We characterize asymptotic collective behaviour of rectangular random matrices, the sizes of which tend to infinity at different rates: when embedded in a space of larger square matrices, independent rectangular random matrices are asymtotically free with amalgamation over a subalgebra. Therefore we can define a “rectangular free convolution”, linearized by cumulants and by an analytic integral...

We characterize asymptotic collective behavior of rectangular random matrices, the sizes of which tend to infinity at different rates: when embedded in a space of larger square matrices, independent rectangular random matrices are asymptotically free with amalgamation over a subalgebra. Therefore we can define a “rectangular free convolution”, linearized by cumulants and by an analytic integral...

Abstract This review/research paper deals with the reduction of nonlinear partial differential equations governing dynamic behavior structural mechanical members emphasis put on theoretical aspects applied methods and signal processing. Owing to rapid development technology, materials science in particular micro/nano systems, there is a need not only revise approaches mathematical modeling vibr...

In this paper, we connect rectangular free probability theory and spherical integrals. We prove the analogue, for rectangular or square non-Hermitian matrices, of a result that Guionnet and Mäıda proved for Hermitian matrices in [12]. More specifically, we study the limit, as n,m tend to infinity, of 1 n logE{exp[nmθXn]}, where θ ∈ R, Xn is the real part of an entry of UnMnVm, Mn is a certain n...

in this article, finite difference method (fdm) is used to solve sixth-order derivatives of differential equations in buckling analysis of nanoplates due to coupled surface energy and non-local elasticity theories. the uniform temperature change is used to study thermal effect. the small scale and surface energy effects are added into the governing equations using eringen’s non-local elasticity...