The article presents the proof of the validity of the generalized Riemann hypothesis on the basis of adjustment and correction of the proof of the Riemanns hypothesis in the work?[1], obtained by a finite exponential ...The article presents the proof of the validity of the generalized Riemann hypothesis on the basis of adjustment and correction of the proof of the Riemanns hypothesis in the work?[1], obtained by a finite exponential functional series and finite exponential functional progression.展开更多
为将Lehmer同余式从模奇质数平方推广至模任意数的平方,Cai等(CAI T X, FU X D, ZHOU X. Acta Aritmetica,2002,103(3):203-214.)定义了广义欧拉函数φ e (n).最近Cai等给出了 e=3,4,6 时广义欧拉函数φ e (n)的计算公式.利用初等数论...为将Lehmer同余式从模奇质数平方推广至模任意数的平方,Cai等(CAI T X, FU X D, ZHOU X. Acta Aritmetica,2002,103(3):203-214.)定义了广义欧拉函数φ e (n).最近Cai等给出了 e=3,4,6 时广义欧拉函数φ e (n)的计算公式.利用初等数论与组合的方法和技巧,完全确定了一类广义欧拉函数的计算公式,即给出当 e 为 n 的特殊正因数时,φ e (n)的准确计算公式,从而推广Cai等的相关主要结果,并由此给出φ e (n)为偶数的一个充分必要条件.展开更多
Positive-instantaneous frequency representation for transient signals has always been a great concern due to its theoretical and practical importance,although the involved concept itself is paradoxical.The desire and ...Positive-instantaneous frequency representation for transient signals has always been a great concern due to its theoretical and practical importance,although the involved concept itself is paradoxical.The desire and practice of uniqueness of such frequency representation(decomposition)raise the related topics in approximation.During approximately the last two decades there has formulated a signal decomposition and reconstruction method rooted in harmonic and complex analysis giving rise to the desired signal representations.The method decomposes any signal into a few basic signals that possess positive instantaneous frequencies.The theory has profound relations to classical mathematics and can be generalized to signals defined in higher dimensional manifolds with vector and matrix values,and in particular,promotes kernel approximation for multi-variate functions.This article mainly serves as a survey.It also gives two important technical proofs of which one for a general convergence result(Theorem 3.4),and the other for necessity of multiple kernel(Lemma 3.7).Expositorily,for a given real-valued signal f one can associate it with a Hardy space function F whose real part coincides with f.Such function F has the form F=f+iHf,where H stands for the Hilbert transformation of the context.We develop fast converging expansions of F in orthogonal terms of the form F=∑k=1^(∞)c_(k)B_(k),where B_(k)'s are also Hardy space functions but with the additional properties B_(k)(t)=ρ_(k)(t)e^(iθ_(k)(t)),ρk≥0,θ′_(k)(t)≥0,a.e.The original real-valued function f is accordingly expanded f=∑k=1^(∞)ρ_(k)(t)cosθ_(k)(t)which,besides the properties ofρ_(k)andθ_(k)given above,also satisfies H(ρ_(k)cosθ_(k))(t)ρ_(k)(t)sinρ_(k)(t).Real-valued functions f(t)=ρ(t)cosθ(t)that satisfy the conditionρ≥0,θ′(t)≥0,H(ρcosθ)(t)=ρ(t)sinθ(t)are called mono-components.If f is a mono-component,then the phase derivativeθ′(t)is defined to be instantaneous frequency of f.The above described positive-instantaneous frequency expansion is a generalization of the Fourier series expansion.Mono-components are crucial to understand the concept instantaneous frequency.We will present several most important mono-component function classes.Decompositions of signals into mono-components are called adaptive Fourier decompositions(AFDs).Wc note that some scopes of the studies on the ID mono-components and AFDs can be extended to vector-valued or even matrix-valued signals defined on higher dimensional manifolds.We finally provide an account of related studies in pure and applied mathematics.展开更多
The main purpose of this paper is to define prime and introduce non-commutative arithmetics based on Thompson's group F.Defining primes in a non-abelian monoid is highly non-trivial,which relies on a concept calle...The main purpose of this paper is to define prime and introduce non-commutative arithmetics based on Thompson's group F.Defining primes in a non-abelian monoid is highly non-trivial,which relies on a concept called"castling".Three types of castlings are essential to grasp the arithmetics.The divisor functionτon Thompson's monoid S satisfiesτ(uv)≤τ(u)τ(v)for any u,v∈S.Then the limitτ_(0)(u)=lim_(n→∞)(τ(u~n))^(1/n)exists.The quantityC(S)=sup_(1≠u∈S)τ_(0)(u)/τ(u)describes the complexity for castlings in S.We show thatC(S)=1.Moreover,the MCbius function on S is calculated.And the Liouville functionCon S is studied.展开更多
文摘The article presents the proof of the validity of the generalized Riemann hypothesis on the basis of adjustment and correction of the proof of the Riemanns hypothesis in the work?[1], obtained by a finite exponential functional series and finite exponential functional progression.
文摘为将Lehmer同余式从模奇质数平方推广至模任意数的平方,Cai等(CAI T X, FU X D, ZHOU X. Acta Aritmetica,2002,103(3):203-214.)定义了广义欧拉函数φ e (n).最近Cai等给出了 e=3,4,6 时广义欧拉函数φ e (n)的计算公式.利用初等数论与组合的方法和技巧,完全确定了一类广义欧拉函数的计算公式,即给出当 e 为 n 的特殊正因数时,φ e (n)的准确计算公式,从而推广Cai等的相关主要结果,并由此给出φ e (n)为偶数的一个充分必要条件.
基金Macao University Multi-Year Research Grant(MYRG)MYRG2016-00053-FSTMacao Government Science and Technology Foundation FDCT 0123/2018/A3.
文摘Positive-instantaneous frequency representation for transient signals has always been a great concern due to its theoretical and practical importance,although the involved concept itself is paradoxical.The desire and practice of uniqueness of such frequency representation(decomposition)raise the related topics in approximation.During approximately the last two decades there has formulated a signal decomposition and reconstruction method rooted in harmonic and complex analysis giving rise to the desired signal representations.The method decomposes any signal into a few basic signals that possess positive instantaneous frequencies.The theory has profound relations to classical mathematics and can be generalized to signals defined in higher dimensional manifolds with vector and matrix values,and in particular,promotes kernel approximation for multi-variate functions.This article mainly serves as a survey.It also gives two important technical proofs of which one for a general convergence result(Theorem 3.4),and the other for necessity of multiple kernel(Lemma 3.7).Expositorily,for a given real-valued signal f one can associate it with a Hardy space function F whose real part coincides with f.Such function F has the form F=f+iHf,where H stands for the Hilbert transformation of the context.We develop fast converging expansions of F in orthogonal terms of the form F=∑k=1^(∞)c_(k)B_(k),where B_(k)'s are also Hardy space functions but with the additional properties B_(k)(t)=ρ_(k)(t)e^(iθ_(k)(t)),ρk≥0,θ′_(k)(t)≥0,a.e.The original real-valued function f is accordingly expanded f=∑k=1^(∞)ρ_(k)(t)cosθ_(k)(t)which,besides the properties ofρ_(k)andθ_(k)given above,also satisfies H(ρ_(k)cosθ_(k))(t)ρ_(k)(t)sinρ_(k)(t).Real-valued functions f(t)=ρ(t)cosθ(t)that satisfy the conditionρ≥0,θ′(t)≥0,H(ρcosθ)(t)=ρ(t)sinθ(t)are called mono-components.If f is a mono-component,then the phase derivativeθ′(t)is defined to be instantaneous frequency of f.The above described positive-instantaneous frequency expansion is a generalization of the Fourier series expansion.Mono-components are crucial to understand the concept instantaneous frequency.We will present several most important mono-component function classes.Decompositions of signals into mono-components are called adaptive Fourier decompositions(AFDs).Wc note that some scopes of the studies on the ID mono-components and AFDs can be extended to vector-valued or even matrix-valued signals defined on higher dimensional manifolds.We finally provide an account of related studies in pure and applied mathematics.
基金Supported by National Natural Science Foundation of China(Grant No.11701549)。
文摘The main purpose of this paper is to define prime and introduce non-commutative arithmetics based on Thompson's group F.Defining primes in a non-abelian monoid is highly non-trivial,which relies on a concept called"castling".Three types of castlings are essential to grasp the arithmetics.The divisor functionτon Thompson's monoid S satisfiesτ(uv)≤τ(u)τ(v)for any u,v∈S.Then the limitτ_(0)(u)=lim_(n→∞)(τ(u~n))^(1/n)exists.The quantityC(S)=sup_(1≠u∈S)τ_(0)(u)/τ(u)describes the complexity for castlings in S.We show thatC(S)=1.Moreover,the MCbius function on S is calculated.And the Liouville functionCon S is studied.
