Numerical Approximation of Real Functions and One Minkowski's Conjecture on Diophintine Approximations
Nikolaj M. Glazunov
Abstract
In this paper I consider the applications of several kinds of approximations of real functions to the problem of verified computation (reliable computing) of the range of implicitly defined real function xn+1 = G(x1, ..., xn), where dependency F(x1, ..., xn+1) = 0 is defined on some compact domain by a sufficiently smooth real function F(x1, ..., xn+1) >. Constructive version of Kolmogorov-Arnold and implicit function theorems, results about floating-point approximation, floating-point approximations which give lower-bound and upper-bound estimates of some real functions, and approximate algebraic computation are used for the purpose. The rigorous theory can be build on the base of analysis on manifolds over floating points domains. In the text we demonstrate our approach on examples from investigation of Minkowski's conjecture on critical determinant of the region x p + y p ≤ 1, p > 1..
Create a lesson
Related papers
Sensitivity Calculus and its Numerical Implementation for Multi-D Hyperbolic Balance Laws
Olivia Dreßen, Michael Herty, Adrian Kolb et al.
A Unified Framework for Wasserstein Convergence of ULMC Methods beyond Log-Concavity: Old and New
Wanjie Lyu, Xiaojie Wang, Bin Yang
Beyond PINNs: A Unified Gauss--Newton and Petrov--Galerkin Framework for Neural and Hybrid PDE Solvers
Nilo Schwencke, Roland Maier
Incremental Column Subset Selection via Conditional Determinantal Point Processes
Laura Grigori, Zhipeng Xue
A Hybrid High-Order Method for the Elasticity Problem with Linear Slip Interface
Erik Burman, Peiqi Huang
Adaptive Sparse-grid Discontinuous Galerkin Approximations the Bhatnagar--Gross--Krook Model
Stefan Schnake, Miroslav Stoyanov, Eirik Endeve et al.