Related papers: Extensional constructive real analysis via locator…
Usually, for extension of local maps, one uses multiplication by so called bump functions. However, majority of infinite-dimensional linear topological spaces do not have smooth bump functions. Therefore, in \cite{BR} we suggested a new…
Real numbers in constructive mathematics have always seemed to require compromises of one form or another. Classical proofs of Cauchy completeness require countable choice, Bishop's setoid construction introduces persistent bookkeeping…
We consider the problem of the observability of positively expansive maps by the time series associated to continuous real functions. For this purpose we prove a general result on the generic observability of a locally injective map of a…
We study a new flexible method to extend linearly the graph of a non-linear, and usually not bijective, function so that the resulting extension is a bijection. Our motivation comes from cryptography. Examples from symmetric cryptography…
Cantor's diagonal method is traditionally used to prove the uncountability of the set of all infinite binary sequences. This paper analyzes the expressive limits of this method. It is shown that under any constructive application --…
Almost all representations considered in computable analysis are partial. We provide arguments in favor of total representations (by elements of the Baire space). Total representations make the well known analogy between numberings and…
This note describes a representation of the real numbers due to Schanuel. The representation lets us construct the real numbers from first principles. Like the well-known construction of the real numbers using Dedekind cuts, the idea is…
We describe the supports of a class of real-valued maps on $C*(X)$ introduced by Radul. Using this description, a characterization of compact-valued retracts of a given space in terms of functional extenders is obtained. For example, if…
Computational interpretations of linear logic allow static control of memory resources: the data produced by the program are endowed through its type with attributes that determine its life cycle, and guarantee safe deallocation. The use of…
In this paper we provide a complete approach to the real numbers via decimal representations. Construction of the real numbers by Dedekind cuts, Cauchy sequences of rational numbers, and the algebraic characterization of the real number…
A countable structure is said to be extendible if it has the same Scott sentence as some uncountable structure. Rigid structures are not extendible. We give an example of an extendible model with a rigid elementary extension.
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…
We provide a numerical method to determine the critical lengths of linear differential operators with constant real coefficients. The need for such a procedure arises when the orders increase. The interest of this article is clearly on the…
We study the properties of the constructive linear programing problems. The parameters of linear functions in such problems are constructive real numbers. To solve such a problem is to find the optimal plan with the constructive real number…
Suppose $p \geq 1$ is a computable real. We extend previous work of Clanin, Stull, and McNicholl by classifying the computable $L^p$ spaces whose underlying measure spaces are atomic but not purely atomic. In addition, we determine the…
In this article, we consider a simple representation for real numbers and propose top-down procedures to approximate various algebraic and transcendental operations with arbitrary precision. Detailed algorithms and proofs are provided to…
Nonlinear causal discovery from observational data imposes strict identifiability assumptions on the formulation of structural equations utilized in the data generating process. The evaluation of structure learning methods under assumption…
A number of constructions in function field arithmetic involve extensions from linear objects using digit expansions. This technique is described here as a method of constructing orthonormal bases in spaces of continuous functions. We…
We consider existential problems over the reals. Extended quantifier elimination generalizes the concept of regular quantifier elimination by providing in addition answers, which are descriptions of possible assignments for the quantified…
Despite the success of Transformers on language understanding, code generation, and logical reasoning, they still fail to generalize over length on basic arithmetic tasks such as addition and multiplication. A major reason behind this…