Posts

Dell'uso dell'inglese nelle aule universitarie

(2018-02-01)

A proposito di lingua inglese per le lezioni/esami (e internalizzazione) È di qualche giorno fa la sentenza del Consiglio di Stato N. 00617/2018REG.PROV.COLL. – N. 05151/2013 REG.RIC, «sul ricorso numero di registro generale 5151 del 2013, proposto da: Ministero dell’Istruzione dell’Università e della Ricerca, in persona del Ministro pro tempore, Politecnico di Milano, in persona del legale rappresentante pro tempore, rappresentati e difesi per legge dall’Avvocatura generale dello Stato. [..omissis..] per la riforma della sentenza 23 maggio 2013, n.

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Gli Esami Non Finiscono Mai

(2017-11-03)

Norme di Riferimento REGIO DECRETO 31 agosto 1933, n. 1592 FONTE Approvazione del testo unico delle leggi sull’istruzione superiore. (033U1592) (GU n.283 del 7-12-1933 - Suppl. Ordinario n. 283 ) note: Entrata in vigore del provvedimento: 22/12/1933 VITTORIO EMANUELE III, PER GRAZIA DI DIO E PER VOLONTA' DELLA NAZIONE RE D’ITALIA Veduto l’art. 70 del R. decreto-legge 4 settembre 1925, n. 1604, col quale il Governo del Re e' autorizzato a riunire in testo unico, provvedendo al loro coordinamento e introducendo, ove occorra, norme integrative; tutte le disposizioni vigenti e quelle da emanarsi eventualmente anche posteriormente alla pubblicazione del decreto stesso in materia d’istruzione superiore e relative a corpi, istituti, stabilimenti, uffici e servizi comunque attinenti all’istruzione e alla cultura superiore;

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Winding Around and Mountains

(2017-10-26)

Just … …reading John Roe’s book Winding Around. Here: For an extended riff on the idea of mathematics as mountaineering see “Into Thin Air” by Colin Adams, on Mathematical Intelligencer. How much about “mathematics as mountaineering”?

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Fixing MarkDown LaTeX underscore problem

(2017-10-04)

How to fix Markdown underscore with LaTeX problem? Test of a displaystyle formula $$ A_k^n = \dfrac{n!}{k!(n-k)!} = \dfrac{n (n-1) \ldots (n-j) \ldots (n-k+1)}{ 1 \ldots j \ldots (k-1) \cdot k } = \dfrac{\prod_ {j=1}^k (n-j+1) }{ \prod_ {j=1}^k j } $$ And the same for an inline sum: $\sum_j j^2$. The answer is: in the preamble markup = "mmark" and then just use $$ .. $$ or $ .. $ apparently.

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Updating my (Xe)LaTeX vim plugin

(2017-07-04)

After some years, it was time to update my LaTeX vim plugins. My setting was: plain vim, a makefile, some configuration lines. Back-forward search, not much completion. Now, with the vundle plugin. Things to do: I had vundle installed already. I needed to add lervag’s vimtex and ajh17’s VimCompletesMe. Hence, in .vimrc: " set the runtime path to include Vundle and initialize set rtp+=~/.vim/bundle/Vundle.vim call vundle#rc() " let Vundle manage Vundle, required Plugin 'VundleVim/Vundle.

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Wielandt's proof of Sylow's Theorem

(2017-04-28)

Wieldandt’s proof of Sylow’s Theorem A brief preparatory combinatorial discussion: $$ \binom{n}{k} = \dfrac{n!}{k!(n-k)!} = \dfrac{n (n-1) \ldots (n-j) \ldots (n-k+1)}{ 1 \ldots j \ldots (k-1) \cdot k } = \dfrac{\prod_ {j=1}^k (n-j+1) }{ \prod_ {j=1}^k j } $$ and therefore $$ \binom{n}{k} = \dfrac{n}{k} \cdot \dfrac{n-1}{1} \ldots \dfrac{n-j}{j} \ldots \dfrac{n-(k-1)}{k-1} = \dfrac{n}{k} \prod_{j=1}^{k-1} \dfrac{n-j}{j} $$ Let $p$ be a prime number. Now, if $n$ is an integer divisible by a power of $p$, say $p^\alpha$, then $n=p^\alpha m$.

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Lettera sulla Polonia (1574-09-25)

(2016-12-27)

Lettera del Cavalier Battista Guarini (1538-1612) al Segretario del Cardinal Luigi d’Este (1538-1586), Monsignor Benedetto Manzuoli (1530-1585). Testo preso dalla raccolta di lettere L’idea del segretario dal signore Bartolomeo Zucchi gentil’huomo di Monza, academico insensato di Perugia. Parte seconda. pubblicato nel 1606 da Bartolomeo Zucchi (1570-1630) a Venezia, presso la Compagnia Minima. ARCHIVE LINK L’ortografia è stata parzialmente adattata a quella moderna. Guarini fu Segretario di Alfonso Secondo d’Este, duca di Ferrara (1533-1597) tra il 1573 e il 1583, e scrive al segretario del Cardinal Luigi d’Este, fratello di Alfonso Secondo d’Este.

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Fundamental group of SO(3) and Dirac's scissors

(2016-11-25)

Topology of $SO(3)$ and fixed-distance pairs of points in $S^2$. Consider the space $X\subset S^2\times S^2$ of all pairs $(A,B)$ of points such that $|A-B|=\sqrt 3$, where the distance is the euclidean distance of $S^2\subset \RR^3$, and $S^2$ is the unit sphere in $\RR^3$. (1) $X\approx SO(3) \approx \PP^3(\RR)$. Proof: The fact that $SO(3) \approx \PP^3(\RR)$ is a not-so-easy elementary exercise (cf. exercise (11.27) and exercise (7.30) of my Geometry Notes, protected by a single-character password: the number of non-zero digits appearing in the password itself; the problem might be ill-posed).

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Homology of (simplicial) spheres and applications

(2016-11-21)

Nullhomotopic odd maps, Borsuk, Lusternik and Schnirelmann Let $\Delta^n$ the standard simplex of dimension $n$ in $\RR^{n+1}$, which is homeomorphic to a $n$-dimensional disk, and $\Sigma^{n}\subset \Delta^{n+1}$ the simplicial $n$-sphere, i.e. the boundary of $\Delta^{n+1}$ (the union of its $n+2$ $n$-dimensional faces), which is homeomorphic to an $n$-dimensional sphere. The homeomorphisms can be more explicit: let $B\in \Delta^{n+1}$ be its barycenter, and $\varphi\from \Sigma^n\subset \Delta^{n+1} \to \RR^{n+2}$ the map defined as $\varphi(x) = \dfrac{x-B}{|x-B|}$.

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Mesh, Barycentric Subdivision and Simplicial Approximation Theorem

(2016-11-18)

Mesh of a euclidean simplicial complex. Let $|K| \subset \RR^k$ an euclidean simplicial complex (i.e., an abstract simplicial complex, where the set of vertices actually is a set of points in $\RR^k$, such that $|K|$ is the disjoint union of the interiors of the convex hulls of the sets of vertices ranging over the set of simplices). A vertex $P$ of $K$ is therefore a point of $\RR^k$. For each simplex $\sigma \subset K$, the diameter of $\sigma$ (or $|\sigma|$ more precisely) is the maximum distance $|x-y|$ as $x,y$ vary in $|\sigma|$.

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Joins and Products of polyhedra

(2016-11-14)

Join $K*L$ of two simplicial complexes Let $K = (X_ K, \Phi_ K)$ and $L = (X_ L, \Phi_ L)$ be two simplicial complexes ($X_ K$ denotes the set of vertices of $K$ and $\Phi_ K$ the set of simplices). Then the join $KL$ is the simplicial complex defined as follows. Its set of vertices $X=X_ K \cup X_ L$ is the union of the sets of vertices of $K$ and of $L$, and a non-empty subset of $X$ is a simplex of $KL$ if and only if it is of type $\sigma \cup \tau$, where $\sigma \in \Phi_ K \cup \{ \emptyset \}$ and $\tau \in \Phi_ L \cup \{\emptyset \}$.

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Ranks, Tensors and Abelian Groups

(2016-11-09)

$\require{AMScd}$ Abelian Groups Given $a\in A$, the homomirphism $i_ a \from \ZZ \to A$ defined by $i_ a(k) = ka$ is mono (injective) iff $a$ has infinite order. Of $S$ is a set, the free Abelian group generated by $S$ is $$ \ZZ[S] = \ZZ S = \bigoplus_ {s \in S} \ZZ, $$ and it is the group of all finite linear combinations (with integer coefficients) of elements in $S$. The elements of $S$ are a basis (base) of $\ZZ[S]$.

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Graphs, trees and simplicial complexes

(2016-11-04)

Graphs and simplicial complexes A one-dimensional simplicial complex is a graph. That is, $K=(X_ K,\Phi_ K)$ where $V(K) = X_ K$ is the set of vertices and $\Phi_ K$ is the set of simplexes, with $\Phi_ K = \Phi^1_ K \cup \Phi^0_ K$, where $E(K) = \Phi^1_ K$ are the 1-dimensional simplexes (termed edges of $K$) and $\Phi^0_ K = \{ \{P\} : P \in X_ K \}$ are the $0$-dimensional simplexes (sometimes termed again vertices of $K$).

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

(2016-10-27)

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] != 0 ) and (abs(M[i][j]) <= valmin) : ijmin = [i,j] valmin = abs(M[i][j]) return ijmin def IdentityMatrix(n): res=[[0 for j in range(n)] for i in range(n)] for i in range(n): res[i][i] = 1 return res def display(M): r="" for x in M: r += "%s\n" % x return r +"" Then, one needs the elementary operations on rows and columns on the matrix $M$: swap (permute) two rows, add to a row an integer multiple of another row, and change sign of a row.

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