Partial Differential Equations-APM346
Term Test- Winter 2017
Problem 1. If the velocity of a one-dimensional fluid is v = x where (t, x) is the density
of the fluid, write the continuity equation and obtain the implicit solution if (0, x) = x.
Show that t
APM236
Tutorial #3 Polyhedra & friends
January 25, 2017
1. For each of the sets below, (a) find the extreme points if there are any, (b) decide if
its a polyhedron or not, and (c) decide if its a convex set or not.
(a) , Z, R, the closed ball B1 (0)
(b) t
APM236
Tutorial #2 Convex costs
January 18, 2017
1. The function f : R R is defined by the following graph:
6
5
4
3
2
1
1
2
3
4
5
6
(a) Express f (x) as a maximum of three functions.
(b) Suppose that d Rn is a fixed vector. Formulate the following problem
University of Toronto
APM236 H1S: Applications of linear programming
2017 Course outline
Welcome to APM236! The main objectives of this course are to train you to:
produce and critically analyze mathematical formulations of simple physical problems.
und
APM236
Tutorial #3 Extreme points & convex combinations
October 12 14, 2016
1. Find the extreme points of the sets below, and decide which ones are polyhedra. Which
of these sets has the property that the convex hull of its extreme points is exactly the
s
APM236
Tutorial #1 Standard form
September 21 23, 2016
1. Write the following linear programs in standard form.
(a) minimize
subject to
(b) maximize
subject to
x y 7z
y+z 1
z9
(c) maximize
subject to
x+y
|x| 1
|y| 3
7x + 10y + z
3x 2y = 2
2y + 13z 8
2z3
y
APM236 study checklist
Disclaimer: This is not a list of things that will be on your test. It is just a handy guideline
to help you gauge what sorts of things you might be able to do to study more effectively.
I have read the assigned sections of B&T and
APM236
Practice Problems #4
October 24 28, 2016
1. B&T exercises 2.4, 2.10, 2.13, 2.14, 2.16, 3.1, 3.2, 3.3, 3.5.
2. In the proof of Theorem 2.7, we set things up so that x = y + (1 )z for some [0, 1]
and some y, z P \ cfw_x . We then obtained the equatio
APM236
Tutorial #2 Convex costs
September 28 30, 2016
1. The function f : R R is defined by the following graph:
6
5
4
3
2
1
1
2
3
4
5
6
(a) Express f (x) as a maximum of three functions.
(b) Let A be an mn matrix, and suppose that b Rm and d Rn are fixed
APM236
Practice Problems #0 Review
September 12 16, 2016
1. Find all solutions to the following linear systems.
(a) x
+ y + 3z = 5
3x 2y + 5z + u = 1
(b)
2x y + 4z = 11
x + y 3z + 2u = 2
y+ z =3
(c)
(e)
(g)
6x + y 4z + 3u = 7
+ 2y + z = 1
(d) x + y 3z + u
APM236
Practice Problems #3
October 17 21, 2016
1. Here are some extra practice problems for you in the aftermath of test 1.
(a) Consider the LPP of optimizing c> x over the polyhedron P = cfw_x Rn : Ax b.
Show that the set S = cfw_c> x : x P of values o
APM236
Practice Problems #2
September 26 30, 2016
1. Bertsimas & Tsitsiklis 1.7 : #1.3, 1.4, 1.10, 1.14, 1.20 and 2.10 : #2.1, 2.2.
2. Let r > 0. Prove that the closed ball Br (x) = cfw_y Rn : kx yk r is a convex set.
3. Show that a hyperplane in Rn is co
APM236
Practice Problems #1
September 19 23, 2016
1. Convert all of the following problems to standard form using matrix notation.
(a) maximize
x + 2y
subject to
3x + 4y 12
(d) minimize
subject to
x, y 0
(b) minimize
subject to
2x1 + 3x2 + x3
x1 + x2 + x3
Problem Sheet 5
APM 384
Autumn 2014
On this sheet, all questions are assessed. Exercises 1-2 are worth 5 points, exercises
3-4 are worth 10 points. Solutions are due in the lecture on Monday 10 November.
1. Consider the partial differential equation
(x)
Problem Sheet 6
APM 384
Autumn 2014
On this sheet, all questions are assessed and have equal weight. Solutions are due
in the lecture on Monday 17 November.
1. Consider non-uniform heat equation
u
u
=
K0 (x)
c(x)(x)
t
x
x
(1)
on the interval [0, L], subje
Problem Sheet 7
APM 384
Autumn 2014
On this sheet, questions 1 - 4 and 8 are assessed and have equal weight. Solutions
are due before the lecture on Friday 28 November.
1. Consider the boundary value problem
u(x, y) = f (x, y)
u
(x, 0) = 0.
y
if y > 0
(a)
APM384
Questions:
1) Existence?
[2) Uniqueness?]
3) Properties
1
Introduction
Definition 1.1
The order of a differential equation is the highest derivative that appears.
2
Eg.
u u
+ =0
2
x t
(second order)
3
u u
=
3
x t
(3rd order)
Definition 1.2
A DE
APM 236H1F term test 1 announcement, 3 October, 2012
1. No calculators are allowed.
2. The test will have 3 questions of approximately equal weight. Total marks = 40.
3. The test will cover Kolman and Beck, all of chapters 0 and 1.
4. Our teaching assista
Credit Models
Exercise
KMV and Merton Model
Exercises and Examples
Portfolio Credit Risk III
Prof. Luis Seco
Prof. Luis Seco
University of Toronto
Department of Mathematics
Department of Mathematical Finance
July 31, 2011
Prof. Luis Seco
Portfolio Credit
A Worked-Out Example
Credit VaR
Goodrich-Rabobank Revisited
Portfolio Credit Risk II
Prof. Luis Seco
Prof. Luis Seco
University of Toronto
Department of Mathematics
Department of Mathematical Finance
July 31, 2011
Prof. Luis Seco
Portfolio Credit Risk II
Review of Basic Concepts
Goodrich-Morgan-Robabank Swap: A Fixed Rate Loan
Credit Loss
Portfolio Credit Risk
Prof. Luis Seco
Prof. Luis Seco
University of Toronto
Department of Mathematics
Department of Mathematical Finance
July 27, 2011
Prof. Luis Seco
Po
Option Pricing
OptionPricing
Prof. Luis Seco
University of Toronto
Department of Mathematics
Department of Mathematical Finance
July 23, 2011
Prof. Luis Seco (University of Toronto)
Option Pricing
July 23, 2011
1 / 26
Table of Contents
1
Example
2
Discoun
APM346 - Partial Dierential Equations - Winter 2016
Term Test 1 - February 4, 2016
Time allotted: 90 minutes
Aids permitted: None
Total marks: 50
Full Name:
Last
First
Student Number:
Email:
Instructions
@mail.utoronto.ca
(READ CAREFULLY)
DO NOT WRITE ON
APM346 - Partial Dierential Equations - Winter 2016
Term Test 1 - February 4, 2016
Time allotted: 90 minutes
Aids permitted: None
Total marks: 50
Full Name:
Last
First
Student Number:
Email:
Instructions
@mail.utoronto.ca
(READ CAREFULLY)
DO NOT WRITE ON
Time is money
The theory of interest rates
The issuing
entity
Coupons are
paid every 6
months. The
amount
represents
the total for
one year
On this date,
the issuer
pays $100 to
the holder of
the bond
Price = - (annual coupon rate)
n
365
+ pi (1 + r )
cle
Risk Management
Bank Risks
Market Risk. The risk in reducing the value of the Banks positions due
to changes in markets.
Credit Risk. The risk in reducing the value of the Banks assets due to
changes in the credit quality of the counterparties.
Counterpar
The Business Case
I am holding a 15-year treasury zero on a $1,000 notional. What is my VaR?
(Note: this instrument does not exist)
1
The Model
No default risk
1 day VaR
95% condence
log-normally distributed yield
Present value: $370.
The yield chan