Solutions to practice problems for Midterm 1
1. Find the gcd of 621 and 483.
Solution: We run the Euclidean algorithm:
621 = 1 483 + 138
483 = 3 138 + 69
138 = 2 69.
So gcd(621, 483) = 69.
2. Find a solution of 621m + 483n = k, where k is the gcd of 621 a
E.1
Normality of Linear Continua
is normal in the order topology.
X
Every linear continuum
Theorem E.1.
For if
and no smallest element.
form a new ordered set
(,1)
by taking the disjoint union of
Y
X, and
and
X.
to be less than every element of
(,1)
decla
6.890: Algorithmic Lower Bounds
Fall 2014
Prof. Erik Demaine
Problem Set 1
Due: Monday, September 22nd, 2014
Problem 1. For each of the following problems, either show that the problem is in P by giving a
polynomial-time algorithm (e.g., by reducing to sh
C.1
The Long Line
We follow the outline of Exercise 12 of 24.
Let
denote the set
L
denote the smallest element of
Let o
Lemma C.1.
[o 0 X 0, ckX 0]
interval
S
of
has the order type of
L
0 in 5
S_.
Sppose the lemma holds for all
We show it holds for 3.
.
4
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Mechanical Engineering
2.004 Dynamics and Control II
Fall 2007
Problem Set #2
Posted: Friday, Sept. 21, 07
Solution
1. In class, we showed in two dierent ways that the torque constant of a DC motor
equal
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Mechanical Engineering
2.004 Dynamics and Control II
Fall 2007
Problem Set #1
Posted: Friday, Sept. 14, 07
Solution
1. For each one of the following systems, argue if in your opinion it is openloop
or cl
Solutions to Problem Set
18.904 Spring 2011
Problem
Statement Let n 1 be an integer. Let CPn denote the set of all lines in Cn 1 passing through
the origin. There is a natural map : Cn 1 \ cfw_0 ! CPn taking a point to the line it spans. We
give CPn the q
2.094
FINITE ELEMENT ANALYSIS OF SOLIDS AND FLUIDS
SPRING 2008
Homework 9
Instructor:
Assigned:
Due:
Prof. K. J. Bathe
04/17/2008
04/24/2008
Problem 1 (10 points):
Complete Exercise 7.4 in the textbook, page 660, but consider only steady-state conditions.
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
2.111J/18.435J/ESD.79
Quantum Computation
Problem 1. Prove that the Fredkin gate is universal. A Fredkin gate accepts three input
bits A, B, and C, and gives three output bits D, E, and F in the following way:
D=A, E=
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
DEPARTMENT OF MECHANICAL ENGINEERING
CAMBRIDGE, MASSACHUSETTS 02139
2.29 NUMERICAL FLUID MECHANICS FALL 2011
QUIZ 2
The goals of this quiz 2 are to: (i) ask some general higher-level questions to ensure you understand
D 1
Countability axioms
We have studied four basic countability properties:
(1)
The first countability axiom.
(2) The second countability axiom,
(3)
(4)
The Lindel6f condition.
tse condition that the space has
a countable dense subset.
We know that condit