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The Vandermonde matrix is the $n\times n$ matrix whose $(i,j)$-th component is $x_j^{i-1}$, where the $x_j$ are indeterminates. It is well known that the determinant of this matrix is $$\prod_{1\leq ...

Let $A_0A_1...A_n$ be a simplex in $\Bbb E^n.$ Let $B_{ij}$ be a point on the edge $A_iA_j,\ 0\le i\not=j\le n.$ Denote by $\beta_i$ the hyperplane passing through the points $B_{i0},$ $B_{i1},$ $B_{...

Given two vector sets, $\vec x_i$ and $\vec y_i$ (for $i$=1,2,...N, but the dimensionality of each vector can be more than N), let their sum set be $\vec z_i = \vec x_i + \vec y_i$. It's easy to ...