Rodríguez 2017a - Revision history

Zarate: Zarate moved page Review Rodríguez 2017a to Rodríguez 2017a

2017-11-28T09:43:04Z

Zarate moved page Review Rodríguez 2017a to Rodríguez 2017a

Zarate at 09:40, 28 November 2017

2017-11-28T09:40:04Z

Zarate at 09:38, 28 November 2017

2017-11-28T09:38:30Z

Zarate at 09:37, 28 November 2017

2017-11-28T09:37:17Z

Zarate at 09:31, 28 November 2017

2017-11-28T09:31:33Z

Zarate at 09:27, 28 November 2017

2017-11-28T09:27:38Z

Zarate at 09:23, 28 November 2017

2017-11-28T09:23:56Z

Zarate at 09:21, 28 November 2017

2017-11-28T09:21:49Z

Zarate at 09:20, 28 November 2017

2017-11-28T09:20:43Z

Zarate at 09:19, 28 November 2017

2017-11-28T09:19:52Z

@@ Line 730: / Line 730: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>ds^2 = - \alpha (r)^2 dt^2+ a(r)^2 dr^2 + r^2 d\theta ^2 + r^2 \sin ^2(\theta ) d\varphi ^2,   </math>
+| style="text-align: center;" | <math>ds^2 = - \alpha (r)^2 dt^2+ a(r)^2 dr^2 + r^2 d\theta ^2 + r^2 \sin ^2(\theta ) d\varphi ^2   </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (35)
@@ Line 743: / Line 743: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\frac{a_{,r}}{a} + \frac{\alpha _{,r}}{\alpha } = \frac{k_{0}}{2} r \frac{a^2 \omega ^2 \phi _{0}^2}{\alpha ^2} + \frac{k_{0}}{2} r \phi _{0,r}^2,   </math>
+| style="text-align: center;" | <math>\frac{a_{,r}}{a} + \frac{\alpha _{,r}}{\alpha } = \frac{k_{0}}{2} r \frac{a^2 \omega ^2 \phi _{0}^2}{\alpha ^2} + \frac{k_{0}}{2} r \phi _{0,r}^2   </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (36)
@@ Line 754: / Line 754: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\frac{a_{,r}}{a} - \frac{\alpha _{,r}}{\alpha } = \frac{k_{0}}{2} r a^2 \phi _{0}^2 m^2 + \frac{k_{0}}{4} r \lambda a^2 \phi _{0}^4 + \frac{1-a^2}{r}.  </math>
+| style="text-align: center;" | <math>\frac{a_{,r}}{a} - \frac{\alpha _{,r}}{\alpha } = \frac{k_{0}}{2} r a^2 \phi _{0}^2 m^2 + \frac{k_{0}}{4} r \lambda a^2 \phi _{0}^4 + \frac{1-a^2}{r}  </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (37)
@@ Line 767: / Line 767: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\phi _{0,rr} + \phi _{0,r} \left(\frac{2}{r} + \frac{\alpha _{,r}}{\alpha } - \frac{a_{,r}}{a} \right)+ \omega ^2 \phi _{0} \frac{a^2}{\alpha ^2}  - a^2 (m^2 + \lambda \phi _{0}^2)  \phi _{0} = 0 \textrm{,}   </math>
+| style="text-align: center;" | <math>\phi _{0,rr} + \phi _{0,r} \left(\frac{2}{r} + \frac{\alpha _{,r}}{\alpha } - \frac{a_{,r}}{a} \right)+ \omega ^2 \phi _{0} \frac{a^2}{\alpha ^2}  - a^2 (m^2 + \lambda \phi _{0}^2)  \phi _{0} = 0    </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (38)

@@ Line 663: / Line 663: @@
 |-
 | style="text-align: center;" | <math>\begin{align} f(t,v,v') & = m v' + k v + m g= 0\\
-v(0) & = v_0,    \end{align}     </math>
+v(0) & = v_0    \end{align}     </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (34)

@@ Line 600: / Line 600: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\lim _{h \rightarrow 0}  \max _{1 \leq n \leq N} |e_n(h)| = 0,  n = 1,2, \dots , N,    </math>
+| style="text-align: center;" | <math>\lim _{h \rightarrow 0}  \max _{1 \leq n \leq N} |e_n(h)| = 0,  n = 1,2, \dots , N    </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (31)
@@ Line 614: / Line 614: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\lim _{h \rightarrow 0}  \max _{1 \leq n \leq N} |x_n - \tilde{x}_n| = 0,  n = 1,2, \dots , N,    </math>
+| style="text-align: center;" | <math>\lim _{h \rightarrow 0}  \max _{1 \leq n \leq N} |x_n - \tilde{x}_n| = 0,  n = 1,2, \dots , N    </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (32)
@@ Line 630: / Line 630: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>|\phi (t,x,h) - \phi (t,y,h)| < L|x-y| + 2 C h,    </math>
+| style="text-align: center;" | <math>|\phi (t,x,h) - \phi (t,y,h)| < L|x-y| + 2 C h    </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (33)

@@ Line 507: / Line 507: @@
 |}
-donde <math display="inline">L</math> es la constante de Lipschitz para <math display="inline">x' = g(t,x)</math>    Además, el valor óptimo para <math display="inline">h</math> (el mínimo error posible) se presenta para un <math display="inline">h</math> aproximado a <math display="inline">h = \sqrt{2 \Delta / M}</math>, o, despejando a <math display="inline">Delta</math>, cuando <math display="inline">\Delta = \frac{M}{2} h^2</math>, es decir, cuando
+donde <math display="inline">L</math> es la constante de Lipschitz para <math display="inline">x' = g(t,x)</math>    Además, el valor óptimo para <math display="inline">h</math> (el mínimo error posible) se presenta para un <math display="inline">h</math> aproximado a <math display="inline">h = \sqrt{2 \Delta / M}</math>, o, despejando a <math display="inline">\Delta</math>, cuando <math display="inline">\Delta = \frac{M}{2} h^2</math>, es decir, cuando
 <span id="eq-25"></span>

@@ Line 415: / Line 415: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>0 \leq |\tilde{x}' - x'| < \frac{t_f - t_0}{2^{N(h)} } =: \delta (h) = \tau (h) - \epsilon (h) \leq \tau (h), \hbox{ con } 0 \leq \epsilon < \tau /2,     </math>
+| style="text-align: center;" | <math>0 \leq |\tilde{x}' - x'| < \frac{t_f - t_0}{2^{N(h)} } =: \delta (h) = \tau (h) - \epsilon (h) \leq \tau (h), \hbox{ con } 0 \leq \epsilon < \tau /2   </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (19)
@@ Line 427: / Line 427: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math> \begin{align} \tilde{x}_0 & = x_0, \\
+| style="text-align: center;" | <math> \begin{align} \tilde{x}_0 & = x_0 \\
 \tilde{x}_{n+1} & = \tilde{x}_n + h {\tilde{x}'}_n        \end{align} </math>
 |}
@@ Line 440: / Line 440: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\begin{align} \tilde{x}_0 & = x_0 + (\delta _{-1} = 0), \\
+| style="text-align: center;" | <math>\begin{align} \tilde{x}_0 & = x_0 + (\delta _{-1} = 0)\\
-\tilde{x}_{n+1} & = \tilde{x}_n + h g(t_n,\tilde{x}_n) + (h \delta _n := \Delta _n), \hbox{ para } n = 0,1 \ldots N-1, \\        \end{align}     </math>
+\tilde{x}_{n+1} & = \tilde{x}_n + h g(t_n,\tilde{x}_n) + (h \delta _n := \Delta _n), \hbox{ para } n = 0,1 \ldots N-1 \\        \end{align}     </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (20)
@@ Line 454: / Line 454: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>0 \leq |\Delta _n| = h |\delta _n| <  h \delta \leq h \tau := \Delta{.}  </math>
+| style="text-align: center;" | <math>0 \leq |\Delta _n| = h |\delta _n| <  h \delta \leq h \tau := \Delta </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (21)
@@ Line 470: / Line 470: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>x'' = \frac{-\frac{\partial f}{\partial t} -\frac{\partial f}{\partial x} x'}{\frac{\partial f}{\partial x'}}.     </math>
+| style="text-align: center;" | <math>x'' = \frac{-\frac{\partial f}{\partial t} -\frac{\partial f}{\partial x} x'}{\frac{\partial f}{\partial x'}}     </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (22)
@@ Line 483: / Line 483: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>|x''| = | \frac{\frac{\partial f}{\partial t} + \frac{\partial f}{\partial x} x'}{\frac{\partial f}{\partial x'}} | \leq M,     </math>
+| style="text-align: center;" | <math>|x''| = | \frac{\frac{\partial f}{\partial t} + \frac{\partial f}{\partial x} x'}{\frac{\partial f}{\partial x'}} | \leq M     </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (23)
@@ Line 502: / Line 502: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>|x_n - \tilde{x}_n| \leq \frac{1}{L} \left(\frac{h M}{2} + \frac{\Delta }{h} \right)(e^{L(t_n - t_0)} - 1) = \frac{1}{L} \left(\frac{h M}{2} + \tau (h) \right)(e^{L(t_n - t_0)} - 1),          </math>
+| style="text-align: center;" | <math>|x_n - \tilde{x}_n| \leq \frac{1}{L} \left(\frac{h M}{2} + \frac{\Delta }{h} \right)(e^{L(t_n - t_0)} - 1) = \frac{1}{L} \left(\frac{h M}{2} + \tau (h) \right)(e^{L(t_n - t_0)} - 1)          </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (24)
@@ Line 515: / Line 515: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\tau = \frac{M}{2} h \approx \delta ,    </math>
+| style="text-align: center;" | <math>\tau = \frac{M}{2} h \approx \delta     </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (25)
@@ Line 554: / Line 554: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\tau = C \cdot h,  </math>
+| style="text-align: center;" | <math>\tau = C \cdot h  </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (28)

← Older revision	Revision as of 09:43, 28 November 2017
(No difference)

@@ Line 270: / Line 270: @@
 | style="text-align: center;" | <math>
-\frac{\partial f}{\partial x'}|_{t_r,x_r} \neq 0 </li>
+\frac{\partial f}{\partial x'}|_{t_r,x_r} \neq 0
-</math>
+</math></li>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (11)

@@ Line 164: / Line 164: @@
 |-
 <ol>
-<li> '''1.''' <math> {\tilde{x}'}_{0} = Biseccion(t_{0},\tilde{x}_{0},\alpha _i,\alpha _d,\tau ) </math></li>
+<li> '''-''' <math> {\tilde{x}'}_{0} = Biseccion(t_{0},\tilde{x}_{0},\alpha _i,\alpha _d,\tau ) </math></li>
-<li> '''2.''' para  <math> (n = 0 \hbox{ hasta } N-1) </math> </li>
+<li> '''-''' para  <math> (n = 0 \hbox{ hasta } N-1) </math> </li>
     '''a)'''   <math> t_{n+1} = t_n + h </math>
     '''b)'''   <math> \tilde{x}_{n+1} = \tilde{x}_n + {\tilde{x}'}_n h </math>
     '''c)'''   <math> \tau = \tau (h) </math>
     '''d)'''   <math>{\tilde{x}'}_{n+1} = Biseccion(t_{n+1},\tilde{x}_{n+1},\alpha _i,\alpha _d,\tau ) </math>
-<li>'''3.'''  fin</li>
+<li>'''-'''  fin</li>
 |}
 </ol>