1 IntroductionIn 1823 Abel made a conjecture in a particular case of Fermat's last theorem:If n>2and x,y and z are nonzero integers such thatx^n+y^n=z^n(1)then none of x,y or z can be a prime power(see P.Ribenb...1 IntroductionIn 1823 Abel made a conjecture in a particular case of Fermat's last theorem:If n>2and x,y and z are nonzero integers such thatx^n+y^n=z^n(1)then none of x,y or z can be a prime power(see P.Ribenboim[1],p.25).It is clear that we may assume,without loss of generality,that(x,y)=(x,z)=(y,z)=1and 0<x<y<z.展开更多
In this paper, the solution, more general than [1], of a weak singular integral equation integral(0)(pi)integral(-infinity)(infinity) p(s,psi)d sk(psi)d psi=F(r,theta), (r,theta)epsilon (Q) over bar=Q+partial derivati...In this paper, the solution, more general than [1], of a weak singular integral equation integral(0)(pi)integral(-infinity)(infinity) p(s,psi)d sk(psi)d psi=F(r,theta), (r,theta)epsilon (Q) over bar=Q+partial derivative Q subject to constraint p(s,psi)=0, for (s,psi)=(r,theta)is not an element of Q={r,theta)/F(r,theta)>c*} is found p=2/pi[root w g'(0)+integral(0)(w) root w-u g '(u)du] where k and F are given continuous functions; (s,psi) is a local polar coordinating with origin at M(r,theta); (r,theta) is the global polar coordinating with origin at O(0,0) F(r,theta)=c* (const.) is the boundary contour partial derivative Q of the considered range Q; g(w)=F(r,theta)/[pi k(psi(0))]; g'=dg/dw; w=N-r(2)sin(2)(theta+psi(0)); psi(0) and N are mean values. The solution shown in type (2.19) of [1] is a special case of the above solution and only suits F(r,theta)=w. The solution of a rigid cone contact with elastic half space, more simple and clear than Love's (1939), is given as an example of application.展开更多
In this work, we use the finiteness of the Mordell-weil group and the Riemann Roch spaces to give a geometric parametrization of the set of algebraic points of any given degree over the field of rational numbers Q on ...In this work, we use the finiteness of the Mordell-weil group and the Riemann Roch spaces to give a geometric parametrization of the set of algebraic points of any given degree over the field of rational numbers Q on curve C<sub>3 </sub>(11): y<sup>11</sup> = x<sup>3</sup> (x-1)<sup>3</sup>. This result is a special case of quotients of Fermat curves C<sub>r,s </sub>(p) : y<sup>p</sup> = x<sup>r</sup>(x-1)<sup>s</sup>, 1 ≤ r, s, r + s ≤ p-1 for p = 11 and r = s = 3. The results obtained extend the work of Gross and Rohrlich who determined the set of algebraic points on C<sub>1</sub>(11)(K) of degree at most 2 on Q.展开更多
文摘1 IntroductionIn 1823 Abel made a conjecture in a particular case of Fermat's last theorem:If n>2and x,y and z are nonzero integers such thatx^n+y^n=z^n(1)then none of x,y or z can be a prime power(see P.Ribenboim[1],p.25).It is clear that we may assume,without loss of generality,that(x,y)=(x,z)=(y,z)=1and 0<x<y<z.
文摘In this paper, the solution, more general than [1], of a weak singular integral equation integral(0)(pi)integral(-infinity)(infinity) p(s,psi)d sk(psi)d psi=F(r,theta), (r,theta)epsilon (Q) over bar=Q+partial derivative Q subject to constraint p(s,psi)=0, for (s,psi)=(r,theta)is not an element of Q={r,theta)/F(r,theta)>c*} is found p=2/pi[root w g'(0)+integral(0)(w) root w-u g '(u)du] where k and F are given continuous functions; (s,psi) is a local polar coordinating with origin at M(r,theta); (r,theta) is the global polar coordinating with origin at O(0,0) F(r,theta)=c* (const.) is the boundary contour partial derivative Q of the considered range Q; g(w)=F(r,theta)/[pi k(psi(0))]; g'=dg/dw; w=N-r(2)sin(2)(theta+psi(0)); psi(0) and N are mean values. The solution shown in type (2.19) of [1] is a special case of the above solution and only suits F(r,theta)=w. The solution of a rigid cone contact with elastic half space, more simple and clear than Love's (1939), is given as an example of application.
文摘In this work, we use the finiteness of the Mordell-weil group and the Riemann Roch spaces to give a geometric parametrization of the set of algebraic points of any given degree over the field of rational numbers Q on curve C<sub>3 </sub>(11): y<sup>11</sup> = x<sup>3</sup> (x-1)<sup>3</sup>. This result is a special case of quotients of Fermat curves C<sub>r,s </sub>(p) : y<sup>p</sup> = x<sup>r</sup>(x-1)<sup>s</sup>, 1 ≤ r, s, r + s ≤ p-1 for p = 11 and r = s = 3. The results obtained extend the work of Gross and Rohrlich who determined the set of algebraic points on C<sub>1</sub>(11)(K) of degree at most 2 on Q.
