The generation of broadband supercontinua (SC) in air-silica microstructured fibers results from a delicate balance of dispersion and nonlinearity. We analyze two models aimed at better understanding SC. In the first one, we characterize linear dispersion in the Fourier domain from the calculated group velocity dispersion (GVD) without using a Taylor approximation for the propagation constant. ...

In this paper, a new algorithm for function approximation is proposed to obtain better generalization performance and faster convergent rate. The new algorithm incorporates the architectural constraints from a priori information of the function approximation problem into Extreme Learning Machine. On one hand, according to Taylor theorem, the activation functions of the hidden neurons in this al...

In this paper, we consider the problem of best approximation in e,(n), 1 5 p 5 co. If h,, 1 5 p < co, denotes the best &-approximation of the element h E W " from a proper affine subspace K of W " , h @ K, then limp,1 h p-h* where h; is a best er-approximation of h from K, the i, so-called natural best er-approximation. We prove that, for every T E A, the best &approximations have a Taylor expa...

With the Gaussian Process model, the predictive distribution of the output corresponding to a new given input is Gaussian. But if this input is uncertain or noisy, the predictive distribution becomes non-Gaussian. We present an analytical approach that consists of computing only the mean and variance of this new distribution (Gaussian approximation). We show how, depending on the form of the co...

In this appendix we show that 1 2 ∆ F (θ)∆ is a second order Taylor approximation of D KL (p(θ)p(θ + ∆)). First, let g q (θ) :=D KL (qp(θ)) = ω∈Ω q(ω) ln q(ω) p(ω|θ). We begin by deriving equations for the Jacobian and Hessian of g q at θ: ∂g q (θ) ∂θ = ω∈Ω q(ω) p(ω|θ) q(ω) ∂ ∂θ q(ω) p(ω|θ) = ω∈Ω q(ω) p(ω|θ) q(ω) −q(ω) ∂p(ω|θ) ∂θ p(ω|θ) 2 = ω∈Ω − q(ω) p(ω|θ) ∂p(ω|θ) ∂θ , (4) and so: ∂ 2 g q (θ)...

In this study, a Taylor method is developed for numerically solving the high-order most general nonlinear Fredholm integro-differential-difference equations in terms of Taylor expansions. The method is based on transferring the equation and conditions into the matrix equations which leads to solve a system of nonlinear algebraic equations with the unknown Taylor coefficients. Also, we test the ...

We study the observable pulse profiles that can be generated from precessing pulsars. A novel coordinate system is defined to aid visualization of the observing geometry. Using this system we explore the different families of profiles that can be generated by simple, circularly symmetric beam shapes. An attempt is then made to fit our model to the observations of relativistic binary PSR B1913+1...

The most popular method for computing the matrix logarithm is a combination of inverse scaling and squaring in conjunction with Padé approximation, sometimes accompanied by Schur decomposition. In this work, we present Taylor series algorithm, based on free-transformation approach technique, that uses recent polynomial formulas evaluating approximation more efficiently than Paterson–Stockmeyer ...

This paper deals with Chebyshev wavelets. We analyze their properties computing Fourier transform. Moreover, we discuss the differential of wavelets due to connection coefficients. Uniform convergence and approximation error allow us provide rigorous proofs. In particular, expand mother wavelet in Taylor series an application both fractional calculus fractal geometry. Finally, give two examples...

This article concerns the numerical approximation of two-dimensional nonstationary Navier–Stokes equations with $$H^1$$ initial data. By utilizing special locally refined temporal stepsizes, we prove that linearly extrapolated Crank–Nicolson scheme, usual stabilized Taylor–Hood finite element method in space, can achieve second-order convergence time and space. Numerical examples are provided t...