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 ...

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

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

Dissymmetric 1,3-diphenylallene derivative 3 connected with 4,5-bis(methyl-thio)tetrathiafulvalenyl and 1-pyrenyl substituents was prepared and characterized. The molecular structure was determined by X-ray crystallographic analysis. Optical resolution was accomplished using a recycling chiral HPLC, and its chiroptical properties were examined with optical rotation and electronic circular dichr...

In X-ray tomography, the Fourier slice theorem provides a relationship between the Fourier components of the object being imaged and the measured projection data. The Fourier slice theorem is the basis for X-ray Fourier-based tomographic inversion techniques. A similar relationship, referred to as the 'Fourier shell identity' has been previously derived for photoacoustic applications. However, ...

Novel derivatives of the anti-HIV-1 agent, TSAO-T, bearing at the N-3 position a variety of polar, lipophilic, or aromatic groups linked to that position through flexible polymethylene linkers of different length, were prepared and evaluated for their anti-HIV activity. Several compounds (within the series of polar bearing groups) exhibited a 2-6-fold improved antiviral potency with regard to t...

Let P(z):= ?n v=0 avzv be a univariate complex coefficient polynomial of degree n. Then as generalization well-known classical inequality Tur?n [25], it was shown by Govil [7] that if P(z) has all its zeros in |z| ? k, k 1, then max |z|=1 |P?(z)| n/1 + kn |P(z)|, whereas, , 0 < again [6] who gave an extension the Erd?s-Lax [13], obtaining provided and |Q?(z)| attain maximum at same point on ...

Let \(P(z)\) be a polynomial of degree \(n\), then concerning the estimate for maximum \(|P'(z)|\) on unit circle, it was proved by S. Bernstein that \(\| P'\|_{\infty}\leq n\| P\|_{\infty}\). Later, Zygmund obtained an \(L_p\)-norm extension this inequality. The polar derivative \(D_{\alpha}[P](z)\) \(P(z)\), with respect to point \(\alpha \in \mathbb{C}\), generalizes ordinary in sense \(\lim...