Function approximation arises in many branches of applied mathematics and computer science, particular numerical analysis, finite element theory more recently data sciences domain. From most common we cite, polynomial, Chebychev Fourier series approximations. In this work establish some approximations a continuous function by activation functions. First, deal with one two dimensional cases. The...

Linear and planar antenna arrays are synthesized to have maximum directivity for a specified sidelobe level. The directivity is maximized subject to a given SLL. The beamwidth and the zeros of array factor are studied as well as the directivity. Maximum directivity-arrays are compared through some examples with super-directive, uniform, Dolph-Chebyshev and Riblet-Chebychev arrays to find a comp...

Theorem 26.1.2 (Chebychev inequality) Let X be a random variable with μx = E[X] and σx be the standard deviation of X. That is σX = E [ (X − μx) ] . Then, Pr [|X − μX | ≥ tσX] ≤ 1 t2 . Proof: Note that Pr [|X − μX | ≥ tσX] = Pr[(X − μX) ≥ t2σ2X] . Set Y = (X − μX). Clearly, E [ Y ] = σX. Now, apply Markov inequality to Y . This work is licensed under the Creative Commons Attribution-Noncommerci...

Uncertainties in an electromagnetic observable, that arise from uncertainties in geometric and electromagnetic parameters of an interaction configuration, are here characterized by combining computable higher-order moments of the observable with higher-order Chebychev inequalities. This allows for the estimation of the range of the observable by rigorous confidence intervals. The estimated rang...

The first two papers in this series reviewed the basic is used to derive the z-transforms of the filters from their concepts which apply to digital filter theory and presented s-plane continuous time descriptions. Recurrence relationdesign techniques based on the z plane pole-zero plot. In ships which may be used to implement filters of van’ous this paper these methods are used to develop digit...

Understanding the physical structures of accreted matter very close to a black hole in quasars and active galactic nucleus (AGN) is an important milestone constrain activities occurring their centers. In this paper, we numerically investigate effects asymptotic velocities on accretion disk around Kerr Einstein–Gauss–Bonnet (EGB) rapidly rotating holes. The Bondi–Hoyle considered with falling ga...

Theorem 27.1.2 (Chebychev inequality) Let X be a random variable with μx = E[X] and σx be the standard deviation of X. That is σX = E [ (X − μx) ] . Then, Pr [|X − μX | ≥ tσX] ≤ 1 t2 . Proof: Note that Pr [|X − μX | ≥ tσX] = Pr[(X − μX) ≥ t2σ2X] . Set Y = (X − μX). Clearly, E [ Y ] = σX. Now, apply Markov inequality to Y . This work is licensed under the Creative Commons Attribution-Noncommerci...

Definition 28.1.2 The variance of a random variable X with expectation μ = E[X] is the quantity V[X] = E [ (X − μ) ] = E [ X2 ] − μ2. The standard deviation of X is σX = √ V[X]. Theorem 28.1.3 (Chebychev inequality) Let X be a random variable with μx = E[X] and σx be the standard deviation of X. That is σX = E [ (X − μx) ] . Then, Pr [|X − μX | ≥ tσX] ≤ 1 t2 . This work is licensed under the Cr...