Related papers: A note on exact approximations
We prove the theorems which are equivalent to the Roland's results such that a new form of them allows to consider some generalizations. In particular, we give generators of primes more than a fixed prime.
In this paper, we examine how well a rational point P on an algebraic variety X can be approximated by other rational points. We conjecture that if P lies on a rational curve, then the best approximations to P on X can be chosen to lie…
We introduce an asymmetric operator of generalised translation, define the generalised modulus of smoothness by its means, and obtain the direct and inverse theorems in approximation theory for it.
We show how rational function approximations to the logarithm, such as $\log z \approx (z^2 - 1)/(z^2 + 6z + 1)$, can be turned into fast algorithms for approximating the determinant of a very large matrix. We empirically demonstrate that…
We introduce a class of real algebraic varieties characterised by a simple rationality condition, which exhibit strong properties regarding approximation of continuous and smooth mappings by regular ones. They form a natural counterpart to…
We strengthen the Weierstrass approximation theorem by proving that any real-valued continuous function on an interval $I \subset \mathbb{R}$ can be uniformly approximated by a real-valued polynomial whose only (possibly complex) critical…
We call an $\alpha \in \mathbb{R}$ regainingly approximable if there exists a computable nondecreasing sequence $(a_n)_n$ of rational numbers converging to $\alpha$ with $\alpha - a_n < 2^{-n}$ for infinitely many $n \in \mathbb{N}$. We…
Inspired by computer assisted proofs in analysis, we present an interval approach to real-number computations.
We prove an easy statement about inhomogeneous approximation in metric theory of Diophantine Approximation.
Exact real computation is an alternative to floating-point arithmetic where operations on real numbers are performed exactly, without the introduction of rounding errors. When proving the correctness of an implementation, one can focus…
We prove a result on approximations to a real number $\theta$ by algebraic numbers of degree $\le 2$ in the case when we have information about the uniform Diophantine exponent $\hat{\omega}$ for the linear form $x_0 +\theta…
Let K be a totally real Galois number field. C. J. Hillar proved that if f in Q[x_1,...,x_n] is a sum of m squares in K[x_1,...,x_n], then f is a sum of N(m) squares in Q[x_1,...,x_n]. Modifying Hillar's proof, we improve the improve the…
We establish a real version of Turrittin's result on polynomial and formal normal forms of linear systems of ODEs with meromorphic coefficients. Both the normal forms or the transformations used have only real coefficients. In order to…
In this note we prove the optimality of a family of known coincidence theorems for absolutely summing multilinear operators. We connect our results with the theory of multiple summing multilinear operators and prove the sharpness of similar…
In this short note we prove a result that is an extension of an old Olympiad problem and is a very simple variant of the question of finding an approximation for $k$, where it is a nonzero constant and it satisfies the equation $a^k+b^k=c$,…
By counting with triangles and the octohedral axiom, we find a direct way to prove the formula of To\"en in \cite{Toen2005} for a triangulated category with (left) homological-finite condition.
Given a single (differential-algebraic) input-output equation, we present a method for finding different representations of the associated system in the form of rational realizations; these are dynamical systems with rational right-hand…
We establish the exact overlaps conjecture for iterated functions systems on the real line with algebraic contractions and arbitrary translations.
We introduce real induction, a proof technique analogous to mathematical induction but applicable to statements indexed by an interval on the real line. More generally we give an inductive principle applicable in any Dedekind complete…
A short proof of the Mazur-Ulam theorem concerning isometries of real normed spaces.