The basic idea is to set a continuous solution of the region with a finite number of discrete points instead of a grid consisting of, these discrete points called grid nodes the solution of a continuous area on a given function of continuous variables used in the definition of discrete grid variable function approximation the original equation and boundary conditions in the micro-business operators to approximate differential, integral with the points and to approximate, so the original differential equations and boundary conditions are replaced by algebraic equations near Side, that is, finite difference equations , solve this equation group can get the original problem in discrete points on the approximate solution.

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The calculated results of the discrete model accord with the experimental data well. Compared with a simple continual model, the discrete model is true of the characteristic of the flow in manifold systems, and it also shows the availability of the way to use BP neural networks to measure the key coefficient of the discrete model.

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By means of the difference discrete variational principle, discrete Legendre transformation and discrete version of the E-L cohomology, for a kind of regular meshes we''ve found there is a kind of symplectic/multisymplectic structures with corresponding symplectic/multisymplectic finite element method/mixed finite element method schemes.

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Combinatorial computational geometry, which deals with collections of discrete objects or defined in discrete terms: points, lines, polygons, polytopes, etc., and algorithms of discrete/combinatorial character are used

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In this thesis, we firstly introduce several class of discrete-time neural network models and the research progress of the neural networks. By Schauder fixed-point principle we prove the existence of an equilibrium (i.e. a fixed point) of a discrete-time neural network with generalized input-output function and by using the converse theorem of Lyapunov function we study the uniformly asymptotical stability of equilibrium in this discrete-time neural network with variable weight and give some sufficient conditions that guarantee the stability of it.

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Proper combinations of number of outcomes, discrete step size, and scaling of the random variable, the discrete cumulate percentage distribution and discrete percentage density will converge to the continuous cumulative probability distribution and probability density function respectively.

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The combinatorial and delimitative algorithm can be used to solve the problem. A software called "Power Discrete" to implement this algorithm is developed by using Microsoft Visual6.0. It is a useful tool for linear programming problem with discrete variables and for discrete optimum design of structures, and in addition, the software also includes other algorithms such as genetic algorithm.

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The underlying uniformity on a discrete metric space is the discrete uniformity, and the underlying topology on a discrete uniform space is the discrete topology.

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Thirdly, on the base of the existent error estimate of elliptical Galerkin projection operator, obtain semi-discrete and complete-discrete error estimates betwent EFG solution and exact solution of parabolic partial differential equation. Semi-discrete error estimate show that the rank of r is accordance with the approximation rank of subspace S_h. Complete-discrete error estimates show that it not only have relation with radius r of domain of influence, but also have relation with the step length of time variable and the way of discrete.Lastly, give typical examples and draw up procedures of MATLAB.

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Using the mathematical tool"DifferenceGeometry",it is possible to define the discrete geometric metrics such as discretefirst fundamental form,discrete second fundamental form,discrete unit mainnormal vector,discrete Gauss curvature,discrete mean curvature and discreteLaplacian.

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