CDS 140a Winter 2013 Homework 3 - Revision history

Macmardg at 21:05, 25 January 2013

2013-01-25T21:05:37Z

2013-01-24T03:12:15Z

← Older revision		Revision as of 03:12, 24 January 2013
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	\aligned		\aligned
	\frac{d\psi}{dt} &= \frac{1}{4 B^2 l_c}\left(\phi - \Phi_T(\psi) \right), \\		\frac{d\psi}{dt} &= \frac{1}{4 B^2 l_c}\left(\phi - \Phi_T(\psi) \right), \\
	\frac{d\phi}{dt} &= \frac{1}{l_c}\left( \~~Phi_c~~(\phi) - \psi + \frac{J}{8}		\frac{d\phi}{dt} &= \frac{1}{l_c}\left( \Psi_c(\phi) - \psi + \frac{J}{8}
	\frac{\partial^2 \Psi_c}{\partial \phi^2} \right), \\		\frac{\partial^2 \Psi_c}{\partial \phi^2} \right), \\
	\frac{dJ}{dt} &= \frac{2}{\mu + m} \left(		\frac{dJ}{dt} &= \frac{2}{\mu + m} \left(
	\frac{\partial \~~Phi_c~~}{\partial \phi} + \frac{J}{8}		\frac{\partial \Psi_c}{\partial \phi} + \frac{J}{8}
	\frac{\partial^3 \~~Phi_c~~}{\partial \phi^3} \right) J,		\frac{\partial^3 \Psi_c}{\partial \phi^3} \right) J,
	\endaligned		\endaligned
	</amsmath></center>		</amsmath></center>

2013-01-24T02:29:14Z

← Older revision		Revision as of 02:29, 24 January 2013
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	</li>		</li>

	<li> '''Perko, Section 2.2, problem 3''' <font color=blue>(if you have not had ACM 125a or CDS 201)</font>: Use the method of successive approximations to show that if $f(x,t$ is continuous in $t$ in some interval containing $t = 0$ and continuously differentiable in $x$ for all $x$ in some open set $E \subset {\mathbb R}^n$ ~~ccontaining~~ $x_0$, then there exists $a > 0$ such that the initial value problem		<li> '''Perko, Section 2.2, problem 3''' <font color=blue>(if you have not had ACM 125a or CDS 201)</font>: Use the method of successive approximations to show that if $f(x,t)$ is continuous in $t$ in some interval containing $t = 0$ and continuously differentiable in $x$ for all $x$ in some open set $E \subset {\mathbb R}^n$ containing $x_0$, then there exists $a > 0$ such that the initial value problem
	<center><amsmath>		<center><amsmath>
	\dot x = f(x, t), \qquad x(0) = x_0		\dot x = f(x, t), \qquad x(0) = x_0

2013-01-23T14:22:54Z

← Older revision		Revision as of 14:22, 23 January 2013
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	~~{{warning\|This homework set is still being written. Do not start working on these problems until this banner is removed.}}~~

	{{CDS homework		{{CDS homework
	\| instructor = R. Murray, D. MacMartin		\| instructor = R. Murray, D. MacMartin

2013-01-18T18:28:24Z

@@ Line 14: / Line 14: @@
 <ol>
-<li> '''Perko, Section 2.2, problem 5''': </li>
+<li>'''Perko, Section 2.2, problem 5''' <font color=blue>(only if you have had AM 125a or CDS 201)</font>: Let $V$ be a normed linear space.  If $T:V \to V$ satisfies
-<li> '''Perko, Section 2.3, problem 1''': </li>
+<center><amsmath>
-<li> '''Perko, Section 2.5, problem 4''': </li>
+\|T(u) - T(v) \| \leq c\|u - v \|
-<li> '''Perko, Section 2.5, problem 5''': </li>
+</amsmath></center>
 <li> Choose ''one'' of the following systems and determine all of the equilibrium points for
 the system, indicating whether each is a sync, source, or saddle. <br>

2013-01-18T18:04:13Z

← Older revision		Revision as of 18:04, 18 January 2013
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	router fills up. Consider the case where $N = 2$ (so that we have		router fills up. Consider the case where $N = 2$ (so that we have
	three states, $x_1$, $x_2$ and $b$) and take $b_{\text{max}} = 1$ Mb		three states, $x_1$, $x_2$ and $b$) and take $b_{\text{max}} = 1$ Mb
	and $c = 2$ Mb~~\per\second~~.		and $c = 2$ Mb/s.
	</li>		</li>
	</ol>		</ol>

2013-01-18T18:03:29Z

← Older revision		Revision as of 18:03, 18 January 2013
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	<center><amsmath>		<center><amsmath>
	\aligned		\aligned
	\frac{dx_i}{dt} &= -b \frac{x_i^2}{2} + (b_{\text{max}} - b), \nl		\frac{dx_i}{dt} &= -b \frac{x_i^2}{2} + (b_{\text{max}} - b), \qquad
	\frac{db}{dt} &= \Bigl( \sum_{i=1}^N x_i \Bigr) - c,		\frac{db}{dt} &= \Bigl( \sum_{i=1}^N x_i \Bigr) - c,
	\endaligned		\endaligned

2013-01-18T18:03:06Z

← Older revision		Revision as of 18:03, 18 January 2013
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	\endaligned		\endaligned
	</amsmath></center>		</amsmath></center>
	where $x_i \in \~~reals~~$, $i = 1, \ldots, N$ are the transmission rates for the		where $x_i \in {\mathbb R}$, $i = 1, \ldots, N$ are the transmission rates for the
	sources of data, $b \in \~~reals~~$ is the current buffer size of the		sources of data, $b \in {\mathbb R}$ is the current buffer size of the
	router, $b_{\text{max}} > 0$ is the maximum buffer size and $c > 0$		router, $b_{\text{max}} > 0$ is the maximum buffer size and $c > 0$
	is the capacity of the link connecting the router to the computers.		is the capacity of the link connecting the router to the computers.

2013-01-18T18:02:01Z

@@ Line 46: / Line 46: @@
 compressor and $J$ representing the amplitude squared of the first
 modal flow perturbation (corresponding to a rotating stall
-disturbance).  (Hint: There is more than one equilibrium point and not
+disturbance).
 </li>
 </ol>
 <hr>
 Notes:
 * The problems are transcribed above in case you don't have access to Perko.  However, in the case of discrepancy, you should use Perko as the definitive source of the problem statement.

2013-01-18T17:55:38Z

← Older revision		Revision as of 17:55, 18 January 2013
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	<li> '''Perko, Section 2.5, problem 5''': </li>		<li> '''Perko, Section 2.5, problem 5''': </li>
	<li> Choose ''one'' of the following systems and determine all of the equilibrium points for		<li> Choose ''one'' of the following systems and determine all of the equilibrium points for
	the system, indicating whether each is a sync, source, or saddle.		the system, indicating whether each is a sync, source, or saddle. <br>

	(a) Moore-Greitzer model: The Moore-Greitzer equations model		(a) Moore-Greitzer model: The Moore-Greitzer equations model
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	\frac{\partial^2 \Psi_c}{\partial \phi^2} \right), \\		\frac{\partial^2 \Psi_c}{\partial \phi^2} \right), \\
	\frac{dJ}{dt} &= \frac{2}{\mu + m} \left(		\frac{dJ}{dt} &= \frac{2}{\mu + m} \left(
	\~~pder~~{\Phi_c}{\phi} + \frac{J}{8}		\frac{\partial \Phi_c}{\partial \phi} + \frac{J}{8}
	\frac{\partial^3 \Phi_c}{\partial \phi^3} \right) J,		\frac{\partial^3 \Phi_c}{\partial \phi^3} \right) J,
	\endaligned		\endaligned