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
Riemann (1859) had proved four theorems: analytic continuation ζ(s), functional equation ξ(z)=G(s)ζ(s)(s=1/2+iz, z=t−i(σ−1/2)), product expression ξ1(z)and Riemann-Siegel formula Z(z), and proposed Riemann conjec...Riemann (1859) had proved four theorems: analytic continuation ζ(s), functional equation ξ(z)=G(s)ζ(s)(s=1/2+iz, z=t−i(σ−1/2)), product expression ξ1(z)and Riemann-Siegel formula Z(z), and proposed Riemann conjecture (RC): All roots of ξ(z)are real. We have calculated ξand ζ, and found that ξ(z)is alternative oscillation, which intuitively implies RC, and the property of ζ(s)is not good. Therefore Riemann’s direction is correct, but he used the same notation ξ(t)=ξ1(t)to confuse two concepts. So the product expression only can be used in contraction. We find that if ξhas complex roots, then its structure is destroyed, so RC holds. In our proof, using Riemann’s four theorems is sufficient, needn’t cite other results. Hilbert (1900) proposed Riemann hypothesis (RH): The non-trivial roots of ζhave real part 1/2. Of course, RH also holds, but can not be proved directly by ζ(s).展开更多
The Berry-Tabor(BT)conjecture is a famous statistical inference in quantum chaos,which not only establishes the spectral fluctuations of quantum systems whose classical counterparts are integrable but can also be used...The Berry-Tabor(BT)conjecture is a famous statistical inference in quantum chaos,which not only establishes the spectral fluctuations of quantum systems whose classical counterparts are integrable but can also be used to describe other wave phenomena.In this paper,the BT conjecture has been extended to Lévy plates.As predicted by the BT conjecture,level clustering is present in the spectra of Lévy plates.The consequence of level clustering is studied by introducing the distribution of nearest neighbor frequency level spacing ratios P(r),which is calculated through the analytical solution obtained by the Hamiltonian approach.Our work investigates the impact of varying foundation parameters,rotary inertia,and boundary conditions on the frequency spectra,and we find that P(r)conforms to a Poisson distribution in all cases.The reason for the occurrence of the Poisson distribution in the Lévy plates is the independence between modal frequencies,which can be understood through mode functions.展开更多
This study introduces the representation of natural number sets as row vectors and pretends to offer a new perspective on the strong Goldbach conjecture. The natural numbers are restructured and expanded with the incl...This study introduces the representation of natural number sets as row vectors and pretends to offer a new perspective on the strong Goldbach conjecture. The natural numbers are restructured and expanded with the inclusion of the zero element as the source of a strong Goldbach conjecture reformulation. A prime Boolean vector is defined, pinpointing the positions of prime numbers within the odd number sequence. The natural unit primality is discussed in this context and transformed into a source of quantum-like indetermination. This approach allows for rephrasing the strong Goldbach conjecture, framed within a Boolean scalar product between the prime Boolean vector and its reverse. Throughout the discussion, other intriguing topics emerge and are thoroughly analyzed. A final description of two empirical algorithms is provided to prove the strong Goldbach conjecture.展开更多
An edge coloring of hypergraph H is a function such that holds for any pair of intersecting edges . The minimum number of colors in edge colorings of H is called the chromatic index of H and is ...An edge coloring of hypergraph H is a function such that holds for any pair of intersecting edges . The minimum number of colors in edge colorings of H is called the chromatic index of H and is denoted by . Erdös, Faber and Lovász proposed a famous conjecture that holds for any loopless linear hypergraph H with n vertices. In this paper, we show that is true for gap-restricted hypergraphs. Our result extends a result of Alesandroni in 2021.展开更多
In this article, I consider the right triangle as the simplex in the Euclidean plane, and extend this definition to higher dimensions. The n-dimensional simplex has one hypotenuse and (n−1)legs (catheti). The (n−1)leg...In this article, I consider the right triangle as the simplex in the Euclidean plane, and extend this definition to higher dimensions. The n-dimensional simplex has one hypotenuse and (n−1)legs (catheti). The (n−1)legs define an orthogonal path of edges in the solid with perpendicular adjacent edges along the path. The length of the hypotenuse and the volume of the solid can be calculated without the Cayley-Menger determinant, by direct extension of the corresponding right triangle formulas. I give a proof of the existence of these shapes, describe the distribution of right angles in them, give an algebraic proof of the Coxeter trisection of a right tetrahedron into three smaller right tetrahedra, and generalize this construction to n-dimensional spaces. Finally, I investigate the connection between the Coxeter partition and the Hadwiger conjecture on the partition of the simplex into orthoschemes, which I call Pythagorean simplexes.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
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
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
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Riemann (1859) had proved four theorems: analytic continuation ζ(s), functional equation ξ(z)=G(s)ζ(s)(s=1/2+iz, z=t−i(σ−1/2)), product expression ξ1(z)and Riemann-Siegel formula Z(z), and proposed Riemann conjecture (RC): All roots of ξ(z)are real. We have calculated ξand ζ, and found that ξ(z)is alternative oscillation, which intuitively implies RC, and the property of ζ(s)is not good. Therefore Riemann’s direction is correct, but he used the same notation ξ(t)=ξ1(t)to confuse two concepts. So the product expression only can be used in contraction. We find that if ξhas complex roots, then its structure is destroyed, so RC holds. In our proof, using Riemann’s four theorems is sufficient, needn’t cite other results. Hilbert (1900) proposed Riemann hypothesis (RH): The non-trivial roots of ζhave real part 1/2. Of course, RH also holds, but can not be proved directly by ζ(s).
基金supported by the National Natural Science Foundation of China(Grant Nos.12261064 and 11861048)the Natural Science Foundation of Inner Mongolia,China(Grant No.2021MS01004)the Innovation Program for Graduate Education of Inner Mongolia University(Grant No.11200-5223737).
文摘The Berry-Tabor(BT)conjecture is a famous statistical inference in quantum chaos,which not only establishes the spectral fluctuations of quantum systems whose classical counterparts are integrable but can also be used to describe other wave phenomena.In this paper,the BT conjecture has been extended to Lévy plates.As predicted by the BT conjecture,level clustering is present in the spectra of Lévy plates.The consequence of level clustering is studied by introducing the distribution of nearest neighbor frequency level spacing ratios P(r),which is calculated through the analytical solution obtained by the Hamiltonian approach.Our work investigates the impact of varying foundation parameters,rotary inertia,and boundary conditions on the frequency spectra,and we find that P(r)conforms to a Poisson distribution in all cases.The reason for the occurrence of the Poisson distribution in the Lévy plates is the independence between modal frequencies,which can be understood through mode functions.
文摘This study introduces the representation of natural number sets as row vectors and pretends to offer a new perspective on the strong Goldbach conjecture. The natural numbers are restructured and expanded with the inclusion of the zero element as the source of a strong Goldbach conjecture reformulation. A prime Boolean vector is defined, pinpointing the positions of prime numbers within the odd number sequence. The natural unit primality is discussed in this context and transformed into a source of quantum-like indetermination. This approach allows for rephrasing the strong Goldbach conjecture, framed within a Boolean scalar product between the prime Boolean vector and its reverse. Throughout the discussion, other intriguing topics emerge and are thoroughly analyzed. A final description of two empirical algorithms is provided to prove the strong Goldbach conjecture.
文摘An edge coloring of hypergraph H is a function such that holds for any pair of intersecting edges . The minimum number of colors in edge colorings of H is called the chromatic index of H and is denoted by . Erdös, Faber and Lovász proposed a famous conjecture that holds for any loopless linear hypergraph H with n vertices. In this paper, we show that is true for gap-restricted hypergraphs. Our result extends a result of Alesandroni in 2021.
文摘In this article, I consider the right triangle as the simplex in the Euclidean plane, and extend this definition to higher dimensions. The n-dimensional simplex has one hypotenuse and (n−1)legs (catheti). The (n−1)legs define an orthogonal path of edges in the solid with perpendicular adjacent edges along the path. The length of the hypotenuse and the volume of the solid can be calculated without the Cayley-Menger determinant, by direct extension of the corresponding right triangle formulas. I give a proof of the existence of these shapes, describe the distribution of right angles in them, give an algebraic proof of the Coxeter trisection of a right tetrahedron into three smaller right tetrahedra, and generalize this construction to n-dimensional spaces. Finally, I investigate the connection between the Coxeter partition and the Hadwiger conjecture on the partition of the simplex into orthoschemes, which I call Pythagorean simplexes.
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.
