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Computing the Smith Normal Form of a matrix, and homology groups?

A simple algorithm for computing the Smith Normal Form of a matrix in $\ZZ$ The proof and the algorithm are the same. First, a few auxiliary functions. Given a matrix $M$, the follow two functions are self-explanatory. def dims(M): num_righe=len(M) num_colonne=len(M[0]) return (num_righe,num_colonne) def MinAij(M,s): num_righe, num_colonne=dims(M) ijmin=[s,s] valmin=max( max([abs(x) for x in M[j][s:]]) for j in range(s,num_righe) ) for i in (range(s,num_righe)): for j in (range(s,num_colonne)): if (M[i][j] !


Császár torus, python and sagemath

The simplicial complex Consider the (abstract) simplicial $T$ defined as follows: vertices are $$ X=\{ 0,1,2,3,4,5,6 \} = \{ j \mod 7 \}. $$ Now, for each $x\in X$ let $\Phi_ x$ denote the following subsets of $2^X$ : $$ \begin{aligned} H_ x &= \{ \{j,j+1\} : j =0 \ldots 6 \} = \{ \{j,j-1\} : j =0 \ldots 6 \} \\ V_ x &= \{ \{ j,j+x \} : j =0 \ldots 6 \} = \{ \{ j,j-x \} : j =0 \ldots 6 \} \\


Categorical products and coproducts

Universal property of coproducts Let $X_1, X_2$ be objects of the category $\vC$. The coproduct of $X_1$ and $X_2$ (if it exists) is a space $X_1 \coprod X_2$ and two morphisms $j_1 \from X_1 \to X_1 \coprod X_2$, $j_2 \from X_2 \to X_1 \coprod X_2$ such that the following diagram commutes (the first is with MathJax AMScd, the second is an embedded SVG produced with xypic): $\require{AMScd}$ \begin{CD} X_1 @>{j_1}>> X_1 \coprod X_2 @<{j_2}<< X_2 \\


A counter-example for the exponentiation / adjoint of a function

The problem: Let $X$, $Y$, and $Z$ be topological spaces. Let $Y^X$ denote the space of all continuous functions $X\to Y$, with the compact-open topology generated by the elements of the sub-basis $W_{K,U} = \{\varphi \in Y^X : \varphi K \subset U\}$, where $K$ ranges over all compact subsets of $X$ and $U$ ranges over over all open subsets of $Y$. Finite intersections of $W_{K,U}$ are a basis for the C.