comparing and analyzing research that has been done relating to irregular and irregular aspects, the analysis process is carried out on irregular n-construction construction with the help of MATLAB for visualization. The results of the research show that one of the requirements for irregular n-terms can be made and contain unit circles found, and the theory test is carried out empirically with the help of MATLAB for visualization. Another result is a theory about the existence of the irregular n-aspect mathematically.

In order to establish that extremal functions in the Bergman space A p act as both expansive multipliers and contractive divisors, Duren, Khavinson, Shapiro and Sundberg made use of an integral formula involving the biharmonic Green function. Using a weighted biharmonic Green function, we derive an analogous integral formula in the standard weighted Bergman space \(A_{\alpha}^{p}\ {\rm when}\ \alpha =1\), and we also discuss how the formula can be established for general α. Moreover, we show that each \(A_{\alpha}^{p}\) inner function acts as a contractive divisor on the invariant subspace which it generates.

The fractional Laplacian operator is a very important fractional operator that is often used to describe several anomalous diffusion phenomena. In this paper, we develop some numerical schemes, including a finite difference scheme and finite volume scheme for the fractional Laplacian operator, and apply the resulting numerical schemes to solve some fractional diffusion equations. First, the fractional Laplacian operator can be characterized as the weak singular integral by an integral operator with zero boundary condition. Second, because the solutions of fractional diffusion equations are usually singular near the boundary, we use a fractional interpolation function in the region near the boundary and use a classical interpolation function in other intervals. Then, we apply a finite difference scheme to the discrete fractional Laplacian operator and fractional diffusion equation with the above fractional interpolation function and classical interpolation function. Moreover, it is found that the differential matrix of the above scheme is a symmetric matrix and strictly row-wise diagonally dominant in special fractional interpolation functions. Third, we show a finite volume scheme for a discrete fractional diffusion equation by fractional interpolation function and classical interpolation function and analyze the properties of the differential matrix. Finally, the numerical experiments are given, and we verify the correctness of the theoretical results and the efficiency of the schemes.

We study the norm of point evaluation at the origin in the Paley–Wiener space PWp for 0 < p < ∞, i.e., we search for the smallest positive constant C, called Cp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathscr{C}}_{p}$$\\end{document}, such that the inequality |f(0)|p≤C∥f∥pp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\vert f(0)\\vert ^{p}\\leq C \\Vert f\\Vert _{p}^{p}$$\\end{document} holds for every f in PWp. We present evidence and prove several results supporting the following monotonicity conjecture: The function p↦Cp/p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p \\mapsto {\\mathscr{C}}_{p}/p$$\\end{document} is strictly decreasing on the half-line (0, ∞). Our main result implies that Cp<p/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathscr{C}}_{p} < p/2$$\\end{document} for 2 < p < ∞, and we verify numerically that Cp>p/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathscr{C}}_{p} > p/2$$\\end{document} for 1 ≤ p < 2. We also estimate the asymptotic behavior of Cp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathscr{C}}_{p}$$\\end{document} as p → ∞ and as p → 0+. Our approach is based on expressing Cp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathscr{C}}_{p}$$\\end{document} as the solution of an extremal problem. Extremal functions exist for all 0 < p < ∞; they are real entire functions with only real zeros, and the extremal functions are known to be unique for 1 ≤ p < ∞. Following work of Hörmander and Bernhardsson, we rely on certain orthogonality relations associated with the zeros of extremal functions, along with certain integral formulas representing respectively extremal functions and general functions at the origin. We also use precise numerical estimates for the largest eigenvalue of the Landau–Pollak–Slepian operator of time-frequency concentration. A number of qualitative and quantitative results on the distribution of the zeros of extremal functions are established. In the range 1 < p < ∞, the orthogonality relations associated with the zeros of the extremal function are linked to a de Branges space. We state a number of conjectures and further open problems pertaining to Cp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathscr{C}}_{p}$$\\end{document} and the extremal functions.

Guest Editorial: Situational awareness of integrated energy systems

The paper looks the work of E.M. Kennaugh in the early development of backscatter radar polarimetry. After some remarks on the history of Kennaugh's dual space approach, we discuss time reversal and the voltage equation. The passage from the linear vector space to the dual linear vector space with the corresponding linear form, i.e.. the voltage equation, used extensively by Kennaugh and others, brought radar polarimetry in contact with branches of rather esoteric pure mathematics, namely the geometry of the classical groups and geometric algebra. This theory was already prospering during all of the last century, with numerous interrelations to other fields of mathematics. Many theorems that were considered as new in radar polarimetry actually have a long and important history in their special mathematical context. Many new discoveries in geometric algebra are waiting to be related to the theory of polarimetry as well as to ellipsometry in optics. These remarks do not diminish the achievements and progress made by Kennaugh in the pioneering stages of radar polarimetry, but honor him. He was the first to clear the undergrowth that covered and surrounded early radar polarimetry and opened the door to other points of view that involve interesting, and by far not yet fully exploited, branches of mathematics.

Two key concepts strongly supported by the NCTM's Curriculum and Evaluation Standards for School Mathematics (1989) are (1) that geometry should be taught from multiple perspectives and (2) that it is necessary to make mathematical connections. Both concepts can be implemented by appropriately selecting topics that can be presented to students from several points of view in different environments and that can also link different branches of mathematics or mathematics and other sciences.