1 5th October Exercise 1.2.3(c) if has exactly one solution, then has exactly one solution, if has innitely many solutions, then has innitely many solutions, and if has no solutions, then has no solutions Proof of Lemma 1.4.2 replace x by y Exercise 2.1.4
Complex Methods Course P3
T. W. Krner o September 18, 2007
Small print The syllabus for the course is dened by the Faculty Board Schedules (which are minimal for lecturing and maximal for examining). I should very much appreciate being told of any correct
CORRECTIONS TO A COMPANION
T.W.KORNER
This rst correction page is based on a long and insightful lists from Bob Burckel and Eric Lw. I should also like to thank Marit Sandstad, Norton Starr, Wan-Teh Chang,Silas Davis and Daniel James for contributions. P
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FURTHER CORRECTIONS TO THE PLEASURES OF COUNTING
T. W. KORNER
Richard Hardwick points out that on page X, email and http have changed from pmms to dpmms. (TWK says, whoops! this should certainly have been corrected in the reprint, but was not.) He also p
201B, Winter 11, Professor John Hunter
Homework 8 Solutions
1. A sequence of bounded linear operators An B (H) on a Hilbert space H is
said to converge to an operator A H: uniformly if An A with respect
to the operator norm on B (H); strongly if An x Ax s
201B, Winter 11, Professor John Hunter
Homework 7 Solutions
1. Let H = L2 (0, 1) with the standard inner product
1
f, g =
f (x)g (x) dx.
0
Dene M : H H by
(M f )(x) = xf (x)
i.e. M is multiplication by x.
(a) Show that M is a bounded self-adjoint linear o
201B, Winter 11, Professor John Hunter
Homework 9 Solutions
1. If A : H H is a bounded, self-adjoint linear operator, show that
An = A
n
for every n N. (You can use the results proved in class.)
Proof.
Let A : H H be a bounded, self-adjoint linear operat
Midterm: Math 201B
Winter, 2011
1. Say if the following Fourier series represent functions or distributions in
C (T), C (T), L2 (T), or D (T):
f (x)
(1)n inx
e;
1 + n4
n=
g (x)
n=
1
2
ein x ;
1 + n2
4
en einx ;
h(x)
n=
4
en einx .
k (x)
n=
You may use
Practice Midterm problems: Math 201B
1. What can you say about the dierentiability of the functions with the
following Fourier series:
f (x)
1
einx ;
(1 + n4 )1/7
n=
ne|n| einx ;
g (x)
n=
h(x)
n=1
1 i2n x
e
.
3n
e.g. How many continuously derivatives c
201B, Winter 11, Professor John Hunter
Homework 6 Solutions
1. Let X be a (real or complex) linear space and P, Q : X X projections.
(a) Show that I P is the projection onto ker P along ran P .
Proof. To show that I P is a projection we need to show that
201B, Winter 11, Professor John Hunter
Homework 5 Solutions
1. Let Td = T T T denote the d-dimensional Torus.
(a) Show that B = ei n x : Zd is an orthogonal set in L2 (Td ) and give an
n
expression for the Fourier coecients f () of a function
n
f ()ei
201B, Winter 11, Professor John Hunter
Homework 1 Solutions
Please note that there are some remarks regarding this homework posted on
Professors Hunter webpage!
1. If 1 p < q < , show that Lp (T) Lq (T). Give an example of a function
in Lp (T) \ Lq (T).
P
Final Solutions: Math 201B
Winter, 2011
1. (a) Show that there is a unique solution G D (T) of the ODE
G + G = ,
where D (T) is the periodic delta-function supported at 0, and compute
the Fourier series of G.
(b) Dene the Sobolev space H s (T) for real nu
201B, Winter 11, Professor John Hunter
Homework 2 Solutions
1. If 1 p < , show the trigonometric polynomials are dense in Lp (T).
Proof. For this problem I am going to give two dierent approches of proving
this problem.
First proof is based on the result
201B, Winter 11, Professor John Hunter
Homework 3 Solutions
1. Suppose that
cn is a series of complex numbers with partial sums
n=0
n
sn =
ck .
k=0
The series is Borel summable with Borel sum s if the following limit exists:
s n xn
n!
x
s = lim e
x+
n=0
.
201B, Winter 11, Professor John Hunter
Homework 4 Solutions
1. Let D R2 be the unit disc and f C (D) a continuous function dened on
the unit circle D. Suppose that u : D R is a function u C 2 C (D) such
that
u = 0 in D
u = f on D
(a) Show that
max u = max
ON BECOMING A QUANT
MARK JOSHI
1. What does a quant do? A quant designs and implements mathematical models for the pricing of derivatives, assessment of risk, or predicting market movements. 2. What sorts of quant are there? (1) (2) (3) (4) (5) (6) Front
Central Spain August 1812
Foreign Oce London Gentlemen Whilst marching from Portugal to a position which commands the approach to Madrid and the French forces, my ocers have been diligently complying with your requests which have been sent by H.M. ship fr
How to Write a Part III Essay
T. W. Krner o Trinity Hall
These unocial notes replace an earlier set by Marj Batchelor which were becoming illegible through repeated photocopying. Many of the key pieces of advice are taken almost word for word from her not
Thoughts on the Essay Question
T. W. Krner o November 5, 2008
Small print The opinions expressed in this note are the authors own. Even the best advice (and there is no reason to suppose that the advice here is the best advice) does not apply to all peopl
Introduction to Functional Analysis Part III, Autumn 2004
T. W. Krner o October 21, 2004
Small print This is just a rst draft for the course. The content of the course will be what I say, not what these notes say. Experience shows that skeleton notes (at
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Plot synopsis of Jaws A group of so-called government funded experts whip up alarmist fears of a killer shark o the coast of Amity, a sea side town. Their goal is to destroy the local tourist industry, send Amity back to the dark ages and thus achieve the
Linear Analysis
T. W. Krner o January 8, 2008
Small print The syllabus for the course is dened by the Faculty Board Schedules (which are minimal for lecturing and maximal for examining). Several of the results are called Exercises. I will do some as part
In Praise of Lectures
T. W. Krner o October 31, 2004
The Ibis was a sacred bird to the Egyptians and worshippers acquired merit by burying them with due ceremony. Unfortunately the number of worshippers greatly exceeded the number of birds dying of natura
RESULTS IN FIRST PART OF METHODS AND CALCULUS
T.W.KORNER
Denition 1. Let an , a Rk . We say that an a as n , if given > 0, we can nd N ( ) such that |an a| < for all n > N ( ). Theorem 2. (i) If an a and bn b in Rk then an + bn a + b as n . (ii) If an a
A Practice Based M Level Course The idea that a degree was formally taken by the applicant showing himself competent for it, may be well illustrated from the quaint ceremony of admitting a Master in Grammar at Cambridge, as described by the Elizabethan Es
CORRECTIONS TO NAIVE DECISION MAKING
T.W.KORNER
This correction page (dated 4th February 2010) is based on corrections by John Haigh and Robert MacKay to whom many thanks Page 4, line -3 Reverse inequality sign. Page 5, line 2 Reverse inequality sign. Pa