Waring-Goldbach problems in short intervals have appealed to many authors and have been investigated, among which the Goldbach-Vinogradov theorem with almost equal prime variables may be the most famous one (see for example [1], [2], [3] and[4]).

A different opinion from the equal symbol of f_s≥2f_m in the Channon's sampling theorem with several practical examples is proposed, and thc fault in the proving method used in the past is pointed out.

An important inequality between standard deviation and mean deviation δ=E|X-EX| is given :σ≥δ,the equality holds if and only if X is one-point distributed or equal probabieity two-point distributed.

The inequalifies are valid if and only if all the positive eigenvalues of matrix ((m ie i,m je j)) N×N are equal, where (m ie i,m je j) denotes inner product and e i is PP i [TX-〗(i=1,2,…,N)。

Suppose that D is a simply connected region whose boundary, the complex function q(z), normal derivative of |q(z)|does not equal to zero for z ∈Γ,and |q(z)|≤1 for z ∈D where equality may hold only at z ∈Γ. Type and boundary correspondence of the homeomorphic solution of the Beltrami equation are discussed under above conditions of complex dilatation q(z).

The method divides eight attribute indicators of the spare parts into inputs and outputs to make the efficiency value of the model equal to the important degree of the spare parts directly.

The 2-distance coloring of a graph G is a proper vertex coloring such that no two vertices at distance less than or equal to 2 in G are assigned the same color.

It is well-known that the ring of invariants associated to a non-modular representation of a finite group is Cohen-Macaulay and hence has depth equal to the dimension of the representation.

The essential dimension is a numerical invariant of the group; it is often equal to the minimal number of independent parameters required to describe all algebraic objects of a certain type.

We also give conditions for the two algebras to be equal, relating equality to good filtrations and saturated subgroups.

We present two generalizations of the orthogonal basis of Malvar and Coifman-Meyer: biorthogonal and equal parity bases.

An important property of this matrix is that the maximum and minimum eigenvalues are equal (in some sense) to the upper and lower frame bounds.

In the statistical theory of superlattices in binary alloys, the dependence of the interaction energy upon atomic arrangements is taken into account by regarding the interaction energy in Bethe's theory as an average quantity depending on the degree of order and the composition of the alloy. Two simple assumptions concerning the functional relationship of the interaction energy with order and composition are made. The first is a linear function of order and composition. The second is a linear function of the...

In the statistical theory of superlattices in binary alloys, the dependence of the interaction energy upon atomic arrangements is taken into account by regarding the interaction energy in Bethe's theory as an average quantity depending on the degree of order and the composition of the alloy. Two simple assumptions concerning the functional relationship of the interaction energy with order and composition are made. The first is a linear function of order and composition. The second is a linear function of the average numbers of pairs of atoms. The result of applying these assumptions to superlattices of the type AB is that the critical temperature as a function of the composition is a maximum for equal sumber of A and B atoms only when a certain relation between the coefficients in the assumed function is satisfied. In the cass of superlattices of type AB3 the theory of Bragg and Williams is used for simplicity. It is shown that when the composition varies, the maximum of the critical temperature may occur at any desired composition by a suitables adjustment of the coefficients in the assumed functions. There is thus a hope of removing the discrepancy between theory and experiment on this line. The anomalous specific heat at the critical temperature is also calculated for different compositions. In the case of the AB type of superlattices, Bethe's formula for the energy is no longer valid, and in order to calculate the specific heat, an approximate formula for the energy is obtained by analogy with the theory of Bragg and Williams. Finally, the problem of separation into more than one phase is briefly discussed.

The thermodynamic functions of an ideal subetance represented by van der Waals equation are obtained with the help of the condition that these functions reduce to those of a perfect gas in the limiting case of vanisning prassure. The volumes of the liquid state and gas state in coexistence as determined by Maxwell's rule of equal areas are expressed in a parametrie form. The nature of the dependence of the constants a and b on the chemical composition of the gas is briefly considered.

The present paper treats the compression of a rectangular block between two parallel rough plates as a problem in the theory of plane strain for perfectly plastic-rigid materials.At first, the plastic-rigid theory of plane strain was outlined, then, the solution to the present problem is briefly surveyed. In section 4, the case that is left out in the present literature, viz. when the width-height ratio lies between 1 and 3.64 for partially rough plates is solved. In this treatment, the coefficient of friction...

The present paper treats the compression of a rectangular block between two parallel rough plates as a problem in the theory of plane strain for perfectly plastic-rigid materials.At first, the plastic-rigid theory of plane strain was outlined, then, the solution to the present problem is briefly surveyed. In section 4, the case that is left out in the present literature, viz. when the width-height ratio lies between 1 and 3.64 for partially rough plates is solved. In this treatment, the coefficient of friction ν is considered as constant along the contact surfaces. For eachμ, a critical value of the ratio w0/h is given. When w/hequal circular arcs are given in Appendix I. The computation was compared to the results obtained by R. Hill (ref. 1) using graphical construction with fairly rough meshes. The comparison shows that the graphical construction used is accurate for all practical purposes. From these expressions we obtain the analytic expression for wo/h in terms of the frictional angle connected with μ(Eq. 11).Finally, a short discussion on the graphical construction used for the case of constant μ is given in Appendix Ⅱ.