On the discretization of backward doubly stochastic differential equations

In this paper, we are dealing with the approximation of the process (Y,Z) solution to the backward doubly stochastic differential equation with the forward process X . After proving the L2-regularity of Z, we use the Euler scheme to discretize X and the Zhang approach in order to give a discretization scheme of the process (Y,Z)

Comment on "Conjectures on exact solution of three-dimensional (3D) simple orthorhombic Ising lattices" [arXiv:0705.1045]

It is shown that a recent article by Z.-D. Zhang [arXiv:0705.1045] is in error and violates well-known theorems.Comment: LaTeX, 3 pages, no figures, submitted to Philosophical Magazine. Expanded versio

A Generalized Positive Energy Theorem for Spaces with Asymptotic SUSY Compactification

A generalized positive energy theorem for spaces with asymptotic SUSY compactification involving non-symmetric data is proved. This work is motivated by the work of Dai [D1][D2], Hertog-Horowitz-Maeda [HHM], and Zhang [Z].Comment: 13 pages, without figures; Some errors are correcte

Continuum limit, Galilean invariance, and solitons in the quantum equivalent of the noisy Burgers equation

A continuum limit of the non-Hermitian spin-1/2 chain, conjectured recently to belong to the universality class of the noisy Burgers or, equivalently, Kardar-Parisi-Zhang equation, is obtained and analyzed. The Galilean invariance of the Burgers equation is explicitly realized in the operator algebra. In the quasi-classical limit we find nonlinear soliton excitations exhibiting the $\omega\propto k^z$ dispersion relation with dynamical exponent $z=3/2$.Comment: 12 pages, latex, no figure

Phenomenology of ageing in the Kardar-Parisi-Zhang equation

We study ageing during surface growth processes described by the one-dimensional Kardar-Parisi-Zhang equation. Starting from a flat initial state, the systems undergo simple ageing in both correlators and linear responses and its dynamical scaling is characterised by the ageing exponents a=-1/3, b=-2/3, lambda_C=lambda_R=1 and z=3/2. The form of the autoresponse scaling function is well described by the recently constructed logarithmic extension of local scale-invariance.Comment: Latex2e, 5 pages, with 4 figures, final for

Explicit eigenvalues of certain scaled trigonometric matrices

In a very recent paper "\emph{On eigenvalues and equivalent transformation of trigonometric matrices}" (D. Zhang, Z. Lin, and Y. Liu, LAA 436, 71--78 (2012)), the authors motivated and discussed a trigonometric matrix that arises in the design of finite impulse response (FIR) digital filters. The eigenvalues of this matrix shed light on the FIR filter design, so obtaining them in closed form was investigated. Zhang \emph{et al.}\ proved that their matrix had rank-4 and they conjectured closed form expressions for its eigenvalues, leaving a rigorous proof as an open problem. This paper studies trigonometric matrices significantly more general than theirs, deduces their rank, and derives closed-forms for their eigenvalues. As a corollary, it yields a short proof of the conjectures in the aforementioned paper.Comment: 7 pages; fixed Lemma 2, tightened inequalitie