LIS202 Homework for Week 9 March 13, 15 Education
solutions
EXAM 2 March 15 (covers materials from Weeks 6-9). See study guide (on
Canvas and on the reverse of this printed page).
Required reading
Cha
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 func
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
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
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.
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 =
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
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 expectatio
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.
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 c
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. Backgrou
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).
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 de
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
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 mo
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
Implicit Solvation Models
Overview
Solvation and Macromolecular Structure The structure and dynamics of biological macromolecules are strongly influenced by water: Electrostatic effects: charges are s
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 r
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
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.
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
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 sem
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 r
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 sem
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).
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
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 mo
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.