Comment on "PT Symmetry as a Generalization of Hermiticity" [arXiv:1002.2676]

We notice that the general PT-symmetric Hamiltonian matrix(N=2) having 6-real parameters fails to reproduce one parameter PT-symmetric matrix.Comment: Title unchanged ,one ref [5] added . Abstract changed . All the previous matrices have been changed consideing the ref[5

Zero energy correction method for non-Hermitian Harmonic oscillator with simultaneous transformation of co-ordinate and momentum: Wave function analysis under Iso-spectral condition

We present a complete analysis on energy and wave function of Harmonic oscillator with simultaneous non-hermitian transformation of co-ordinate ($(x \rightarrow \frac{(x+ i\lambda p)}{\sqrt{(1+\beta \lambda)}}$ and momentum $(p \rightarrow \frac{(p+ i\beta x)}{\sqrt{(1+\beta \lambda)}}$ for getting energy eigenvalue using perturbation theory under iso-spectral condition. Further we notice that two different frequency of oscillation ($w_{1}, w_{2}$)correspond to same energy eigenvalue, which can also be verified using Lie algebraic approach [Zhang et.al J.Math.Phys 56 ,072103 (2015)]. Interestingly wave function analysis using similarity transformation [F.M. Fernandez, Int. J. Theo. Phys. (2015)(in Press)] refers to a very special case.Comment: This paper for replacement .(i) Minor change in title reflecting wave function analysis(ii) Abstract-chaed suitably to refect wave function (iii) Text original work with information on wave function ,comparison and slight modification in references.Kindly accep

The Shapley-Folkman Theorem and the Range of a Bounded Measure: An Elementary and Unified Treatment

We present proofs, based on the Shapley-Folkman theorem, of the convexity of the range of a strongly continuous, finitely additive measure, as well as that of an atomless, countably additive measure. We also present proofs, based on diagonalization and separation arguments respectively, of the closure of the range of a purely atomic or purely nonatomic countably additive measure. A combination of these results yields Lyapunov's celebrated theorem on the range of a countably additive measure. We also sketch, through a comprehensive bibliography, the pervasive diversity of the applications of the Shapley-Folkman theorem in mathematical economics.

Si3N4 single-crystal nanowires grown from silicon micro and nanoparticles near the threshold of passive oxidation

A simple and most promising oxide-assisted catalyst-free method is used to prepare silicon nitride nanowires that give rise to high yield in a short time. After a brief analysis of the state of the art, we reveal the crucial role played by the oxygen partial pressure: when oxygen partial pressure is slightly below the threshold of passive oxidation, a high yield inhibiting the formation of any silica layer covering the nanowires occurs and thanks to the synthesis temperature one can control nanowire dimensions