This paper is concerned with the Diophantine properties of the sequence {ξθn}, where 1 ≤ξ 〈 θ and θ is a rational or an algebraic integer. We establish a combinatorial proposition which can be used to study suc...This paper is concerned with the Diophantine properties of the sequence {ξθn}, where 1 ≤ξ 〈 θ and θ is a rational or an algebraic integer. We establish a combinatorial proposition which can be used to study such two cases in the same manner. It is shown that the decay rate of the Fourier transforms of self-similar measures μλ with λ = θ-1 as the uniform contractive ratio is logarithmic. This generalizes some results of Kershner and Bufetov-Solomyak, who consider the case of Bernoulli convolutions. As an application, we prove that μλ ahaost every x is normal to any base b ≥ 2, which implies that there exist infinitely many absolute normal numbers on the corresponding self-similar set. This can be seen as a complementary result of the well-known Cassels-Schmidt theorem.展开更多
A fundamental problem in four dimensional differential topology is to find a surface with minimal genus which represents a given homology class. This problem was considered by many people for closed 4 manifolds. In th...A fundamental problem in four dimensional differential topology is to find a surface with minimal genus which represents a given homology class. This problem was considered by many people for closed 4 manifolds. In this paper,we consider this problem for four manifold with boundary.展开更多
It is difficult to trace the range of available divinatory numbers from the statistics of digital hexagrams. Now, however, we have definite proofs about the numerical ranges of Shifa （a new unearthed divination metho...It is difficult to trace the range of available divinatory numbers from the statistics of digital hexagrams. Now, however, we have definite proofs about the numerical ranges of Shifa （a new unearthed divination method） and about the Dayan divination method （the orthodox divination method of Zhouyi）. As Shifa is closely related to Guicang （Reverting to the Hidden, Yi of Shang dynasty）, SEVEN could be the key divinatory number in analyzing the numerical range of Guicang. Therefore, relationships among number groups of different divination methods could be distinct. The annotation ＂It is divined by the fixity of SEVEN and EIGHT of Yi in Xia and Shang dynasties＂ implies that Jia Gongyan （a famous confucian of the Tang dynasty） had misused the Dayan divination method. It could be certified by the odd-even analysis of Guicang of Qin Bamboo Slip Manuscripts. This study also reveals that the divinatory numbers of unearthed dice correspond to Shifa rather than the original report.展开更多
文摘This paper is concerned with the Diophantine properties of the sequence {ξθn}, where 1 ≤ξ 〈 θ and θ is a rational or an algebraic integer. We establish a combinatorial proposition which can be used to study such two cases in the same manner. It is shown that the decay rate of the Fourier transforms of self-similar measures μλ with λ = θ-1 as the uniform contractive ratio is logarithmic. This generalizes some results of Kershner and Bufetov-Solomyak, who consider the case of Bernoulli convolutions. As an application, we prove that μλ ahaost every x is normal to any base b ≥ 2, which implies that there exist infinitely many absolute normal numbers on the corresponding self-similar set. This can be seen as a complementary result of the well-known Cassels-Schmidt theorem.
文摘A fundamental problem in four dimensional differential topology is to find a surface with minimal genus which represents a given homology class. This problem was considered by many people for closed 4 manifolds. In this paper,we consider this problem for four manifold with boundary.
文摘It is difficult to trace the range of available divinatory numbers from the statistics of digital hexagrams. Now, however, we have definite proofs about the numerical ranges of Shifa （a new unearthed divination method） and about the Dayan divination method （the orthodox divination method of Zhouyi）. As Shifa is closely related to Guicang （Reverting to the Hidden, Yi of Shang dynasty）, SEVEN could be the key divinatory number in analyzing the numerical range of Guicang. Therefore, relationships among number groups of different divination methods could be distinct. The annotation ＂It is divined by the fixity of SEVEN and EIGHT of Yi in Xia and Shang dynasties＂ implies that Jia Gongyan （a famous confucian of the Tang dynasty） had misused the Dayan divination method. It could be certified by the odd-even analysis of Guicang of Qin Bamboo Slip Manuscripts. This study also reveals that the divinatory numbers of unearthed dice correspond to Shifa rather than the original report.
