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.
(ii) Fi
Honors Algebra
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
Solutions
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
Honors Algebra
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
Honors Algebra
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.
123
123
123
123
Solution: The ips
,
Honors Algebra
Assignment 1
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
more dicult.
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
multipl