Math 490 Intro to Topology Basics
Injective - f(x)=f(y) x=y
Surjective - For all yY, xX s.t. f(x)=y.
Equivalence Relation - reflexivity, symmetry, transitivity
Order Relation - Comparability: xy x<y or y<x
Non-reflexivity: no x s.t. xx
Transitivity
Immedi
An Introduction to Stochastic Calculus
Haijun Li
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Department of Mathematics
Washington State University
Week 9
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An Introduction to Stochastic Calculus
Week 9
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Outline
1
The Stratonovich and Other Integrals
The Stratonovich I
An Introduction to Stochastic Calculus
Haijun Li
[email protected]
Department of Mathematics
Washington State University
Week 13
Haijun Li
An Introduction to Stochastic Calculus
Week 13
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Outline
1
Numerical Solutions
The Euler Approximation
The Milst
An Introduction to Stochastic Calculus
Haijun Li
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Department of Mathematics
Washington State University
Week 2-3
Haijun Li
An Introduction to Stochastic Calculus
Week 2-3
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Outline
1
Brownian Motion
Basic Properties
Processes Related
An Introduction to Stochastic Calculus
Haijun Li
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Department of Mathematics
Washington State University
Weeks 1-2
Haijun Li
An Introduction to Stochastic Calculus
Weeks 1-2
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Outline
1
Basic Concepts from Probability Theory
Notations
An Introduction to Stochastic Calculus
Haijun Li
[email protected]
Department of Mathematics
Washington State University
Week 11
Haijun Li
An Introduction to Stochastic Calculus
Week 11
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Outline
1
The Black-Scholes Option Pricing Formula
A Short Excu
An Introduction to Stochastic Calculus
Haijun Li
[email protected]
Department of Mathematics
Washington State University
Week 10
Haijun Li
An Introduction to Stochastic Calculus
Week 10
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Outline
1
Solving It Differential Equations via Stratonovich Ca
An Introduction to Stochastic Calculus
Haijun Li
[email protected]
Department of Mathematics
Washington State University
Week 5
Haijun Li
An Introduction to Stochastic Calculus
Week 5
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Outline
1
Martingales
Filtration
Martingale
Martingale Transform
An Introduction to Stochastic Calculus
Haijun Li
[email protected]
Department of Mathematics
Washington State University
Week 7
Haijun Li
An Introduction to Stochastic Calculus
Week 7
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Outline
1
The It Stochastic Integrals
Simple Processes
The Genera
An Introduction to Stochastic Calculus
Haijun Li
[email protected]
Department of Mathematics
Washington State University
Week 8
Haijun Li
An Introduction to Stochastic Calculus
Week 8
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Outline
1
The It Lemma: Stochastic Analogue of the Chain Rule
A S
An Introduction to Stochastic Calculus
Haijun Li
[email protected]
Department of Mathematics
Washington State University
Weeks 3-4
Haijun Li
An Introduction to Stochastic Calculus
Weeks 3-4
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Outline
1
Conditional Expectation
A Motivating Example
-Fie
Math 490 Intro to Topology Elementary Topologies
What does physical topology of a network refer to? - The configuration of cables, computers, and
other peripherals.
What is Logical Topology? - The method used to pass information between workstations.
What
Math 490 Intro to Topology Chapters 1-4 Notes
There must be a reversible transformation between them that is continuous - Topologically
equivalent (homeomorphic)
u is an upper bound for a set A of real numbers provided that a<= u for all a in A - Upper bo
p. 1
Math 490-01 notes N1: Chapter 1 Stu
Topology depends heavily on set theory, which is introduced in chapter 1 of Munkres. You
should read section 1.1 for basic information and notation about sets and set operations.
Well begin our lectures with sectio
p. 1
Math 490 Notes 13
Here are some of the topologies on R that weve played with so far:
d = 2R
d
( discrete; all sets open )
s
coc
s
( Sorgenfrey; basic open sets [a, b) )
u
( usual topology; basic open sets (a, b) )
u
coc
( cocountable; open sets and R
Math 490 Intro to Topology Vocabulary for Exam 1
Convergence and continuity are. - .dependent on choice of topology.
In the discrete topology. - .all functions are continuous.
The final topology of the Heaviside function has. - .four open set.
Equicontinu
Math 490 Intro to Topology Terms to Know
Statement - A meaningful declarative sentence that is either true or false
Negation - An operation that makes a statement what it is not. If P is false then ~P is true
Converse - Result of reversing two parts of a
An Introduction to Stochastic Calculus
Haijun Li
[email protected]
Department of Mathematics
Washington State University
Week 6
Haijun Li
An Introduction to Stochastic Calculus
Week 6
1 / 13
Outline
1
The It Integral
Why does the Riemann-Stieltjes Approach