Author

# I.M. Ryzhik

Bio: I.M. Ryzhik is an academic researcher. The author has contributed to research in topic(s): Table (landform) & Elementary function. The author has an hindex of 12, co-authored 19 publication(s) receiving 51170 citation(s).

Topics: Table (landform), Elementary function, Special functions, Series (mathematics), Carlson symmetric form

##### Papers

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01 Jan 2007

4,897 citations

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3 citations

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6,027 citations

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01 Jan 1980Abstract: This chapter discusses Fourier and Laplace transforms. The inversion of the Laplace transform is accomplished for analytic functions f (p) of order O (p -k ) with k > 1 by means of the inversion integral. The inversion of the Fourier transform is accomplished by means of the inversion integral. The Fourier sine and cosine transforms of the function f(x), denoted by F s (ξ) and F c (ξ), respectively, are defined by the integrals. The functions f(x) and F s ( ξ ) are called a Fourier sine transform pair, and the functions f(x) and F c ( ξ ) a Fourier cosine transform pair, and knowledge of either F s ( ξ ) or F c ( ξ ) enables f(x) to be recovered. The inversions of the Fourier sine and Fourier cosine transform are accomplished by means of the inversion integral.

30 citations

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01 Jan 1980Abstract: This chapter discusses the ordinary differential equations. Some growth estimates for solutions of second-order equations suppose that G(x) > 0 are continuous in (– ∞, ∞). The real function φ (t) is said to be an approximate solution, to within the error θ, of the differential equation except at points of discontinuity of the derivative. The Cauchy problem for the system is the problem of existence and uniqueness of the solution to this system satisfying the initial vector condition. A fundamental system of solutions of a homogeneous second-order linear differential equation in the canonical form is a system of two linearly independent solutions. Equations whose solutions possess an infinite number of zeros in the interval (0, ∞) are said to have oscillatory solutions.

1 citations

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Abstract: A considerable body of literature has been recently devoted to the inference problem of the reliability parameter R = P ( X > Y ) based on record data. In this article, we consider the records as well as the corresponding inter-record times to develop inference procedures for R assuming X and Y come from Weibull distribution. The maximum likelihood estimator of R and its corresponding confidence interval are determined. Bayesian analyses involving Tierney and Kadane’s approximation and Metropolis-Hasting samplers are used to estimate R based on LINEX and square error loss functions. In addition, the estimation problem of R is discussed in the models with known shape parameters. To compare all methods developed here, numerical simulation is carried out. Finally, different real data sets are analyzed.

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Abstract: A mean zero spatial process X ( t ) on the n -dimensional Euclidean space R n is isotropic if its covariance function (c.f.) is of the form: R n ( τ ) = Cov [ X ( t ) , X ( s ) ] , where τ = | t − s | , t , s ∈ R n , and R n is an admissible function on R 1 . An isotropic spatial process X has a bounded range of dependence if sup { τ : R n ( τ ) ≠ 0 } ∞ . Here we consider a class of isotropic c.f.’s R n ( τ ) , n = 1 , 2 , … with bounded ranges of dependence, among which there are R 1 , the classical triangular c.f. on the real line ( n = 1 ), and R 3 , the spherical c.f. in dimension three ( n = 3 ). For each dimension n ≥ 1 , the admissibility of R n ( τ ) as a c.f. in higher dimensions is studied. While it is well known that for each n ≥ 1 , R n is a legitimate c.f. on R m for all m ≤ n but it is shown that the considered R n is not a legitimate c.f. on R m when m > n . Thus the spherical c.f. R 3 cannot be a c.f. on R n when n > 3 . The issue of recognition of an isotropic c.f. on R n is discussed, and simple procedures of constructing isotropic c.f.’s on R n for every n ≥ 1 are given. This article serves as one more reminder that caution must be taken concerning the legitimacy of a selected c.f. in the corresponding spatial dimensions.

1 citations

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Abstract: The main goal in this paper is to study asymptotic behavior in L p ( R N ) for the solutions of the fractional version of the discrete in time N-dimensional diffusion equation, which involves the Caputo fractional h-difference operator. The techniques to prove the results are based in new subordination formulas involving the discrete in time Gaussian kernel, and which are defined via an analogue in discrete time setting of the scaled Wright functions. Moreover, we get an equivalent representation of that subordination formula by Fox H-functions.

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Abstract: This study investigates the problem of sound radiation from a thin vibrating annular plate. The plate is supported elastically at its circumference and embedded into the bottom of a circular cavity in a rigid plane. The selected model provides results covering the occurrence of sound radiation from the thin annular plates with any arbitrary boundary configurations. The cavity and the half-space are filled with air, and the air in the cavity plays the role of the compatibility layer. Therefore, the vibrations of the plate are coupled with that of air. Application of the compatibility layer allows the decomposition of the coupled problem of sound radiation to two problems: one for the vibration of the plate coupled with the cavity and the other for the sound radiation from the cavity to the half-space. Consequently, the plate is not coupled to the half-space directly. Therefore, the modal impedance coefficient of the plate is not required to solve the problem. The coefficients of the cavity are sufficient for the calculations, and they can be calculated significantly faster than in the case of performing a numerical integration while simultaneously satisfying the assumed accuracy by using radial polynomials. This also makes it possible to simplify numerical calculations. Finally, the total acoustic power, the acoustic pressure, and the vibration velocity have been obtained using the proposed approach and analyzed numerically. In the specific case when the cavity depth is reduced to zero, the nondimensionalized added virtual mass incremental as well as the modal impedance coefficients of the plate are calculated. Using the obtained results, the resonant frequencies can be accurately determined within a broad frequency band. The described method can be used for calculating the sound radiation for any arbitrary boundary configuration of the thin annular plates. In addition, the proposed approximation reduces the time required for the numerical calculations by roughly 2000 times compared to the time required for the numerical integration.

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Abstract: In this paper, we study the ergodic capacity (EC) and average bit error rate (BER) of spatial diversity underwater wireless optical communications (UWOC) over the generalized gamma (GG) fading channels using quadrature amplitude modulation (QAM) direct current-biased optical orthogonal frequency division multiplexing (DCO-OFDM). We derive closed-form expressions of the EC and BER for the spatial diversity UWOC with the equal gain combining (EGC) at receivers based on the approximation of the sum of independent identical distributed (i.i.d) GG random variables (RVs). Numerical results of EC and BER for QAM DCO-OFDM spatial diversity systems over GG fading channels are presented. The numerical results are shown to be closely matched by the Monte Carlo simulations, verifying the analysis. The study clearly shows the adverse effect of turbulence on the EC & BER and advantage of EGC to overcome the turbulence effect.

1 citations