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115 files changed, 694 insertions, 402 deletions
diff --git a/aalg/n1.lyx b/aalg/n1.lyx index 520ce4b..a783d88 100644 --- a/aalg/n1.lyx +++ b/aalg/n1.lyx @@ -1235,7 +1235,7 @@ Demostración: en común, los tres puntos estarían alineados. Así, podemos tomar -\begin_inset Formula $\{O\}:=m\cap m'$ +\begin_inset Formula $\{O\}\mid =m\cap m'$ \end_inset y entonces @@ -2296,7 +2296,7 @@ hemisferio norte \end_inset de la hipérbola ( -\begin_inset Formula $\{(x,y)\in{\cal H}:y\geq0\}$ +\begin_inset Formula $\{(x,y)\in{\cal H}\mid y\geq0\}$ \end_inset ), dado por diff --git a/aalg/n2.lyx b/aalg/n2.lyx index 94fb772..d6c0241 100644 --- a/aalg/n2.lyx +++ b/aalg/n2.lyx @@ -338,7 +338,7 @@ Los vectores propios de . Así, -\begin_inset Formula $V_{\lambda}=\text{Nuc}(f-\lambda Id)=\{v\in V:(f-\lambda Id)(v)=0\}=\{v\in V:f(v)=\lambda v\}$ +\begin_inset Formula $V_{\lambda}=\text{Nuc}(f-\lambda Id)=\{v\in V\mid (f-\lambda Id)(v)=0\}=\{v\in V\mid f(v)=\lambda v\}$ \end_inset es el diff --git a/aalg/n3.lyx b/aalg/n3.lyx index d0e0932..a6df369 100644 --- a/aalg/n3.lyx +++ b/aalg/n3.lyx @@ -1883,7 +1883,7 @@ Sean \end_inset y -\begin_inset Formula ${\cal L}:=\{(x,y)\in\mathbb{A}^{2}(\mathbb{K}):f(x,y)=0\}$ +\begin_inset Formula ${\cal L}:=\{(x,y)\in\mathbb{A}^{2}(\mathbb{K})\mid f(x,y)=0\}$ \end_inset , llamamos @@ -1899,7 +1899,7 @@ completación proyectiva \end_inset a -\begin_inset Formula $\overline{{\cal L}}:=\{<(x,y,z)>\in\mathbb{P}^{2}(\mathbb{K}):f^{*}(x,y,z)=0\}$ +\begin_inset Formula $\overline{{\cal L}}:=\{<(x,y,z)>\in\mathbb{P}^{2}(\mathbb{K})\mid f^{*}(x,y,z)=0\}$ \end_inset , y para @@ -1915,7 +1915,7 @@ parte afín \end_inset es -\begin_inset Formula $\hat{{\cal L}}^{\text{afín}}:=\{(x,y)\in\mathbb{A}^{2}(\mathbb{K}):<(x,y,1)>\in\hat{{\cal L}}\}$ +\begin_inset Formula $\hat{{\cal L}}^{\text{afín}}:=\{(x,y)\in\mathbb{A}^{2}(\mathbb{K})\mid <(x,y,1)>\in\hat{{\cal L}}\}$ \end_inset . @@ -1928,12 +1928,12 @@ parte afín \end_inset , -\begin_inset Formula $\hat{{\cal L}}^{\text{afín}}=\{(x,y):F(x,y,1)=0\}=\{(x,y):F_{*}(x,y)=0\}$ +\begin_inset Formula $\hat{{\cal L}}^{\text{afín}}=\{(x,y)\mid F(x,y,1)=0\}=\{(x,y)\mid F_{*}(x,y)=0\}$ \end_inset . Entonces -\begin_inset Formula $\overline{\hat{{\cal L}}^{\text{afín}}}=\{<(a,b,c)>:(F_{*})^{*}(a,b,c)=0\}=\hat{{\cal L}}\cup\{<(x,y,0)>:F(x,y,0)=0\}$ +\begin_inset Formula $\overline{\hat{{\cal L}}^{\text{afín}}}=\{<(a,b,c)>\mid (F_{*})^{*}(a,b,c)=0\}=\hat{{\cal L}}\cup\{<(x,y,0)>\mid F(x,y,0)=0\}$ \end_inset , y si diff --git a/aalg/n4.lyx b/aalg/n4.lyx index 11a1a77..96b456a 100644 --- a/aalg/n4.lyx +++ b/aalg/n4.lyx @@ -827,7 +827,7 @@ subespacio ortogonal \end_inset al subespacio -\begin_inset Formula $E^{\bot}:=\{v\in V:\forall e\in E,\langle v,e\rangle=0\}$ +\begin_inset Formula $E^{\bot}:=\{v\in V\mid \forall e\in E,\langle v,e\rangle=0\}$ \end_inset . @@ -3827,7 +3827,7 @@ cónica proyectiva \end_inset , o de formas cuadráticas no nulas de dimensión 3, bajo la relación -\begin_inset Formula $q\sim q':\iff\exists\lambda\in\mathbb{K}\backslash\{0\}:q'=\lambda q$ +\begin_inset Formula $q\sim q':\iff\exists\lambda\in\mathbb{K}\backslash\{0\}\mid q'=\lambda q$ \end_inset . @@ -3975,7 +3975,7 @@ recta polar \end_inset a -\begin_inset Formula $r_{P}:=\{X\in\mathbb{P}^{2}(\mathbb{K}):[P]^{t}\overline{A}[X]=0\}$ +\begin_inset Formula $r_{P}:=\{X\in\mathbb{P}^{2}(\mathbb{K})\mid [P]^{t}\overline{A}[X]=0\}$ \end_inset , y decimos que @@ -799,7 +799,7 @@ status open \backslash -begin{reminder}{ga} +begin{reminder}{GyA} \end_layout \end_inset @@ -3379,7 +3379,7 @@ Dado un espacio topológico \end_inset , -\begin_inset Formula $\{f\in\mathbb{R}^{X}:f\text{ continua}\}$ +\begin_inset Formula $\{f\in\mathbb{R}^{X}\mid f\text{ continua}\}$ \end_inset es un subanillo de @@ -3404,7 +3404,7 @@ Dado un espacio vectorial \end_inset , -\begin_inset Formula $\{f\in V^{V}:f\text{ lineal}\}$ +\begin_inset Formula $\{f\in V^{V}\mid f\text{ lineal}\}$ \end_inset es un subanillo de @@ -3433,7 +3433,7 @@ Dado un anillo \end_inset , -\begin_inset Formula $\{f\in A^{X}:f\text{ constante}\}$ +\begin_inset Formula $\{f\in A^{X}\mid f\text{ constante}\}$ \end_inset es un subanillo de @@ -3893,7 +3893,7 @@ ideal de a \begin_inset Formula \[ -(S)\coloneqq\bigcap\{I\trianglelefteq A:S\subseteq I\}=\{a_{1}s_{1}+\dots+a_{n}s_{n}\}_{n\in\mathbb{N},a\in A^{n},s\in S^{n}}, +(S)\coloneqq\bigcap\{I\trianglelefteq A\mid S\subseteq I\}=\{a_{1}s_{1}+\dots+a_{n}s_{n}\}_{n\in\mathbb{N},a\in A^{n},s\in S^{n}}, \] \end_inset @@ -3912,7 +3912,7 @@ conjunto generador . En efecto, -\begin_inset Formula $\bigcap\{I\trianglelefteq A:S\subseteq I\}$ +\begin_inset Formula $\bigcap\{I\trianglelefteq A\mid S\subseteq I\}$ \end_inset es un ideal de @@ -5609,7 +5609,7 @@ Dado un homomorfismo de anillos , la extensión es una biyección \begin_inset Formula \[ -\{I\trianglelefteq A:\ker f\subseteq I\}\to\{J\trianglelefteq\text{Im}f\}, +\{I\trianglelefteq A\mid\ker f\subseteq I\}\to\{J\trianglelefteq\text{Im}f\}, \] \end_inset @@ -5715,7 +5715,7 @@ Si es la proyección canónica, \begin_inset Formula \[ -\rho:\{J\trianglelefteq A:I\subseteq J\}\to\{K\trianglelefteq A/I\} +\rho:\{J\trianglelefteq A\mid I\subseteq J\}\to\{K\trianglelefteq A/I\} \] \end_inset @@ -5821,7 +5821,7 @@ Hay tantos ideales de \end_inset y -\begin_inset Formula $\{I\trianglelefteq\mathbb{Z}:(n)\subseteq I\}$ +\begin_inset Formula $\{I\trianglelefteq\mathbb{Z}\mid(n)\subseteq I\}$ \end_inset , pero @@ -6810,11 +6810,11 @@ espectro maximal \end_inset , la biyección -\begin_inset Formula $\{J\in{\cal L}(A):I\subseteq J\}\to{\cal L}(A/I)$ +\begin_inset Formula $\{J\in{\cal L}(A)\mid I\subseteq J\}\to{\cal L}(A/I)$ \end_inset del teorema de la correspondencia se restringe a una biyección -\begin_inset Formula $\{J\in\text{MaxSpec}(A):I\subseteq J\}\to\text{MaxSpec}(A/I)$ +\begin_inset Formula $\{J\in\text{MaxSpec}(A)\mid I\subseteq J\}\to\text{MaxSpec}(A/I)$ \end_inset . @@ -6911,7 +6911,7 @@ Si Demostración: \series default Sea -\begin_inset Formula $\Omega\coloneqq\{J\triangleleft A:I\subseteq J\}$ +\begin_inset Formula $\Omega\coloneqq\{J\triangleleft A\mid I\subseteq J\}$ \end_inset , @@ -7037,7 +7037,7 @@ radical de Jacobson \end_inset a -\begin_inset Formula $\text{Jac}(A)\coloneqq\bigcap\text{MaxSpec}(A)=\{a\in A:1+(a)\subseteq A^{*}\}$ +\begin_inset Formula $\text{Jac}(A)\coloneqq\bigcap\text{MaxSpec}(A)=\{a\in A\mid1+(a)\subseteq A^{*}\}$ \end_inset . @@ -7543,11 +7543,11 @@ espectro primo \end_inset , la biyección -\begin_inset Formula $\{J\in{\cal L}(A):I\subseteq J\}\to{\cal L}(A/I)$ +\begin_inset Formula $\{J\in{\cal L}(A)\mid I\subseteq J\}\to{\cal L}(A/I)$ \end_inset se restringe a una biyección -\begin_inset Formula $\{J\in\text{Spec}(A):I\subseteq J\}\to\text{Spec}(A/I)$ +\begin_inset Formula $\{J\in\text{Spec}(A)\mid I\subseteq J\}\to\text{Spec}(A/I)$ \end_inset . @@ -7992,7 +7992,7 @@ primo minimal sobre Demostración: \series default Sea -\begin_inset Formula $\Omega\coloneqq\{P\trianglelefteq_{\text{p}}A:I\subseteq P\subseteq Q\}$ +\begin_inset Formula $\Omega\coloneqq\{P\trianglelefteq_{\text{p}}A\mid I\subseteq P\subseteq Q\}$ \end_inset , @@ -8363,7 +8363,7 @@ Lema de Krull: \end_layout \begin_layout Enumerate -\begin_inset Formula ${\cal L}_{I,S}\coloneqq\{J\trianglelefteq A:I\subseteq J,J\cap S=\emptyset\}$ +\begin_inset Formula ${\cal L}_{I,S}\coloneqq\{J\trianglelefteq A\mid I\subseteq J,J\cap S=\emptyset\}$ \end_inset es un conjunto inductivo no vacío. @@ -8490,7 +8490,7 @@ radical a \begin_inset Formula \[ -\sqrt{I}\coloneqq\{x\in A:\exists n\in\mathbb{N}:x^{n}\in I\}=\bigcap\{J\trianglelefteq_{\text{r}}A:I\subseteq J\}=\bigcap\{J\trianglelefteq_{\text{p}}A:I\subseteq J\}, +\sqrt{I}\coloneqq\{x\in A\mid\exists n\in\mathbb{N}\mid x^{n}\in I\}=\bigcap\{J\trianglelefteq_{\text{r}}A\mid I\subseteq J\}=\bigcap\{J\trianglelefteq_{\text{p}}A\mid I\subseteq J\}, \] \end_inset @@ -8785,7 +8785,7 @@ euclídea \end_layout \begin_layout Enumerate -\begin_inset Formula $\forall a\in D,b\in D\setminus\{0\},\exists q,r\in D:(a=bq+r\land(r=0\lor\delta(r)<\delta(b)))$ +\begin_inset Formula $\forall a\in D,b\in D\setminus\{0\},\exists q,r\in D\mid(a=bq+r\land(r=0\lor\delta(r)<\delta(b)))$ \end_inset . @@ -9338,11 +9338,11 @@ polinomios constantes \end_inset , -\begin_inset Formula $\{a_{0}+a_{1}X+\dots+a_{n}X^{n}\in A[X]:a_{0}\in I\}$ +\begin_inset Formula $\{a_{0}+a_{1}X+\dots+a_{n}X^{n}\in A[X]\mid a_{0}\in I\}$ \end_inset e -\begin_inset Formula $I[X]:=\{a_{0}+a_{1}X+\dots+a_{n}X^{n}\in A[X]:a_{0},\dots,a_{n}\in I\}$ +\begin_inset Formula $I[X]:=\{a_{0}+a_{1}X+\dots+a_{n}X^{n}\in A[X]\mid a_{0},\dots,a_{n}\in I\}$ \end_inset son ideales de @@ -9366,7 +9366,7 @@ grado \end_inset a -\begin_inset Formula $\text{gr}(p):=\max\{k\in\mathbb{N}:p_{k}\neq0\}$ +\begin_inset Formula $\text{gr}(p):=\max\{k\in\mathbb{N}\mid p_{k}\neq0\}$ \end_inset , @@ -9916,7 +9916,7 @@ Para \end_inset , existe -\begin_inset Formula $m:=\max\{k\in\mathbb{N}:(X-a)^{k}\mid f\}$ +\begin_inset Formula $m:=\max\{k\in\mathbb{N}\mid(X-a)^{k}\mid f\}$ \end_inset . @@ -10448,7 +10448,7 @@ Definimos \end_inset , -\begin_inset Formula $c(p):=\{x:x=\text{mcd}_{k\geq0}p_{k}\}$ +\begin_inset Formula $c(p):=\{x\mid x=\text{mcd}_{k\geq0}p_{k}\}$ \end_inset , y para @@ -771,7 +771,7 @@ Para \end_inset , los -\begin_inset Formula $I_{n}\coloneqq\{a:\forall k>n,a_{k}=0\}$ +\begin_inset Formula $I_{n}\coloneqq\{a\mid \forall k>n,a_{k}=0\}$ \end_inset cumplen @@ -779,7 +779,7 @@ Para \end_inset y los -\begin_inset Formula $J_{n}\coloneqq\{a:\forall k<n,a_{k}=0\}$ +\begin_inset Formula $J_{n}\coloneqq\{a\mid \forall k<n,a_{k}=0\}$ \end_inset cumplen @@ -1333,7 +1333,7 @@ Dados \end_inset , llamamos -\begin_inset Formula $(I:S)=\{a\in A:aS\subseteq I\}$ +\begin_inset Formula $(I:S)=\{a\in A\mid aS\subseteq I\}$ \end_inset . @@ -1491,7 +1491,7 @@ anulador \end_inset a -\begin_inset Formula $\text{ann}_{A}(X)\coloneqq(0:X)=\{a\in A:aX=0\}$ +\begin_inset Formula $\text{ann}_{A}(X)\coloneqq(0:X)=\{a\in A\mid aX=0\}$ \end_inset , y entonces @@ -1719,7 +1719,7 @@ Claramente \end_layout \begin_layout Standard -\begin_inset Formula $(P:(a))=\{c\in A:c(a)=(ca)\subseteq P\}=\{c\in A:ac\in P\}$ +\begin_inset Formula $(P:(a))=\{c\in A\mid c(a)=(ca)\subseteq P\}=\{c\in A\mid ac\in P\}$ \end_inset , y entonces @@ -2218,7 +2218,7 @@ dimensión de Krull es \begin_inset Formula \[ -\dim A\coloneqq\text{Kdim}A\coloneqq\sup\{n\in\mathbb{N}:\exists P_{0},\dots,P_{n}\trianglelefteq_{\text{p}}A:P_{0}\subsetneq\dots\subsetneq P_{n}\}\in\mathbb{N}\cup\{\infty\}, +\dim A\coloneqq\text{Kdim}A\coloneqq\sup\{n\in\mathbb{N}\mid \exists P_{0},\dots,P_{n}\trianglelefteq_{\text{p}}A:P_{0}\subsetneq\dots\subsetneq P_{n}\}\in\mathbb{N}\cup\{\infty\}, \] \end_inset @@ -2443,7 +2443,7 @@ Dado . Si no lo fuera, -\begin_inset Formula $\Omega\coloneqq\{K\trianglelefteq A:KI\neq0\}\neq\emptyset$ +\begin_inset Formula $\Omega\coloneqq\{K\trianglelefteq A\mid KI\neq0\}\neq\emptyset$ \end_inset , pues @@ -304,7 +304,7 @@ anulador \end_inset a -\begin_inset Formula $\text{ann}_{M}(X)\coloneqq\{m\in M:Xm=0\}\leq_{A}M$ +\begin_inset Formula $\text{ann}_{M}(X)\coloneqq\{m\in M\mid Xm=0\}\leq_{A}M$ \end_inset . @@ -339,7 +339,7 @@ externa ) \begin_inset Formula \[ -\bigoplus_{i\in I}M_{i}\coloneqq\left\{ x\in\prod_{i\in I}M_{i}:\{i\in I:x_{i}\neq0\}\text{ finito}\right\} . +\bigoplus_{i\in I}M_{i}\coloneqq\left\{ x\in\prod_{i\in I}M_{i}\;\middle|\;\{i\in I\mid x_{i}\neq0\}\text{ finito}\right\} . \] \end_inset @@ -645,7 +645,7 @@ Si \end_inset , -\begin_inset Formula $\{f\in A[X]:\text{gr}f\leq n\}$ +\begin_inset Formula $\{f\in A[X]\mid\text{gr}f\leq n\}$ \end_inset es un submódulo de @@ -1296,7 +1296,7 @@ Si \end_inset , -\begin_inset Formula $_{A/I}\text{Mod}\equiv\{M\in_{A}\text{Mod}:IM=0\}$ +\begin_inset Formula $_{A/I}\text{Mod}\equiv\{M\in_{A}\text{Mod}\mid IM=0\}$ \end_inset por la biyección diff --git a/aec/n.pdf b/aec/n.pdf Binary files differdeleted file mode 100755 index d5293be..0000000 --- a/aec/n.pdf +++ /dev/null diff --git a/aed1/graph.eps b/aed1/graph.eps Binary files differnew file mode 100644 index 0000000..79fafd8 --- /dev/null +++ b/aed1/graph.eps diff --git a/aed1/n2.lyx b/aed1/n2.lyx index 26201e5..22a657a 100644 --- a/aed1/n2.lyx +++ b/aed1/n2.lyx @@ -303,7 +303,7 @@ Diccionario[ (k,v,d)\overset{k\notin\text{Dom}(d)}{\mapsto}D\cup\{(k,v)\} & (k,d)\overset{k\in\text{Dom}(d)}{\mapsto}d(k)\\ \mathsf{}\\ \mathsf{Suprime}:T_{k}\times D\rightarrow D & \mathsf{Vacío}:\rightarrow D\\ -(k,d)\mapsto\{(a,b)\in d:a\neq k\} & \mapsto\emptyset +(k,d)\mapsto\{(a,b)\in d\mid a\neq k\} & \mapsto\emptyset \end{array} \] @@ -373,7 +373,7 @@ Abierta cubetas \series default , que contienen los elementos -\begin_inset Formula $\{e\in c:h(e)=k\}$ +\begin_inset Formula $\{e\in c\mid h(e)=k\}$ \end_inset , siendo diff --git a/aed1/n4.lyx b/aed1/n4.lyx index db68fda..3e1e0aa 100644 --- a/aed1/n4.lyx +++ b/aed1/n4.lyx @@ -107,7 +107,7 @@ nodos aristas \series default -\begin_inset Formula $E\subseteq\{(a,b)\in V\times V:a\neq b\}$ +\begin_inset Formula $E\subseteq\{(a,b)\in V\times V\mid a\neq b\}$ \end_inset , mientras que uno @@ -123,7 +123,7 @@ no dirigido \end_inset y -\begin_inset Formula $E\subseteq\{x\in{\cal P}(V):|x|=2\}$ +\begin_inset Formula $E\subseteq\{x\in{\cal P}(V)\mid |x|=2\}$ \end_inset . @@ -136,7 +136,7 @@ bucles \end_inset para que el grafo sea dirigido o que -\begin_inset Formula $E\subseteq\{x\in{\cal P}(V):|x|\in\{1,2\}\}$ +\begin_inset Formula $E\subseteq\{x\in{\cal P}(V)\mid |x|\in\{1,2\}\}$ \end_inset para que sea no dirigido. @@ -374,7 +374,7 @@ grado \end_inset es el número de arcos adyacentes a él ( -\begin_inset Formula $|\{X\in E:v\in X\}|$ +\begin_inset Formula $|\{X\in E\mid v\in X\}|$ \end_inset ), mientras que en uno dirigido @@ -390,7 +390,7 @@ grado de entrada \end_inset como -\begin_inset Formula $|\{(a,b)\in A:b=v\}|$ +\begin_inset Formula $|\{(a,b)\in A\mid b=v\}|$ \end_inset y el @@ -398,7 +398,7 @@ grado de entrada grado de salida \series default como -\begin_inset Formula $|\{(a,b)\in A:a=v\}|$ +\begin_inset Formula $|\{(a,b)\in A\mid a=v\}|$ \end_inset . @@ -419,7 +419,7 @@ Operaciones elementales: ((V,E),v)\mapsto(V\cup\{v\},E) & ((V,E),(a,b))\overset{a,b\in V}{\mapsto}(V,E\cup\{e\})\\ \\ \mathsf{EliminarNodo}:G\times{\cal U}\rightarrow G & \mathsf{EliminarArista}:G\times({\cal U}\times{\cal U})\rightarrow G\\ -((V,E),v)\mapsto(V\backslash\{e\},\{(a,b)\in E:a,b\neq v\}) & ((V,E),e)\mapsto(V,E\backslash\{e\})\\ +((V,E),v)\mapsto(V\backslash\{e\},\{(a,b)\in E\mid a,b\neq v\}) & ((V,E),e)\mapsto(V,E\backslash\{e\})\\ \\ \mathsf{ConsultarArista}:G\times({\cal U}\times{\cal U})\rightarrow B\\ ((V,E),(a,b))\mapsto(a,b)\in A @@ -456,8 +456,9 @@ status open \begin_layout Plain Layout \align center -\begin_inset Graphics - filename graph.svg +\begin_inset External + template VectorGraphics + filename graph.eps scale 60 \end_inset @@ -508,7 +509,7 @@ En un ordenador podemos representar un grafo finito \end_inset o -\begin_inset Formula $(V:=\{1,\dots,n\},E,\sigma:E\rightarrow X)$ +\begin_inset Formula $(V:=\{1,\dots,n\},E,\sigma\mid E\rightarrow X)$ \end_inset mediante: @@ -594,12 +595,12 @@ Listas de adyacencia (representados como listas enlazadas en una lista contigua) de los que -\begin_inset Formula $C_{i}=\{j:(i,j)\in E\}$ +\begin_inset Formula $C_{i}=\{j\mid(i,j)\in E\}$ \end_inset . Si el grafo es etiquetado, -\begin_inset Formula $C_{i}=\{(j,\sigma(i,j)):(i,j)\in E\}$ +\begin_inset Formula $C_{i}=\{(j,\sigma(i,j))\mid(i,j)\in E\}$ \end_inset . @@ -617,7 +618,7 @@ Listas de adyacencia \begin_layout Standard En adelante, salvo que se indique lo contrario, suponemos un grafo -\begin_inset Formula $(V:=\{1,\dots,n\},E,\sigma:E\rightarrow X)$ +\begin_inset Formula $(V:=\{1,\dots,n\},E,\sigma\mid E\rightarrow X)$ \end_inset , y que las variables en pseudocódigo se inicializan con su valor por defecto. @@ -2586,7 +2587,7 @@ grafo reducido \end_inset y -\begin_inset Formula $E_{R}:=\{(A,B)\in V_{R}:\exists a\in A,b\in B:(a,b)\in E\}$ +\begin_inset Formula $E_{R}:=\{(A,B)\in V_{R}\mid \exists a\in A,b\in B:(a,b)\in E\}$ \end_inset . diff --git a/aed2/n.pdf b/aed2/n.pdf Binary files differdeleted file mode 100644 index a7d382d..0000000 --- a/aed2/n.pdf +++ /dev/null @@ -258,7 +258,7 @@ espacio normado \end_inset , y llamamos -\begin_inset Formula $B_{X}\coloneqq B[0,1]=\overline{B(0,1)}=\{x\in X:\Vert x\Vert\leq1\}$ +\begin_inset Formula $B_{X}\coloneqq B[0,1]=\overline{B(0,1)}=\{x\in X\mid \Vert x\Vert\leq1\}$ \end_inset y conjunto de @@ -266,7 +266,7 @@ espacio normado vectores unitarios \series default a -\begin_inset Formula $S_{X}\coloneqq\partial B(0,1)=\{x\in X:\Vert x\Vert=1\}$ +\begin_inset Formula $S_{X}\coloneqq\partial B(0,1)=\{x\in X\mid \Vert x\Vert=1\}$ \end_inset . @@ -2655,7 +2655,7 @@ topología cociente \end_inset a -\begin_inset Formula $\{V\subseteq(X/\sim):p^{-1}(V)\in{\cal T}\}$ +\begin_inset Formula $\{V\subseteq(X/\sim)\mid p^{-1}(V)\in{\cal T}\}$ \end_inset , donde @@ -3615,7 +3615,7 @@ Por isomorfismo podemos suponer que el dominio es \end_inset , -\begin_inset Formula $\sup_{x\in S_{\ell_{n}^{1}}}\Vert T(x)\Vert=\sup_{\{x\in\mathbb{K}^{n}:\sum_{i}x_{i}=1\}}\left\Vert \sum_{i}x_{i}a_{i}\right\Vert =\sup_{i=1}^{n}a_{i}<\infty$ +\begin_inset Formula $\sup_{x\in S_{\ell_{n}^{1}}}\Vert T(x)\Vert=\sup_{\{x\in\mathbb{K}^{n}\mid \sum_{i}x_{i}=1\}}\left\Vert \sum_{i}x_{i}a_{i}\right\Vert =\sup_{i=1}^{n}a_{i}<\infty$ \end_inset . diff --git a/algl/n1.lyx b/algl/n1.lyx index 47d85c9..da5b457 100644 --- a/algl/n1.lyx +++ b/algl/n1.lyx @@ -242,7 +242,7 @@ unidad: Inverso para el producto: \series default -\begin_inset Formula $\forall a\in K\backslash\{0\},\exists!a'':a\cdot a''=1$ +\begin_inset Formula $\forall a\in K\backslash\{0\},\exists!a''\mid a\cdot a''=1$ \end_inset ; @@ -1062,7 +1062,7 @@ Si \end_inset , el conjunto -\begin_inset Formula $\mathcal{F}(\mathcal{S},K)=\{f:\mathcal{S}\rightarrow K\}$ +\begin_inset Formula $\mathcal{F}(\mathcal{S},K)=\{f\mid \mathcal{S}\rightarrow K\}$ \end_inset , formado por todas las aplicaciones de @@ -1087,7 +1087,7 @@ Si -espacio vectorial. Con estas mismas operaciones, el conjunto -\begin_inset Formula $\mathcal{C}([a,b],\mathbb{R})=\{f:[a,b]\rightarrow\mathbb{R}|f\text{ continua}\}$ +\begin_inset Formula $\mathcal{C}([a,b],\mathbb{R})=\{f\mid [a,b]\rightarrow\mathbb{R}|f\text{ continua}\}$ \end_inset es un @@ -1320,7 +1320,7 @@ Los subconjuntos . También lo es -\begin_inset Formula $U_{a,b}=\{f\in\mathcal{C}([a,b],\mathbb{R}):f(a)=f(b)\}$ +\begin_inset Formula $U_{a,b}=\{f\in\mathcal{C}([a,b],\mathbb{R})\mid f(a)=f(b)\}$ \end_inset respecto de @@ -1738,7 +1738,7 @@ base canónica \end_inset y 0 en el resto, entonces -\begin_inset Formula $\{A_{ij}:1\leq i\leq m,1\leq j\leq n\}$ +\begin_inset Formula $\{A_{ij}\mid 1\leq i\leq m,1\leq j\leq n\}$ \end_inset es base de diff --git a/algl/n4.lyx b/algl/n4.lyx index cf26416..a0f5c5f 100644 --- a/algl/n4.lyx +++ b/algl/n4.lyx @@ -1095,7 +1095,7 @@ Llamamos filas o columnas: \begin_inset Formula \[ -\chi_{r}=\{(i_{1},\dots,i_{r}):1\leq i_{1}<\dots<i_{r}\leq n\} +\chi_{r}=\{(i_{1},\dots,i_{r})\mid 1\leq i_{1}<\dots<i_{r}\leq n\} \] \end_inset diff --git a/algl/n5.lyx b/algl/n5.lyx index bb844d5..963ebd6 100644 --- a/algl/n5.lyx +++ b/algl/n5.lyx @@ -526,7 +526,7 @@ Los vectores propios de . Así, -\begin_inset Formula $V_{\lambda}=\text{Nuc}(f-\lambda Id)=\{v\in V:(f-\lambda Id)(v)=0\}=\{v\in V:f(v)=\lambda v\}$ +\begin_inset Formula $V_{\lambda}=\text{Nuc}(f-\lambda Id)=\{v\in V\mid (f-\lambda Id)(v)=0\}=\{v\in V\mid f(v)=\lambda v\}$ \end_inset es el @@ -1519,7 +1519,7 @@ Queremos ver que . Si -\begin_inset Formula $E_{k-1}^{\bot}:=\{v\in V:v\bot E_{k-1}\}$ +\begin_inset Formula $E_{k-1}^{\bot}:=\{v\in V\mid v\bot E_{k-1}\}$ \end_inset , basta ver que para todo subespacio @@ -1860,7 +1860,7 @@ Sea \begin_deeper \begin_layout Standard -\begin_inset Formula $\sup\{\Vert Ax\Vert:\Vert x\Vert=1\}=\sup\{\sum_{k}|Ax|_{k}:\sum_{k}|x_{k}|=1\}=\sup\{\sum_{k,i}|a_{ki}||x_{i}|:\sum_{i}|x_{i}|=1\}$ +\begin_inset Formula $\sup\{\Vert Ax\Vert\mid\Vert x\Vert=1\}=\sup\{\sum_{k}|Ax|_{k}\mid\sum_{k}|x_{k}|=1\}=\sup\{\sum_{k,i}|a_{ki}||x_{i}|\mid\sum_{i}|x_{i}|=1\}$ \end_inset . @@ -1889,7 +1889,7 @@ Sea \end_inset luego -\begin_inset Formula $\sup\{\sum_{k,i}|a_{ki}||x_{i}|:\sum_{i}|x_{i}|=1\}=\max_{i}\sum_{k}|a_{ki}|$ +\begin_inset Formula $\sup\{\sum_{k,i}|a_{ki}||x_{i}|\mid\sum_{i}|x_{i}|=1\}=\max_{i}\sum_{k}|a_{ki}|$ \end_inset . @@ -1922,10 +1922,14 @@ luego \begin_deeper \begin_layout Standard -\begin_inset Formula $\Vert A\Vert_{2}^{2}=\sup\left\{ \frac{\Vert Ax\Vert_{2}^{2}}{\Vert x\Vert_{2}^{2}}:\Vert x\Vert_{2}=1\right\} =\sup\left\{ \frac{\langle Ax,Ax\rangle}{\langle x,x\rangle}=\frac{\langle A^{*}Ax,x\rangle}{\langle x,x\rangle}=R_{A^{*}A}(x):\Vert x\Vert_{2}=1\right\} $ +\begin_inset Formula +\[ +\Vert A\Vert_{2}^{2}=\sup\left\{ \frac{\Vert Ax\Vert_{2}^{2}}{\Vert x\Vert_{2}^{2}}\;\middle|\;\Vert x\Vert_{2}=1\right\} =\sup\left\{ \frac{\langle Ax,Ax\rangle}{\langle x,x\rangle}=\frac{\langle A^{*}Ax,x\rangle}{\langle x,x\rangle}=R_{A^{*}A}(x)\;\middle|\;\Vert x\Vert_{2}=1\right\} , +\] + \end_inset -, pero si + pero si \begin_inset Formula $\lambda_{1},\dots,\lambda_{m}\geq0$ \end_inset @@ -1938,7 +1942,7 @@ luego \end_inset son los subespacios propios asociados, -\begin_inset Formula $\rho(A^{*}A)=\max\{\lambda_{1},\dots,\lambda_{m}\}=\max_{k=1}^{m}\max\{R_{A^{*}A}(v):v\in E_{k}\setminus\{0\}\}=\max\{R_{A^{*}A}(v):v\neq0\}$ +\begin_inset Formula $\rho(A^{*}A)=\max\{\lambda_{1},\dots,\lambda_{m}\}=\max_{k=1}^{m}\max\{R_{A^{*}A}(v)\mid v\in E_{k}\setminus\{0\}\}=\max\{R_{A^{*}A}(v)\mid v\neq0\}$ \end_inset , y como @@ -1950,7 +1954,7 @@ R_{A^{*}A}(v)=\frac{\langle Av,v\rangle}{\langle v,v\rangle}=\left\langle A\frac \end_inset queda -\begin_inset Formula $\rho(A^{*}A)=\max\{R_{A^{*}A}(v):v\neq0\}=\max\{R_{A^{*}A}(v):\Vert v\Vert_{2}=1\}=\Vert A\Vert_{2}^{2}$ +\begin_inset Formula $\rho(A^{*}A)=\max\{R_{A^{*}A}(v)\mid v\neq0\}=\max\{R_{A^{*}A}(v)\mid\Vert v\Vert_{2}=1\}=\Vert A\Vert_{2}^{2}$ \end_inset . @@ -1968,8 +1972,8 @@ queda \begin_layout Standard \begin_inset Formula \begin{align*} -\Vert A\Vert_{\infty} & =\sup\{\Vert Ax\Vert_{\infty}:\Vert x\Vert_{\infty}=1\}=\sup\{\max_{k}|Ax|_{k}:\max_{k}|x_{k}|=1\}=\\ - & =\sup\left\{ \max_{k}\left|\sum_{i}a_{ki}x_{i}\right|:\max_{i}|x_{i}|=1\right\} =\max_{k}\sup\left\{ \left|\sum_{i}a_{ki}x_{i}\right|:\max_{i}|x_{i}|=1\right\} . +\Vert A\Vert_{\infty} & =\sup\{\Vert Ax\Vert_{\infty}\mid\Vert x\Vert_{\infty}=1\}=\sup\{\max_{k}|Ax|_{k}\mid\max_{k}|x_{k}|=1\}=\\ + & =\sup\left\{ \max_{k}\left|\sum_{i}a_{ki}x_{i}\right|\;\middle|\;\max_{i}|x_{i}|=1\right\} =\max_{k}\sup\left\{ \left|\sum_{i}a_{ki}x_{i}\right|\;\middle|\;\max_{i}|x_{i}|=1\right\} . \end{align*} \end_inset @@ -1995,7 +1999,7 @@ queda \end_inset , con lo que -\begin_inset Formula $\sup\{|\sum_{i}a_{ki}x_{i}|:\max_{i}|x_{i}|=1\}=\left|\sum_{i}|a_{ki}|\right|=\sum_{i}|a_{ki}|$ +\begin_inset Formula $\sup\{|\sum_{i}a_{ki}x_{i}|\mid\max_{i}|x_{i}|=1\}=\left|\sum_{i}|a_{ki}|\right|=\sum_{i}|a_{ki}|$ \end_inset , luego @@ -2211,7 +2215,7 @@ La diagonal no cambia, la matriz sigue siendo triangular superior y, para \end_deeper \begin_layout Standard De aquí que -\begin_inset Formula $\rho(A)=\inf\{\Vert A\Vert:\Vert\cdot\Vert\text{ es una norma matricial en }{\cal M}_{n}(\mathbb{K})\}$ +\begin_inset Formula $\rho(A)=\inf\{\Vert A\Vert\mid\Vert\cdot\Vert\text{ es una norma matricial en }{\cal M}_{n}(\mathbb{K})\}$ \end_inset . @@ -2722,7 +2722,7 @@ Si Demostración: \series default Sea -\begin_inset Formula $K:=\{g\in G:\Vert f-g\Vert\leq\Vert f\Vert\}$ +\begin_inset Formula $K:=\{g\in G\mid \Vert f-g\Vert\leq\Vert f\Vert\}$ \end_inset , @@ -907,7 +907,7 @@ Demostración: . En dimensión finita, -\begin_inset Formula $\Vert M^{-1}N\Vert_{A}=\max\{\Vert M^{-1}Nv\Vert_{A}:\Vert v\Vert_{A}=1\}$ +\begin_inset Formula $\Vert M^{-1}N\Vert_{A}=\max\{\Vert M^{-1}Nv\Vert_{A}\mid \Vert v\Vert_{A}=1\}$ \end_inset . @@ -1769,7 +1769,7 @@ A \emph default es vector, devuelve una matriz diagonal con elementos del vector en la diagonal -\begin_inset Formula $\{(i,j):i+k=j\}$ +\begin_inset Formula $\{(i,j)\mid i+k=j\}$ \end_inset , y de lo contrario devuelve un vector con los elementos de dicha diagonal @@ -1722,7 +1722,7 @@ nodos hiperarcos \series default -\begin_inset Formula $H\subseteq\{(A,B)\in{\cal P}(V)\times{\cal P}(V):A,B\neq\emptyset\}$ +\begin_inset Formula $H\subseteq\{(A,B)\in{\cal P}(V)\times{\cal P}(V)\mid A,B\neq\emptyset\}$ \end_inset . diff --git a/ar/n.pdf b/ar/n.pdf Binary files differdeleted file mode 100755 index ea5d2a3..0000000 --- a/ar/n.pdf +++ /dev/null @@ -234,12 +234,16 @@ Claves Ajenas \family sans (lista numerada de +\lang english + \begin_inset Quotes cld \end_inset ( \emph on -atributo +a +\lang spanish +tributo \emph default , ...) Referencia_a \emph on @@ -249,7 +253,9 @@ NOMBRE_TABLA \emph on atributo_clave \emph default -, ...) +, ... +\lang english +) \begin_inset Quotes crd \end_inset @@ -269,18 +275,23 @@ Derivado \family sans (lista numerada de +\lang english + \begin_inset Quotes cld \end_inset \emph on -atributo +a +\lang spanish +tributo \emph default = \emph on -fórmula +fórmul \emph default - +\lang english +a \begin_inset Quotes crd \end_inset @@ -4639,7 +4639,7 @@ condición \end_inset es una condición, -\begin_inset Formula $\sigma_{C}(R):=(\{r\in R:C(r)\},T,N)$ +\begin_inset Formula $\sigma_{C}(R):=(\{r\in R\mid C(r)\},T,N)$ \end_inset , donde @@ -4787,7 +4787,7 @@ El producto cartesiano ampliado y la reunión son asociativas, y son conmutativa Reunión natural \series default : Sea -\begin_inset Formula $\{j_{1},\dots,j_{p}\}:=\{j:M_{j}\notin\{N_{i}\}\}$ +\begin_inset Formula $\{j_{1},\dots,j_{p}\}\mid =\{j\mid M_{j}\notin\{N_{i}\}\}$ \end_inset , si para @@ -4805,7 +4805,7 @@ Reunión natural , entonces \begin_inset Formula \[ -R\hexstar S:=(\{r*(s_{j_{1}},\dots,s_{j_{p}}):r\in R,s\in S,\forall i,j,(N_{i}=M_{j}\implies r_{i}=s_{j})\},T*U,N*M). +R\hexstar S:=(\{r*(s_{j_{1}},\dots,s_{j_{p}})\mid r\in R,s\in S,\forall i,j,(N_{i}=M_{j}\implies r_{i}=s_{j})\},T*U,N*M). \] \end_inset @@ -4836,7 +4836,7 @@ reunión externa izquierda \end_inset como -\begin_inset Formula $R]\bowtie_{C}S:=R\bowtie_{C}S\cup(\{r\in R:\nexists s\in S:C(r,s)\}\times N_{m})$ +\begin_inset Formula $R]\bowtie_{C}S:=R\bowtie_{C}S\cup(\{r\in R\mid \nexists s\in S\mid C(r,s)\}\times N_{m})$ \end_inset , la @@ -4844,7 +4844,7 @@ reunión externa izquierda reunión externa derecha \series default como -\begin_inset Formula $R\bowtie[_{C}S:=R\bowtie_{C}S\cup(N_{n}\times\{s\in S:\nexists r\in R:C(r,s)\})$ +\begin_inset Formula $R\bowtie[_{C}S:=R\bowtie_{C}S\cup(N_{n}\times\{s\in S\mid \nexists r\in R\mid C(r,s)\})$ \end_inset y la @@ -4870,7 +4870,7 @@ División , entonces \begin_inset Formula \[ -R\div S:=(\{r:\forall s\in S,r*s\in R\},(T_{1},\dots,T_{n}),(N_{1},\dots,N_{n})). +R\div S:=(\{r\mid \forall s\in S,r*s\in R\},(T_{1},\dots,T_{n}),(N_{1},\dots,N_{n})). \] \end_inset @@ -5220,7 +5220,7 @@ segura \end_inset se refiere al conjunto -\begin_inset Formula $\{T:t_{1},\dots,t_{n}\in\bigcup_{n\in\mathbb{N}}D^{n}\land\text{COND}(t_{1},\dots,t_{n})\}$ +\begin_inset Formula $\{T\mid t_{1},\dots,t_{n}\in\bigcup_{n\in\mathbb{N}}D^{n}\land\text{COND}(t_{1},\dots,t_{n})\}$ \end_inset . @@ -917,6 +917,8 @@ sideways false status open \begin_layout Plain Layout + +\lang english \begin_inset ERT status open @@ -1092,6 +1094,8 @@ in C$}{rehacer $e$} \end_inset +\lang spanish + \begin_inset Caption Standard \begin_layout Plain Layout @@ -901,7 +901,7 @@ Una forma sentencial \series default es un elemento de -\begin_inset Formula $D(G):=\{\alpha\in(V_{N}\cup V_{T})^{*}:S\Rightarrow^{*}\alpha\}$ +\begin_inset Formula $D(G):=\{\alpha\in(V_{N}\cup V_{T})^{*}\mid S\Rightarrow^{*}\alpha\}$ \end_inset , y una @@ -742,7 +742,7 @@ Dada una GLC como \begin_inset Formula \[ -\mathsf{PRIMERO}(\alpha):=\{a\in V_{T}:\exists\beta:\alpha\Rightarrow^{*}a\beta\}\cup\{\lambda:\alpha\Rightarrow^{*}\lambda\}. +\mathsf{PRIMERO}(\alpha):=\{a\in V_{T}\mid \exists\beta:\alpha\Rightarrow^{*}a\beta\}\cup\{\lambda\mid \alpha\Rightarrow^{*}\lambda\}. \] \end_inset @@ -986,7 +986,7 @@ noprefix "false" \begin_inset Formula \begin{multline*} \mathsf{PRIMERO}(X_{1}\cdots X_{n})=\\ -=\bigcup_{i=1}^{\min(\{i:X_{1}\cdots X_{i}\nRightarrow^{*}\lambda\}\cup\{n\})}(\sigma(X_{i})\setminus\{\lambda\})\cup\{\lambda:X_{1}\cdots X_{n}\Rightarrow^{*}\lambda\}. +=\bigcup_{i=1}^{\min(\{i\mid X_{1}\cdots X_{i}\nRightarrow^{*}\lambda\}\cup\{n\})}(\sigma(X_{i})\setminus\{\lambda\})\cup\{\lambda\mid X_{1}\cdots X_{n}\Rightarrow^{*}\lambda\}. \end{multline*} \end_inset @@ -1250,7 +1250,7 @@ Definimos como \begin_inset Formula \[ -\mathsf{SIGUIENTE}(A):=\{a\in V_{T}:\exists\alpha,\beta:S\Rightarrow^{+}\alpha Aa\beta\}\cup\{\$:\exists\alpha:S\Rightarrow^{*}\alpha A\}, +\mathsf{SIGUIENTE}(A):=\{a\in V_{T}\mid \exists\alpha,\beta:S\Rightarrow^{+}\alpha Aa\beta\}\cup\{\$\mid \exists\alpha\mid S\Rightarrow^{*}\alpha A\}, \] \end_inset @@ -3251,7 +3251,7 @@ Si, para \end_inset , -\begin_inset Formula $\rho(I):=\{R:\exists a\in V_{T}:[R,a]\in I\}$ +\begin_inset Formula $\rho(I):=\{R\mid \exists a\in V_{T}\mid [R,a]\in I\}$ \end_inset , para @@ -5179,7 +5179,7 @@ tabla de análisis \end_inset dada por -\begin_inset Formula $M(A,a):=\{A\to\alpha\in P:a\in\mathsf{Predict}(A\to\alpha)\}$ +\begin_inset Formula $M(A,a):=\{A\to\alpha\in P\mid a\in\mathsf{Predict}(A\to\alpha)\}$ \end_inset , que a cada no terminal a derivar y terminal siguiente en la entrada le diff --git a/cn/n.pdf b/cn/n.pdf Binary files differdeleted file mode 100755 index 6264062..0000000 --- a/cn/n.pdf +++ /dev/null @@ -608,11 +608,11 @@ Una familia de conjuntos es una colección Unión arbitraria: \series default -\begin_inset Formula $\cup{\cal C}=\{x|\exists A\in{\cal C}:x\in A\}$ +\begin_inset Formula $\cup{\cal C}=\{x|\exists A\in{\cal C}\mid x\in A\}$ \end_inset ; -\begin_inset Formula $\cup_{i\in I}A_{i}=\{x|\exists i\in I:x\in A_{i}\}$ +\begin_inset Formula $\cup_{i\in I}A_{i}=\{x|\exists i\in I\mid x\in A_{i}\}$ \end_inset @@ -624,11 +624,11 @@ Unión arbitraria: Intersección arbitraria: \series default -\begin_inset Formula $\cap{\cal C}=\{x|\forall A\in{\cal C}:x\in A\}$ +\begin_inset Formula $\cap{\cal C}=\{x|\forall A\in{\cal C}\mid x\in A\}$ \end_inset ; -\begin_inset Formula $\cap_{i\in I}A_{i}=\{x|\forall i\in I:x\in A_{i}\}$ +\begin_inset Formula $\cap_{i\in I}A_{i}=\{x|\forall i\in I\mid x\in A_{i}\}$ \end_inset @@ -888,7 +888,7 @@ Conjunto final: Dominio: \series default -\begin_inset Formula $\text{Dom}R=\{a\in A|\exists b\in B:(a,b)\in R\}$ +\begin_inset Formula $\text{Dom}R=\{a\in A|\exists b\in B\mid (a,b)\in R\}$ \end_inset . @@ -900,7 +900,7 @@ Dominio: Imagen: \series default -\begin_inset Formula $\text{Im}R=\{b\in B|\exists a\in A:(a,b)\in R\}$ +\begin_inset Formula $\text{Im}R=\{b\in B|\exists a\in A\mid (a,b)\in R\}$ \end_inset . @@ -121,7 +121,7 @@ aplicación \end_inset , de modo que -\begin_inset Formula $f=\{(n,n^{2}):n\in\mathbb{N}\}$ +\begin_inset Formula $f=\{(n,n^{2})\mid n\in\mathbb{N}\}$ \end_inset . @@ -221,7 +221,7 @@ imagen directa \end_inset : -\begin_inset Formula $\text{Im}f=f(A)=\{b\in B:\exists a:f(a)=b\}\subseteq B$ +\begin_inset Formula $\text{Im}f=f(A)=\{b\in B\mid\exists a\mid f(a)=b\}\subseteq B$ \end_inset . @@ -1359,7 +1359,7 @@ producto directo como el conjunto \begin_inset Formula \[ -\prod_{i\in I}A_{i}=\left\{ f:I\rightarrow\cup_{i\in I}:f(i)\in A_{i}\forall i\in I\right\} +\prod_{i\in I}A_{i}=\left\{ f\mid I\rightarrow\bigcup_{i\in I}\;\middle|\;f(i)\in A_{i}\forall i\in I\right\} \] \end_inset @@ -1383,7 +1383,7 @@ Si es finito y se escribe como una lista, podemos escribir el conjunto como -\begin_inset Formula $A_{1}\times\cdots\times A_{n}=\{(x_{1},\dots,x_{n}):x_{i}\in A_{i},i=1,\dots,n\}$ +\begin_inset Formula $A_{1}\times\cdots\times A_{n}=\{(x_{1},\dots,x_{n})\mid x_{i}\in A_{i},i=1,\dots,n\}$ \end_inset . @@ -1420,7 +1420,7 @@ Sean \end_inset y un conjunto de biyecciones -\begin_inset Formula $\{f_{i}:A_{i}\rightarrow B_{\sigma(i)}\}_{i\in I}$ +\begin_inset Formula $\{f_{i}\mid A_{i}\rightarrow B_{\sigma(i)}\}_{i\in I}$ \end_inset , entonces existe una biyección @@ -125,7 +125,7 @@ Sea \end_inset , su clase de equivalencia es -\begin_inset Formula $[a]=\{b\in A:a\sim b\}$ +\begin_inset Formula $[a]=\{b\in A\mid a\sim b\}$ \end_inset . @@ -2100,7 +2100,7 @@ raíz Así, todo número complejo tiene \begin_inset Formula \[ -\phi(n)=|\{m\in\{1,\dots,n-1\}:\text{mcd}(m,n)=1\}| +\phi(n)=|\{m\in\{1,\dots,n-1\}\mid \text{mcd}(m,n)=1\}| \] \end_inset @@ -201,7 +201,7 @@ Demostración: \end_inset y -\begin_inset Formula $R=\{x\in\mathbb{Z}|x\geq0\land\exists n\in\mathbb{Z}:x=a-bn\}\subseteq\mathbb{N}$ +\begin_inset Formula $R=\{x\in\mathbb{Z}|x\geq0\land\exists n\in\mathbb{Z}\mid x=a-bn\}\subseteq\mathbb{N}$ \end_inset . @@ -512,7 +512,7 @@ Dados máximo común divisor \series default es -\begin_inset Formula $\text{mcd}(a,b)=\max\{d\in\mathbb{Z}:d|a\land d|b\}$ +\begin_inset Formula $\text{mcd}(a,b)=\max\{d\in\mathbb{Z}\mid d|a\land d|b\}$ \end_inset (excepción: @@ -792,7 +792,7 @@ El máximo común divisor de \end_inset es -\begin_inset Formula $\text{mcd}(a_{1},\dots,a_{n})=\max\{d\in\mathbb{Z}:\forall i,d|a_{i}\}$ +\begin_inset Formula $\text{mcd}(a_{1},\dots,a_{n})=\max\{d\in\mathbb{Z}\mid \forall i,d|a_{i}\}$ \end_inset . @@ -1071,7 +1071,7 @@ Dados mínimo común múltiplo \series default es -\begin_inset Formula $\text{mcm}(a,b)=\min\{m\in\mathbb{Z}^{+}:a|m\land b|m\}$ +\begin_inset Formula $\text{mcm}(a,b)=\min\{m\in\mathbb{Z}^{+}\mid a|m\land b|m\}$ \end_inset . @@ -1215,7 +1215,7 @@ El mínimo común múltiplo de \end_inset es -\begin_inset Formula $\text{mcm}(a_{1},\dots,a_{n})=\min\{m\in\mathbb{Z}^{+}:\forall i,a_{i}|m\}$ +\begin_inset Formula $\text{mcm}(a_{1},\dots,a_{n})=\min\{m\in\mathbb{Z}^{+}\mid \forall i,a_{i}|m\}$ \end_inset . @@ -453,7 +453,7 @@ divisor \end_layout \begin_layout Enumerate -\begin_inset Formula $A|B\land B|A\implies\exists\mu\in K\backslash\{0\}:A=\mu B$ +\begin_inset Formula $A|B\land B|A\implies\exists\mu\in K\backslash\{0\}\mid A=\mu B$ \end_inset . diff --git a/ealg/n1.lyx b/ealg/n1.lyx index 7068e05..a5d022d 100644 --- a/ealg/n1.lyx +++ b/ealg/n1.lyx @@ -223,7 +223,7 @@ grado \end_inset a -\begin_inset Formula $\text{gr}(p):=\max\{k\in\mathbb{N}:p_{k}\neq0\}$ +\begin_inset Formula $\text{gr}(p):=\max\{k\in\mathbb{N}\mid p_{k}\neq0\}$ \end_inset , @@ -831,7 +831,7 @@ euclídea \end_layout \begin_layout Enumerate -\begin_inset Formula $\forall a\in D,b\in D\setminus\{0\},\exists q,r\in D:(a=bq+r\land(r=0\lor\delta(r)<\delta(b)))$ +\begin_inset Formula $\forall a\in D,b\in D\setminus\{0\},\exists q,r\in D\mid (a=bq+r\land(r=0\lor\delta(r)<\delta(b)))$ \end_inset . @@ -968,7 +968,7 @@ Para \end_inset , existe -\begin_inset Formula $m:=\max\{k\in\mathbb{N}:(X-a)^{k}\mid f\}$ +\begin_inset Formula $m:=\max\{k\in\mathbb{N}\mid (X-a)^{k}\mid f\}$ \end_inset . @@ -1875,7 +1875,7 @@ teorema \end_inset ], -\begin_inset Formula $c(p):=\{x:x=\text{mcd}_{k\geq0}p_{k}\}$ +\begin_inset Formula $c(p):=\{x\mid x=\text{mcd}_{k\geq0}p_{k}\}$ \end_inset , y [...] si @@ -3967,11 +3967,11 @@ Queremos ver que, para . Con esto, sean -\begin_inset Formula $A:=\{i\in\mathbb{N}^{n}:a_{i}\neq0\}$ +\begin_inset Formula $A:=\{i\in\mathbb{N}^{n}\mid a_{i}\neq0\}$ \end_inset , -\begin_inset Formula $B:=\{j\in\mathbb{N}^{n}:b_{j}\neq0\}$ +\begin_inset Formula $B:=\{j\in\mathbb{N}^{n}\mid b_{j}\neq0\}$ \end_inset , diff --git a/ealg/n2.lyx b/ealg/n2.lyx index a006108..cbcd97d 100644 --- a/ealg/n2.lyx +++ b/ealg/n2.lyx @@ -4611,7 +4611,7 @@ clausura algebraica es \begin_inset Formula \[ -\overline{K}_{L}:=\{\alpha\in L:\alpha\text{ es algebraico sobre }K\}. +\overline{K}_{L}:=\{\alpha\in L\mid \alpha\text{ es algebraico sobre }K\}. \] \end_inset diff --git a/ealg/n4.lyx b/ealg/n4.lyx index e9f8c50..4a46a08 100644 --- a/ealg/n4.lyx +++ b/ealg/n4.lyx @@ -1089,7 +1089,7 @@ grupo de Galois \end_inset lleva raíces a raíces y por tanto -\begin_inset Formula $\sigma|_{\{\alpha_{1},\dots,\alpha_{n}\}}:\{\alpha_{1},\dots,\alpha_{n}\}\to\{\alpha_{1},\dots,\alpha_{n}\}$ +\begin_inset Formula $\sigma|_{\{\alpha_{1},\dots,\alpha_{n}\}}\mid \{\alpha_{1},\dots,\alpha_{n}\}\to\{\alpha_{1},\dots,\alpha_{n}\}$ \end_inset es inyectiva por serlo @@ -1491,7 +1491,7 @@ teorema \end_inset , -\begin_inset Formula $K(\{\alpha\in\overline{K}:\exists f\in{\cal P}:f(\alpha)=0\})$ +\begin_inset Formula $K(\{\alpha\in\overline{K}\mid \exists f\in{\cal P}:f(\alpha)=0\})$ \end_inset , por lo que existe un cuerpo de descomposición de @@ -2010,7 +2010,7 @@ Para cada \end_inset elementos y viene dado por -\begin_inset Formula $\mathbb{F}_{p^{n}}:=\{\alpha\in\overline{\mathbb{Z}_{p}}:\alpha^{p^{n}}=\alpha\}$ +\begin_inset Formula $\mathbb{F}_{p^{n}}:=\{\alpha\in\overline{\mathbb{Z}_{p}}\mid \alpha^{p^{n}}=\alpha\}$ \end_inset . @@ -2019,7 +2019,7 @@ Para cada \begin_deeper \begin_layout Standard Sea -\begin_inset Formula $S:=\{\alpha\in\overline{\mathbb{Z}_{p}}:\alpha^{p^{n}}=\alpha\}$ +\begin_inset Formula $S:=\{\alpha\in\overline{\mathbb{Z}_{p}}\mid \alpha^{p^{n}}=\alpha\}$ \end_inset el conjunto de raíces de diff --git a/ealg/n5.lyx b/ealg/n5.lyx index a3eaed8..18c97fd 100644 --- a/ealg/n5.lyx +++ b/ealg/n5.lyx @@ -112,7 +112,7 @@ de uno , y llamamos \begin_inset Formula \[ -{\cal U}_{n}(K):=\{\xi\in K:\xi^{n}=1\}=\{\xi\in K:o_{K^{*}}(\xi)\mid n\}. +{\cal U}_{n}(K):=\{\xi\in K\mid \xi^{n}=1\}=\{\xi\in K\mid o_{K^{*}}(\xi)\mid n\}. \] \end_inset diff --git a/ealg/n6.lyx b/ealg/n6.lyx index 343a1ac..fd441a7 100644 --- a/ealg/n6.lyx +++ b/ealg/n6.lyx @@ -243,7 +243,7 @@ Demostración: \end_inset y -\begin_inset Formula $R:=\{\alpha_{1}:=\alpha,\dots,\alpha_{m}\}$ +\begin_inset Formula $R:=\{\alpha_{1}\mid =\alpha,\dots,\alpha_{m}\}$ \end_inset el conjunto de las raíces de @@ -354,7 +354,7 @@ teorema \end_inset Sean -\begin_inset Formula ${\cal P}:=\{f_{\alpha}:=\text{Irr}(\alpha,K)\}_{\alpha\in L}\subseteq K[X]\setminus0$ +\begin_inset Formula ${\cal P}:=\{f_{\alpha}\mid =\text{Irr}(\alpha,K)\}_{\alpha\in L}\subseteq K[X]\setminus0$ \end_inset y @@ -1107,7 +1107,7 @@ clausura normal , y viene dada por \begin_inset Formula \[ -N:=\bigcap\{E\text{ intermedio en }L\subseteq\overline{L}:K\subseteq E\text{ normal}\}. +N:=\bigcap\{E\text{ intermedio en }L\subseteq\overline{L}\mid K\subseteq E\text{ normal}\}. \] \end_inset diff --git a/ealg/n7.lyx b/ealg/n7.lyx index 2faa1a1..f5f15b6 100644 --- a/ealg/n7.lyx +++ b/ealg/n7.lyx @@ -83,7 +83,7 @@ \begin_layout Standard \begin_inset Formula \[ -\text{Gal}(K(X)/K)=\bigg\{\sigma\,\Big\vert\,\exists a,b,c,d\in K:\bigg(ad-bc\neq0\land\sigma(X)=\frac{aX+b}{cX+d}\bigg)\bigg\}. +\text{Gal}(K(X)/K)=\bigg\{\sigma\,\Big\vert\,\exists a,b,c,d\in K\mid \bigg(ad-bc\neq0\land\sigma(X)=\frac{aX+b}{cX+d}\bigg)\bigg\}. \] \end_inset @@ -139,8 +139,8 @@ conexión de Galois dado por \begin_inset Formula \begin{align*} -f(F):=F' & :=\{\sigma\in G:\forall\alpha\in F,\sigma(\alpha)=\alpha\}=\text{Gal}(L/F),\\ -g(H):=H' & :=\{\alpha\in L:\forall\sigma\in H,\sigma(\alpha)=\alpha\}=\bigcap_{\sigma\in H}\text{Fix}\sigma. +f(F):=F' & :=\{\sigma\in G\mid \forall\alpha\in F,\sigma(\alpha)=\alpha\}=\text{Gal}(L/F),\\ +g(H):=H' & :=\{\alpha\in L\mid \forall\sigma\in H,\sigma(\alpha)=\alpha\}=\bigcap_{\sigma\in H}\text{Fix}\sigma. \end{align*} \end_inset @@ -150,7 +150,7 @@ En particular, para \end_inset , -\begin_inset Formula $K(\beta)'=\{\sigma\in G:\sigma(\beta)=\beta\}$ +\begin_inset Formula $K(\beta)'=\{\sigma\in G\mid \sigma(\beta)=\beta\}$ \end_inset , y para diff --git a/edo/n.pdf b/edo/n.pdf Binary files differdeleted file mode 100644 index 88ff256..0000000 --- a/edo/n.pdf +++ /dev/null diff --git a/epe/n.pdf b/epe/n.pdf Binary files differdeleted file mode 100644 index d992aea..0000000 --- a/epe/n.pdf +++ /dev/null diff --git a/fc/AND_ANSI_Labelled.svg b/fc/AND_ANSI_Labelled.svg Binary files differindex ee294dc..5ee5c9c 100644 --- a/fc/AND_ANSI_Labelled.svg +++ b/fc/AND_ANSI_Labelled.svg diff --git a/fc/NAND_ANSI_Labelled.svg b/fc/NAND_ANSI_Labelled.svg Binary files differindex 7f97027..719786a 100644 --- a/fc/NAND_ANSI_Labelled.svg +++ b/fc/NAND_ANSI_Labelled.svg diff --git a/fc/NOR_ANSI_Labelled.svg b/fc/NOR_ANSI_Labelled.svg Binary files differindex 0fd18f9..01f63e4 100644 --- a/fc/NOR_ANSI_Labelled.svg +++ b/fc/NOR_ANSI_Labelled.svg diff --git a/fc/Not-gate-en.svg b/fc/Not-gate-en.svg Binary files differindex daf957b..523d62d 100644 --- a/fc/Not-gate-en.svg +++ b/fc/Not-gate-en.svg diff --git a/fc/OR_ANSI_Labelled.svg b/fc/OR_ANSI_Labelled.svg Binary files differindex 6275ef9..05b61be 100644 --- a/fc/OR_ANSI_Labelled.svg +++ b/fc/OR_ANSI_Labelled.svg diff --git a/fc/XOR_ANSI.svg b/fc/XOR_ANSI.svg Binary files differindex 6f14e5b..4981dec 100644 --- a/fc/XOR_ANSI.svg +++ b/fc/XOR_ANSI.svg diff --git a/fc/Xnor-gate-en.svg b/fc/Xnor-gate-en.svg Binary files differindex b205563..2a18ed0 100644 --- a/fc/Xnor-gate-en.svg +++ b/fc/Xnor-gate-en.svg @@ -134,9 +134,33 @@ esquema de Von Neumann \end_layout \begin_layout Standard +\begin_inset Float figure +wide false +sideways false +status open + +\begin_layout Plain Layout +\align center \begin_inset Graphics filename buses.png - width 100text% + width 90text% + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Caption Standard + +\begin_layout Plain Layout +Esquema von Neumann en un ordenador moderno. +\end_layout + +\end_inset + + +\end_layout \end_inset @@ -749,10 +773,33 @@ Unified Extensible Firmware Interface \end_layout \begin_layout Standard +\begin_inset Float figure +wide false +sideways false +status open + +\begin_layout Plain Layout \align center \begin_inset Graphics filename image.TZVI9Y.png - width 100text% + width 90text% + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Caption Standard + +\begin_layout Plain Layout +Placa base de un ordenador de escritorio típico. +\end_layout + +\end_inset + + +\end_layout \end_inset @@ -393,8 +393,9 @@ Puertas lógicas \begin_inset Text \begin_layout Plain Layout -\begin_inset Graphics - filename AND_ANSI_Labelled.svg +\begin_inset External + template VectorGraphics + filename AND_ANSI_Labelled.pdf height 14pt \end_inset @@ -541,8 +542,9 @@ AND \begin_inset Text \begin_layout Plain Layout -\begin_inset Graphics - filename OR_ANSI_Labelled.svg +\begin_inset External + template VectorGraphics + filename OR_ANSI_Labelled.pdf height 14pt \end_inset @@ -689,8 +691,9 @@ OR \begin_inset Text \begin_layout Plain Layout -\begin_inset Graphics - filename XOR_ANSI.svg +\begin_inset External + template VectorGraphics + filename XOR_ANSI.pdf height 14pt \end_inset @@ -730,8 +733,9 @@ XOR \begin_inset Text \begin_layout Plain Layout -\begin_inset Graphics - filename NAND_ANSI_Labelled.svg +\begin_inset External + template VectorGraphics + filename NAND_ANSI_Labelled.pdf height 14pt \end_inset @@ -767,8 +771,9 @@ NAND \begin_inset Text \begin_layout Plain Layout -\begin_inset Graphics - filename NOR_ANSI_Labelled.svg +\begin_inset External + template VectorGraphics + filename NOR_ANSI_Labelled.pdf height 14pt \end_inset @@ -804,8 +809,9 @@ NOR \begin_inset Text \begin_layout Plain Layout -\begin_inset Graphics - filename Xnor-gate-en.svg +\begin_inset External + template VectorGraphics + filename Xnor-gate-en.pdf height 14pt \end_inset @@ -909,8 +915,9 @@ XNOR \begin_inset Text \begin_layout Plain Layout -\begin_inset Graphics - filename Not-gate-en.svg +\begin_inset External + template VectorGraphics + filename Not-gate-en.pdf height 14pt \end_inset @@ -1047,6 +1054,12 @@ Circuito con \end_layout \begin_layout Standard +\begin_inset Float figure +wide false +sideways false +status open + +\begin_layout Plain Layout \align center \begin_inset Graphics filename image.RAWR9Y.png @@ -1057,6 +1070,23 @@ Circuito con \end_layout +\begin_layout Plain Layout +\begin_inset Caption Standard + +\begin_layout Plain Layout +Codificador. +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + \begin_layout Subsection Decodificador \end_layout @@ -1075,6 +1105,12 @@ Circuito con \end_layout \begin_layout Standard +\begin_inset Float figure +wide false +sideways false +status open + +\begin_layout Plain Layout \align center \begin_inset Graphics filename image.V5MB9Y.png @@ -1085,6 +1121,23 @@ Circuito con \end_layout +\begin_layout Plain Layout +\begin_inset Caption Standard + +\begin_layout Plain Layout +Decodificador. +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + \begin_layout Standard Podemos implementar una función con un decodificador conectando las salidas correspondientes a un @@ -1113,9 +1166,33 @@ Circuito con \end_layout \begin_layout Standard +\begin_inset Float figure +wide false +sideways false +status open + +\begin_layout Plain Layout +\align center \begin_inset Graphics filename image.0PXO9Y.png - width 100text% + width 90text% + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Caption Standard + +\begin_layout Plain Layout +Multiplexores. +\end_layout + +\end_inset + + +\end_layout \end_inset @@ -1186,9 +1263,33 @@ anchura \end_layout \begin_layout Standard +\begin_inset Float figure +wide false +sideways false +status open + +\begin_layout Plain Layout +\align center \begin_inset Graphics filename image.Y3EN9Y.png - width 100text% + width 90text% + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Caption Standard + +\begin_layout Plain Layout +Memoria ROM. +\end_layout + +\end_inset + + +\end_layout \end_inset @@ -253,7 +253,7 @@ En relaciones con aridad dominio \series default como -\begin_inset Formula $\text{Dom}(R)=\{(x_{1},\dots,x_{n-1})|\exists x_{n}:(x_{1},\dots,x_{n})\in R\}$ +\begin_inset Formula $\text{Dom}(R)=\{(x_{1},\dots,x_{n-1})|\exists x_{n}\mid (x_{1},\dots,x_{n})\in R\}$ \end_inset (si la aridad es @@ -261,7 +261,7 @@ dominio \end_inset , entonces -\begin_inset Formula $\text{Dom}(R)=\{x|\exists y:xRy\}$ +\begin_inset Formula $\text{Dom}(R)=\{x|\exists y\mid xRy\}$ \end_inset ), y el @@ -269,7 +269,7 @@ dominio rango \series default como -\begin_inset Formula $\text{Ran}(R)=\{x_{n}|\exists(x_{1},\dots,x_{n-1}):(x_{1},\dots,x_{n})\in R\}$ +\begin_inset Formula $\text{Ran}(R)=\{x_{n}|\exists(x_{1},\dots,x_{n-1})\mid (x_{1},\dots,x_{n})\in R\}$ \end_inset (si la aridad es @@ -277,7 +277,7 @@ rango \end_inset , entonces -\begin_inset Formula $\text{Ran}(R)=\{y|\exists x:xRy\}$ +\begin_inset Formula $\text{Ran}(R)=\{y|\exists x\mid xRy\}$ \end_inset . diff --git a/fuvr1/n1.lyx b/fuvr1/n1.lyx index c26556f..fe23ed5 100644 --- a/fuvr1/n1.lyx +++ b/fuvr1/n1.lyx @@ -269,7 +269,7 @@ Pongamos que existe otro Inverso para el producto: \series default -\begin_inset Formula $\forall a\in\mathbb{K}\backslash\{0\},\exists!a'':a\cdot a''=1$ +\begin_inset Formula $\forall a\in\mathbb{K}\backslash\{0\},\exists!a''\mid a\cdot a''=1$ \end_inset ; @@ -893,7 +893,7 @@ bicho \end_inset -\begin_inset Formula $\bigcap\{I:I\text{ es un conjunto inductivo de }\mathbb{R}\}$ +\begin_inset Formula $\bigcap\{I\mid I\text{ es un conjunto inductivo de }\mathbb{R}\}$ \end_inset , la intersección de todos los conjuntos inductivos y por tanto el más pequeño @@ -960,7 +960,7 @@ Para . Entonces -\begin_inset Formula $S=\{n\in\mathbb{N}:1<n<2\}\neq\emptyset\land r\in s$ +\begin_inset Formula $S=\{n\in\mathbb{N}\mid 1<n<2\}\neq\emptyset\land r\in s$ \end_inset . @@ -1023,11 +1023,11 @@ Demostrar resto de propiedades cuando las estudiemos, si no como ejercicio. \begin_layout Standard Definimos -\begin_inset Formula $\mathbb{Z}:=\{0\}\cup\{n\in\mathbb{R}:n\in\mathbb{N}\text{ o }-n\in\mathbb{N}\}$ +\begin_inset Formula $\mathbb{Z}:=\{0\}\cup\{n\in\mathbb{R}\mid n\in\mathbb{N}\text{ o }-n\in\mathbb{N}\}$ \end_inset y -\begin_inset Formula $\mathbb{Q}:=\{m\cdot n^{-1}:m\in\mathbb{Z},n\in\mathbb{N}\}$ +\begin_inset Formula $\mathbb{Q}:=\{m\cdot n^{-1}\mid m\in\mathbb{Z},n\in\mathbb{N}\}$ \end_inset . @@ -1098,7 +1098,7 @@ Dado un número natural \end_inset , un conjunto -\begin_inset Formula $S\subseteq\{n\in\mathbb{N}:n\geq N\}\subseteq\mathbb{N}$ +\begin_inset Formula $S\subseteq\{n\in\mathbb{N}\mid n\geq N\}\subseteq\mathbb{N}$ \end_inset nos sirve para realizar demostraciones para los naturales a partir de un @@ -1145,7 +1145,7 @@ Teorema Fundamental de la Aritmética Demostración: \series default Sea -\begin_inset Formula $A=\{2\leq n\in\mathbb{N}:n\text{ cumple el Teorema Fund. de la Aritmética}\}$ +\begin_inset Formula $A=\{2\leq n\in\mathbb{N}\mid n\text{ cumple el Teorema Fund. de la Aritmética}\}$ \end_inset . @@ -1233,7 +1233,7 @@ propiedad arquimediana: Demostración: \series default De no ser así, -\begin_inset Formula $A:=\{ny:n\in\mathbb{N}\}$ +\begin_inset Formula $A:=\{ny\mid n\in\mathbb{N}\}$ \end_inset estaría acotado superiormente por @@ -1405,7 +1405,7 @@ Demostremos que existe. \end_inset , se tiene que el conjunto -\begin_inset Formula $\{n\in\mathbb{N}:n>x\}\neq\emptyset$ +\begin_inset Formula $\{n\in\mathbb{N}\mid n>x\}\neq\emptyset$ \end_inset , por lo que tiene un primer elemento @@ -1542,7 +1542,7 @@ raíz cuadrada Definimos \begin_inset Formula \[ -\sqrt{x}:=\sup\{0\leq r\in\mathbb{Q}:r^{2}<x\} +\sqrt{x}:=\sup\{0\leq r\in\mathbb{Q}\mid r^{2}<x\} \] \end_inset @@ -1805,7 +1805,7 @@ Ahora veremos que esto también se cumple con si \end_layout \begin_layout Standard -\begin_inset Formula $\exists\alpha\in\mathbb{R}\backslash\mathbb{Q}:(\alpha^{2}=2\land\alpha=\sup\{0\leq r\in\mathbb{Q}:r^{2}<2\})$ +\begin_inset Formula $\exists\alpha\in\mathbb{R}\backslash\mathbb{Q}:(\alpha^{2}=2\land\alpha=\sup\{0\leq r\in\mathbb{Q}\mid r^{2}<2\})$ \end_inset . @@ -1821,7 +1821,7 @@ status open Demostración: \series default Sea -\begin_inset Formula $A=\{0\leq r\in\mathbb{Q}:r^{2}<2\}$ +\begin_inset Formula $A=\{0\leq r\in\mathbb{Q}\mid r^{2}<2\}$ \end_inset . @@ -1950,7 +1950,7 @@ Sea . También podemos probar que -\begin_inset Formula $\forall x\in\mathbb{R},x=\sup\{r:r\in\mathbb{Q},r<x\}$ +\begin_inset Formula $\forall x\in\mathbb{R},x=\sup\{r\mid r\in\mathbb{Q},r<x\}$ \end_inset , pues si @@ -2235,7 +2235,7 @@ Sea \end_inset ; -\begin_inset Formula $\alpha=\sup\{r\in\mathbb{Q}:r^{p}<x\}$ +\begin_inset Formula $\alpha=\sup\{r\in\mathbb{Q}\mid r^{p}<x\}$ \end_inset . @@ -2266,7 +2266,7 @@ raíz Lo escribimos como \begin_inset Formula \[ -x^{\frac{1}{p}}:=\sqrt[p]{x}:=\sup\{r:r\in\mathbb{Q},r^{p}<x\} +x^{\frac{1}{p}}:=\sqrt[p]{x}:=\sup\{r\mid r\in\mathbb{Q},r^{p}<x\} \] \end_inset diff --git a/fuvr1/n2.lyx b/fuvr1/n2.lyx index bb73cad..6312a4f 100644 --- a/fuvr1/n2.lyx +++ b/fuvr1/n2.lyx @@ -369,7 +369,7 @@ intervalo cerrado \end_inset al conjunto -\begin_inset Formula $[a,b]:=\{x\in\mathbb{R}:a\leq x\leq b\}$ +\begin_inset Formula $[a,b]:=\{x\in\mathbb{R}\mid a\leq x\leq b\}$ \end_inset , @@ -377,7 +377,7 @@ intervalo cerrado intervalo abierto \series default a -\begin_inset Formula $(a,b):=\{x\in\mathbb{R}:a<x<b\}$ +\begin_inset Formula $(a,b):=\{x\in\mathbb{R}\mid a<x<b\}$ \end_inset e @@ -385,11 +385,11 @@ intervalo abierto intervalos semiabiertos \series default por la derecha e izquierda, respectivamente, a -\begin_inset Formula $[a,b):=\{x\in\mathbb{R}:a\leq x<b\}$ +\begin_inset Formula $[a,b):=\{x\in\mathbb{R}\mid a\leq x<b\}$ \end_inset y -\begin_inset Formula $(a,b]:=\{x\in\mathbb{R}:a<x\leq b\}$ +\begin_inset Formula $(a,b]:=\{x\in\mathbb{R}\mid a<x\leq b\}$ \end_inset . @@ -415,7 +415,7 @@ bola cerrada \end_inset al conjunto -\begin_inset Formula $B[x_{0},r]:=\{x\in K:|x-x_{0}|\leq r\}$ +\begin_inset Formula $B[x_{0},r]:=\{x\in K\mid |x-x_{0}|\leq r\}$ \end_inset , y @@ -423,7 +423,7 @@ bola cerrada bola abierta \series default a -\begin_inset Formula $B(x_{0},r):=\{x\in K:|x-x_{0}|<r\}$ +\begin_inset Formula $B(x_{0},r):=\{x\in K\mid |x-x_{0}|<r\}$ \end_inset . @@ -504,7 +504,7 @@ Demostración: \begin_layout Standard Toda sucesión convergente es acotada, es decir -\begin_inset Formula $\{a_{n}:n\in\mathbb{N}\}$ +\begin_inset Formula $\{a_{n}\mid n\in\mathbb{N}\}$ \end_inset es un conjunto acotado. @@ -1567,11 +1567,11 @@ Demostración: . Entonces uno de los conjuntos -\begin_inset Formula $\{n\in\mathbb{N}:a_{n}\in[c_{0},m_{0}]\}$ +\begin_inset Formula $\{n\in\mathbb{N}\mid a_{n}\in[c_{0},m_{0}]\}$ \end_inset o -\begin_inset Formula $\{n\in\mathbb{N}:a_{n}\in[m_{0},d_{0}]\}$ +\begin_inset Formula $\{n\in\mathbb{N}\mid a_{n}\in[m_{0},d_{0}]\}$ \end_inset es infinito. @@ -2744,7 +2744,7 @@ Demostración: \end_inset y sea -\begin_inset Formula $A:=\{z\in\mathbb{R}:a^{z}\leq x\}$ +\begin_inset Formula $A:=\{z\in\mathbb{R}\mid a^{z}\leq x\}$ \end_inset , que sabemos acotado superiormente. diff --git a/fuvr1/n3.lyx b/fuvr1/n3.lyx index 95517f3..e8b4534 100644 --- a/fuvr1/n3.lyx +++ b/fuvr1/n3.lyx @@ -1431,7 +1431,7 @@ Existen \end_inset Si -\begin_inset Formula $\alpha:=\sup\{f(x):x\in[a,b]\}$ +\begin_inset Formula $\alpha:=\sup\{f(x)\mid x\in[a,b]\}$ \end_inset , existe diff --git a/fuvr2/n1.lyx b/fuvr2/n1.lyx index a8766da..b840f8f 100644 --- a/fuvr2/n1.lyx +++ b/fuvr2/n1.lyx @@ -1141,7 +1141,7 @@ Sea . Sea -\begin_inset Formula $A:=\{z\in(x,y]:f(x)\leq f(z)\}$ +\begin_inset Formula $A:=\{z\in(x,y]\mid f(x)\leq f(z)\}$ \end_inset , como diff --git a/fuvr2/n2.lyx b/fuvr2/n2.lyx index 9d5d103..b0dcf59 100644 --- a/fuvr2/n2.lyx +++ b/fuvr2/n2.lyx @@ -263,7 +263,7 @@ de Darboux ), respectivamente, a \begin_inset Formula \begin{eqnarray*} -\underline{\int_{a}^{b}}f:=\sup\{s(f,\pi)\}_{\pi\in{\cal P}([a,b])} & \text{ y } & \overline{\int_{a}^{b}}f:=\inf\{S(f,\pi)\}_{\pi\in{\cal P}([a,b])} +\underline{\int_{a}^{b}}f:=\sup\{s(f,\pi)\}_{\pi\in{\cal P}([a,b])} & \text{ y } & \overline{\int_{a}^{b}}f\mid =\inf\{S(f,\pi)\}_{\pi\in{\cal P}([a,b])} \end{eqnarray*} \end_inset diff --git a/fuvr2/n3.lyx b/fuvr2/n3.lyx index 5d9c1ab..9db6cf2 100644 --- a/fuvr2/n3.lyx +++ b/fuvr2/n3.lyx @@ -489,7 +489,7 @@ Vemos que \begin_layout Standard El conjunto -\begin_inset Formula $\{x>0:\cos x=0\}$ +\begin_inset Formula $\{x>0\mid \cos x=0\}$ \end_inset es no vacío y de hecho tiene un primer elemento, que se denota @@ -91,7 +91,7 @@ Teorema de Cauchy-Goursat: \end_inset y -\begin_inset Formula $\Delta(a,b,c):=\{\mu a+\lambda b+\gamma c:\mu+\lambda+\gamma=1;\mu,\lambda,\gamma\geq0\}\subseteq\Omega$ +\begin_inset Formula $\Delta(a,b,c):=\{\mu a+\lambda b+\gamma c\mid \mu+\lambda+\gamma=1;\mu,\lambda,\gamma\geq0\}\subseteq\Omega$ \end_inset , entonces @@ -1583,7 +1583,7 @@ Sean \end_inset y -\begin_inset Formula $H:=\{z\in\mathbb{C}:d(z,K)\leq\rho\}$ +\begin_inset Formula $H:=\{z\in\mathbb{C}\mid d(z,K)\leq\rho\}$ \end_inset , con lo que @@ -87,7 +87,7 @@ Sean \end_inset y -\begin_inset Formula $Z(f):=\{z\in\Omega:f(z)=0\}$ +\begin_inset Formula $Z(f):=\{z\in\Omega\mid f(z)=0\}$ \end_inset , @@ -210,7 +210,7 @@ status open \end_inset Sea -\begin_inset Formula $A:=\{z\in\Omega:\forall k\in\mathbb{N},f^{(k)}(z)=0\}\neq\emptyset$ +\begin_inset Formula $A:=\{z\in\Omega\mid \forall k\in\mathbb{N},f^{(k)}(z)=0\}\neq\emptyset$ \end_inset , pues @@ -221,7 +221,7 @@ status open Como \begin_inset Formula \[ -A=\bigcap_{k=0}^{\infty}\{z\in\Omega:f^{(k)}(z)=0\}, +A=\bigcap_{k=0}^{\infty}\{z\in\Omega\mid f^{(k)}(z)=0\}, \] \end_inset @@ -337,7 +337,7 @@ principio de identidad para funciones holomorfas \end_inset no es idénticamente nula, entonces todo punto de -\begin_inset Formula $Z(f):=\{z\in\Omega:f(z)=0\}$ +\begin_inset Formula $Z(f):=\{z\in\Omega\mid f(z)=0\}$ \end_inset es aislado y @@ -377,7 +377,7 @@ cero orden \series default -\begin_inset Formula $\min\{n\in\mathbb{N}:f^{(n)}(a)\neq0\}$ +\begin_inset Formula $\min\{n\in\mathbb{N}\mid f^{(n)}(a)\neq0\}$ \end_inset . @@ -968,7 +968,7 @@ f'(z) & \text{si }z=w. \end_inset es continua en -\begin_inset Formula $\{(z,w)\in\Omega\times\Omega:z\neq w\}$ +\begin_inset Formula $\{(z,w)\in\Omega\times\Omega\mid z\neq w\}$ \end_inset . @@ -1083,7 +1083,7 @@ Ahora bien, fijado \begin_layout Standard Sea -\begin_inset Formula $\Omega_{0}:=\{z\in\mathbb{C}\setminus\Gamma^{*}:\text{Ind}_{\Gamma}(z)=0\}$ +\begin_inset Formula $\Omega_{0}:=\{z\in\mathbb{C}\setminus\Gamma^{*}\mid \text{Ind}_{\Gamma}(z)=0\}$ \end_inset , que es abierto por ser unión de componentes conexas de @@ -1834,7 +1834,7 @@ Sean \end_inset , entonces -\begin_inset Formula $\{a\in S:\text{Ind}_{\Gamma}(a)\neq0\}$ +\begin_inset Formula $\{a\in S\mid \text{Ind}_{\Gamma}(a)\neq0\}$ \end_inset es finito y @@ -1854,7 +1854,7 @@ Sean Demostración: \series default Sea -\begin_inset Formula $\Omega_{0}=\{z\in\mathbb{C}\setminus\Gamma^{*}:\text{Ind}_{\Gamma}(z)=0\}$ +\begin_inset Formula $\Omega_{0}=\{z\in\mathbb{C}\setminus\Gamma^{*}\mid \text{Ind}_{\Gamma}(z)=0\}$ \end_inset , que es abierto por ser unión de componentes conexas de @@ -1886,7 +1886,7 @@ status open . Sea -\begin_inset Formula $K:=\mathbb{C}\setminus\Omega_{0}=\Gamma^{*}\cup\{z\in\mathbb{C}\setminus\Gamma^{*}:\text{Ind}_{\Gamma}(z)\neq0\}$ +\begin_inset Formula $K:=\mathbb{C}\setminus\Omega_{0}=\Gamma^{*}\cup\{z\in\mathbb{C}\setminus\Gamma^{*}\mid \text{Ind}_{\Gamma}(z)\neq0\}$ \end_inset , que es cerrado por ser complementario de un abierto y acotado porque no @@ -1896,7 +1896,7 @@ status open , luego es compacto. Si -\begin_inset Formula $S\cap K=\{a\in S:\text{Ind}_{\Gamma}(z)\neq0\}$ +\begin_inset Formula $S\cap K=\{a\in S\mid \text{Ind}_{\Gamma}(z)\neq0\}$ \end_inset no fuera finito, tendría un punto de acumulación que, por compacidad, debería diff --git a/fvv1/n1.lyx b/fvv1/n1.lyx index 41aa455..e3422b2 100644 --- a/fvv1/n1.lyx +++ b/fvv1/n1.lyx @@ -163,7 +163,7 @@ Ejemplos de normas en . Además, -\begin_inset Formula $V:={\cal C}[a,b]:=\{f:[a,b]\rightarrow\mathbb{R}\text{ continua}\}$ +\begin_inset Formula $V:={\cal C}[a,b]:=\{f\mid [a,b]\rightarrow\mathbb{R}\text{ continua}\}$ \end_inset con @@ -706,7 +706,7 @@ teorema , que es continua por ser composición de dos funciones continuas (la identidad es continua por la otra cota y la demostración del teorema anterior), entonces -\begin_inset Formula $S:=\{x\in\mathbb{R}^{n}:\Vert x\Vert_{1}=1\}$ +\begin_inset Formula $S:=\{x\in\mathbb{R}^{n}\mid \Vert x\Vert_{1}=1\}$ \end_inset es cerrado dentro del compacto diff --git a/fvv1/n2.lyx b/fvv1/n2.lyx index 14dd50a..1b761e2 100644 --- a/fvv1/n2.lyx +++ b/fvv1/n2.lyx @@ -897,7 +897,7 @@ to por abiertos de \end_inset y -\begin_inset Formula $\{B_{i}\}_{i=1}^{k}:=\{B(x_{i},\frac{\delta_{x_{i}}}{2})\}_{i=1}^{k}$ +\begin_inset Formula $\{B_{i}\}_{i=1}^{k}\mid =\{B(x_{i},\frac{\delta_{x_{i}}}{2})\}_{i=1}^{k}$ \end_inset un subrecubrimiento finito del que suponemos que no podemos quitar ninguna diff --git a/fvv1/n3.lyx b/fvv1/n3.lyx index 776351a..91f5019 100644 --- a/fvv1/n3.lyx +++ b/fvv1/n3.lyx @@ -840,7 +840,7 @@ suponiendo . Pero -\begin_inset Formula $\frac{x-a}{\Vert x-a\Vert}\in\{y\in\mathbb{R}^{m}:\Vert y\Vert=1\}=:K$ +\begin_inset Formula $\frac{x-a}{\Vert x-a\Vert}\in\{y\in\mathbb{R}^{m}\mid \Vert y\Vert=1\}=:K$ \end_inset , que es compacto por ser cerrado y acotado, y diff --git a/fvv1/n4.lyx b/fvv1/n4.lyx index 07fa28a..f95baae 100644 --- a/fvv1/n4.lyx +++ b/fvv1/n4.lyx @@ -104,7 +104,7 @@ implícita un abierto. La región -\begin_inset Formula $A=\{(x_{1},\dots,x_{n})\in{\cal U}:f(x_{1},\dots,x_{n})=0\}$ +\begin_inset Formula $A=\{(x_{1},\dots,x_{n})\in{\cal U}\mid f(x_{1},\dots,x_{n})=0\}$ \end_inset está @@ -459,7 +459,7 @@ Si \end_inset está dado en forma implícita como -\begin_inset Formula $\{x\in{\cal U}:g(x)=0\}$ +\begin_inset Formula $\{x\in{\cal U}\mid g(x)=0\}$ \end_inset , donde diff --git a/fvv2/n1.lyx b/fvv2/n1.lyx index 7f67d1f..e7eda47 100644 --- a/fvv2/n1.lyx +++ b/fvv2/n1.lyx @@ -208,7 +208,7 @@ gráfica a \begin_inset Formula \[ -\text{graf}(f):=\{(x_{1},\dots,x_{n},y)\in\mathbb{R}^{n+1}:(x_{1},\dots,x_{n})\in[a_{1},b_{1}]\times\dots\times[a_{n},b_{n}]\land y=f(x_{1},\dots,x_{n})\} +\text{graf}(f):=\{(x_{1},\dots,x_{n},y)\in\mathbb{R}^{n+1}\mid (x_{1},\dots,x_{n})\in[a_{1},b_{1}]\times\dots\times[a_{n},b_{n}]\land y=f(x_{1},\dots,x_{n})\} \] \end_inset @@ -221,7 +221,7 @@ subgrafo \begin_inset Formula \begin{multline*} \text{subgraf}(f):=\\ -\{(x_{1},\dots,x_{n},y)\in\mathbb{R}^{n+1}:(x_{1},\dots,x_{n})\in[a_{1},b_{1}]\times\dots\times[a_{n},b_{n}]\land0\leq y\leq f(x_{1},\dots,x_{n})\} +\{(x_{1},\dots,x_{n},y)\in\mathbb{R}^{n+1}\mid (x_{1},\dots,x_{n})\in[a_{1},b_{1}]\times\dots\times[a_{n},b_{n}]\land0\leq y\leq f(x_{1},\dots,x_{n})\} \end{multline*} \end_inset @@ -1452,7 +1452,7 @@ Sea \end_inset , -\begin_inset Formula $B:=\{x\in A:\text{osc}(f,x)\geq\varepsilon\}$ +\begin_inset Formula $B:=\{x\in A\mid \text{osc}(f,x)\geq\varepsilon\}$ \end_inset es cerrado. @@ -1539,7 +1539,7 @@ teorema de Lebesgue de caracterización de las funciones integrables \end_inset si y sólo si -\begin_inset Formula $B:=\{x\in R:f\text{ no es continua en }x\}$ +\begin_inset Formula $B:=\{x\in R\mid f\text{ no es continua en }x\}$ \end_inset tiene medida nula. @@ -1559,7 +1559,7 @@ status open \end_inset Sea -\begin_inset Formula $B_{k}:=\{x\in R:o(f,x)\geq\frac{1}{k}\}$ +\begin_inset Formula $B_{k}:=\{x\in R\mid o(f,x)\geq\frac{1}{k}\}$ \end_inset , basta probar que cada diff --git a/fvv2/n2.lyx b/fvv2/n2.lyx index 56b1b12..bd555e8 100644 --- a/fvv2/n2.lyx +++ b/fvv2/n2.lyx @@ -654,7 +654,7 @@ espacio de medida \end_inset -finita si y sólo si -\begin_inset Formula $\{x\in\Omega:f(x)>0\}$ +\begin_inset Formula $\{x\in\Omega\mid f(x)>0\}$ \end_inset es numerable. @@ -889,7 +889,7 @@ medida exterior como \begin_inset Formula \[ -\lambda_{n}^{*}(B):=\inf\left\{ \sum_{k\in\mathbb{N}}v([a_{k},b_{k})):B\subseteq\dot{\bigcup_{k\in\mathbb{N}}}[a_{k},b_{k})\right\} +\lambda_{n}^{*}(B):=\inf\left\{ \sum_{k\in\mathbb{N}}v([a_{k},b_{k}))\mid B\subseteq\dot{\bigcup_{k\in\mathbb{N}}}[a_{k},b_{k})\right\} \] \end_inset @@ -1146,7 +1146,7 @@ Para \end_inset , y por tanto -\begin_inset Formula $\lambda_{n}^{*}(S)=\inf\{\lambda_{n}^{*}(A):A\supseteq S\text{ abierto}\}$ +\begin_inset Formula $\lambda_{n}^{*}(S)=\inf\{\lambda_{n}^{*}(A)\mid A\supseteq S\text{ abierto}\}$ \end_inset . diff --git a/fvv2/n3.lyx b/fvv2/n3.lyx index a35f67f..11ac40c 100644 --- a/fvv2/n3.lyx +++ b/fvv2/n3.lyx @@ -172,7 +172,7 @@ status open \end_inset Sea -\begin_inset Formula ${\cal A}:=\{E\in\Sigma':f^{-1}(E)\in\Sigma\}$ +\begin_inset Formula ${\cal A}:=\{E\in\Sigma'\mid f^{-1}(E)\in\Sigma\}$ \end_inset , vemos que @@ -627,7 +627,7 @@ Una función \end_inset y la notación -\begin_inset Formula $\{f\bullet a\}:=\{\omega\in\Omega:f(\omega)\bullet a\}$ +\begin_inset Formula $\{f\bullet a\}\mid =\{\omega\in\Omega\mid f(\omega)\bullet a\}$ \end_inset . @@ -1554,7 +1554,7 @@ Sea \end_inset y -\begin_inset Formula ${\cal S}(\Omega)^{+}:=\{h\in{\cal S}(\Omega):h\geq0\}$ +\begin_inset Formula ${\cal S}(\Omega)^{+}:=\{h\in{\cal S}(\Omega)\mid h\geq0\}$ \end_inset , llamamos @@ -1719,7 +1719,7 @@ Para medible, se define \begin_inset Formula \[ -\int f\,d\mu:=\sup\left\{ \int s\,d\mu:s\in{\cal S}(\Omega)\land0\leq s\leq f\right\} +\int f\,d\mu:=\sup\left\{ \int s\,d\mu\mid s\in{\cal S}(\Omega)\land0\leq s\leq f\right\} \] \end_inset @@ -2236,7 +2236,7 @@ Una función medible \end_inset , si y sólo si -\begin_inset Formula $f^{+}=\max\{f,0\},f^{-}=\min\{f,0\}:\Omega\rightarrow[-\infty,+\infty]$ +\begin_inset Formula $f^{+}=\max\{f,0\},f^{-}=\min\{f,0\}\mid \Omega\rightarrow[-\infty,+\infty]$ \end_inset son integrables, y definimos @@ -3315,11 +3315,11 @@ Demostración: \end_inset es continua, y como -\begin_inset Formula $\delta:=\min\{d(x,K):x\notin A\}>0$ +\begin_inset Formula $\delta:=\min\{d(x,K)\mid x\notin A\}>0$ \end_inset , -\begin_inset Formula $A_{0}:=\{x:d(x,K)<\frac{\delta}{2}\}$ +\begin_inset Formula $A_{0}:=\{x\mid d(x,K)<\frac{\delta}{2}\}$ \end_inset es un abierto acotado con @@ -3328,7 +3328,7 @@ Demostración: . Tomando -\begin_inset Formula $F_{0}:=\mathbb{R}^{n}\backslash A_{0}=\{x:d(x,K)\geq\frac{\delta}{2}\}$ +\begin_inset Formula $F_{0}:=\mathbb{R}^{n}\backslash A_{0}=\{x\mid d(x,K)\geq\frac{\delta}{2}\}$ \end_inset , podemos definir diff --git a/fvv2/n4.lyx b/fvv2/n4.lyx index 6628b45..2db00c5 100644 --- a/fvv2/n4.lyx +++ b/fvv2/n4.lyx @@ -360,7 +360,7 @@ teorema \end_inset es acotada y -\begin_inset Formula $D(f):=\{x\in[a,b]:f\text{ es discontinua en }x\}$ +\begin_inset Formula $D(f):=\{x\in[a,b]\mid f\text{ es discontinua en }x\}$ \end_inset , entonces diff --git a/fvv3/n.pdf b/fvv3/n.pdf Binary files differdeleted file mode 100644 index 397a272..0000000 --- a/fvv3/n.pdf +++ /dev/null @@ -2271,7 +2271,7 @@ Dado un espacio topológico \end_inset , -\begin_inset Formula $\{f\in\mathbb{R}^{X}:f\text{ continua}\}$ +\begin_inset Formula $\{f\in\mathbb{R}^{X}\mid f\text{ continua}\}$ \end_inset es un subanillo de @@ -2287,7 +2287,7 @@ Dado un espacio vectorial \end_inset , -\begin_inset Formula $\{f\in V^{V}:f\text{ lineal}\}$ +\begin_inset Formula $\{f\in V^{V}\mid f\text{ lineal}\}$ \end_inset es un subanillo de @@ -2307,7 +2307,7 @@ Dado un anillo \end_inset , -\begin_inset Formula $\{f\in A^{X}:f\text{ constante}\}$ +\begin_inset Formula $\{f\in A^{X}\mid f\text{ constante}\}$ \end_inset es un subanillo de @@ -3944,7 +3944,7 @@ Demostración: \end_inset , pues -\begin_inset Formula $\pi^{-1}(J/I)=\{x:\pi(x)=[x]\in J/I\}$ +\begin_inset Formula $\pi^{-1}(J/I)=\{x\mid\pi(x)=[x]\in J/I\}$ \end_inset , pero si @@ -4005,7 +4005,7 @@ Ahora vemos que, dado un ideal \end_inset , -\begin_inset Formula $\pi^{-1}(X)=\{x:[x]\in X\}\ni0$ +\begin_inset Formula $\pi^{-1}(X)=\{x\mid[x]\in X\}\ni0$ \end_inset ; para @@ -4058,7 +4058,7 @@ Ahora vemos que, dado un ideal . Además, -\begin_inset Formula $\pi^{-1}(X)/I=\{x:[x]\in X\}/I=\{[x]:[x]\in X\}=X$ +\begin_inset Formula $\pi^{-1}(X)/I=\{x\mid[x]\in X\}/I=\{[x]\mid[x]\in X\}=X$ \end_inset . @@ -4185,8 +4185,8 @@ La intersección de una familia de ideales de , definimos los ideales \begin_inset Formula \begin{eqnarray*} -\sum_{x\in X}I_{x} & := & \left\{ \sum_{x\in S}a_{x}:S\subseteq X\text{ finito},a_{x}\in I_{x}\right\} ,\\ -\prod_{x\in X}I_{x} & := & \left\{ \sum_{k=1}^{n}\prod_{x\in S}a_{kx}:n\in\mathbb{N},S\subseteq X\text{ finito},a_{kx}\in I_{x}\right\} . +\sum_{x\in X}I_{x} & := & \left\{ \sum_{x\in S}a_{x}\;\middle|\;S\subseteq X\text{ finito},a_{x}\in I_{x}\right\} ,\\ +\prod_{x\in X}I_{x} & := & \left\{ \sum_{k=1}^{n}\prod_{x\in S}a_{kx}\;\middle|\;n\in\mathbb{N},S\subseteq X\text{ finito},a_{kx}\in I_{x}\right\} . \end{eqnarray*} \end_inset @@ -4257,7 +4257,7 @@ En efecto, \end_inset , -\begin_inset Formula $(n)\cap(m)=\{k\in\mathbb{Z}:n,m|k\}=\{k:\text{mcm}(n,m)|k\}=(\text{mcm}(n,m))$ +\begin_inset Formula $(n)\cap(m)=\{k\in\mathbb{Z}\mid n,m|k\}=\{k\mid\text{mcm}(n,m)|k\}=(\text{mcm}(n,m))$ \end_inset y @@ -2668,7 +2668,7 @@ Si . Veamos que -\begin_inset Formula $\mathbb{Z}[\sqrt{m}]^{*}=\{x:|N(x)|=1\}$ +\begin_inset Formula $\mathbb{Z}[\sqrt{m}]^{*}=\{x\mid |N(x)|=1\}$ \end_inset . @@ -3376,7 +3376,7 @@ euclídea \end_layout \begin_layout Enumerate -\begin_inset Formula $\forall a\in D,b\in D\setminus\{0\},\exists q,r\in D:(a=bq+r\land(r=0\lor\delta(r)<\delta(b)))$ +\begin_inset Formula $\forall a\in D,b\in D\setminus\{0\},\exists q,r\in D\mid (a=bq+r\land(r=0\lor\delta(r)<\delta(b)))$ \end_inset . @@ -169,11 +169,11 @@ polinomios constantes \end_inset , -\begin_inset Formula $\{a_{0}+a_{1}X+\dots+a_{n}X^{n}\in A[X]:a_{0}\in I\}$ +\begin_inset Formula $\{a_{0}+a_{1}X+\dots+a_{n}X^{n}\in A[X]\mid a_{0}\in I\}$ \end_inset e -\begin_inset Formula $I[X]:=\{a_{0}+a_{1}X+\dots+a_{n}X^{n}\in A[X]:a_{0},\dots,a_{n}\in I\}$ +\begin_inset Formula $I[X]:=\{a_{0}+a_{1}X+\dots+a_{n}X^{n}\in A[X]\mid a_{0},\dots,a_{n}\in I\}$ \end_inset son ideales de @@ -197,7 +197,7 @@ grado \end_inset a -\begin_inset Formula $\text{gr}(p):=\max\{k\in\mathbb{N}:p_{k}\neq0\}$ +\begin_inset Formula $\text{gr}(p):=\max\{k\in\mathbb{N}\mid p_{k}\neq0\}$ \end_inset , @@ -1570,7 +1570,7 @@ Para \end_inset , existe -\begin_inset Formula $m:=\max\{k\in\mathbb{N}:(X-a)^{k}\mid f\}$ +\begin_inset Formula $m:=\max\{k\in\mathbb{N}\mid (X-a)^{k}\mid f\}$ \end_inset @@ -3473,7 +3473,7 @@ Definimos \end_inset , -\begin_inset Formula $c(p):=\{x:x=\text{mcd}_{k\geq0}p_{k}\}$ +\begin_inset Formula $c(p):=\{x\mid x=\text{mcd}_{k\geq0}p_{k}\}$ \end_inset , y para @@ -4641,7 +4641,7 @@ Demostración: \end_inset , luego existe -\begin_inset Formula $i:=\min\{j:p\nmid b_{j}\}$ +\begin_inset Formula $i:=\min\{j\mid p\nmid b_{j}\}$ \end_inset y entonces @@ -745,7 +745,7 @@ Si \end_inset es una familia de grupos, -\begin_inset Formula $\bigoplus_{i\in I}G_{i}:=\{(g_{i})_{i\in I}\in\prod_{i\in I}G_{i}:\{i\in I:g_{i}\ne1\}\text{ es finito}\}$ +\begin_inset Formula $\bigoplus_{i\in I}G_{i}:=\{(g_{i})_{i\in I}\in\prod_{i\in I}G_{i}\mid \{i\in I\mid g_{i}\ne1\}\text{ es finito}\}$ \end_inset es un subgrupo de @@ -773,7 +773,7 @@ centralizador \end_inset es el subgrupo -\begin_inset Formula $C_{G}(x):=\{g\in G:gx=xg\}$ +\begin_inset Formula $C_{G}(x):=\{g\in G\mid gx=xg\}$ \end_inset , y el @@ -785,7 +785,7 @@ centro \end_inset es el subgrupo abeliano -\begin_inset Formula $Z(G):=\{g\in G:\forall x\in G,gx=xg\}=\bigcap_{x\in X}C_{G}(x)$ +\begin_inset Formula $Z(G):=\{g\in G\mid \forall x\in G,gx=xg\}=\bigcap_{x\in X}C_{G}(x)$ \end_inset . @@ -2973,7 +2973,7 @@ estabilizador \end_inset a -\begin_inset Formula $\text{Estab}_{G}(x):=\{g\in G:g\cdot x=x\}$ +\begin_inset Formula $\text{Estab}_{G}(x):=\{g\in G\mid g\cdot x=x\}$ \end_inset . @@ -3014,7 +3014,7 @@ estabilizador \end_inset a -\begin_inset Formula $\text{Estab}_{G}(x):=\{g\in G:x\cdot g=x\}$ +\begin_inset Formula $\text{Estab}_{G}(x):=\{g\in G\mid x\cdot g=x\}$ \end_inset . @@ -3050,7 +3050,7 @@ acción por translación a la izquierda y \begin_inset Formula \[ -\text{Estab}_{G}(xH)=\{g\in G:gxH=xH\}=\{g\in G:x^{-1}gx\in H\}=xHx^{-1}=H^{x^{-1}}. +\text{Estab}_{G}(xH)=\{g\in G\mid gxH=xH\}=\{g\in G\mid x^{-1}gx\in H\}=xHx^{-1}=H^{x^{-1}}. \] \end_inset @@ -3170,7 +3170,7 @@ normalizador \end_inset es -\begin_inset Formula $N_{G}(H):=\text{Estab}_{G}(H)=\{g\in G:H^{g}=H\}$ +\begin_inset Formula $N_{G}(H):=\text{Estab}_{G}(H)=\{g\in G\mid H^{g}=H\}$ \end_inset , el mayor subgrupo de @@ -3393,12 +3393,12 @@ status open \begin_layout Plain Layout Si la acción es por la izquierda, -\begin_inset Formula $\text{Estab}_{G}(x)^{g^{-1}}=\{ghg^{-1}:h\cdot x=x\}=\{p\in G:g^{-1}pg\cdot x=x\}=\{p\in G:p\cdot(g\cdot x)=g\cdot x\}=\text{Estab}_{G}(g\cdot x)$ +\begin_inset Formula $\text{Estab}_{G}(x)^{g^{-1}}=\{ghg^{-1}\mid h\cdot x=x\}=\{p\in G\mid g^{-1}pg\cdot x=x\}=\{p\in G\mid p\cdot(g\cdot x)=g\cdot x\}=\text{Estab}_{G}(g\cdot x)$ \end_inset . Si es por la derecha, -\begin_inset Formula $\text{Estab}_{G}(x)^{g}=\{g^{-1}hg:x\cdot h=x\}=\{p\in G:x\cdot gpg^{-1}=x\}=\{p\in G:(x\cdot g)\cdot p=x\cdot g\}$ +\begin_inset Formula $\text{Estab}_{G}(x)^{g}=\{g^{-1}hg\mid x\cdot h=x\}=\{p\in G\mid x\cdot gpg^{-1}=x\}=\{p\in G\mid (x\cdot g)\cdot p=x\cdot g\}$ \end_inset . @@ -3606,7 +3606,7 @@ status open Demostración: \series default Sea -\begin_inset Formula $X:=\{(g_{1},\dots,g_{p})\in G^{p}:g_{1}\cdots g_{p}=1\}$ +\begin_inset Formula $X:=\{(g_{1},\dots,g_{p})\in G^{p}\mid g_{1}\cdots g_{p}=1\}$ \end_inset , @@ -98,7 +98,7 @@ suma \end_inset a -\begin_inset Formula $\sum_{i\in I}B_{i}:=\{\sum_{i\in I}b_{i}:b_{i}\in B_{i},\{i\in I:b_{i}\neq0\}\text{ es finito}\}$ +\begin_inset Formula $\sum_{i\in I}B_{i}:=\{\sum_{i\in I}b_{i}\mid b_{i}\in B_{i},\{i\in I\mid b_{i}\neq0\}\text{ es finito}\}$ \end_inset . @@ -453,7 +453,7 @@ Para \end_inset con -\begin_inset Formula $\{i\in I:b_{i}\neq0\}$ +\begin_inset Formula $\{i\in I\mid b_{i}\neq0\}$ \end_inset finito. @@ -704,7 +704,7 @@ subgrupo de es \begin_inset Formula \[ -t_{p}(A):=\{a\in A:\exists n\in\mathbb{N}:p^{n}a=0\}=\{a\in A:|a|\text{ es potencia de }p\}. +t_{p}(A):=\{a\in A\mid \exists n\in\mathbb{N}\mid p^{n}a=0\}=\{a\in A\mid |a|\text{ es potencia de }p\}. \] \end_inset @@ -168,7 +168,7 @@ mueve \series default en caso contrario. Llamamos -\begin_inset Formula $M(\sigma):=\{i\in\mathbb{N}_{n}:\sigma(i)\neq i\}$ +\begin_inset Formula $M(\sigma):=\{i\in\mathbb{N}_{n}\mid \sigma(i)\neq i\}$ \end_inset , y es claro que @@ -718,7 +718,7 @@ punto fijo , y definimos \begin_inset Formula \[ -\text{Fix}(f):=\{Q\in{\cal E}:f(Q)=Q\} +\text{Fix}(f):=\{Q\in{\cal E}\mid f(Q)=Q\} \] \end_inset @@ -755,7 +755,7 @@ vectores invariantes o asociado al autovalor 1, \begin_inset Formula \[ -\text{Inv}(\phi):=\text{Nuc}(\phi-id_{V})=\{\vec{v}\in V:\phi(\vec{v})=\vec{v}\} +\text{Inv}(\phi):=\text{Nuc}(\phi-id_{V})=\{\vec{v}\in V\mid \phi(\vec{v})=\vec{v}\} \] \end_inset @@ -771,7 +771,7 @@ opuestos , \begin_inset Formula \[ -\text{Opp}(\phi):=\text{Nuc}(\phi+id_{V})=\{\vec{v}\in V:\phi(\vec{v})=-\vec{v}\} +\text{Opp}(\phi):=\text{Nuc}(\phi+id_{V})=\{\vec{v}\in V\mid \phi(\vec{v})=-\vec{v}\} \] \end_inset @@ -1647,11 +1647,11 @@ distancia orientada \end_inset en dos semiplanos -\begin_inset Formula $H^{+}:=\{p:\text{dist}(p,\ell)\geq0\}$ +\begin_inset Formula $H^{+}:=\{p\mid \text{dist}(p,\ell)\geq0\}$ \end_inset y -\begin_inset Formula $H^{-}:=\{p:\text{dist}(p,\ell)\leq0\}$ +\begin_inset Formula $H^{-}:=\{p\mid \text{dist}(p,\ell)\leq0\}$ \end_inset , de modo que @@ -2984,7 +2984,7 @@ Sean \end_inset y -\begin_inset Formula $J:=\{t\in I:\alpha(t)\in V\}$ +\begin_inset Formula $J:=\{t\in I\mid \alpha(t)\in V\}$ \end_inset , entonces @@ -4304,7 +4304,7 @@ Sean \end_inset y -\begin_inset Formula $A:=\{p\in S:f(p)=a\}\neq\emptyset$ +\begin_inset Formula $A:=\{p\in S\mid f(p)=a\}\neq\emptyset$ \end_inset , pues @@ -4698,7 +4698,7 @@ Dados \end_inset , el cilindro -\begin_inset Formula $C:=\{(x,y,z):x^{2}+y^{2}=r^{2}\}$ +\begin_inset Formula $C:=\{(x,y,z)\mid x^{2}+y^{2}=r^{2}\}$ \end_inset y la parametrización @@ -472,7 +472,7 @@ Sea \end_inset es la superficie de nivel -\begin_inset Formula $\{p:f(p)=r^{2}\}$ +\begin_inset Formula $\{p\mid f(p)=r^{2}\}$ \end_inset , luego admite la orientación @@ -1018,7 +1018,7 @@ Los cilindros se obtienen por un movimiento de \end_inset , -\begin_inset Formula $N(S_{r})=\{\frac{1}{r}(x,y,0):x^{2}+y^{2}=r^{2}\}=\{(x,y,0):x^{2}+y^{2}=1\}$ +\begin_inset Formula $N(S_{r})=\{\frac{1}{r}(x,y,0)\mid x^{2}+y^{2}=r^{2}\}=\{(x,y,0)\mid x^{2}+y^{2}=1\}$ \end_inset . @@ -2275,7 +2275,7 @@ El cilindro \begin_deeper \begin_layout Standard Sean -\begin_inset Formula $C:=\{x^{2}+y^{2}=r^{2}\}=\{X(u,v):=(r\cos u,r\sin u,v)\}_{u,v\in\mathbb{R}}$ +\begin_inset Formula $C:=\{x^{2}+y^{2}=r^{2}\}=\{X(u,v)\mid =(r\cos u,r\sin u,v)\}_{u,v\in\mathbb{R}}$ \end_inset , @@ -2635,7 +2635,7 @@ status open \begin_layout Plain Layout La superficie es el grafo -\begin_inset Formula $S:=\{X(u,v):=(u,v,(u^{2}+v^{2})^{2}\}_{u,v\in\mathbb{R}}$ +\begin_inset Formula $S:=\{X(u,v)\mid =(u,v,(u^{2}+v^{2})^{2}\}_{u,v\in\mathbb{R}}$ \end_inset , de modo que @@ -569,7 +569,7 @@ intervalo maximal de existencia Demostración: \series default Sea -\begin_inset Formula ${\cal J}_{p,v}:=\{(I,\alpha):\alpha:I\to S\text{ geodésica},0\in I,\alpha(0)=p,\alpha'(0)=v\}$ +\begin_inset Formula ${\cal J}_{p,v}:=\{(I,\alpha)\mid \alpha\mid I\to S\text{ geodésica},0\in I,\alpha(0)=p,\alpha'(0)=v\}$ \end_inset . @@ -669,7 +669,7 @@ Sean ahora es abierto y, por el teorema del peine, también conexo, luego es un intervalo. Sea -\begin_inset Formula $A:=\{t\in I_{1}\cap I_{2}:\alpha_{1}(t)=\alpha_{2}(t),\alpha'_{1}(t)=\alpha'_{2}(t)\}$ +\begin_inset Formula $A:=\{t\in I_{1}\cap I_{2}\mid \alpha_{1}(t)=\alpha_{2}(t),\alpha'_{1}(t)=\alpha'_{2}(t)\}$ \end_inset , y queremos ver que @@ -1401,7 +1401,7 @@ geodésicamente completa \begin_layout Enumerate Dado el plano -\begin_inset Formula $S=\{p\in\mathbb{R}^{3}:\langle p,a\rangle=c\}$ +\begin_inset Formula $S=\{p\in\mathbb{R}^{3}\mid \langle p,a\rangle=c\}$ \end_inset , la geodésica maximal de @@ -1579,7 +1579,7 @@ Sean \end_inset , -\begin_inset Formula $S:=\{(x,y,z)\in\mathbb{R}^{3}:x^{2}+y^{2}=r^{2}\}$ +\begin_inset Formula $S:=\{(x,y,z)\in\mathbb{R}^{3}\mid x^{2}+y^{2}=r^{2}\}$ \end_inset un cilindro, @@ -110,7 +110,7 @@ aplicación exponencial \end_inset donde -\begin_inset Formula ${\cal D}_{p}:=\{v\in T_{p}S:1\in I_{v}\}$ +\begin_inset Formula ${\cal D}_{p}:=\{v\in T_{p}S\mid 1\in I_{v}\}$ \end_inset . @@ -909,7 +909,7 @@ Sean \end_inset tal que -\begin_inset Formula ${\cal D}(0,r):=\{v\in T_{p}S:\Vert v\Vert<r\}\subseteq{\cal D}_{p}$ +\begin_inset Formula ${\cal D}(0,r):=\{v\in T_{p}S\mid \Vert v\Vert<r\}\subseteq{\cal D}_{p}$ \end_inset , llamamos @@ -933,7 +933,7 @@ disco geodésico \end_inset cumple que -\begin_inset Formula ${\cal S}(0,r):=\{v\in T_{p}S:\Vert v\Vert=r\}\subseteq{\cal D}_{p}$ +\begin_inset Formula ${\cal S}(0,r):=\{v\in T_{p}S\mid \Vert v\Vert=r\}\subseteq{\cal D}_{p}$ \end_inset , llamamos @@ -1099,7 +1099,7 @@ Sean \end_inset , luego -\begin_inset Formula $t_{0}=\max\{t\in[a,b]:\alpha(t)=p_{0}\}<b$ +\begin_inset Formula $t_{0}=\max\{t\in[a,b]\mid \alpha(t)=p_{0}\}<b$ \end_inset (pues @@ -1422,7 +1422,7 @@ Finalmente, sea es \begin_inset Formula \[ -A:=\{t\in(a,b):\Vert\tilde{\alpha}(t)\Vert=r^{*}\}=\{t\in[a,b]:\alpha(t)\in S(p_{0},r^{*})\}\neq\emptyset. +A:=\{t\in(a,b)\mid \Vert\tilde{\alpha}(t)\Vert=r^{*}\}=\{t\in[a,b]\mid \alpha(t)\in S(p_{0},r^{*})\}\neq\emptyset. \] \end_inset @@ -258,7 +258,7 @@ Demostración: \begin_layout Standard Primero vemos que -\begin_inset Formula $A:=\{q\in S:\Omega(p,q)\neq\emptyset\}=S$ +\begin_inset Formula $A:=\{q\in S\mid \Omega(p,q)\neq\emptyset\}=S$ \end_inset viendo que es abierto, cerrado y no vacío. @@ -750,7 +750,7 @@ Queremos ver que \end_inset , existe -\begin_inset Formula $t^{*}:=\inf\{t\in[a,b]:\alpha(t)\notin D(p,r^{*})\}$ +\begin_inset Formula $t^{*}:=\inf\{t\in[a,b]\mid \alpha(t)\notin D(p,r^{*})\}$ \end_inset , pero @@ -229,7 +229,7 @@ Demostración: \end_inset y -\begin_inset Formula $A:=\{t\in[0,1]:\tilde{\alpha}(t)=tw\}$ +\begin_inset Formula $A:=\{t\in[0,1]\mid \tilde{\alpha}(t)=tw\}$ \end_inset , queremos ver que @@ -273,7 +273,7 @@ soporte \end_inset es -\begin_inset Formula $\text{sop}f:=\overline{\{x\in D:f(x)\neq0\}}$ +\begin_inset Formula $\text{sop}f:=\overline{\{x\in D\mid f(x)\neq0\}}$ \end_inset . diff --git a/graf/n1.lyx b/graf/n1.lyx index c547ff0..921c7d8 100644 --- a/graf/n1.lyx +++ b/graf/n1.lyx @@ -119,7 +119,7 @@ grafo no dirigido \end_inset definido de forma similar, pero -\begin_inset Formula $E\subseteq\{S\in{\cal P}(V):|S|\in\{1,2\}\}$ +\begin_inset Formula $E\subseteq\{S\in{\cal P}(V)\mid |S|\in\{1,2\}\}$ \end_inset es un conjunto de @@ -136,7 +136,7 @@ ejes \end_inset a uno dirigido -\begin_inset Formula $(V,\{(i,j)\in V\times V:i,j\in E\})$ +\begin_inset Formula $(V,\{(i,j)\in V\times V\mid i,j\in E\})$ \end_inset . @@ -340,7 +340,7 @@ grafo complementario es \begin_inset Formula \[ -G^{\complement}:=(V,E^{\complement}):=(V,\{S\in{\cal P}(V):|S|=2,S\notin E\}). +G^{\complement}:=(V,E^{\complement}):=(V,\{S\in{\cal P}(V)\mid |S|=2,S\notin E\}). \] \end_inset @@ -408,7 +408,7 @@ inducido \end_inset , donde -\begin_inset Formula $E_{V'}:=\{S\in E:S\subseteq V'\}$ +\begin_inset Formula $E_{V'}:=\{S\in E\mid S\subseteq V'\}$ \end_inset , y @@ -680,7 +680,7 @@ teorema pues \begin_inset Formula \[ -\sum_{v\in V}o(v)=\sum_{v\in V}|\{S\in E:v\in S\}|=\sum_{S\in E}|S|=2|E|. +\sum_{v\in V}o(v)=\sum_{v\in V}|\{S\in E\mid v\in S\}|=\sum_{S\in E}|S|=2|E|. \] \end_inset diff --git a/graf/n2.lyx b/graf/n2.lyx index eb5661f..9d905d7 100644 --- a/graf/n2.lyx +++ b/graf/n2.lyx @@ -2145,7 +2145,7 @@ grafo en línea \end_inset y -\begin_inset Formula $E^{L}:=\{(e,f):e\neq f,e\cap f\neq\emptyset\}$ +\begin_inset Formula $E^{L}:=\{(e,f)\mid e\neq f,e\cap f\neq\emptyset\}$ \end_inset . diff --git a/graf/n4.lyx b/graf/n4.lyx index 6674531..5334582 100644 --- a/graf/n4.lyx +++ b/graf/n4.lyx @@ -1782,11 +1782,11 @@ Si . Sean ahora -\begin_inset Formula $X:=\{i\in\{2,\dots,n-2\}:(u_{i},v)\in E_{k}\}$ +\begin_inset Formula $X:=\{i\in\{2,\dots,n-2\}\mid (u_{i},v)\in E_{k}\}$ \end_inset e -\begin_inset Formula $Y:=\{i\in\{2,\dots,n-2\}:(u_{i+1},u)\in E_{k}\}$ +\begin_inset Formula $Y:=\{i\in\{2,\dots,n-2\}\mid (u_{i+1},u)\in E_{k}\}$ \end_inset , se tiene diff --git a/graf/n6.lyx b/graf/n6.lyx index e296d0b..6bf574a 100644 --- a/graf/n6.lyx +++ b/graf/n6.lyx @@ -222,11 +222,11 @@ teorema \end_inset , -\begin_inset Formula $P:=\{[x,y]\in\mathbb{R}^{p+q}:Ax+Gy\leq b\}$ +\begin_inset Formula $P:=\{[x,y]\in\mathbb{R}^{p+q}\mid Ax+Gy\leq b\}$ \end_inset y -\begin_inset Formula $S:=\{[x,y]\in P:x\in\mathbb{Z}^{p}\}$ +\begin_inset Formula $S:=\{[x,y]\in P\mid x\in\mathbb{Z}^{p}\}$ \end_inset , existen @@ -242,7 +242,7 @@ teorema \end_inset tales que -\begin_inset Formula $\text{ec}S=\{[x,y]:A'x+G'y\leq b'\}$ +\begin_inset Formula $\text{ec}S=\{[x,y]\mid A'x+G'y\leq b'\}$ \end_inset . @@ -253,11 +253,11 @@ teorema Demostración: \series default Sean -\begin_inset Formula $S:=\{(x,y)\in\mathbb{Z}^{2}:y\leq\sqrt{2}x,x\geq0,y\geq0\}$ +\begin_inset Formula $S:=\{(x,y)\in\mathbb{Z}^{2}\mid y\leq\sqrt{2}x,x\geq0,y\geq0\}$ \end_inset y -\begin_inset Formula $C:=\{(x,y):y<\sqrt{2}x,x\geq0,y\geq0\}\cup\{0\}$ +\begin_inset Formula $C:=\{(x,y)\mid y<\sqrt{2}x,x\geq0,y\geq0\}\cup\{0\}$ \end_inset . @@ -406,7 +406,7 @@ Sean \end_inset y -\begin_inset Formula $P:=\{x\in\mathbb{R}^{n}:Ax\leq b\}$ +\begin_inset Formula $P:=\{x\in\mathbb{R}^{n}\mid Ax\leq b\}$ \end_inset , si @@ -706,7 +706,7 @@ Lema de Veinott-Dantzig: \end_inset , -\begin_inset Formula $Q:=\{x\in\mathbb{R}^{n}:Ax=b,x\geq0\}$ +\begin_inset Formula $Q:=\{x\in\mathbb{R}^{n}\mid Ax=b,x\geq0\}$ \end_inset es entero. @@ -913,7 +913,7 @@ Teorema de Hoffman-Kruskal: \end_inset , el poliedro -\begin_inset Formula $\{x\in\mathbb{R}^{n}:Ax\leq b,x\geq0\}$ +\begin_inset Formula $\{x\in\mathbb{R}^{n}\mid Ax\leq b,x\geq0\}$ \end_inset es entero. @@ -978,7 +978,7 @@ Dada una submatriz \end_inset es unimodular, con lo que -\begin_inset Formula $Q:=\{[x,y]\in\mathbb{R}^{n+m}:Ax+Iy=b,[x,y]\geq0\}$ +\begin_inset Formula $Q:=\{[x,y]\in\mathbb{R}^{n+m}\mid Ax+Iy=b,[x,y]\geq0\}$ \end_inset es entero. @@ -1003,7 +1003,7 @@ Dada una submatriz \end_inset es -\begin_inset Formula $P:=\{x\in\mathbb{R}^{n}:b=b,x\geq0,b-Ax\geq0\}=\{Ax\leq b,x\geq0\}$ +\begin_inset Formula $P:=\{x\in\mathbb{R}^{n}\mid b=b,x\geq0,b-Ax\geq0\}=\{Ax\leq b,x\geq0\}$ \end_inset . @@ -1069,11 +1069,11 @@ Sean \end_inset , -\begin_inset Formula $P:=\{x:Ax\leq b,x\geq0\}$ +\begin_inset Formula $P:=\{x\mid Ax\leq b,x\geq0\}$ \end_inset , -\begin_inset Formula $Q:=\{[x,y]:Ax+y=b,[x,y]\geq0\}$ +\begin_inset Formula $Q:=\{[x,y]\mid Ax+y=b,[x,y]\geq0\}$ \end_inset y @@ -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 $R:=(V:=\{0,\dots,n-1\},E:=\{\{i,j\}\}_{i,j\in V,i\neq j},d)$ +\begin_inset Formula $R:=(V:=\{0,\dots,n-1\},E\mid =\{\{i,j\}\}_{i,j\in V,i\neq j},d)$ \end_inset , existen varias formulaciones: @@ -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\}:(i,j)\in E\\ + & & u_{i}-u_{j}+nx_{ij} & \leq n-1 & & \forall i,j\in\{1,\dots,n-1\}\mid (i,j)\in E\\ & & x_{ij} & \in\{0,1\} & & \forall i,j\\ & & u_{i} & \in\mathbb{R}^{>0} & & \forall i \end{alignat*} diff --git a/graf/n7.lyx b/graf/n7.lyx index 04fd675..dc0abb4 100644 --- a/graf/n7.lyx +++ b/graf/n7.lyx @@ -850,7 +850,7 @@ regla de Bland: \end_inset , -\begin_inset Formula $F:=\{x:Ax=b,x\geq0\}$ +\begin_inset Formula $F:=\{x\mid Ax=b,x\geq0\}$ \end_inset y @@ -888,7 +888,7 @@ Si [...] \end_inset es la matriz formada por las columnas añadidas, escribimos -\begin_inset Formula $F^{*}:=\{[x,x^{*}]\in\mathbb{R}^{n+p}:Ax+Tx^{*}=b,[x,x^{*}]\geq0\}$ +\begin_inset Formula $F^{*}:=\{[x,x^{*}]\in\mathbb{R}^{n+p}\mid Ax+Tx^{*}=b,[x,x^{*}]\geq0\}$ \end_inset y vemos que @@ -921,7 +921,7 @@ vector de variables artificiales Método de las dos fases: \series default ] La primera fase consiste en hallar -\begin_inset Formula $\min\{\sum_{i}x_{i}^{*}:Ax+Tx^{*}=b,[x,x^{*}]\geq0\}$ +\begin_inset Formula $\min\{\sum_{i}x_{i}^{*}\mid Ax+Tx^{*}=b,[x,x^{*}]\geq0\}$ \end_inset . @@ -188,6 +188,13 @@ TerminateProcess en Windows. \end_layout +\begin_layout Standard +\begin_inset Newpage pagebreak +\end_inset + + +\end_layout + \begin_layout Section Estados \end_layout @@ -359,7 +366,32 @@ Bloqueado \begin_inset Quotes frd \end_inset -, si el proces, ueado suspendido +, si el proceso hace una llamada al sistema que no se puede responder inmediatam +ente. +\end_layout + +\begin_layout Itemize +De +\begin_inset Quotes cld +\end_inset + +Bloqueado +\begin_inset Quotes crd +\end_inset + + a +\begin_inset Quotes cld +\end_inset + +Listo +\begin_inset Quotes crd +\end_inset + + o de +\begin_inset Quotes cld +\end_inset + +Bloqueado suspendido \begin_inset Quotes frd \end_inset @@ -435,6 +467,18 @@ Implementación \end_layout \begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +begin{samepage} +\end_layout + +\end_inset + El SO mantiene una \series bold tabla de procesos @@ -464,6 +508,22 @@ administración de procesos usado de CPU. \end_layout +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +end{samepage} +\end_layout + +\end_inset + + +\end_layout + \begin_layout Itemize Para \series bold @@ -882,6 +942,22 @@ Diagrama de Gantt. \end_layout \begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +begin{samepage} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard Algoritmos no apropiativos: \end_layout @@ -972,6 +1048,22 @@ maduración \end_layout \begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +end{samepage} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard Algoritmos apropiativos: \end_layout @@ -489,7 +489,7 @@ Sean \end_inset y -\begin_inset Formula $F'\coloneqq\{r\in Q':r\cap F\neq\emptyset\}$ +\begin_inset Formula $F'\coloneqq\{r\in Q'\mid r\cap F\neq\emptyset\}$ \end_inset . @@ -1807,7 +1807,7 @@ Sean \[ \delta'(q,r)\coloneqq\begin{cases} \epsilon, & (q,r)=(q_{0},q_{1})\lor(q\in F\land r=q_{\text{F}});\\ -a_{1}\mid\dots\mid a_{k}, & \{a\in\Sigma:r\in\delta(q,a)\}=\{a_{1},\dots,a_{k}\}\neq\emptyset;\\ +a_{1}\mid\dots\mid a_{k}, & \{a\in\Sigma\mid r\in\delta(q,a)\}=\{a_{1},\dots,a_{k}\}\neq\emptyset;\\ \emptyset, & \text{en otro caso}. \end{cases} \] @@ -602,7 +602,7 @@ variable inicial \end_inset , donde -\begin_inset Formula $\{w_{1},\dots,w_{n}\}=\{w:(T,w)\in V\}$ +\begin_inset Formula $\{w_{1},\dots,w_{n}\}=\{w\mid (T,w)\in V\}$ \end_inset . @@ -668,7 +668,7 @@ lenguaje generado \end_inset es -\begin_inset Formula $L(G)\coloneqq\{w\in\Sigma^{*}:S\Rightarrow^{*}w\}$ +\begin_inset Formula $L(G)\coloneqq\{w\in\Sigma^{*}\mid S\Rightarrow^{*}w\}$ \end_inset . @@ -439,7 +439,7 @@ input \end_inset que reconoce -\begin_inset Formula $K\coloneqq\{\langle{\cal A},w\rangle:\text{la MT \ensuremath{{\cal A}} acepta \ensuremath{w}}\}$ +\begin_inset Formula $K\coloneqq\{\langle{\cal A},w\rangle\mid \text{la MT \ensuremath{{\cal A}} acepta \ensuremath{w}}\}$ \end_inset . @@ -1953,7 +1953,7 @@ Algunos lenguajes decidibles: \end_layout \begin_layout Enumerate -\begin_inset Formula $\text{Acc}^{\text{DFA}}\coloneqq\{\langle{\cal A},w\rangle:\text{el DFA \ensuremath{{\cal A}} acepta la cadena \ensuremath{w}}\}$ +\begin_inset Formula $\text{Acc}^{\text{DFA}}\coloneqq\{\langle{\cal A},w\rangle\mid \text{el DFA \ensuremath{{\cal A}} acepta la cadena \ensuremath{w}}\}$ \end_inset . @@ -2044,7 +2044,7 @@ fun m q0 finals w -> contains (==) (sim m w q0) finals \end_deeper \begin_layout Enumerate -\begin_inset Formula $\text{Acc}^{\text{NFA}}\coloneqq\{\langle{\cal A},w\rangle:\text{el NFA \ensuremath{{\cal A}} acepta la cadena \ensuremath{w}}\}$ +\begin_inset Formula $\text{Acc}^{\text{NFA}}\coloneqq\{\langle{\cal A},w\rangle\mid \text{el NFA \ensuremath{{\cal A}} acepta la cadena \ensuremath{w}}\}$ \end_inset . @@ -2275,7 +2275,7 @@ fun (states, syms, m, r0, finals) -> \end_layout \begin_layout Enumerate -\begin_inset Formula $\text{Acc}^{\text{PDA}}\coloneqq\{\langle{\cal A},w\rangle:\text{el PDA \ensuremath{{\cal A}} acepta la cadena \ensuremath{w}}\}$ +\begin_inset Formula $\text{Acc}^{\text{PDA}}\coloneqq\{\langle{\cal A},w\rangle\mid \text{el PDA \ensuremath{{\cal A}} acepta la cadena \ensuremath{w}}\}$ \end_inset . @@ -2322,7 +2322,7 @@ forma normal de Chomsky \end_layout \begin_layout Enumerate -\begin_inset Formula $\text{Empty}^{\text{DFA}}\coloneqq\{\langle{\cal A}\rangle:\text{el DFA }{\cal A}\text{ no acepta ninguna cadena}\}$ +\begin_inset Formula $\text{Empty}^{\text{DFA}}\coloneqq\{\langle{\cal A}\rangle\mid \text{el DFA }{\cal A}\text{ no acepta ninguna cadena}\}$ \end_inset . @@ -2433,7 +2433,7 @@ fun (trans, q0, finals) -> anystring trans finals nil (cons q0 nil) \end_deeper \begin_layout Enumerate -\begin_inset Formula $\text{Empty}^{\text{NFA}}\coloneqq\{\langle{\cal A}\rangle:\text{el NFA }{\cal A}\text{ no acepta ninguna cadena}\}$ +\begin_inset Formula $\text{Empty}^{\text{NFA}}\coloneqq\{\langle{\cal A}\rangle\mid \text{el NFA }{\cal A}\text{ no acepta ninguna cadena}\}$ \end_inset . @@ -2446,7 +2446,7 @@ Análogo. \end_deeper \begin_layout Enumerate -\begin_inset Formula $\text{Empty}^{\text{PDA}}\coloneqq\{\langle{\cal A}\rangle:\text{el PDA }{\cal A}\text{ no acepta ninguna cadena}\}$ +\begin_inset Formula $\text{Empty}^{\text{PDA}}\coloneqq\{\langle{\cal A}\rangle\mid \text{el PDA }{\cal A}\text{ no acepta ninguna cadena}\}$ \end_inset . @@ -2610,7 +2610,7 @@ Demostración: \end_inset , sea -\begin_inset Formula $B\coloneqq\{x\in A:x\notin f(x)\}$ +\begin_inset Formula $B\coloneqq\{x\in A\mid x\notin f(x)\}$ \end_inset , existe @@ -2767,7 +2767,7 @@ status open \begin_layout Standard \begin_inset Formula \[ -K\coloneqq\{\langle{\cal M},w\rangle:\text{la MT }{\cal M}\text{ acepta con entrada }w\}\in{\cal RE}\setminus{\cal DEC}. +K\coloneqq\{\langle{\cal M},w\rangle\mid \text{la MT }{\cal M}\text{ acepta con entrada }w\}\in{\cal RE}\setminus{\cal DEC}. \] \end_inset @@ -2806,7 +2806,7 @@ Demostración: \end_inset que decide -\begin_inset Formula $\{\langle{\cal M}\rangle:{\cal H}\text{ rechaza }\langle{\cal M},\langle{\cal M}\rangle\rangle\}$ +\begin_inset Formula $\{\langle{\cal M}\rangle\mid {\cal H}\text{ rechaza }\langle{\cal M},\langle{\cal M}\rangle\rangle\}$ \end_inset , pero entonces @@ -327,7 +327,7 @@ Problema de la parada. \begin_inset Formula \[ -\text{HALT}^{\text{MT}}\coloneqq\{\langle{\cal M},w\rangle:{\cal M}\text{ es una MT que para con entrada }w\}\notin{\cal DEC}. +\text{HALT}^{\text{MT}}\coloneqq\{\langle{\cal M},w\rangle\mid {\cal M}\text{ es una MT que para con entrada }w\}\notin{\cal DEC}. \] \end_inset @@ -380,7 +380,7 @@ mapping \end_deeper \begin_layout Enumerate -\begin_inset Formula $\text{EMPTY}^{\text{MT}}\coloneqq\{\langle{\cal M}\rangle:{\cal M}\text{ es una MT que no acepta ninguna cadena}\}\notin{\cal DEC}$ +\begin_inset Formula $\text{EMPTY}^{\text{MT}}\coloneqq\{\langle{\cal M}\rangle\mid {\cal M}\text{ es una MT que no acepta ninguna cadena}\}\notin{\cal DEC}$ \end_inset . @@ -454,7 +454,7 @@ mapping \end_deeper \begin_layout Enumerate -\begin_inset Formula $\text{Pass}\coloneqq\{\langle{\cal M},w,q\rangle:{\cal M}\text{ es una MT que, con entrada }w\text{, pasa por el estado \ensuremath{q}}\}\notin{\cal DEC}$ +\begin_inset Formula $\text{Pass}\coloneqq\{\langle{\cal M},w,q\rangle\mid {\cal M}\text{ es una MT que, con entrada }w\text{, pasa por el estado \ensuremath{q}}\}\notin{\cal DEC}$ \end_inset . @@ -674,7 +674,7 @@ Teorema de Rice: no trivial, \begin_inset Formula \[ -{\cal L}_{P}\coloneqq\{\langle{\cal M}\rangle:{\cal M}\text{ es una MT con }L(M)\in P\}\notin{\cal DEC}. +{\cal L}_{P}\coloneqq\{\langle{\cal M}\rangle\mid {\cal M}\text{ es una MT con }L(M)\in P\}\notin{\cal DEC}. \] \end_inset @@ -1113,7 +1113,7 @@ Están en \end_layout \begin_layout Enumerate -\begin_inset Formula $\text{RELPRIM}\coloneqq\{\langle x,y\rangle:x,y\in\mathbb{N}\text{ son primos relativos}\}$ +\begin_inset Formula $\text{RELPRIM}\coloneqq\{\langle x,y\rangle\mid x,y\in\mathbb{N}\text{ son primos relativos}\}$ \end_inset . @@ -1192,7 +1192,7 @@ noprefix "false" \end_deeper \begin_layout Enumerate -\begin_inset Formula $\text{PATH}\coloneqq\{\langle G,s,t\rangle:G\text{ es un grafo dirigido con un camino de }s\text{ a }t\}$ +\begin_inset Formula $\text{PATH}\coloneqq\{\langle G,s,t\rangle\mid G\text{ es un grafo dirigido con un camino de }s\text{ a }t\}$ \end_inset . @@ -1251,7 +1251,7 @@ Se añade el nodo \end_deeper \begin_layout Enumerate -\begin_inset Formula $\text{4-CLIQUE}\coloneqq\{\langle G\rangle:G\text{ es un grafo no dirigido con una 4-clique}\}$ +\begin_inset Formula $\text{4-CLIQUE}\coloneqq\{\langle G\rangle\mid G\text{ es un grafo no dirigido con una 4-clique}\}$ \end_inset . @@ -1287,7 +1287,7 @@ Si \end_deeper \begin_layout Enumerate -\begin_inset Formula $\text{EULCYCLE}\coloneqq\{\langle G\rangle:G\text{ es un grafo dirigido con un ciclo euleriano}\}$ +\begin_inset Formula $\text{EULCYCLE}\coloneqq\{\langle G\rangle\mid G\text{ es un grafo dirigido con un ciclo euleriano}\}$ \end_inset . @@ -1317,7 +1317,7 @@ Un teorema de Euler dice que un grafo dirigido \end_deeper \begin_layout Enumerate -\begin_inset Formula $\text{2-COLOR}\coloneqq\{\langle G\rangle:G\text{ es un grafo no dirigido bipartito}\}$ +\begin_inset Formula $\text{2-COLOR}\coloneqq\{\langle G\rangle\mid G\text{ es un grafo no dirigido bipartito}\}$ \end_inset . @@ -1562,7 +1562,7 @@ verificador \end_inset tal que -\begin_inset Formula $L=\{w:\exists c:V\text{ acepta }\langle w,c\rangle\}$ +\begin_inset Formula $L=\{w\mid \exists c\mid V\text{ acepta }\langle w,c\rangle\}$ \end_inset . @@ -408,7 +408,7 @@ satisfacible Definimos \begin_inset Formula \[ -\text{SAT}\coloneqq\text{SAT}_{0}\coloneqq\text{SAT}_{\text{LP}}\coloneqq\{\langle\Phi\rangle:\Phi\text{ es una fórmula booleana satisfacible}\}. +\text{SAT}\coloneqq\text{SAT}_{0}\coloneqq\text{SAT}_{\text{LP}}\coloneqq\{\langle\Phi\rangle\mid \Phi\text{ es una fórmula booleana satisfacible}\}. \] \end_inset @@ -1039,7 +1039,7 @@ Son \end_layout \begin_layout Enumerate -\begin_inset Formula $\text{CLIQUE}\coloneqq\{\langle G,k\rangle:G\text{ es grafo no dirigido con }k\text{-clique}\}$ +\begin_inset Formula $\text{CLIQUE}\coloneqq\{\langle G,k\rangle\mid G\text{ es grafo no dirigido con }k\text{-clique}\}$ \end_inset . @@ -1209,7 +1209,7 @@ La función de conversión de \end_deeper \begin_layout Enumerate -\begin_inset Formula $\text{HAMPATH}\coloneqq\{\langle G,s,t\rangle:G\text{ es un grafo dirigido con camino hamiltoniano de }s\text{ a }t\}$ +\begin_inset Formula $\text{HAMPATH}\coloneqq\{\langle G,s,t\rangle\mid G\text{ es un grafo dirigido con camino hamiltoniano de }s\text{ a }t\}$ \end_inset . @@ -1607,7 +1607,7 @@ La función de conversión de \end_deeper \begin_layout Enumerate -\begin_inset Formula $\text{HAMCYCLE}\coloneqq\{\langle G\rangle:G\text{ es un grafo dirigido con un ciclo hamiltoniano}\}$ +\begin_inset Formula $\text{HAMCYCLE}\coloneqq\{\langle G\rangle\mid G\text{ es un grafo dirigido con un ciclo hamiltoniano}\}$ \end_inset . @@ -1765,7 +1765,7 @@ La función de conversión de \end_deeper \begin_layout Enumerate -\begin_inset Formula $\text{UHAMCYCLE}\coloneqq\{\langle G\rangle:G\text{ es un grafo no dirigido con un ciclo hamiltoniano}\}$ +\begin_inset Formula $\text{UHAMCYCLE}\coloneqq\{\langle G\rangle\mid G\text{ es un grafo no dirigido con un ciclo hamiltoniano}\}$ \end_inset . @@ -2011,7 +2011,7 @@ Claramente la función de conversión de \end_deeper \begin_layout Enumerate -\begin_inset Formula $\text{COLOR}\coloneqq\{\langle G,k\rangle:G\text{ es un grafo no dirigido }k\text{-coloreable}\}$ +\begin_inset Formula $\text{COLOR}\coloneqq\{\langle G,k\rangle\mid G\text{ es un grafo no dirigido }k\text{-coloreable}\}$ \end_inset . @@ -2277,7 +2277,7 @@ Un ciclo hamiltoniano en \end_deeper \begin_layout Enumerate -\begin_inset Formula $\text{SUBSET-SUM}\coloneqq\{\langle S,t\rangle:S\text{ es una lista de naturales con una subsecuencia que suma }t\}.$ +\begin_inset Formula $\text{SUBSET-SUM}\coloneqq\{\langle S,t\rangle\mid S\text{ es una lista de naturales con una subsecuencia que suma }t\}.$ \end_inset @@ -2605,7 +2605,7 @@ ión, pero calcular las potencias de 10 corresponde a multiplicar por 10 \end_deeper \begin_layout Enumerate -\begin_inset Formula $\text{VERTEX-COVER}\coloneqq\{\langle G,k\rangle:G\text{ es un grafo no dirigido con una }k\text{-cobertura}\}$ +\begin_inset Formula $\text{VERTEX-COVER}\coloneqq\{\langle G,k\rangle\mid G\text{ es un grafo no dirigido con una }k\text{-cobertura}\}$ \end_inset . @@ -241,7 +241,7 @@ con para dicho problema con redondeo, dado por \begin_inset Formula \[ -\left\{ \begin{aligned}\omega_{0} & :=x_{0}+\delta_{0},\\ +\left\{ \begin{aligned}\omega_{0} & \mid =x_{0}+\delta_{0},\\ \omega_{i+1} & :=\omega_{i}+hf(t_{i},\omega_{i})+\delta_{i+1}, \end{aligned} \right. @@ -240,7 +240,7 @@ región de estabilidad absoluta \end_inset , -\begin_inset Formula $R=\{z\in\mathbb{C}:|Q(z)|<1\}$ +\begin_inset Formula $R=\{z\in\mathbb{C}\mid |Q(z)|<1\}$ \end_inset , y para uno multipaso que converge cuando cada @@ -248,7 +248,7 @@ región de estabilidad absoluta \end_inset , es -\begin_inset Formula $R=\{z\in\mathbb{C}:|\beta_{i}|<1,\forall i\}$ +\begin_inset Formula $R=\{z\in\mathbb{C}\mid |\beta_{i}|<1,\forall i\}$ \end_inset . @@ -272,7 +272,7 @@ Hay que tener en cuenta la región de estabilidad antes de considerar un A-estable \series default si -\begin_inset Formula $\{z\in\mathbb{C}:\text{Re}z<0\}\subseteq R$ +\begin_inset Formula $\{z\in\mathbb{C}\mid \text{Re}z<0\}\subseteq R$ \end_inset . diff --git a/pcd/n.pdf b/pcd/n.pdf Binary files differdeleted file mode 100644 index 1b7678e..0000000 --- a/pcd/n.pdf +++ /dev/null @@ -421,7 +421,7 @@ Dada una asociación \end_inset es el conjunto de posibles valores de -\begin_inset Formula $|\{a_{i}\in C_{i}:(a_{1},\dots,a_{n})\in R\}|$ +\begin_inset Formula $|\{a_{i}\in C_{i}\mid (a_{1},\dots,a_{n})\in R\}|$ \end_inset para cada diff --git a/rc/n.pdf b/rc/n.pdf Binary files differdeleted file mode 100644 index e01c446..0000000 --- a/rc/n.pdf +++ /dev/null @@ -269,7 +269,7 @@ Y-O \end_inset , sea -\begin_inset Formula $N:=\{S\subseteq V:(u,S)\in A\}$ +\begin_inset Formula $N:=\{S\subseteq V\mid (u,S)\in A\}$ \end_inset , @@ -315,7 +315,7 @@ primitiva árbol Y/O \series default es un grafo Y/O para el que el grafo no dirigido -\begin_inset Formula $(V,\{(u,v)\in V\times V:\exists(u,S)\in A:v\in S\})$ +\begin_inset Formula $(V,\{(u,v)\in V\times V\mid \exists(u,S)\in A\mid v\in S\})$ \end_inset es acíclico. @@ -145,7 +145,7 @@ Podemos representar un problema de búsqueda en un espacio de estados como \end_inset , -\begin_inset Formula $\{w\in V:(v,w)\in A\}$ +\begin_inset Formula $\{w\in V\mid (v,w)\in A\}$ \end_inset es finito y recursivamente enumerable a partir de @@ -1224,7 +1224,7 @@ Podemos representar un problema de reducción como una tupla \end_inset contable y tanto -\begin_inset Formula $\{S\subseteq V:(u,S)\in V\}$ +\begin_inset Formula $\{S\subseteq V\mid (u,S)\in V\}$ \end_inset como cada uno de sus elementos finito y recursivamente enumerable a partir @@ -685,7 +685,7 @@ En lógica de predicados, a todo predicado \end_inset le corresponde un conjunto -\begin_inset Formula $\{x\in U:P(x)\}$ +\begin_inset Formula $\{x\in U\mid P(x)\}$ \end_inset y una @@ -449,7 +449,7 @@ soporte \end_inset es -\begin_inset Formula $s(Z):=\frac{|\{e\in D:Z\subseteq e\}|}{|D|}$ +\begin_inset Formula $s(Z):=\frac{|\{e\in D\mid Z\subseteq e\}|}{|D|}$ \end_inset ; la @@ -490,7 +490,7 @@ cobertura . Las diapositivas usan la notación de mierda -\begin_inset Formula $|X|:=|\{e\in D:X\subseteq e\}|$ +\begin_inset Formula $|X|:=|\{e\in D\mid X\subseteq e\}|$ \end_inset . @@ -406,7 +406,7 @@ La topología cofinita \series default : -\begin_inset Formula ${\cal T}_{CF}=\{\emptyset\}\cup\{A\subseteq X:X\backslash A\text{ es finito}\}$ +\begin_inset Formula ${\cal T}_{CF}=\{\emptyset\}\cup\{A\subseteq X\mid X\backslash A\text{ es finito}\}$ \end_inset . @@ -1381,7 +1381,7 @@ círculo \end_inset es el conjunto -\begin_inset Formula $C_{d}(p;r):=C(p;r):=\{x\in X:d(p,x)=r\}$ +\begin_inset Formula $C_{d}(p;r):=C(p;r):=\{x\in X\mid d(p,x)=r\}$ \end_inset . @@ -1402,7 +1402,7 @@ bola abierta \end_inset es el conjunto -\begin_inset Formula $B_{d}(p;r):=B(p;r):=\{x\in X:d(p,x)<r\}$ +\begin_inset Formula $B_{d}(p;r):=B(p;r):=\{x\in X\mid d(p,x)<r\}$ \end_inset , y la @@ -1422,7 +1422,7 @@ bola cerrada \end_inset es el conjunto -\begin_inset Formula $\overline{B}_{d}(p;r):=\overline{B}(p;r):=B[p;r]:=\{x\in X:d(p,x)\leq r\}$ +\begin_inset Formula $\overline{B}_{d}(p;r):=\overline{B}(p;r):=B[p;r]:=\{x\in X\mid d(p,x)\leq r\}$ \end_inset . @@ -110,7 +110,7 @@ adherencia denota \begin_inset Formula \[ -\overline{S}:=\text{cl}(S):=\text{ad}(S):=\bigcap\{C\in{\cal C}_{{\cal T}}:S\subseteq C\} +\overline{S}:=\text{cl}(S):=\text{ad}(S):=\bigcap\{C\in{\cal C}_{{\cal T}}\mid S\subseteq C\} \] \end_inset @@ -709,7 +709,7 @@ interior , y se denota \begin_inset Formula \[ -\mathring{S}:=\text{int}S:=\bigcup\{A\in{\cal T}:A\subseteq S\} +\mathring{S}:=\text{int}S:=\bigcup\{A\in{\cal T}\mid A\subseteq S\} \] \end_inset @@ -1160,7 +1160,7 @@ Sea \end_inset , entonces -\begin_inset Formula $x\in\overline{S}\iff\exists\{x_{n}\}_{n=1}^{\infty}\subseteq S:x_{n}\rightarrow x$ +\begin_inset Formula $x\in\overline{S}\iff\exists\{x_{n}\}_{n=1}^{\infty}\subseteq S\mid x_{n}\rightarrow x$ \end_inset . @@ -1249,7 +1249,7 @@ Así pues, en un espacio métrico \end_inset si y sólo si -\begin_inset Formula $\forall x\in X,\exists\{x_{n}\}_{n=1}^{\infty}\subseteq S:x_{n}\rightarrow x$ +\begin_inset Formula $\forall x\in X,\exists\{x_{n}\}_{n=1}^{\infty}\subseteq S\mid x_{n}\rightarrow x$ \end_inset , y @@ -1257,7 +1257,7 @@ Así pues, en un espacio métrico \end_inset si y sólo si -\begin_inset Formula $\exists\{x_{n}\}_{n=1}^{\infty}\subseteq S,\{y_{n}\}_{n=1}^{\infty}\subseteq X\backslash S:x_{n},y_{n}\rightarrow x$ +\begin_inset Formula $\exists\{x_{n}\}_{n=1}^{\infty}\subseteq S,\{y_{n}\}_{n=1}^{\infty}\subseteq X\backslash S\mid x_{n},y_{n}\rightarrow x$ \end_inset . @@ -245,7 +245,7 @@ De aquí que Demostración: \series default Tomando -\begin_inset Formula ${\cal B}(p)=\{B(p;\delta):\delta>0\}$ +\begin_inset Formula ${\cal B}(p)=\{B(p;\delta)\mid \delta>0\}$ \end_inset y @@ -369,7 +369,7 @@ Demostración: \end_inset y definimos -\begin_inset Formula $G=\{x\in[a,b]|\exists\{A_{i_{1}},\dots,A_{i_{n}}\}\in{\cal P}_{0}({\cal A}):[a,x]\subseteq A_{i_{1}}\cup\dots\cup A_{i_{n}}\}$ +\begin_inset Formula $G=\{x\in[a,b]|\exists\{A_{i_{1}},\dots,A_{i_{n}}\}\in{\cal P}_{0}({\cal A})\mid [a,x]\subseteq A_{i_{1}}\cup\dots\cup A_{i_{n}}\}$ \end_inset . @@ -268,7 +268,7 @@ entorno \end_inset es un elemento de -\begin_inset Formula ${\cal E}(x):=\{U\in{\cal T}:x\in{\cal U}\}$ +\begin_inset Formula ${\cal E}(x):=\{U\in{\cal T}\mid x\in{\cal U}\}$ \end_inset . @@ -459,7 +459,7 @@ abierta a \begin_inset Formula \[ -B_{d}(x,\delta):=\{y\in X:d(x,y)<\varepsilon\}. +B_{d}(x,\delta):=\{y\in X\mid d(x,y)<\varepsilon\}. \] \end_inset @@ -485,7 +485,7 @@ inducida \end_inset a la topología -\begin_inset Formula ${\cal T}_{d}:=\{A\in X:\forall x\in A,\exists\delta>0:B_{d}(x,\delta)\subseteq A\}$ +\begin_inset Formula ${\cal T}_{d}:=\{A\in X\mid \forall x\in A,\exists\delta>0\mid B_{d}(x,\delta)\subseteq A\}$ \end_inset . @@ -578,7 +578,7 @@ La -esfera \series default , -\begin_inset Formula $\mathbb{S}^{n}(r):=\{(x_{1},\dots,x_{n+1})\in\mathbb{R}^{n+1}:x_{1}^{2}+\dots+x_{n+1}^{2}=r^{2}\}$ +\begin_inset Formula $\mathbb{S}^{n}(r):=\{(x_{1},\dots,x_{n+1})\in\mathbb{R}^{n+1}\mid x_{1}^{2}+\dots+x_{n+1}^{2}=r^{2}\}$ \end_inset . @@ -630,7 +630,7 @@ El cilindro \series default , -\begin_inset Formula $C:=\{(x,y,z)\in\mathbb{R}^{3}:x^{2}+y^{2}=1,0\leq z\leq1\}$ +\begin_inset Formula $C:=\{(x,y,z)\in\mathbb{R}^{3}\mid x^{2}+y^{2}=1,0\leq z\leq1\}$ \end_inset , cono de rotación sobre el eje @@ -666,7 +666,7 @@ El toro \series default , -\begin_inset Formula $\mathbb{T}:=\{(x,y,z)\in\mathbb{R}^{3}:x^{2}+y^{2}+z^{2}-4\sqrt{x^{2}+y^{2}}+3=0\}$ +\begin_inset Formula $\mathbb{T}:=\{(x,y,z)\in\mathbb{R}^{3}\mid x^{2}+y^{2}+z^{2}-4\sqrt{x^{2}+y^{2}}+3=0\}$ \end_inset , cono de rotación sobre el eje @@ -674,7 +674,7 @@ toro \end_inset de -\begin_inset Formula $\{(x,0,z):(x-2)^{2}+z^{2}=1\}$ +\begin_inset Formula $\{(x,0,z)\mid (x-2)^{2}+z^{2}=1\}$ \end_inset . @@ -695,7 +695,7 @@ status open \end_inset Tenemos -\begin_inset Formula $\{(x,0,z):(x-2)^{2}+z^{2}=1\}=\{\alpha(s):=(\cos s+2,0,\sin s)\}_{s\in[0,2\pi]}$ +\begin_inset Formula $\{(x,0,z)\mid (x-2)^{2}+z^{2}=1\}=\{\alpha(s)\mid =(\cos s+2,0,\sin s)\}_{s\in[0,2\pi]}$ \end_inset , luego el cono de rotación es @@ -1056,7 +1056,7 @@ Como los abiertos en \end_inset , -\begin_inset Formula $s^{-1}((a,b))=\{(x,y):a<s(x,y)=x+y<b\}=\{(x,y):a-x<y<b-x\}$ +\begin_inset Formula $s^{-1}((a,b))=\{(x,y)\mid a<s(x,y)=x+y<b\}=\{(x,y)\mid a-x<y<b-x\}$ \end_inset . @@ -1135,7 +1135,7 @@ Dado \end_inset , queremos ver que -\begin_inset Formula $p^{-1}((a,b))=\{(x,y):a<p(x,y)=xy<b\}$ +\begin_inset Formula $p^{-1}((a,b))=\{(x,y)\mid a<p(x,y)=xy<b\}$ \end_inset es abierto. @@ -1217,7 +1217,7 @@ Basta ver que, dada una bola , su inversa es un abierto. Tenemos -\begin_inset Formula $d^{-1}(B_{d_{\infty}}(y,r))=\{x:d_{\infty}((x,\dots,x),y)<r\}=\{t:|x-y_{1}|,\dots,|x-y_{n}|<r\}$ +\begin_inset Formula $d^{-1}(B_{d_{\infty}}(y,r))=\{x\mid d_{\infty}((x,\dots,x),y)<r\}=\{t\mid |x-y_{1}|,\dots,|x-y_{n}|<r\}$ \end_inset , pero @@ -2043,7 +2043,7 @@ topología generada \end_inset a -\begin_inset Formula ${\cal T}_{{\cal B}}:=\{U\subseteq X:\forall x\in U,\exists B\in{\cal B}:x\in B\subseteq U\}$ +\begin_inset Formula ${\cal T}_{{\cal B}}:=\{U\subseteq X\mid \forall x\in U,\exists B\in{\cal B}\mid x\in B\subseteq U\}$ \end_inset , y se tiene que @@ -2456,7 +2456,7 @@ Dada una base \end_inset numerable, -\begin_inset Formula ${\cal B}_{x}:=\{B\in{\cal B}:x\in B\}$ +\begin_inset Formula ${\cal B}_{x}:=\{B\in{\cal B}\mid x\in B\}$ \end_inset es base de entornos de @@ -1125,7 +1125,7 @@ Ejemplos de conexión \begin_layout Enumerate La hipérbola -\begin_inset Formula $\{(x,y)\in\mathbb{R}^{2}:x^{2}-y^{2}=1\}$ +\begin_inset Formula $\{(x,y)\in\mathbb{R}^{2}\mid x^{2}-y^{2}=1\}$ \end_inset no es conexa. @@ -1134,11 +1134,11 @@ status open \begin_layout Plain Layout Sean -\begin_inset Formula $U:=\{(x,y):x>0\}$ +\begin_inset Formula $U:=\{(x,y)\mid x>0\}$ \end_inset , -\begin_inset Formula $V:=\{(x,y):x<0\}$ +\begin_inset Formula $V:=\{(x,y)\mid x<0\}$ \end_inset e @@ -1150,7 +1150,7 @@ Sean \end_inset , luego -\begin_inset Formula $Y\subseteq U\cap V=\{(x,y):x\neq0\}$ +\begin_inset Formula $Y\subseteq U\cap V=\{(x,y)\mid x\neq0\}$ \end_inset ; @@ -1351,7 +1351,7 @@ La función status open \begin_layout Plain Layout -\begin_inset Formula ${\cal GL}(3,\mathbb{R})=\{A\in{\cal M}_{3}(\mathbb{R}):\det A\neq0\}$ +\begin_inset Formula ${\cal GL}(3,\mathbb{R})=\{A\in{\cal M}_{3}(\mathbb{R})\mid \det A\neq0\}$ \end_inset , luego existe la función continua @@ -1372,7 +1372,7 @@ status open . -\begin_inset Formula ${\cal O}(3,\mathbb{K})=\{A\in{\cal M}_{3}(\mathbb{R}):\det A\in\{-1,1\}\}$ +\begin_inset Formula ${\cal O}(3,\mathbb{K})=\{A\in{\cal M}_{3}(\mathbb{R})\mid \det A\in\{-1,1\}\}$ \end_inset , luego @@ -2393,7 +2393,7 @@ Sea . Ahora bien, -\begin_inset Formula $\{U_{\delta}:=(-\infty,z-\delta)\cup(z+\delta,+\infty)\}_{\delta>0}$ +\begin_inset Formula $\{U_{\delta}\mid =(-\infty,z-\delta)\cup(z+\delta,+\infty)\}_{\delta>0}$ \end_inset es un recubrimiento de @@ -2750,7 +2750,7 @@ Sea \end_inset continua, -\begin_inset Formula $\text{fix}f:=\{x\in X:f(x)=x\}$ +\begin_inset Formula $\text{fix}f:=\{x\in X\mid f(x)=x\}$ \end_inset es cerrado en @@ -309,7 +309,7 @@ Sean status open \begin_layout Plain Layout -\begin_inset Formula $\mathbb{S}^{n}\setminus\{N:=(0,\dots,0,1)\}$ +\begin_inset Formula $\mathbb{S}^{n}\setminus\{N\mid =(0,\dots,0,1)\}$ \end_inset y @@ -736,7 +736,7 @@ unión disjunta \end_inset son espacios topológicos, definimos la topología -\begin_inset Formula ${\cal T}_{X\amalg Y}:=\{U\subseteq X\amalg Y:\{x:(x,0)\in U\}\in{\cal T}_{X}\land\{y:(y,1)\in U\}\in{\cal T}_{Y}\}$ +\begin_inset Formula ${\cal T}_{X\amalg Y}:=\{U\subseteq X\amalg Y\mid \{x\mid (x,0)\in U\}\in{\cal T}_{X}\land\{y\mid (y,1)\in U\}\in{\cal T}_{Y}\}$ \end_inset . @@ -934,7 +934,7 @@ Sea \end_inset , -\begin_inset Formula $\{U_{i}:=\{x:(x,0)\in A_{i}\}\}_{i\in I}$ +\begin_inset Formula $\{U_{i}\mid =\{x\mid (x,0)\in A_{i}\}\}_{i\in I}$ \end_inset lo es de @@ -947,7 +947,7 @@ Sea . Del mismo modo -\begin_inset Formula $\{V_{j}:=\{y:(y,1)\in A_{i}\}\}_{j\in I}$ +\begin_inset Formula $\{V_{j}\mid =\{y\mid (y,1)\in A_{i}\}\}_{j\in I}$ \end_inset admite un subrecubrimiento finito @@ -1122,11 +1122,11 @@ Sean \end_inset disjuntos, y basta tomar -\begin_inset Formula $\{x:(x,0)\in U\}$ +\begin_inset Formula $\{x\mid (x,0)\in U\}$ \end_inset y -\begin_inset Formula $\{x:(x,0)\in V\}$ +\begin_inset Formula $\{x\mid (x,0)\in V\}$ \end_inset . @@ -1449,7 +1449,7 @@ Dado un abierto \end_inset , -\begin_inset Formula $a^{-1}(U)=\{x\in X:a(x)\in U\}=f^{-1}(U\times Y)$ +\begin_inset Formula $a^{-1}(U)=\{x\in X\mid a(x)\in U\}=f^{-1}(U\times Y)$ \end_inset , que es abierto por la hipótesis. @@ -1479,7 +1479,7 @@ Dado un elemento básico \end_inset , -\begin_inset Formula $f^{-1}(U\times)=\{x\in X:a(x)\in U,b(x)\in V\}=a^{-1}(U)\cap b^{-1}(V)$ +\begin_inset Formula $f^{-1}(U\times)=\{x\in X\mid a(x)\in U,b(x)\in V\}=a^{-1}(U)\cap b^{-1}(V)$ \end_inset , que es abierto. @@ -2269,7 +2269,7 @@ Sean \end_inset , sea -\begin_inset Formula $I_{x}:=\{i\in I:x\in U_{i}\}$ +\begin_inset Formula $I_{x}:=\{i\in I\mid x\in U_{i}\}$ \end_inset , @@ -2360,7 +2360,7 @@ topología cociente \end_inset a -\begin_inset Formula $\{V\subseteq(X/\sim):p^{-1}(V)\in{\cal T}\}$ +\begin_inset Formula $\{V\subseteq(X/\sim)\mid p^{-1}(V)\in{\cal T}\}$ \end_inset , donde @@ -2832,7 +2832,7 @@ Si \end_inset es Hausdorff si y sólo si -\begin_inset Formula $\{(x,y)\in X\times X:x\sim y\}$ +\begin_inset Formula $\{(x,y)\in X\times X\mid x\sim y\}$ \end_inset es cerrado en @@ -747,7 +747,7 @@ El recíproco no se cumple: \begin_layout Enumerate La corona circular -\begin_inset Formula $\{(x,y)\in\mathbb{R}^{2}:x^{2}+y^{2}\in[0,1]\}$ +\begin_inset Formula $\{(x,y)\in\mathbb{R}^{2}\mid x^{2}+y^{2}\in[0,1]\}$ \end_inset es homotópicamente equivalente, pero no homeomorfa, a @@ -258,7 +258,7 @@ envoltura convexa , \begin_inset Formula \[ -\text{conv}W=\left\{ t_{1}v_{1}+\dots+t_{k}v_{k}:\sum_{i=1}^{k}t_{i}=1,t_{i}\in[0,1]\right\} . +\text{conv}W=\left\{ t_{1}v_{1}+\dots+t_{k}v_{k}\;\middle|\;\sum_{i=1}^{k}t_{i}=1,t_{i}\in[0,1]\right\} . \] \end_inset @@ -520,6 +520,22 @@ dimensión \end_layout \begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +begin{samepage} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard Ejemplos: \end_layout @@ -578,6 +594,22 @@ Añadir dibujos. \end_layout +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +end{samepage} +\end_layout + +\end_inset + + +\end_layout + \begin_layout Section Número de Euler \end_layout |
