latex_formula
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\begin{align*}
\langle b_{i}\ |\ ab_{j}\rangle b&=\langle b_{i}\ |\ ab_{j}b\rangle \\
&=\langle b_{i}\ |\ bab_{j}\rangle\\
&=\langle b^{*}b_{i}\ |\ ab_{j}\rangle\\
&=\langle b_{i}b^{*}\ |\ ab_{j}\rangle\\
&=b \langle b_{i}\ |\ ab_{j}\rangle
\end{align*} |
\begin{align*}
\langle u^{F,G}_{X,Y}(\sum b_i\boxtimes c_{j} a_{ij})\ |\ u^{F,G}_{X,Y}(\sum b_{i}\boxtimes c_{j} a_{ij})\rangle &=\langle \sum c_{j}\boxtimes_{A} b_{i}\ a_{ij} | \sum c_{j}\boxtimes_{A} b_{i} a_{ij}\rangle\\
&=\sum a^{*}_{ij}\langle b_i\ | \langle c_{j} | c_{k}\rangle b_l \rangle a_{lk}\\
&=\sum a^{*}_{ij}\ \langle b_i\ |\ b_{l}\rangle\ \langle c_{j}\ | c_{k}\rangle\ a_{lk}\\
&=\sum \langle c_{j} a_{ij}\ | \langle b_i\ |\ b_{l}\rangle c_{k} a_{lk} \rangle\\
&=\langle \sum b_i\boxtimes c_{j}\ a_{ij}\ |\ \sum b_{i}\boxtimes c_{j}\ a_{ij}\rangle
\end{align*} |
\begin{align*}
u^{F,G}_{X,Y}(b^{\prime}_{i}\boxtimes c^{\prime}_{j})&=u^{F,G}_{X,Y}\left(\sum_{l,k} b_{l} \langle b_{l}\ |\ b^{\prime}_{i}\rangle \boxtimes c_{k} \langle c_{k}\ |\ c^{\prime}_{j}\rangle \right)\\
&=u^{F,G}_{X,Y}\left(\sum_{l,k} b_{l}\boxtimes c_{k}\ \langle b_{l}\ |\ b^{\prime}_{i}\rangle \langle c_{k}\ |\ c^{\prime}_{j}\rangle\right)\\
&=\sum_{l,k} c_{k}\boxtimes b_{l}\ \langle b_{l}\ |\ b^{\prime}_{i}\rangle \langle c_{k}\ |\ c^{\prime}_{j}\rangle\\
&=c^{\prime}_{j}\boxtimes b^{\prime}_{i}.\\
\end{align*} |
\begin{align*}
u^{F^{\prime},G^{\prime}}_{X,Y}(b_{i}\boxtimes c_{j})&=u_{F^{\prime},G^{\prime}}\left(\sum_{l,k} b^{\prime}_{l} \langle b^{\prime}_{l}\ |\ b_{i}\rangle \boxtimes c^{\prime}_{k} \langle c^{\prime}_{k}\ |\ c_{j}\rangle \right)\\
&=\sum_{l,k} c^{\prime}_{k}\boxtimes b^{\prime}_{l}\ \langle b^{\prime}_{l}\ |\ b_{i}\rangle \langle c^{\prime}_{k}\ |\ c_{j}\rangle\\
&=c_{j}\boxtimes b_{i}=u^{F,G}_{X,Y}(b_{i}\boxtimes c_{j}),\\
\end{align*} |
\begin{align*}
u_{X,Y}(ab_{i}\boxtimes c_{k})&=u_{X,Y}(\sum_{j}b_{j}\langle b_{j}\ | ab_{i}\rangle\boxtimes c_{k})\\
&=u_{X,Y}(\sum_{j} b_{j}\boxtimes c_{k} \langle b_{j}\ | ab_{i}\rangle)\\
&=u_{F,G}(\sum_{j} b_{j}\boxtimes c_{k}) \langle b_{j}\ | ab_{i}\rangle\\
&=\sum_{j} c_{k}\boxtimes b_{j}\langle b_{j}\ | ab_{i}\rangle\\
&=c_{k}\boxtimes ab_{i}\\
&=ac_{k}\boxtimes b_{i}\\
&=au_{X,Y}(b_{i}\boxtimes c_{k})
\end{align*} |
\begin{align*}
u_{X^{\prime},Y^{\prime}}\circ (f\boxtimes g)(b_{i}\boxtimes c_{j})&=\sum_{l,k} u_{X^{\prime},Y^{\prime}}\left(b^{\prime}_{l}\langle b^{\prime}_{l}\ |\ f(b_{i})\rangle \boxtimes c^{\prime}_{k}\langle c^{\prime}_{k}\ |\ g(c_{j})\rangle\right)\\
&=\sum_{l,k} u_{X^{\prime},Y^{\prime}}(b^{\prime}_{l}\boxtimes c^{\prime}_{k}) \langle b^{\prime}_{l}\ |\ f(b_{i})\rangle \langle c^{\prime}_{k}\ |\ g(c_{j})\rangle\\
&=\sum_{l,k} c^{\prime}_{k}\langle c^{\prime}_{k}\ |\ g(c_{j})\rangle\boxtimes b^{\prime}_{l} \langle b^{\prime}_{l}\ |\ f(b_{i})\rangle\\
&=g(c_j)\boxtimes f(b_i)\\
&=(g\boxtimes f)\circ u_{X,Y}(b_{i}\boxtimes c_{j})
\end{align*} |
\begin{align*}
(1_{Y}\boxtimes u_{X,Z})\circ (u_{X,Y}\boxtimes 1_{Z})(b_{i}\boxtimes c_{j}\boxtimes d_{k})&=c_{j}\boxtimes d_{k}\boxtimes b_{i}\\
&=u^{F,K}_{X, Y\boxtimes Z}(b_{i}\boxtimes c_{j}\boxtimes d_{k})\\
&=u_{X,Y\boxtimes Z}(b_{i}\boxtimes c_{j}\boxtimes d_{k}),
\end{align*} |
\begin{align*}
(\mu^{\alpha}_{X,Y})^{*}\circ \alpha_{*}(u_{X,Y})\circ \mu^{\alpha}_{X,Y}(b_{i}\boxtimes_{B} c_{j})&= (\mu^{\alpha}_{X,Y})^{*}(u^{F,G}_{X,Y}(b_i \boxtimes_{A} c_{j}))\\
&=(\mu^{\alpha}_{X,Y})^{*}(c_{j}\boxtimes_{A} b_{i})\\
&=c_{j}\boxtimes_{B} b_{i}\\
&=u^{F^{\prime}, G^{\prime}}_{\alpha_{*}(X),\alpha_{*}(Y)}(b_{i}\boxtimes_{B} c_{j})\\
&=u_{\alpha_{*}(X),\alpha_{*}(Y)}(b_{i}\boxtimes_{B} c_{j})
\end{align*} |
\begin{align*}
&\sum_{i} | 1_{X^{k}}\otimes b_{i}\otimes 1_{X^{k}}\rangle_{A_{[a-k,b+k]}}\ \langle 1_{X^{k}}\otimes b_{i}\otimes 1_{X^{k}}|\\
&= \sum_{i} (1_{X^{k}}\otimes b_{i}\otimes 1_{X^{k}})\circ (1_{X^{k}}\otimes b^{*}_{i}\otimes 1_{X^{k}})\\
&=1_{X^{2k+b-a+1}\otimes Z}=id_{F^{k}_{[a,b]}(Z,c)}
\end{align*} |
\begin{align*}
i_{c,d}(x)\triangleright j_{a,b}(b_{i})&=j_{a-k,b+k}(x\otimes 1_{X^{b-d+k}}\triangleright 1_{X^{k}}\otimes b_{i} \otimes 1_{X^{k}})\\
&=j_{a-k,b+k}(x\otimes 1_{X^{a-d}}\otimes 1_{X^{b-c}}\triangleright 1_{X^{k}}\otimes b_{i} \otimes 1_{X^{k}})\\
&=j_{a-k,b+k}((x\otimes 1_{X^{a-d}}\otimes 1_{X^{b-c}})\circ (1_{X^{k}}\otimes b_{i} \otimes 1_{X^{k}}))\\
&=j_{a-k,b+k}( (1_{X^{k}}\otimes b_{i} \otimes 1_{X^{k}})\circ (x\otimes 1_{X^{a-d}}\otimes 1_{X^{b-c}}\circ)\\
&=j_{a,b}(b_{i})\triangleleft i_{c,d}(x)
\end{align*} |
\begin{align*}
i_{c,d}(x)\triangleright j_{a,b}(b_{i})&=j_{a-k,b+k}(1_{X^{\otimes c-a+k}}\otimes x\triangleright 1_{X^k}\otimes b_{i}\otimes 1_{X^{k}})\\
&=j_{a-k,b+k}(1_{X^{\otimes c-a+k}}\otimes x \triangleright 1_{X^{d-b}}\otimes b_{i} \otimes 1_{X^{d-b-c}})\\
&=(1_{X^{k+b-a}}\otimes c^{*}_{Z,X^{k}})\circ (1_{X^{c-a+k}}\otimes x) \circ (1_{X^{k+b-a}}\otimes c_{Z,X^{k}})\circ(1_{X^{k}}\otimes b_i \otimes 1_{X^{k}})\\
&=j_{a-k,b+k}( (1_{X^{k}}\otimes b_{i} \otimes 1_{X^{k}})\circ (x\otimes 1_{X^{c-a+k}}))\\
&=j_{a,b}(b_{i})\triangleleft i_{c,d}(x)
\end{align*} |
\begin{align}\mu^{[a,b]}_{(Z,c),(W,d)}\circ u_{F_{a,b}(Z,c),F_{a,b}(W,d)}(j_{a,b}(e_{i})\boxtimes j_{a,b}(f_{j}))= j_{a,b}\left(1_{X^{b-a}}\otimes c_{Z, W})\circ (e_{j}\otimes 1_{W})\circ f_{j}\right)
\end{align} |
\begin{align*}
&u_{F_{a,b}(Z,c),F_{a,b}(W,d)}(j_{a,b}(e_{i})\boxtimes j_{a,b}(f_{j}))\\
&=u_{F_{a,b}(Z,c),F_{a,b}(W,d)}\left(\sum_{l} j_{a,b}(e_{i})\boxtimes \kappa_{0}(f^{\prime}_{l}) \langle k_{0}(f^{\prime}_{l})\ | j_{a,b}(f_{j}))\rangle\right)\\
&=\sum_{l} \kappa_{0}(f^{\prime}_{l}) \boxtimes j_{a,b}(e_{i})\langle \kappa_{0}(f^{\prime}_{l})\ | j_{a,b}(f_{j}))\rangle\\
&=\sum_{l,s} j_{a,b}(f_{s})\langle j_{a,b}(f_s)\ |\kappa_{0}(f^{\prime}_{l})\rangle\ \boxtimes\ j_{a,b}(e_{i})\langle \kappa_{0}(f^{\prime}_{l})\ | j_{a,b}(f_{j}))\rangle\\
&=\sum_{s,l} j_{a,b}(f_{s})\ \boxtimes\ \langle j_{a,b}(f_s)\ |\kappa_{0}(f^{\prime}_{l})\rangle j_{a,b}(e_{i})\langle \kappa_{0}(f^{\prime}_{l})\ | j_{a,b}(f_{j}))\rangle
\end{align*} |
\begin{align*}
&\sum_{l}\langle j_{a,b}(f_s)\ |\kappa_{0}(f^{\prime}_{l})\rangle j_{a,b}(e_{i})\langle \kappa_{0}(f^{\prime}_{l})\ | j_{a,b}(f_{j}))\rangle\\
&=j_{a,b}\left( (f^{*}_{s}\otimes 1_{Z}) \circ 1_{X^{b-a}}\otimes c_{Z,W})\circ (e_{i}\otimes 1_{W})\circ f_{j}\right)
\end{align*} |
\begin{align*}
&\mu^{[a,b]}_{(Z,c),(W,d)}\circ u_{F_{a,b}(Z,c),F_{a,b}(W,d)}(j_{a,b}(e_{i})\boxtimes j_{a,b}(f_{j}))\\
&=j_{a,b}\circ \mu^{0;[a,b]}_{(Z,c),(W,d)}\left( \sum_{s} f_{s}\boxtimes (f^{*}_{s}\otimes 1_{Z}) \circ 1_{X^{b-a}}\otimes c_{Z,W})\circ (e_{i}\otimes 1_{W})\circ f_{j}\right)\\
&=j_{a,b}\left((1_{X^{b-a}}\otimes c_{Z, W})\circ (e_{j}\otimes 1_{W})\circ f_{j}\right)
\end{align*} |
\begin{align*}
u^{F,G}_{X,Y}(a(b_{i}\boxtimes c_{j}))&=\sum_{k} c_{k}\boxtimes b_{i}\langle c_{k}\ |\ ac_{j}\rangle\\
&=\sum_{k} c_{k}\boxtimes \langle c_{k}\ |\ ac_{j}\rangle b_{i}\\
&=\sum_{k} c_{k}\langle c_{k}\ |\ ac_{j}\rangle\boxtimes b_{i}\\
&=ac_{j}\boxtimes b_{i}=a(c_{j}\boxtimes b_i)\\
&=au^{F,G}_{X,Y}(b_{i}\boxtimes c_{j}).
\end{align*} |
\begin{align*}
u^{F,G}_{X,Y}(a(b_{i}\boxtimes c_{j}))&=\sum_{l}c_{j}\boxtimes b_{l}\langle b_{l}\ |\ ab_{i}\rangle\\
&=c_{j}\boxtimes ab_{i}=c_{j}a\boxtimes b_{i}\\
&=ac_{j}\boxtimes b_{i}=a(c_{j}\boxtimes b_{i})\\
&=au^{F,G}_{X,Y}(b_{i}\boxtimes c_{j}).
\end{align*} |
\begin{equation}
R_1 \times R_2 \ = \ \{ (x_1,x_2),(y_1,y_2)) : (x_1,y_1) \in R_1 \ \ \text{and} \ \ (x_2,y_2) \in R_2 \}.
\end{equation} |
\begin{equation}
R \cap R^{-1} \ = \ 1_X, \text{and} R \cup R^{-1} \ = \ X \times X.
\end{equation} |
\begin{align}\begin{split}
x \ \ \text{is right balanced} &\Longleftrightarrow \overline{R^{\circ}(x)} = R(x) \\
x \ \ \text{is left balanced} &\Longleftrightarrow \overline{R^{\circ -1}(x)} = R^{-1}(x)\\
x \ \ \text{is balanced} &\Longleftrightarrow x \ \ \text{is both left and right balanced}.
\end{split}\end{align} |
\begin{align}\begin{split}
X_2 \ = \ X_1 \times \{ Y_x \} \ = \ &\ \bigcup_{x \in X} \ \{ x \} \times Y_x, \\
R_2 \ = \ R_1 \ltimes \{ S_x \} \text{where} \ &\text{for} \ (x_1,y_1), (x_2,y_2) \in X_2, \\
((x_1,y_1), (x_2,y_2)) \in R_2 \Longleftrightarrow &\begin{cases} (x_1,x_2) \in R_1^{\circ} \text{or}\\
x_1 = x_2 \ \text{and} \ (y_1,y_2) \in S_{x_1}. \end{cases}
\end{split} \end{align} |
\begin{equation}
X \ = \ \{ x \in \prod_{i \in \N} \ X_i : f_i(x_{i+1}) \ = \ x_i \ \ \text{for all} \ \ i \in \N \},
\end{equation} |
\begin{align}\begin{split}
R \cap R^{-1} \ = \ &\bigcap_{i} \ (\pi_i \times \pi_i)^{-1}(R_i \cap R_i^{-1}) \\
= \ \bigcap_{i} \ &(\pi_i \times \pi_i)^{-1}(1_{X_i}) \ = \ 1_X.
\end{split}\end{align} |
\begin{equation}
(y,z) \in S^{\circ} \Longleftrightarrow (y_i,z_i) \in S_i \ \ \text{for} \ \ i = \min \{ j : y_j \not= z_j \},
\end{equation} |
\begin{equation}
q(x)_{i} \ = \ g_{i+1}(\pi_{i+1}(x)), \ \ \text{for} \ \ i \in \Z_+
\end{equation} |
\begin{equation}
A \cap A^{-1} \ = \ \{ e \}, \text{and} A \cup A^{-1} \ = \ G.
\end{equation} |
\begin{align} \begin{split}
\widehat{A} \ = \ \{ (x,y) : x^{-1}y &\in A \} \text{so that} \ \ \widehat{A}^{-1} \ = \ \widehat{A^{-1}},\\
\text{and so} \ \ \widehat{A}^{\circ} \ &= \ \{ (x,y) : x^{-1}y \in A^{\circ} \}.
\end{split}\end{align} |
\begin{align} \begin{split}
h_x(y) \ = \ &x y^{-1} x \text{so that} \ \ h_x(x) \ = \ x, \\
\text{and} &x^{-1}h_x(y) \ = \ y^{-1} x, \\
\text{and} &h_x \circ h_x = 1_G.
\end{split}\end{align} |
\begin{equation}
\om(x) \ = \ \bigcap_{n \in \N} \ \overline{\{ x^i : i \ge n \}}
\end{equation} |
\begin{equation}
A \ = \ \bigcap_{i \in \N} \ \pi_{i}^{-1}(A_i) \ = \ \overleftarrow{Lim} \{ A_i \}
\end{equation} |
\begin{align} \begin{split}
a_i, b_i \in &U_{i-1},\overline{U_{i}} \subset U_{i-1}, \\
\{ a_i, z, b_i \} \ \text{ is a } \ &3-\text{cycle, for all} \ z \in \overline{U_i}.
\end{split}\end{align} |
\begin{equation}
(x_n)_A \ = \ \begin{cases} 1 \ \ \text{if} \ \ n \in A, \\ 0 \ \ \text{if} \ \ n \not\in A. \end{cases}
\end{equation} |
\begin{equation}
((x,t),(y,s)) \in R_2^{\circ} \Longleftrightarrow \begin{cases} (x,y) \in L_1^{\circ}, \text{or} \\
x = y \ \ \text{and} \ \ (t,s) \in S_x^{\circ}. \end{cases}
\end{equation} |
\begin{equation}
Q \ = \ \bigcup_{n=2}^{\infty} \ ( \{ t \} \times A_n) \times (\{ t_n \} \times A_1).
\end{equation} |
\begin{align}\begin{split}
Q \ = \ \{ (x,y,z) : &(x,z), (z,y) \in R \} \ \cup \\ &\{ (x,y,z) : (y,z),(z,x) \in R \}, \\
Q^{\circ} \ = \ \{ (x,y,z) : &(x,z), (z,y) \in R^{\circ} \} \ \cup \\ &\{ (x,y,z) : (y,z),(z,x) \in R^{\circ} \}
\end{split}\end{align} |
\begin{align} \begin{split}
\{ x \} \ = \ \ &Q(x,x), \emptyset \ = \ Q^{\circ}(x,x). \\
\{ x, y \} \ \subset \ \ &Q(x,y) \ = \ Q(\{ x,y \} \times \{ x,y \}). \\
Q(x,y) \setminus \{ x,y \} \ = \ \ &Q^{\circ}(x,y) \ = \ Q^{\circ}(\{ x,y \} \times \{ x,y \}).
\end{split}
\end{align} |
\begin{equation}
f'_i \circ q_{i+1} \ = \ q_i \circ f_i, \text{and} q_i \circ h_i \ = \ h_i'.
\end{equation} |
\begin{equation}
h_{n+1}(x) \ = \ (h_n(x),\pi_{nh_n(x)}(x)) \text{for all} \ \ x \in X.
\end{equation} |
\begin{align} \begin{split}
(a_1, b_1), \ \ (b_2, a_2), (a_1, b_2), &\ \ (a_2, b_1), (a_1, a_2), \ \ (b_1, b_2), \\
(c, a_1), \ \ (c, a_2), &(b_1,c), \ \ (b_2,c).
\end{split}\end{align} |
\begin{align} \begin{split}
a \in J &\Longrightarrow a- \ha a+, \ a+ \ha 0,\ 0 \ha a- \ \ \text{in} \ 2P. \\
a \ha b \ \ \text{in} \ P &\Longrightarrow a+ \ha b+, \ a- \ha b-, \ b+ \ha a-, \ b- \ha a+\ \ \text{in} \ 2P.
\end{split}\end{align} |
\begin{equation}
i \ha j \Longleftrightarrow i < j \text{including } \ \ j = \infty.
\end{equation} |
\begin{equation}
i+1 \ha i \ha j \Longleftrightarrow i+1 < j \text{including } \ \ j = \infty.
\end{equation} |
\begin{align} \begin{split}
\N+ \ = \ (2L)^{\circ -1}(\infty), &\N- \ = \ (2L)^{\circ}(\infty), \\
(i-, i+) \ \in \ 2L &\text{for all} \ \ i \in \N.
\end{split}\end{align} |
\begin{equation}
i- \ha i+, (i+2)+ i+ \ha j- \text{for all} \ \ j \not= i, i-2.
\end{equation} |
\begin{equation}
i- \ha i+, i+ \ha j- \text{for all} \ \ j \not= i.
\end{equation} |
\begin{equation}
\tilde \pi(0,x) \ = \ x, \text{and} \tilde \pi(1,x) \ = \ \infty.
\end{equation} |
\begin{align}\begin{split}
(Y_{i+},S_{i+}) \ = \ &(Z+,P+), \\ (Y_{i-},S_{i-}) \ = \ &(Z-,P-)
\end{split}\end{align} |
\begin{equation}
R'|X \ = \ R, \text{and} R'^{\circ}(u) \ = \ F, \ \ R'^{\circ -1}(u) \ = \ E.
\end{equation} |
\begin{equation}
E_1 \ \not= \ E_2 \Leftrightarrow F_1 \ \not= \ F_2 \Leftrightarrow [(E_1 \cap F_2) \cup (E_2 \cap F_1)] \ \not= \ \emptyset.
\end{equation} |
\begin{equation}
T \ = \ S \cup R \cup ( \bigcup_i [(E_i \times C_i) \cup (C_i \times F_i)]).
\end{equation} |
\begin{equation}
C_1 \ = \ E_1 \ = \ F_2 \ = \ E, \text{and} C_2 \ = \ F_1 \ = \ E_2 \ = \ F.
\end{equation} |
\begin{equation}
\s(y) \ = \ \begin{cases} \ \ y/2 \ \text{if} \ y \ \text{is even}, \\ \ (y - \1 )/2 \ \text{if} \ y \ \text{is odd}.\end{cases}
\end{equation} |
\begin{align}\begin{split}
(i) & 0^{j-1}1x + 0^{i-1}1 z \ = \ 0^{i-1}1 y, \text{with} \ \ z + 0^{j-i-1}1x = y, \\
(ii) &0^{i-1}10x + 0^{i-1}10z \ = \ 0^i1y \ \text{with} \ \ z + x = y, \\
(iii) &0^{i-1}11x + 0^{i-1}11z \ = \ 0^i1y \ \text{with} \ \ z + x + \1 = y, \\
(iv) &0^{i-1}10x + 0^{i}1\ep z \ = \ 0^{i-1}11y \ \text{with} \ \ \ep z + x = y \ (\ep = 0,1).
\end{split}\end{align} |
\begin{align}\begin{split}
x \in A_i \ \ \ \text{and} \ \ x' \in A_j \cup - A_j \ \ &\Rightarrow \ \ (x', x) \in \widehat{A}^{\circ}, \\
x \in -A_i \ \ \text{and} \ \ x' \in A_j \cup - A_j \ &\Rightarrow \ \ (x, x') \in \widehat{A}^{\circ}, \\
x \in A_i \ \ \text{and} \ \ x' \in A_{i+1} \cup - A_{i+1} \ \ &\Rightarrow \ \ (x, x') \in \widehat{A}^{\circ}, \\
x \in -A_i \ \ \text{and} \ \ x' \in A_{i+1} \cup - A_{i+1} \ \ &\Rightarrow \ \ (x, x') \in \widehat{A}^{\circ}.
\end{split}\end{align} |
\begin{equation}
(x,x') \in \widehat{A}^{\circ} \ \ \text{if } \ \ x_{i+2} = x'_{i+2} \ \ \text{and} \ \ (x',x) \in \widehat{A}^{\circ} \ \ \text{otherwise}.
\end{equation} |
\begin{equation}
D_j \ = \ \{ x \in \Z[2] : x_j = 0 \}, \bar D_j \ = \ \{ x \in \Z[2] : x_j = 1 \}.
\end{equation} |
\begin{equation}
\t_j(w0x) \ = \ w1(x+ \1 ), \t_j(w1x) \ = \ w0x \text{with} \ \ w \in \{0, 1 \}^{j-1}, x \in \Z[2].
\end{equation} |
\begin{align}\begin{split}
P[j,k] \ = \ \widehat{A}+ \ &\cup \ \widehat{A}- \ \cup\\
[(D_j- \times D_j+) \cup (\bar D_j- \times \bar D_j+)] \ &\cup \ [(D_j+ \times \bar D_j-) \cup (\bar D_j+ \times D_j-)].
\end{split}\end{align} |
\begin{align}\begin{split}
\t_{j,k}(z+) \ = \ (\t_j(z))+, &\t_{j,k}(z-) \ = \ (\t_k(z))-, \\
\r_{j,k}(z+) \ = \ z-, &\r_{j,k}(z-) \ = \ (\t_j(z))+.
\end{split}\end{align} |
\begin{equation}
P[j,k]^{\circ}(z+) \ = \ \bar D_k- \ \cup \ [\bigcup_{i \in \N} \ (z + A_i)+].
\end{equation} |
\begin{equation}
Z_{z+} \ = \ \{ z+ \} \ \cup \ \{ \bar d_k - \} \ \cup \ \{ i+ : i \ge j-1 \} \ \cup \ [\bigcup_{i=1}^{j-2} \ (z + A_i)+].
\end{equation} |
\begin{align}\begin{split}
P[j] \ = \ \widehat{A}+ \ &\cup \ \widehat{A}- \ \cup\\
(D_j- \times D_j+) \ \cup \ (\bar D_j- \times \bar D_j+) \ &\cup \ (D_j+ \times \bar D_j-) \ \cup \ (\bar D_j+ \times D_j-).
\end{split}\end{align} |
\begin{align}\begin{split}
T[\th](x) \ = \ (\{ i- \} \times &\widehat{A}(z)) \ \cup (\{ i+ \} \times D_j) \ \cup\\
(\{ (i+1)- \} \ &\cup \ \{ k- : k < i-1 \}) \times \Z[2]).
\end{split}\end{align} |
\begin{align}\begin{split}
T[\th](x) \ = \ (\{ i+ \} \times &\widehat{A}(z)) \ \cup (\{ i- \} \times \bar D_j) \ \cup \
\{ (i+1)- \} \times \Z[2] \ \cup \\
[\ \bigcup_{k < i} \ \{ j- \} \times \Z[2]\ ] \ &\cup \ [\bigcup_{k > i+1} \ \{ k-, k+ \} \times \Z[2]\ ] \ \cup \ \{ \infty \}.
\end{split}\end{align} |
\begin{align}\begin{split}
K'_1 \ &= \ (\bigcup_{k > i+1} \ \{ k-, k+ \} \times \Z[2]) \cup \{ \infty \}, \\
K'_2 \ &= \ \{ 1-, 2-, \dots, (i-1)-, (i- , \bar d), (i+1)-,\\ &((i+2)-,a), ((i+3)-,a), \dots, \infty \}. \\
K'_3 \ &= \ \{ (i-, \bar d) \} \cup \{ (i+,k) : j-1 \le k \le \infty \} \cup (\bigcup_{k < j-1} \ \{ i+ \} \times \{ z + A_k \}).
\end{split}\end{align} |
\begin{equation}
R \ = \ S \ \cup \ P \ \cup \ \
\bigcup_i [H_i \times (E_i \cup G_i) \ \cup \ F_i \times H_i].
\end{equation} |
\begin{align}\begin{split}
R \ = \ &2L \cup \ P \ \cup \\
\bigcup_i [\{ i+ \} \times &(E_i \cup G_{i})] \cup [F_i \times \{ i+ \}] \cup \\
\bigcup_i [\{ i- \} \times &E_i] \cup [(F_i \cup G_i) \times \{ i- \}].
\end{split} \end{align} |
\begin{align}\begin{split}
A+ \ = \ \{ i+ : i &\leq n \}, A- \ = \ \{ i- : i \leq n \}, \\
B+ \ = \ \{ i+ : n < i &\leq n + m \}, B- \ = \ \{ i- : n < i \leq n + m \}, \\
C \ = \ K \setminus (A+ \cup \ A- \cup \ &B+ \cup B-) \ = \ \{ i+, i- : n + m < i \} \cup \{ \infty \}.
\end{split} \end{align} |
\begin{equation}
A(\ep)_i \ = \ \begin{cases} -A_i \ \ \text{ when} \ \ \ep_i = 1, \\
\ \ \ A_i \ \ \text{ when} \ \ \ep_i = 0. \end{cases} \text{and} A(\ep) = \{ \0 \} \cup (\bigcup_{i} A(\ep)_i).
\end{equation} |
\begin{equation} \begin{split}
0x \in A(\ep)_{i+1} = (-1)^{\ep_{i+1}}A_{i+1} \ \ \Longleftrightarrow \\ x \in (-1)^{\ep_{i+1}}A_{i} = (-1)^{(\s(\ep)_{i}}A_{i} = A(\s(\ep))_i.
\end{split}\end{equation} |
\begin{align}\begin{split}
x \in A_i(\ep) \ \ \text{and} \ \ x' \in A_j(\ep) \cup (- A_j(\ep)) \ \ &\Rightarrow \ \ (x', x) \in \widehat{A(\ep)}^{\circ}, \\
x \in -A_i(\ep) \ \ \text{and} \ \ x' \in A_j(\ep) \cup (- A_j(\ep)) \ \ &\Rightarrow \ \ (x, x') \in \widehat{A(\ep)}^{\circ}, \\
x \in A_i(\ep) \ \ \text{and} \ \ x' \in A_{i+1}(\ep) \cup (- A_{i+1}(\ep)) \ \ &\Rightarrow \ \ (x, x') \in \widehat{A(\ep)}^{\circ}, \\
x \in -A_i(\ep) \ \ \text{and} \ \ x' \in A_j(\ep) \cup (- A_j(\ep)) \ \ &\Rightarrow \ \ (x, x') \in \widehat{A(\ep)}^{\circ}.
\end{split}\end{align} |
\begin{align}\begin{split}
x \in A_i(\ep) \ \ \text{and} \ \ x' \in -A_i(\ep) \ \ &\Rightarrow \ \ (x, x') \in \widehat{A(\ep)}^{\circ} \\
\text{if either } \ x_{i+2} = x'_{i+2} \ \text{and} \ \ep_i = \ep_{i+1} \ \ &\text{or} \
\ x_{i+2} = \bar x'_{i+2} \ \text{and} \ \ep_i = \bar \ep_{i+1}, \\
\text{ and} \ \ (x', x) \in &\widehat{A(\ep)}^{\circ} \ \ \text{otherwise}.
\end{split}\end{align} |
\begin{align}\begin{split}
\text{For} \ \ (\ep_1,\ep_2) = (0,0), &10z \mapsto 10 h[\s^2(\ep)](z), \ \ 11x \mapsto 11 h[\s^2(\ep)](z). \\
\text{For} \ \ (\ep_1,\ep_2) = (0,1), &10z \mapsto 10 (h[\s^2(\ep)](z) + \1 ), \ \ 11x \mapsto 11 h[\s^2(\ep)](z). \\
\text{For} \ \ (\ep_1,\ep_2) = (1,0), &10z \mapsto 11 (h[\s^2(\ep)](z) + \1 ), \ \ 11z \mapsto 10 h[\s^2(\ep)](z). \\
\text{For} \ \ (\ep_1,\ep_2) = (1,1), &10z \mapsto 11 h[\s^2(\ep)](z), \ \ 11z \mapsto 10 h[\s^2(\ep)](z).
\end{split}\end{align} |
\begin{align}\begin{split}
x \ \cong \ x' \ (mod \ 2^k) \ &\Longleftrightarrow \ z \ \cong \ z' \ (mod \ 2^{k-1})
\Longleftrightarrow \\
h(\s(\ep))(z) \ \cong \ &h(\s(\ep'))(z') \ (mod \ 2^{k-1}) \Longleftrightarrow \\
h[\ep](x) = 0h(\s(\ep)(z) &\ \cong \ 0h(\s(\ep')(z') = h[\ep'](x') \ (mod \ 2^k).
\end{split}\end{align} |
\begin{align}\begin{split}
\max ( \bar u(Hx, Hy), \bar u(Hy, Hz)) \ = \ &\max( u(x_1, y_1),u(y_1, z_1))\\
\ge \ u(x_1,z_1) \ \ge \ &\bar u(Hx, Hz).
\end{split}\end{align} |
\begin{equation}
dt= r\, d\tau \Longleftrightarrow \frac{d\tau}{dt}=\frac 1 r,
\end{equation} |
\begin{equation}
\frac{dx^i}{dt}=X^i(x^1,\ldots,x^n), i=1,\ldots,n,
\end{equation} |
\begin{equation}
dt=f(x^1,\ldots,x^n)\ d\tau, f(x^1,\ldots,x^n)>0,
\end{equation} |
\begin{equation}
\frac{dx^i}{d\tau}=f(x^1,\ldots,x^n)\ X^i(x^1,\ldots,x^n),i=1,\ldots,n.
\end{equation} |
\begin{equation}
X= \sum_{i=1}^nX^i(x^1,\ldots,x^n)\pd{}{x^i},
\end{equation} |
\begin{equation}
\frac{d\tau}{dt}=\frac 1{ f(\gamma(t))},
\end{equation} |
\begin{equation}
\Delta(q,v)= \sum_{i=1}^nv^i\pd{}{v^i},
\end{equation} |
\begin{equation}
S= \sum_{i=1}^n\pd{}{v^i}\otimes dq^i.
\end{equation} |
\begin{equation}
D(q,v)= \sum_{i=1}^nf^i(q,v)\pd{}{v^i},
\end{equation} |
\begin{equation}
\mathcal{L}_\Delta S=-S.
\end{equation} |
\begin{equation}\left\{
\begin{array}{rcl} \dfrac{dq^i}{dt}&=&v^i\\\dfrac{dv^i}{dt}&=&f^i(q,v)\end{array}
\right. i=1,\ldots,n,
\end{equation} |
\begin{equation}
\theta_L= \sum_{i=1}^n\pd L{v^i}\, dq^i,E_L= \sum_{i=1}^nv^i\pd L{v^i}-L.
\end{equation} |
\begin{equation}
i(\Gamma)\omega_L=dE_L.
\end{equation} |
\begin{equation} \mathcal{L}_\Gamma\theta_L=dL,
\end{equation} |
\begin{equation}
\Delta=\sum_{\alpha=1}^{n-k}y^\alpha\pd{}{y^\alpha}.
\end{equation} |
\begin{equation}
N_T(X_1,X_2)=[T(X_1),T(X_2)]+T^2([X_1,X_2])- T([T(X_1),X_2])-T([X_1,T(X_2)]),\ \forall X_1,X_2\in\mathfrak{X}(M),
\end{equation} |
\begin{equation}
S=\sum_{i=1}^n\pd{}{u^i}\otimes dx^i.
\end{equation} |
\begin{equation}
\bar X(x)=\sum_{i=1}^n f^i(x)\pd{}{x^i},
\end{equation} |
\begin{equation}
X(x,u)=\sum_{i=1}^n f^i(x)\pd{}{x^i}+\sum_{i,j=1}^n\pd {f^i}{x^j}u^j\pd{}{u^i},
\end{equation} |
\begin{equation}
\Delta = \sum_{i=1}^nu^i\pd{}{u^i}.
\end{equation} |
\begin{equation}
\frac{d^2x^i}{dt^2}=X^i(x^1,\ldots,x^n,\dot x^1,\ldots,\dot x^n), i=1,\ldots,n,
\end{equation} |
\begin{equation}\left\{
\begin{array}{rcl}
{\displaystyle \frac{dx^i}{dt}}&=&v^i\\ {\displaystyle\frac{dv^i}{dt}}&=&X^i(x^1,\ldots,x^n,v^1,\ldots, v^n)
\end{array}\right. i=1,\ldots,n,
\end{equation} |
\begin{equation}
\Gamma= \sum_{i=1}^n\left(v^i\pd{}{x^i}+X^i(x,v)\pd{}{v^i}\right),
\end{equation} |
\begin{equation}
\begin{array}{rcl}[\tilde \Delta,\Gamma]&=&{\displaystyle\left[ \sum_{i=1}^n\left(x^i\pd{}{x^i}+v^i\pd{}{v^i}\right), \sum_{j=1}^n\left(v^j\pd{}{x^j}+X^j(x , v)\pd{}{v^j} \right)\right] }= \\
&=& {\displaystyle\sum_{j, k=1}^n\left(x^j\pd{X^k}{x^j}\pd{}{v^k}+v^j\pd{X^k}{v^j}\pd{}{v^k}\right)-\sum_{j=1}^nX^j(x , v)\pd{}{v^j}},
\end{array}
\end{equation} |
\begin{equation}X^i(x,v)= \sum_{j=1}^n(A^i\,_j\,x^j+B^i\,_j \, v^j),i=1,\ldots,n.
\end{equation} |