MATH 425/525 - University of Arizona - Module 1
1
MATH 425/525 A
U NIVERSITY OF A RIZONA
M ODULE 1
IN THIS MODULE we begin the study of the real number system. The concepts discussed here
will be used throughout the course.
SECTION 1 deals with the axiom

SIE 550 (Linear) Systems Theory
Homework #5 Due date: Thursday, April 21, 2016
Problem 1: Textbook (Linear System Theory) Chapter 6, Problem 1 (page 322)
Problem 2: Textbook (Linear System Theory) Chapter 6, Problem 2 (page 322)
Problem 3: Textbook (Linea

SIE 550 (Linear) Systems Theory
Homework #1 Due date: Monday, February 1, 2016
Preliminaries: In , we define the canonical basis to be the basis formed by the following n
vectors
= [1,0, . ,0]
= [0,1, . ,0]
= [0,0, . ,1]
Problem 1: Given the vectors in

SIE 550 (Linear) Systems Theory
Homework #3 Due date: Monday, March 7 2016
Problem 1: Consider the following non-linear autonomous system
1 = 2 3 + 1
cfw_ 2 = 1 3 2
3 = 32 (1 3 )
1. Show that the system has a unique equilibrium point
2. Using linearizat

SIE 550 (Linear) Systems Theory
Homework #6 Due date: Wednesday, May 4, 2016
Problem 1: Textbook (Linear System Theory) Chapter 9, Problem 6 (page 458). Confirm your
hand calculation via MATLAB (use place or acker command depending on the situation).
Assu

SIE 520 Homework 1
due Friday, Feb 12, at the beginning of the class
1. Let X and Y be two random variables and define
X Y = mincfw_X, Y ,
and X Y = maxcfw_X, Y .
Show that, analogous to probability law P(A B) = P(A) + P(B) P(A B), we have
E(X Y ) = EX +

SIE 550 (Linear) Systems Theory
Homework #2 Due date: Thursday, February 18 2016
Problem 1: Consider the following Linear Time-Independent (LTI) system:
=
(0) = 0
Where the dynamics is characterized by the following matrix
2
0
A=
0
0
1
2
0
0
0
0
3
0
0
0

SIE 550 (Linear) Systems Theory
Homework #4 Due date: Tuesday, April 5, 2016
Problem 1: Consider the following non-linear autonomous system
cfw_
1 = |1 |0.5 (1 ) 15 + 2
2 = |2 |0.5 (2 ) 25 1
Study the Finite-Time Stability (FTS) of the origin and answer

SIE 520 Stochastic Modeling
Homework 2
due Wednesday, March 9, at the beginning of the class
1. Consider a Markov chain cfw_ X n : n 0 with state space S and transition
matrix P. Suppose that p ii > 0 and let i represent the exit time from
state i, i.e.,

SIE 520 Homework 3
due Friday, Apr 25, at the beginning of the class
1. For a pure renewal process cfw_N (t) : t 0 with inter-renewal times 1 , 2 , . . . , and 0 <
E[1 ] < , prove the following:
(a) N (t) = 1 + maxcfw_n : Sn t.
(b) Sn = as n goes to infin

SIE 520 Stochastic Modeling
Homework 4
due Monday, May 2, at the beginning of the class
1. Consider a two-server system where interarrival times are iid exponential
with parameter . Service time at server i is exponential with parameter
i , i = 1, 2, and