NOTES FOR APM503/MAT570, APPLIED/REAL ANALYSIS, FALL 2012
JACK SPIELBERG
Contents
Part 1. Metric spaces and continuity
1. Metric spaces
2. The topology of metric spaces
3. Sequences
4. Continuous functions
5. Bounded linear maps
6. Cauchy sequences and co
APM504, Spring 2013. Dr. Vladislav Vysotsky.
HW1 due January 24.
Sec 2.1, Problem 1 Let = cfw_r : r [0, 1] be the set of rational points of [0, 1]. A
the algebra of sets each of which is a nite sum of disjoint sets A of one of the forms
cfw_r : a < r < b,
APM504, Spring 2013. Dr. Vladislav Vysotsky.
HW2 due February 12.
Sec 2.6, Problem 1 Prove that the expectation E of a nonnegative random variable
satises:
E = sup Es,
cfw_sS:s
where S is the set of simple random variables.
We already know that E = limn
APM504, Spring 2013. Dr. Vladislav Vysotsky.
HW3 due February 26.
P
P
Sec 2.10, Ex. 4 Let n , n , and let and be equivalent (P( = ) = 0). Show
that
Pcfw_| n n | 0, n
The triangle inequality gives
a.s.
| n n | | n | + | | + | n | = | n | + | n |,
Since |
APM504, Spring 2013. Dr. Vladislav Vysotsky.
HW4 due April 2.
Sec 2.12, Ex. 9 Let be an integer-valued random variable and (t) be its characteristic
function. Show that:
P( = k ) =
1
2
eitk (t)dt,
k = 0, 1, 2,
Since is integer-valued, we rewrite integral
APM504, Spring 2013. Dr. Vladislav Vysotsky.
HW5 due April 25.
Problem 10 Let X1 , X2 , be a Markov chain.
a. Prove that for any positive integer n 3, 3 k n, and i1 < < in , it holds that
P ( Xi n = x i n , , Xi k = x i k | Xi k 1 = x i k 1 , , Xi 1 = x i
APPLIED PROBABILITY AND STOCHASTIC PROCESSES APM504
SPRING 2013
DR. VLADISLAV VYSOTSKY
Abstract. This course is a classical measure-theory based introduction to probability theory including expectation, notions of convergence, weak and strong laws of larg
Quantum Chemistry
Theory
Computational Chemistry
Ab initio methods seek to solve the Schrdinger equation. o
Molecular orbital theory expresses the solution as a linear combination of atomic orbitals. Density functional theory (DFT) attempts to solve for t
Multiscale model of the lac operon.
Villa et al. (2005) Structural dynamics of the lac repressor-DNA complex revealed by a multiscale simulation. PNAS 102: 6783-6788. Background: The lac operon is a cluster of genes in the E. coli genome that encode prote
3D Geometry of the Human Genome
Background
Lieberman-Aiden et al. (2009) Comprehensive Mapping of Long-Range Interactions Reveals Folding Principles of the Human Genome. Science 326: 289-293. Background: Understanding the 3D conformation of the genome can
Rare Event Sampling
Importance Sampling
Statement of Problem Suppose that X is an (E , d)-valued random variable with distribution and that we need to calculate the expectation E [f (X )] = f (x)(dx).
E
Unfortunately, analytical and numerical evaluations
Langevin and Brownian Dynamics
Overview
Langevin Dynamics of a Single Particle We consider a spherical particle of radius r immersed in a viscous fluid and suppose that the dynamics of the particle depend on two forces: A drag force arising from friction
Stochastic Calculus
The Normal Distribution
Preliminaries: Normal Random Variables Definition: A random variable Z with values in R is said to be normally distributed with mean and variance 2 > 0 if Z has density 1 2 2 p(z) = e -(z-) /2 . 2 In this case,
Symplectic Integration
Introduction
Realistic Objectives for Molecular Dynamics Simulations In general, the aim of a MD simulation is to identify qualitative and statistical properties of molecular motions rather than to reproduce a precise trajectory. La
Molecular Dynamics
Introduction
Motivation Most questions in molecular biology are concerned with dynamic processes in macromolecules. Levinthal's paradox: How does protein folding happen quickly on a high-dimensional energy landscape? How does protein st
Implicit Solvation Models
Overview
Solvation and Macromolecular Structure The structure and dynamics of biological macromolecules are strongly influenced by water: Electrostatic effects: charges are screened by water molecules and counterions. Hydrophobic
Non-bonded Interactions
Overview
Computation of the Non-Bonded Potential Recall that the non-bonded contribution to the potential function takes the form qi qj Aij Bij Vnb (R) = - 6 + 12 + . rij rij rij
i<j i<j
Since each of these sums contains O(N 2 ) te
Molecular Mechanics
Overview
Molecular Mechanics Molecular mechanics can be used to study molecules that are too large for quantum mechanical models. Molecules are treated as simple mechanical systems governed by Newtonian mechanics. The potential energy
Quantum Chemistry
Theory
Computational Chemistry
Ab initio methods seek to solve the Schrdinger equation. o
Molecular orbital theory expresses the solution as a linear combination of atomic orbitals. Density functional theory (DFT) attempts to solve for t
Molecular Structure
Background
Valence Each element tends to form a fixed number of bonds that depends on the number of electrons in its outer shell:
Carbon and phosphorus have valence 4 Nitrogen has valence 3 Oxygen and sulphur have valence 2 Hydrogen ha
APM 530 - Mathematical Models of Cell Physiology Jay Taylor
Course web page at http:/math.asu.edu/jtaylor syllabus lecture notes readings
Jay Taylor (ASU)
APM 530 - Lecture 1
Fall 2010
1 / 44
This semester will focus on nucleic acid structure and function
Non-bonded Interactions
Overview
Computation of the Non-Bonded Potential Recall that the non-bonded contribution to the potential function takes the form qi qj Aij Bij Vnb (R) = - 6 + 12 + . rij rij rij
i<j i<j
Since each of these sums contains O(N 2 ) te
APM 530 - Mathematical Models of Cell Physiology Jay Taylor
Course web page at http:/math.asu.edu/jtaylor syllabus lecture notes readings
Jay Taylor (ASU)
APM 530 - Lecture 1
Fall 2010
1 / 44
This semester will focus on nucleic acid structure and function
Rare Event Sampling
Importance Sampling
Statement of Problem Suppose that X is an (E , d)-valued random variable with distribution and that we need to calculate the expectation E [f (X )] = f (x)(dx).
E
Unfortunately, analytical and numerical evaluations
Molecular Mechanics
Overview
Molecular Mechanics Molecular mechanics can be used to study molecules that are too large for quantum mechanical models. Molecules are treated as simple mechanical systems governed by Newtonian mechanics. The potential energy
Symplectic Integration
Introduction
Realistic Objectives for Molecular Dynamics Simulations In general, the aim of a MD simulation is to identify qualitative and statistical properties of molecular motions rather than to reproduce a precise trajectory. La
Quantum Chemistry
Theory
Computational Chemistry
Ab initio methods seek to solve the Schrdinger equation. o
Molecular orbital theory expresses the solution as a linear combination of atomic orbitals. Density functional theory (DFT) attempts to solve for t