In recent work with Michael Albert and Nik Ruškuc, a family of words has arisen which is counted by the Catalan numbers. I've looked at Richard Stanley's Catalan exercises in EC2 a …

If $a$ and $b$ are natural numbers, then $a-b$ is an integer and so the square $(a-b)^2$ is a natural number. In particular $$ (a-b)^2 \geq 0. \qquad (1)$$ Combining this fact …

Let $C$ be a nonsingular projective curve defined over $\mathbb{Q}$, which does not admit a map of degree 1 or 2 to $\mathbb{P}^1$ or to an elliptic curve. It is then a consequence …

Is it true that for a given nonnegative number, there exists a measure-theoretical entropy value (supremum of entropies of all partitions under a measure-preserving transformation) …

Let $X\subseteq\mathbb{C}P^n$ be s smooth variety. Let $C\subseteq X$ be an algebraic rational curve [i.e. an algebraic curve which is birational to $\mathbb{C}P^1$]. In what fol …

Consider a nonsingular projective variety $X$ over an algebraically closed field $k$ and let $Y \subseteq X$ be a nonsingular closed subvariety. Let $\mathcal I \subseteq \mathcal …

I'd like to start by noting that for some fixed natural $N$ basis functions for my system will be generated by $f(k,x)$ as defined and explained here or in numerous other sources: …

Let $s \in R^{n}$ (meaning $s$ is $n \times 1$ vector), where $n$ is the dimension of the vector. The ideal sliding term, $\nu$ is taken to be: \begin{equation} \nu = \frac …

My question is: usually, a partial differential equation, for example, those coming from physics, is written in a lauguage of vector calculus in a local coordinate, is there anyway …

I read the article in wikipedia, but I didn't find it totally illuminating. As far as I've understood, essentially you have a morphism (in some probably geometrical category) $Y \r …

This is a somewhat imprecise question, as I am not sure how exactly how to formalise how to do mathematics "without" a certain key tool, but hopefully the intent of the question wi …

There are several occasions in the study of dynamical systems that are called phase transitions. For example consider a homeomorphism $f:X\to X$ and a potential function $\phi\in C …

suppose sensors of homogeneous radius r are dropped by Poisson distribution on a straight line of length L. how to calculate that the straight line is covered by sensors with prob …

I am looking for the original reference for Ostrowski's theorem of 1916 that the only valuations on the rational numbers are the trivial, Archimedean, and p-adic valuations. http: …

Consider the “product” $\gamma = \alpha \times \beta$ of two binary strings $\alpha$, $\beta$ $\in \lbrace 0,1\rbrace^+$ which one gets by replacing every 1 in $\beta$ …