The buying and selling commission schedule shown in the table is from an online discount brokerage firm. Taking into consideration the buying and selling commissions in this schedule, find the annual compound rate of interest earned by each investment in Problems 95 - 98 . An investor purchases 200 shares of stock at $28 per share, holds the stock for 4 years, and then sells the stock for $55 a share.
The buying and selling commission schedule shown in the table is from an online discount brokerage firm. Taking into consideration the buying and selling commissions in this schedule, find the annual compound rate of interest earned by each investment in Problems 95 - 98 . An investor purchases 200 shares of stock at $28 per share, holds the stock for 4 years, and then sells the stock for $55 a share.
Solution Summary: The author calculates the annual compound rate of interest earned when an investor purchases 200 shares of a stock. The commission schedule represents the buying and selling of stocks from an online discount brokerage firm.
The buying and selling commission schedule shown in the table is from an online discount brokerage firm. Taking into consideration the buying and selling commissions in this schedule, find the annual compound rate of interest earned by each investment in Problems
95
-
98
.
An investor purchases
200
shares of stock at
$28
per share, holds the stock for
4
years, and then sells the stock for
$55
a share.
3. (a) Let A be an algebra. Define the notion of an A-module M. When is a module M
a simple module?
(b) State and prove Schur's Lemma for simple modules.
(c) Let AM(K) and M = K" the natural A-module.
(i) Show that M is a simple K-module.
(ii) Prove that if ƒ € Endд(M) then ƒ can be written as f(m) = am, where a
is a matrix in the centre of M, (K).
[Recall that the centre, Z(M,(K)) == {a Mn(K) | ab
M,,(K)}.]
= ba for all bЄ
(iii) Explain briefly why this means End₁(M) K, assuming that Z(M,,(K))~
K as K-algebras.
Is this consistent with Schur's lemma?
(a) State, without proof, Cauchy's theorem, Cauchy's integral formula and Cauchy's
integral formula for derivatives. Your answer should include all the conditions
required for the results to hold.
(8 marks)
(b) Let U{z EC: |z| -1}. Let 12 be the triangular contour with vertices at
0, 2-2 and 2+2i, parametrized in the anticlockwise direction. Calculate
dz.
You must check the conditions of any results you use.
(d) Let U C. Calculate
Liz-1ym dz,
(z - 1) 10
(5 marks)
where 2 is the same as the previous part. You must check the conditions of any
results you use.
(4 marks)
(a) Suppose a function f: C→C has an isolated singularity at wЄ C. State what it
means for this singularity to be a pole of order k.
(2 marks)
(b) Let f have a pole of order k at wЄ C. Prove that the residue of f at w is given
by
1
res (f, w):
=
Z
dk
(k-1)! >wdzk−1
lim
-
[(z — w)* f(z)] .
(5 marks)
(c) Using the previous part, find the singularity of the function
9(z) =
COS(πZ)
e² (z - 1)²'
classify it and calculate its residue.
(5 marks)
(d) Let g(x)=sin(211). Find the residue of g at z = 1.
(3 marks)
(e) Classify the singularity of
cot(z)
h(z) =
Z
at the origin.
(5 marks)
Chapter 3 Solutions
Finite Mathematics for Business, Economics, Life Sciences and Social Sciences
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