1
00:00:00,199 -> 00:00:03,238
Announcer: Previously on
AMC's The Walking Dead.
2
00:00:03,358 -> 00:00:05,859
Rick: That exit sends
them right at us.
3
00:00:05,960 -> 00:00:07,561
We force them west here.
4
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Away from the com
5.
Brownian motion
Section 1. The de nition and some simple properties. Section 2. Visualizing Brownian motion. Discussion and demysti cation of some strange and scary pathologies. Section 3. The re ection principle. Section 4. Conditional distribution of
MAL 245: TUTORIAL SHEET 8
Abstract. bounded linear functionals, operators, Hahn-Banach Theorem, dual.
Part A
Introduction to Topology and Modern Analysis, by G.F. Simmons.
Exercises 47.1,2,3,6,7; 48.1, 2; 49.2,3
Part B
(1) Let T : B C be a linear mapping
MAL 245: TUTORIAL SHEET 7
Abstract. normed linear spaces, Banach spaces.
Part A
Introduction to Topology and Modern Analysis, by G.F. Simmons.
Exercises 42.1,2; 46.1,2,3; 14.1,2,3,4; 15.1,2,3,4.
(1)
(2)
(3)
(4)
(5)
(6)
Part B
Let x 1 and x 2 be norms on a
MAL 245: TUTORIAL SHEET 6
Abstract. regular spaces, normal spaces.
(1) Prove or disprove: If there exists continuous map from X onto
Y and X is regular, normal; then so is Y .
(2) Construct a non-Hausdor regular topology on X = cfw_a, b, c.
(3) On reals,
MAL 245: TUTORIAL SHEET 5
Abstract. Locally compact and compactications.
Part A
Introduction to Topology and Modern Analysis, by G.F. Simmons.
Exercises 23.3,4;
Part B
(1) Prove or disprove: R and Q are locally compact.
(2) Every topology on X = cfw_a, b,
MAL 245: TUTORIAL SHEET 4
Abstract. compactness, connectedness.
(1) Which of the following subsets of R are compact:
1
cfw_1 n : n N
[0, 1] [3, 4]
rationals in [0, 1]
cfw_x : f (x) = 0, f : R R is continuous
cfw_x : x f 1 [0, 1], f : R R is continuou
MAL 245: TUTORIAL SHEET 2
Abstract. Topology, basis, subbasis, Hausdor spaces, products.
Part A
Introduction to Topology and Modern Analysis, by G.F. Simmons.
Exercises 16.6,7; 17.8; 18.3, 6, 7, 8
Part B
I. State Whether The Following Statements Are TRUE/
MAL 245: TUTORIAL SHEET 1
Abstract. metric spaces, Cauchy sequences, complete metric spaces, continuity, uniform
continuity, dense subsets, Baire Category.
Part A
Introduction to Topology and Modern Analysis, by G.F. Simmons.
Exercises 12.1,2,7; 13.6, 7,
2
Brownian Motion
We begin with Brownian motion for two reasons. First, it is an essential ingredient in the denition of the Schramm-Loewner evolution. Second, it is a relatively simple example of several of the key ideas in the course - scaling limits, u
Optimization Methods in Finance
(EPFL, Fall 2010)
Lecture 8: Fundamental Theorem of Asset Pricing
10.11.2010
Lecturer: Prof. Friedrich Eisenbrand
Scribe: Parmeet Singh Bhatia
Financial background
In the last lecture we have been through basics of arbitrag
BROWNIAN MOTION
A tutorial
Krzysztof Burdzy
University of Washington
A paradox
f : [0,1] R,
sup | f (t ) | <
t[ 0 ,1]
P ( f (t ) < Bt < f (t ) + , 0 < t < 1)
11
2
c( ) exp ( f (t ) dt
2
0
(*)
(*) is maximized by f(t) = 0, t>0
The most likely (?!?) sha
4 Brownian motion
4.1 De nition of the process
Our discrete models are only a crude approximation to the way in which stock markets actually move. A better model would be one in which stock prices can change at any instant. As early as 1900 Bachelier, in
Brownian Motion
Draft version of May 25, 2008
Peter Mrters and Yuval Peres
o
1
Contents
Foreword
7
List of frequently used notation
9
Chapter 0. Motivation
13
Chapter 1. Denition and rst properties of Brownian motion
1. Paul Lvys construction of Brownian
Session 4 - Brownian Motion
Reading Assignment: Sections 4.1 - 4.5. (Correspondence with McDonald
Chapter 20, up to Page 665.)
4.0. Some basic stu:
The normal distribution:
The symbol, X N (a, b) stands for the statement, X has a
normal distribution wit
Stochastic Programming and Financial Analysis IE447 Lecture 6
Dr. Ted Ralphs
IE447 Lecture 6
1
Reading for This Lecture
C&T Chapter 4
IE447 Lecture 6
2
Derivative Securities
A derivative security is one whose price depends on the value of an underlying
Finite Element Method
The essence of the finite element method can be known
by considering various ways of representing a function
f(x) on an interval a x b. In the finite difference
method, the function is defined only on a set of grid
points, i.e., f(x
Numerical Techniques used in NWP
In Numerical Weather Prediction, the initial values of the
meteorological parameters are given at a time t and by
time stepping procedure the coupled equations are solved
to give the values at future time. The commonly use
Numerical Techniques used in NWP
In Numerical Weather Prediction, the initial values of the
meteorological parameters are given at a time t and by
time stepping procedure the coupled equations are solved
to give the values at future time. The commonly use
M is eliminated by the initial condition. The 2nd
constant E will be determined in the first time step
in the following way. The simplest procedure is to
take a forward step in time while retaining a
F
centered difference for x .
Fm + 1,0 Fm 1,0
Fm,1 Fm,0
Numerical Method
Finite difference scheme
Taylors series
x 3
x 2
+ f ' ' ' (x)
+ .(1)
f(x + x) = f(x) + f ' (x) x + f ' ' (x)
3!
2!
x 2
x 3
f ' ' ' (x)
+ .(2)
f(x x) = f(x) f ' (x) x + f ' ' (x)
2!
3!
where x > 0
f(x + x) f(x)
+R
f ' (x) =
from (1),
x
f(
Numerical Method
Finite difference scheme
Taylors series
x 3
x 2
+ f ' ' ' (x)
+ .(1)
f(x + x) = f(x) + f ' (x) x + f ' ' (x)
3!
2!
x 2
x 3
f ' ' ' (x)
+ .(2)
f(x x) = f(x) f ' (x) x + f ' ' (x)
2!
3!
where x > 0
f(x + x) f(x)
+R
f ' (x) =
from (1),
x
f(
Solenoidal term in the
circulation theorem
dp
If the fluid is barotropic (density a function of pressure alone) this term
becomes the closed line integral of an exact differential, and so is zero. This
argument is good for any given moment at which the fl
Twisting or tilting term: Second term in the RHS of equation (3)
represents vertical vorticity which is generated by the tilting of
horizontally oriented components of vorticity into the vertical by a non
uniform vertical motion field.
Solenoidal term: I
Use of Vorticity and Divergence equations
in NWP
The horizontal components of wind u&v are not
pure scalars due to nearly spherical shape of the
earths surface and singularities at the poles.
Those are pseudo-scalars.
Whereas vertical component of vortici
Relationship between circulation and vorticity (vertical component )
rr
lim U kdA
rr
U A0 A
=k
A
rr
r
Use Stokes theorem U kdl = U dl
A
r
lim U dl
C
= A0
= lim
A0 A
A
Physical interpretation of Vorticity
Circulation about the infinitesimal contour
V
C =
Relationship between circulation and vorticity (vertical component )
rr
lim U kdA
rr
U A0 A
=k
A
rr
r
Use Stokes theorem U kdl = U dl
A
r
lim U dl
C
= A0
= lim
A0 A
A
Physical interpretation of Vorticity
Circulation about the infinitesimal contour
V
C =