Related papers: The reverse mathematics of Cousin's lemma
Compactness is one of the core notions of analysis: it connects local properties to global ones and makes limits well-behaved. We study the computational properties of the compactness of Cantor space $2^{\mathbb{N}}$ for uncountable covers.…
The strong continuity principle reads "every pointwise continuous function from a complete separable metric space to a metric space is uniformly continuous near each compact image." We show that this principle is equivalent to the fan…
We show that for $\Pi_2$-properties of second or third order arithmetic as formalized in appropriate natural signatures the apparently weaker notion of forcibility overlaps with the standard notion of consistency (assuming large cardinal…
We give a short, self-contained, and elementary proof of the strong law of large numbers under a power law decay hypothesis for joint second moments. The result is related to the classical one by Lyons. However, we also provide a rate of…
A Lemma of Riemann--Lebesgue type for Fourier--Jacobi coefficients is derived. Via integral representations of Dirichlet--Mehler type for Jacobi polynomials its proof directly reduces to the classical Riemann--Lebesgue Lemma for Fourier…
Integration, just as much as differentiation, is a fundamental calculus tool that is widely used in many scientific domains. Formalizing the mathematical concept of integration and the associated results in a formal proof assistant helps in…
The Riemann-Lebesgue Lemma says that the Fourier transform of an absolutely integrable function on the real line tends to zero as the transform parameter tends to infinity. When the integral is allowed to converge conditionally, the…
Since being isolated by Viale and Weiss in 2009, the Guessing Model Property has emerged as a particularly prominent and powerful consequence of the Proper Forcing Axiom. In this paper, we investigate connections between variations of the…
This paper is an overview of results that show the Brouwer fixed-point theorem (BFPT) to be essentially non-constructive and non-computable. The main results, the counter-examples of Orevkov and Baigger, imply that there is no procedure for…
Reverse differentiation is an essential operation for automatic differentiation. Cartesian reverse differential categories axiomatize reverse differentiation in a categorical framework, where one of the primary axioms is the reverse chain…
In reverse mathematics, is is possible to have a curious situation where we know that an implication does not reverse, but appear to have no information on on how to weaken the assumption while preserving the conclusion. A main cause of…
This paper continues to study the connection between reverse mathematics and Weihrauch reducibility. In particular, we study the problems formed from Maltsev's theorem on the order types of countable ordered groups. Solomon showed that the…
The Riemann-Lebesque Theorem is commonly proved in a few strokes using the theory of Lebesque integration. Here, the upper bound $2\pi|c_k(f)|\le S_k(f)-s_k(f)$ for the Fourier coefficients $c_k$ is proved in terms of majoring and minoring…
We consider algebras of languages over the signature of reversible Kleene lattices, that is the regular operations (empty and unit languages, union, concatenation and Kleene star) together with intersection and mirror image. We provide a…
This paper establishes the equivalence of the Aubin property and the strong regularity for generalized equations over $C^2$-cone reducible sets. This result resolves a long-standing question in variational analysis and extends the…
The aim of this note is to present a self-contained proof of the fact that a function can be approximated using a linear combination of Gaussian coherent states, with a number of terms controlled in terms of the smoothness and of the decay…
Reflection principles (or dually speaking, compactness principles) often give rise to combinatorial guessing principles. Uniformization properties, on the other hand, are examples of anti-guessing principles. We discuss the tension and the…
Reverse mathematics studies which subsystems of second order arithmetic are equivalent to key theorems of ordinary, non-set-theoretic mathematics. The main philosophical application of reverse mathematics proposed thus far is foundational…
In the spirit of the famous KOML\'OS (1967) theorem, every sequence of nonnegative, measurable functions $\{ f_n \}_{n \in \N}$ on a probability space, contains a subsequence which - along with all its subsequences - converges a.e. in…
We construct a canonical pseudofunctor ^# on the category of finite-type maps of (say) connected noetherian universally catenary finite-dimensional separated schemes, taking values in the category of Cousin complexes. This pseudofunctor is…