The main purpose of this article is to establish an effective version of the Grunwald-Wang theorem,which asserts that given a family of local characters χvof K *vof exponent m, where v ∈ S for a finite set S of prim...The main purpose of this article is to establish an effective version of the Grunwald-Wang theorem,which asserts that given a family of local characters χvof K *vof exponent m, where v ∈ S for a finite set S of primes of K, there exists a global character χ of the idele class group CK of exponent m(unless some special case occurs, when it is 2m) whose local component at v is χv. The effectiveness problem for this theorem is to bound the norm N(χ) of the conductor of χ in terms of K, m, S and N(χv)(v ∈ S). The Kummer case(when K contains μm) is easy since it is almost an application of the Chinese remainder theorem. In this paper, we solve this problem completely in general case, and give three versions of bound, one is with GRH, and the other two are unconditional bounds. These effective results have some interesting applications in concrete situations. To give a simple example, if we fix p and l, one gets a good least upper bound for N such that p is not an l-th power mod N. One also gets the least upper bound for N such that lr| φ(N) and p is not an l-th power mod N.Some part of this article is adopted(with some revision) from the unpublished thesis by Wang(2001).展开更多
In this paper, two numerical methods are proposed for solving distributed-order fractional Bagley-Torvik equation.This equation is used in modeling the motion of a rigid plate immersed in a Newtonian fluid with respec...In this paper, two numerical methods are proposed for solving distributed-order fractional Bagley-Torvik equation.This equation is used in modeling the motion of a rigid plate immersed in a Newtonian fluid with respect to the nonnegative density function. Using the composite Boole's rule the distributedorder Bagley-Torvik equation is approximated by a multi-term time-fractional equation, which is then solved by the GrunwaldLetnikov method(GLM) and the fractional differential transform method(FDTM). Finally, we compared our results with the exact results of some cases and show the excellent agreement between the approximate result and the exact solution.展开更多
We obtain an upper bound for the average error of the quasi-Grünwald interpo- lation based on the zeros of Chebyshev polynomial of the second kind in the Wiener space.
This paper gives the weighted Lp convergence rate estimations of the Grunwald interpolatory polynomials based on the zeros of Chebyshev polynomials of the first kind, and proves that the order of the estimations is op...This paper gives the weighted Lp convergence rate estimations of the Grunwald interpolatory polynomials based on the zeros of Chebyshev polynomials of the first kind, and proves that the order of the estimations is optimal for p≥1.展开更多
基金supported by National Basic Research Program of China(973 Program)(Grant No.2013CB834202)National Natural Science Foundation of China(Grant No.11321101)the One Hundred Talent’s Program from Chinese Academy of Sciences
文摘The main purpose of this article is to establish an effective version of the Grunwald-Wang theorem,which asserts that given a family of local characters χvof K *vof exponent m, where v ∈ S for a finite set S of primes of K, there exists a global character χ of the idele class group CK of exponent m(unless some special case occurs, when it is 2m) whose local component at v is χv. The effectiveness problem for this theorem is to bound the norm N(χ) of the conductor of χ in terms of K, m, S and N(χv)(v ∈ S). The Kummer case(when K contains μm) is easy since it is almost an application of the Chinese remainder theorem. In this paper, we solve this problem completely in general case, and give three versions of bound, one is with GRH, and the other two are unconditional bounds. These effective results have some interesting applications in concrete situations. To give a simple example, if we fix p and l, one gets a good least upper bound for N such that p is not an l-th power mod N. One also gets the least upper bound for N such that lr| φ(N) and p is not an l-th power mod N.Some part of this article is adopted(with some revision) from the unpublished thesis by Wang(2001).
文摘In this paper, two numerical methods are proposed for solving distributed-order fractional Bagley-Torvik equation.This equation is used in modeling the motion of a rigid plate immersed in a Newtonian fluid with respect to the nonnegative density function. Using the composite Boole's rule the distributedorder Bagley-Torvik equation is approximated by a multi-term time-fractional equation, which is then solved by the GrunwaldLetnikov method(GLM) and the fractional differential transform method(FDTM). Finally, we compared our results with the exact results of some cases and show the excellent agreement between the approximate result and the exact solution.
基金Foundation item： Supported bv the National Natural Science Foundation of China（10471010）
文摘We obtain an upper bound for the average error of the quasi-Grünwald interpo- lation based on the zeros of Chebyshev polynomial of the second kind in the Wiener space.
文摘This paper gives the weighted Lp convergence rate estimations of the Grunwald interpolatory polynomials based on the zeros of Chebyshev polynomials of the first kind, and proves that the order of the estimations is optimal for p≥1.
