Related papers: Elementary real analysis without compactness argum…

We give a new proof of a partition theorem popularly known as Elder's theorem, but which is also credited to Stanley and Fine. We extend the theorem to the context of colored partitions (or prefabs). More specifically, we give analogous…

The author wrote this note after being asked about the existence of compactifications of algebraic spaces. Subsequent to posting the article to the math arXiv, the author learned from Yutakaa Matsuura that the results of this paper had been…

In the 1930s Tarski showed that real quantifier elimination was possible, and in 1975 Collins gave a remotely practicable method, albeit with doubly-exponential complexity, which was later shown to be inherent. We discuss some of the recent…

One of the most famous results in Complex Analysis is the Little Picard Theorem, that characterizes the image set of an arbitrary entire function. Specifically, the theorem states that this image set is either the whole complex plane or the…

The main purpose of this paper is to provide explicit computations of the fundamental group of several algebras. For this purpose, given a $k$-algebra $A$, we consider the category of all connected gradings of $A$ by a group $G$ and we…

The geometric and algebraic theory of monomial ideals and multigraded modules is initiated over real-exponent polynomial rings and, more generally, monoid algebras for real polyhedral cones. The main results include the generalization of…

Section 10.4 of the 1998 Springer-Verlag book {\em Complexity and Real Computation}, by Blum, Cucker, Shub, and Smale, contains a particularly elegant proof of the Fundamental Theorem of Algebra: The central idea of the proof naturally…

The paper is devoted to graded algebras having a single homogeneous relation. Using Gerasimov's theorem, a criterion to be N-Koszul is given, providing new examples. An alternative proof of Gerasimov's theorem for N=2 is given. Some related…

For more than a century, Cantor's theory of transfinite numbers has played a pivotal role in set theory, with ramifications that extend to many areas of mathematics. This article extends earlier findings with a fresh look at the critical…

This expository article proves some results of Ferguson, on the approximation of continuous functions on a compact subset of R by polynomials with integral coefficients.

The compactness phenomenon is one of the featured aspects of structuralism in mathematics. In simple and broad words, a compactness property holds in a structure if a related property is satisfied by sufficiently many substructures of that…

Euler's theorem asserts that $A(n)=B(n)$ where $A(n)$ is the number of partitions of $n$ into distinct parts and $B(n)$ is the number of partitions of $n$ into odd parts. In this paper, it is proved that for $n>0$, \begin{align*}…

We analyse partitions of products with two ordered factors in two classes where both factors are countable or well-ordered and at least one of them is countable. This relates the partition properties of these products to cardinal…

Using methods from coarse topology we show that fundamental classes of closed enlargeable manifolds map non-trivially both to the rational homology of their fundamental groups and to the K-theory of the corresponding reduced C*-algebras.…

We report an ongoing work on clustering algorithms for complex roots of a univariate polynomial $p$ of degree $d$ with real or complex coefficients. As in their previous best subdivision algorithms our root-finders are robust even for…

We present a proof of the sufficiency of Rado's condition for the partition regularity of linear Diophantine equations that avoids any use of van der Waerden's theorem. The proof is based on fundamental properties that are common knowledge…

In answer to a question of Andrews about finding combinatorial proofs of two identities in Ramanujan's "Lost" Notebook, we obtain weighted forms of Euler's theorem on partitions with odd parts and distinct parts. This work is inspired by…

We compare two methods of proving separable reduction theorems in functional analysis -- the method of rich families and the method of elementary submodels. We show that any result proved using rich families holds also when formulated with…

In this paper we refine a weighted partition identity of Alladi. We write explicit formulas of generating functions for the number of partitions grouped with respect to a partition statistic other than the norm. We tie our weighted results…

We prove Cuntz-Krieger and graded uniqueness theorems for Steinberg algebras. We also show that a Steinberg algebra is basically simple if and only if its associated groupoid is both effective and minimal. Finally we use results of…