We present a method for the derivation of the generating function and computation of critical exponents for several cluster models (staircase, bar-graph, and directed columnconvex polygons, as well as partially directed self-avoiding walks), starting with nonlinear functional equations for the generating function. By linearising these equations, we first give a derivation of the generating func...

Converting complex equations into simpler, more tractable equations usually involves approximation. Approximation is usually done by identifying and removing insignii-cant terms, while retaining signiicant ones. The signiicance of a term can be determined by order of magnitude reasoning. In this paper we describe NAPIER, an implemented order of magnitude reasoning system. NAPIER deenes the orde...

A unified approach is given for the analysis of the weak error of spatially semidiscrete finite element methods for linear stochastic partial differential equations driven by additive noise. An error representation formula is found in an abstract setting based on the semigroup formulation of stochastic evolution equations. This is then applied to the stochastic heat, linearized Cahn-Hilliard, a...

The aim of this short note is twofold. First, we give a sketch of the proof of a recent result proved by the authors in the paper [7] concerning existence and uniqueness of renormalized solutions of continuity equations with unbounded damping coe cient. Second, we show how the ideas in [7] can be used to provide an alternative proof of the result in [6, 9, 12], where the usual requirement of bo...

abstract. the main contribution of the current paper is to propose a new effective numerical method for solving the first-order linear matrix differential equations. properties of the legendre basis operational matrix of integration together with a collocation method are applied to reduce the problem to a coupled linear matrix equations. afterwards, an iterative algorithm is examined for solvin...

The method of matched asymptotic expansions is used to study a contact problem for a system consisting of a large number of small punches situated along a given curve on the boundary of an elastic half-space. The cases of cylindrical punches (the linear problem) and spherical punches (the structurally non-linear contact problem) are considered. In the linear case the reduced logarithmic capacit...