Solutions for Homework 4
1 Solution 1: The hour hand is a uniform oscillator with period T = 1 = 60 minutes. The minute hand is a uniform oscillator with period T = 1 minute. The two oscillators are described by 2 1 = 60 and 2 = 2
if time is measured in
Solutions for Homework 3
1. 3.4.16 a) The potential is x3 . 3 You could add a constant C, but that would not make any essential difference; see below. Here is the graph of V for r = 2, r = 0, and r = 2: V (x) = rx +
r=2 5 0 5 4 2 5 0 5 4 2
r=0 5 0 5 4 2
r
Solutions for Homework 1
1. 2.1.1 Fixed points are points where sin x = 0, that is, integer multiples of . 2.1.2 The greatest velocity to the right occurs where sin x is the greatest. That is, where x is /2 plus an integer multiple of 2. 2.1.3 a) x = sin
In this document:
1) covering points with disks
2) Ramsey number lower bound
3) Tournaments
4) Large bipartite subgraphs
5) Large independent sets (weak Turan theorem)
6) Large dominating sets
TO DO: hypergraph coloring
Treat this as a zero
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From Euler formula on planar graph E 3V 6, claim that at least 80% of vertices have degree less than 30.
Suppose by contradiction, if more than 20% of vertices have degree 30, then E (30 V 0.2)/2 = 3V (We have to divide
by 2 because in the smallest case
1. Saddle-node bifurcations in 1D:
problems 3.1.1 through 3.1.4
2. Pitchfork bifurcations in 1D:
problems 3.4.1 through 3.4.4
3. Nonuniform flows on the circle:
problems 4.3.5, 4.3.6
4. Classification of fixed points for linear systems:
5.2.3 through 5.2.
Solutions for the second midterm exam
1. (a) The rate at which people who are not immune get infected is proportional to x (the number of such people) and y (the number of sick people). That is Eq. (1). The number of sick people changes because health peo
Math 150, Fall 2009, Midterm Exam 2 The exam is due on Friday, November 20, at the start of class. When you hand in the exam, please keep this sheet. Please write your name on the paper you hand in, and sign. With your signature, you pledge that you have
Homework 8, Solutions page 285, problem 8.1.10: a) The average size S of the trees grows logistically, but with a carrying capacity that depends on the health of the forest: The carrying capacity is E KS , KE where KS denotes the equilibrium value of S wh
Solutions for Homework 7
3. The equations can be written like this: V I = 0 1/C 1/L 0 V I .
Armed with Math 38 knowledge (or equivalent), you can write down the solution explicitly: V (t ) = V (0) cos I (t ) = I (0) cos t LC + L t , I (0) sin C LC t C V (
Homework 6, Solutions a) = aL and = bK . Eq. (3) is obtained from (1) by dividing both sides by K . Eq. (4) is obtained from (2) by dividing both sides by L. b) (x , y ) is a xed point of (3), (4) if and only if x = 0 and y = 0, or or or 1 x y = 0 and y =
Homework 5, Solutions 3. p. 143, problem 5.2.13. a) As a two-dimensional linear system, the equations look like this: x = y, b k y = y x. m m The matrix is A= b) The trace of A is = and the determinant is b < 0, m 1 m 0m k b
k > 0. m This implies that the
Solutions for Homework 2
1. 3.1.1 The graph of f (x) = 1 + rx + x2 is a parabola, opening upwards. The minimum occurs where f (x) = 0, that is, at r x0 = . 2 At this point, the value of f is f (x0 ) = 1 r2 r2 r2 + = 1 . 2 4 4
If |r| < 2, then f (x0 ) > 0,
Mathematical Aspects of Data Analysis
Christoph B
orgers
Mathematics Department
Tufts University
Spring 2017
Part I
Dimension Reduction Via PCA
1. The principal direction of a cloud of
points in the plane
Think about n points in R2 .
1
0
-1
-1
0
1
CLOUD O