We obtain an explicit Iwasawa decomposition of the symplectic matrices, complex or real, in terms of the Cholesky factorization for positive definite n×n matrices. We also provide a MATLAB program to compute the decomposition. 1. Iwasawa decomposition of the symplectic groups Let G be the real (noncompact) symplectic group [3, p.129] (the notation there is Spn), [4, p.265] G := Spn(R) = {g ∈ SL...

Trong bM n~y, chang toi trlnh b~y mot cOng tr~nh v~ ffch lu~ cdc ta.p h.op tit v.tmg, bao gbm s.~ thu h'Oi cdc thi nguyen tit die'n v~ s.tr s~n sinh cdc t.~p h.op tit v.trng m6i. Phttong phdp thu h'Oi s{t du.ng mot bo. djch theo ca phdp phi nggt cdnh vh cdc k~ thu.at s~a ddi van bhn, da chang tO d~ ddng, linh ho.at vh hiCu quh. Trdn ca s(r phan rich cdc lo.ai hinh sdn sinh cdc t.~p h.op tit v.~...

We find that the fundamental quadratic form of classical string propagation in 2 + 1 dimensional constant curvature spacetimes solves the Sinh-Gordon equation, the Cosh-Gordon equation or the Liouville equation. We show that in both de Sitter and anti de Sitter spacetimes (as well as in the 2 + 1 black hole anti de Sitter spacetime), all three equations must be included to cover the generic str...

We present results for ground state structures of small SinH (2≤n ≤10) clusters using the Car-Parrinello molecular dynamics. In particular, we focus on how the addition of a hydrogen atom affects the ground state geometry, total energy and the first excited electronic level gap of an Sin cluster. We discuss the nature of bonding of hydrogen in these clusters. We find that hydrogen bonds with tw...

Duality relations for the 2D nonhomogeneous Ising model on the finite square lattice wrapped on the torus are obtained. The partition function of the model on the dual lattice with arbitrary combinations of the periodical and antiperiodical boundary conditions along the cycles of the torus is expressed through some specific combination of the partition functions of the model on the original lat...

The ‘tanh-coth expansion method’ for finding solitary travelling-wave solutions to nonlinear evolution equations has been used extensively in the literature. It is a natural extension to the basic tanh-function expansion method which was developed in the 1990s. It usually delivers three types of solution, namely a tanh-function expansion, a coth-function expansion, and a tanh-coth expansion. It...