Circular nodes in neural networks.

The purpose of this text is to present some new results in the spectral theory of orthogonal polynomials and Schrodinger operators. These results concern perturbations of the free Schrodinger operator and of the free case for orthogonal polynomials on the unit circle (which corresponds to Verblunsky coefficients equal to 0) and the real line (which corresponds to off-diagonal Jacobi coefficients equal to 1 and diagonal Jacobi coefficients equal to 0). The condition central to our results is that of generalized bounded variation. This class consists of finite linear combinations of sequences of rotated bounded variation with an L^1 perturbation. This generalizes both usual bounded variation and expressions of the form lambda(x) cos(phi x + alpha) with lambda(x) of bounded variation (and, in particular, with lambda(x) = x^gamma, Wigner-von Neumann potentials) as well as their finite linear combinations. Assuming generalized bounded variation and an L^p condition (with any finite p) on the perturbation, our results show preservation of absolutely continuous spectrum, absence of singular continuous spectrum, and that embedded pure points in the continuous spectrum can only occur in an explicit finite set.

The mixed problem for the telegraph equation well-known in electrical engineering and electronics, provided that the line is free from distortions, is reduced to a similar problem for one-dimensional inhomogeneous wave equation. An effective way to solve this problem is based on the use of special functions – polylogarithms, which are complex power series with power coefficients, converging in the unit circle. The exact solution of the problem is expressed in integral form in terms of the imaginary part of the first-order polylogarithm on the unit circle, and the approximate one – in the form of a finite sum in terms of the real part of the dilogarithm and the imaginary part of the third-order polylogarithm. All the indicated parts of the polylogarithms are periodic functions that have polynomial expressions of the corresponding degrees on an interval of length in the period, which makes it possible to obtain a solution to the problem in elementary functions. In the paper, we study a mixed problem for the telegrapher’s equation which is well-known in applications. This problem of linear substitution of the desired function witha time-exponential coefficient is reduced to a similar problem for the Klein – Gordon equation. The solution of the latter can be found by dividing the variables in the form of a series of trigonometric functions of a line point with time-dependent coefficients. Such a solution is of little use for practical application, since it requires the calculation of a large number of coefficients-integrals and it is difficult to estimate the error of calculations. In the present paper, we propose another way to solve this problem, based on the use of special He-functions, which are complex power series of a certain type that converge in the unit circle. The exact solution of the problem is presented in integral form in terms of second-order He-functions on the unit circle. The approximate solution is expressed in the final form in terms of third-order He-functions. The paper also proposes a simple and effective estimate of the error of the approximate solution of the problem. It is linear in relation to the line splitting step with a time-exponential coefficient. An example of solving the problem for the Klein – Gordon equation in the way that has been developed is given, and the graphs of exact and approximate solutions are constructed.

УДК 517.5 Якщо неперервна на дійсній осі 2 π -періодична функція f змінює свій знак у 2 s , s ∈ ℕ , точках y i : - π ≤ y 2 s < y 2 s - 1 < … < y 1 < π , а для інших i ∈ ℤ точки y i визначаються періодично, то для кожного натурального n , більшого за деяку сталу N ( k , y i ) , що залежить тільки від k ∈ ℕ і min i = 1 , … ,2 s { y i - y i + 1 } , знайдено тригонометричний поліном P n порядку не вищого за n такий, що P n має скрізь той самий знак, що і f , за винятком, можливо, маленьких околів точок y i : ( y i - π / n , y i + π / n ) , і ‖ f - P n ‖ ≤ c ( k , s ) ω k ( f , π / n ) , де c ( k , s ) — стала, що залежить тільки від k і s , ω k ( f , ⋅ ) — модуль гладкості k -го порядку функції f і ‖ ⋅ ‖ — max-норма.

The digital control of a nonlinear system is analyzed in general terms. The analysis considers the case of a pair of branches of the root loci crossing the unit circle. The crossing occurs as the eigenvalues of the Jacobian of the discrete-time system are varied depending on, for example, the sampling period, and the crossing is assumed to take place anywhere on the unit circle except on the real axis. Results are presented for a particular example. >

The authors suggest a modification of the Toeplitz approximation method for estimating frequencies of multiple sinusoids from covariance measurements. The method constructs a state-feedback matrix following a low-rank approximation of the Toeplitz covariance matrix via singular-value decomposition. Ideally, the eigenvalues of this state-feedback matrix will be on the unit circle in the complex plane, and the angles that they make with the real axis will be equal to the unknown sinusoid frequencies. The modification proposed here exploits this prior knowledge of the modulus of the eigenvalues and guarantees that even in the presence of noise the eigenvalues of the estimated state-feedback matrix will lie on the unit circle. >

How to count the number of zeros that a polynomial has on the unit circle?

RII type recurrence, generalized eigenvalue problem and orthogonal polynomials on the unit circle

A recently introduced fast algorithm for the computation of the first N terms in an expansion of an analytic function into ultraspherical polynomials consists of three steps: Firstly, each expansion coefficient is represented as a linear combination of derivatives; secondly, it is represented, using the Cauchy integral formula, as a contour integral of the function multiplied by a kernel; finally, the integrand is transformed to accelerate the convergence of the Taylor expansion of the kernel, allowing for rapid computation using Fast Fourier Transform. In the current paper we demonstrate that the first two steps remain valid in the general setting of orthogonal polynomials on the real line with finite support, orthogonal polynomials on the unit circle and Laurent orthogonal polynomials on the unit circle.

ABSTRACTA stochastic representation is derived for a vector random field that is stationary, isotropic, and mean square continuous on a sphere or unit circle. The established stochastic representation is an infinite series involving the sequence of ultraspherical or Gegenbauer's polynomials, looks like a mimic of the series representation of the covariance matrix function of the isotropic vector random field, but differs from the spectral representation in terms of the ordinary spherical harmonics. It is also shown in this paper that some isotropic and continuous covariance matrix functions on the real line or , if they are compactly supported, can be adopted as covariance matrix functions on the unit circle or .

Matrix methods for quadrature formulas on the unit circle. A survey

In this chapter we introduce shift-invariant algebras, the main objects of this book. These are uniform algebras on a compact connected group G, consisting of continuous functions on G, whose spectrum is contained in a semigroup S of the dual group Ĝ. If Ĝ is a subgroup of ℝ, and S ⊂ ℝ+, then the maximal ideal space of the corresponding shift-invariant algebra is the G-disc, or, big disc over G. In this chapter we describe two important models of shift-invariant algebras, namely, by the means of almost periodic functions on ℝ, and by the means of H∞-functions on the unit circle. The set of automorphisms, and the peak groups of shift-invariant algebras are also characterized. Extensions on G-discs and groups of several classical theorems of Complex Analysis, such as Radó’s theorem for null-sets and the Riemann theorem for removable singularities of analytic functions, are stated and proved. It is shown that these extensions hold for some semigroups, while in general they fail. In principle we state all results for general shift-invariant algebras AS, though they apply automatically to the particular cases of algebras APS of almost periodic functions, and of H S ∞ -algebras. Asymptotically almost periodic functions combine the properties of classical almost periodic functions on ℝ, and of continuous functions on ℝ that vanish at infinity.

The purpose of this work is to study remotely almost periodic functions defined on the semi-axis ℝ + := [ 0 , + ∞ ) {\mathbb{R}_{+}:=[0,+\infty)} (respectively, on the real axis ℝ := ( - ∞ , + ∞ ) {\mathbb{R}:=(-\infty,+\infty)} ) with values in the Banach space 𝔅 {\mathfrak{B}} . We establish a relation between remotely almost periodic movements in the shift dynamical system ( C ( 𝕋 , 𝔅 ) , 𝕋 {(C(\mathbb{T},\mathfrak{B}),\mathbb{T}} , σ ) {\sigma)} ( 𝕋 = ℝ + {\mathbb{T}=\mathbb{R}_{+}} or ℝ {\mathbb{R}} ) in the space of continuous functions C ⁢ ( 𝕋 , 𝔅 ) C(\mathbb{T},\mathfrak{B}) and remotely almost periodic functions. Using this connection, we prove some important analytical and topological properties of remotely almost periodic functions.

Optimal Approximation of Periodic Analytic Functions with Integrable Boundary Values

To enhance the poor diagnosing system for detecting COVID pneumonia using AI and to compare the algorithms and their performances in diagnosis prediction. Materials and methods: The technique of Convolutional Neural Network (CNN) with forward propagation for the feature extraction from X-ray images of patients diagnosis. The implementation has been carried out with the help of Tensorflow and Keras. The algorithms tested over 700 sets of chest X-ray images which has been utilized for test and train dataset images respectively, that are user chest X-rays classified and predicted with the novel diagnosis. Result: The analysis on the dataset and test cases has been performed successfully and acquired 91% accuracy for the CNN with forward propagation algorithm and compared to the artificial neural network algorithm (ANN), which gave 73% accuracy. With the level of significance (p<0.005), the resultant data depicts the reliability in independent sample t tests. Conclusion: On the whole process of prediction of accuracy for the image recognition through the CNN with forward propagation gives significantly better performance when compared to ANN.

In this chapter, we will discuss the basic architecture of neural networks including activation functions, forward propagation, and backpropagation. We will also create a simple neural network model from scratch using the sigmoid activation function. In particular, this chapter will discuss: Neural network architecture. Activation functions used in neural networks. Forward propagation. Loss function of neural networks. Backpropagation.