Related papers: How strong are single fixed points of normal funct…

We extend the well-known Shannon decomposition of Boolean functions to more general classes of functions. Such decompositions, which we call pivotal decompositions, express the fact that every unary section of a function only depends upon…

Approximate $p$-point Leibniz derivation formulas as well as interpolatory Simpson quadrature sums adapted to oscillatory functions are discussed. Both theoretical considerations and numerical evidence concerning the dependence of the…

We characterize rotation equivariant bounded linear operators from $C(\mathbb{S}^{n-1})$ to $C^2(\mathbb{S}^{n-1})$ by the mass distribution of the spherical Laplacian of their kernel function on small polar caps. Using this…

Motivated by the problem of the dynamics of point-particles in high post-Newtonian (e.g. 3PN) approximations of general relativity, we consider a certain class of functions which are smooth except at some isolated points around which they…

Boyer and Moore have discussed a recursive function that puts conditional expressions into normal form [1]. It is difficult to prove that this function terminates on all inputs. Three termination proofs are compared: (1) using a measure…

We prove a structure theorem for stable functions on amenable groups, which extends the arithmetic regularity lemma for stable subsets of finite groups. Given a group $G$, a function $f\colon G\to [-1,1]$ is called stable if the binary…

Gromov showed that for fixed, arbitrarily large C, any uniformly C-Lipschitz affine action of a random group in his graph model on a Hilbert space has a fixed point. We announce a theorem stating that more general affine actions of the same…

We introduce a strategy to tackle some known obstructions of current approaches to the Fourier uniformity conjecture. Assuming GRH, we then show the conjecture holds for intervals of length at least $(\log X)^{\psi(X)}$, with $\psi(X)…

Bent functions are Boolean functions in an even number of variables that are indicators of Hadamard difference sets in elementary abelian 2-groups. A bent function in m variables is said to be normal if it is constant on an affine space of…

We study the existence of fixed points for continuous maps $f$ from an $n$-ball $X$ in $\mathbb R^n$ to $\mathbb R^n$ with $n\geq 1$. We show that $f$ has a fixed point if, for some absolute retract $Y\subset\partial X$, $f(Y)\subset X$ and…

In this paper we consider the question of smoothness of slowly varying functions satisfying the modern definition that, in the last two decades, gained prevalence in the applications concerning function spaces and interpolation. We show,…

This paper introduces the ring of all real valued Baire one functions, denoted by $B_1(X)$ and also the ring of all real valued bounded Baire one functions, denoted by $B_1^*(X)$. Though the resemblance between $C(X)$ and $B_1(X)$ is the…

We analyze Ekeland's variational principle in the context of reverse mathematics. We find that that the full variational principle is equivalent to $\Pi^1_1$-${\sf CA}_0$, a strong theory of second-order arithmetic, while natural…

A general fixed point theorem for isometries in terms of metric functionals is proved under the assumption of the existence of a conical bicombing. It is new even for isometries of Banach spaces as well as for non-locally compact…

Let $\operatorname{Lip}_0(M)$ be the space of Lipschitz functions on a complete metric space $M$ that vanish at a base point. We show that every normal functional in $\operatorname{Lip}_0(M)^\ast$ is weak$^*$ continuous, answering a…

We prove that if a continuous piecewise-smooth map on $\mathbb{R}^n$ is comprised of two linear functions, has a bounded orbit, and satisfies a certain non-degeneracy condition, then it has a fixed point. The result has important…

Inductive and coinductive specifications are widely used in formalizing computational systems. Such specifications have a natural rendition in logics that support fixed-point definitions. Another useful formalization device is that of…

Adding a column of numbers produces "carries" along the way. We show that random digits produce a pattern of carries with a neat probabilistic description: the carries form a one-dependent determinantal point process. This makes it easy to…

We present a simplified and streamlined characterisation of provably total computable functions of the theory ID_1 of non-iterated inductive definitions. The idea of the simplification is to employ the method of operator-controlled…

We study membership of rational inner functions in Dirichlet-type spaces in polydisks. In particular, we prove a theorem relating such inclusions to $H^p$ integrability of partial derivatives of a RIF, and as a corollary we prove that all…