Let $S$ be an arbitrary finite spanning subset of $\mathbb{R}^d$ of cardinality $N$. Let $W(S)$ be the formal $\mathbb{R}$-vector space generated by all $d$-dimensional simplices ( …

Suppose you start from partial differential equations and functional analysis (on $\mathbb R^n$ and on real manifolds, which prominent example problems lead you to work with Pseudo …

Let $S$ be a patch of a smooth 2-manifold in $\mathbb{R}^3$, and pick two distinct points $a,\ b \in S$. Let $c$ be the set of points on $S$ equidistant to $a$ and $b$, where dista …

I am looking for a quick definition of the tangent space $T_p M$, where $p$ is a point of a smooth manifold $M$. I mean a definition that allows easily to prove that: (1) $T_p M$ i …

It is known that any closed topological manifold is homotopy equivalent to an open smooth manifold. Question 1. What can be said about the smallest dimension of a smooth manifol …

Consider a commutative ring $A= ( \mathbb{R}^n , + , \times) $, where $+$ is the usual one. Assume further that $ \times $ is continuous (with respect to the usual topology). Let $ …

Various questions have been posted on MO (some answered, some not) involving the character table of a finite group $G$ over a splitting field such as $\mathbb{C}$ of characteristic …

Recall that in the $n$-simplex $\Delta[n]$, we have a combinatorially crucial bijection between facets, (codimension $1$ faces) and vertices, where the $i$th face of a simplex corr …

Recall that we call a category rigid if it contains no non-identity isomorphisms. Let $\mathbf{rig}$ denote the full 2-subcategory of $\mathbf{Cat}$ spanned by the small rigid cate …

Is there some simple upper bound on $||(B^{-1}+A^{-1})^{-1}||$, where $A,B$ are $n \times n$ symmetric matrices?

While browsing the Net for some articles related to the history of the Whittaker-Shannon sampling theorem, so important to our digital world today, I came across this passage by H. …

Is there any length function on additive group of $\mathcal{Q}$ such that $\mathcal{Q}$ is of polynomial growth WRT this length function? What about the multiplicative group of $\ …

Recently I began to consider algebraic surfaces, that is, the zero set of a polynomial in 3 (or more variables). My algebraic geometry background is poor, and I'm more used to diff …

Hi all -- what would be good methods (and/or software packages) to try for solving a problem minimizing a quadratic function $f(x) = \sum_{i=1}^N{(x_i - y_i)^2}$, where some constr …

Hi fellows, Does anyone know the number of holes of a level 2 Menger Sponge ?