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 + + a2 x2 + a1 x + a0 where
a0 , a1 , . . . , an Z. Prov
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 principle. Let A2 = A1 \cfw_e1 and e2 to
be the smallest
MarkIII
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All Aboard Florida
Ridership and
Revenue Study
Prepared for:
Prepared by:
May 7, 2015
All Aboard Florida
Ridership and Revenue Study
This Report was prepared by The Louis Berger Group, Inc. (LBG) for the benefit of Florida East Coast Industries, LLC (Clie
Introduction to Difference Equations
Allan Brunner
Macroeconometrics
Johns Hopkins University
Overview
Time-Series Models
Difference Equations and Their Solutions
Brute Force: Solution by Iteration
2
A. Time-Series Models
3
Macro-Econometrics
This cou
All Aboard Florida
Ridership and
Revenue Study
SUMMARY REPORT
Prepared for:
Prepared by:
Proprietary and Confidential
September 2013
All Aboard Florida
Ridership and Revenue Study - Summary Report
This Summary Report was prepared by The Louis Berger Group
All Aboard Florida
Final Environmental Impact Statement and Section 4(f) Determination Appendices
Appendix 3.2.1-B
Ridership and Revenue Study (2015)
Appendices
All Aboard Florida
Ridership and
Revenue Study
Prepared for:
Prepared by:
May 7, 2015
All Aboa
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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 number:
Let x be a real number, we x a Cauchy sequence
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 , . . . , an Z. Prove that the set of algebraic numbers i
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 the following statements/ denitions:
a) A set is open i
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 an interval containing the point aj and the length
1
of
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)dx
|f (x)|dx.
a
a
x
Problem 4: Let F (x) = a f (t)dt th
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 derivatives, there
must be an x0 between x1 and x2 such
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). Then, take m = maxcfw_m1 , m2 . When |x x0 | < m, we kno
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 subsequence converging to limxx+ f (x), the limit of f (x)
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 many of them contains a point of A.
Now, assume that the i
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 of the US PPI (unknown DGP)
Forecasting with ARMA Mode