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
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
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
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
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.
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.
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.
