Related papers: Cutting Cakes Correctly
We give a simple proof of the "tree-width duality theorem" of Seymour and Thomas that the tree-width of a finite graph is exactly one less than the largest order of its brambles.
Product structure theorems are a collection of recent results that have been used to resolve a number of longstanding open problems on planar graphs and related graph classes. One particularly useful version states that every planar graph…
We prove an analytic KAM-Theorem, which is used in [1], where the differential part of KAM-theory is discussed. Related theorems on analytic KAM-theory exist in the literature (e. g., among many others, [7], [8], [13]). The aim of the…
A seminal result of Koml\'os, S\'ark\"ozy, and Szemer\'edi states that any n-vertex graph G with minimum degree at least (1/2 + {\alpha})n contains every n-vertex tree T of bounded degree. Recently, Pham, Sah, Sawhney, and Simkin extended…
The divisor theory of the complete graph $K_n$ is in many ways similar to that of a plane curve of degree $n$. We compute the splitting types of all divisors on the complete graph $K_n$. We see that the possible splitting types of divisors…
Counterfactual frameworks have grown popular in machine learning for both explaining algorithmic decisions but also defining individual notions of fairness, more intuitive than typical group fairness conditions. However, state-of-the-art…
The impossibility theorem of fairness is a foundational result in the algorithmic fairness literature. It states that outside of special cases, one cannot exactly and simultaneously satisfy all three common and intuitive definitions of…
The allocation of resources among multiple agents is a fundamental problem in both economics and computer science. In these settings, fairness plays a crucial role in ensuring social acceptability and practical implementation of resource…
An error in the paper [J. Math. Phys. 43, 6343 (2002); math-ph/0207009] is corrected. Further explanation is given.
This paper has been withdrawn by the corresponding author because the newest version is now published in Discrete Applied Mathematics.
We consider the classic cake cutting problem in the Robertson-Webb model, with the objective of proportional fairness. We show that any randomized algorithm must use $\Omega(n \log n)$ queries.
The paper disproves a basic theorem on quasi-birth-and-death processes given in [M. F. Neuts (1995). Matrix Geometric Solutions in Stochastic Models: An Algorithmic Approach. Dover, New York].
To divide a cake into equal sized pieces most people use a knife and a mixture of luck and dexterity. These attempts are often met with varying success. Through precise geometric constructions performed with the knife replacing Euclid's…
We study splittings, or lack of them, in lattices of subvarieties of some logic-related varieties. We present a general lemma, the Non-Splitting Lemma, which when combined with some variety-specific constructions, yields each of our…
Double twist knots $K_{m, n}$ are known to be rationally slice if $mn = 0$, $n = -m\pm 1$, or $n = -m$. In this paper, we prove the converse. It is done by showing that infinitely many prime power-fold cyclic branched covers of the other…
An error is spotted in the statement of Theorem~1.3 of our published article titled "On oriented cliques with respect to push operation" (Discrete Applied Mathematics 2017). The theorem provided an exhaustive list of 16 minimal (up to…
Some mistaken reasonings at the end of the paper omitted.
In 2012, Andrews and Merca obtained a truncated version of Euler's pentagonal number theorem and showed the nonnegativity related to partition functions. Meanwhile, Andrews-Merca and Guo-Zeng independently conjectured that the truncated…
In this paper we present an explicit counterexample of degree $n=7$, which shows that the conjecture proposed by Li et al. \cite{Li2013} regarding the first derivative bounds for rational B\'ezier curves is generally false. We further…
This study proposes a new efficiency requirement, a minimal almost weak Pareto principle, which says that x is socially better than y whenever the only one individual never prefers y to x, and all the others prefers x to y. Then, I show…