Given two graphs G and H,the Ramsey number R(G,H)is the smallest positive integer N such that every 2-coloring of the edges of K_(N)contains either a red G or a blue H.Let K_(N-1)■K_(1,k)be the graph obtained from K_...Given two graphs G and H,the Ramsey number R(G,H)is the smallest positive integer N such that every 2-coloring of the edges of K_(N)contains either a red G or a blue H.Let K_(N-1)■K_(1,k)be the graph obtained from K_(N-1)by adding anew vertexνconnecting k vertices of K_(N-1).A graph G withχ(G)=k+1 is called edge-critical if G contains an edge e such thatχ(G-e)=k.A considerable amount of research has been conducted by previous scholars on Ramsey numbers ofgraphs.In this study,we show that for an edge-critical graph G with x(G)=k+1,when k≥2,1≥2,and n is sufficiently large,R(G,K_(1)+nK_(t))=knt+1 and r,(G,K_(1)+nK_(t))=(k-1)nt+1.展开更多
In 1694,Gregory and Newton proposed the problem to determine the kissing number of a rigid material ball.This problem and its higher dimensional generalization have been studied by many mathematicians,including Minkow...In 1694,Gregory and Newton proposed the problem to determine the kissing number of a rigid material ball.This problem and its higher dimensional generalization have been studied by many mathematicians,including Minkowski,van der Waerden,Hadwiger,Swinnerton-Dyer,Watson,Levenshtein,Odlyzko,Sloane and Musin.In this paper,we introduce and study a further generalization of the kissing numbers for convex bodies and obtain some exact results,in particular for balls in dimensions three,four and eight.展开更多
In the quantum Monte Carlo(QMC)method,the pseudo-random number generator(PRNG)plays a crucial role in determining the computation time.However,the hidden structure of the PRNG may lead to serious issues such as the br...In the quantum Monte Carlo(QMC)method,the pseudo-random number generator(PRNG)plays a crucial role in determining the computation time.However,the hidden structure of the PRNG may lead to serious issues such as the breakdown of the Markov process.Here,we systematically analyze the performance of different PRNGs on the widely used QMC method known as the stochastic series expansion(SSE)algorithm.To quantitatively compare them,we introduce a quantity called QMC efficiency that can effectively reflect the efficiency of the algorithms.After testing several representative observables of the Heisenberg model in one and two dimensions,we recommend the linear congruential generator as the best choice of PRNG.Our work not only helps improve the performance of the SSE method but also sheds light on the other Markov-chain-based numerical algorithms.展开更多
This paper presents a low power,truly random number generator （TRNG） based on a simple chaotic map of the Bernoulli shift,which is extended to remain robustness in implementation. The map is realized by switched-cur...This paper presents a low power,truly random number generator （TRNG） based on a simple chaotic map of the Bernoulli shift,which is extended to remain robustness in implementation. The map is realized by switched-current techniques that can fully integrate it in a cryptosystem on a chip. A pipelined architecture post-processed by a simple XOR circuit is used to improve the entropy. The TRNG is fabricated in an HJTC 0.18μm CMOS mixed signal process,and the statistical properties are investigated by measurement results. The power consumption is only 1.42mW and the truly random output bit rate is 10Mbit/s.展开更多
This paper proposes a novel single electron random number generator （RNG）. The generator consists of multiple tunneling junctions （MTJ） and a hybrid single electron transistor （SET）/MOS output circuit. It is an ...This paper proposes a novel single electron random number generator （RNG）. The generator consists of multiple tunneling junctions （MTJ） and a hybrid single electron transistor （SET）/MOS output circuit. It is an oscillator-based RNG. MTJ is used to implement a high-frequency oscillator, which uses the inherent physical randomness in tunneling events of the MTJ to achieve large frequency drift. The hybrid SET and MOS output circuit is used to amplify and buffer the output signal of the MTJ oscillator. The RNG circuit generates high-quality random digital sequences with a simple structure. The operation speed of this circuit is as high as 1GHz. The circuit also has good driven capability and low power dissipation. This novel random number generator is a promising device for future cryptographic systems and communication applications.展开更多
The purpose of this paper is to define the generalized Euler numbers and the generalized Euler numbers of higher order, their recursion formula and some properties were established, accordingly Euler numbers and Euler...The purpose of this paper is to define the generalized Euler numbers and the generalized Euler numbers of higher order, their recursion formula and some properties were established, accordingly Euler numbers and Euler numbers of higher order were extended.展开更多
Quantum random number generators adopting single negligible dead time of avalanche photodiodes （APDs） photon detection have been restricted due to the non- We propose a new approach based on an APD array to improve...Quantum random number generators adopting single negligible dead time of avalanche photodiodes （APDs） photon detection have been restricted due to the non- We propose a new approach based on an APD array to improve the generation rate of random numbers significantly. This method compares the detectors＇ responses to consecutive optical pulses and generates the random sequence. We implement a demonstration experiment to show its simplicity, compactness and scalability. The generated numbers are proved to be unbiased, post-processing free, ready to use, and their randomness is verified by using the national institute of standard technology statistical test suite. The random bit generation efficiency is as high as 32.8% and the potential generation rate adopting the 32× 32 APD array is up to tens of Gbits/s.展开更多
The purpose of this paper is to give the extensions of some identities involving generalized Fibonacci and Lucas numbers with binomial coefficients.These results generalize the identities by Gulec,Taskara and Uslu in ...The purpose of this paper is to give the extensions of some identities involving generalized Fibonacci and Lucas numbers with binomial coefficients.These results generalize the identities by Gulec,Taskara and Uslu in Appl.Math.Lett.23(2010)68-72 and Appl.Math.Comput.220(2013)482-486.展开更多
Montgomery modular multiplication in the residue number system （RNS） can be applied for elliptic curve cryptography. In this work, unified modular multipliers over generalized Mersenne numbers are proposed for RNS M...Montgomery modular multiplication in the residue number system （RNS） can be applied for elliptic curve cryptography. In this work, unified modular multipliers over generalized Mersenne numbers are proposed for RNS Montgomery modular multiplication, which enables efficient elliptic curve point multiplication （ECPM）. Meanwhile, the elliptic curve arithmetic with ECPM is performed by mixed coordinates and adjusted for hardware implementation. In addition, the conversion between RNS and the binary number system is also discussed. Compared with the results in the literature, our hardware architecture for ECPM demonstrates high performance. A 256-bit ECPM in Xilinx XC2VP100 field programmable gate array device （FPGA） can be performed in 1.44 ms, costing 22147 slices, 45 dedicated multipliers, and 8.25K bits of random access memories （RAMs）.展开更多
Random numbers play an increasingly important role in secure wire and wireless communication. Thus the design quality of random number generator(RNG) is significant in information security. A novel pseudo RNG is propo...Random numbers play an increasingly important role in secure wire and wireless communication. Thus the design quality of random number generator(RNG) is significant in information security. A novel pseudo RNG is proposed for improving the security of network communication. The back propagation neural network(BPNN) is nonlinear, which can be used to improve the traditional RNG. The novel pseudo RNG is based on BPNN techniques. The result of test suites standardized by the U.S shows that the RNG can satisfy the security of communication.展开更多
The generation of good pseudo-random numbers is the base of many important fields in scientific computing, such as randomized algorithms and numerical solution of stochastic differential equations. In this paper, a cl...The generation of good pseudo-random numbers is the base of many important fields in scientific computing, such as randomized algorithms and numerical solution of stochastic differential equations. In this paper, a class of random number generators （RNGs） based on Weyl sequence is proposed. The uniformity of those RNGs is proved theoretically. Statistical and numerical computations show the efficiency of the methods.展开更多
In this paper, we observe the generalized Harmonic numbers H<sub>n,k,r</sub> (α,β). Using generating function, we investigate some new identities involving generalized Harmonic numbers H<sub>n,k,r&...In this paper, we observe the generalized Harmonic numbers H<sub>n,k,r</sub> (α,β). Using generating function, we investigate some new identities involving generalized Harmonic numbers H<sub>n,k,r</sub> (α,β) with Changhee sequences, Daehee sequences, Degenerate Changhee-Genoocchi sequences, Two kinds of degenerate Stirling numbers. Using Riordan arrays, we explore interesting relations between these polynomials, Apostol Bernoulli sequences, Apostol Euler sequences, Apostol Genoocchi sequences.展开更多
In this paper, we consider <i>r</i>-generalization of the central factorial numbers with odd arguments of the first and second kind. Mainly, we obtain various identities and properties related to these num...In this paper, we consider <i>r</i>-generalization of the central factorial numbers with odd arguments of the first and second kind. Mainly, we obtain various identities and properties related to these numbers. Matrix representation and the relation between these numbers and Pascal matrix are given. Furthermore, the distributions of the signless r-central factorial numbers are derived. In addition, connections between these numbers and the Legendre-Stirling numbers are given.展开更多
In recent years, various chaotic equation based pseudorandom number generators have been proposed. However, the chaotic equations are all defined in the real number field. In this paper, an equation is proposed and pr...In recent years, various chaotic equation based pseudorandom number generators have been proposed. However, the chaotic equations are all defined in the real number field. In this paper, an equation is proposed and proved to be chaotic in the imaginary axis. And a pseudorandom number generator is constructed based on the chaotic equation. The alteration of the definitional domain of the chaotic equation from the real number field to the complex one provides a new approach to the construction of chaotic equations, and a new method to generate pseudorandorn number sequences accordingly. Both theoretical analysis and experimental results show that the sequences generated by the proposed pseudorandom number generator possess many good properties.展开更多
In this paper, we define the generalized r-Whitney numbers of the first and second kind. Moreover, we drive the generalized Whitney numbers of the first and second kind. The recurrence relations and the generating fun...In this paper, we define the generalized r-Whitney numbers of the first and second kind. Moreover, we drive the generalized Whitney numbers of the first and second kind. The recurrence relations and the generating functions of these numbers are derived. The relations between these numbers and generalized Stirling numbers of the first and second kind are deduced. Furthermore, some special cases are given. Finally, matrix representation of the relations between Whitney and Stirling numbers is given.展开更多
This paper proposes a well-performing hybrid-type truly quantum random number generator based on the time interval between two independent single-photon detection signals, which is practical and intuitive, and generat...This paper proposes a well-performing hybrid-type truly quantum random number generator based on the time interval between two independent single-photon detection signals, which is practical and intuitive, and generates the initial random number sources from a combination of multiple existing random number sources. A time-to-amplitude converter and multichannel analyzer are used for qualitative analysis to demonstrate that each and every step is random. Furthermore, a carefully designed data acquisition system is used to obtain a high-quality random sequence. Our scheme is simple and proves that the random number bit rate can be dramatically increased to satisfy practical requirements.展开更多
We present componentwise condition numbers for the problems of MoorePenrose generalized matrix inversion and linear least squares. Also, the condition numbers for these condition numbers are given.
A simple recursive algorithm to generate the set of natural numbers, based on Mersenne numbers: M<sub>N</sub> = 2<sup>N</sup> – 1, is used to count the number of prime numbers within the preci...A simple recursive algorithm to generate the set of natural numbers, based on Mersenne numbers: M<sub>N</sub> = 2<sup>N</sup> – 1, is used to count the number of prime numbers within the precise Mersenne natural number intervals: [0;M<sub>N</sub>]. This permits the formulation of an extended twin prime conjecture. Moreover, it is found that the prime numbers subsets contained in Mersenne intervals have cardinalities strongly correlated with the corresponding Mersenne numbers.展开更多
This article presents very original and relatively brief or very brief proofs about of two famous problems: 1) Are there any odd perfect numbers? and 2) “Fermat’s last theorem: A new proof of theorem and its general...This article presents very original and relatively brief or very brief proofs about of two famous problems: 1) Are there any odd perfect numbers? and 2) “Fermat’s last theorem: A new proof of theorem and its generalization”. They are achieved with elementary mathematics. This is why these proofs can be easily understood by any mathematician or anyone who knows basic mathematics. Note that, in both problems, proof by contradiction was used as a method of proof. The first of the two problems to date has not been resolved. Its proof is completely original and was not based on the work of other researchers. On the contrary, it was based on a simple observation that all natural divisors of a positive integer appear in pairs. The aim of the first work is to solve one of the unsolved, for many years, problems of the mathematics which belong to the field of number theory. I believe that if the present proof is recognized by the mathematical community, it may signal a different way of solving unsolved problems. For the second problem, it is very important the fact that it is generalized to an arbitrarily large number of variables. This generalization is essentially a new theorem in the field of the number theory. To the classical problem, two solutions are given, which are presented in the chronological order in which they were achieved. Note that the second solution is very short and does not exceed one and a half pages. This leads me to believe that Fermat, as a great mathematician was not lying and that he had probably solved the problem, as he stated in his historic its letter, with a correspondingly brief solution. To win the bet on the question of whether Fermat was telling truth or lying, go immediately to the end of this article before the General Conclusions.展开更多
基金supported by the National Key Research and Development Program of China(2023YFA1010200,2020YFA0713100)the National Natural Science Foundation of China(12071453)the Innovation Program for Quantum Science and Technology(2021ZD0302902).
文摘Given two graphs G and H,the Ramsey number R(G,H)is the smallest positive integer N such that every 2-coloring of the edges of K_(N)contains either a red G or a blue H.Let K_(N-1)■K_(1,k)be the graph obtained from K_(N-1)by adding anew vertexνconnecting k vertices of K_(N-1).A graph G withχ(G)=k+1 is called edge-critical if G contains an edge e such thatχ(G-e)=k.A considerable amount of research has been conducted by previous scholars on Ramsey numbers ofgraphs.In this study,we show that for an edge-critical graph G with x(G)=k+1,when k≥2,1≥2,and n is sufficiently large,R(G,K_(1)+nK_(t))=knt+1 and r,(G,K_(1)+nK_(t))=(k-1)nt+1.
基金supported by the National Natural Science Foundation of China(12226006,11921001)the Natural Key Research and Development Program of China(2018YFA0704701).
文摘In 1694,Gregory and Newton proposed the problem to determine the kissing number of a rigid material ball.This problem and its higher dimensional generalization have been studied by many mathematicians,including Minkowski,van der Waerden,Hadwiger,Swinnerton-Dyer,Watson,Levenshtein,Odlyzko,Sloane and Musin.In this paper,we introduce and study a further generalization of the kissing numbers for convex bodies and obtain some exact results,in particular for balls in dimensions three,four and eight.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.12274046,11874094,and 12147102)Chongqing Natural Science Foundation(Grant No.CSTB2022NSCQ-JQX0018)Fundamental Research Funds for the Central Universities(Grant No.2021CDJZYJH-003).
文摘In the quantum Monte Carlo(QMC)method,the pseudo-random number generator(PRNG)plays a crucial role in determining the computation time.However,the hidden structure of the PRNG may lead to serious issues such as the breakdown of the Markov process.Here,we systematically analyze the performance of different PRNGs on the widely used QMC method known as the stochastic series expansion(SSE)algorithm.To quantitatively compare them,we introduce a quantity called QMC efficiency that can effectively reflect the efficiency of the algorithms.After testing several representative observables of the Heisenberg model in one and two dimensions,we recommend the linear congruential generator as the best choice of PRNG.Our work not only helps improve the performance of the SSE method but also sheds light on the other Markov-chain-based numerical algorithms.
文摘This paper presents a low power,truly random number generator （TRNG） based on a simple chaotic map of the Bernoulli shift,which is extended to remain robustness in implementation. The map is realized by switched-current techniques that can fully integrate it in a cryptosystem on a chip. A pipelined architecture post-processed by a simple XOR circuit is used to improve the entropy. The TRNG is fabricated in an HJTC 0.18μm CMOS mixed signal process,and the statistical properties are investigated by measurement results. The power consumption is only 1.42mW and the truly random output bit rate is 10Mbit/s.
文摘This paper proposes a novel single electron random number generator （RNG）. The generator consists of multiple tunneling junctions （MTJ） and a hybrid single electron transistor （SET）/MOS output circuit. It is an oscillator-based RNG. MTJ is used to implement a high-frequency oscillator, which uses the inherent physical randomness in tunneling events of the MTJ to achieve large frequency drift. The hybrid SET and MOS output circuit is used to amplify and buffer the output signal of the MTJ oscillator. The RNG circuit generates high-quality random digital sequences with a simple structure. The operation speed of this circuit is as high as 1GHz. The circuit also has good driven capability and low power dissipation. This novel random number generator is a promising device for future cryptographic systems and communication applications.
基金Supported by the NNSF of China(10001016) SF for the Prominent Youth of Henan Province
文摘The purpose of this paper is to define the generalized Euler numbers and the generalized Euler numbers of higher order, their recursion formula and some properties were established, accordingly Euler numbers and Euler numbers of higher order were extended.
基金Supported by the Chinese Academy of Sciences Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics,Shanghai Branch,University of Science and Technology of Chinathe National Natural Science Foundation of China under Grant No 11405172
文摘Quantum random number generators adopting single negligible dead time of avalanche photodiodes （APDs） photon detection have been restricted due to the non- We propose a new approach based on an APD array to improve the generation rate of random numbers significantly. This method compares the detectors＇ responses to consecutive optical pulses and generates the random sequence. We implement a demonstration experiment to show its simplicity, compactness and scalability. The generated numbers are proved to be unbiased, post-processing free, ready to use, and their randomness is verified by using the national institute of standard technology statistical test suite. The random bit generation efficiency is as high as 32.8% and the potential generation rate adopting the 32× 32 APD array is up to tens of Gbits/s.
基金Supported by the Youth Backbone Teacher Foundation of Henan's University(Grant No.2016GGJS-117)Supported by the National Natural Science Foundation of China(Grant No.11871258)。
文摘The purpose of this paper is to give the extensions of some identities involving generalized Fibonacci and Lucas numbers with binomial coefficients.These results generalize the identities by Gulec,Taskara and Uslu in Appl.Math.Lett.23(2010)68-72 and Appl.Math.Comput.220(2013)482-486.
基金supported by the National Natural Science Foundation of China under Grant No. 61073173
文摘Montgomery modular multiplication in the residue number system （RNS） can be applied for elliptic curve cryptography. In this work, unified modular multipliers over generalized Mersenne numbers are proposed for RNS Montgomery modular multiplication, which enables efficient elliptic curve point multiplication （ECPM）. Meanwhile, the elliptic curve arithmetic with ECPM is performed by mixed coordinates and adjusted for hardware implementation. In addition, the conversion between RNS and the binary number system is also discussed. Compared with the results in the literature, our hardware architecture for ECPM demonstrates high performance. A 256-bit ECPM in Xilinx XC2VP100 field programmable gate array device （FPGA） can be performed in 1.44 ms, costing 22147 slices, 45 dedicated multipliers, and 8.25K bits of random access memories （RAMs）.
基金National Natural Science Foundation of China(60363087 ,90104005 and 60473023)
文摘Random numbers play an increasingly important role in secure wire and wireless communication. Thus the design quality of random number generator(RNG) is significant in information security. A novel pseudo RNG is proposed for improving the security of network communication. The back propagation neural network(BPNN) is nonlinear, which can be used to improve the traditional RNG. The novel pseudo RNG is based on BPNN techniques. The result of test suites standardized by the U.S shows that the RNG can satisfy the security of communication.
基金Supported by National Natural Science Foundation of China （19871047）and National Key Basic Research Special Fund（1998020306）.
文摘The generation of good pseudo-random numbers is the base of many important fields in scientific computing, such as randomized algorithms and numerical solution of stochastic differential equations. In this paper, a class of random number generators （RNGs） based on Weyl sequence is proposed. The uniformity of those RNGs is proved theoretically. Statistical and numerical computations show the efficiency of the methods.
文摘In this paper, we observe the generalized Harmonic numbers H<sub>n,k,r</sub> (α,β). Using generating function, we investigate some new identities involving generalized Harmonic numbers H<sub>n,k,r</sub> (α,β) with Changhee sequences, Daehee sequences, Degenerate Changhee-Genoocchi sequences, Two kinds of degenerate Stirling numbers. Using Riordan arrays, we explore interesting relations between these polynomials, Apostol Bernoulli sequences, Apostol Euler sequences, Apostol Genoocchi sequences.
文摘In this paper, we consider <i>r</i>-generalization of the central factorial numbers with odd arguments of the first and second kind. Mainly, we obtain various identities and properties related to these numbers. Matrix representation and the relation between these numbers and Pascal matrix are given. Furthermore, the distributions of the signless r-central factorial numbers are derived. In addition, connections between these numbers and the Legendre-Stirling numbers are given.
基金Project supported by the National Natural Science Foundation of China (Grant No. 60973162)the Natural Science Foundation of Shandong Province, China (Grant No. ZR2009GM037)+1 种基金the Science and Technology of Shandong Province, China(Grant No. 2010GGX10132)the Key Program of the Natural Science Foundation of Shandong Province, China (Grant No. Z2006G01)
文摘In recent years, various chaotic equation based pseudorandom number generators have been proposed. However, the chaotic equations are all defined in the real number field. In this paper, an equation is proposed and proved to be chaotic in the imaginary axis. And a pseudorandom number generator is constructed based on the chaotic equation. The alteration of the definitional domain of the chaotic equation from the real number field to the complex one provides a new approach to the construction of chaotic equations, and a new method to generate pseudorandorn number sequences accordingly. Both theoretical analysis and experimental results show that the sequences generated by the proposed pseudorandom number generator possess many good properties.
文摘In this paper, we define the generalized r-Whitney numbers of the first and second kind. Moreover, we drive the generalized Whitney numbers of the first and second kind. The recurrence relations and the generating functions of these numbers are derived. The relations between these numbers and generalized Stirling numbers of the first and second kind are deduced. Furthermore, some special cases are given. Finally, matrix representation of the relations between Whitney and Stirling numbers is given.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.61178010 and 11374042)the Fund of State Key Laboratory of Information Photonics and Optical Communications(Beijing University of Posts and Telecommunications),Chinathe Fundamental Research Funds for the Central Universities of China(Grant No.bupt2014TS01)
文摘This paper proposes a well-performing hybrid-type truly quantum random number generator based on the time interval between two independent single-photon detection signals, which is practical and intuitive, and generates the initial random number sources from a combination of multiple existing random number sources. A time-to-amplitude converter and multichannel analyzer are used for qualitative analysis to demonstrate that each and every step is random. Furthermore, a carefully designed data acquisition system is used to obtain a high-quality random sequence. Our scheme is simple and proves that the random number bit rate can be dramatically increased to satisfy practical requirements.
基金the NSF of China under grant 10471027 and Shanghai Education Commission.
文摘We present componentwise condition numbers for the problems of MoorePenrose generalized matrix inversion and linear least squares. Also, the condition numbers for these condition numbers are given.
文摘A simple recursive algorithm to generate the set of natural numbers, based on Mersenne numbers: M<sub>N</sub> = 2<sup>N</sup> – 1, is used to count the number of prime numbers within the precise Mersenne natural number intervals: [0;M<sub>N</sub>]. This permits the formulation of an extended twin prime conjecture. Moreover, it is found that the prime numbers subsets contained in Mersenne intervals have cardinalities strongly correlated with the corresponding Mersenne numbers.
文摘This article presents very original and relatively brief or very brief proofs about of two famous problems: 1) Are there any odd perfect numbers? and 2) “Fermat’s last theorem: A new proof of theorem and its generalization”. They are achieved with elementary mathematics. This is why these proofs can be easily understood by any mathematician or anyone who knows basic mathematics. Note that, in both problems, proof by contradiction was used as a method of proof. The first of the two problems to date has not been resolved. Its proof is completely original and was not based on the work of other researchers. On the contrary, it was based on a simple observation that all natural divisors of a positive integer appear in pairs. The aim of the first work is to solve one of the unsolved, for many years, problems of the mathematics which belong to the field of number theory. I believe that if the present proof is recognized by the mathematical community, it may signal a different way of solving unsolved problems. For the second problem, it is very important the fact that it is generalized to an arbitrarily large number of variables. This generalization is essentially a new theorem in the field of the number theory. To the classical problem, two solutions are given, which are presented in the chronological order in which they were achieved. Note that the second solution is very short and does not exceed one and a half pages. This leads me to believe that Fermat, as a great mathematician was not lying and that he had probably solved the problem, as he stated in his historic its letter, with a correspondingly brief solution. To win the bet on the question of whether Fermat was telling truth or lying, go immediately to the end of this article before the General Conclusions.
