We study the global dynamics of a rational diference equation with higher order,which includes many rational diference equations as its special cases.By some complicate computations and mathematical skills,we show tha...We study the global dynamics of a rational diference equation with higher order,which includes many rational diference equations as its special cases.By some complicate computations and mathematical skills,we show that its unique nonnegative fixed point is globally attractive.As application,our results not only improve many known ones,but also solve several“Open Problems and Conjectures”given by Professors Ladas and Camouzis,et al.展开更多
BEIC （Bayesian equilibrium by iterative conjectures） analyzes games with players forming their conjectures about what other players will do through iterative reasoning starting with first order uninformative conject...BEIC （Bayesian equilibrium by iterative conjectures） analyzes games with players forming their conjectures about what other players will do through iterative reasoning starting with first order uninformative conjectures and keep updating their conjectures iteratively with game theoretic reasoning until a convergence of conjectures is achieved. In a BEIC, beliefs about the other players＇ strategies are specified and they are consistent with the equilibrium strategies they supported. A BEIC is therefore a perfect Bayesian equilibrium and hence a refinement of Nash equilibrium. Through six examples, the BE1C solutions are compared with those obtained by the other refining criteria of payoff-dominance, risk-dominance, iterated admissibility, subgame perfect equilibrium, Bayesian Nash equilibrium, perfect Bayesian equilibrium and the intuitive criterion. The outstanding results from the comparisons are that the BEIC approach is able to pick the natural focal point of a game when the iterated admissibility criterion fails to, the BEIC approach rules out equilibrium depending upon non credible threat, and that in simultaneous and sequential games of incomplete information, the BEIC approach not only normally narrows down the equilibriums to one but it also picks the most compelling equilibrium compare with Bayesian Nash equilibrium or perfect Bayesian equilibrium or intuitive criterion.展开更多
Recently proposed two swampland criteria that arising from string theory landscape leads to the important challenge of the realization of single-field inflationary models. Especially one of swampland criteria which im...Recently proposed two swampland criteria that arising from string theory landscape leads to the important challenge of the realization of single-field inflationary models. Especially one of swampland criteria which implies a large tensor-to-scalar ratio is strongly in tension with recent observational results. In this paper, we explore the possibility the swampland conjectures could be compatible with single-field inflationary scenarios if the effects due to the quantum theory of gravity are considered. We show that the quantum gravitational effects due to the nonlinear dispersion relation provides significant modifications on the amplitude of both the scalar and tensor perturbation spectra. Such modifications could be either raise or reduce the perturbation spectra depending on the values of the parameters in the nonlinear terms of the dispersion relations. Therefore, these effects can reduce the tensor-to-scalar ratio to a smaller value, which helps to relax the tension between the swampland conjecture and observational data.展开更多
In this paper, we investigate the growth of the meromorphic solutions of the following nonlinear difference equations F（Z）N＋pN-1（F）=0,where n ≥ 2 and small functions as proposed by Yang than 1. Pn-1（f） is a di...In this paper, we investigate the growth of the meromorphic solutions of the following nonlinear difference equations F（Z）N＋pN-1（F）=0,where n ≥ 2 and small functions as proposed by Yang than 1. Pn-1（f） is a difference polynomial of degree at most n - 1 in f with coefficients. Moreover, we give two examples to show that one conjecture and Laine [2] does not hold in general if the hyper-order of f（z） is no less展开更多
It is generally known that under the generalized Riemann hypothesis one could establish the twin primes conjecture by the circle method, provided one could obtain the estimate o (nlog-2 n)?for the integral of the repr...It is generally known that under the generalized Riemann hypothesis one could establish the twin primes conjecture by the circle method, provided one could obtain the estimate o (nlog-2 n)?for the integral of the representation function over the minor arcs. One of the new results here is that the assumption of GRH can be removed. We compare this and other such sufficiency results with similar results for the Goldbach conjecture.展开更多
In the space C[-1,1],G G Lorentz proposed four conjectures on the properties of the polynomials of the best approximation in 1977,1978 and 1980.The present paper transplants the four conjectures in the space Lρ2[-a,a...In the space C[-1,1],G G Lorentz proposed four conjectures on the properties of the polynomials of the best approximation in 1977,1978 and 1980.The present paper transplants the four conjectures in the space Lρ2[-a,a] and proves them being all right in only one theorem under the corresponding conditions,although each of the original conjectures is very difficulty.展开更多
In this paper, we prove Legendre’s conjecture: There is a prime number between n<sup>2</sup> and (n +1)<sup>2</sup> for every positive integer n. We also prove three related conjectures. The m...In this paper, we prove Legendre’s conjecture: There is a prime number between n<sup>2</sup> and (n +1)<sup>2</sup> for every positive integer n. We also prove three related conjectures. The method that we use is to analyze binomial coefficients. It is developed by the author from the method of analyzing binomial central coefficients, that was used by Paul Erdős in his proof of Bertrand’s postulate - Chebyshev’s theorem.展开更多
适逢Wang-Zahl[Wang H,Zahl J.Volume estimates for unions of convex sets,and the Kakeya set conjecture in three dimensions[J/OLl.arXiv:2502.17655,2025.]宣布解决三维Kakeya几何猜想之际,撰写此综述文章介绍调和分析及相关领...适逢Wang-Zahl[Wang H,Zahl J.Volume estimates for unions of convex sets,and the Kakeya set conjecture in three dimensions[J/OLl.arXiv:2502.17655,2025.]宣布解决三维Kakeya几何猜想之际,撰写此综述文章介绍调和分析及相关领域中的公开问题.围绕Kakeya猜想(源于几何测度论,分析版本对应Kakeya极大猜想)、限制性猜想、Bochner-Riesz猜想、局部光滑性猜想等四大猜想的研究,发展了诸如解析插值方法、正交性与双线性方法,Heisenberg不确定原理与局部化方法、微局部分析与驻相分析,催生了波包分解与尺度归纳,多线性理论、Bourgain-Guth的broad-narrow分析、关联几何及多项式方法,特别是"Wolff及Bourgain-Demeter等发展的解耦方法,不仅推动了调和分析中四大猜想的研究,同时也为解决其他数学领域的重要问题提供了一系列强有力工具.展开更多
成果名称:Shapley's Conjecture on the Cores of Abstract Market Games主要作者:曹志刚,秦承忠,杨晓光奖项类别:著作论文奖获奖等级:二等奖获奖论文《Shapley's Conjecture on the Cores of Abstract Market Games》发表于博...成果名称:Shapley's Conjecture on the Cores of Abstract Market Games主要作者:曹志刚,秦承忠,杨晓光奖项类别:著作论文奖获奖等级:二等奖获奖论文《Shapley's Conjecture on the Cores of Abstract Market Games》发表于博弈论领域顶级期刊《Games and Economic Behavior》2018年第2期。论文研究成果初步解决了诺贝尔经济学奖获得者罗伊德·沙普利(Lloyd S. Shapley)提出的抽象市场博弈核非空的猜想。展开更多
A nowhere-zero k-flow on a graph G=(V(G),E(G))is a pair(D,f),where D is an orientation on E(G)and f:E(G)→{±1,±2,,±(k-1)}is a function such that the total outflow equals to the total inflow at each vert...A nowhere-zero k-flow on a graph G=(V(G),E(G))is a pair(D,f),where D is an orientation on E(G)and f:E(G)→{±1,±2,,±(k-1)}is a function such that the total outflow equals to the total inflow at each vertex.This concept was introduced by Tutte as an extension of face colorings,and Tutte in 1954 conjectured that every bridgeless graph admits a nowhere-zero 5-flow,known as the 5-Flow Conjecture.This conjecture is verified for some graph classes and remains unresolved as of today.In this paper,we show that every bridgeless graph of Euler genus at most 20 admits a nowhere-zero 5-flow,which improves several known results.展开更多
We present a proof of the Strominger-Yau-Zaslow (SYZ) conjecture by demonstrating that mirror symmetry fundamentally represents an equivalence of computational structures between Calabi-Yau manifolds. Through developm...We present a proof of the Strominger-Yau-Zaslow (SYZ) conjecture by demonstrating that mirror symmetry fundamentally represents an equivalence of computational structures between Calabi-Yau manifolds. Through development of a rigorous quantum complexity operator formalism, we show that mirror pairs must have equivalent complexity spectra and that the SYZ fibration naturally preserves these computational invariants while implementing the required geometric transformations. Our proof proceeds by first establishing a precise mathematical framework connecting quantum complexity with geometric structures, then demonstrating that the special Lagrangian torus fibration preserves computational complexity at both local and global levels, and finally proving that this preservation necessarily implies the geometric correspondences required by the SYZ conjecture. This approach not only resolves the conjecture but reveals deeper insights about the relationship between computation and geometry in string theory. We introduce new complexity-based invariants for studying mirror symmetry and demonstrate how our framework extends naturally to related geometric structures.展开更多
Let A be a finite set of integers. For any integer h ≥ 2, let hA and h∧A be the sets of all sums of h elements of A and all sums of h distinct elements of A, respectively.In this paper, we survey some significant ad...Let A be a finite set of integers. For any integer h ≥ 2, let hA and h∧A be the sets of all sums of h elements of A and all sums of h distinct elements of A, respectively.In this paper, we survey some significant advances in additive combinatorics concerning the size and structure of sumsets of finite integer sets.展开更多
This article delves Chern's conjecture for hypersurfaces with constant second fundamental form squared length S in the spherical space.At present,determining whether the third gap point of S is 2n remains unsolved...This article delves Chern's conjecture for hypersurfaces with constant second fundamental form squared length S in the spherical space.At present,determining whether the third gap point of S is 2n remains unsolved yet.First,we investigate the height functions and their properties of the position vector and normal vector in natural coordinate vectors,and then prove the existence of a Simons-type integral formula on the hypersurface that simultaneously includes the first,second,and third gap point terms of S.These results can provide new avenues of thought and methods for solving Chern's conjecture.展开更多
基金Supported by the National Natural Science Foundation of China(61473340)Distinguished Professor Foundation of Qianjiang Scholar in Zhejiang Province(F703108L02)。
文摘We study the global dynamics of a rational diference equation with higher order,which includes many rational diference equations as its special cases.By some complicate computations and mathematical skills,we show that its unique nonnegative fixed point is globally attractive.As application,our results not only improve many known ones,but also solve several“Open Problems and Conjectures”given by Professors Ladas and Camouzis,et al.
文摘BEIC （Bayesian equilibrium by iterative conjectures） analyzes games with players forming their conjectures about what other players will do through iterative reasoning starting with first order uninformative conjectures and keep updating their conjectures iteratively with game theoretic reasoning until a convergence of conjectures is achieved. In a BEIC, beliefs about the other players＇ strategies are specified and they are consistent with the equilibrium strategies they supported. A BEIC is therefore a perfect Bayesian equilibrium and hence a refinement of Nash equilibrium. Through six examples, the BE1C solutions are compared with those obtained by the other refining criteria of payoff-dominance, risk-dominance, iterated admissibility, subgame perfect equilibrium, Bayesian Nash equilibrium, perfect Bayesian equilibrium and the intuitive criterion. The outstanding results from the comparisons are that the BEIC approach is able to pick the natural focal point of a game when the iterated admissibility criterion fails to, the BEIC approach rules out equilibrium depending upon non credible threat, and that in simultaneous and sequential games of incomplete information, the BEIC approach not only normally narrows down the equilibriums to one but it also picks the most compelling equilibrium compare with Bayesian Nash equilibrium or perfect Bayesian equilibrium or intuitive criterion.
基金Supported by National Natural Science Foundation of China under Grant No.11675143the Fundamental Research for the Provincial Universities of Zhejiang in China under Grant No.RF-A2019015
文摘Recently proposed two swampland criteria that arising from string theory landscape leads to the important challenge of the realization of single-field inflationary models. Especially one of swampland criteria which implies a large tensor-to-scalar ratio is strongly in tension with recent observational results. In this paper, we explore the possibility the swampland conjectures could be compatible with single-field inflationary scenarios if the effects due to the quantum theory of gravity are considered. We show that the quantum gravitational effects due to the nonlinear dispersion relation provides significant modifications on the amplitude of both the scalar and tensor perturbation spectra. Such modifications could be either raise or reduce the perturbation spectra depending on the values of the parameters in the nonlinear terms of the dispersion relations. Therefore, these effects can reduce the tensor-to-scalar ratio to a smaller value, which helps to relax the tension between the swampland conjecture and observational data.
基金supported by the NNSF of China(11171013,11371225,11201014)the YWF-14-SXXY-008 of Beihang Universitythe Fundamental Research Funds for the Central University
文摘In this paper, we investigate the growth of the meromorphic solutions of the following nonlinear difference equations F（Z）N＋pN-1（F）=0,where n ≥ 2 and small functions as proposed by Yang than 1. Pn-1（f） is a difference polynomial of degree at most n - 1 in f with coefficients. Moreover, we give two examples to show that one conjecture and Laine [2] does not hold in general if the hyper-order of f（z） is no less
文摘It is generally known that under the generalized Riemann hypothesis one could establish the twin primes conjecture by the circle method, provided one could obtain the estimate o (nlog-2 n)?for the integral of the representation function over the minor arcs. One of the new results here is that the assumption of GRH can be removed. We compare this and other such sufficiency results with similar results for the Goldbach conjecture.
文摘In the space C[-1,1],G G Lorentz proposed four conjectures on the properties of the polynomials of the best approximation in 1977,1978 and 1980.The present paper transplants the four conjectures in the space Lρ2[-a,a] and proves them being all right in only one theorem under the corresponding conditions,although each of the original conjectures is very difficulty.
文摘In this paper, we prove Legendre’s conjecture: There is a prime number between n<sup>2</sup> and (n +1)<sup>2</sup> for every positive integer n. We also prove three related conjectures. The method that we use is to analyze binomial coefficients. It is developed by the author from the method of analyzing binomial central coefficients, that was used by Paul Erdős in his proof of Bertrand’s postulate - Chebyshev’s theorem.
文摘适逢Wang-Zahl[Wang H,Zahl J.Volume estimates for unions of convex sets,and the Kakeya set conjecture in three dimensions[J/OLl.arXiv:2502.17655,2025.]宣布解决三维Kakeya几何猜想之际,撰写此综述文章介绍调和分析及相关领域中的公开问题.围绕Kakeya猜想(源于几何测度论,分析版本对应Kakeya极大猜想)、限制性猜想、Bochner-Riesz猜想、局部光滑性猜想等四大猜想的研究,发展了诸如解析插值方法、正交性与双线性方法,Heisenberg不确定原理与局部化方法、微局部分析与驻相分析,催生了波包分解与尺度归纳,多线性理论、Bourgain-Guth的broad-narrow分析、关联几何及多项式方法,特别是"Wolff及Bourgain-Demeter等发展的解耦方法,不仅推动了调和分析中四大猜想的研究,同时也为解决其他数学领域的重要问题提供了一系列强有力工具.
文摘成果名称:Shapley's Conjecture on the Cores of Abstract Market Games主要作者:曹志刚,秦承忠,杨晓光奖项类别:著作论文奖获奖等级:二等奖获奖论文《Shapley's Conjecture on the Cores of Abstract Market Games》发表于博弈论领域顶级期刊《Games and Economic Behavior》2018年第2期。论文研究成果初步解决了诺贝尔经济学奖获得者罗伊德·沙普利(Lloyd S. Shapley)提出的抽象市场博弈核非空的猜想。
文摘A nowhere-zero k-flow on a graph G=(V(G),E(G))is a pair(D,f),where D is an orientation on E(G)and f:E(G)→{±1,±2,,±(k-1)}is a function such that the total outflow equals to the total inflow at each vertex.This concept was introduced by Tutte as an extension of face colorings,and Tutte in 1954 conjectured that every bridgeless graph admits a nowhere-zero 5-flow,known as the 5-Flow Conjecture.This conjecture is verified for some graph classes and remains unresolved as of today.In this paper,we show that every bridgeless graph of Euler genus at most 20 admits a nowhere-zero 5-flow,which improves several known results.
文摘We present a proof of the Strominger-Yau-Zaslow (SYZ) conjecture by demonstrating that mirror symmetry fundamentally represents an equivalence of computational structures between Calabi-Yau manifolds. Through development of a rigorous quantum complexity operator formalism, we show that mirror pairs must have equivalent complexity spectra and that the SYZ fibration naturally preserves these computational invariants while implementing the required geometric transformations. Our proof proceeds by first establishing a precise mathematical framework connecting quantum complexity with geometric structures, then demonstrating that the special Lagrangian torus fibration preserves computational complexity at both local and global levels, and finally proving that this preservation necessarily implies the geometric correspondences required by the SYZ conjecture. This approach not only resolves the conjecture but reveals deeper insights about the relationship between computation and geometry in string theory. We introduce new complexity-based invariants for studying mirror symmetry and demonstrate how our framework extends naturally to related geometric structures.
基金Supported by National Natural Science Foundation of China(Grant No.12371003)。
文摘Let A be a finite set of integers. For any integer h ≥ 2, let hA and h∧A be the sets of all sums of h elements of A and all sums of h distinct elements of A, respectively.In this paper, we survey some significant advances in additive combinatorics concerning the size and structure of sumsets of finite integer sets.
文摘This article delves Chern's conjecture for hypersurfaces with constant second fundamental form squared length S in the spherical space.At present,determining whether the third gap point of S is 2n remains unsolved yet.First,we investigate the height functions and their properties of the position vector and normal vector in natural coordinate vectors,and then prove the existence of a Simons-type integral formula on the hypersurface that simultaneously includes the first,second,and third gap point terms of S.These results can provide new avenues of thought and methods for solving Chern's conjecture.
