![](https://crypto4nerd.com/wp-content/uploads/2024/04/1712778899_15EyQc-m3dOkCJyMbdLfL2w.jpeg)
- An Approximation of Local Antiderivatives of Relative Differential on Arithmetic Surface(arXiv)
Author : Yuhan Zha
Abstract : Let ω be a relative differential on aithmetic surface X. We construct a family of rational functions Gx on X⊗C, which can approximate local antiderivatives of ω over an open set on X⊗C. From this family of functions, we construct a rational function G2 on X. The function G2 can generate an element in the ring of integers of a number field, which can approximate an inner product produced by ω and the conjugate of ω over an open set on X⊗C. This will give a relation between the height of a rational curve EP on X and the canonical norm of ω on X⊗C. This relation will give an upper bound for the height of EP under a few assumptions
2. Abstract convex optimal antiderivatives(arXiv)
Author : Sedi Bartz, Simeon Reich
Abstract : Having studied families of antiderivatives and their envelopes in the setting of classical convex analysis, we now extend and apply these notions and results in settings of abstract convex analysis. Given partial data regarding a c-subdifferential, we consider the set of all c-convex c-antiderivatives that comply with the given data. Under a certain assumption, this set is not empty and contains both its lower and upper envelopes. We represent these optimal antiderivatives by explicit formulae. Some well known functions are, in fact, optimal c-convex c-antiderivatives. In one application, we point out a natural minimality property of the Fitzpatrick function of a c-monotone mapping, namely that it is a minimal antiderivative. In another application, in metric spaces, a constrained Lipschitz extension problem fits naturally the convexity notions we discuss here. It turns out that the optimal Lipschitz extensions are precisely the optimal antiderivatives. This approach yields explicit formulae for these extensions, the most particular case of which recovers the well known extensions due to McShane and Whitney