Local boundedness property for parabolic BVP's and the gaussian upper bound for their Green functions

In the present note, we give a concise proof for the equivalence between the local boundedness property for parabolic Dirichlet BVP's and the gaussian upper bound for their Green functions. The parabolic equations we consider are of general divergence form and our proof is essentially based on the gaussian upper bound by Daners \cite{Da} and a Caccioppoli's type inequality. We also show how the same analysis enables us to get a weaker version of the local boundedness property for parabolic Neumann BVP's assuming that the corresponding Green functions satisfy a gaussian upper bound

A note on a relationship between the inverse eigenvalue problems for nonnegative and doubly stochastic matrices and some applications

In this note, we establish some connection between the nonnegative inverse eigenvalue problem and that of doubly stochastic one. More precisely, we prove that if $(r; {\lambda}_2, ..., {\lambda}_n)$ is the spectrum of an $n\times n$ nonnegative matrix A with Perron eigenvalue r, then there exists a least real number $k_A\geq -r$ such that $(r+\epsilon; {\lambda}_2, ..., {\lambda}_n)$ is the spectrum of an $n\times n$ nonnegative generalized doubly stochastic matrix for all $\epsilon\geq k_A.$ As a consequence, any solutions for the nonnegative inverse eigenvalue problem will yield solutions to the doubly stochastic inverse eigenvalue problem. In addition, we give a new sufficient condition for a stochastic matrix A to be cospectral to a doubly stochastic matrix B and in this case B is shown to be the unique closest doubly stochastic matrix to A with respect to the Frobenius norm. Some related results are also discussed.Comment: 8 page

Investigative powers of the Egyptian Competition Authority: a guide for companies in the Egyptian market

This article explores the investigative powers of the Egyptian Competition Authority vis-à-vis companies under scrutiny and discerns whether the documents/correspondences exchanged between these companies, their in-house legal departments and; external consultants are legally privileged under relevant Egyptian laws and regulations

Continuous spin and tensionless strings

A classical action is proposed which upon quantisation yields massless particles belonging to the continuous spin representation of the Poincar\'e group. The string generalisation of the action is identical to the tensionless extrinsic curvature action proposed by Savvidy. We show that the BRST quantisation of the string action gives a critical dimension of 28. The constraints result in a number of degrees of freedom which is the same as the bosonic string.Comment: 13 pages; v2 added references; v3 discussion adde