typo

html/algoritmi.html

@@ -285,9 +285,9 @@ <h1 class="title"> </h1>
 </ul></li>
 <li><span class="math inline">r_n \mid d</span>
 <ul>
-<li><span class="math inline">\left \{ \begin{array}{l} r_{n + 1} = 0 \\
-0 \equiv r_{n + 1} \equiv r_{n - 1} \ (\bmod \ r_n) \iff \exists q_n \in
-\mathbb{Z} \mid r_{n- 1} = r_nq_n \end{array} \iff r_n \mid r_{n -
+<li><span class="math inline">\left \{ \begin{array}{l} r_{n + 1} = 0
+\\  0 \equiv r_{n + 1} \equiv r_{n - 1} \ (\bmod \ r_n) \iff \exists q_n
+\in \mathbb{Z} \mid r_{n- 1} = r_nq_n \end{array} \iff r_n \mid r_{n -
 1}\right.</span></li>
 <li><span class="math inline">r_n \equiv r_{n - 2} \ (\bmod r_{n - 1})
 \iff \exists q_{n - 1} \in \mathbb{Z} \mid r_n =r_{n - 1} q_{n - 1} +
@@ -364,11 +364,11 @@ <h1 class="title"> </h1>
 <ul>
 <li><span class="math inline">\left \{ \begin{array}{l} (p-1) \mid
 \lambda(n) \\ (q - 1) \mid \lambda(n) \end{array} \right. \implies
-\exists i, j \in \mathbb{Z} \mid \lambda(n) = (p-1) \cdot i = (q - 1)
+\exists i, j \in \mathbb{Z} \mid \lambda(n) = (p-1) \cdot i  = (q - 1)
 \cdot j</span></li>
 <li><span class="math inline">\textrm{MCD}(m, n) = 1 \implies p \nmid m
 \land q \nmid m \implies \left \{ \begin{array}{l}p \nmid m \implies m^p
-\equiv m \iff m^{p-1} \equiv 1 \implies m^{\lambda(n)} \equiv
+\equiv m \iff m^{p-1} \equiv 1  \implies m^{\lambda(n)} \equiv
 m^{(p-1)\cdot i} \equiv 1 \ (\bmod \ p) \\ q \nmid m \implies m^q \equiv
 m \iff m^{q-1} \equiv 1 \implies m^{\lambda(n)} \equiv m^{(q-1)\cdot j}
 \equiv 1 \ (\bmod \ q) \end{array} \right. \iff m^{\lambda(n)} \equiv 1
@@ -407,7 +407,7 @@ <h1 class="title"> </h1>
 <li><strong>Alg</strong>
 <ul>
 <li><span class="math inline">\forall i \in [0, n] \quad p_i(x) :=
-\displaystyle \prod_{\begin{subarray}{c}0 \le j \le n \\ i \neq\ j
+\displaystyle \prod_{\begin{subarray}{c}0 \le  j \le n \\ i \neq\ j
 \end{subarray}}{\dfrac{x - b_j}{b_i - b_j}}</span></li>
 <li><span class="math inline">p(x) := c_0p_0(x) + \ldots + c_n
 p_n(x)</span></li>

html/coefficienti-binomiali.html

@@ -360,15 +360,12 @@ <h1 class="title"> </h1>
 class="math inline">\left[a_{1}\right]^{p}=\left[a_{1}\right]^{p}</span>
 per dimostrazione precedente</li>
 <li><span class="math inline">n&gt;1
-\implies\left(\left[a_{1}\right]+\ldots+\left[a_{n}\right]+\left[a_{n+1}\right]\right)^{p}=
-\left[a_{1}\right]^{p}+\ldots+\left[a_{n}\right]^{p}+\left[a_{n+1}\right]^{p}</span>
+\implies\left(\left[a_{1}\right]+\ldots+\left[a_{n}\right]+\left[a_{n+1}\right]\right)^{p}=  \left[a_{1}\right]^{p}+\ldots+\left[a_{n}\right]^{p}+\left[a_{n+1}\right]^{p}</span>
 <ul>
 <li>per ipotesi induttiva, <span
-class="math inline">\left[a_{1}\right]^{p}+\ldots+\left[a_{n}\right]^{p}+\left[a_{n+1}\right]^{p}=
-\left(\left[a_{1}\right]+\ldots+\left[a_{n}\right]\right)^{p}+\left[a_{n+1}\right]^{p}</span></li>
+class="math inline">\left[a_{1}\right]^{p}+\ldots+\left[a_{n}\right]^{p}+\left[a_{n+1}\right]^{p}=  \left(\left[a_{1}\right]+\ldots+\left[a_{n}\right]\right)^{p}+\left[a_{n+1}\right]^{p}</span></li>
 <li>allora, ancora per ipotesi induttiva <span
-class="math inline">\left(\left[a_{1}\right]+\ldots+\left[a_{n}\right]\right)^{p}+\left[a_{n+1}\right]^{p}=
-\left(\left[a_{1}\right]+\ldots+\left[a_{n+1}\right]\right)^{p}</span></li>
+class="math inline">\left(\left[a_{1}\right]+\ldots+\left[a_{n}\right]\right)^{p}+\left[a_{n+1}\right]^{p}=  \left(\left[a_{1}\right]+\ldots+\left[a_{n+1}\right]\right)^{p}</span></li>
 </ul></li>
 </ul></li>
 </ul>

html/determinante.html

@@ -207,7 +207,7 @@ <h1 class="title"> </h1>
 \rightarrow W:(v_1, \ldots, v_n) \rightarrow w</span></li>
 <li><span class="math inline">f</span> è detta
 <strong>multilineare</strong> <span class="math inline">\iff \forall i
-\in [1, n], v_1 , \ldots, v_n \in V_1 \times \ldots \times V_n, v_i,
+\in [1, n],  v_1 , \ldots, v_n \in V_1 \times \ldots \times V_n, v_i,
 v_i&#39; \in V_i, \lambda, \mu \in \mathbb{K} \quad f(v_1, \ldots,
 \lambda v_i+\mu v_i&#39;, \ldots, v_n) = \lambda f(v_1, \ldots, v_i,
 \ldots, v_n) + \mu f(v_1, \ldots, v_i&#39;, \ldots, v_n)</span>
@@ -316,16 +316,16 @@ <h1 class="title"> </h1>
 class="math inline">\det(A_1, \ldots, A_i + A_j, \ldots, A_j + A_i,
 \ldots, A_n) = 0</span></li>
 <li>allora, per multilinearità di <span class="math inline">\det</span>
-si ha che <span class="math inline">\det</span> si ha che <span
-class="math inline">0 =\det(A_1, \ldots, A_i + A_j, \ldots, A_j + A_i,
-\ldots, A_n) = \det(A_1, \ldots, A_i, \ldots, A_j + A_i, \ldots, A_n) +
-\det(A_1, \ldots, A_j, \ldots, A_j + A_i, \ldots, A_n) =\det(A_1,
-\ldots, A_i, \ldots, A_j, \ldots, A_n)+\det(A_1, \ldots, A_i, \ldots,
-A_i, \ldots, A_n) + \det(A_1, \ldots, A_j, \ldots,A_j, \ldots, A_n) +
-\det(A_1, \ldots, A_j, \ldots, A_i, \ldots, A_n) = \det(A_1, \ldots,
-A_i, \ldots, A_j, \ldots, A_n) + 0 + 0 + \det(A_1, \ldots, A_j , \ldots,
-A_i, \ldots, A_n) \iff \det(A_1, \ldots, A_i, \ldots, A_j, \ldots, A_n)
-= -\det(A_1,\ldots, A_j, \ldots, A_i, \ldots, A_n)</span></li>
+si ha che <span class="math inline">0 =\det(A_1, \ldots, A_i + A_j,
+\ldots, A_j + A_i, \ldots, A_n) = \det(A_1, \ldots, A_i, \ldots, A_j +
+A_i, \ldots, A_n) + \det(A_1, \ldots, A_j, \ldots, A_j + A_i, \ldots,
+A_n) =\det(A_1, \ldots, A_i, \ldots, A_j, \ldots, A_n)+\det(A_1, \ldots,
+A_i, \ldots, A_i, \ldots, A_n) + \det(A_1, \ldots, A_j, \ldots,A_j,
+\ldots, A_n) + \det(A_1, \ldots, A_j, \ldots, A_i, \ldots, A_n) =
+\det(A_1, \ldots, A_i, \ldots, A_j, \ldots, A_n) + 0 + 0 + \det(A_1,
+\ldots, A_j , \ldots, A_i, \ldots, A_n) \iff \det(A_1, \ldots, A_i,
+\ldots, A_j, \ldots, A_n) = -\det(A_1,\ldots, A_j, \ldots, A_i, \ldots,
+A_n)</span></li>
 <li>si noti che la tesi è verificata sia per righe che per colonne, per
 definizione di <span class="math inline">\det</span></li>
 </ul></li>
@@ -549,9 +549,9 @@ <h1 class="title"> </h1>
 I_n</span></li>
 <li>allora <span class="math inline">I_n = A \cdot B = (A \cdot B^1,
 \ldots, A \cdot B^n) = (\mathscr{L}_A(B^1), \ldots, \mathscr{L}_A(B^n))
-\iff \left \{ \begin{array}{c} \mathscr{L}_A(B^1) = e_1 \\ \vdots \\
-\mathscr{L}_A(B^n) = e_n \end{array} \right.\iff e_1, \ldots, e_n \in
-\textrm{im}(\mathscr{L}_A) \implies \textrm{span}(e_1, \ldots, e_n)
+\iff \left \{ \begin{array}{c} \mathscr{L}_A(B^1) = e_1 \\  \vdots
+\\  \mathscr{L}_A(B^n) = e_n \end{array} \right.\iff e_1, \ldots, e_n
+\in \textrm{im}(\mathscr{L}_A) \implies \textrm{span}(e_1, \ldots, e_n)
 \subseteq \textrm{im}(\mathscr{L}_A)</span></li>
 <li><span class="math inline">e_1, \ldots, e_n</span> base canonica di
 <span class="math inline">\mathbb{K}^n \implies \dim(\textrm{span}(e_1,
@@ -889,8 +889,8 @@ <h1 class="title"> </h1>
 I_n})</span></li>
 <li>allora, per il teorema del rango <span
 class="math inline">\textrm{rk}(\mathscr{L}_{A - \lambda \cdot I_n}) = n
-- \dim(\ker(\mathscr{L}_{A - \lambda \cdot I_n})) \iff
-\dim(\ker(\mathscr{L}_{A - \lambda \cdot I_n})) = n -
+- \dim(\ker(\mathscr{L}_{A - \lambda \cdot I_n}))
+\iff  \dim(\ker(\mathscr{L}_{A - \lambda \cdot I_n})) = n -
 \textrm{rk}(\mathscr{L}_{A - \lambda \cdot I_n}) =
 \dim(\textrm{E}_\lambda(A)) =: \nu(\lambda)</span></li>
 </ul></li>

html/everything.html

@@ -984,7 +984,7 @@ <h1 class="title"> </h1>
 </ul></li>
 <li><strong>Th</strong>
 <ul>
-<li><span class="math inline">\exists d \in I \mid I = I(d)</span>, o
+<li><span class="math inline">\exists  d  \in I \mid I = I(d)</span>, o
 equivalentemente, in <span class="math inline">\mathbb{Z}</span> ogni
 ideale è principale</li>
 </ul></li>
@@ -997,7 +997,7 @@ <h1 class="title"> </h1>
 <ul>
 <li><span class="math inline">a_{1}, \ldots , a_{n} \in
 \mathbb{Z}</span></li>
-<li><span class="math inline">\exists !d \in \mathbb{N} \mid
+<li><span class="math inline">\exists !d \in \mathbb{N}  \mid
 I\left(a_{1}, \ldots , a_{n}\right)=I(d)</span>, ed è detto
 <strong>massimo comun divisore degli <span class="math inline">a_1,
 \ldots, a_n</span></strong>
@@ -1113,7 +1113,7 @@ <h1 class="title"> </h1>
 <ul>
 <li><span class="math inline">a_{1}, \ldots, a_{n} \in
 \mathbb{Z}</span></li>
-<li><span class="math inline">\displaystyle \exists ! m \in \mathbb{N}
+<li><span class="math inline">\displaystyle \exists ! m  \in \mathbb{N}
 \mid I(m) = I(a_1) \cap \ldots \cap I(a_n) =
 \bigcap_{i=1}^{n}{I(a_i)}</span>, ed è detto <strong>minimo comune
 multiplo degli <span class="math inline">a_1, \ldots,
@@ -1178,7 +1178,7 @@ <h1 class="title"> </h1>
 commutativo</li>
 <li><span class="math inline">I, J \subset A</span> ideali</li>
 <li><span class="math inline">I \cdot J = \{i_1 j_1 + \ldots + i_k j_k
-\mid k \ge 1, i_1 , \ldots , i_k \in I, j_1 , \ldots , j_k \in J
+\mid k \ge  1, i_1 , \ldots , i_k \in I, j_1 , \ldots , j_k \in J
 \}</span> è detto <strong>prodotto tra <span
 class="math inline">I</span> e <span
 class="math inline">J</span></strong></li>
@@ -1376,7 +1376,7 @@ <h1 class="title"> </h1>
 </ul></li>
 <li><strong>Th</strong>
 <ul>
-<li><span class="math inline">x \equiv y \ (\bmod \ d)</span></li>
+<li><span class="math inline">x \equiv y \ (\bmod  \ d)</span></li>
 </ul></li>
 </ul>
 <a href="#teorema-43"><h2 id="teorema-43">Teorema 43</h2></a>
@@ -1993,7 +1993,7 @@ <h1 class="title"> </h1>
 <li><span class="math inline">\textrm{sgn}(\sigma) :=
 (-1)^{|\textrm{Inv}(\sigma)|} =</span><span
 class="math inline">\left\{\begin{array}{ll}+1 &amp;
-|\operatorname{Inv}(\sigma)| \equiv 0 \ (\bmod \ 2) \\ -1 &amp;
+|\operatorname{Inv}(\sigma)| \equiv 0 \ (\bmod  \ 2) \\ -1 &amp;
 |\operatorname{Inv}(\sigma)| \equiv 1 \ (\bmod \
 2)\end{array}\right.</span>
 <ul>
@@ -2769,7 +2769,7 @@ <h1 class="title"> </h1>
 </ul></li>
 <li><strong>Th</strong>
 <ul>
-<li><span class="math inline">\exists p(x) \in I \mid I =
+<li><span class="math inline">\exists  p(x) \in I \mid I =
 I(p(x))</span>, o equivalentemente, in <span
 class="math inline">\mathbb{K}[x]</span> ogni ideale è principale</li>
 </ul></li>
@@ -4711,7 +4711,7 @@ <h1 class="title"> </h1>
 <li><span class="math inline">V(b_0, \ldots, b_n) := \left (
 \begin{array}{cccc} b_0^0 &amp; b_0^1 &amp; \cdots &amp; b_0^n \\ b_1^0
 &amp; b_1^1 &amp; \cdots &amp; b_1^n \\ \vdots &amp; \ddots &amp; &amp;
-\vdots \\\vdots &amp; &amp;\ddots &amp; \vdots \\ b_n^0 &amp; b_n^1
+\vdots \\\vdots &amp; &amp;\ddots  &amp; \vdots \\ b_n^0 &amp; b_n^1
 &amp; \cdots &amp; b_n^n\end{array}\right)</span> è detta
 <strong>matrice di Vandermonde a coefficienti <span
 class="math inline">b_0, \ldots, b_n</span></strong></li>
@@ -4748,7 +4748,7 @@ <h1 class="title"> </h1>
 \rightarrow W:(v_1, \ldots, v_n) \rightarrow w</span></li>
 <li><span class="math inline">f</span> è detta
 <strong>multilineare</strong> <span class="math inline">\iff \forall i
-\in [1, n], v_1 , \ldots, v_n \in V_1 \times \ldots \times V_n, v_i,
+\in [1, n],  v_1 , \ldots, v_n \in V_1 \times \ldots \times V_n, v_i,
 v_i&#39; \in V_i, \lambda, \mu \in \mathbb{K} \quad f(v_1, \ldots,
 \lambda v_i+\mu v_i&#39;, \ldots, v_n) = \lambda f(v_1, \ldots, v_i,
 \ldots, v_n) + \mu f(v_1, \ldots, v_i&#39;, \ldots, v_n)</span>
@@ -5532,7 +5532,7 @@ <h1 class="title"> </h1>
 </ul></li>
 <li><strong>Th</strong>
 <ul>
-<li><span class="math inline">z^{n}=|z|^{n} e^{i \theta n} \quad \arg
+<li><span class="math inline">z^{n}=|z|^{n} e^{i  \theta n} \quad \arg
 \left( z^{n} \right)=n \arg (z)</span></li>
 </ul></li>
 </ul>
@@ -5793,7 +5793,7 @@ <h1 class="title"> </h1>
 <li><strong>Alg</strong>
 <ul>
 <li><span class="math inline">\forall i \in [0, n] \quad p_i(x) :=
-\displaystyle \prod_{\begin{subarray}{c}0 \le j \le n \\ i \neq\ j
+\displaystyle \prod_{\begin{subarray}{c}0 \le  j \le n \\ i \neq\ j
 \end{subarray}}{\dfrac{x - b_j}{b_i - b_j}}</span></li>
 <li><span class="math inline">p(x) := c_0p_0(x) + \ldots + c_n
 p_n(x)</span></li>
@@ -5940,9 +5940,9 @@ <h1 class="title"> </h1>
 <ul>
 <li><strong>Hp</strong>
 <ul>
-<li><span class="math inline">a_1, \ldots, a_n \ge 2 \in \mathbb{Z} \mid
-\textrm{MCD}(a_i, a_j) = 1 \quad \forall i, j \in [1, n] : i \neq
-j</span></li>
+<li><span class="math inline">a_1, \ldots, a_n \ge 2 \in
+\mathbb{Z}  \mid \textrm{MCD}(a_i, a_j) = 1 \quad \forall i, j \in [1,
+n] : i \neq j</span></li>
 <li><span class="math inline">m := \textrm{mcm}(a_1, \ldots,
 a_n)</span></li>
 </ul></li>
@@ -5986,7 +5986,7 @@ <h1 class="title"> </h1>
 <ul>
 <li><span class="math inline">\exists ! x \ (\bmod \ m) \mid</span>
 <span class="math inline">\left\{\begin{array}{c}x \equiv b_{1}\
-\left(\bmod \ a_{1}\right) \\ \vdots \\ x \equiv b_{n}\ \left(\bmod \
+\left(\bmod  \ a_{1}\right) \\ \vdots \\ x \equiv b_{n}\ \left(\bmod  \
 a_{n}\right)\end{array}\right.</span></li>
 </ul></li>
 </ul>

html/gruppi-e-anelli.html

@@ -929,7 +929,7 @@ <h1 class="title"> </h1>
 <ul>
 <li><span class="math inline">0 \in \mathbb{Z} \land g^{0}=e \implies 0
 \in I(g)</span></li>
-<li><span class="math inline">m, n \in \mathbb{Z} \mid g^{m}=g^{n}=e
+<li><span class="math inline">m, n \in \mathbb{Z} \mid  g^{m}=g^{n}=e
 \implies g^{m} \cdot g^{n}= g^{m + n} \iff e \cdot e = e \implies m +n
 \in I(g)</span> per definizoine di <span
 class="math inline">I(g)</span>, quindi <span class="math inline">I(g) +
@@ -1033,7 +1033,7 @@ <h1 class="title"> </h1>
 <li><strong>Dim</strong>
 <ul>
 <li><span class="math inline">I(d) = I(g)</span>, allora <span
-class="math inline">d \in I(d) \implies d \in I(g) \implies g^d =
+class="math inline">d \in I(d) \implies d \in  I(g) \implies g^d =
 e</span></li>
 <li><span class="math inline">d = o(g) = |H(g)| \bigg\vert |G|</span>
 per il teorema di Lagrange, e dunque <span class="math inline">\exists k

html/ideali.html

@@ -358,9 +358,9 @@ <h1 class="title"> </h1>
 <li><span class="math inline">(I(a_1, \ldots, a_n) , +) \leqslant (A,
 +)</span>
 <ul>
-<li><span class="math inline">0 = a_1 \cdot 0 + \ldots + a_n \cdot 0 \in
-I(a_1, \ldots a_n)</span>, dunque <span class="math inline">0</span> è
-l’elemento neutro</li>
+<li><span class="math inline">0 = a_1 \cdot 0  + \ldots + a_n \cdot 0
+\in I(a_1, \ldots a_n)</span>, dunque <span class="math inline">0</span>
+è l’elemento neutro</li>
 <li><span class="math inline">\forall x, y \in I(a_1, \ldots, a_n) \quad
 x = a_1b_1 + \ldots +a_nb_n \land y = a_1c_1 + \ldots+ a_nc_n \implies
 x+ y = a_1b_1 + \ldots + a_nb_n + a_1c_1 + \ldots +a_nc_n</span>, che è
@@ -399,7 +399,7 @@ <h1 class="title"> </h1>
 </ul></li>
 <li><strong>Th</strong>
 <ul>
-<li><span class="math inline">\exists d \in I \mid I = I(d)</span>, o
+<li><span class="math inline">\exists  d  \in I \mid I = I(d)</span>, o
 equivalentemente, in <span class="math inline">\mathbb{Z}</span> ogni
 ideale è principale</li>
 </ul></li>
@@ -477,7 +477,7 @@ <h1 class="title"> </h1>
 <ul>
 <li><span class="math inline">a_{1}, \ldots , a_{n} \in
 \mathbb{Z}</span></li>
-<li><span class="math inline">\exists !d \in \mathbb{N} \mid
+<li><span class="math inline">\exists !d \in \mathbb{N}  \mid
 I\left(a_{1}, \ldots , a_{n}\right)=I(d)</span>, ed è detto
 <strong>massimo comun divisore degli <span class="math inline">a_1,
 \ldots, a_n</span></strong>
@@ -528,7 +528,7 @@ <h1 class="title"> </h1>
 <ul>
 <li><span class="math inline">d</span> è il massimo tra i divisori
 comuni se <span class="math inline">\forall k \in \mathbb{Z}: k \mid
-a_1, \ldots, a_n \quad k \mid d</span></li>
+a_1, \ldots, a_n \quad  k \mid d</span></li>
 <li><span class="math inline">\forall i \in [1, n] \quad k \mid a_i \iff
 \exists x_i \in \mathbb{Z} \mid kx_i = a_i</span></li>
 <li><span class="math inline">d \in I(d) = I(a_1, \ldots, a_n) \iff d
@@ -666,7 +666,7 @@ <h1 class="title"> </h1>
 <ul>
 <li><span class="math inline">a_{1}, \ldots, a_{n} \in
 \mathbb{Z}</span></li>
-<li><span class="math inline">\displaystyle \exists ! m \in \mathbb{N}
+<li><span class="math inline">\displaystyle \exists ! m  \in \mathbb{N}
 \mid I(m) = I(a_1) \cap \ldots \cap I(a_n) =
 \bigcap_{i=1}^{n}{I(a_i)}</span>, ed è detto <strong>minimo comune
 multiplo degli <span class="math inline">a_1, \ldots,
@@ -715,7 +715,7 @@ <h1 class="title"> </h1>
 <ul>
 <li><span class="math inline">m</span> è il minimo tra i multipli comuni
 se <span class="math inline">\forall k \in \mathbb{Z} : a_1, \ldots, a_n
-\mid k \quad m \mid k</span></li>
+\mid k \quad  m \mid k</span></li>
 <li><span class="math inline">\forall i \in [1, n] \quad a_i \mid k \iff
 \exists x_i \in \mathbb{Z} \mid a_ix_i = k \iff k \in I(a_i)</span>,
 allora <span class="math inline">k \in I(a_1) \cap \ldots \cap I(a_n) =
@@ -772,7 +772,7 @@ <h1 class="title"> </h1>
 commutativo</li>
 <li><span class="math inline">I, J \subset A</span> ideali</li>
 <li><span class="math inline">I \cdot J = \{i_1 j_1 + \ldots + i_k j_k
-\mid k \ge 1, i_1 , \ldots , i_k \in I, j_1 , \ldots , j_k \in J
+\mid k \ge  1, i_1 , \ldots , i_k \in I, j_1 , \ldots , j_k \in J
 \}</span> è detto <strong>prodotto tra <span
 class="math inline">I</span> e <span
 class="math inline">J</span></strong></li>