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Barnette猜想指出,每个三次、二分、平面且3-连通的图都是哈密顿图。 Goodey验证了Barnette猜想,对于所有面大小不超过6的图。 我们通过证明三次、二分、平面且(2-)连通的图在面大小不超过8的情况下具有哈密顿性,大大加强了Goodey的结果。 证明的部分是计算性的,包括区分339,068,624种情况。
Barnette's conjecture states that every cubic, bipartite, planar and 3-connected graph is Hamiltonian. Goodey verified Barnette's conjecture for all graphs with faces of size up to 6. We substantially strengthen Goodey's result by proving Hamiltonicity for cubic, bipartite, planar and (2-)connected graphs with faces of size up to 8. Parts of the proof are computational, including a distinction of 339.068.624 cases.
使用局部化技术,我们证明了在本质上不分支的$\mathsf{CD}^{\star}(K,N)$空间中的度量球的第一个狄利克雷特征值的精确上界。 这将 Cheng 的一个著名结果扩展到了通过最优传输满足合成意义下里奇曲率下界的度量测度空间的非光滑情形。 提供了$\mathsf{RCD}^{\star}(K,N)$空间的刚性与稳定性陈述;即使对于光滑黎曼流形,这种稳定性似乎也是新的。 然后我们提出了一些数学和物理应用:在前者中,我们得到了本质上不分支的$\mathsf{CD}^{\star}(K,N)$空间中$j^{th}$拉普拉斯特征值的上界以及非紧致$\mathsf{RCD}^{\star}(K,N)$空间中本质谱的界限;在后者中,特征值界限对应于高维引力理论的一般扭曲紧化周围自旋-2 卡鲁扎-克莱因激发态质量的一般上界。
Using the localization technique, we prove a sharp upper bound on the first Dirichlet eigenvalue of metric balls in essentially non-branching $\mathsf{CD}^{\star}(K,N)$ spaces. This extends a celebrated result of Cheng to the non-smooth setting of metric measure spaces satisfying Ricci curvature lower bounds in a synthetic sense, via optimal transport. Rigidity and stability statements are provided for $\mathsf{RCD}^{\star}(K,N)$ spaces; the stability seems to be new even for smooth Riemannian manifolds. We then present some mathematical and physical applications: in the former, we obtain an upper bound on the $j^{th}$ Neumann eigenvalue in essentially non-branching $\mathsf{CD}^{\star}(K,N)$ spaces and a bound on the essential spectrum in non-compact $\mathsf{RCD}^{\star}(K,N)$ spaces; in the latter, the eigenvalue bounds correspond to general upper bounds on the masses of the spin-2 Kaluza-Klein excitations around general warped compactifications of higher-dimensional theories of gravity.
我们一直对图以及将它们作为显式实代数函数的Reeb图感兴趣。 可微函数的Reeb图是定义域流形的商空间,被视为所有单点的原像的所有连通分支组成的空间。 自20世纪上半叶Morse函数理论诞生以来,Reeb图一直是流形几何中的基本而强大的工具。 我们可以很容易地看出,维度至少为$2$的单位球面的自然高度的Reeb图是一个恰好有一条边和两个顶点的图。 我们关注的是可以被良好分解为树的图的实现,每个顶点对应一个恰好有一条边和两个顶点的图或者一个恰好有两条边且与圆同胚的图。
We have been interested in graphs and realizing them as Reeb graphs of explicit real algebraic functions. The Reeb graph of a differentiable function is the quotient space of the manifold of the domain, regarded as the space consisting of all components of preimages of all single points. Reeb graphs have been fundamental and strong tools in geometry of manifolds since the birth of theory of Morse functions, in the former half of the 20th century. We can easily see that the Reeb graph of the natural height of the unit sphere whose dimension is at least $2$ is a graph with exactly one edge and two edges. We are concerned with realizations of graphs decomposed into trees nicely, each vertex of which corresponds to a graph with exactly one edge and two edges or a graph with exactly two edges homeomorphic to a circle.