MASSACHUSETTS INSTITUTE OF TECHNOLOGY
DEPARTMENT OF MECHANICAL ENGINEERING
CAMBRIDGE, MASSACHUSETTS 02139
2.29 NUMERICAL FLUID MECHANICS FALL 2011
QUIZ 1
Monday, October 24, 2011
The goals of quiz 1 are to: (i) ask some general higher-level questions to e
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H.1
Tychonoff via well-ordering,
We present a proof of the Tychonoff theorem that uses the well-ordering
theorem rather than Zorn's lemma.
Lemma H.1.
A be a collection of basis elements for the topology
Let
of the product space
If
of
yA
X
It follows the o
G.1
Normality of quotient spaces
For a quotient space, the separation axioms-even the
are difficult to verify.
ausdorff property-
We give here three situations in which the quotient
space is not only Hausdorff, but normal.
Theorem
G.1.
is normal, then
Pro
F.l
The separation axioms
We give two examples of spaces that satisfy a given separation
axiom but not the next stronger one.
Te first
is a familiar space,
and the second is not.
Teorem F.1. If
not normal.
Proof.
completel
of R J
is uncountable,
is of cou
2.094
FINITE ELEMENT ANALYSIS OF SOLIDS AND FLUIDS
SPRING 2008
Homework 10
Instructor:
Assigned:
Due:
Prof. K. J. Bathe
05/06/2008
05/13/2008
Problem 1 (10 points):
Consider page 683 in the textbook. Show that using the principle of virtual temperatures t
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Mechanical Engineering
2.004 Dynamics and Control II
Fall 2007
Problem Set #4
Posted: Friday, Oct. 5, 07
Solution
1. (a) Problem 21(a) from Nise textbook, Chapter 2 (page 113). (b) After you
nd the trans
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Mechanical Engineering
2.004 Dynamics and Control II
Fall 2007
Problem Set #3
Solution
Posted: Friday, Sept. 28, 07
1. A secondorder system has the step response shown below.1 Determine its transfer func
18.781: Solution to Practice Questions for Final Exam
1. Find three solutions in positive integers of |x2 6y 2 | = 1 by rst calculating the continued
fraction expansion of 6.
Solution: We have
1
6 = [2,
]
62
6+2
= [2,
]
2
1
2
= [2, 2,
] = [2, 2,
] = [2
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
2.111J/18.435J/ESD.79
Quantum Computation
Problem 1. For any unit vector j = ( jx , jy , jz ) we can define the following operator
j = jx X + jy Y + jz Z
which corresponds to a -radian rotation about j-axis.
(a) show
6 856 Randomized Algorithms
David Karger
Handout #23, Nov. 28th, 2002 Homework 13 (Last One), Due 12/6
Answers to some of these problems are in textbooks. Dont use them. There is no
collaboration permitted on this problem set.
1. We are going to analyze a