We study the asymptotic behavior of solutions to a semilinear elliptic equation associated with the critical nonlinear growth in two dimensions. { −∆u = λueu , u > 0 in Ω, u = 0 on ∂Ω, (1.1) where Ω is a unit disk in R2 and λ denotes a positive parameter. We show that for a radially symmetric solution of (1.1) satisfies ∫ D |∇u| dx → 4π, λ ↘ 0. Moreover, by using the Pohozaev identity to the re...

An explicit a priori bound for the condition number associated to each of the following problems is given: general linear equation solving, least squares, non-symmetric eigenvalue problems, solving univariate polynomials, and solving systems of multivariate polynomials. It is assumed that the input has integer coefficients and is not on the degeneracy locus of the respective problem (i.e., the ...

Electric magnetic duality symmetry is well understood in vacuum. For light propagating through a medium this symmetry is typically broken. We investigate under what conditions duality symmetry is preserved in a linear medium and employ these conditions to generalise the definition of optical helicity to a general linear medium. We will discuss some unique properties duality symmetric media poss...

We investigate the properties of localized waves in cubic nonlinear materials with a symmetric nonlocal nonlinear response of arbitrary shape and degree of nonlocality, described by a general nonlocal nonlinear Schrödinger type equation. We prove rigorously by bounding the Hamiltonian that nonlocality of the nonlinearity prevents collapse in, e.g., Bose-Einstein condensates and optical Kerr med...

The extreme ranks, i.e., the maximal and minimal ranks, are established for the general Hermitian solution as well as the general skew-Hermitian solution to the classical matrix equation AXA +BY B = C over the quaternion algebra. Also given in this paper are the formulas of extreme ranks of real matrices Xi, Yi, i = 1, · · · , 4, in a pair (skew-)Hermitian solution X = X1 + X2i + X3j + X4k, Y =...

The extreme ranks, i.e., the maximal and minimal ranks, are established for the general Hermitian solution as well as the general skew-Hermitian solution to the classical matrix equation AXA +BY B = C over the quaternion algebra. Also given in this paper are the formulas of extreme ranks of real matrices Xi, Yi, i = 1, · · · , 4, in a pair (skew-)Hermitian solution X = X1 + X2i + X3j + X4k, Y =...

A matrix A = (aij) ∈ Rn×n is said to be symmetric and antipersymmetric matrix if aij = aji = −an−j+1,n−i+1 for all 1 ≤ i, j ≤ n. Peng gave the bisymmetric solutions of the matrix equation A1X1B1+A2X2B2+. . .+AlXlBl = C, where [X1, X2, . . . , Xl] is a real matrices group. Based on this work, an adjusted iterative method is proposed to find the symmetric and antipersymmetric solutions of the abo...