In this paper,almost continuous concepts are given,and we have proved that {almost everywhere continuous functions } { almost continuous functions} {fundamental conti nuous functions } are proper inclusions.

Based on that Lebesgue measurable function is almost everywhere equal to the almost everywhere continuous function on ,an equivalent definition and several properties on Lebesgue measurable function are given,and so is the tentative plan on Lebesgue measurable function to enter the mathematics teaching of the engineering course.

The theory of Riemann type integrals is established on compact Hausdorff measure spaces. It is proved that a function is Riemann integrable if and only if it is continuous almost everywhere.

Because this method makes the lateral displacement function continuous in deformed zone, the coordination of lateral displacements between strips is ensured and the theory of strip element method is further developed.

Harvey and J. Porking′s methods and traditional methods, we define the current Cauchy principal values in this paper by using homotopy formula and integral transformations. We study the boundary value of Weil type polyhedron integrals and obtain Plemelj formulas, which are different from the methods usually in the studies of boundary value problems.

It is shown that the monotone function acting between Euclidean Spaces R n and R m is continuous almost everywhere with respect to the Lebegue measure on R n.

This paper has studied the graph of one kind of function that is continuous on (-∞,∞) and is not differentiable on (-∞,∞)by Mathematica4.0, and carried on comparative analysis with differentiable function in the local range, and found that this kind of function possesses the obvious characteristic of fractal in the local range.

We describe a class of locally convex spaces on which there exist everywhere infinitely b-differentiable real functions which are not everywhere continuous (and so are not everywhere HL-differentiable).

The method produces spinors which are everywhere continuous with continuous derivates which is particularly important for the relativistic case.

They are e.g.λn-almost everywhere continuous and therefore show satisfactorystability behaviour w.r.t.

Homomorphisms of topological measure spaces had been defined in [5] to be measure-preserving and almost everywhere continuous mappings; this induces a concept of isomorphic topological measures.

Thus constrained optimization algorithms cannot assume an everywhere continuous null space basis.

In this paper an infinitely wide plate under pure plastic bending is discussed. The distribution ot stress and the relation between the couple acting on the plate and the corresponding deformation are found under the assumption that the relation between the intensity of shearing stress and shearing strain has the exponential form.The method suggested by the author is also compared with that proposed by R. Hill under the assumption that the plate is ideally plastic. Obviously the two have markeddifference. The...

In this paper an infinitely wide plate under pure plastic bending is discussed. The distribution ot stress and the relation between the couple acting on the plate and the corresponding deformation are found under the assumption that the relation between the intensity of shearing stress and shearing strain has the exponential form.The method suggested by the author is also compared with that proposed by R. Hill under the assumption that the plate is ideally plastic. Obviously the two have markeddifference. The latter gives rise to the discontinuity of stress in the neighbourhood ofthe "neutral layer", while the former always gives a continuous variation of stress.Also, the elastic recovery of plastic strain at the removal of the load is discussed. The result is compared with B. V. Ryabinin's experiment and is found to be in close agreement.It is believed that the present problem has its application in the cold working ofmetals.

There are some common difficulties encountered in elastic-plastic impact codes such as EPIC[1][2], NONSAP[3], etc. Most of these codes use the simple linear functions usually taken from static problems to represent the displacement components. In such finite element formulation, the strain and stress components are constants in every element. In the equations of motion, the stress components in general appear in the form of their space derivatives. Thus, if we use such form functions to represent the displacement...

There are some common difficulties encountered in elastic-plastic impact codes such as EPIC[1][2], NONSAP[3], etc. Most of these codes use the simple linear functions usually taken from static problems to represent the displacement components. In such finite element formulation, the strain and stress components are constants in every element. In the equations of motion, the stress components in general appear in the form of their space derivatives. Thus, if we use such form functions to represent the displacement components, the effect of internal stresses to the equations of motion vanishes identically. The usual practice to overcome such difficulties is to establish a self-equilibrium system of internal forces acting on various nodal points by means of transforming equations of motion into varia-tional form of energy relation through the application of virtual displacement principle. The nodal acceleration is then calculated from the total forces acting on this node from all the neighbouring elements. The transformation of virtual displacement principle into the varia-tional energy form is performed on the bases of continuity conditions of stress and displacement throughout the integrated space. That is to say, on the interface boundary of finite element, the assumed displacement and stress functions should be conformed. However, it is easily seen that, for linear form function of finite element calculation, the displacement continues everywhere, but not the stress components. Thus, the convergence of such kind of finite element computation is open to question. This kind of treatment has never been justified even in approximation sense. Furthermore, the calculation of acceleration of nodal points needs a rule to calculate the mass matrix. There are two ways to establish mass matrix, namely, lumped mass method and consistent mass method.141 The consistent mass matrix can be obtained naturally through finite element formulation, which is consistent to the assumed form functions. However, the resulting consistent mass matrix is not in diago-nalized form, which is inconvenient for numerical computation. For most codes, the lumped mass matrix is used, and in this case, the element mass is distributed in certain assumed proportions to all the nodal points of this element. The lumped mass matrix is diagonalized with the diagonal terms composed of the nodal masses. However, the lumped mass assumption has never been justified. All these difficulties are originated from the simple linear form functions usually used in static problems.

In this paper, we present an exact solution for nonlinear shallow water on a rotating planet. It is a kind of solitary waves with always negative wave height and a celerity smaller than linear shallow water propagation speed . In fact, it propagates with a speed equal to (1+ a/h) where a is the negative wave height. The lowest point of the water surface is a singular point where the first order derivative has a, discontinuity of the first kind. The horizontal scale of the wave has actually no connection with...

In this paper, we present an exact solution for nonlinear shallow water on a rotating planet. It is a kind of solitary waves with always negative wave height and a celerity smaller than linear shallow water propagation speed . In fact, it propagates with a speed equal to (1+ a/h) where a is the negative wave height. The lowest point of the water surface is a singular point where the first order derivative has a, discontinuity of the first kind. The horizontal scale of the wave has actually no connection with the water depth.