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It can easy be shown by induction that the determinant of the $(N-1)\times (N-1)$ matrix $$\begin{pmatrix} 2 & -1 & & \\ -1 & 2 & \ddots & \\ & \ddots & \ddots & -1\\ & …

Let $\|\cdot \|$ be some matrix norm on the space of $n \times n$ matrices. Denote $$ M(r) := \{ A \in \mathrm{Mat}_{n \times n}(\mathbb{Z}) \mid \| M \| \leq r \}.$$ Denote by $p(r)$ the fraction of …

Let $A$ be an algebra over $\mathbb{C}$. Let $M$ be a left $A$-module, let $N$ be a right $A$-module and consider the tensor product $N \otimes_A M$, which is a complex vector space. Q1: Can this ten …

For appropriate choices of $i$, $j$ (e.g. $i+j \neq n$), $\Omega^i(M)$ and $\Omega^j(M)$ have different ranks as $C^\infty(M)$ module, so at least they cannot be isomorphic as modules. Maybe they coul …

Sorry for the confusion from earlier. I tried to fix the thread. The old version can be found below. For $6$-dimensional subspaces $V$ of the space $\mathbb{R}^{3\times 3}$ of real three-times-three …

Let $V$ be a Euclidean vector space and let $V^{\mathbb{C}} = V \oplus V$ be its complexification, with complex structure $$J = \begin{pmatrix} 0 & -\mathrm{id}\\ \mathrm{id} & 0 \end{pmatrix}.$$ Of c …