Weak convergence of linear forms in D[0, 1]

Convergence in probability of the linear forms Σ_k=1^∞a_nkX_k is obtained in the space D[0, 1], where (X_k) are random elements in D[0, 1] and (a_nk) is an array of real numbers. These results are obtained under varying hypotheses of boundedness conditions on the moments and conditions on the mean oscillation of the random elements (X_n) on subintervals of a partition of [0, 1]. Since the hypotheses are in general much less restrictive than tightness (or convex tightness), these results represent significant improvements over existing weak laws of large numbers and convergence results for weighted sums of random elements in D[0, 1]. Finally, comparisons to classical hypotheses for Banach space and real-valued results are included.     

Domains of attraction of nonnormal operator-stable laws

A sequence of independent, identically distributed random vectors X₁, X₂, X₃,… is said to belong to the domain of attraction of a random vector Y is there exist linear operators A_n and constant vectors b_n such that A_n(X₁,…, X_n)+b_n converges in distribution to Y. We present a simple, necessary, and sufficient condition for the existence of such A_n, B_n in the case where Y has no normal component.     

Evaluation of membrane fouling models based on bench-scale experiments: A comparison between constant flowrate blocking laws and artificial neural network (ANNs) model

Both the blocking laws and ANNs models have been used frequently in membrane filtration process in recent years. However, specially applying these models in constant flowrate condition with synthetic surface water is rare case. Further, there has no literature been found to compare the performance of purely mathematic and mechanical model (blocking laws) with the so-called “black-box” ANNs model. In this study, the performance of both blocking laws and ANNs model is evaluated. The dominant fouling mechanisms in specific experimental period also is identified based on the individual model fitting performance.     

A comparison of numerical methods applied to non-linear adsorption columns

Eight numerical schemes (first-order upstream finite difference, MacCormack, explicit Taylor–Galerkin, random choice, flux-corrected transport, ENO, TVD, and Euler–Lagrange methods) are compared on the basis of their computational efficiency for one-dimensional non-linear convection–diffusion problems. For the ideal chromatographic equation for which an exact solution exists, errors plotted against computational times show that the best methods are the random choice, Euler–Lagrange and flux-corrected MacCormack methods. Even when significant diffusion is added to the model, steep gradients are possible because of non-linearities. In such an instance, the random choice and flux-corrected transport methods give the best performance. One can now tackle more complicated problems and refer to this comparative study in order to choose an adequate numerical method which will provide sufficiently accurate results at a reasonable cost.     

On the design of an upwind scheme for compressible flow on general triangulations

We discuss the design features and mathematical background of an explicit upwind finite-volume method to simulate non-stationary flow of a compressible, inviscid fluid. One of the design goals was the rigorous mathematical justification of each ingredient of the method. The method itself contains elements from finite-difference methods as well as finite-element methods and is formulated in a finite volume framework. The use of well-known algorithmic ingredients in a new framework results in a robust time-accurate scheme. To be able to easily handle complex geometries as well as adaption algorithms a tringale-based formulation was chosen. Numerical tests for two-dimensional flow are presented.     

Asymptotically degenerate maximum eigenvalues of the eight-vertex model transfer matrix and interfacial tension

We obtain complex nonlinear integral equations for the two asymptotically degenerate maximum eigenvalues of the transfer matrix of the eight-vertex model. These are exact for a lattice of a finite numberN of columns. Solving the equations recursively gives an expansion of the eigenvalues aboutN = . Thus we can obtain the interfacial tension of the model, as well as rederiving our previous result for the free energy.     

Empirical distribution functions and functions of order statistics for mixing random variables

Some fundamental properties of the empirical distribution functions are derived in the case of mixing random variables. These properties are then utilized to study asymptotic normality and strong laws of large numbers for functions of order statistics.     

A three-component Camassa-Holm system with cubic nonlinearity and peakons

In this paper, we propose a three-component Camassa-Holm (3CH) system with cubic nonlinearity and peaked solitons (peakons). The 3CH model is proven to be integrable in the sense of Lax pair, Hamiltonian structure, and conservation laws. We show that this system admits peakons and multi-peakon solutions. Additionally, reductions of the 3CH system are investigated so that a new integrable perturbed CH equation with cubic nonlinearity is generated to possess peakon solutions.     

On the Higher Poisson Structures of the Camassa–Holm Hierarchy

We find a generating series for the higher Poisson structures of the nonlocal Camassa–Holm hierarchy, following the method used by Enriques, Orlov, and third author for the KdV case.      

Exact solutions using symmetry methods and conservation laws for the viscous flow through expanding–contracting channels

We present a range of definitions and methods dealing with the reduction of partial differential equations on the basis of the underlying symmetry structure, conservation laws and a combination of these. The method is used to reduce a complex system to an easy-to-handle second-order ordinary differential equation system independent of restrictions on any physical parameters. In particular, we construct exact solutions of a system modelling viscous flow between slowly expanding and contracting walls.