We derive iterative methods for computing the Fréchet derivative of the map which 4 sends a full-rank matrix A to the factor U in its polar decomposition A = UH, where U has 5 orthonormal columns and H is Hermitian positive definite. The methods apply to square matrices 6 as well as rectangular matrices having more rows than columns. Our derivation relies on a novel 7 identity that relates the ...

In this paper, the fractional partial differential equations are defined by modified Riemann-Liouville fractional derivative. With the help of fractional derivative and fractional complex transform, these equations can be converted into the nonlinear ordinary differential equations. By using solitay wave ansatz method, we find exact analytical solutions of the space-time fractional Zakharov Kuz...

We discuss the estimation of derivatives of a performance measure using the likelihood ratio method in simulations of highly reliable Markovian systems. We compare the difficulties of estimating the performance measure and of estimating its partial derivatives with respect to component failure rates as the component failure rates tend to 0 and the component repair rates remain fixed. We first c...

A simple and reliable precolumn derivatization liquid chromatography method with ultraviolet detection has been developed and validated for the analysis of glucosamine (GS) in various dietary supplement formulations and raw materials. Additionally, the proposed method was used for analysis of carisoprodol (CR) found in ternary mixture with paracetamol (PR) and caffeine (CF). The linearity range...

Suppose that K is a nonempty closed convex subset of a real uniformly convex and smooth Banach space E with P as a sunny nonexpansive retraction. Let T1,T2 : K → E be two weakly inward and asymptotically nonexpansive mappings with respect to P with sequences {Kn},{ln} ⊂ [1,∞), limn→∞kn = 1, limn→∞ln = 1, F(T1)∩ F(T2) = {x ∈ K : T1x = T2x = x} =∅, respectively. Suppose that {xn} is a sequence in...

In this paper, we apply spectral method based on the Bernstein polynomials for solving a class of optimal control problems with Jumarie’s modified Riemann-Liouville fractional derivative. In the first step, we introduce the dual basis and operational matrix of product based on the Bernstein basis. Then, we get the Bernstein operational matrix for the Jumarie’s modified Riemann-Liouville fractio...

In this paper, the average level crossing rate of the ratio of product of two k-μ random variables and k-μ random variable will be analyzed. The joint probability density function of the ratio of product of two k-μ random variables and k-μ random variable and its first derivative will be calculated. By using this expression, the average level crossing rate of the ratio of product of two k-μ ran...