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
New better estimates, which are given in terms of elementary functions, for the function r →（2/π）（1 -r2）κ（r）κ′（r） ＋ logr appearing in Hubner＇s sharp upper bound for the Hersch-Pfluger distortion functio...New better estimates, which are given in terms of elementary functions, for the function r →（2/π）（1 -r2）κ（r）κ′（r） ＋ logr appearing in Hubner＇s sharp upper bound for the Hersch-Pfluger distortion function are obtained. With these estimates, some known bounds for the Hersch-Pfluger distortion function in quasiconformal theory are improved, thus. improving the explicit quasiconformal Schwarz lemma and some known estimates for the solutions to the Ramanujan modular equations.展开更多
The authors prove some monotonicity properties of functions involving the generalized Agard distortion function ηg（a,t）, and obtain some inequalities for ηk（a, t） and relative distortion functions.
In this paper, the so-called approximate convexity and concavity properties of generalized Groetzsch ring function μa （r） by studying the monotonieity,convexity or concavity of certain composites of μa（r） are ob...In this paper, the so-called approximate convexity and concavity properties of generalized Groetzsch ring function μa （r） by studying the monotonieity,convexity or concavity of certain composites of μa（r） are obtained.展开更多
In this paper, we establish several inequalities for the the generalized linear distortion function λ（a, K） by using the monotonicity and convexity of certain combinations λ（a, K）.
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
Some properties and asymptotically sharp bounds are obtained for singdar values of Ramanujan’s generalized modular equation. from which infinite-product representations of the Hersch-Pfluger ?dimtortion function ? K ...Some properties and asymptotically sharp bounds are obtained for singdar values of Ramanujan’s generalized modular equation. from which infinite-product representations of the Hersch-Pfluger ?dimtortion function ? K (r) and the Agard η-distortion function η K (t) follow. By these results, the explicit quasiconformal Schwan lemma is improved, several properties are obtained for the Schottky upper bound, and a conjecture on the linear distortion function λ (K) is proved to be true.展开更多
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.
基金Supported by National 973 Project of China(2006CB708304)National Natural Science Foundation of China(10771195)
文摘New better estimates, which are given in terms of elementary functions, for the function r →（2/π）（1 -r2）κ（r）κ′（r） ＋ logr appearing in Hubner＇s sharp upper bound for the Hersch-Pfluger distortion function are obtained. With these estimates, some known bounds for the Hersch-Pfluger distortion function in quasiconformal theory are improved, thus. improving the explicit quasiconformal Schwarz lemma and some known estimates for the solutions to the Ramanujan modular equations.
基金supported by the Natural Science Foundation of China(11071069 and 11171307)the Natural Science Foundation of the Department of Education of Zhejiang Province(Y201328799)
文摘The authors prove some monotonicity properties of functions involving the generalized Agard distortion function ηg（a,t）, and obtain some inequalities for ηk（a, t） and relative distortion functions.
文摘In this paper, the so-called approximate convexity and concavity properties of generalized Groetzsch ring function μa （r） by studying the monotonieity,convexity or concavity of certain composites of μa（r） are obtained.
基金Supported by the National Natural Science Foundation of China(11071069, 11171307)the Natural Science Foundation of Hunan Province(09JJ6003)
文摘In this paper, we establish several inequalities for the the generalized linear distortion function λ（a, K） by using the monotonicity and convexity of certain combinations λ（a, K）.
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.
文摘Some properties and asymptotically sharp bounds are obtained for singdar values of Ramanujan’s generalized modular equation. from which infinite-product representations of the Hersch-Pfluger ?dimtortion function ? K (r) and the Agard η-distortion function η K (t) follow. By these results, the explicit quasiconformal Schwan lemma is improved, several properties are obtained for the Schottky upper bound, and a conjecture on the linear distortion function λ (K) is proved to be true.
