相关论文: A simpler proof of Jensen's coding theorem
Taking up the challenge McConnell laid down at the end of his proof of the law of cosines, we give a completely visual dissection proof of this theorem, which applies to any triangle. In order to avoid the trigonometric expressions of…
This paper presents a formalized proof of a discrete form of the Jordan Curve Theorem. It is based on a hypermap model of planar subdivisions, formal specifications and proofs assisted by the Coq system. Fundamental properties are proven by…
Classical coding theory contains several techniques to obtain new codes from other codes, including puncturing and shortening. For quantum codes, a form of puncturing is known, but its description is based on the code space rather than its…
We consider a generalization of the classic Sperner lemma. This lemma states that every Sperner coloring of a triangulation of a simplex contains a fully colored simplex. We found a weaker assumption than Sperner's coloring. It is also…
The purpose of this article is to prove that the forcing axiom for completely proper forcings is inconsistent with the Continuum Hypothesis. This answers a longstanding problem of Shelah. The corresponding completely proper forcing which…
The (.)_reg construction was introduced in order to make an arbitrary semigroup S divide a regular semigroup (S)_reg which shares some important properties with S (e.g., finiteness, subgroups, torsion bounds, J-order structure). We show…
The minimal excludant of a partition $\lambda$, $\rm{mex}(\lambda)$, is the smallest positive integer that is not a part of $\lambda$. For a positive integer $n$, $ \sigma\, \rm{mex}(n)$ denotes the sum of the minimal excludants of all…
We present a self-contained elementary and detailed exposition of Mertens' own proof of his theorem on the divergence of the series of the reciprocals of the primes and compare it with the modern proofs. His proof contains explicit…
We introduce the notion of confined subalgebras in the context of the group von Neumann algebra. We also define Uniformly Recurrent States -- an operator-algebraic analog of Uniformly Recurrent Subgroups. Using this framework, we show that…
Cut-elimination theorems constitute one of the most important classes of theorems of proof theory. Since Gentzen's proof of the cut-elimination theorem for the system $\mathbf{LK}$, several other proofs have been proposed. Even though the…
We study the list-chromatic number and the coloring number of graphs, especially uncountable graphs. We show that the coloring number of a graph coincides with its list-chromatic number provided that the diamond principle holds. Under the…
This paper is the second in a series of two papers about generalizing Quillen's Theorem A to strict $\infty$-categories. In the first one, we presented a proof of this Theorem A of a simplicial nature, direct but somewhat ad hoc. In the…
Suppose $J = (f_1, \dots, f_n)$ is an $n$-generated ideal in any ring $R$. We prove a general Brian\c{c}on-Skoda-type containment relating the integral closure $\overline{J^{n+k-1}}$ with ordinary powers $J^k$. We prove that our result…
Simon's factorization theorem is a celebrated tool in algebraic automata theory, providing bounded-depth decompositions of words with respect to morphisms into finite semigroups. We develop an analogue of Simon's theorem for \emph{forests}…
We derive a simple lower bound for the multi-version coding problem formulated in [1]. We also propose simple algorithms that almost match the lower bound derived. Another lower bound is proven for an extended version of the multi-version…
Gentzen's 1936 proof of the consistency of Peano Arithmetic was a significant result in the foundations of mathematics. We provide here a modified version of the proof, based on G\"{o}del's reformulation, and including additional details…
We introduce a property of C*-correspondences, which we call Condition (S), to serve as an analogue of Condition (L) of graphs. We use Condition (S) to prove a Cuntz-Krieger Uniqueness Theorem for Cuntz-Pimsner algebras and obtain…
In this paper, we provide an easy proof of the Four-colour Theorem in a special case indeed.
Shoenfield's completeness theorem (1959) states that every true first order arithmetical sentence has a recursive $\omega$-proof encodable by using recursive applications of the $\omega$-rule. For a suitable encoding of Gentzen style…
The purpose of this article is to give new constructions of linear orders which are minimal with respect to being non-$\sigma$-scattered. Specifically, we will show that Jensen's principle $\diamondsuit$ implies that there is a minimal…