Calculating the inverse of an automorphism of a formal power series ring presents a frequent challenge in a myriad of mathematical inquiries, especially in the realm of singularity theory. In instances involving non-linear and multivariable contexts, S. S. Abhyankar pioneered a methodology to tackle this problem. However, calculating the expressions up to a certain order using this method requires calculating higher-order terms and then carry out the selection, which leads to redundant computations in practice. This article introduces two novel approaches for determining the inverse of an automorphism of a formal power series ring over an arbitrary commutative ring with unit, grounded in the newly developed higher order Jacobian matrix theory. These approaches can be conceived as non-linear extensions of the inverse matrix method and the Gaussian elimination method respectively. They avoid redundant computations above. For the two new methods, we also give the application in calculating the explicit expression for the implicit function theorem.
We present two characterizations of smooth compact Ricci flow solutions solely in terms of metrics and measures (one of them only works under positive scalar curvature along the flow); thus, provide weak formulations that are generalized to the singular setting in a straightforward manner. These formulations are achieved by weakly formulating super Ricci flows and imposing a saturation condition (solely in terms of metric and measure) to ensure the super Ricci flow inequality is an equality.
We prove the local Lipschitz continuity of viscosity solutions for two-phase free boundary problems for the p-Laplacian with nonzero right hand side, where p is an element of (1, infinity). This is the optimal regularity for the problem. We also obtain the local Ho & uml;lder continuity for a larger class of problems. The results introduced here are new even in the homogeneous situation, that is, when the right hand side is zero. Our work applies to merely viscosity solutions, which allows a wide applicability.
