This paper presents a new and simple scheme to describe the convex hull in R^d,which only uses three kinds of the faces of the convex hull,i.e.,the d-1-faces,d-2-faces and 0-faces.Thus,we develop an efficient new algo...This paper presents a new and simple scheme to describe the convex hull in R^d,which only uses three kinds of the faces of the convex hull,i.e.,the d-1-faces,d-2-faces and 0-faces.Thus,we develop an efficient new algorithm for constructing the convex hull of a finite set of points incrementally. This algorithm employs much less storage and time than that of the previously-existing approaches.The analysis of the running time as well as the storage for the new algorithm is also theoretically made.The algorithm is optimal in the worst case for even d.展开更多
By applying the Fourier slice theorem, Sθ（λ） =∫^∞-∞Pθ（t）e^-iλt=F（λcosθ,λsinθ）,where Pθ（t） is a projection of f（x,p）=^∞∫∫-∞F（u,v）e^i（uz＋up） dudv along lines of constant, to the Wigner ...By applying the Fourier slice theorem, Sθ（λ） =∫^∞-∞Pθ（t）e^-iλt=F（λcosθ,λsinθ）,where Pθ（t） is a projection of f（x,p）=^∞∫∫-∞F（u,v）e^i（uz＋up） dudv along lines of constant, to the Wigner operator we are naturally led to a projection operator （pure state）, which results in a new complete representation. The Weyl orderimg formalism of the Wigner operator is used in the derivation.展开更多
A variant of Fermat’s last Diophantine equation is proposed by adjusting the number of terms in accord with the power of terms and a theorem describing the solubility conditions is stated. Numerically obtained primit...A variant of Fermat’s last Diophantine equation is proposed by adjusting the number of terms in accord with the power of terms and a theorem describing the solubility conditions is stated. Numerically obtained primitive solutions are presented for several cases with number of terms equal to or greater than powers. Further, geometric representations of solutions for the second and third power equations are devised by recasting the general equation in a form with rational solutions less than unity. Finally, it is suggested to consider negative and complex integers in seeking solutions to Diophantine forms in general.展开更多
基金This work is supported by the National Natural Science Foundation of China.
文摘This paper presents a new and simple scheme to describe the convex hull in R^d,which only uses three kinds of the faces of the convex hull,i.e.,the d-1-faces,d-2-faces and 0-faces.Thus,we develop an efficient new algorithm for constructing the convex hull of a finite set of points incrementally. This algorithm employs much less storage and time than that of the previously-existing approaches.The analysis of the running time as well as the storage for the new algorithm is also theoretically made.The algorithm is optimal in the worst case for even d.
基金Supported by National Natural Science Foundation of China under Grant No.10874174
文摘By applying the Fourier slice theorem, Sθ（λ） =∫^∞-∞Pθ（t）e^-iλt=F（λcosθ,λsinθ）,where Pθ（t） is a projection of f（x,p）=^∞∫∫-∞F（u,v）e^i（uz＋up） dudv along lines of constant, to the Wigner operator we are naturally led to a projection operator （pure state）, which results in a new complete representation. The Weyl orderimg formalism of the Wigner operator is used in the derivation.
文摘A variant of Fermat’s last Diophantine equation is proposed by adjusting the number of terms in accord with the power of terms and a theorem describing the solubility conditions is stated. Numerically obtained primitive solutions are presented for several cases with number of terms equal to or greater than powers. Further, geometric representations of solutions for the second and third power equations are devised by recasting the general equation in a form with rational solutions less than unity. Finally, it is suggested to consider negative and complex integers in seeking solutions to Diophantine forms in general.
