Related papers: Cousin's lemma in second-order arithmetic
We prove a version of Gauss's Lemma. It recursively constructs polynomials {c_k} for k=0,1,...,m+n, in Z[a_i,A_i,b_j,B_j] for i=0,...,m, and j=0,1,...,n, having degree at most (m+n choose m) in each of the four variable sets, such that…
An open question in reverse mathematics is whether the cohesive principle, $\COH$, is implied by the stable form of Ramsey's theorem for pairs, $\SRT^2_2$, in $\omega$-models of $\RCA$. One typical way of establishing this implication would…
A theorem of Lusin states that every Borel function on $R$ is equal almost everywhere to the derivative of a continuous function. This result was later generalized to $R^n$ in works of Alberti and Moonens-Pfeffer. In this note, we prove…
We use the anti-equivalence between Cohen-Macaulay complexes and coherent sheaves on formal schemes to shed light on some older results and prove new results. We bring out the relations between a coherent sheaf M satisfying an S_2 condition…
In this paper we demonstrate that the class of basic feasible functionals has recursion theoretic properties which naturally generalize the corresponding properties of the class of feasible functions. We also improve the Kapron - Cook…
Let $R$ be an associative ring with unit $1$, and $a, b, c\in R$ satisfy $a(ba)^{2}=abaca=acaba=(ac)^{2}a$, this paper proves that $1-ac$ has generalized Drazin inverse (Drazin inverse, pseudo Drazin inverse, respectively) if and only if…
We present a constructive proof of Brouwer's fixed point theorem for uniformly continuous and sequentially locally non-constant functions based on the existence of approximate fixed points. And we will show that Brouwer's fixed point…
We study equilibrium statistical mechanics of classical point counter-ions, formulated on 2D Euclidean space with logarithmic Coulomb interactions (infinite number of particles) or on the cylinder surface (finite particle numbers), in the…
The notion of well order admits an alternative definition in terms of embeddings between initial segments. We use the framework of reverse mathematics to investigate the logical strength of this definition and its connection with…
Proof search has been used to specify a wide range of computation systems. In order to build a framework for reasoning about such specifications, we make use of a sequent calculus involving induction and co-induction. These proof principles…
On a suitable category of formal schemes equipped with codimension functions we construct a canonical pseudofunctor (-)^# taking values in the corresponding categories of Cousin complexes. Cousin complexes on such a formal scheme X…
We propose a new method for constructing Turing ideals satisfying principles of reverse mathematics below the Chain-Antichain Principle (CAC). Using this method, we are able to prove several new separations in the presence of Weak Konig's…
Let ${\mathcal U}(\lambda)$ denote the family of analytic functions $f(z)$, $f(0)=0=f'(0)-1$, in the unit disk $\ID$, which satisfy the condition $\big |\big (z/f(z)\big )^{2}f'(z)-1\big |<\lambda $ for some $0<\lambda \leq 1$. The…
Lebesgue integration is a well-known mathematical tool, used for instance in probability theory, real analysis, and numerical mathematics. Thus its formalization in a proof assistant is to be designed to fit different goals and projects.…
Gowers norms have been studied extensively both in the direct sense, starting with a function and understanding the associated norm, and in the inverse sense, starting with the norm and deducing properties of the function. Instead of…
We introduce and study a new type of compactness principle for strong logics that, roughly speaking, infers the consistency of a theory from the consistency of its small fragments in certain outer models of the set-theoretic universe. We…
We consider fragments of uniform reflection for formulas in the analytic hierarchy over theories of second order arithmetic. The main result is that for any second order arithmetic theory $T_0$ extending ${\sf RCA}_0$ and axiomatizable by a…
The Bernstein approximation problem is to determine whether or not the space of all polynomials is dense in a given weighted $C_0$-space on the real line. A theorem of L. de Branges characterizes non--density by existence of an entire…
An elementary application of Fatou's lemma gives a strengthened version of the monotone convergence theorem. We call this the convergence from below theorem. We make the case that this result should be better known, and deserves a place in…
We propose a conceptually economical and computationally tractable completion of the foundations of gauge theory on quantum principal bundles \`{a} la Brzezi\'{n}ski--Majid to the case of general differential calculi and strong bimodule…