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Keywords = Hilbert’s tenth problem

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12 pages, 271 KiB  
Article
On Productiveness and Complexity in Computable Analysis Through Rice-Style Theorems for Real Functions
by Jingnan Xie, Harry B. Hunt and Richard E. Stearns
Mathematics 2024, 12(20), 3248; https://doi.org/10.3390/math12203248 - 17 Oct 2024
Cited by 1 | Viewed by 984
Abstract
This paper investigates the complexity of real functions through proof techniques inspired by formal language theory. Productiveness, which is a stronger form of non-recursive enumerability, is employed to analyze the complexity of various problems related to real functions. Our work provides a deep [...] Read more.
This paper investigates the complexity of real functions through proof techniques inspired by formal language theory. Productiveness, which is a stronger form of non-recursive enumerability, is employed to analyze the complexity of various problems related to real functions. Our work provides a deep reexamination of Hilbert’s tenth problem and the equivalence to the identically 0 function problem, extending the undecidability results of these problems into the realm of productiveness. Additionally, we study the complexity of the equivalence to the identically 0 function problem over different domains. We then construct highly efficient many-one reductions to establish Rice-style theorems for the study of real functions. Specifically, we show that many predicates, including those related to continuity, differentiability, uniform continuity, right and left differentiability, semi-differentiability, and continuous differentiability, are as hard as the equivalence to the identically 0 function problem. Due to their high efficiency, these reductions preserve nearly any level of complexity, allowing us to address both complexity and productiveness results simultaneously. By demonstrating these results, which highlight a more nuanced and potentially more intriguing aspect of real function theory, we provide new insights into how various properties of real functions can be analyzed. Full article
(This article belongs to the Section E1: Mathematics and Computer Science)
9 pages, 241 KiB  
Article
New Numerical Method for Solving Tenth Order Boundary Value Problems
by Ali Akgül, Esra Karatas Akgül, Dumitru Baleanu and Mustafa Inc
Mathematics 2018, 6(11), 245; https://doi.org/10.3390/math6110245 - 8 Nov 2018
Cited by 6 | Viewed by 3331
Abstract
In this paper, we implement reproducing kernel Hilbert space method to tenth order boundary value problems. These problems are important for mathematicians. Different techniques were applied to get approximate solutions of such problems. We obtain some useful reproducing kernel functions to get approximate [...] Read more.
In this paper, we implement reproducing kernel Hilbert space method to tenth order boundary value problems. These problems are important for mathematicians. Different techniques were applied to get approximate solutions of such problems. We obtain some useful reproducing kernel functions to get approximate solutions. We obtain very efficient results by this method. We show our numerical results by tables. Full article
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