Related papers: Defining Real Numbers as Oracles
Let a quantified inequality constraint over the reals be a formula in the first-order predicate language over the structure of the real numbers, where the allowed predicate symbols are $\leq$ and $<$. Solving such constraints is an…
Let $R$ be an order in an algebraic number field. If $R$ is a principal order, then many explicit results on its arithmetic are available. Among others, $R$ is half-factorial if and only if the class group of $R$ has at most two elements.…
The method of brackets is an efficient method for the evaluation of a large class of definite integrals on the half-line. It is based on a small collection of rules, some of which are heuristic. The extension discussed here is based on the…
An interval matrix is a matrix whose entries are intervals in the set of real numbers. Let $p , q $ be nonzero natural numbers and let $\mu =( [m_{i,j}, M_{i,j}])_{i,j}$ be a $p \times q$ interval matrix; given a $p \times q$ matrix $A$…
A dual approach to defining the triangle sequence (a type of multidimensional continued fraction algorithm, initially developed in NT/9906016) for a pair of real numbers is presented, providing a new, clean geometric interpretation of the…
This survey article is the outgrowth of two talks given at the Journ\'ees X-UPS "P\'eriodes et transcendance" at \'Ecole polytechnique. Periods are complex numbers whose real and imaginary parts can be written as integrals of rational…
We tackle the problem of estimating a regression function observed in an instrumental regression framework. This model is an inverse problem with unknown operator. We provide a spectral cut-off estimation procedure which enables to derive…
The article is devoted to the investigation of representation of rational numbers by Cantor series. Necessary and sufficient conditions for a rational number to be representable by a positive Cantor series are formulated for the case of an…
In this work we study the space complexity of computable real numbers represented by fast convergent Cauchy sequences. We show the existence of families of trascendental numbers which are logspace computable, as opposed to algebraic…
The concept of nearest integer is used to derive theorems and algorithms for the best approximations of an irrational by rational numbers, which are improved with the pigeonhole principle and used to offer an informed presentation of the…
We suggest a new optical solution for solving the YES/NO version of the Exact Cover problem by using the massive parallelism of light. The idea is to build an optical device which can generate all possible solutions of the problem and then…
Algebraic specifications of data types provide a natural basis for testing data types implementations. In this framework, the conformance relation is based on the satisfaction of axioms. This makes it possible to formally state the…
The differences between the sets in which ideal arithmetics takes place and the sets of floating point numbers are outlined. A set of classical problems in correct numerical evaluation is presented, to increase the awareness of newcomers to…
Let $n$ be a positive integer, and let $R$ be a (possibly infinite dimensional) finitely presented algebra over a computable field of characteristic zero. We describe an algorithm for deciding (in principle) whether $R$ has at most finitely…
In physics, Lie groups represent the algebraic structure that describes symmetry transformations of a given system. Then, the descending Lie algebra of those groups are necessarily real. In most cases, the complexification of those Lie…
We show how to extract a monotonic learning algorithm from a classical proof of a geometric statement by interpreting the proof by means of interactive realizability, a realizability sematics for classical logic. The statement is about the…
A group of matrices $G$ with entries in a number field $K$ is defined to be numerical if $G$ has a finite index subgroup of matrices whose entries are algebraic integers. It is shown that an irreducible or completely reducible subgroup of…
A typical oracle problem is finding which software program is installed on a computer, by running the computer and testing its input-output behaviour. The program is randomly chosen from a set of programs known to the problem solver. As…
In this article we will discuss a few aspects of the spacetime description of matter and fields. In Section:1 we will discuss the completeness of real numbers in the context of an alternate definition of the straight line as a geometric…
Interval arithmetic libraries provide the four elementary arithmetic operators for operand intervals bounded by floating-point numbers. Actual implementations need to make a large case analysis that considers, e.g., magnitude relations…