Related papers: Continuity Assumptions in Cake-Cutting
The standard axioms of set theory, the Zermelo-Fraenkel axioms (ZFC), do not suffice to answer all questions in mathematics. While this follows abstractly from Kurt G\"odel's famous incompleteness theorems, we nowadays know numerous…
Many algorithms for inferring causality rely heavily on the faithfulness assumption. The main justification for imposing this assumption is that the set of unfaithful distributions has Lebesgue measure zero, since it can be seen as a…
We study several properties of equi-Baire 1 families of functions between metric spaces. We consider the related equi-Lebesgue property for such families. We examine the behaviour of equi-Baire 1 and equi-Lebesgue families with respect to…
We study classes of Borel subsets of the real line $\mathbb{R}$ such as levels of the Borel hierarchy and the class of sets that are reducible to the set $\mathbb{Q}$ of rationals, endowed with the Wadge quasi-order of reducibility with…
We give a constructive proof that any $\sigma$-porous subset of a Hilbert space has Lebesgue measure zero on typical $C^{1}$ curves. Further, we discover that this result does not extend to all forms of porosity; we find that even power-$p$…
We introduce an interesting method of proving separable reduction theorems - the method of elementary submodels. We are studying whether it is true that a set (function) has given property if and only if it has this property with respect to…
The linear continuity of a function defined on a vector space means that its restriction on every affine line is continuous. For functions defined on $\mathbb R^m$ this notion is near to the separate continuity for which it is required only…
In this article we suggest a model of computation for the cake cutting problem. In this model the mediator can ask the same queries as in the Robertson-Webb model but he or she can only perform algebraic operations as in the Blum-Shub-Smale…
A conjecture of Erd\H{o}s states that for any infinite set $A \subseteq \mathbb R$, there exists $E \subseteq \mathbb R$ of positive Lebesgue measure that does not contain any nontrivial affine copy of $A$. The conjecture remains open for…
This paper deals with connections on $p$-adic analytic curves, in the sense of Berkovich. The curves must be compact but the connections are allowed to have a finite number of meromorphic singularities on them. For any choice of a…
This work develops algorithmic results for the classic cake-cutting problem in which a divisible, heterogeneous resource (modeled as a cake) needs to be partitioned among agents with distinct preferences. We focus on a standard formulation…
In this paper we introduce the concept of completeness of sets. We study this property on the set of integers. We examine how this property is preserved as we carry out various operations compatible with sets. We also introduce the problem…
In this article we treat a notion of continuity for a multi-valued function F and we compute the descriptive set-theoretic complexity of the set of all x for which F is continuous at x. We give conditions under which the latter set is…
The concept of uniform distribution in $[0,1]$ is extended for a certain strictly separated maximal (in the sense of cardinality) family $(\lambda_t)_{t \in [0,1]}$ of invariant extensions of the linear Lebesgue measure $\lambda$ in…
Let $f\colon\mathbb{R}^2\to\mathbb{R}$. The notions of feebly continuity and very feebly continuity of $f$ at a point $\langle x,y\rangle\in\mathbb{R}^2$ were considered by I. Leader in 2009. We study properties of the sets $FC(f)$…
We consider a differential geometric setting on power sets and Borel algebras. Our chosen framework is based on diffeologies, and we make a link between the various diffeological structures that we propose, having in mind set-valued maps,…
In this paper we look for the existence of large linear and algebraic structures of sequences of measurable functions with different modes of convergence. Concretely, the algebraic size of the family of sequences that are convergent in…
Boykin and Jackson recently introduced a property of countable Borel equivalence relations called Borel boundedness, which they showed is closely related to the union problem for hyperfinite equivalence relations. In this paper, we…
We extend Michel's theorem on the geometry of symmetry breaking [L. Michel, {\it Comptes Rendus Acad. Sci. Paris} {\bf 272-A} (1971), 433-436] to the case of pure gauge theories, i.e. of gauge-invariant functionals defined on the space…
A result of P. Tukia from 1989 says that Lebesgue measure on $\mathbb{R}$ has conformal dimension zero: for every $\epsilon > 0$, there is a Borel set $G \subset \mathbb{R}$ of full Lebesgue measure, and a quasisymmetric homeomorphism $f…