Related papers: Regainingly approximable numbers and sets
Literature considers under the name \emph{unimaginable numbers} any positive integer going beyond any physical application, with this being more of a vague description of what we are talking about rather than an actual mathematical…
Generalizations of linear numeration systems in which the set of natural numbers is recognizable by finite automata are obtained by describing an arbitrary infinite regular language following the lexicographic ordering. For these systems of…
We introduce and consider a certain probability question involving elementary number theory and the likelihood that a fixed prime will appear in a certain recursively defined factorization of an integer. We derive several convergent…
What sets A \subset Z^n can be written in the form (K-K) \cap Z^n, where K is a compact subset of R^n such that K+Z^n=R^n? Such sets A are called achievable, and it is known that if A is achievable, then < A >=Z^n. This condition completely…
It is shown that under standard hypotheses, if stochastic approximation iterates remain tight, they converge with probability one to what their o.d.e. limit suggests. A simple test for tightness (and therefore a.s. convergence) is provided.…
Numerical approximate computation can solve large and complex problems fast. It has the advantage of high efficiency. However it only gives approximate results, whereas we need exact results in many fields. There is a gap between…
In this paper, a sample-based procedure for obtaining simple and computable approximations of chance-constrained sets is proposed. The procedure allows to control the complexity of the approximating set, by defining families of…
Generalizations of numeration systems in which N is recognizable by a finite automaton are obtained by describing a lexicographically ordered infinite regular language L over a finite alphabet A. For these systems, we obtain a…
The notion of a k-automatic set of integers is well-studied. We develop a new notion - the k-automatic set of rational numbers - and prove basic properties of these sets, including closure properties and decidability.
We consider approximation or recovery of functions based on a finite number of function evaluations. This is a well-studied problem in optimal recovery, machine learning, and numerical analysis in general, but many fundamental insights were…
Many practical problems need the output of a machine learning model to satisfy a set of constraints, $K$. Nevertheless, there is no known guarantee that classical neural network architectures can exactly encode constraints while…
Proper continued fractions are generalized continued fractions with positive integer numerators $a_i$ and integer denominators with $b_i\geq a_i$. In this paper we study the strength of approximation of irrational numbers to their…
The kernel of analysis, to me anyway, is the following idea: A point is arbitrarily close to a set if every neighborhood of the point intersects the set. Defining ``arbitrarily close'' in this way provides a foundation for classical results…
We call a subset of an ordinal $\lambda$ recognizable if it is the unique subset $x$ of $\lambda$ for which some Turing machine with ordinal time and tape, which halts for all subsets of $\lambda$ as input, halts with the final state $0$.…
We develop a theory of real numbers as rational Cauchy sequences, in which any two of them, $(a_n)$ and $(b_n)$, are equal iff $\lim\,(a_n-b_n)=0$. We need such reals in the Countable Mathematical Analysis ([4]) which allows to use only…
Speedable numbers are real numbers which are algorithmically approximable from below and whose approximations can be accelerated nonuniformly. We begin this article by answering a question of Barmpalias by separating a strict subclass that…
For any prime number p and any positive real number {\alpha}, we construct a finitely generated group {\Gamma} with p-gradient equal to {\alpha}. This construction is used to show that there exist uncountably many pairwise non-commensurable…
A real X is defined to be relatively c.e. if there is a real Y such that X is c.e.(Y) and Y does not compute X. A real X is relatively simple and above if there is a real Y <_T X such that X is c.e.(Y) and there is no infinite subset Z of…
We seek random versions of some classical theorems on complex approximation by polynomials and rational functions, as well as investigate properties of random compact sets in connection to complex approximation.
In many instances in first order logic or computable algebra, classical theorems show that many problems are undecidable for general structures, but become decidable if some rigidity is imposed on the structure. For example, the set of…