Related papers: Cousin's lemma in second-order arithmetic
We consider the problem of approximation of a continuous function $f$ defined on a compact metric space $X$ by elements from a sum of two algebras. We prove a de la Vall\'{e}e Poussin type theorem, which estimates the approximation error…
This paper uses the framework of reverse mathematics to investigate the strength of two recurrence theorems of topological dynamics. It establishes that one of these theorems, the existence of an almost periodic point, lies strictly between…
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…
Gauge symmetries play a fundamental role in Physics, as they provide a mathematical justification for the fundamental forces. Usually, one starts from a non-interactive theory which governs `matter', and features a global symmetry. One then…
We present several new examples of reflection principles which apply to both class groups of number fields and picard groups of of curves over $\mathbb{P}^{1}/\mathbb{F}_{p}$. This proves a conjecture of Lemmermeyer about equality of 2-rank…
By a classical theorem of Koksma the sequence of fractional parts $(\{x^n\})_{n \geq 1}$ is uniformly distributed for almost all values of $x$. In the present paper we obtain an exact quantitative version of Koksma's theorem, by calculating…
This paper presents a reformulation of the Leibniz product rule as a finite sum that expresses the fractional derivative of the product of two differentiable functions. This paper then proves the cases for when the product consists of an…
We construct a family of closeness functions on the space of finite volume Lorentzian geometries using the abundance of discrete intervals in the underlying random causal sets. Although strictly weaker than a Lorentzian Gromov-Hausdorff…
The unified approach of Feldman and Cousins allows for exact statistical inference of small signals that commonly arise in high energy physics. It has gained widespread use, for instance, in measurements of neutrino oscillation parameters…
The quantum adversary method is a versatile method for proving lower bounds on quantum algorithms. It yields tight bounds for many computational problems, is robust in having many equivalent formulations, and has natural connections to…
We develop and investigate a general theory of representations of second-order functionals, based on a notion of a right comodule for a monad on the category of containers. We show how the notion of comodule representability naturally…
Within the gossamer numbers which extend the real numbers to include infinitesimals and infinities we prove the Fundamental Theorem of Calculus (FTC). Riemann sums are also considered in the gossamer number system, and their non-uniqueness…
We show that there is a $\beta$-model of second-order arithmetic in which the choice scheme holds, but the dependent choice scheme fails for a $\Pi^1_2$-assertion, confirming a conjecture of Stephen Simpson. We obtain as a corollary that…
The paper gives a unified and simple proof of both theorems and Cousin's theorem.
There have been, over the last 8 years, a number of far reaching extensions of the famous original F. and M. Riesz's uniqueness theorem that states that if a bounded analytic function in the unit disc of the complex plane $\Bbb C$ has the…
The substitution lemma is a renowned theorem within the realm of lambda-calculus theory and concerns the interactional behaviour of the metasubstitution operation. In this work, we augment the lambda-calculus's grammar with an uninterpreted…
Our objective is to extend the standard results of preservation and reflection of properties by bisimulations to the coalgebraic setting, as well as to study under what conditions these results hold for simulations. The notion of…
In this paper we consider Kakutani's extension of the Brouwer fixed point theorem within the framework of Bishop's constructive mathematics. Kakutani's fixed point theorem is classically equivalent to Brouwer's fixed point theorem. The…
The system of undetermined coefficients of a bifurcation problem G[z]=0 in Banach spaces is investigated for proving the existence of families of solution curves by use of the implicit function theorem. The main theorem represents an…
For integral representations of associated Legendre functions in terms of modified Bessel functions, we establish justification for differentiation under the integral sign with respect to parameters. With this justification, derivatives for…