Given a positive integer n and the residue class ring Z_(n)=Z/nZ,we set Z_(n)^(x)to be the group of units in Z_(n),i.e.,Z_(n)^(x)={x∈Z_(n):ged(x,n)=1}.Let N_(m)(n)be the number of solutions of x_(1)^(4)+…+x_(m)^(4)...Given a positive integer n and the residue class ring Z_(n)=Z/nZ,we set Z_(n)^(x)to be the group of units in Z_(n),i.e.,Z_(n)^(x)={x∈Z_(n):ged(x,n)=1}.Let N_(m)(n)be the number of solutions of x_(1)^(4)+…+x_(m)^(4)≡0(mod n)with x_(1),…,x_(m)∈Z_(n)^(x).In this note,we determine an explicit expression of N_(m)(n).This extends the results of Sun and Yang in 2014.展开更多
The stress rate integral equations of elastoplasticity are deduced based on Ref. [1] by consistent methods. The point at which the stresses and/or displacements are calculated can be in the body or on the boundary, an...The stress rate integral equations of elastoplasticity are deduced based on Ref. [1] by consistent methods. The point at which the stresses and/or displacements are calculated can be in the body or on the boundary, and in the plastic region or elastic one. The existence of the principal value integral in the plastic region is demonstrated strictly, and the theoretical basis is presented for the paticular solution method by unit initial stress fields. In the present method, programming is easy and general, and the numerical results are excellent.展开更多
基金Supported by the Natural Science Foundation of Henan Province(232300420123)the National Natural Science Foundation of China(12026224)。
文摘Given a positive integer n and the residue class ring Z_(n)=Z/nZ,we set Z_(n)^(x)to be the group of units in Z_(n),i.e.,Z_(n)^(x)={x∈Z_(n):ged(x,n)=1}.Let N_(m)(n)be the number of solutions of x_(1)^(4)+…+x_(m)^(4)≡0(mod n)with x_(1),…,x_(m)∈Z_(n)^(x).In this note,we determine an explicit expression of N_(m)(n).This extends the results of Sun and Yang in 2014.
基金The project supported by the National Natural Science Foundation of China
文摘The stress rate integral equations of elastoplasticity are deduced based on Ref. [1] by consistent methods. The point at which the stresses and/or displacements are calculated can be in the body or on the boundary, and in the plastic region or elastic one. The existence of the principal value integral in the plastic region is demonstrated strictly, and the theoretical basis is presented for the paticular solution method by unit initial stress fields. In the present method, programming is easy and general, and the numerical results are excellent.
