In this paper, we give the definition of the height of a valuation and the definition of the big field ? p,G , where p is a prime and G ? ? is an additive subgroup containing 1. We conclude that ? p,G is a field and ?...In this paper, we give the definition of the height of a valuation and the definition of the big field ? p,G , where p is a prime and G ? ? is an additive subgroup containing 1. We conclude that ? p,G is a field and ? p,G is algebraically closed. Based on this the author obtains the complete classification of valuations on arithmetic surfaces. Furthermore, for any m ? p,G n ∈ ?, let V m,n be an ∝-vector space of dimension n - m + 1, whose coordinates are indexed from m to n. We generalize the definition of ? p,G , where p is a prime and G ? V m,n is an additive subgroup containing 1. We also conclude that ? p,G is a field if m ? 0 ? n.展开更多
We prove some transcendence results for the sums of some multivariate serms of the form ∑j1,j2,...,jm=0 ^∞Cj1j2...jm（r1^j1r2^j2...rm^jm） for n = 1, 2, where Cj1j2...jm are some rational functions of j1 ＋ j2 ＋ ....We prove some transcendence results for the sums of some multivariate serms of the form ∑j1,j2,...,jm=0 ^∞Cj1j2...jm（r1^j1r2^j2...rm^jm） for n = 1, 2, where Cj1j2...jm are some rational functions of j1 ＋ j2 ＋ ... ＋ jm.展开更多
Let ω1,..., ωs be a set of real transcendental numbers satisfying a certain Diophantine inequality. The upper bound for the discrepancy of the Kronecker sequence （{nω1},..., {nωs}）（1 ≤ n ≤ N） is given. In pa...Let ω1,..., ωs be a set of real transcendental numbers satisfying a certain Diophantine inequality. The upper bound for the discrepancy of the Kronecker sequence （{nω1},..., {nωs}）（1 ≤ n ≤ N） is given. In particular, some low-discrepancy sequences are constructed.展开更多
Γ(x):=integral fromn=0 to ∞(e^(-t)t^(x-1)dt),x>0为gamma函数。设f(x):=logΓ(x)+logΓ(1-x),x∈Q(0,12]。证明如果存在有理数y0∈Q(0,12],使得f(y0)=logΓ(y0)+logΓ(1-y0)∈Q,则集合{eαπ|α∈珚Q}中恰好有一个代数数,即e-f(...Γ(x):=integral fromn=0 to ∞(e^(-t)t^(x-1)dt),x>0为gamma函数。设f(x):=logΓ(x)+logΓ(1-x),x∈Q(0,12]。证明如果存在有理数y0∈Q(0,12],使得f(y0)=logΓ(y0)+logΓ(1-y0)∈Q,则集合{eαπ|α∈珚Q}中恰好有一个代数数,即e-f(y0)π,且e-f(y0)π=sinπy0。展开更多
The aim of this paper is to investigate the size properties of a planar set whose distance set has some prescribed arithmetic combinatorics. Such research is motivated by the conjecture that the disk has no more than ...The aim of this paper is to investigate the size properties of a planar set whose distance set has some prescribed arithmetic combinatorics. Such research is motivated by the conjecture that the disk has no more than 3 orthogonal exponentials. By proving a shifted version of ErdSs-Solymosi＇s theorem on the distance sets, we give some grounds on the conjecture. The results obtained here extend the corresponding results of Iosevich and Jaming in a simple manner.展开更多
文摘In this paper, we give the definition of the height of a valuation and the definition of the big field ? p,G , where p is a prime and G ? ? is an additive subgroup containing 1. We conclude that ? p,G is a field and ? p,G is algebraically closed. Based on this the author obtains the complete classification of valuations on arithmetic surfaces. Furthermore, for any m ? p,G n ∈ ?, let V m,n be an ∝-vector space of dimension n - m + 1, whose coordinates are indexed from m to n. We generalize the definition of ? p,G , where p is a prime and G ? V m,n is an additive subgroup containing 1. We also conclude that ? p,G is a field if m ? 0 ? n.
基金Acknowledgements The first author＇s research was supported by the Natural Science Foundation Project of Chongqing （Grant No. cstc2012jjA00007） and the second author＇s research was supported by NSERC of Canada.
文摘We prove some transcendence results for the sums of some multivariate serms of the form ∑j1,j2,...,jm=0 ^∞Cj1j2...jm（r1^j1r2^j2...rm^jm） for n = 1, 2, where Cj1j2...jm are some rational functions of j1 ＋ j2 ＋ ... ＋ jm.
基金Project supported by National Natural Science Foundation of China(No.10571180)
文摘Let ω1,..., ωs be a set of real transcendental numbers satisfying a certain Diophantine inequality. The upper bound for the discrepancy of the Kronecker sequence （{nω1},..., {nωs}）（1 ≤ n ≤ N） is given. In particular, some low-discrepancy sequences are constructed.
文摘Γ(x):=integral fromn=0 to ∞(e^(-t)t^(x-1)dt),x>0为gamma函数。设f(x):=logΓ(x)+logΓ(1-x),x∈Q(0,12]。证明如果存在有理数y0∈Q(0,12],使得f(y0)=logΓ(y0)+logΓ(1-y0)∈Q,则集合{eαπ|α∈珚Q}中恰好有一个代数数,即e-f(y0)π,且e-f(y0)π=sinπy0。
基金Supported by Key Project of Ministry of Education of China （Grant No. 108117） and National Natural Science Foundation of China （Grant No. 10871123）
文摘The aim of this paper is to investigate the size properties of a planar set whose distance set has some prescribed arithmetic combinatorics. Such research is motivated by the conjecture that the disk has no more than 3 orthogonal exponentials. By proving a shifted version of ErdSs-Solymosi＇s theorem on the distance sets, we give some grounds on the conjecture. The results obtained here extend the corresponding results of Iosevich and Jaming in a simple manner.
