From 09dbcb32a4d53925915c7e3256c2251352f924e2 Mon Sep 17 00:00:00 2001 From: mark Date: Sun, 8 Oct 2023 20:50:15 -0700 Subject: [PATCH] Post-class edits --- Advanced/Random Walks/parts/0 random.tex | 12 +++- Advanced/Random Walks/parts/1 circuits.tex | 71 +++++++++++++------ Advanced/Random Walks/parts/2 equivalence.tex | 18 ++--- Advanced/Random Walks/parts/3 effective.tex | 10 +-- 4 files changed, 74 insertions(+), 37 deletions(-) diff --git a/Advanced/Random Walks/parts/0 random.tex b/Advanced/Random Walks/parts/0 random.tex index 2e881f7..10c2e70 100644 --- a/Advanced/Random Walks/parts/0 random.tex +++ b/Advanced/Random Walks/parts/0 random.tex @@ -6,8 +6,8 @@ We would like to compute the probability of our particle stopping at node $A$. \ \vspace{2mm} -In other words, we want a function $P(n): N \to [0, 1]$ that returns the probability that our particle stops at $A$, -where $N$ is the set of nodes in $G$. +In other words, we want a function $P: \text{Nodes} \to [0, 1]$ that maps each node of the graph +to the probability that our particle stops at $A$. \begin{center} \begin{tikzpicture} @@ -92,11 +92,17 @@ Find $P(x)$ in terms of $P(v_1), P(v_2), ..., P(v_n)$. \problem{} -How can we use \ref{oneunweighted} to find $P(n)$ for any $n$? +In general, how do we find $P(n)$ for any node $n$? \begin{solution} If we write an equation for each node other than $A$ and $B$, we have a system of $|N| - 2$ linear equations in $|N| - 2$ variables. + + \vspace{2mm} + + We still need to show that this system is nonsingular, but + that's outside the scope of this handout. This could + be offered as a bonus problem. \end{solution} \vfill diff --git a/Advanced/Random Walks/parts/1 circuits.tex b/Advanced/Random Walks/parts/1 circuits.tex index 82b3681..38cb0ec 100644 --- a/Advanced/Random Walks/parts/1 circuits.tex +++ b/Advanced/Random Walks/parts/1 circuits.tex @@ -1,10 +1,12 @@ \section{Circuits} An \textit{electrical circuit} is a graph with a few extra properties, -called \textit{current}, \textit{voltage}, and \textit{resistance}. +called \textit{current}, \textit{voltage}, and \textit{resistance}. \par +In the definitions below, let $X$ be the set of nodes in a circuit. + \begin{itemize}[itemsep=3mm] - \item \textbf{Voltage} is a function $V(n): N \to \mathbb{R}$ that assigns a number to each node of our graph. \par + \item \textbf{Voltage} is a function $V: X \to \mathbb{R}$ that assigns a number to each node of our graph. \par In any circuit, we pick a \say{ground} node, and define the voltage\footnotemark{} there as 0. \par We also select a \say{source} node, and define its voltage as 1. \par @@ -15,15 +17,18 @@ called \textit{current}, \textit{voltage}, and \textit{resistance}. \footnotetext{ In the real world, voltage is always measured \textit{between two points} on a circuit. Voltage is defined as the \textit{difference} in electrical charge between two points. - Here, all voltages are measured with respect to our \say{ground} node. + Hence, voltage is a function of two nodes. - This detail isn't directly relevant to the problems in this handout, so you mustn't worry about it today. \par - Just remember that the electrical definitions here are a significant oversimplification of reality. + \vspace{2mm} + + Note that this is different than current and resistance, which aren't functions + of two arbitrary nodes --- rather, they are functions of \textit{edges} + (i.e, two adjecent nodes). } - \item \textbf{Current} is a function $I(e^\rightarrow): N \times N \to \mathbb{R}$ that assigns a number to each - \textit{oriented edge} $e^\rightarrow$ in our graph. An \say{oriented edge} is just an ordered pair of nodes $(n_1, n_2)$. \par + \item \textbf{Current} is a function $I: X^2 \to \mathbb{R}$ that assigns a number to each + \textit{oriented edge} in our graph. An \say{oriented edge} is just an ordered pair of nodes $(n_1, n_2)$. \par \vspace{1mm} @@ -31,7 +36,7 @@ called \textit{current}, \textit{voltage}, and \textit{resistance}. Naturally, $I(a, b) = -I(b, a)$. - \item \textbf{Resistance} is a function $R(e): N \times N \to \mathbb{R}^+_0$ that represents a certain edge's + \item \textbf{Resistance} is a function $R: X^2 \to \mathbb{R}^+_0$ that represents a certain edge's resistance to the flow of current through it. \par Resistance is a property of each \textit{link} between nodes, so order doesn't matter: $R(a, b) = R(b, a)$. \end{itemize} @@ -50,10 +55,14 @@ the case! Any circuit obeys \textit{Ohm's law}, stated below: $$ V(a, b) = I(a,b) \times R(a,b) $$ -\note{ - $V(a, b)$ is the voltage between nodes $a$ and $b$. If this doesn't make sense, read the footnote below. \\ - In this handout, it will be convenient to write $V(a, b)$ as $V(a) - V(b)$. -} + +This handout uses two notations for voltage: two-variable $V(a, b)$ and one-variable $V(a)$. \par +The first represents the voltage between points $a$ and $b$, better reflecting reality (see the footnote below). +The second measures the voltage between $a$ and ground, and is more convenient to use in equations. +\textbf{Try to use the single-variable notation in your equations.} +Convince yourself that $V(a, b) = V(a) - V(b)$. + +\vfill \definition{Kirchoff's law} @@ -64,25 +73,46 @@ Formally, we can state this as follows: \vspace{2mm} -Let $x$ be a node in our circuit and $B_x$ the set of its neighbors. We than have +Let $x$ be a node in our circuit and $N_x$ the set of its neighbors. We than have $$ - \sum_{b \in B_x} I(x, b) = 0 + \sum_{b \in N_x} I(x, b) = 0 $$ which must hold at every node \textbf{except the source and ground vertices.} \par \hint{Keep this exception in mind, it is used in a few problems later on.} - \vfill \pagebreak +\begin{instructornote} + Be aware that some students may not be comfortable with these concepts from physics, + nor with the circuit notation on the next page. + + \vspace{2mm} + + It may be a good idea to give the class a quick lecture on this topic, + explaining the basics of electonic circuits and circuit diagrams. + + \vspace{2mm} + + Things to cover: + \begin{itemize} + \item All the definitions on the previous page, in detail. + \item What's an Ohm, an Amp, a Volt? + \item Measuring voltage. Why is $V(a, b) = V(a) - V(b)$? + \item What does the $\Omega$ in the picture below mean? + \item Circuit symbols in the diagram below. + \end{itemize} + + \vspace{2mm} + + You could also draw connections to the graph flow handout, + if the class covered it before. +\end{instructornote} - - - -Consider the circuit below. This the graph from \ref{firstgraph}, turned into a circuit by: +Consider the circuit below. \textbf{This the graph from \ref{firstgraph}}, turned into a circuit by: \begin{itemize} \item Replacing all edges with $1\Omega$ resistors \item Attaching a 1 volt battery between $A$ and $B$ @@ -107,7 +137,8 @@ It exists only to create a potential difference between the two nodes. \problem{} From the circuit diagram above, we immediatly know that $V(A) = 1$ and $V(B) = 0$. \par -What equations related to the currents out of $x$ and $y$ does Kirchoff's law give us? +What equations related to the currents out of $x$ and $y$ does Kirchoff's law give us? \par +\hint{Current into $x$ = current out of $x$} \vfill diff --git a/Advanced/Random Walks/parts/2 equivalence.tex b/Advanced/Random Walks/parts/2 equivalence.tex index fe6d070..9162825 100644 --- a/Advanced/Random Walks/parts/2 equivalence.tex +++ b/Advanced/Random Walks/parts/2 equivalence.tex @@ -22,12 +22,12 @@ $$ \problem{} Let $x$ be a node in a graph. \par -Let $B_x$ be the set of $x$'s neighbors, $w(x, y)$ the weight of the edge between nodes $x$ and $y$, and $W_x$ +Let $N_x$ be the set of $x$'s neighbors, $w(x, y)$ the weight of the edge between nodes $x$ and $y$, and $W_x$ the sum of the weights of all edges connected to $x$. We saw earlier that the probability function $P$ satisfies the following sum: $$ - P(x) = \sum_{b \in B_x} \biggl( P(b) \times \frac{w(x, b)}{W_x} \biggr) + P(x) = \sum_{b \in N_x} \biggl( P(b) \times \frac{w(x, b)}{W_x} \biggr) $$ \note{This was never explicitly stated, but is noted in \ref{weightedgraph}.} @@ -36,7 +36,7 @@ $$ Use Ohm's and Kirchoff's laws to show that the voltage function $V$ satisfies a similar sum: $$ - V(x) = \sum_{b \in B_x} \biggl( V(b) \times \frac{C(x, b)}{C_x} \biggr) + V(x) = \sum_{b \in N_x} \biggl( V(b) \times \frac{C(x, b)}{C_x} \biggr) $$ where $C(x, b)$ is the conductance of edge $(x, b)$ and $C_x$ is the sum of the conductances of all edges connected to $x$. @@ -44,16 +44,16 @@ where $C(x, b)$ is the conductance of edge $(x, b)$ and $C_x$ is the sum of the \begin{solution} First, we know that $$ - \sum_{b \in B_x} I(x, b) = 0 + \sum_{b \in N_x} I(x, b) = 0 $$ for all nodes $x$. Now, substitute $I(x, b) = \frac{V(x) - V(b)}{R(x, y)}$ and pull out $V(x)$ terms to get $$ - V(x) \sum_{b \in B_x} \frac{1}{R(x, b)} - \sum_{b \in B_x} \frac{V(b)}{R(x, b)} = 0 + V(x) \sum_{b \in N_x} \frac{1}{R(x, b)} - \sum_{b \in N_x} \frac{V(b)}{R(x, b)} = 0 $$ Rearranging and replacing $R(x, b)^{-1}$ with $C(x, b)$ and $\sum C(x, b)$ with $C_x$ gives us $$ - V(x) = \sum_{b \in B_x} V(b) \frac{C(x, b)}{C_x} + V(x) = \sum_{b \in N_x} V(b) \frac{C(x, b)}{C_x} $$ \end{solution} @@ -71,7 +71,7 @@ two problems. \problem{} Let $q$ be a solution to the following equations, where $x \neq a, b$. $$ - q(x) = \sum_{b \in B_x} \biggl( q(b) \times \frac{w(x, b)}{W_x} \biggr) + q(x) = \sum_{b \in N_x} \biggl( q(b) \times \frac{w(x, b)}{W_x} \biggr) $$ Show that the maximum and minimum of $q$ are $q(a)$ and $q(b)$ (not necessarily in this order). @@ -80,7 +80,7 @@ Show that the maximum and minimum of $q$ are $q(a)$ and $q(b)$ (not necessarily \vspace{2mm} - Since $q(x)$ is a weighted average of all $q(b), ~b \in B_x$, there exist $y, z \in B_x$ satisfying + Since $q(x)$ is a weighted average of all $q(b), ~b \in N_x$, there exist $y, z \in N_x$ satisfying $q(y) \leq q(x) \leq q(z)$. Therefore, none of these can be an extreme point. \vspace{2mm} @@ -104,7 +104,7 @@ and that $p(x) - q(x) = 0$ for every $x$. \note{Note that $p(x) - q(x) = 0 ~ \fo \begin{solution} The equations in \ref{generaleq} for $p$ and $q$ directly imply that $$ - [p - q](x) = \sum_{b \in B_x} \biggl( [p - q](b) \times \frac{w(x, b)}{W_x} \biggr) + [p - q](x) = \sum_{b \in N_x} \biggl( [p - q](b) \times \frac{w(x, b)}{W_x} \biggr) $$ Which are the equations from \ref{generaleq} for $(p - q)$. diff --git a/Advanced/Random Walks/parts/3 effective.tex b/Advanced/Random Walks/parts/3 effective.tex index 71b06b3..46e89bf 100644 --- a/Advanced/Random Walks/parts/3 effective.tex +++ b/Advanced/Random Walks/parts/3 effective.tex @@ -38,20 +38,20 @@ out of $A$ is equal to the current flowing into $B$. \problem{} Using Kirchoff's law, show that the following equality holds. \par Remember that we assumed Kirchoff's law holds only at nodes other than $A$ and $B$. \par -\note[Note]{As before, $B_x$ is the set of neighbors of $x$. Naturally, $B_B$ is the set of neighbors of $B$.} +\note[Note]{As before, $N_x$ is the set of neighbors of $x$.} $$ - \sum_{b \in B_A} I(S, b) = \sum_{b \in B_B} I(b, B) + \sum_{b \in N_A} I(A, b) = \sum_{b \in N_B} I(b, B) $$ \begin{solution} Add Kirchoff's law for all vertices $x \neq A$ to get $$ - \sum_{\forall x} \biggl( ~ \sum_{b \in B_x } I(x, b) \biggr) = 0 + \sum_{\forall x} \biggl( ~ \sum_{b \in N_x } I(x, b) \biggr) = 0 $$ This sum counts both $I(x, y)$ and $I(x, y)$ for all edges $x, y$, except $I(x, y)$ when $x$ is $A$ or $B$. Since $I(a, b) + I(b, a) = 0$, these cancel out, leaving us with $$ - \sum_{b \in B_A} I(A, b) + \sum_{b \in B_B} I(B, b) = 0 + \sum_{b \in N_A} I(A, b) + \sum_{b \in N_B} I(B, b) = 0 $$ \vspace{2mm} @@ -61,7 +61,7 @@ $$ \vfill -If we call this current $I_A = \sum_{b \in B_A} I(A, b)$, we can pretend that the box contains only one resistor, +If we call this current $I_A = \sum_{b \in N_A} I(A, b)$, we can pretend that the box contains only one resistor, carrying $I_A$ units of current. Using this information and Ohm's law, we can calculate the \textit{effective resistance} of the box.