MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Mechanical Engineering
2.004 Dynamics and Control II
Fall 2007
Problem Set #5
Solution
Posted: Friday, Oct. 19, 07
1. Modify the Laplacedomain KVL equation (2) to include the inductance. Argue
that (4),
I.1
Locally Euclidean Spaces
The basic objects of study in differential geometry are certain topological
spaces called manifolds.
Oe crucial property that manifolds possess is
that they are locally just like euclidean space.
Formally,
that for each such s
J.1
The Pruifer Manifold.
;The so-called Prufer mnanifold is a space that is locally 2-euclidean
and Hausdorff, but not normal.
In discussing it, we follow the outline of
Exercise 6 on p. 317.
Definition.
Le-t
A
be! the following subspace of
gx, y,)
A=
Gi
6 856 Randomized Algorithms
David Karger
Handout #19, November 8, 2002 Homework 10, Due 11/13
1. A ow in an undirected graph is a set of edge-disjoint paths from a source vertex s
to a sink vertex t. The value of the ow is the number of edge disjoint path
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Mechanical Engineering
2.004 Dynamics and Control II
Fall 2007
Problem Set #6
Solution
Posted: Friday, Oct. 26, 07
1. Starting with the Supplement to Lecture 12 (Supplement for short,) modify the
timedom
18.781 Practice Questions for Midterm 2
Note: The actual exam will be shorter (about 10 of these questions), in case you are timing yourself.
1. Find a primitive root modulo 343 = 73 .
Solution: We start with a primitive root modulo 7, for example 3. The
2.094
FINITE ELEMENT ANALYSIS OF SOLIDS AND FLUIDS
SPRING 2008
Quiz #1
Instructor: Prof. K. J. Bathe
Date:
04/01/2008
Problem 1 (10 points)
Calculate the nodal point forces corresponding to the surface loading on the axisymmetric element shown
(consider 1
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
2.111J/18.435J/ESD.79
Quantum Computation
2n 1
1
Problem 1. For the state =
n /2
2
(1)f (x )
x
1
x =0
g(x ) 2 , where g(x ) is a 1-1
function, find the partial trace 1 tr2 ( ) and calculate
n
+ 1 +
n
.
Problem 2. Fin
K.1
Compactly generated spaces
A space
Definition.
the following condition:
X
is said to be compactly generated if it satisfies
A set
for each compact subspace
C
A
of
is open in
X
if
An C
is open in
C
X.
Said differently, a space is compactly generated