Wed Dec 13 08:16:28 GMT 2017

Author: dalcde

IB_M/methods.tex

@@ -1647,7 +1647,7 @@ \subsubsection*{Energetics and uniqueness}
   \]
   Since $\frac{\partial^2 \phi}{\partial t^2} = c^2 \nabla^2 \phi$ by the wave equation, and $\phi$ is constant on $\partial \Omega$, we know that
   \[
-    \frac{\d E_\phi}{\d t} = 0.
+    \frac{\d E_\phi}{\d t} = 0.\qedhere
   \]
 \end{proof}
 

III_M/advanced_probability.tex

@@ -3020,7 +3020,7 @@ \subsubsection*{Variational formulation}
 \end{enumerate}
 We will need Lagrangian multipliers to impose these constraints, and so the augmented Lagrangian is
 \[
-  \mathcal{G} = \frac{\bra \mathbf{q}_T, \mathbf{q}_T\ket}{\bra \mathbf{q}_0, \mathbf{q}_0\ket} - \int_0^T \bra \tilde{\mathbf{q}},  (\mathcal{D}_t - \mathcal{L}) \mathbf{q}\ket \;\d t + \bra \tilde{\mathbf{q}}_0, \mathbf{q}(0) - \mathbf{q}_0\ket.
+  \mathcal{G} = \frac{\bra \mathbf{q}_T, \mathbf{q}_T\ket}{\bra \mathbf{q}_0, \mathbf{q}_0\ket} - \int_0^T \bra \tilde{\mathbf{q}}, (\mathcal{D}_t - \mathcal{L}) \mathbf{q}\ket \;\d t + \bra \tilde{\mathbf{q}}_0, \mathbf{q}(0) - \mathbf{q}_0\ket.
 \]
 Taking variations with respect to $\tilde{\mathbf{q}}$ and $\tilde{\mathbf{q}}_0$ recover the evolution equation and the initial condition. The integral can then be written as
 \[
@@ -3072,7 +3072,7 @@ \subsubsection*{Variational formulation}
 
 When we set $p = 1$, i.e.\ we use the usual energy norm, then we get a beautiful example of the Orr mechanism, in perfect agreement with what SVD tells us. For, say, $p = 50$, we get more exotic center/wall modes. % insert pictures ?
 
-%Replacing the energy term in $\mathcal{G}$ with $\mathcal{J}$, we obtain 
+%Replacing the energy term in $\mathcal{G}$ with $\mathcal{J}$, we obtain
 %Then we have
 %\[
 %  \mathcal{M} = \mathcal{J} - \int_0^T \left\bra \mathbf{u}^\dagger, \left(\frac{\partial \mathbf{u}}{\partial t} + \mathbf{u} \cdot \nabla \mathbf{u} + \mathbf{u} \cdot \nabla \mathbf{U} + \nabla p - \frac{1}{Re} \nabla^2 \mathbf{u}\right)\right\ket - \bra \mathbf{p}^\dagger, \nabla \cdot \mathbf{u}\ket \;\d t - \bra \mathbf{u}^\dagger_0, \mathbf{u}_0 - \mathbf{u}(0)\ket.

III_M/analysis_of_partial_differential_equations.tex

@@ -2732,7 +2732,7 @@ \subsection{Multiple extensions}
 \begin{prop}
   Let $M/L/K$ be finite extensions of local fields, and $M/K$ Galois. Then
   \[
-    G_s(N/K) \cap \Gal(M/L) = G_s(M/L).
+    G_s(M/K) \cap \Gal(M/L) = G_s(M/L).
   \]
 \end{prop}
 
@@ -3034,7 +3034,7 @@ \subsection{Multiple extensions}
     \frac{G^t(M/K)H}{H} &= \frac{G_{\psi_{M/K}(t)}(M/K) H}{H} \\
     &\cong G_{\eta_{M/L}(\psi_{M/K}(t))}(L/K)\\
     &= G_s(L/K)\\
-    &= G^t(L/K).
+    &= G^t(L/K).\qedhere
   \end{align*}
 \end{proof}
 

III_M/hydrodynamic_stability.tex

@@ -1218,7 +1218,7 @@ \subsection{Random variables}
   \[
     (\Omega, \mathcal{F}, \P) = ((0, 1), \mathcal{B}(0, 1), \text{Lebesgue}).
   \]
-  Given any sequence $(F_n)$ of distribution functions, there is a sequence $(X_n)$ of random variables with $F_{X_n} = F_n$ for all $n$.
+  Given any sequence $(F_n)$ of distribution functions, there is a sequence $(X_n)$ of independent random variables with $F_{X_n} = F_n$ for all $n$.
 \end{prop}
 
 \begin{proof}

III_M/local_fields.tex

@@ -58,9 +58,16 @@ Currently, the notes for the following subjects are available.
 
 ## Part III
 ### Michaelmas Term
+ - Advanced Probability (2017, M. Lis)
  - Algebraic Topology (2016, O. Randal-Williams)
+ - Analysis of Partial Differential Equations (2017, C. Warnick)
+ - Combinatorics (2017, B. Bollobas)
  - Differential Geometry (2016, J. A. Ross)
+ - Extremal Graph Theory (2017, A. G. Thomason)
+ - Hydrodynamic Stability (2017, C. P. Caulfield)
  - Local Fields (2016, H. C. Johansson)
+ - Modern Statistical Methods (2017, R. D. Shah)
+ - Percolation and Random Walks on Graphs (2017, P. Sousi)
  - Quantum Computation (2016, R. Jozsa)
  - Quantum Field Theory (2016, B. Allanach)
  - Symmetries, Fields and Particles (2016, N. Dorey)
@@ -78,5 +85,8 @@ Currently, the notes for the following subjects are available.
  - Classical and Quantum Solitons (2017, N. S. Manton and D. Stuart)
 
 ## Part IV
+### Michaelmas Term
+ - Topics in Geometric Group Theory (2017, H. Wilton)
+
 ### Easter Term
  - Bounded Cohomology (2017, M. Burger)