We introduce the notion of pseudo-Hermiticity and show that every Hamiltonian with a real spectrum is pseudo-Hermitian. We point out that all the PT -symmetric non-Hermitian Hamiltonians studied in the literature are pseudo-Hermitian and argue that the structure responsible for the reality of the spectrum of these Hamiltonian is their pseudo-Hermiticity not PT -symmetry. We explore the basic pr...

For solving large sparse non-Hermitian positive definite linear equations, Bai et al. proposed the Hermitian and skew-Hermitian splitting methods (HSS). They recently generalized this technique to the normal and skew-Hermitian splitting methods (NSS). In this paper, we present an accelerated normal and skew-Hermitian splitting methods (ANSS) which involve two parameters for the NSS iteration. W...

In this paper, we first construct a preconditioned two-parameter generalized Hermitian and skew-Hermitian splitting (PTGHSS) iteration method based on the two-parameter generalized Hermitian and skew-Hermitian splitting (TGHSS) iteration method for non-Hermitian positive definite linear systems. Then a class of PTGHSSbased iteration methods are proposed for solving weakly nonlinear systems base...

Non-Hermitian band theory distinguishes between line gaps and point gaps. While can give rise to intrinsic non-Hermitian topology without Hermitian counterparts, line-gapped systems always be adiabatically deformed a limit. Here we show that line-gap point-gap intricately connected: topological in $d$ dimensions induce nontrivial on their $(d-1)$-dimensional boundaries when suitable internal sp...

We extend the recently proved holomorphic quantization formula of Teleman to cases of singular reductions. §0. Introduction and the statement of main results Let (M,ω, J) be a compact Kähler manifold with the Kähler form ω and the complex structure J . Let g denote the corresponding Kähler metric. We make the assumption that there exists a Hermitian line bundle L over M admitting a Hermitian co...

The Lax formulation of the hyper-Hermiticity condition in four dimensions is used to derive a potential that generalises Plebanski’s second heavenly equation for hyper-Kahler 4-manifolds. A class of examples of hyper-Hermitian metrics which depend on two arbitrary functions of two complex variables is given. The twistor theory of four-dimensional hyper-Hermitian manifolds is formulated as a com...

We study the convergence of P -regular splitting iterative methods for non-Hermitian positive definite linear systems. Our main result is that P -regular splittings of the form A = M−N , where N = N, are convergent. Natural examples of splittings satisfying the convergence conditions are constructed, and numerical experiments are performed to illustrate the convergence results obtained.

In this paper we consider enveloping C *-algebras of *-algebras given by generators and defining relations of the following form A = CX, X * | XX * = f (X * X), where f is a Hermitian mapping. Some properties of these algebras associated with simple dynamical systems (f, R) are studied. As an example quadratic dynamical systems are considered.

We will introduce a method to get all universal Hermitian lattices over imaginary quadratic fields over Q( √ −m) for all m. For each imaginary quadratic field Q( √ −m), we obtain a criterion on universality of Hermitian lattices: if a Hermitian lattice L represents 1, 2, 3, 5, 6, 7, 10, 13, 14 and 15, then L is universal. We call this the fifteen theorem for universal Hermitian lattices. Note t...