Diophantine equations have always fascinated mathematicians about existence, finitude, and the calculation of possible solutions. Among these equations, one of them will be the object of our research. This is the Pyth...Diophantine equations have always fascinated mathematicians about existence, finitude, and the calculation of possible solutions. Among these equations, one of them will be the object of our research. This is the Pythagoras’- Fermat’s equation defined as follows. (1) when , it is well known that this equation has an infinity of solutions but has none (non-trivial) when . We also know that the last result, named Fermat-Wiles theorem (or FLT) was obtained at great expense and its understanding remains out of reach even for a good fringe of professional mathematicians. The aim of this research is to set up new simple but effective tools in the treatment of Diophantine equations and that of Pythagoras-Fermat. The tools put forward in this research are the properties of the quotients and the Diophantine remainders which we define as follows. Let a non-trivial triplet () solution of Equation (1) such that . and are called the Diophantine quotients and remainders of solution . We compute the remainder and the quotient of b and c by a using the division algorithm. Hence, we have: and et with . We prove the following important results. if and only if and if and only if . Also, we deduce that or for any hypothetical solution . We illustrate these results by effectively computing the Diophantine quotients and remainders in the case of Pythagorean triplets using a Python program. In the end, we apply the previous properties to directly prove a partial result of FLT. .展开更多
This paper is a logical continuation of my recently-published paper. Security of modern communication based on RSA cryptographic protocols and their analogues is as crypto-immune as integer factorization (iFac) is dif...This paper is a logical continuation of my recently-published paper. Security of modern communication based on RSA cryptographic protocols and their analogues is as crypto-immune as integer factorization (iFac) is difficult. In this paper are considered enhanced algorithms for the iFac that are faster than the algorithm proposed in the previous paper. Among these enhanced algorithms is the one that is based on the ability to count the number of integer solutions on quadratic and bi-quadratic modular equations. Therefore, the iFac complexity is at most as difficult as the problem of counting. Properties of various modular equations are provided and confirmed in numerous computer experiments. These properties are instrumental in the proposed factorization algorithms, which are numerically illustrated in several examples.展开更多
In this paper, we establish a general theta function identity. It is a common origin of many theta function identities. From which many classical and new modular equations are derived. All the proofs are elementary.
This paper is a review, a thesis, of some interesting results that have been obtained in various research concerning the “brane collisions in string and M-theory” (Cyclic Universe), p-adic inflation and p-adic cosmo...This paper is a review, a thesis, of some interesting results that have been obtained in various research concerning the “brane collisions in string and M-theory” (Cyclic Universe), p-adic inflation and p-adic cosmology. In Section 2, we have described some equations concerning cosmic evolution in a Cyclic Universe. In Section 3, we have described some equations concerning the cosmological perturbations in a Big Crunch/Big Bang space-time, the M-theory model of a Big Crunch/Big Bang transition and some equations concerning the solution of a braneworld Big Crunch/Big Bang Cosmology. In Section 4, we have described some equations concerning the generating ekpyrotic curvature perturbations before the Big Bang, some equations concerning the effective five-dimensional theory of the strongly coupled heterotic string as a gauged version of N=1five-dimensional supergravity with four-dimensional boundaries, and some equations concerning the colliding branes and the origin of the Hot Big Bang. In Section 5, we have described some equations regarding the “null energy condition” violation concerning the inflationary models and some equations concerning the evolution to a smooth universe in an ekpyrotic contracting phase with w>1. In Section 6, we have described some equations concerning the approximate inflationary solutions rolling away from the unstable maximum of p-adic string theory. In Section 7, we have described various equations concerning the p-adic minisuperspace model, zeta strings, zeta nonlocal scalar fields and p-adic and adelic quantum cosmology. In Section 8, we have shown various and interesting mathematical connections between some equations concerning the p-adic inflation, the p-adic quantum cosmology, the zeta strings and the brane collisions in string and M-theory. Furthermore, in each section, we have shown the mathematical connections with various sectors of Number Theory, principally the Ramanujan’s modular equations, the Aurea Ratio and the Fibonacci’s numbers.展开更多
与常规直流相比,永富直流逆变站存在功率全送和功率分送运行方式,而其处于分网接入方式时电网换相换流器高压直流输电(line commutated converter based high voltage directcurrent,LCC-HVDC)系统的交互振荡模式及特征尚不明确。针对...与常规直流相比,永富直流逆变站存在功率全送和功率分送运行方式,而其处于分网接入方式时电网换相换流器高压直流输电(line commutated converter based high voltage directcurrent,LCC-HVDC)系统的交互振荡模式及特征尚不明确。针对这一特殊运行方式,采用模块化建模的思路建立可以反映系统电气/控制回路间交互耦合路径的运动方程模型。在此基础上,依据系统整流侧-逆变侧、正极-负极间的交互耦合路径分解得到影响系统主导模式稳定性的3条扰动传递路径,即整流侧内部自稳性路径、逆变侧内部自稳性路径、双极交互作用致稳性路径。最后,设置不同工况下的案例,量化评估不同作用路径提供的阻尼大小,并通过仿真验证运动方程模型及扰动传递路径分析结果的正确性,为后续研究分网接入方式下LCC-HVDC系统交互振荡模式的阻尼特征提供模型基础。展开更多
Finding the solution to a general multivariate modular linear equation plays an important role in cryptanalysis field. Earlier results show that obtaining a relatively short solution is possible in polynomial time. Ho...Finding the solution to a general multivariate modular linear equation plays an important role in cryptanalysis field. Earlier results show that obtaining a relatively short solution is possible in polynomial time. However, one problem arises here that if the equation has a short solution in given bounded range, the results outputted by earlier algorithms are often not the ones we are interested in. In this paper, we present a probability method based on lattice basis reduction to solve the problem. For a general multivariate modular linear equation with short solution in the given bounded range, the new method outputs this short solution in polynomial time, with a high probability. When the number of unknowns is not too large （smaller than 68）, the probability is approximating 1. Experimental results show that Knapsack systems and Lu-Lee type systems are easily broken in polynomial time with this new method.展开更多
In this paper,we prove quasi-modularity property for the twisted Gromov–Witten theory of O(3)over P^2.Meanwhile,we derive its holomorphic anomaly equation.
文摘Diophantine equations have always fascinated mathematicians about existence, finitude, and the calculation of possible solutions. Among these equations, one of them will be the object of our research. This is the Pythagoras’- Fermat’s equation defined as follows. (1) when , it is well known that this equation has an infinity of solutions but has none (non-trivial) when . We also know that the last result, named Fermat-Wiles theorem (or FLT) was obtained at great expense and its understanding remains out of reach even for a good fringe of professional mathematicians. The aim of this research is to set up new simple but effective tools in the treatment of Diophantine equations and that of Pythagoras-Fermat. The tools put forward in this research are the properties of the quotients and the Diophantine remainders which we define as follows. Let a non-trivial triplet () solution of Equation (1) such that . and are called the Diophantine quotients and remainders of solution . We compute the remainder and the quotient of b and c by a using the division algorithm. Hence, we have: and et with . We prove the following important results. if and only if and if and only if . Also, we deduce that or for any hypothetical solution . We illustrate these results by effectively computing the Diophantine quotients and remainders in the case of Pythagorean triplets using a Python program. In the end, we apply the previous properties to directly prove a partial result of FLT. .
文摘This paper is a logical continuation of my recently-published paper. Security of modern communication based on RSA cryptographic protocols and their analogues is as crypto-immune as integer factorization (iFac) is difficult. In this paper are considered enhanced algorithms for the iFac that are faster than the algorithm proposed in the previous paper. Among these enhanced algorithms is the one that is based on the ability to count the number of integer solutions on quadratic and bi-quadratic modular equations. Therefore, the iFac complexity is at most as difficult as the problem of counting. Properties of various modular equations are provided and confirmed in numerous computer experiments. These properties are instrumental in the proposed factorization algorithms, which are numerically illustrated in several examples.
基金Supported by the National Natural Science Foundation of China（11071107, 11371184）
文摘In this paper, we establish a general theta function identity. It is a common origin of many theta function identities. From which many classical and new modular equations are derived. All the proofs are elementary.
文摘This paper is a review, a thesis, of some interesting results that have been obtained in various research concerning the “brane collisions in string and M-theory” (Cyclic Universe), p-adic inflation and p-adic cosmology. In Section 2, we have described some equations concerning cosmic evolution in a Cyclic Universe. In Section 3, we have described some equations concerning the cosmological perturbations in a Big Crunch/Big Bang space-time, the M-theory model of a Big Crunch/Big Bang transition and some equations concerning the solution of a braneworld Big Crunch/Big Bang Cosmology. In Section 4, we have described some equations concerning the generating ekpyrotic curvature perturbations before the Big Bang, some equations concerning the effective five-dimensional theory of the strongly coupled heterotic string as a gauged version of N=1five-dimensional supergravity with four-dimensional boundaries, and some equations concerning the colliding branes and the origin of the Hot Big Bang. In Section 5, we have described some equations regarding the “null energy condition” violation concerning the inflationary models and some equations concerning the evolution to a smooth universe in an ekpyrotic contracting phase with w>1. In Section 6, we have described some equations concerning the approximate inflationary solutions rolling away from the unstable maximum of p-adic string theory. In Section 7, we have described various equations concerning the p-adic minisuperspace model, zeta strings, zeta nonlocal scalar fields and p-adic and adelic quantum cosmology. In Section 8, we have shown various and interesting mathematical connections between some equations concerning the p-adic inflation, the p-adic quantum cosmology, the zeta strings and the brane collisions in string and M-theory. Furthermore, in each section, we have shown the mathematical connections with various sectors of Number Theory, principally the Ramanujan’s modular equations, the Aurea Ratio and the Fibonacci’s numbers.
文摘与常规直流相比,永富直流逆变站存在功率全送和功率分送运行方式,而其处于分网接入方式时电网换相换流器高压直流输电(line commutated converter based high voltage directcurrent,LCC-HVDC)系统的交互振荡模式及特征尚不明确。针对这一特殊运行方式,采用模块化建模的思路建立可以反映系统电气/控制回路间交互耦合路径的运动方程模型。在此基础上,依据系统整流侧-逆变侧、正极-负极间的交互耦合路径分解得到影响系统主导模式稳定性的3条扰动传递路径,即整流侧内部自稳性路径、逆变侧内部自稳性路径、双极交互作用致稳性路径。最后,设置不同工况下的案例,量化评估不同作用路径提供的阻尼大小,并通过仿真验证运动方程模型及扰动传递路径分析结果的正确性,为后续研究分网接入方式下LCC-HVDC系统交互振荡模式的阻尼特征提供模型基础。
基金Supported by the National Natural Science Foundation of China (Grant Nos. 60873249, 60973142)the National High-Tech Research & Development Program of China (Grant Nos. 2008AA10Z419, 2009AA011906)the Project Funded by Basic Research Foundation of School of Information Science and Technology of Tsinghua University
文摘Finding the solution to a general multivariate modular linear equation plays an important role in cryptanalysis field. Earlier results show that obtaining a relatively short solution is possible in polynomial time. However, one problem arises here that if the equation has a short solution in given bounded range, the results outputted by earlier algorithms are often not the ones we are interested in. In this paper, we present a probability method based on lattice basis reduction to solve the problem. For a general multivariate modular linear equation with short solution in the given bounded range, the new method outputs this short solution in polynomial time, with a high probability. When the number of unknowns is not too large （smaller than 68）, the probability is approximating 1. Experimental results show that Knapsack systems and Lu-Lee type systems are easily broken in polynomial time with this new method.
基金Supported by NSFC(Grant No.11601279)by Shandong Provincial Natural Science Foundation,China(Grant No.ZR2016AQ05)
文摘In this paper,we prove quasi-modularity property for the twisted Gromov–Witten theory of O(3)over P^2.Meanwhile,we derive its holomorphic anomaly equation.
