As is well known,the definitions of fractional sum and fractional difference of f(z)on non-uniform lattices x(z)=c1z^(2)+c2z+c3 or x(z)=c1q^(z)+c2q^(-z)+c3 are more difficult and complicated.In this article,for the fi...As is well known,the definitions of fractional sum and fractional difference of f(z)on non-uniform lattices x(z)=c1z^(2)+c2z+c3 or x(z)=c1q^(z)+c2q^(-z)+c3 are more difficult and complicated.In this article,for the first time we propose the definitions of the fractional sum and fractional difference on non-uniform lattices by two different ways.The analogue of Euler’s Beta formula,Cauchy’Beta formula on non-uniform lattices are established,and some fundamental theorems of fractional calculas,the solution of the generalized Abel equation on non-uniform lattices are obtained etc.展开更多
If Goldbach’s conjecture is true, then for each prime number p there is at least one pair of primes symmetric with respect to p and whose sum is 2p. In the multiplicative number theory, covering the positive integers...If Goldbach’s conjecture is true, then for each prime number p there is at least one pair of primes symmetric with respect to p and whose sum is 2p. In the multiplicative number theory, covering the positive integers with primes, during the prime factorization, may be viewed as being the outcome of a parallel system which functions properly if and only if Euler’s formula of the product of the reciprocals of the primes is true. An exact formula for the number of primes less than or equal to an arbitrary bound is given. This formula may be implemented using Wolfram’s computer package Mathematica.展开更多
In this paper, a two-dimensional nanometer scale tip-plate discharge model has been employed to study nanoscale electrical discharge in atmospheric conditions. The field strength dis- tributions in a nanometer scale t...In this paper, a two-dimensional nanometer scale tip-plate discharge model has been employed to study nanoscale electrical discharge in atmospheric conditions. The field strength dis- tributions in a nanometer scale tip-to-plate electrode arrangement were calculated using the finite element analysis （FEA） method, and the influences of applied voltage amplitude and frequency as well as gas gap distance on the variation of effective discharge range （EDR） on the plate were also investigated and discussed. The simulation results show that the probe with a wide tip will cause a larger effective discharge range on the plate; the field strength in the gap is notably higher than that induced by the sharp tip probe; the effective discharge range will increase linearly with the rise of excitation voltage, and decrease nonlinearly with the rise of gap length. In addition, probe dimension, especially the width/height ratio, affects the effective discharge range in different manners. With the width/height ratio rising from 1 ： 1 to 1 ： 10, the effective discharge range will maintain stable when the excitation voltage is around 50 V. This will increase when the excitation voltage gets higher and decrease as the excitation voltage gets lower. Fhrthermore, when the gap length is 5 nm and the excitation voltage is below 20 V, the diameter of EDR in our simulation is about 150 nm, which is consistent with the experiment results reported by other research groups. Our work provides a preliminary understanding of nanometer scale discharges and establishes a predictive structure-behavior relationship.展开更多
The purpose of this paper is to establish, paralleling a well-known result for definite integrals, the conditional convergence of a family of trigonometric sine series. The fundamental idea is to group appropriately t...The purpose of this paper is to establish, paralleling a well-known result for definite integrals, the conditional convergence of a family of trigonometric sine series. The fundamental idea is to group appropriately the terms of the series in order to show absolute divergence of the series, given the well-established result that the series as it stands is convergent.展开更多
It is a fact that imaginary numbers do not have practical significance. But the role of imaginary numbers is very broad and enormous, due to the existence of Euler’s formula. Due to Euler’s formula, imaginary number...It is a fact that imaginary numbers do not have practical significance. But the role of imaginary numbers is very broad and enormous, due to the existence of Euler’s formula. Due to Euler’s formula, imaginary numbers have been applied in many theoretical theories. One of the biggest functions of imaginary numbers is to represent changes in phase, which is indispensable in signal analysis theory. The imaginary numbers in quantum mechanics pose a greater mystery: do the imaginary numbers really exist? This question still needs further scientific development to be answered.展开更多
We modified Mulikens overlap population n(A, B)=2∑Bλ∑Aμ∑in_ic~*_A,μic_B,λis_Aμ,Bλ and obtained an experiential formula N(A,B)=N_A(A,B)+N_B(A,B) of judging bond strength, where N_A(A, B)=\{Z_AN^2_A(2∑Bλ∑...We modified Mulikens overlap population n(A, B)=2∑Bλ∑Aμ∑in_ic~*_A,μic_B,λis_Aμ,Bλ and obtained an experiential formula N(A,B)=N_A(A,B)+N_B(A,B) of judging bond strength, where N_A(A, B)=\{Z_AN^2_A(2∑Bλ∑Aμ∑i\}n_ic~*_A,μic_B,λis_Aμ,Bλ), N_B(A, B)=Z_BN^2_B(2∑Bλ∑Aμ∑in_ic~*_A,μic_B,λis_Aμ,Bλ). Twenty-eight bonds calculated by IEHM method and 11 monohydrides calculated by using 6-31G~** basis sets at Hartree-Fock level, electronic correlation effects are also considered through MP2/6-31G~**, were used to verify our experiential formula. Compared with the judgment of chemical bond strength by means of Mulikens overlap population, our experiential formula has a more obvious improvement as a judgment of bond strength than Mulikens overlap population. As a judgment of chemical bond strength between atoms in molecules, the experiential formula has conquered some limitations of Mulikens overlap population, and accorded with the experimental results.展开更多
基金Supported by the National Natural Science Foundation Fujian province of China(2016J01032).
文摘As is well known,the definitions of fractional sum and fractional difference of f(z)on non-uniform lattices x(z)=c1z^(2)+c2z+c3 or x(z)=c1q^(z)+c2q^(-z)+c3 are more difficult and complicated.In this article,for the first time we propose the definitions of the fractional sum and fractional difference on non-uniform lattices by two different ways.The analogue of Euler’s Beta formula,Cauchy’Beta formula on non-uniform lattices are established,and some fundamental theorems of fractional calculas,the solution of the generalized Abel equation on non-uniform lattices are obtained etc.
文摘If Goldbach’s conjecture is true, then for each prime number p there is at least one pair of primes symmetric with respect to p and whose sum is 2p. In the multiplicative number theory, covering the positive integers with primes, during the prime factorization, may be viewed as being the outcome of a parallel system which functions properly if and only if Euler’s formula of the product of the reciprocals of the primes is true. An exact formula for the number of primes less than or equal to an arbitrary bound is given. This formula may be implemented using Wolfram’s computer package Mathematica.
基金supported in part by External Cooperation Program of Chinese Academy of Sciences(No.GJHZ1218)National Natural Science Foundation of China(No.61004133)SSSTC JRP awards 2011(IZLCZ2 138953)
文摘In this paper, a two-dimensional nanometer scale tip-plate discharge model has been employed to study nanoscale electrical discharge in atmospheric conditions. The field strength dis- tributions in a nanometer scale tip-to-plate electrode arrangement were calculated using the finite element analysis （FEA） method, and the influences of applied voltage amplitude and frequency as well as gas gap distance on the variation of effective discharge range （EDR） on the plate were also investigated and discussed. The simulation results show that the probe with a wide tip will cause a larger effective discharge range on the plate; the field strength in the gap is notably higher than that induced by the sharp tip probe; the effective discharge range will increase linearly with the rise of excitation voltage, and decrease nonlinearly with the rise of gap length. In addition, probe dimension, especially the width/height ratio, affects the effective discharge range in different manners. With the width/height ratio rising from 1 ： 1 to 1 ： 10, the effective discharge range will maintain stable when the excitation voltage is around 50 V. This will increase when the excitation voltage gets higher and decrease as the excitation voltage gets lower. Fhrthermore, when the gap length is 5 nm and the excitation voltage is below 20 V, the diameter of EDR in our simulation is about 150 nm, which is consistent with the experiment results reported by other research groups. Our work provides a preliminary understanding of nanometer scale discharges and establishes a predictive structure-behavior relationship.
文摘The purpose of this paper is to establish, paralleling a well-known result for definite integrals, the conditional convergence of a family of trigonometric sine series. The fundamental idea is to group appropriately the terms of the series in order to show absolute divergence of the series, given the well-established result that the series as it stands is convergent.
文摘It is a fact that imaginary numbers do not have practical significance. But the role of imaginary numbers is very broad and enormous, due to the existence of Euler’s formula. Due to Euler’s formula, imaginary numbers have been applied in many theoretical theories. One of the biggest functions of imaginary numbers is to represent changes in phase, which is indispensable in signal analysis theory. The imaginary numbers in quantum mechanics pose a greater mystery: do the imaginary numbers really exist? This question still needs further scientific development to be answered.
文摘We modified Mulikens overlap population n(A, B)=2∑Bλ∑Aμ∑in_ic~*_A,μic_B,λis_Aμ,Bλ and obtained an experiential formula N(A,B)=N_A(A,B)+N_B(A,B) of judging bond strength, where N_A(A, B)=\{Z_AN^2_A(2∑Bλ∑Aμ∑i\}n_ic~*_A,μic_B,λis_Aμ,Bλ), N_B(A, B)=Z_BN^2_B(2∑Bλ∑Aμ∑in_ic~*_A,μic_B,λis_Aμ,Bλ). Twenty-eight bonds calculated by IEHM method and 11 monohydrides calculated by using 6-31G~** basis sets at Hartree-Fock level, electronic correlation effects are also considered through MP2/6-31G~**, were used to verify our experiential formula. Compared with the judgment of chemical bond strength by means of Mulikens overlap population, our experiential formula has a more obvious improvement as a judgment of bond strength than Mulikens overlap population. As a judgment of chemical bond strength between atoms in molecules, the experiential formula has conquered some limitations of Mulikens overlap population, and accorded with the experimental results.
