The eigenvalues of the isotropic magnetic interaction system extraction from Internal Field Approximation (IFA) q e computed. on the basis of RKKY for the complicated magnetic behaviour of Gd, X system X= (Au , Pt , Al , In ) as well as Gd. From the strength and slope of eigenvalue curves in different ranges of interaction the strength and sharpness of the magnetic transition can be suggest...

* Correspondence: [email protected] Department of Mathematics, University of Karachi, University Road, Karachi, Pakistan Full list of author information is available at the end of the article Abstract In this article, we give more generalized results than in Anwar et al. (2010) and Latif and Pečarić (2010) in new direction by using second-order divided difference. We investigate the expone...

Rényi divergence is related to Rényi entropy much like information divergence (also called Kullback-Leibler divergence or relative entropy) is related to Shannon’s entropy, and comes up in many settings. It was introduced by Rényi as a measure of information that satisfies almost the same axioms as information divergence. We review the most important properties of Rényi divergence, including it...

Recently a majorization method for optimizing partition functions of log-linear models was proposed alongside a novel quadratic variational upper-bound. In the batch setting, it outperformed state-of-the-art firstand second-order optimization methods on various learning tasks. We propose a stochastic version of this bound majorization method as well as a low-rank modification for highdimensiona...

Many inequality relations between real vector quantities can be succinctly expressed as “weak (sub)majorization” relations using the symbol ≺w. We explain these ideas and apply them in several areas: angles between subspaces, Ritz values, and graph Laplacian spectra, which we show are all surprisingly related. Let Θ(X ,Y) be the vector of principal angles in nondecreasing order between subspace...

We present a straightforward linear algebraic model of greed, based only on extensions of classical majorization and convexity theory. This gives an alternative to other models of greedy-solvable problems such as matroids, greedoids, submodular functions, etc., and it is able to express established examples of greedy-solvable optimization problems that they cannot. The linear algebraic approach...

Let Mn,m be the set of all n × m matrices with entries in F, where F is the field of real or complex numbers. A matrix R ∈ Mn with the property Re=e, is said to be a g-row stochastic (generalized row stochastic) matrix. Let A,B∈ Mn,m, so B is said to be gw-majorized by A if there exists an n×n g-row stochastic matrix R such that B=RA. In this paper we characterize all linear operators that stro...

This paper considers the problem of constructing a normatively significant multidimensional Gini index of relative inequality. The social evaluation relation (SER) from which the index is derived is required to satisfy a weak version of the Pigou-Dalton Bundle Principle (WPDBP) (rather than Uniform Majorization or similar conditions). It is also desired to satisfy a weak form of the condition o...

Herpes simplex virus 1 mutants lacking the gene encoding glycoprotein D (gD) and the gD normally present in the envelope of the virus (gD(-/-) stocks) or mutants lacking the gD gene but containing trans-induced gD in their envelopes (gD(-/+)) cause apoptosis in human SK-N-SH cells. The gD(-/-) virions are taken up by endocytosis and are degraded, whereas gD(-/+) viruses replicate but produce gD...