Fundamental Locally Solid Riesz Spaces

In this paper we focus ourselves on the positive cone of the locally solid Riesz spaces to characterize the fundamentality. From one example the article indicates that the fundamentality of the locally solid Riesz space is independent from the Lebesgue property.

REMARKS ON THE REPRESENTATION THEOREM FOR RIESZ SPACES POSSESSING WEAK UNITS

The author obtains an algebraic version of the main result from his previous paper 'A characterization of Riesz spaces which are Riesz isomorphic to C(X) for some completely regular space X', and also studies the relations among some conditions used therein.

Amarts on Riesz Spaces

The concepts of conditional expectations, martingales and stopping times were extended to the Riesz space context by Kuo, Labuschagne and Watson （Discrete time stochastic processes on Riesz spaces, Indag. Math.,15（2004）, 435-451）. Here we extend the definition of an asymptotic martingale （amart） to the Riesz spaces context, and prove that Riesz space amarts can be decomposed into the sum of a martingale and an adapted sequence convergent to zero. Consequently an amart convergence theorem is deduced.

Probabilistic Normed Riesz Spaces

In this paper, the concepts of probabilistic normed Riesz space and probabilistic Banach lattice are introduced, and their basic properties are studied. In this context, some continuity and convergence theorems are proved.

Notes on Some Convergences in Riesz Space

An equivalent description of u-uniform convergence is presented first. Then the relations among the order convergence, u-uniform convergence and norm convergence of sequences are discussed in Riesz spaces. An equivalence of the three convergences is brought forward; namely, {fn} is a u-uniform Cauchy sequence. Finally the relations among the three convergences of sequences are also extended to the relations among the convergences of nets in Riesz spaces.

Calculus of Generalized Quasi-differentiable Functions I: Some Results on the Space of Pairs of Convex-Set Collections

A Riesz space K1 whose elements are pairs of convex-set collections is presented for the study on the calculus of generalized quasi-differentiable functions. The space K1 is constructed by introducing a well-defined equivalence relation among pairs of collections of convex sets. Some important properties on the norm and operations in K1 are given.

Riesz Product Spaces and Representation Theory

Let {Ei:i∈I}be a family of Archimedean Riesz spaces.The Riesz product space is denoted by Πi∈I Ei.The main result in this paper is the following conclusion:There exists a completely regular Hausdorff space X such that Πi∈I Ei is Riesz isomorphic to C（X）if and only if for every i ∈ I there exists a completely regular Hausdorff space Xi such that Ei is Riesz isomorphic to C（Xi）.

BANDED M-MATRIX SPLITTING PRECONDITIONER FOR RIESZ SPACE FRACTIONAL REACTION-DISPERSION EQUATION

Based on the Crank-Nicolson and the weighted and shifted Grunwald operators,we present an implicit difference scheme for the Riesz space fractional reaction-dispersion equations and also analyze the stability and the convergence of this implicit difference scheme.However,after estimating the condition number of the coefficient matrix of the discretized scheme,we find that this coefficient matrix is ill-conditioned when the spatial mesh-size is sufficiently small.To overcome this deficiency,we further develop an effective banded M-matrix splitting preconditioner for the coefficient matrix.Some properties of this preconditioner together with its preconditioning effect are discussed.Finally,Numerical examples are employed to test the robustness and the effectiveness of the proposed preconditioner.

Numerical Methods for Semilinear Fractional Diffusion Equations with Time Delay

In this paper,we consider the numerical solutions of the semilinear Riesz space-fractional diffusion equations(RSFDEs)with time delay,which constitute an important class of differential equations of practical significance.We develop a novel implicit alternating direction method that can effectively and efficiently tackle the RSFDEs in both two and three dimensions.The numerical method is proved to be uniquely solvable,stable and convergent with second order accuracy in both space and time.Numerical results are presented to verify the accuracy and efficiency of the proposed numerical scheme.