110.405 Spring 2013
Practice Midterm
1.
(a) State the denition of a countable set.
(b) A real number x is called algebraic if it is a root of a polynomial with
integer coecients that is p(x) = an xn +
110.405 Analysis
HW 1
Solutions
Section 1.2.3
Exercise 1: Let E N which is proper (i.e. E = N) and not nite. We dene A1 = E and e1 to
be the smallest integer in A1 . Such e1 exists by well-ordering pr
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All Aboard Florida
Ridership and
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All Aboard Florida
Ridership and Revenue Study
This Report was prepared by The Louis Berger Group, Inc. (LBG) for
Introduction to Difference Equations
Allan Brunner
Macroeconometrics
Johns Hopkins University
Overview
Time-Series Models
Difference Equations and Their Solutions
Brute Force: Solution by Iteration
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Ridership and
Revenue Study
SUMMARY REPORT
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Proprietary and Confidential
September 2013
All Aboard Florida
Ridership and Revenue Study - Summary Report
Thi
All Aboard Florida
Final Environmental Impact Statement and Section 4(f) Determination Appendices
Appendix 3.2.1-B
Ridership and Revenue Study (2015)
Appendices
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Ridership and
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110.405 Analysis
HW 2
Solutions
Section 2.1.3
Problem 1: We can show this by establishing an injection from the power set of the natural
numbers into the set of Cauchy sequences converging to a given
110.405 Analysis
HW 3
Solutions
Section 2.2.4
Problem 12: This follows from repeated iterations of Denition 2.2.2 on page 41: We know that
if x1 , x2 , . . . represents x, then x1 x1 , x2 x2 , . . . r
110.405 Analysis
HW 4
Solutions
Section 3.1.3
Problem 5:. This follows almost immediately from the denitions:
lim supcfw_xn + yn = lim
j
lim
j
= lim
j
supcfw_xn + yn
n>j
supcfw_xn + supcfw_yn
110.405 Spring 2013
Practice Midterm
Solutions
1. A real number x is called algebraic if it is a root of a polynomial
with integer coecients that is p(x) = an xn + + a2 x2 + a1 x + a0
where a0 , a1 ,
110.405 Analysis
Spring 2013 Midterm
Name:
Let R, Z, N denotes the set of Real numbers, Integers, Natural numbers respectively.
Show all your work and fully justify your answers.
1. [20 pts] Complete
HW 10 Solutions
Section 6.2.4
Problem 6
Again in this problem it is implicitly assumed that f and g are bounded functions. Let a1 , a2 , . . . , aN
be the points for which f (aj ) = g (aj ) and Ij be
HW 9 SOLUTIONS
Section 6.1.5
Problem 2: Since f is continuous by Theorem 4.2.1 |f | = max(f, f ) is also
continuous. Moreover, since
|f (x)| f (x) |f (x)|
for every x we have
b
b
|f (x)|dx
a
b
f (x)d
HW 8 Solutions
Section 5.3.4
Problem 5: Assume otherwise: there exists at least one point x1 where f (x1 ) > 0 and at least
one point x2 where f (x2 ) < 0. Then, by the Intermediate Value Theorem for
HOMEWORK 7 SOLUTIONS
Section 5.1.3
Problem 2: Use the denition of big Oh to nd C1 > 0 and m1 such that
1
|f (x)| < C1 |x x0 |k when |x x0 | < m . Similarly, nd the corresponding C2 and
m2 for g (x). T
HW 6 SOLUTIONS
Section 4.1.5
Problem 6: We know that each element xk in the sequence is either greater
than x0 or less than x0 . If there are innitely many terms greater than x0 , this
gives us a subs
110.405 Analysis
HW 5
Solutions
Section 3.3
2. Let A be a compact set. We will show that A has nite intersection property: let B be a
collection of closed sets such that intersection of any nitely man
ARMA ModelsModel Selection
Allan Brunner
Macroeconometrics
Johns Hopkins University
Overview
Estimation and Model Selection
Some Simulation Results
Some Pitfalls of Estimating ARMA Models
A Model