Related papers: On relative computability for curves

There is a one-sided central limit theorem for the logarithms of $L$-derivatives of a fixed rational non-CM elliptic curve $E$ over imaginary quadratic fields of rank one, analogous to a result of Radziwi\l\l\ and Soundararajan. There are…

By finding all integral points on certain elliptic and hyperelliptic curves we completely solve the Diophantine equation $\binom{n}{k}=\binom{m}{l}+d$ for $-3\leq d\leq 3$ and $(k,l)\in\{(2,3),\; (2,4),\;(2,5),\; (2,6),\; (2,8),\; (3,4),\;…

The goal of this paper consists of developing a new (more physical and numerical in comparison with standard and non-standard analysis approaches) point of view on Calculus with functions assuming infinite and infinitesimal values. It uses…

In this paper, elliptic curves theory is used for solving the Diophantine equations X^3+Y^3+Z^3+aU^k=a_0U_0^{t_0}+...+a_nU_n^{t_n}, k=3,4 where n, ti are natural numbers and a, a_i are fixed arbitrary rational numbers. We try to transform…

Dummett's argument for intuitionism is well known. There is a concern that the argument proves too much, specifically, that it supports the extreme and apparently incoherent position of strict finitism. The central question is how to…

The purpose of this note is to provide a transparent and unified retelling of both Skvortsov's proof of the structural completeness of Medvedev's logic of finite problems, which is a classical result originally due to Prucnal, and of…

We consider the problem of approximating the reachability probabilities in Markov decision processes (MDP) with uncountable (continuous) state and action spaces. While there are algorithms that, for special classes of such MDP, provide a…

Inverse limits, unlike direct limits, can in general be void, [1]. The existence of fixed points for arbitrary mappings $T : X \longrightarrow X$ is conjectured to be equivalent with the fact that related direct limits of all finite…

We prove the universality of the regular realizability problems for several classes of filters. The filters are encodings of finite relations on the set of non-negative integers in the format proposed by P. Wolf and H. Fernau. The…

Let $X,Y$ be two irreducible subvarieties of the projective space $\mathbb{P}^n$, and $d\geq 1$ an integer number. The main result of this paper is an algorithm to construct {\bf explicitly}, in terms of $d$ and the ideals defining $X$ and…

Are minds subject to laws of physics? Are the laws of physics computable? Are conscious thought processes computable? Currently there is little agreement as to what are the right answers to these questions. Penrose goes one step further and…

We consider the problem of characterizing solutions in $(x, y)$ to the equation ${x \choose y}={{x-a} \choose {y+b}}$ in terms of $a$ and $b$. We obtain one simple result which allows the determination of a ratio in terms of $a$ and $b$…

The computational complexity of reasoning within the Dempster-Shafer theory of evidence is one of the main points of criticism this formalism has to face. To overcome this difficulty various approximation algorithms have been suggested that…

For a tuple $(\theta_1,..,\theta_M)$ of complex number, buliding on the approximation techniques in earlier papers of this series, this paper engages in deducing lower estimates on the transcendence degree of the field generated by…

Generalizing an argument of Matiyasevich, we illustrate a method to generate infinitely many diophantine equations whose solutions can be completely described by linear recurrences. In particular, we provide an integer-coefficient…

This article presents a general solution to the problem of computational complexity. First, it gives a historical introduction to the problem since the revival of the foundational problems of mathematics at the end of the 19th century.…

Beginning with Turing's seminal work in 1950, artificial intelligence proposes that consciousness can be simulated by a Turing machine. This implies a potential theory of everything where the universe is a simulation on a computer, which…

We overview a series of recent works addressing numerical simulations of partial differential equations in the presence of some elements of randomness. The specific equations manipulated are linear elliptic, and arise in the context of…

We study the finiteness of low degree points on certain modular curves and their Atkin--Lehner quotients, and, as an application, prove the modularity of elliptic curves over all but finitely many totally real fields of degree $5$. On the…

The higher-dimensional version of Kannan and Lipton's Orbit Problem asks whether it is decidable if a target subspace can be reached from a starting point under repeated application of a linear transformation. Similarly, the continuous…