diff options
Diffstat (limited to 'graf')
| -rw-r--r-- | graf/n1.lyx | 32 | ||||
| -rw-r--r-- | graf/n2.lyx | 24 | ||||
| -rw-r--r-- | graf/n3.lyx | 32 | ||||
| -rw-r--r-- | graf/n4.lyx | 22 | ||||
| -rw-r--r-- | graf/n5.lyx | 40 | ||||
| -rw-r--r-- | graf/n6.lyx | 74 | ||||
| -rw-r--r-- | graf/n7.lyx | 26 |
7 files changed, 127 insertions, 123 deletions
diff --git a/graf/n1.lyx b/graf/n1.lyx index 921c7d8..0f2f84f 100644 --- a/graf/n1.lyx +++ b/graf/n1.lyx @@ -209,7 +209,7 @@ Dados un grafo \end_inset y -\begin_inset Formula $e:=(i,j)\in E$ +\begin_inset Formula $e\coloneqq (i,j)\in E$ \end_inset , @@ -346,7 +346,7 @@ G^{\complement}:=(V,E^{\complement}):=(V,\{S\in{\cal P}(V)\mid |S|=2,S\notin E\} \end_inset Un grafo -\begin_inset Formula $G':=(V',E')$ +\begin_inset Formula $G'\coloneqq (V',E')$ \end_inset es un @@ -354,7 +354,7 @@ Un grafo subgrafo \series default de -\begin_inset Formula $G:=(V,E)$ +\begin_inset Formula $G\coloneqq (V,E)$ \end_inset si @@ -404,11 +404,11 @@ inducido \end_inset a -\begin_inset Formula $G_{V'}:=(V',E_{V'})$ +\begin_inset Formula $G_{V'}\coloneqq (V',E_{V'})$ \end_inset , donde -\begin_inset Formula $E_{V'}:=\{S\in E\mid S\subseteq V'\}$ +\begin_inset Formula $E_{V'}\coloneqq \{S\in E\mid S\subseteq V'\}$ \end_inset , y @@ -462,7 +462,7 @@ independiente \end_inset , -\begin_inset Formula $G-v:=G-\{v\}$ +\begin_inset Formula $G-v\coloneqq G-\{v\}$ \end_inset , y si @@ -470,7 +470,7 @@ independiente \end_inset , -\begin_inset Formula $G-e:=G-\{e\}$ +\begin_inset Formula $G-e\coloneqq G-\{e\}$ \end_inset . @@ -504,11 +504,11 @@ maximal \begin_layout Standard Dos grafos -\begin_inset Formula $G:=(V,E)$ +\begin_inset Formula $G\coloneqq (V,E)$ \end_inset y -\begin_inset Formula $G':=(V',E')$ +\begin_inset Formula $G'\coloneqq (V',E')$ \end_inset son @@ -540,7 +540,7 @@ Grado de un nodo \begin_layout Standard Dado un grafo -\begin_inset Formula $G:=(V,E)$ +\begin_inset Formula $G\coloneqq (V,E)$ \end_inset , llamamos @@ -627,11 +627,11 @@ eje colgante \series default . Llamamos -\begin_inset Formula $\delta_{G}:=\min_{v\in V}o(v)$ +\begin_inset Formula $\delta_{G}\coloneqq \min_{v\in V}o(v)$ \end_inset y -\begin_inset Formula $\Delta_{G}:=\max_{v\in V}o(v)$ +\begin_inset Formula $\Delta_{G}\coloneqq \max_{v\in V}o(v)$ \end_inset . @@ -749,7 +749,7 @@ Teorema de Erdös y Gallai : \series default Una secuencia -\begin_inset Formula $S:=(d_{1},\dots,d_{n})$ +\begin_inset Formula $S\coloneqq (d_{1},\dots,d_{n})$ \end_inset monótona decreciente de naturales es una secuencia gráfica si y sólo si @@ -1598,7 +1598,7 @@ status open \end_inset Sea -\begin_inset Formula $G:=(\{1,\dots,n\},E)$ +\begin_inset Formula $G\coloneqq (\{1,\dots,n\},E)$ \end_inset un grafo con @@ -2008,7 +2008,7 @@ Representaciones matriciales \begin_layout Standard Dado un grafo no dirigido -\begin_inset Formula $G:=(\{1,\dots,n\},E)$ +\begin_inset Formula $G\coloneqq (\{1,\dots,n\},E)$ \end_inset , la @@ -2020,7 +2020,7 @@ matriz de adyacencia \end_inset es la matriz -\begin_inset Formula $A:=(a_{ij})_{ij}\in{\cal M}_{n}(\mathbb{Z})$ +\begin_inset Formula $A\coloneqq (a_{ij})_{ij}\in{\cal M}_{n}(\mathbb{Z})$ \end_inset dada por diff --git a/graf/n2.lyx b/graf/n2.lyx index 9d905d7..dd4d85b 100644 --- a/graf/n2.lyx +++ b/graf/n2.lyx @@ -513,7 +513,7 @@ Recorrido de componentes conexas \begin_layout Standard Sean -\begin_inset Formula $G:=(V,E)$ +\begin_inset Formula $G\coloneqq (V,E)$ \end_inset un grafo de orden @@ -1224,7 +1224,7 @@ Demostración: \end_inset , -\begin_inset Formula $Y:=[V_{1},V\setminus V_{1}]$ +\begin_inset Formula $Y\coloneqq [V_{1},V\setminus V_{1}]$ \end_inset es un corte contenido estrictamente en @@ -1336,7 +1336,7 @@ Si \end_inset Sean -\begin_inset Formula $[V_{1},V_{2}]:=\{e\}$ +\begin_inset Formula $[V_{1},V_{2}]\coloneqq \{e\}$ \end_inset , @@ -1653,7 +1653,7 @@ Demostración: \end_inset y -\begin_inset Formula $q:=|V_{1}|$ +\begin_inset Formula $q\coloneqq |V_{1}|$ \end_inset . @@ -2013,7 +2013,7 @@ Si . Si -\begin_inset Formula $e:=(u,v)\in E$ +\begin_inset Formula $e\coloneqq (u,v)\in E$ \end_inset , como @@ -2137,15 +2137,15 @@ grafo en línea \end_inset es -\begin_inset Formula $L(G):=(V^{L},E^{L})$ +\begin_inset Formula $L(G)\coloneqq (V^{L},E^{L})$ \end_inset dado por -\begin_inset Formula $V^{L}:=E$ +\begin_inset Formula $V^{L}\coloneqq E$ \end_inset y -\begin_inset Formula $E^{L}:=\{(e,f)\mid e\neq f,e\cap f\neq\emptyset\}$ +\begin_inset Formula $E^{L}\coloneqq \{(e,f)\mid e\neq f,e\cap f\neq\emptyset\}$ \end_inset . @@ -2197,15 +2197,15 @@ Demostración: \end_inset y -\begin_inset Formula $G':=(V',E')$ +\begin_inset Formula $G'\coloneqq (V',E')$ \end_inset dado por -\begin_inset Formula $V':=V^{L}\dot{\cup}\{x,y\}$ +\begin_inset Formula $V'\coloneqq V^{L}\dot{\cup}\{x,y\}$ \end_inset y -\begin_inset Formula $E':=E^{L}\cup\{((i,u),x)\}_{(i,u)\in E}\cup\{((j,v),y)\}_{(j,v)\in E}$ +\begin_inset Formula $E'\coloneqq E^{L}\cup\{((i,u),x)\}_{(i,u)\in E}\cup\{((j,v),y)\}_{(j,v)\in E}$ \end_inset . @@ -2586,7 +2586,7 @@ Si . Si -\begin_inset Formula $e:=(u,v)\in E$ +\begin_inset Formula $e\coloneqq (u,v)\in E$ \end_inset , como diff --git a/graf/n3.lyx b/graf/n3.lyx index ed819e4..b2cfb39 100644 --- a/graf/n3.lyx +++ b/graf/n3.lyx @@ -155,11 +155,11 @@ teorema . Sean -\begin_inset Formula $u_{0}:=v_{0}:=u$ +\begin_inset Formula $u_{0}\coloneqq v_{0}\coloneqq u$ \end_inset , -\begin_inset Formula $u_{p}:=u_{q}:=v$ +\begin_inset Formula $u_{p}\coloneqq u_{q}\coloneqq v$ \end_inset e @@ -228,7 +228,7 @@ teorema \end_inset , sea -\begin_inset Formula $e:=(u,v)\in E$ +\begin_inset Formula $e\coloneqq (u,v)\in E$ \end_inset , @@ -447,7 +447,7 @@ teorema \end_inset el ciclo que se forma al añadir -\begin_inset Formula $e:=(u,v)$ +\begin_inset Formula $e\coloneqq (u,v)$ \end_inset a @@ -753,7 +753,7 @@ La altura de status open \begin_layout Plain Layout -\begin_inset Formula $\lg x:=\log_{2}x$ +\begin_inset Formula $\lg x\coloneqq \log_{2}x$ \end_inset . @@ -792,15 +792,15 @@ Todos los niveles hasta el \end_inset se alcanza en -\begin_inset Formula $T':=(V',E')$ +\begin_inset Formula $T'\coloneqq (V',E')$ \end_inset dado por -\begin_inset Formula $V':=\{b_{0},a_{1},b_{1},\dots,a_{h},b_{h}\}$ +\begin_inset Formula $V'\coloneqq \{b_{0},a_{1},b_{1},\dots,a_{h},b_{h}\}$ \end_inset y -\begin_inset Formula $E':=\{(a_{k},b_{k-1}),(b_{k},b_{k-1})\}_{k\in\{1,\dots,h\}}$ +\begin_inset Formula $E'\coloneqq \{(a_{k},b_{k-1}),(b_{k},b_{k-1})\}_{k\in\{1,\dots,h\}}$ \end_inset . @@ -833,15 +833,15 @@ n\leq2^{h+1}-1\iff n+1\leq2^{h+1}\iff\lg(n+1)-1\leq h\overset{h\in\mathbb{Z}}{\i \end_inset La igualdad se alcanza en -\begin_inset Formula $T':=(V',E')$ +\begin_inset Formula $T'\coloneqq (V',E')$ \end_inset con -\begin_inset Formula $V':=\{1,\dots,n\}$ +\begin_inset Formula $V'\coloneqq \{1,\dots,n\}$ \end_inset y -\begin_inset Formula $E':=\{(k,\lfloor\frac{k}{2}\rfloor)\}_{k\in\{2,\dots,n\}}$ +\begin_inset Formula $E'\coloneqq \{(k,\lfloor\frac{k}{2}\rfloor)\}_{k\in\{2,\dots,n\}}$ \end_inset . @@ -1025,7 +1025,7 @@ mínimo \end_inset tales que -\begin_inset Formula $a:=(u,v)\in E$ +\begin_inset Formula $a\coloneqq (u,v)\in E$ \end_inset , si @@ -1095,7 +1095,7 @@ mínimo \end_inset y -\begin_inset Formula $S:=(V,E_{0}\cup\{e\})$ +\begin_inset Formula $S\coloneqq (V,E_{0}\cup\{e\})$ \end_inset , como @@ -1115,7 +1115,7 @@ mínimo \end_inset y -\begin_inset Formula $T_{1}:=(V,E_{1}:=E_{0}\cup\{e\}\setminus\{a\})$ +\begin_inset Formula $T_{1}\coloneqq (V,E_{1}\coloneqq E_{0}\cup\{e\}\setminus\{a\})$ \end_inset tiene menor o igual (en concreto igual) peso que @@ -1371,7 +1371,9 @@ Mientras{$|V_1|<|V|$}{ \backslash in V_1$ y $v_2 \backslash -in V_2$ con $e:=(v_1,v_2) +in V_2$ con $e +\backslash +coloneqq (v_1,v_2) \backslash in E$ de peso mínimo \backslash diff --git a/graf/n4.lyx b/graf/n4.lyx index 5334582..3506c7c 100644 --- a/graf/n4.lyx +++ b/graf/n4.lyx @@ -90,7 +90,7 @@ Dada una red \end_inset y un camino -\begin_inset Formula $P:=v_{0}e_{1}v_{1}\cdots e_{k}v_{k}$ +\begin_inset Formula $P\coloneqq v_{0}e_{1}v_{1}\cdots e_{k}v_{k}$ \end_inset en @@ -198,7 +198,7 @@ Como teorema \series default , sean -\begin_inset Formula $(V:=\{1,\dots,n\},E,\ell)$ +\begin_inset Formula $(V\coloneqq \{1,\dots,n\},E,\ell)$ \end_inset una red conexa, @@ -311,7 +311,7 @@ status open \end_inset Sea -\begin_inset Formula $P:=si_{1}\cdots i_{k}$ +\begin_inset Formula $P\coloneqq si_{1}\cdots i_{k}$ \end_inset un camino, y queremos ver que @@ -403,7 +403,7 @@ Si \begin_deeper \begin_layout Standard Sean -\begin_inset Formula $P:=st_{1}\cdots t_{p}j$ +\begin_inset Formula $P\coloneqq st_{1}\cdots t_{p}j$ \end_inset un camino de @@ -423,11 +423,11 @@ Sean \end_inset y -\begin_inset Formula $t_{k:=i+1},\dots,t_{p},j\in R$ +\begin_inset Formula $t_{k\coloneqq i+1},\dots,t_{p},j\in R$ \end_inset , entonces -\begin_inset Formula $P':=st_{1}\cdots t_{i}t_{k}$ +\begin_inset Formula $P'\coloneqq st_{1}\cdots t_{i}t_{k}$ \end_inset cumple @@ -1761,7 +1761,7 @@ Si \end_inset tal que -\begin_inset Formula $G_{i}:=(V,E_{i}):=G+\{e_{1},\dots,e_{i}\}$ +\begin_inset Formula $G_{i}\coloneqq (V,E_{i})\coloneqq G+\{e_{1},\dots,e_{i}\}$ \end_inset es hamiltoniano si y sólo si @@ -1769,7 +1769,7 @@ Si \end_inset , por lo que existe un camino hamiltoniano -\begin_inset Formula $(u=:u_{1})u_{2}\cdots(u_{n}:=v)$ +\begin_inset Formula $(u=:u_{1})u_{2}\cdots(u_{n}\coloneqq v)$ \end_inset en @@ -1777,16 +1777,16 @@ Si \end_inset , con -\begin_inset Formula $n:=|V|$ +\begin_inset Formula $n\coloneqq |V|$ \end_inset . Sean ahora -\begin_inset Formula $X:=\{i\in\{2,\dots,n-2\}\mid (u_{i},v)\in E_{k}\}$ +\begin_inset Formula $X\coloneqq \{i\in\{2,\dots,n-2\}\mid(u_{i},v)\in E_{k}\}$ \end_inset e -\begin_inset Formula $Y:=\{i\in\{2,\dots,n-2\}\mid (u_{i+1},u)\in E_{k}\}$ +\begin_inset Formula $Y\coloneqq \{i\in\{2,\dots,n-2\}\mid(u_{i+1},u)\in E_{k}\}$ \end_inset , se tiene diff --git a/graf/n5.lyx b/graf/n5.lyx index 4ae5cf7..5914553 100644 --- a/graf/n5.lyx +++ b/graf/n5.lyx @@ -447,7 +447,7 @@ Sea \end_inset la partición, definimos -\begin_inset Formula $f(v):=0$ +\begin_inset Formula $f(v)\coloneqq 0$ \end_inset para @@ -455,7 +455,7 @@ Sea \end_inset y -\begin_inset Formula $f(v):=1$ +\begin_inset Formula $f(v)\coloneqq 1$ \end_inset para @@ -502,7 +502,7 @@ Se tiene \end_inset dada por -\begin_inset Formula $f(v):=[n(v)]_{2}$ +\begin_inset Formula $f(v)\coloneqq [n(v)]_{2}$ \end_inset es una coloración de @@ -556,7 +556,7 @@ ciclo \begin_deeper \begin_layout Standard Como -\begin_inset Formula $C_{n}:=(V:=\{0,\dots,n-1\},\{\{i,[i+1]_{n}\}\}_{i\in V})$ +\begin_inset Formula $C_{n}\coloneqq (V\coloneqq \{0,\dots,n-1\},\{\{i,[i+1]_{n}\}\}_{i\in V})$ \end_inset tiene ejes, @@ -614,7 +614,7 @@ Como \end_inset , y tomamos -\begin_inset Formula $f(i):=[i]_{2}$ +\begin_inset Formula $f(i)\coloneqq [i]_{2}$ \end_inset para @@ -622,7 +622,7 @@ Como \end_inset y -\begin_inset Formula $f(0):=2$ +\begin_inset Formula $f(0)\coloneqq 2$ \end_inset . @@ -711,7 +711,7 @@ Si \begin_deeper \begin_layout Standard Sean -\begin_inset Formula $k:=\chi(G-v)$ +\begin_inset Formula $k\coloneqq \chi(G-v)$ \end_inset y @@ -735,7 +735,7 @@ Sean \end_inset dada por -\begin_inset Formula $g(i):=f(i)$ +\begin_inset Formula $g(i)\coloneqq f(i)$ \end_inset para @@ -743,7 +743,7 @@ Sean \end_inset y -\begin_inset Formula $g(v):=k+1$ +\begin_inset Formula $g(v)\coloneqq k+1$ \end_inset es una @@ -1124,7 +1124,7 @@ Si todos los vértices de \end_inset con -\begin_inset Formula $\chi(H_{0}:=G_{0}-e_{1})=\chi(G_{0})$ +\begin_inset Formula $\chi(H_{0}\coloneqq G_{0}-e_{1})=\chi(G_{0})$ \end_inset . @@ -1172,7 +1172,7 @@ teorema Demostración: \series default Sea -\begin_inset Formula $k:=\chi(G)$ +\begin_inset Formula $k\coloneqq \chi(G)$ \end_inset y supongamos @@ -1308,11 +1308,11 @@ Si \end_inset y -\begin_inset Formula $e:=(u,v)$ +\begin_inset Formula $e\coloneqq (u,v)$ \end_inset , llamamos -\begin_inset Formula $G+e:=(V,E\cup\{e\})$ +\begin_inset Formula $G+e\coloneqq (V,E\cup\{e\})$ \end_inset , y si @@ -1357,7 +1357,7 @@ Teorema de reducción: Demostración: \series default Sea -\begin_inset Formula $(u,v):=e$ +\begin_inset Formula $(u,v)\coloneqq e$ \end_inset , las coloraciones @@ -1377,7 +1377,7 @@ Demostración: \end_inset haciendo -\begin_inset Formula $f(*):=f(u)=f(v)$ +\begin_inset Formula $f(*)\coloneqq f(u)=f(v)$ \end_inset , y las coloraciones @@ -1576,7 +1576,7 @@ planar \end_inset tales que, para -\begin_inset Formula $e:=(u,v)\in E$ +\begin_inset Formula $e\coloneqq (u,v)\in E$ \end_inset , @@ -1663,11 +1663,11 @@ estrella \end_inset , llamamos -\begin_inset Formula $f(v_{0}):=0$ +\begin_inset Formula $f(v_{0})\coloneqq 0$ \end_inset , -\begin_inset Formula $f(v_{i}):=(\cos i/n,\sin i/n)$ +\begin_inset Formula $f(v_{i})\coloneqq (\cos i/n,\sin i/n)$ \end_inset para @@ -1675,7 +1675,7 @@ estrella \end_inset y -\begin_inset Formula $g(v_{0},v_{i})(t):=tv_{i}$ +\begin_inset Formula $g(v_{0},v_{i})(t)\coloneqq tv_{i}$ \end_inset . @@ -1915,7 +1915,7 @@ Demostración: \end_inset y -\begin_inset Formula $c:=|F|$ +\begin_inset Formula $c\coloneqq |F|$ \end_inset , como toda diff --git a/graf/n6.lyx b/graf/n6.lyx index 6bf574a..c3d6148 100644 --- a/graf/n6.lyx +++ b/graf/n6.lyx @@ -158,7 +158,7 @@ Si \end_inset , llamamos -\begin_inset Formula $[x,y]:=(x_{1},\dots,x_{m},y_{1},\dots,y_{n})\in\mathbb{R}^{n+m}$ +\begin_inset Formula $[x,y]\coloneqq (x_{1},\dots,x_{m},y_{1},\dots,y_{n})\in\mathbb{R}^{n+m}$ \end_inset ; si @@ -170,11 +170,11 @@ Si \end_inset , llamamos -\begin_inset Formula $[A,B]:=(c_{ij})\in{\cal M}_{n\times(p+q)}(\mathbb{R})$ +\begin_inset Formula $[A,B]\coloneqq (c_{ij})\in{\cal M}_{n\times(p+q)}(\mathbb{R})$ \end_inset dada por -\begin_inset Formula $c_{ij}:=a_{ij}$ +\begin_inset Formula $c_{ij}\coloneqq a_{ij}$ \end_inset para @@ -182,7 +182,7 @@ Si \end_inset y -\begin_inset Formula $c_{ij}:=b_{i(j-p)}$ +\begin_inset Formula $c_{ij}\coloneqq b_{i(j-p)}$ \end_inset para @@ -190,7 +190,7 @@ Si \end_inset , y escribimos -\begin_inset Formula $[x_{1},\dots,x_{n}]:=[x_{1},[x_{2},\dots,x_{n}]]$ +\begin_inset Formula $[x_{1},\dots,x_{n}]\coloneqq [x_{1},[x_{2},\dots,x_{n}]]$ \end_inset para @@ -198,7 +198,7 @@ Si \end_inset y -\begin_inset Formula $[x_{1}]:=x_{1}$ +\begin_inset Formula $[x_{1}]\coloneqq x_{1}$ \end_inset . @@ -222,11 +222,11 @@ teorema \end_inset , -\begin_inset Formula $P:=\{[x,y]\in\mathbb{R}^{p+q}\mid Ax+Gy\leq b\}$ +\begin_inset Formula $P\coloneqq \{[x,y]\in\mathbb{R}^{p+q}\mid Ax+Gy\leq b\}$ \end_inset y -\begin_inset Formula $S:=\{[x,y]\in P\mid x\in\mathbb{Z}^{p}\}$ +\begin_inset Formula $S\coloneqq \{[x,y]\in P\mid x\in\mathbb{Z}^{p}\}$ \end_inset , existen @@ -253,11 +253,11 @@ teorema Demostración: \series default Sean -\begin_inset Formula $S:=\{(x,y)\in\mathbb{Z}^{2}\mid y\leq\sqrt{2}x,x\geq0,y\geq0\}$ +\begin_inset Formula $S\coloneqq \{(x,y)\in\mathbb{Z}^{2}\mid y\leq\sqrt{2}x,x\geq0,y\geq0\}$ \end_inset y -\begin_inset Formula $C:=\{(x,y)\mid y<\sqrt{2}x,x\geq0,y\geq0\}\cup\{0\}$ +\begin_inset Formula $C\coloneqq \{(x,y)\mid y<\sqrt{2}x,x\geq0,y\geq0\}\cup\{0\}$ \end_inset . @@ -283,7 +283,7 @@ Demostración: \end_inset y -\begin_inset Formula $p:=(1-t)a+tb$ +\begin_inset Formula $p\coloneqq (1-t)a+tb$ \end_inset , si uno de @@ -406,11 +406,11 @@ Sean \end_inset y -\begin_inset Formula $P:=\{x\in\mathbb{R}^{n}\mid Ax\leq b\}$ +\begin_inset Formula $P\coloneqq \{x\in\mathbb{R}^{n}\mid Ax\leq b\}$ \end_inset , si -\begin_inset Formula $P_{I}:=\text{ec}(P\cap\mathbb{Z}^{n})\neq\emptyset$ +\begin_inset Formula $P_{I}\coloneqq \text{ec}(P\cap\mathbb{Z}^{n})\neq\emptyset$ \end_inset , para @@ -552,11 +552,11 @@ variable básica \end_inset , llamamos -\begin_inset Formula $x_{B}:=(x_{s_{1}},\dots,x_{s_{m}})$ +\begin_inset Formula $x_{B}\coloneqq (x_{s_{1}},\dots,x_{s_{m}})$ \end_inset , -\begin_inset Formula $x_{N}:=(x_{t_{1}},\dots,x_{t_{n-m}})$ +\begin_inset Formula $x_{N}\coloneqq (x_{t_{1}},\dots,x_{t_{n-m}})$ \end_inset , @@ -564,7 +564,7 @@ variable básica \end_inset y -\begin_inset Formula $\mathbf{n}(x_{1},\dots,x_{n-m}):=\sum_{k}e_{t_{k}}x_{k}$ +\begin_inset Formula $\mathbf{n}(x_{1},\dots,x_{n-m})\coloneqq \sum_{k}e_{t_{k}}x_{k}$ \end_inset , @@ -609,7 +609,7 @@ factible \begin_layout Standard Dado -\begin_inset Formula $F:=\{Ax=b,x\geq0\}$ +\begin_inset Formula $F\coloneqq \{Ax=b,x\geq0\}$ \end_inset , @@ -706,7 +706,7 @@ Lema de Veinott-Dantzig: \end_inset , -\begin_inset Formula $Q:=\{x\in\mathbb{R}^{n}\mid Ax=b,x\geq0\}$ +\begin_inset Formula $Q\coloneqq \{x\in\mathbb{R}^{n}\mid Ax=b,x\geq0\}$ \end_inset es entero. @@ -804,11 +804,11 @@ Sea \end_inset tal que -\begin_inset Formula $z:=y+(B^{-1})_{i}\geq0$ +\begin_inset Formula $z\coloneqq y+(B^{-1})_{i}\geq0$ \end_inset y -\begin_inset Formula $b:=Bz=By+e_{i}$ +\begin_inset Formula $b\coloneqq Bz=By+e_{i}$ \end_inset , @@ -828,11 +828,11 @@ Sea \end_inset todos los coeficientes enteros, luego -\begin_inset Formula $Q:=\{Ax=b,x\geq0\}$ +\begin_inset Formula $Q\coloneqq \{Ax=b,x\geq0\}$ \end_inset es entero y -\begin_inset Formula $x:=\mathbf{b}z=\mathbf{b}B^{-1}b$ +\begin_inset Formula $x\coloneqq \mathbf{b}z=\mathbf{b}B^{-1}b$ \end_inset es una solución básica factible de @@ -978,7 +978,7 @@ Dada una submatriz \end_inset es unimodular, con lo que -\begin_inset Formula $Q:=\{[x,y]\in\mathbb{R}^{n+m}\mid Ax+Iy=b,[x,y]\geq0\}$ +\begin_inset Formula $Q\coloneqq \{[x,y]\in\mathbb{R}^{n+m}\mid Ax+Iy=b,[x,y]\geq0\}$ \end_inset es entero. @@ -1003,7 +1003,7 @@ Dada una submatriz \end_inset es -\begin_inset Formula $P:=\{x\in\mathbb{R}^{n}\mid b=b,x\geq0,b-Ax\geq0\}=\{Ax\leq b,x\geq0\}$ +\begin_inset Formula $P\coloneqq \{x\in\mathbb{R}^{n}\mid b=b,x\geq0,b-Ax\geq0\}=\{Ax\leq b,x\geq0\}$ \end_inset . @@ -1016,7 +1016,7 @@ Dada una submatriz \end_inset es un punto extremo, pues si no lo fuera existirían -\begin_inset Formula $U:=[u,b-Au],V:=[v,b-Av]\in Q$ +\begin_inset Formula $U\coloneqq [u,b-Au],V\coloneqq [v,b-Av]\in Q$ \end_inset distintos y @@ -1069,11 +1069,11 @@ Sean \end_inset , -\begin_inset Formula $P:=\{x\mid Ax\leq b,x\geq0\}$ +\begin_inset Formula $P\coloneqq \{x\mid Ax\leq b,x\geq0\}$ \end_inset , -\begin_inset Formula $Q:=\{[x,y]\mid Ax+y=b,[x,y]\geq0\}$ +\begin_inset Formula $Q\coloneqq \{[x,y]\mid Ax+y=b,[x,y]\geq0\}$ \end_inset y @@ -1293,7 +1293,7 @@ teorema \end_inset tal que, si -\begin_inset Formula $F_{2}:=F\setminus F_{1}$ +\begin_inset Formula $F_{2}\coloneqq F\setminus F_{1}$ \end_inset , para @@ -1496,7 +1496,7 @@ Si las tareas se pueden hacer a la vez, lo que queremos minimizar es \begin_layout Standard Sean ahora -\begin_inset Formula $R:=(V:=\{1,\dots,n\},E,\omega)$ +\begin_inset Formula $R\coloneqq (V\coloneqq \{1,\dots,n\},E,\omega)$ \end_inset una red y @@ -1595,7 +1595,7 @@ Para obtener el árbol generador minimal de \end_inset , llamamos -\begin_inset Formula $x_{ij}:=\chi_{E_{T}}(i,j)$ +\begin_inset Formula $x_{ij}\coloneqq \chi_{E_{T}}(i,j)$ \end_inset para @@ -1643,7 +1643,7 @@ Otra posible formulación, con las mismas variables resulta de cambiar la \begin_layout Standard Para el problema del viajante de comercio sobre una red completa -\begin_inset Formula $R:=(V:=\{0,\dots,n-1\},E\mid =\{\{i,j\}\}_{i,j\in V,i\neq j},d)$ +\begin_inset Formula $R\coloneqq (V\coloneqq \{0,\dots,n-1\},E\coloneqq \{\{i,j\}\}_{i,j\in V,i\neq j},d)$ \end_inset , existen varias formulaciones: @@ -1774,7 +1774,7 @@ es . Llamando -\begin_inset Formula $n:=|V|$ +\begin_inset Formula $n\coloneqq |V|$ \end_inset : @@ -1783,7 +1783,7 @@ es & \min & {\textstyle \sum}_{ij}d_{ij}x_{ij}\\ & & {\textstyle \sum_{(i,j)\in E}}x_{ij} & =1 & & \forall i\\ & & {\textstyle \sum_{(k,i)\in E}}x_{ki} & =1 & & \forall i\\ - & & u_{i}-u_{j}+nx_{ij} & \leq n-1 & & \forall i,j\in\{1,\dots,n-1\}\mid (i,j)\in E\\ + & & u_{i}-u_{j}+nx_{ij} & \leq n-1 & & \forall i,j\in\{1,\dots,n-1\}:(i,j)\in E\\ & & x_{ij} & \in\{0,1\} & & \forall i,j\\ & & u_{i} & \in\mathbb{R}^{>0} & & \forall i \end{alignat*} @@ -1830,7 +1830,7 @@ Sea \end_inset la representación por variables de un ciclo hamiltoniano, llamamos -\begin_inset Formula $u_{i}:=t$ +\begin_inset Formula $u_{i}\coloneqq t$ \end_inset si @@ -1928,7 +1928,7 @@ Dadas dos variables \end_inset , para definir una variable -\begin_inset Formula $y:=[x_{1}>x_{2}]$ +\begin_inset Formula $y\coloneqq [x_{1}>x_{2}]$ \end_inset ( @@ -2053,7 +2053,7 @@ Si \end_inset , para definir -\begin_inset Formula $y:=\min\{x_{1},x_{2}\}$ +\begin_inset Formula $y\coloneqq \min\{x_{1},x_{2}\}$ \end_inset añadimos @@ -2073,7 +2073,7 @@ Si \end_inset , y para definir -\begin_inset Formula $y:=\max\{x_{1},x_{2}\}$ +\begin_inset Formula $y\coloneqq \max\{x_{1},x_{2}\}$ \end_inset añadimos diff --git a/graf/n7.lyx b/graf/n7.lyx index dc0abb4..fbf8456 100644 --- a/graf/n7.lyx +++ b/graf/n7.lyx @@ -850,11 +850,11 @@ regla de Bland: \end_inset , -\begin_inset Formula $F:=\{x\mid Ax=b,x\geq0\}$ +\begin_inset Formula $F\coloneqq \{x\mid Ax=b,x\geq0\}$ \end_inset y -\begin_inset Formula $P:=\{c\cdot x\}_{x\in F}$ +\begin_inset Formula $P\coloneqq \{c\cdot x\}_{x\in F}$ \end_inset . @@ -888,7 +888,7 @@ Si [...] \end_inset es la matriz formada por las columnas añadidas, escribimos -\begin_inset Formula $F^{*}:=\{[x,x^{*}]\in\mathbb{R}^{n+p}\mid Ax+Tx^{*}=b,[x,x^{*}]\geq0\}$ +\begin_inset Formula $F^{*}\coloneqq \{[x,x^{*}]\in\mathbb{R}^{n+p}\mid Ax+Tx^{*}=b,[x,x^{*}]\geq0\}$ \end_inset y vemos que @@ -957,11 +957,11 @@ Método de penalización: \end_inset lo suficientemente grande, definimos -\begin_inset Formula $P_{M}:=\{c\cdot x+M\sum_{i}x_{i}^{*}\}_{[x,x^{*}]\in F^{*}}$ +\begin_inset Formula $P_{M}\coloneqq \{c\cdot x+M\sum_{i}x_{i}^{*}\}_{[x,x^{*}]\in F^{*}}$ \end_inset si estamos minimizando o -\begin_inset Formula $P_{-M}:=\{c\cdot x-M\sum_{i}x_{i}^{*}\}_{[x,x^{*}]\in F^{*}}$ +\begin_inset Formula $P_{-M}\coloneqq \{c\cdot x-M\sum_{i}x_{i}^{*}\}_{[x,x^{*}]\in F^{*}}$ \end_inset [si maximizamos]. @@ -1103,7 +1103,9 @@ to \backslash dots,m \backslash -}$ tal que $B:=[A_{ +}$ tal que $B +\backslash +coloneqq [A_{ \backslash sigma(1)}, \backslash @@ -1335,7 +1337,7 @@ Supongamos que tenemos una tabla de símplex óptima con parámetros , podemos añadir la restricción directamente a la tabla añadiendo lo siguiente, donde -\begin_inset Formula $t:=\beta-\sum_{j=1}^{n}\alpha_{j}x_{j}$ +\begin_inset Formula $t\coloneqq \beta-\sum_{j=1}^{n}\alpha_{j}x_{j}$ \end_inset y @@ -1501,7 +1503,7 @@ desigualdad válida factibles. Se puede usar para mejorar las cotas en los nodos del árbol de ramificación. Llamamos -\begin_inset Formula $[[x]]:=x-\lfloor x\rfloor\in[0,1)$ +\begin_inset Formula $[[x]]\coloneqq x-\lfloor x\rfloor\in[0,1)$ \end_inset . @@ -1522,7 +1524,7 @@ Dados un problema entero puro \end_inset tal que -\begin_inset Formula $x_{k':=\sigma(k)}^{*}\notin\mathbb{Z}$ +\begin_inset Formula $x_{k'\coloneqq \sigma(k)}^{*}\notin\mathbb{Z}$ \end_inset , entonces @@ -1599,11 +1601,11 @@ Dados un problema entero puro \end_inset tal que -\begin_inset Formula $k':=\sigma(k)\in I$ +\begin_inset Formula $k'\coloneqq \sigma(k)\in I$ \end_inset y -\begin_inset Formula $x_{k':=\sigma(k)}^{*}\notin\mathbb{Z}$ +\begin_inset Formula $x_{k'\coloneqq \sigma(k)}^{*}\notin\mathbb{Z}$ \end_inset , entonces @@ -1754,7 +1756,7 @@ Desigualdades de Chvátal-Gomory \begin_layout Standard Dado un problema entero puro con conjunto factible -\begin_inset Formula $P:=\{Ax\leq b,x\in\mathbb{N}^{n}\}$ +\begin_inset Formula $P\coloneqq \{Ax\leq b,x\in\mathbb{N}^{n}\}$ \end_inset , donde |