MATH-UA 263 PARTIAL DIFFERENTIAL EQUATIONS
HOMEWORK 4 (DUE 10/9)
(1) (a) Consider the eigenvalue problem with Robin BCs at both ends:
X = X
X (0) a0 X(0) = 0, X (l) + al X(l) = 0
(i) Show that = 0 is an eigenvalue if and only if a0 + al = a0 al l.
Assignment 4 Solutions
1. In this problem, assume (important!) that G is an Abelian. Set H =
cfw_g G : g5 = e. (Warning: Expressions such as x1/5 are not well
dened. Do not use them!)
(a) Show H is a subgroup of G. Point out where the assum
Algebra , Assignment 6
1. Let : G G be an automorphism of G. Let x, y G with y = (x).
(a) Assume xs = e. Prove y s = e.
Solution:y s = (x)s = (xs ) = (e) = e
(b) Assume xs = e. Prove y s = e.
Solution:As above, y s = (xs ). An automorphism is in
Assignment 3 Solutions
1. Let v be any nonzero vector in Rn and set
H = cfw_A GLn (R) : Av = v for some positive
Prove that H is a subgroup of GLn (R).
There are three parts:
Identity: Iv = v so I H (with = 1)
Multiplication: If A, B H then
Assignment 2 Solutions
1. In S3 (reminder, this is our standard notation for the permutations
on cfw_1, 2, 3) show that there are four elements x satisfying x2 = e and
three elements x satisfying x3 = e.
Solution: The ips
1. List with description the symmetries of the square. (There are eight
of them.) Give the table for the products of the symmetries. Give the
inverse of each symmetry.
Solution: (Please save this as this will be a standard exam
HONORS ALGEBRA MIDTERM
Do all problems. Maximal Score: 100. Problems marked with (*) are
Notations: Re is the reals, Z the integers, Q the rational numbers. Re
is the nonzero reals under multiplication. Re+ is the positive reals under