论文标题
超出纤维方程溶液的共形平均值定理
A Conformal Mean Value Theorem for Solutions of the Ultrahyperbolic Equation
论文作者
论文摘要
Asgeirsson的定理建立了超出纤维方程解决方案解决方案的平均值属性。在四个变量的情况下，它指出，在某些成对的共轭圆圈上的解决方案的积分是相同的。在本文中，使用签名2+2的伪 - 欧几里得空间的共形图中的四个维超纤维方程的不变性用于获得平均值定理的最通用版本。 名称非脱位共轭圆锥用于最通用的曲线对，在该曲线上，超出纤维方程的解决方案享有平均值属性。事实证明，这些是圆锥形部分对，因此除了已知存在的共轭圆外，图片还可以通过在共轭体重，结合抛物线和线路上找到平均值定理和线条。 此外，弗里茨·约翰（Fritz John）建立了共轭圈子与革命倍增的两个裁决之间的联系。双线统治表面的线发病率特性用于证明非脱位的共轭锥与欧几里得三个空间中偶有统治表面的裁决之间的一对一对应关系。
Asgeirsson's theorem establishes a mean value property for solutions of the ultrahyperbolic equation. In the case of four variables, it states that the integrals of a solution over certain pairs of conjugate circles are the same. In this paper, the invariance of the four dimensional ultrahyperbolic equation under conformal maps of the pseudo-Euclidean space of signature 2+2 is used to get the most general version of the mean value theorem. The name non-degenerate conjugate conics is used for the most general pairs of curves over which solutions of the ultrahyperbolic equation enjoy the mean value property. These are proven to be pairs of conic sections, so that, in addition to conjugate circles which were known to exist, the picture is completed by finding mean value theorems over conjugate hyperbolae, conjugate parabolae, and line-empty pairs. In addition, Fritz John established a link between conjugate circles and the two rulings of a hyperboloid of revolution. The line incidence property of doubly ruled surfaces is used to prove a one-to-one correspondence between non-degenerate conjugate conics and pairs of rulings of doubly ruled surfaces in Euclidean 3-space.
