MATH 8401, Midterm 1
1. Let A be a n n real symmetric matrix and B a n n real symmetric
positive denite matrix. Consider the problem:
Maximize
Ax, x
for x Rn , x = 0,
B x, x
where , is the standard inner product in Rn . Suppose the maximum
is attained at
MATH 8401, Homework 8 (Due 12/11/2013)
Problems from the book: A Guide to Distribution Theory and Fourier Transforms
1. Problems from Chapter 1: 1, 2, 6, 7, 12. Note: In problem 12, smooth
curve is not dened, and some natural denitions of a smooth curve
m
Mathematical Modeling and Methods of Applied Mathematics
MATH 8401

Fall 2015
38
The smart grid and the promise of
demandside management
The next generation of DSM technologies will enable customers to make more
informed decisions about their energy consumption, adjusting both when they use
electricity and how much they use.
Brand
Mathematical Modeling and Methods of Applied Mathematics
MATH 8401

Fall 2015
IE 1101, Fall 2015, Homework 1
Instructions:
This homework is due on Sep 24 at the beginning of class.
You must turn in a hard copy at the beginning of class.
Your work must be legible and your answers must be clearly marked, e.g. by boxing in an answe
Mathematical Modeling and Methods of Applied Mathematics
MATH 8401

Fall 2015
Gasoline Tax Dec
2009
(per
liter in home
Country
currency)
0.318
Canada
0.812
France
0.862
Germany
0.777
Italy
61.840
Japan
0.584
Spain
0.703
UK
0.106
USA
Sources
Exchange
Rate Dec 2009
Gasoline Price
($US to buy
Dec 2009
one unit of
(per liter in
home
ho
MATH 8401, Takehome nal
1. For a vector in Cn , the norm in the following is the Euclidean norm
whereas for a matrix n n matrix A, it is the norm induced by this
vector norm:
A = max Av .
(1)
v =1
(a) Let be a diagonal matrix with complex entries, and le
Elementary Hilbert Space Theory
Yoichiro Mori
November 5, 2013
1
Orthogonal Complement, Riesz Representation
A Hilbert Space is an inner product space that is complete. Let H be a
Hilbert space and for f, g H , let f , g b e the inner product and
f, f .
f
MATH 8401, Homework 2 (Due 10/28/2013)
In the following problems, Pn (f ) refers to the Fourier partial sum dened
as follows. For f a function dened on T = R/2 Z (2 periodic functions),
Pn (f ) =
1
f (k)k , k = exp(ikx),
2
k n
where
f, g =
T
f gdx, f (k
MATH 8401, Homework 1 (Due 9/18/2013)
1. Problems from textbook:
(a) Problems from Section 1.2: 3, 9, 10
(b) Problems from Section 1.3: 2, 3, 4 (It may be better to attempt
some of these problems after Problem 4 below and read the example in p.2122 in th
MATH 8401, Homework 3 (Due 10/7/2013)
1. Consider the matrix:
A=
1a
01
where a = 0. Compute the condition number of the eigenvalues of A
with respect to perturbations in the (2, 1) component of this matrix
(i.e., the value 0 is perturbed in the above matr
MATH 8401, Homework 2 (Due 9/27/2013)
1. Problems from textbook:
(a) Section 1.4: 2, 3, 4.
(b) Section 1.5: 5, 6, 11.
2. For a m n matrix A denote by 1 (A) 2 (A) r (A) > 0 the
singular values of A.
(a) Let Tp be the set of all m n matrices of rank B . Sho
MATH 8401, Homework 6 (Due 11/15/2013)
1. Suppose uk is bounded and converges weakly to u and wk converges
strongly to w. Show that uk , wk converges to u, w .
2. Suppose uk is a bounded sequence converging weakly to u and uk
u . Then show that uk is in
MATH 8401, Homework 5 (Due 11/4/2013)
1. Section 3.1, problem 1, 2
2. Section 3.2, problem 2, 3a), 3c).
3. Section 3.4, problem 1a), 1b), 6.
4. Check that the right and leftshift operators in the notes are adjoints
of each other. Check the The Fredholm a