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# Study Notes

## Putting It All Into Practice

How do we put all this into practice? It is daunting. At the first level, focus on two principles: Learning to look and making safety....

## Yeah, But …

This chapter has several common scenarios and advice on each one. Worth a read.

## Move To Action

Once everyone has added meaning to the pool (after reaching safety, etc), there is still a need for action. What should we do with this...

## Explore Others’ Path

Sometimes the problem is that the other person is resorting to silence/violence. What then? One option is to simply make it the other’s...

## State My Path

Share Risky Meaning When it comes to touchy information, the worst is to alternate between blunt and silence (Fool’s Choice). The less...

## Master My Stories

Claim One: Others don’t make you mad. You make you mad. Claim Two: Once you’re upset, there are only two routes: Act on them or be acted...

## Make It Safe

If someone says something sarcastic or a side remark, that means they need safety. Do not ask “What is that supposed to mean?” It can...

## Learn To Look

Most crucial conversations are about two things: Content and emotions. When we get deeply involved in one, we usually focus on only one...

The problem is not that our behavior degenerates. It is that our motives do. In crucial conversations, our motive often changes to...

## Mastering Crucial Conversations

The Fool’s Choice The Fool’s Choice: When you feel you have only two choices: Silently accept the problem Speak about the problem but...

## Expressive Appreciation in Nonviolent Communication

The Intention Behind The Appreciation Positive judgments are still problematic judgments. Example: “It was kind of you to…” “You are a...

## Liberating Ourselves and Counseling Others

Resolving Internal Conflicts Depression is indicative of a state of alienation from our own needs. Internal thoughts like “should”,...

## The Protective Use of Force

When The Use of Force is Unavoidable Dialog is great but sometimes the opportunity for dialog may not exist, and force is necessary. The...

## Conflict Resolution and Mediation

Human Connection Creating a connection between the people who are in conflict is the most important thing. The parties need to know from...

## Expressing Anger Fully

Hurting people is one of the weakest ways to express our anger. The goal of nonviolence is not to stifle anger, but to express it in...

## Connecting Compassionately With Ourselves

Evaluating Ourselves When We’ve Been Less Than Perfect We chastise ourselves in non-constructive ways when we screw up. Shame and guilt...

## The Power of Empathy

Empathy That Heals Listening empathically to someone (no advice, etc) can really relieve the other person’s tension. Ending the...

## $$z$$ Tests and Confidence Intervals for a Difference Between Two Population Means

## Poisson Distribution Tests

## Some Comments on Selecting a Test Procedure

## p-Values

## Tests Concerning a Population Proportion

## Tests About a Population Mean

## Hypotheses and Test Procedures

## A Confidence Interval For The Median

## Confidence Intervals For The Variance and Standard Deviation of a Normal Distribution

## Intervals for Non-Normal Distributions

## A Prediction Interval for a Single Future Value For a Normal Distribution

## Intervals Based on a Normal Population Distribution: The T-Distribution

## Large Sample Confidence Intervals for a Population Mean and Proportion

## Basic Properties of Confidence Intervals

## Estimating the Mean of a Symmetric Distribution

## Use Power To Educate

Some people just want to win, and their measure of success is how much you lose. If your goal becomes victory, realize you’re stuck in a...

## Build Them a Golden Bridge

Obstacles to Agreement The other side may stall: Lack of interest, vague statements, delays, breaking agreements, a flat No! There are...

## Reframe

When the other side doesn’t budge, resist the temptation to counter with your position. “That’s interesting. Why do you want that?” “You...

## Step To Their Side

Do not ignore the emotions while focusing on the problems. You need to defuse them, and you do it by surprise. If they stonewall, they...

## Go To The Balcony

Three Natural Reactions We have 3 natural reactions: Striking back Giving in and yielding to pressure Cutting off the relationship Don’t...

## Prepare, Prepare, Prepare

Before every meeting, prepare. After every meeting, assess and re-prepare for the next meeting. Do not try to wing it. Mapping Out The...

## Getting Past No: Overview

Five Barriers To Cooperation Your Reaction: You either strike back or give in. Both are damaging. Their Emotion: Anger, hostility, fear,...

## Methods of Point Estimation

## Some General Concepts of Point Estimation

## The Distribution of a Linear Combination

## The Distribution of the Sample Mean and Sum

Let $$X_{1}\dots X_{n}$$ be a random sample from a distribution with mean $$\mu$$ and standard deviation $$\sigma$$. Then:...

## Statistics and Their Distributions

When we take a sample and calculate its mean and standard deviation, this is treated as a random variable for the population...

## Expected Values, Covariance and Correlation

## Jointly Distributed Random Variables

Probability Given two random variables $$X,Y$$, the joint pdf is given by $$p(x,y)=P(X=x,Y=y)$$. Let $$A$$ be an event. Then the joint...

## Probability Plots

Sample Percentiles Calculating sample percentiles is challenging. What is the 23rd percentile of 10 points? One rule: Order the $$n$$...

## Extreme Value Distribution

For Weibull, let $$Y=\ln(X)$$. This has both scale and location parameters. Location is $$\theta_{1}=\ln(\beta)$$ and scale is...

## The Beta Distribution

The beta distribution The parameters are $$\alpha,\beta>0$$, and $$A,B$$ with $$B\ge A$$. \begin{equation*}...

## The Lognormal Distribution

If $$Y=\ln X$$ is a normal distribution, then $$X$$ is log-normal. \begin{equation*} f(x;\mu,\sigma)=\frac{1}{\sqrt{2\pi\sigma}...

## The Weibull Distribution

Probability Density Function Let $$\alpha,\beta>0$$ \begin{equation*}...

## The Erlang Distribution

If the time between successive events is independent each with an exponential distribution with $$\lambda$$, then the total time $$X$$...

## The Chi-Squared Distribution

Chi-squared distribution Probability Density Function The parameter is $$\nu$$ and it is a positive integer. It is the gamma...

## The Exponential Distribution

Exponential distribution Probability Density Function Let $$\lambda>0$$ \begin{equation*} f(x;\lambda)=\lambda e^{-\lambda x}...

## The Gamma Distribution

The problem with the normal distribution is that it is symmetric. The Gamma distribution is useful for skewed distributions....

## The Normal Distribution

Probability Distribution Function The parameters are $$-\infty<\mu<\infty$$ and $$\sigma>0$$. \begin{equation*}...

## Pareto Distribution

The Pareto distribution is good for approximating income distributions or population sizes. The pdf is given by: \begin{equation*}...

## Continuous Random Variables and Probability Distributions

Continuous distributions are given by a probability density function (pdf): \begin{equation*} P\left(a\le X\le...

## Zipf Distribution

Suppose you have a library with $$M$$ items and you want to sort them by popularity. The parameters are $$M$$ and $$\alpha$$. The domain...

## Scarcity

People are motivated more by the thought of losing something than the thought of gaining something of the same value. Frame something in...

## Receiving Empathically

Don’t Just Do Something. Stand There! Empathy is emptying our mind and listening with our whole being. The presence that empathy...

## Requesting That Which Would Enrich Life

. Use positive language when making requests! Example of negative language: Asking someone not to do something. Instead, request what...

## Taking Responsibility For Our Feelings

Hearing a Negative Message: Four Options What others say or do may be the stimulus of our feelings, but not the cause. When someone...

## Identifying and Expressing Feelings

“I feel that…” When you use “that” after feel, you are likely not revealing your feelings. Adequately expressing feelings is...

## Observing Without Evaluating

Separate the observation from the evaluation. If you say both in one go, people are more likely to interpret it as criticism and resist....

## Communication That Blocks Compassion

Moralistic Judgments Moralistic judgments are “life-alienating” communications. The focus tends to be on analyzing wrongness then...

## Giving From The Heart

The NVC Components: Observations: State the facts Feelings: State your feelings Needs: State your needs connected to your feelings Make...

## Authority

Titles Study: A person is introduced to various groups as either a student, lecturer, professor, etc. After he leaves the room, people...

## Poisson Distribution

$$X$$ is of a Poisson distribution if its pmf is $$p(x;\lambda)=\frac{e^{-\lambda}\lambda^{x}}{x!}$$ where $$x$$ is 0, 1, 2, etc....

## Negative Binomial Distribution

The experiment requires: The trials be independent The outcome is binary (success or failure). The probability of success or failure is...

## Hypergeometric Distribution

The hypergeometric experiment requires: The population is finite, with $$N$$ individuals. The outcome of each trial is binary (success...

## Binomial Distribution

Bernoulli Distribution A Bernoulli random variable is one whose only possible values are 0 and 1. $$P(X=1)=p$$ $$E[X]=p,V[X]=p(1-p)$$...

## Discrete Random Variables

Random Variables A discrete random variable is one whose set of possible values is countable. Probability Distributions for Discrete...

## Read From UIUC

Read hypothesis testing in Chapter 2 (ML rules). Reliability in Chapter 2

## Probability

Sample Spaces and Events The sample space $$\mathcal{S}$$ is a set. An event is a subset of the sample space. A simple event is an event...

## Liking

A common tactic: A salesperson who has completed a sale with you will ask for a list of friends, and call them up and talk about how...

## Measures of Location and Variability

Measures of Location When reporting a sample mean, use one extra significant digit. Sample mean: $$\bar{x}$$ Population mean: $$\mu$$...

## Pictorial and Tabular Methods in Descriptive Statistics

Pictorial and Tabular Methods in Descriptive Statistics Stem and Leaf Display Stem and leaf display Advantages: Useful for displaying...

## Statistics: Overview

Populations, Samples and Processes A census means you poll every member of the population. Univariate vs bivariate or multivariate:...

## Frequently Asked Questions

What If The Other Person is a Bully, Lying, or Trying to Derail While it does happen, most of our evaluations about the other person are...

## Putting It All Together

Step One: Prepare By Walking Through The Three Conversations Stories What’s my story? (information, past experiences, rules?) What’s his...

## Problem Solving: Take The Lead

Most people don’t have the conversation skills. You’ll talk about understanding and they’ll talk about who’s right. You’ll talk about...

## Expression: Speak For Yourself With Clarity

Don’t just listen - you need to share your side of the story. Orators Need Not Apply Do not try to be eloquent or witty. Do not try to...

## Learning: Listen From The Inside Out

People have a deep desire: To be heard To know that others care enough to listen Listening Transforms The Conversation Mother has...

## Getting Started: Begin From The Third Story

A bad beginning can crater the conversation. But a good one helps you immensely. Why Our Typical Openings Don’t Help Don’t dive in with...

## What’s Your Purpose: When to Raise It and When To Let Go

Pick your battles! You can’t have every difficult conversation you come across. Deciding whether to raise an issue or not torments us....

## Ground Your Identity: Ask Yourself What’s At Stake

Difficult Conversations Threaten Our Identity Sometimes the anxiety is not due to facing others, but facing ourselves. The conversation...

## Have Your Feelings Or They Will Have You

Don’t focus on just the objective facts. Feelings have to be part of the conversation. At the same time, having them explode is...

## Abandon Blame: Map The Contribution

Someone screws up, which causes you to screw up. You can blame him/her explicitly (“You screwed up!”) or implicitly (“Let’s do better...

## Social Proof

Social proof is also a convenient shortcut for decision making. Kids who are scared to play with dogs become willing to merely by...

## Commitment and Consistency

We have a deep desire to be/appear consistent. This is why when people bet on a horse, they are more confident it will win after they’ve...

## Reciprocation

We try to repay, in kind, what someone else has given us - even we didn’t want it or take it! We feel obligated. Society provides a very...

## Influence: Weapons of Influence

A group of people are waiting in line to use a photocopying machine. Someone says “May I use the machine because I am in a rush?” to...

## Don’t Assume They Meant It

Intentions strongly influence our judgments. Watch your sentences and notice how often you are attributing intentions to the...

## Stop Arguing About Who’s Right

At the heart of the What Happened conversation is a disagreement. Argument is the natural outcome of disagreement. But it is...

## The Three Conversations

You need to understand what people are thinking and feeling, but not saying. Every difficult conversation is really a mix of...

## Difficult Conversations

What makes a conversation difficult? Usually it is the fear of consequences. If we avoid the conversation, we feel taken advantage of,...

## Start Your Own Support Group

You can start a support group yourself. Rule: The group has no “leaders”. All the members own it. Meeting Format (Suggestion) Run a...

## A Little Structure Goes a Long Way

The principles in the book can be applied informally, but success is much higher if you formalize it. (My idea): To an extent, perhaps...

## Step Nine: Fake It Till You Make It

Making a commitment is like making a resolution. If you don’t stick to it, it’s useless. Don’t fear the mistakes you’ll make along...

## Step Eight: Commit To Improvement

Benefits of making a commitment: Liberating to admit your problems By committing publicly, you are more likely to follow through. Public...

## Step Seven: Diagnose Your Weakness

Common problems: The struggler: Craves attention. Seeks turmoil and impossible situations. The conflict avoider: Downplays...

## Step Six: Learn to Fight

Learn to spar with your team: Sparring is where they can grill you - hold you accountable, but not in a way that causes injury. It is...

## Step Five: Create Your Personal Success Wheel

Take a week off in December to reassess everything. Use the Personal Success Wheel or make your own. The elements in the wheel are not...

## Step Four: Broaden Your Goal Setting Strategy

Goal setting should be done be a team! Don’t insist on setting all your goals yourself. Let the team help you. His opinion: Goals may be...

## Step Three: Practicing The Art of The Long Slow Dinner

Once identified, the lifeline needs to be eased into the process. The benefit of a long, slow dinner is that the clock stops. No rush to...

## Step Two: Find Your Lifeline Relationships

Sources of Lifelines Your confidant need not be someone close to you - sometimes those close to you are very poor at it. We have close...

## Step One: Activate Your Vision

His view: People spend too much time meticulously recycling vs living up to their talents and abilities. You get only one life. What...

Advice from his mentor: “What do you really want? Define your greatness. Then be convinced of it and start acting like it. Someone will...

## 4th Mindset: Accountability

Accountability is the 4th mindset. We all make efforts to change (e.g. corporate workshops, trainings, etc), but most efforts don’t have...

## 3rd Mindset: Candor

Realize that opening yourself to get feedback from others is risk free! They will have their perceptions of you regardless of your...

## 2nd Mindset: Vulnerability

When introducing yourself, don’t just list accomplishments. Tell a story about your dreams, how you’re getting there, and what...

## 1st Mindset: Generosity

What do I have to offer others? There are two types of generosity currency: Universal and personal. Universal currency is very...

## Who’s Got Your Back

Everyone needs confidants for their professional work. People who care about your success and will not let you fail. Most people have...

## Liberals vs Conservatives

Haidt claims that genes account for whether you are left/right a lot more than upbringing does. A single gene doesn’t predict much. But...

## The Definition of Morality

For Western/modern society, the author favors rule utilitarianism (best rules that will produce the greatest good) as opposed to act...

## Religion and Charity

Is God a Force of Good or Evil? Religious people generally give more in charity, but usually to their own kind or through their own...

## Religion and the New Atheists

You cannot understand religion by looking at the individual. You must look at the collective. The Lone Believer Critics of religion...

## Hives

The hive switch: The ability to temporarily transcend self-interest and lose ourselves in something bigger. It is essentially a...

## Group Selection

Group Selection: If a trait benefits the group against other groups, but not the individual against other individuals in the same group,...

## Punishment for Improving Cooperation

Reciprocal altruism theory explains why altruism/fairness behavior develops among pairs (tit for tat) but does not explain it...

## Hierarchy vs Egalitarianism

Republicans appeal to all 5 moral tastes than Democrats, who appeal to only 1 or 2. This is why the former tend to have an edge...

## Moral Foundations

Notable was Haidt’s criticism of amateur evolutionary theorists: They pick a trait and see if they can find a story that explains it. A...

## Bentham and Kant: Bias Towards Reason as a Root of Morality

Moral monism is the reduction of morality to a single principle (e.g. don’t do harm). Haidt argues this is unhealthy for society. Is...

## The Ethic of Divinity

Psychologists have a tendency to study the mind independent of culture. Anthropologists tend to study the culture and not the mind....

## WEIRD People

Most research in psychology is conducted on WEIRD people. WEIRD stands for Western, Educated, Industrialized, Rich and Democratic. But...

## The Rationalist Delusion

So do good reasoning skills lead to more moral behavior (the atheist’s dream)? A study found that moral philosophers behave just like...

## Self-Esteem, Cheating, Voting and Confirmation Bias

Self-Esteem Is self-esteem about how we perceive ourselves, or about how we perceive other’s perceptions about ourselves? Levy Study:...

## The Search For Truth?

There has existed for a long time a dilemma: Is it better to be just and honest in a world that will never recognize you as such, vs...

## When Does an Elephant Listen to Reason?

Primarily by engaging with others. On can change via introspection, but others are more likely to give us non-confirmatory evidence....

## Deontologists vs Utilitarianists

Utilitarianism (save the most lives) vs Deontologism (cannot impinge on individual rights - even to save many lives). Both claim their...

## Infants’ Sense of Morality

Infants do not appear to be born with a blank slate. They have certain expectations on the physical laws, as well as the social laws. By...

## Psychopaths

1% of males are psychopaths. Fewer females are. Most are nonviolent. But virtually half of crimes like serial rape and killing,...

## Cleanliness and Morality

The sense of smell impacts moral judgments. Asking participants to fill out a survey on moral scenarios in the presence of fart smell...

## Implicit Association Test

Affective Priming: Flash words like sunshine or cancer to the screen. You need to judge whether the word is positive or negative. If you...

## Brains Evaluate Instantly and Constantly

Our brains evaluate everything instantly in terms of potential threat or benefit, and adjusts behavior to get more of the good and less...

## Thalia’s Hypnosis Study

Hypnotized one set of students and made them dislike the word “take”. Hypnotized another set and made them dislike “often”. Gave them 6...

## Does Reason Exist to Serve The Passions? Or Vice Versa?

The rationalist delusion: The worship of reason and the distrust of passions. It is a delusion because once you declare it to be an...

## Negotiations - The Coursera Treatment 2: Seidel

Should I Negotiate? Consider the time spent in preparing and executing the negotiation, and the expected gain. Is the time spent worth...

## Sociocentrism vs Individualism Part 2

To tackle Turiel’s concern that the “harmless” stories could be interpreted as harmful, Haidt decided to do a similar story, but added a...

## Sociocentrism vs Individualism

Most societies have a need to “resolve” the tension between individuals and the society. Most societies are sociocentric: The group has...

## The Anthropologist View

Anthropologists had studied and documented that many unrelated cultures had: Witchcraft and sorcery themes - with a great deal of...

## Morality vs Social Conventions

To gauge a child’s developmental level, it is important that the method of gauging doesn’t rely on the child’s verbal abilities - this...

## How Do Kids Learn About Morality? Role-playing

How do children learn right from wrong? There used to be 2 theories: Nature: Nativist (be it through God or evolution) Nurture:...

## The Righteous Mind

Introduction Intuitions come first, strategic reasoning comes second. Moral intuitions arise automatically - long before moral reason....

## Mutation Via References

val x = ref 42 val y = ref 42 (* Although the number is the same, x and y do NOT point to the same thing! *) val z = x (* z is now a...

## First Class Functions and Functional Programming

Functions can take other functions as arguments. Consider: fun compose (f, n, x) = if n = 0 then x else f(compose(f, n-1, x)) This calls...

## Tail Recursion

A tail call is when the callee’s result is merely returned without being used (i.e. it is not multiplied or added to anything, etc). In...

## Exceptions

exception MyException exception AnException of int*int raise MyException e1 handle ex => e2 e1 handle ex(x,y) => e2 (* More generally *)...

## Nested Patterns

We can nest patterns as deep as we like. Anywhere we put a variable in our pattern, we can put another pattern. exception BadTriple fun...

## Polymorphic Types and Equality Types

The type ''a represents an equality type: It means anything that can be compared with the equality operator. Not everything can be...

## Functions and Type Inference

If you use patterns to access records or function arguments, you usually do not need to specify their types. They will be inferred from...

## Functions, Pattern Matching And Val Bindings

The pattern {f1=x1, ..., fn=xn} will match against a record (assuming all the types match) {f1=v1,...,fn=vn}. Note that since all tuples...

## Case Expressions

Example (newtype is defined in a previous post). fun f (x : newtype) = case x of Pizza => 3 | Str s => 8 | TwoInts(i1, i2) => i1 + i2 To...

## Datatype Bindings

Example: datatype newtype = TwoInts of int*int | Str of string | Pizza newtype is a new type that is added to the environment. TwoInts,...

## Type Synonyms

type aname = t aname becomes an alias for type t. Useful for giving meaningful names: type date = int*int*int Think of it like typedef in C++.

## Records

Example: val x = {k1=3, k2=false} Ordering of keys does not matter. You can nest records. The type is: {k1:int, k2:bool}. You use #k1 to...

## Immutability

Most of Standard ML has immutability. Behind the scenes, references may be in use for optimization purposes, but the language guarantees...

## Options

Consider this motivating example: I want to write a function that takes a list of integers, and returns its sum. What should it return...

## Let Expressions

Syntax let b1 b2 ... bn in e end The keywords are let, in, end. bi are bindings. e is an expression. Note that often bi are each on...

## Tuples and Lists

Tuples Syntax (e1, e2, ..., en) Note that the length of a tuple is known in advance. Type Checking ei will have type ti (the types need...

## Functions

Defining Functions Look at the following example to declare a function: fun pow(x:int, y:int) = if y = 0 then 1 else x * pow(x, y-1) If...

## Standard ML - Basics

Comments Comments begin with (* and end with *). You can nest comments. Statement Ends In the REPL, each statement must end in a...

## Ten Questions People Ask About Getting To Yes

Much of the material with the tag gty comes from Getting To Yes. Does Positional Bargaining Ever Make Sense Principled negotiation is...

## What If They Use Dirty Tactics?

Much of the material with the tag gty comes from Getting To Yes. Examples: Deception Throwing Off Balance Nibbling (asking for stuff...

## What If They Won’t Play?

Much of the material with the tag gty comes from Getting To Yes. They may stick to their position. They may attack you instead of...

## What If They Are More Powerful? Protecting Yourself

Much of the material with the tag gty comes from Getting To Yes. If you are trying to make a deal that is very important to you, it is...

## Insist On Using Objective Criteria

Much of the material with the tag gty comes from Getting To Yes. Deciding On The Basis of Will is Costly Examples of negotiating on the...

## Invent Options For Mutual Gain

Much of the material with the tag gty comes from Getting To Yes. Diagnosis There are typically 4 reasons that prevent the invention...

## Focus on Interests, Not Positions

Much of the material with the tag gty comes from Getting To Yes. The basic problem in a negotiation is not in conflicting positions, but...

## Separate The People From The Problem

Much of the material with the tag gty comes from Getting To Yes. Do not ignore the human factor, with all its emotions. Do not just...

## Don’t Bargain Over Positions

Much of the material with the tag gty comes from Getting To Yes. A wise agreement is one that: Meets the legitimate interests of each...

## Introduction

Much of the material with the tag gty comes from Getting To Yes. Negotiation is easier if both sides know how to perform it.

## Negotiations (Getting To Yes)

Much of the material with the tag gty comes from Getting To Yes.

## Story Writing Checklist

Is there something wrong in the third act? Then it is wrong in the first act. Are you writing because you have something to say, or...

## Handling Critique

People’s critiques will be very subjective and very related to themselves. They may hate a character because it reminds them of someone...

## Show Them Once So They Know

In Jurassic Park, a character diverts a dinosaur’s attention by throwing a flare away from him. The dinosaur chases the flare while they...

## Superior Position

Superior position is when the audience knows something the characters do not. It is a useful tool for suspense, but can serve other...

## Dialogue

With dialogue, it is important to set up the subtext. If you have established that two people hate each other, and then you have them in...

## Subplots or Supporting Plots?

Do not use subplots to make the world more complex. Use them only to support the main plot. You are a slave to the story. You do not...

## God From the Machine

Deus Ex Machina is horrible. Do not use it to solve problems. However, feel free to use it to create problems. No one minds it, and it...

## Climax

The climax should bring masculine and feminine elements together. It shows how a character changed (or didn’t).

## Genre

Genre is visible (e.g. western, horror, etc). Do not think in terms of genre. Focus on the story, and then choose an appropriate genre....

## Show, Don’t Tell: Teaching vs Preaching

A storyteller teaches, not preaches. If you talk about what you want to say, you are preaching. If you show, you are teaching. The...

## The Masculine and the Feminine

Consider this quote: “The King died and then the Queen died” is a story. “The King died and then the Queen died of grief” is a plot. -...

## Tell the Truth

Truth is not about the facts. In a horror movie, a girl going down a scary basement is not the truth. Real scared women will not do...

## Character Development

Ritual Pain Think of rites of passage. In many cultures, one has to undergo severe pain to become a man (not for women). Gang...

## The Use of Clones

A clone in a story is a tool for showing, not telling. They are characters that represent what could, should, or might happen to the...

## The Armature

People tell stories to teach something. It is likely the origin of stories - a way to teach something beyond simply using boring facts....

## The Recipe for All Narratives

Less is more. Don’t go for a complex plot. Simplicity rules. There are seven steps in all narratives: Once upon a time And every day...

## Negotiations - The Coursera Treatment

The Pie The first principle in negotiation: Figure out what the pie you are fighting over is. It may not be obvious, and the parties may...

## Common Tactics Others Use

Much of the materials with the tag bfa comes from the Bargaining for Advantage book. Lies about the bottom line and alternatives: Never...

## Bargaining With the Devil

Much of the materials with the tag bfa comes from the Bargaining for Advantage book. The more ethical you are, the higher the cost...

## Closing and Giving Commitment

Much of the materials with the tag bfa comes from the Bargaining for Advantage book. Scarcity Effect: People will value (or overvalue) a...

## Opening and Making Concessions

Much of the materials with the tag bfa comes from the Bargaining for Advantage book. If your opponent is competitive, you may need to...

## Exchanging Information

Much of the materials with the tag bfa comes from the Bargaining for Advantage book. 3 aspects to it: Rapport development Discovering...

## Preparing Your Strategy

Much of the materials with the tag bfa comes from the Bargaining for Advantage book. There are 4 steps to negotiations: Preparation...

## Leverage

Much of the materials with the tag bfa comes from the Bargaining for Advantage book. Leverage gives you power to reach an agreement on...

## The Other Party’s Interests

Much of the materials with the tag bfa comes from the Bargaining for Advantage book. It is imperative to ask yourself how achieving your...

## Relationships

Much of the materials with the tag bfa comes from the Bargaining for Advantage book. Reciprocity is the key to sustaining trust...

## Authoritative Standards and Norms

Much of the materials with the tag bfa comes from the Bargaining for Advantage book. There exist several norms and standards. Examples:...

## Your Goals and Expectations

Much of the materials with the tag bfa comes from the Bargaining for Advantage book. Research shows: The more specific your vision of...

## Your Bargaining Style

Much of the materials with the tag bfa comes from the Bargaining for Advantage book. There are typically 4 steps in negotiation:...

## Bargaining For Advantage - Introduction

Much of the materials with the tag bfa comes from the Bargaining for Advantage book. When negotiating, it is vital to be yourself: If...

## Burrows-Wheeler Transform

The method: Given a string of, say, length 8, form all 8 rotations of it, and then sort them lexicographically. Then form the 8 letter...

## LZW Compression

The idea: First assign a number to all the letters of the alphabet. Say A is 41, B is 42, etc. Now let’s say we’re compressing the...

## Huffman Encoding

The problem with run length encoding was that our chunk size (e.g. 4 bits) is fixed. This is nonoptimal. With Huffman encoding, we can...

## Run Length Encoding

We have a large string (e.g. text file). Usually, ASCII is not the optimal space allocation for the file. There are various schemes for...

## Substring Matching

We have a string of length $$N$$, and wish to find an $$M$$ character string in it. Brute Force Keep moving through the string till the...

## Tries

An R-way trie is a tree where each node has $$R$$ children. It’s best described by an example: Each node consists only of two...

## Symbol Tables

A symbol table is an associative array. It’s a good idea to make the keys orderable for performance reasons (it also opens up more...

## Bloom Filters

Bloom filters are an application of hashing. It is a data structure that allows us to check if an object has already been seen (without...

## Hash Functions

We often want to store an object for quick retrieval, or see if a given object has been observed before in $$O(1)$$ time. Let $$U$$ be...

## Suffix Arrays

A suffix array is an array formed by taking a string, and storing pointers to all the suffixes in the array. As an example, if the...

Least Significant Digit Suppose we have $$N$$ strings of length $$W$$. The idea behind the LSD sort is to first sort the last “digit”...

## Counting Sort

Counting Sort Suppose we have $$N$$ elements, but we know the elements are drawn from a fixed, finite list (called the alphabet in this...

## Max Flows and Min Cuts

Mincut Problem An s-t cut is a cut with $$s$$ in one set and $$t$$ in the other. Call the sets $$A$$ and $$B$$. Given a digraph, with...

## Shortest Paths Summary

There are many “kinds” of shortest paths. Algorithm Constraint Typical Worst Case Topological Sort No directed cycles $$N+M$$ $$N+M$$...

## Negative Cycles

A negative cycle is a directed cycle where the sum of the edge weights is negative. A shortest paths tree exists if and only if there...

## Shortest Paths

Given a digraph, we may wish to find the shortest distance between two nodes, or between a node and all other nodes, etc. Each edge has...

## Minimum Spanning Trees

Given a connected graph $$G$$, with positive edge weights, the spanning tree is a subgraph $$T$$ that is both a tree (connected and...

## Finding the Strongly Connected Components of a Digraph

Applications When looking at code/library dependencies, one may want to see which code pieces form strongly connected components, and...

## Topological Sort

Topological Sort A topological sort is a traversal of a digraph with ordering constraints. Let each node have some value/weight. It is a...

## Minimum Cut of a Graph

A cut of a graph is a partitioning of the nodes into two nonempty sets. A minimum cut is a cut that minimizes the number of edges...

## Karatsuba Multiplication

In grade school we learn an algorithm for multiplying two $$N$$ digit integers. Let our basic operation to be the multiplication or...

## Rectangle Intersection

Given $$N$$ rectangles, find all the intersections. This algorithm is very similar to the 1-D line intersection search. The difference...

Suppose we have $$N$$ 1-D intervals of the form $$(a,b)$$ where $$a ## 2-D Range Search, Points in a Box Search and K-D Trees This is like the 1-D range search, but where the keys can now have two coordinates. It’s finding the number of points in a rectangle... ## 1-D Range Search and Line Intersection Search Sample problems we want to solve: What are all the keys between two keys? (Relevant to databases) How many keys exist between two keys?... ## Red Black Trees We introduced 2-3 trees with highly desirable properties. Here we’ll convert them into binary trees while maintaining all the... ## Asymptotic Expansions A decreasing sequence \(g_{k}(N)$$ is called an asymptotic scale if $$g_{k+1}(N)=o(g_{k}(N))$$. $$f(N)\sim... ## The Average Number of Leaves in a Binary Tree of \(N$$ Internal Nodes

Let’s use generating functions to count the average number of leaves in a binary tree with $$N$$ internal nodes. Definitions Let $$P$$...

## The Average Number of 1’s in a Binary String of Length $$N$$

Let’s use generating functions to count the average number of 1’s in a binary string of length $$N$$. Definitions Let $$P$$ be the set...

## Counting With Generating Functions

Generating functions can be used for counting. Consider the problem of finding the number of trees with internal nodes. Let $$T$$ be the...

## 2-3 Trees

A 2-node is like a regular node in a binary search tree: It has one key and two children. A 3-node has 2 keys and 3 children. The middle...

## Exponential Generating Functions

To form a generating function, we multiplied by $$z^{N}$$ and summed. This is not the only way to form a generating function. Another...

## Catalan Numbers

The Catalan numbers satisfy the recurrence relation: \begin{equation*} T_{n}=\delta_{n,0}+\sum_{k=0}^{n-1}T_{k}T_{N-k-1} \end{equation*}...

## Solving Recurrence Relations With Ordinary Generating Functions

Technique The basic technique is to multiply the equation with $$z^{N}$$, sum over all $$N$$, recast in terms of the generating function...

## Ordinary Generating Functions

Definition An ordinary generating function of a sequence $$\left\{a_{k}\right\}$$ is: \begin{equation*} A(z) = \sum_{k\ge 0}a_{k}z^{k}...

## Motivation for Generating Functions (Classical Approach)

Problem How many bits are needed to represent a binary tree with $$n$$ internal nodes (I think leaf nodes are not internal)? These trees...

## Binary Search Trees

A binary search tree is a fairly useful structure for storing ordered data. The invariant for a binary search tree is straightforward:...

## Heap Sort

One way to sort a list is to use the heap data structure. Given an array of $$N$$ unsorted elements, build a max heap in place. Start...

## Binary Heaps & Priority Queues

At times we want a structure similar to a queue or stack, but instead of LIFO or FIFO, we want it to pop the maximum element each time....

## K-th Largest Element

Sometimes we wish to find the k-th largest element in a list. Brute Force A brute force way to do this would be to sort the list and...

## Quick Sort

A quick sort works recursively: Shuffle the list. This protects against a carefully crafted input that will make the algorithm...

## Convex Hull Problem

Say you have $$N$$ points on a Euclidean plane. You want to find the polygon with the smallest area that can be formed connecting a...

## Sorting Complexity Analysis

Any comparison based sorting algorithm must use at least $$\lg(N!)\sim N\lg N$$ comparisons for the worst case. This is absent any...

## Finding the Closest Pairs in a Plane

Suppose we have $$N$$ points in a plane, and we wish to find the closest pair. The brute force algorithm takes $$\Theta(N^{2})$$. We can...

## Counting Inversions

Why would we want to count the number of inversions? One application may be to see how close two different people’s rankings of the same...

## Dealing With Recurrence Relations

Useful Identities \begin{equation*} \sum_{k=m}^{n}\dbinom{k}{m}=\dbinom{n+1}{m+1} \end{equation*} The mnemonic for this is: Suppose you...

## Merge Sort

A merge sort works recursively: Split the list into two roughly equal halves. Sort each half using mergesort. Merge the two...

## Shuffling Caveats

When using a random number generator for shuffling, be wary of simple issues. Using cards as an example: If the seed is 32 bits, you’ll...

## Shell Sort

A shell sort works somewhat recursively. It first sorts every $$k$$ elements (i.e. the sublist formed by elements 0, $$k$$, $$2k$$,...

## Insertion Sort

An insertion sort starts from the first element in the list, and then for each element will check with the previous one to see if it is...

## Inversions & Partial Sortedness

An inversion in a list of elements is a pair of elements where the 2nd element is smaller than the first (i.e. the 2nd element would be...

## Selection Sort

A selection sort starts out by traversing the whole list, finding the minimal element, and swapping it with the first element. It then...

## Loop Invariants

A loop invariant is a statement that holds true just before and after each iteration. It may be violated within the iteration, but the...

## Graphs Using NetworkX

NetworkX is a Python library for handling graphs. In [1]: %matplotlib inline In [14]: import networkx as nx import pylab as plt In [3]:...

## Breadth First Search And Finding the Distance Between Two Nodes

We wish to find the distance between two nodes $$a$$ and $$b$$. Let’s generalize this further and find the distance from $$a$$ to all...

## Depth First Search

The depth first search algorithm is a way to traverse all the nodes in a graph. The algorithm is trivial: Start at a given node. Mark it...

## Queues

A queue is a data structure for holding objects. It is a FIFO structure. It supports two operations: enqueue and dequeue dequeue returns...

## Graphs

An undirected graph is defined by $$(V, E)$$ where $$V$$ is a set of nodes and $$E$$ is a set of edges (each edge is an unordered set of...

## Stacks

A stack is a data structure for holding objects. It is a LIFO structure. It supports two operations: pop and push. pop returns (and...

## 2-Sum and 3-Sum

2-Sum Problem Statement Given a set of $$N$$ integers, find how many pairs sum to 0. Brute Force Sum all pairs and count whichever sum...

## Percolation: An Application of Union Find

Consider an $$N\times N$$ square grid, where all the grids are black. Now let each square in the grid be white with probability $$p$$....

## Union-Find Algorithm

Dynamic Connectivity & Disjoint Sets Given $$N$$ nodes, some connected amongst themselves, does a path exist between any two...

## Asymptotic Notation

An asymptotically positive function $$f(n)$$ is one that is always positive for sufficiently large $$n$$. A similar definition holds for...

## Probability Review

Linearity of expectation: $$E(aX+bY+c)=aE(x)+bE(Y)+c$$. This holds even when $$X$$ and $$Y$$ are dependent. Exploit this! Independent...

## Useful Inequalities

\begin{equation*} \sum_{k=2}^{n}\frac{1}{k}\le\ln n \end{equation*} To see this, look at the curve $$1/x$$, and integrate it from 1 to...


text:
# Poisson's equation in 2 dimensions

1. Oct 13, 2007

### ehrenfest

1. The problem statement, all variables and given/known data
If we lived in a world with 2 spatial dimensions, would the solution to Poisson's equation for a singular charge be the same i.e. would it be 1/r radial dependence?
I am doing a computer simulation and if I take out the third dimension, the results are manifestly different.

2. Relevant equations

3. The attempt at a solution

2. Oct 14, 2007

### Dick

It would be manifestly different. Look at Gauss's law for flux integral vs source charge. In two dimensions, it's a log.

3. Oct 14, 2007

### ehrenfest

Sorry. Could you explain that more.

In 3 dimensions, Gauss's law says that the flux of E through a surface around a charge q is given by

$$\int _{S} \vec{E} \cdot d\vec{a} = \frac{q}{\epsilon_0}$$

In two dimensions, it is just over a closed path instead of circle and you replace the da by a d(normal vector to the curve) or something.

Last edited: Oct 14, 2007
4. Oct 14, 2007

### Dick

The 1/r dependence in 3d is for the potential U. In your 2d example you can conclude that (take a circle for the curve) E*length of curve is proportional to charge. So E is proportional to 1/r. But E=-dU/dr. So U must be like log(r).

Last edited: Oct 14, 2007
5. Oct 14, 2007

### ehrenfest

But then the potential is infinity at infinity? I think maybe Gauss's law would be different in 2 dimensions.

Last edited: Oct 14, 2007
6. Oct 14, 2007

### Dick

Yes. If you integrate a 1/r force from something to infinity, it diverges. Try one dimension as well. The force is a constant independent of distance. Potential is a linear function of r. Also infinite at infinity. Remind you of anything?

7. Oct 14, 2007

### ehrenfest

Not really.

But try inserting ln( sqrt(x^2 +y^2)) into the two dimensional poisson equation i.e. the gradient only has two components. I get 1/sqrt(x^2 + y^2) = rho/epsilon_0, which is not a solution.

But I just tried the 3 dimensional solution U = 1/sqrt(x^2 +y^2 +z^2) and I get 0 = rho/epsilon_0 .

So, what could be wrong? Now I am really confused. I am using a point charge at 0, so the RHS of Poisson's equation should be a delta function?

8. Oct 14, 2007

### Dick

You should get 0 in both cases. Your source is only at the origin so the laplacian is correctly giving you the charge density=0, at the origin it's singular. To test that your solution is a delta function you have to integrate it against a test function and use integration by parts.

9. Oct 14, 2007

### ehrenfest

Aha, I see now.

In my model, I was putting the charge in a square with a boundary condition that the sides of the square have a potential of 0. In 3 dimensions it made sense to make the sides of the box around the charge zero as an approximation of infinity, but in 2 dimensions that breaks down apparently. I am not really sure how you would make an accurate numerical model of the potential in two dimensions then since I can't have a boundary condition of infinity.

Maybe I need to use Neumann boundary conditions or something.

Thanks.

Last edited: Oct 14, 2007

format: plain


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# Logistic Regression with Household Fixed Effects, SPSS

I have cross-sectional data on households and individuals for several countries.

I am interested in the marginal effect of a particular dichotomous individual variable $D_i$ (dummy denoting whether individual $i$ has some condition or not) on an individual outcome $Y_i$ (dummy denoting success).

Hence, I estimated the logistic regression

$\text{Pr}(Y_i=1)=\text{logit}^{-1}(\beta \boldsymbol{X}_i+\gamma D_i)$

where $\boldsymbol{X}_i$ is a vector of individual, household, and community control variables, e.g., age, age squared, gender, residential location, etc.

Now I wish to account for potential endogeneity of certain control variables on $D_i$ and $Y_i$ (i.e., some $x_i$ in $\boldsymbol{X}_i$ simultaneously affecting the likelihood of an individual experiencing condition $D$ and their outcome $Y_i$), by adopting a household fixed-effects approach.

A previous paper I am following simply re-estimated the regression on a subsample of individuals living in households where at least one individual had the condition and one did not, replacing all household and community variables with dummy variables for each household minus one.

However, reading suggests I cannot simply estimate the revised logistic regression

$\text{Pr}(Y_i=1)=\text{logit}^{-1}(\beta' \boldsymbol{X'}_i+\gamma' D_i+\sum\delta_j {H_j}_i)$

where ${H_j}_i$ is a dummy denoting whether individual $i$ is resident in household $j$ or not, because serious coefficient bias will arise given the small number of observations per household (similar to having a small number of $T$ in a panel fixed-effects modeL).

Instead, it has been advised to use a conditional logit fixed effects estimator.

(1) Is this possible in SPSS, and if so, how?

(2) Will this allow computation of the marginal effect of $D_i$?

(3) If not, is it preferable to adopt instead a linear probability model?


text:
Now showing items 21-39 of 61

• #### 0630 - Geometric Group Theory, Hyperbolic Dynamics and Symplectic Geometry

[OWR-2006-33] (2006) - (23 Jul - 29 Jul 2006)
Geometric Group theory, Hyperbolic Dynamics and Symplectic Geometry are young and rapidly developing fields of mathematics. The scientific goal of the workshop was twofold: first, to present some recent breakthroughs, and, ...
• #### 0837 - Geometric Group Theory, Hyperbolic Dynamics and Symplectic Geometry

[OWR-2008-42] (2008) - (07 Sep - 13 Sep 2008)
• #### 1229 - Geometric Group Theory, Hyperbolic Dynamics and Symplectic Geometry

[OWR-2012-35] (2012) - (15 Jul - 21 Jul 2012)
The workshop brought together experts in the young and rapidly developing mathematical fields of Geometric group theory, hyperbolic dynamics and symplectic geometry. The activities included talks on recent achievements and ...
• #### 1920 - Geometry and Physics of Higgs Bundles

[OWR-2019-23] (2019) - (12 May - 18 May 2019)
This workshop focused on interactions between the various perspectives on the moduli space of Higgs bundles over a Riemann surface. This subject draws on algebraic geometry, geometric topology, geometric analysis and ...
• #### 1334 - Group Theory, Measure, and Asymptotic Invariants

[OWR-2013-42] (2013) - (18 Aug - 24 Aug 2013)
The workshop ‘Group Theory, Measure, and Asymptotic Invariants’ organized by Miklos Abert (Budapest), Damien Gaboriau (Lyon) and Andreas Thom (Leipzig) was held 18 - 24 August 2013. The event was a continuation of the ...
• #### 1721 - Harmonic Analysis and the Trace Formula

[OWR-2017-25] (2017) - (21 May - 27 May 2017)
The purpose of this workshop was to discuss recent results in harmonic analysis that arise in the study of the trace formula. This theme is common to different directions of research on automorphic forms such as representation ...
• #### 0742 - Harmonische Analysis und Darstellungstheorie Topologischer Gruppen

[OWR-2007-49] (2007) - (14 Oct - 20 Oct 2007)
This was a meeting in the general area of representation theory and harmonic analysis on reductive groups.
• #### Holomorphic automorphic forms and cohomology

[OWP-2014-07] (Mathematisches Forschungsinstitut Oberwolfach, 2014-04-25)
We investigate the correspondence between holomorphic automorphic forms on the upper half-plane with complex weight and parabolic cocycles. For integral weights at least 2 this correspondence is given by the Eichler integral. ...
• #### 1027 - Homogeneous Dynamics and Number Theory

[OWR-2010-29] (2010) - (04 Jul - 10 Jul 2010)
The theory of flows on homogeneous spaces of Lie groups has emerged as a distinct, rapidly advancing subject over the last few decades incorporating ergodic theory, geometry and number theory. The workshop showcased the ...
• #### 0915a - Homotopy Theory of Function Spaces and Related Topics

[OWR-2009-19] (2009) - (05 Apr - 11 Apr 2009)
This workshop brought together researchers studying a variety of problems related to the homotopy theory of function spaces. Topics covered included: evaluation maps and Gottlieb groups, the classiﬁcation of gauge groups ...
• #### 1046a - Infinite Dimensional Lie Theory

[OWR-2010-51] (2010) - (14 Nov - 20 Nov 2010)
The workshop focussed on recent developments in infinite-dimensional Lie theory. The talks covered a broad range of topics, such as structure and classification theory of infinite-dimensional Lie algebras, geometry of ...
• #### 0650 - Infinite Dimensional Lie Theory

[OWR-2006-55] (2006) - (10 Dec - 16 Dec 2006)
The workshop focussed on recent developments in infinite-dimensional Lie theory. The talks covered a broad range of topics, such as structure and classification theory of infinite-dimensional Lie algebras, geometry of ...
• #### Invariant Four-forms and Symmetric Pairs

[OWP-2012-03] (Mathematisches Forschungsinstitut Oberwolfach, 2012-04-24)
We give criteria for real, complex and quaternionic representations to define $s$-representations, focusing on exceptional Lie algebras defined by spin representations. As applications, we obtain the classification of ...
• #### 1603 - Lattices and Applications in Number Theory

[OWR-2016-3] (2016) - (17 Jan - 23 Jan 2016)
This is a report on the workshop on Lattices and Applications in Number Theory held in Oberwolfach, from January 17 to January 23, 2016. The workshop brought together people working in various areas related to the field: ...
• #### 1635 - Measured Group Theory

[OWR-2016-41] (2016) - (28 Aug - 03 Sep 2016)
The workshop aimed to study discrete and Lie groups and their actions using measure theoretic methods and their asymptotic invariants, such as $\ell^2$-invariants, the rank gradient, cost, torsion growth, entropy-type ...
• #### 1215c - Mini-Workshop: Endomorphisms, Semigroups and C*-Algebras of Rings

[OWR-2012-20] (2012) - (08 Apr - 14 Apr 2012)
The main aim of the workshop was to explore recent progress in the study of endomorphisms of $C*$-algebras, semigroup crossed products, graph algebras, ring $C*$-algebras, purely infinite $C*$-algebras and related algebraic ...
• #### 1606a - Mini-Workshop: Operator Spaces and Noncommutative Geometry in Interaction

[OWR-2016-6] (2016) - (07 Feb - 13 Feb 2016)
In recent years, operator space theory has made remarkable appearances in noncommutative geometry, notably in the study of $C^*$-algebras of real reductive groups and the unbounded picture of Kasparov theory. In both these ...
• #### 1915b - Mini-Workshop: Reflection Groups in Negative Curvature

[OWR-2019-17] (2019) - (07 Apr - 13 Apr 2019)
Discrete groups generated by reflections constitute an important source of examples of lattices in simple Lie groups of real rank $1$ (whose associated symmetric spaces are negatively curved). Yet a classification for ...
• #### 1449c - Mini-Workshop: Reflection Positivity in Representation Theory, Stochastics and Physics

[OWR-2014-55] (2014) - (30 Nov - 06 Dec 2014)
The central focus of the workshop was reflection positivity, its occurrence in physics, representation theory, abstract harmonic analysis, and stochastic analysis. The program was intrinsically interdisciplinary and included ...


text:
nLab locally compact topological group

Context

Topology

topology

algebraic topology

group theory

Contents

Definition

A priori a locally compact topological group is a topological group $G$ whose underlying topological space is locally compact.

Typically it is also assumed that $G$ is Hausdorff. (Notice that if not, then $G/\overline{\{1\}}$ is Hausdorff.).

One often says just “locally compact group”.

Properties

In harmonic analysis

We take here locally compact groups $G$ to be also Hausdorff.

Locally compact topological groups are the standard object of study in classical abstract harmonic analysis. The crucial properties of locally compact groups is that they posses a left (right) Haar measure $\rho$ and that $L^1(\rho)$ has a structure of a Banach $*$-algebra.

A left (right) Haar measure on a locally compact topological group is a nonzero Radon measure which is invariant under the left (right) multiplications by elements in the group. A topological subgroup $H$ of a locally compact topological group $G$ is itself locally compact (in induced topology) iff it is closed in $G$.

References

• Gerald B. Folland, A course in abstract harmonic analysis, Studies in Adv. Math. CRC Press 1995

Revised on February 2, 2012 11:50:53 by Urs Schreiber (82.169.65.155)


text:
# Tag Info

## Hot answers tagged random-walk

2

Counterexample Let $$S_n := \sum_{j=1}^n Y_j, \qquad n \in \mathbb{N} \tag{1}$$ a simple random walk on $\mathbb{Z}$, i.e. $Y_j \sim \frac{1}{2} (\delta_1+\delta_{-1})$ independent identically distributed random variables. By Stirling's formula, we have $$\mathbb{P}(S_{2n}=0) = 2^{-2n} {2n \choose n} \sim \frac{1}{\sqrt{\pi n}} \qquad \text{and} \qquad ... 2 As you noted, away from zero the bias is uniformly negative. This suffices to guarantee recurrence, that is, that the process hits zero with full probability. In a nutshell, choose N\gt2(k+1)c, and replace the dynamics on b\geqslant N by b\to b+k with probability c/N and b\to b-1 with probability 1-c/N. Starting from any b(0)\geqslant N, this ... 1 Yes, As long as there are white balls and black balls, the probability of increasing whites W/(B+W) * B/(B+W-1) is equal to probability of increasing blacks B/(B+W) * W/(B+W-1), so the process is as likely to go toward W as to go toward B. But once we reach a state of only one color the probabilities are 1 (for no change) and 0 (for change). The ... 1 For every |z|\leqslant1, the generating function u(x)=E_x(z^\tau) for the random walk starting at x is the unique solution of the integral identity$$ u(x)=z\cdot\int_\mathbb Ru(x+y)g(y)\mathrm dy, $$where g is the standard normal density, with the boundary condition that u(x)=1 for every x\geqslant C. 1$$\sum_{n=2N}^{\infty}p_{ii}^{(n)} = \sum_{n=N}^\infty {2n \choose n} p^n (1-p)^n$$Via the Stirling formula:$${2n \choose n}\sim \frac{2^{2n}}{\sqrt{\pi n}}$$so$${2n \choose n} p^n (1-p)^n \sim \frac{(4p(1-p))^{n}}{\sqrt{\pi n}}$$As p\neq \frac 12, 0\le 4p(1-p)<1 and the series$$\sum {2n \choose n} p^n (1-p)^n and ...

Only top voted, non community-wiki answers of a minimum length are eligible


text:
## Foiled!

March 16, 2010

QuickSort, due to Hoare, is one of the best comparison-based sorting algorithms around. There are a number of variations, but the vanilla QuickSort does a great job… most of the times.

Indeed, there are occasions where QuickSort leaves its expected $O(n \lg n)$ run-time to reach $O(n^2)$ run-time, its worst case. But what does it take to push QuickSort into its worst behaviour?

## Sorting Lists (part II)

June 23, 2009

Last week I showed you the radix sort on simple linked lists. This week, I will present a version of QuickSort modified to sort simply linked lists.


text:
Pełnotekstowe zasoby PLDML oraz innych baz dziedzinowych są już dostępne w nowej Bibliotece Nauki.
Zapraszamy na https://bibliotekanauki.pl

PL EN

Preferencje
Język
Widoczny [Schowaj] Abstrakt
Liczba wyników
• # Artykuł - szczegóły

## Annales Polonici Mathematici

2002 | 78 | 3 | 277-289

## Laurent series expansion for solutions of hypoelliptic equations

EN

### Abstrakty

EN
We prove that any zero solution of a hypoelliptic partial differential operator can be expanded in a generalized Laurent series near a point singularity if and only if the operator is semielliptic. Moreover, the coefficients may be calculated by means of a Cauchy integral type formula. In particular, we obtain explicit expansions for the solutions of the heat equation near a point singularity. To prove the necessity of semiellipticity, we additionally assume that the index of hypoellipticity with respect to some variable is 1.

277-289

wydano
2002

### Twórcy

autor
• Department of Mathematics, University of Oldenburg, D-26111 Oldenburg, Germany


text:
# Why are options studied separately from a portfolio of stocks and bonds?

I have just started studying finance and stochastic calculus so apologies if this question is too naive.

I was first introduced to stocks and bonds as risk and riskless investment assets. Then a new idea is introduced, that of a call option. At first sight, this seems like a new independent investment product, and that you wouldn't be able to create such a portfolio using stocks and bonds only.

But now to calculate the price of this call option, we assume no-arbitrage in our model (Binomial and Black Scholes are the two I have studied so far) and hedge using the stocks and bonds. In other words, just using stocks and bonds we create a portfolio that is the same as a call option. This was surprising to me in a nice way.

But at the same time, it has left me confused. If I could achieve such a portfolio using stocks/bonds then why are call options studied separately?

In fact, it seems to me like introducing call options as an additional investment asset class adds to your "risk dimension". In the sense that, when someone sells a call option, they need to hedge against this risk. So why should someone sell a call option, instead of shorting some stocks or bonds? I agree that options risk will still exist implicitly, but introducing them as a separate asset class just adds to the risk dimension that I would need to hedge.

Also, are there any other merits of options that stocks/bonds alone cannot provide, perhaps in the real world?

"But at the same time, it has left me confused. If I could achieve such a portfolio using stocks/bonds then why are call options studied separately?"

Good question! The short answer: because it turns out options cannot be synthesized using stocks and bonds only, except in the highly idealized case of Black-Scholes which assumes constant volatility, constant interest rates, infinite liquidity, no transaction costs to name but a few sources of replication "slippage".

Hence options should really be regarded as a new source of risk, even perhaps a different asset class.

The uses of options are speculation and/or taking a view without the need to buy the underlying security, but for a large part also hedging the risk of other (more complex) options such as options embedded in structured notes and life insurance guarantees, and to create bespoke investment vehicles that address investors needs for particular risk profiles.

• So if I understand correctly, options cannot be synthesized using stocks/bonds in the real world. But to price them, we create a model where we can synthesize them and this gives us an "approximation" for its price which works out well in the real world? Oct 7, 2021 at 13:38
• Yes you need a model to price them. Blck Scholes is the 'simplest', and you will encounter others such as stochastic vol models etc. All parametric models, no matter how sophisticated, are approximations. Hence, if possible, you'd like model free results. But that's another topic. Oct 7, 2021 at 13:42
• For vanilla options the market price is what is observed in the market. For exotic options it's more difficult to answer. You will see that under the Black Scholes model there is no way you can match market price of vanillas. However if you "abuse" the BS call/put price formula by assigning a different volatility (called implied volatility) to each option strike you can match the market price of vanillas. Feel free of course to post another question or search existing questions and answers when you arrive at the wonderful world of implied vol. Oct 7, 2021 at 13:50
• "because it turns out options cannot be synthesized using stocks and bonds only", I think this can be misinterpreted. It can be, but this synthetization does not work so well in the real world as it does in theory because of the things you mentioned $-$ I would particularly highlight transaction costs and non-constant volatility. Oct 7, 2021 at 14:01
• Indeed an interesting question that is related to the Hakansson's paradox. Oct 7, 2021 at 14:20

Yes, conceptually, you can replicate options with stocks and bonds, but why would you when options are readily available?

Put another way, you can replicate a Coke with the right amount of sugar, water, flavorings, etc., but why would you do that when you can just buy a bottle. How easy are those ingredients to get individually (and in the minute quantities that you need)?

It's impractical to replicate an option exactly with the right about of stocks and bonds. That replication is great for creating pricing models, but is not practical to do in reality.

• Because someone has to start selling coke bottles... someone has to start selling options. Why would they sell an option when they can create an equivalent portfolio for themself using stocks/bonds. Oct 7, 2021 at 13:42
• Again, it's highly impractical to replicate options (whether buying or selling) with stocks and bonds. Replication is much more a theoretical exercise than a practical one. Oct 7, 2021 at 13:58


text:
[Skip Navigation] [CSUSB] / [CNS] / [Comp Sci & Eng ] / [R J Botting] / [ ] / intro_records
[Contents] [Source Text] || [Notation] || [Copyright] || [Contact] [Search ]
Tue May 13 07:34:16 PDT 2008

# Records and Labeled Tuples

## Basics of records and structures

Records/tuples/structures/object_states are shown using the notation of Ada. Here is a record with fields a,b,... and values xa,xb, respectively:
           (a=>xa, b=>xb, ...).
The set of all objects with fields a,b,... of type A,B, .. (respectively) is shown like this:
 	$Net{ a:A, b:B, ... }. Usually however it pays to give a name to the abstract structure  RECORD::=Net{ a:A, b:B, ... }. This allows the structure itself to be referred to, extended, used, inherritted in the definitions of other structures. For complex record structures/classes/signatures/... the long form can be used  RECORD::=following  .Net  ...  a::A.  ...  b::B.  ...  .Close.Net (You can add constraints and comments to these descriptions...) The dollar sign is put in front of the name to indicate the set of object that fit the named structure: $ $RECORD ## Example Class or Set:  phones::=$ Net{name:Strings, number:Strings}.

Object in phones:

 	(name=>"John Smith", number=>"714-123-5678") in phones.

Formally such a set of structures defines (and is defined by) a collection of functions: one for each component. If X=$Net{a:A,b:B,c:C,... } then  a in X -> A,  b in X -> B,  c in X -> C, and if x:X then  a(x)=x.a=the a of x,  b(x)=x.b=the b of x,  c(x)=x.c=the c of x. ## Example - Phone numbers (name=>"John Smith", number=>"714-123-5678").name ="John Smith" (name=>"John Smith", number=>"714-123-5678").number ="714-123-5678" or name(name=>"John Smith", number=>"714-123-5678") ="John Smith" number(name=>"John Smith",number=>"714-123-5678") ="714-123-5678" ## UML and MATHS Nets In UML terms the field names (a,b,c,...) are role names in the class X which would be linked to classes representing A,B,C, ... The link normally has cardinality (0..*) at the X end. The link must have cardinallity 1 at the other end. The reverse role is not (usually) named in MATHS. There is a slight abuse of notation in that the name of the role is also used as the name of the relationship. At the conceptual level it is best to not indicate whether the links are aggregations, generalisations, etc. These physical properties are not shown in the simpler MATHS model. ## Example - pairs 1. A><B::=$ Net{1st:A, 2nd:B}, (a,b) ::=(1st=>a, 2nd=>b), So we can deduce the following formulae must also hold:
2. (a,b).1st=a,
3. (a,b).2nd=b.
4. 1st in A><B->A,
5. 2nd in A><B->B.

## Example - LISP functions

6. CAR::=1st,
7. CDR::=2nd,
8. CONS::=fun[a,b](a,b).

12. $Net{X, P(x,y,z) } can be written for the set of tuples satisfying P. ## Example - a set of tuples 13. circle1::=$ Net{x,y:Real, r:Real&Positive, x^2+y^2<=r^2}.

It is convenient to give a name to a collection of declarations, predicates and definitions. There are two forms - short and long:

## Short form

 S::=Net{ a:A, b:B, c:C, ..., P, Q, ..., x:=e,... }

In this short form all the types can be omitted and then default to being "Sets":

	REA::=Net{ Resources, Entities, Agents }.

## Long Form

 S::=following
 .Net
 	a::A,
 	b::B,
 	c::C,
 	...
 	|-P,
 	|-Q,
 	...,
 	x::=e,
 	...
 .Close.Net

Comments and theorems can also be included in the long form.

 		(reasons)|- Theorem.
Anything that is not recognised as a declaraction, definition, assumption(axiom), formula, or theorem is a comment.

## Example - CIRCLE

14. LONG_CIRCULAR::=following
Net
1. x::Real.
2. y::Real.
3. r::Real & Positive.
4. |- (circle_axiom): x^2+y^2 <= r^2.
5. (above)|- (circle_theorem): if r=0 then x=y=0.

(End of Net)

15. CIRCULAR::=Net{x,y:Real, r:Real&Positive, x^2+y^2<=r^2},
16. circle::=$CIRCULAR. So this definition of circle has the same effect as the definition of circle1 above. We can even prove this be substituting equal eaxpressions: 17. (circle, circle1)|-circle1 = circle. ## Multiple parts in a Structure or Record MATHS doesn't provide a special notation for optional and multiple parts in a structure/signature/class/record. Instead there are types and standard sets that can be used. ## Example -- Days in a Month A month can have a number of days between 27 and 31 -- in the Western calendar. Here are some ways of noting this structure 18. MONTH_HAS_SET_OF_DAYS::= Net{ ..., day: @DAY, 27<= Card(day) <=31, ....}. Days in a month are a set with between 27..31 DAYs. This means that the days are not in order but don't repeat. 19. MONTH_HAS_A_SEQUENCE_OF_DAYS::= Net{ ..., day: #DAY, 27<= Card(day) <=31, ....}. Days in a month are a sequence with between 27..31 DAYs. This means that the days are in order and are numbered 1,2,3,... 20. MONTH_HAS_AN_ARRAY_OF_DAYS::= Net{ ..., ndays:27..31, day: 0..ndays-1>->$ DAY, ....}. Days in a month are an array (map) indexed by a number from 0 to the number of days.

You can abbreviate this to

21. MONTH_HAS_AN_ARRAY_OF_DAYS::= Net{ ..., ndays:27..31, day: $DAYS^0..ndays-1, ....}. ## Example -- 7 days in a week  WEEK::= Net{... day: Days^7, ..., 1st(day)=sunday, ...}. ## Optional parts in a Structure or Record MATHS doesn't provide a special notation for optional parts in a structure/signature/class/record. Instead there are types and standard sets that can be used. When defining data the notation defined in XBNF for option is very helpful: 22. For X:@strings, O(X) = {""} | X. In less concrete cases then one can use a set with no more than one element. ## Example -- optional area code in a phone number  phones::=$ Net{name:Strings, number:Strings, area_code: O(Strings)}.
 	(name=>"John Doe", number=>"123-4567", area_code=>"") \in phones.
 	(name=>"John Doe", number=>"123-4567", area_code=>"909") \in phones.

## Example -- optional air bag in an automobile

 	AUTO ::= Net{ ..., air_bag: @ABU, Card(air_bag) in 0..1, ...}.
 	the AUTO ( make=Ford, model=Escort, air_bag=>{}, ...).
 	the AUTO ( make=Ford, model=Escort, air_bag=>{ford_airbag_17}, ...).

Notice in the above examples the "optional" component is not omitted but is given a null value -- "" or {}. There is a way to define a special kind of object that has extra components. See Inheritance below.

## Inheritance

There is a simple notation when we want to say that one set is a subset of another set, but with certain extra properties:
 		Extended_set::= Super_set with{ Extra_component and properties}

For example:

 	AUTO::= Net{ make:Manufacters, model=Models, ...}.
 	AUTO_WITH_BAG::= AUTO with { air_bag: ABU }.

## Specifying Unique Objects with a Structure

Given a structure S=$N, where  N=Net{a:A, b:B, c:C, ..., P, Q, R, x:=e,...} it is not always necessary to specify all the components to determine an unique object. For example if N=Net{ a,b:Real, a=2*b+1} then there is exactly one object in$ N which has a=3, namely the one with b=1. So
 	(a=>3, b=>1) = the n:N( n.a=3 ).
A shorthand notation can be used:
 	the N(a=3) = (a=>3, b=>1).
In general
 	the N(W) = the $(N and W) = the N with(W). whenever there is exactly one object in$(N and W).

 	the AUTO ( make=Ford, model=Escort, ...) \in $AUTO.  the AUTO_WITH_BAG ( make=Ford, model=Escort, air_bag=>{ford_airbag_17}, ...) \in$ AUTO_WITH_BAG.

However, notice that an AUTO_WITH_BAG is not an AUTO. It contains an AUTO with in it. So, if a ∈ $AUTO_WITH_BAG then, a.AUTO ∈$ AUTO.

## Constrained Structures are Subsets of Type

MATHS lets you include constraints in a structure definition. For example the mathematical structure of a point in a plane as something that has both Cartesian(x,y) and Polar(arg,abs) coordinates, as long as they both refer to the same point:

23. POINT::=following
Net
x,y::Real.
1. arg::Angle.
2. abs::Positive&Real.

3. |-abs^2=x^2+y^2.
4. |-tan(arg)=x/y.

(End of Net)

The symbol
 		|-
used above include an unproved assumption into the meaning of POINT.

This means that

	the POINT(x=>1,y=>1) = the POINT(abs=>1, arg=45.degrees).

Thus at the conceptual, logical, and mathematical level we can document properties that must be true - without describing any means for this to be done.

## Either-Or Structures

There is another way to make structures from components - the discriminated union. This allows an object to have different structures.

 U::=\/{ a:A, b:B, c:C,.... z:Z}
means that for any x:A, (a=>x) in U, for any x in B, (b=>x) in U, and so on. Further, if x:U then
 	type(x) = the Tag t such that (t=>x.t)=x.
Also if S is one of the sets A,B,C,....Z and s:S then
 	s.U = ( t=>s) in U for the 'tag' t.

## More

The above techniques and some other more advanced techniques makes it possible to document the properties of real life entities.

Here are the links into the formal documentation of th notations

Lexemes and Basic Syntax [ notn_10_Lexicon.html ] [ notn_11_Names.html ] [ notn_12_Expressions.html ] [ notn_9_Tables.html ] [ notn_8_Evidence.html ]

Documentation and Ontologies [ notn_13_Docn_Syntax.html ] [ notn_14_Docn_Semantics.html ] [ notn_15_Naming_Documentn.html ] [ notn_16_Classification.html ]

Advanced techniques [ notn_2_Structure.html ] [ notn_3_Conveniences.html ] [ notn_4_Re_Use_.html ] [ notn_5_Form.html ] [ notn_6_Algebra.html ] [ notn_7_OO_vs_Algebra.html ]

. . . . . . . . . ( end of section Records and Labeled Tuples) <<Contents | End>>

# Notes on MATHS Notation

Special characters are defined in [ intro_characters.html ] that also outlines the syntax of expressions and a document.

Proofs follow a natural deduction style that start with assumptions ("Let") and continue to a consequence ("Close Let") and then discard the assumptions and deduce a conclusion. Look here [ Block Structure in logic_25_Proofs ] for more on the structure and rules.

The notation also allows you to create a new network of variables and constraints. A "Net" has a number of variables (including none) and a number of properties (including none) that connect variables. You can give them a name and then reuse them. The schema, formal system, or an elementary piece of documentation starts with "Net" and finishes "End of Net". For more, see [ notn_13_Docn_Syntax.html ] for these ways of defining and reusing pieces of logic and algebra in your documents. A quick example: a circle = Net{radius:Positive Real, center:Point}.

For a complete listing of pages in this part of my site by topic see [ home.html ]

# Notes on the Underlying Logic of MATHS

The notation used here is a formal language with syntax and a semantics described using traditional formal logic [ logic_0_Intro.html ] plus sets, functions, relations, and other mathematical extensions.

For a more rigorous description of the standard notations see

1. STANDARD::= See http://www.csci.csusb.edu/dick/maths/math_11_STANDARD.html

# Glossary

2. above::reason="I'm too lazy to work out which of the above statements I need here", often the last 3 or 4 statements. The previous and previous but one statments are shown as (-1) and (-2).
3. given::reason="I've been told that...", used to describe a problem.
4. given::variable="I'll be given a value or object like this...", used to describe a problem.
5. goal::theorem="The result I'm trying to prove right now".
6. goal::variable="The value or object I'm trying to find or construct".
7. let::reason="For the sake of argument let...", introduces a temporary hypothesis that survives until the end of the surrounding "Let...Close.Let" block or Case.
8. hyp::reason="I assumed this in my last Let/Case/Po/...".
9. QED::conclusion="Quite Easily Done" or "Quod Erat Demonstrandum", indicates that you have proved what you wanted to prove.
10. QEF::conclusion="Quite Easily Faked", -- indicate that you have proved that the object you constructed fitted the goal you were given.
11. RAA::conclusion="Reducto Ad Absurdum". This allows you to discard the last assumption (let) that you introduced.


text:
# Convergence in probability of solution of Geometric Brownian Motion

I am working on the following problem

Given the solution to the Geometric Brownian Motion $$S_t=S(0)\exp\Big[(\mu-\frac{1}{2}\sigma^2)t+\sigma B_t\Big]$$ Where $$\{B_t:t\geq 0\}$$ is a Brownian motion.

a) Show that for $$\mu>\frac{1}{2}\sigma^2$$, we have that $$S(t)\rightarrow\infty$$ as $$t\rightarrow\infty$$.

b) Show that for $$\mu<\frac{1}{2}\sigma^2$$, we have that $$S(t)\rightarrow 0$$ as $$t\rightarrow\infty$$.

I know multiple things that will probably help me in finding the solution, but I'm not able to connect the dots. For question a), we know that the term in the exponent can be split into an exponent with a strictly positive term, and an exponent with the Brownian Motion.

I can't figure out why the exponent with the Brownian motion would go to infinity in question a), but not in question b). In other words, I cannot see why we would have that $$S(0)\exp\Big[ct+\sigma B_t\Big]\rightarrow\infty ,\quad c>0$$But also $$S(0)\exp\Big[dt+\sigma B_t\Big]\rightarrow\infty ,\quad d<0$$ What is the convergence behavior of Brownian Motion in the exponent?

Moreover, we know that the definition of convergence in probability of some sequence $$(X_n)$$ is that $$\forall\epsilon>0:\mathbb{P}(|X_n-X|>\epsilon)\rightarrow 0$$

Any tips are highly appreciated!

• Your solution $S_t$ is incorrect. It should read $S_t=S_0\exp\left[(\mu-0.5\sigma^2)t +\sigma B_t\right]$. Now does the limiting behavior make more sense? – LoveTooNap29 Mar 4 at 18:04
• @LoveTooNap29 Ah, thanks for the correction. It makes slightly more sense that way but the thing that is confusing me is what exactly happens with the term $$exp(\sigma B_t)$$ as $t\rightarrow\infty$? I imagine this term does not converge at all but then again I am not sure. – S. Crim Mar 4 at 18:42
• It isn't necessary that that term alone converge. We just care about the combination of the terms $\exp\left(ct + \sigma B_t\right)$ (where $c < 0$). So perhaps you could start by analysing what happens with $ct + \sigma B_t$ as $t\to \infty$, if $c <0$. – Minus One-Twelfth Mar 4 at 19:17
• @MinusOne-Twelfth I see. I think I am able to solve the problem now. Could you explain why or why not we only have convergence in probability and not also convergence almost surely? – S. Crim Mar 5 at 8:00
• Possibly this page can be of help: math.stackexchange.com/questions/1241149/… – Minus One-Twelfth Mar 5 at 8:34


text:
Solving ODE having function which changes conditionally

I want to solve the following set of ODE with a piecewise function A[t] which changes if a certain condition is satisfied.

eq1 = y1''[t] + 0.5 y2'[t] + 0.7 y1[t] + A[t] == 0;

eq2 = y2''[t] + 0.3 y1'[t] + 1.3 y2[t] == 0;

A[t]=0 when y1[t]<=y2[t] for all t

A[t]=Sin[t] when y1[t]>y2[t] for all t

y1'[0] == 1, y1[0] == 0, y2'[0] == 1, y2[0] == 0

I have tried with WhenEvent inside NDSolve, but it only gives correct result if A[t] is purely discrete using 'DiscreteVariables' option. The following is the code for that.

ti = 0; tf = 30;

eq1 = y1''[t] + 0.5 y2'[t] + 0.7 y1[t] + A[t] == 0;

eq2 = y2''[t] + 0.3 y1'[t] + 1.3 y2[t] == 0;

sol = NDSolve[{eq1, eq2, y1'[ti] == 1, y1[ti] == 0, y2'[ti] == 1, y2[ti] == 0,
A[ti] == If[y1[ti] <= y2[ti], 0, 10], {WhenEvent[y1[t] <= y2[t],
A[t] -> 0], WhenEvent[y1[t] > y2[t], A[t] -> 10]}}, {y1, y2, A}, {t,
ti, tf}, DiscreteVariables -> A][[1]];

Plot[Evaluate[{y1[t], y2[t], A[t]} /. sol], {t, ti, tf},
PlotRange -> All, PlotLegends -> Automatic]

So, please help and suggest if I want to use A[t] as mentioned earlier.

• Why not use Piecewise[]? – J. M. is away May 18 '15 at 13:23
• @Guesswhoitis. Piecewise[] is also not working. I tried with this:--- ti = 0; tf = 30; A[t_] := Piecewise[{0, y1[t] <= y2[t]}, {10*Sin[t], y1[t] > y2[t]}] eq1 = y1''[t] + 0.5 y2'[t] + 0.7 y1[t] + A[t] == 0; eq2 = y2''[t] + 0.3 y1'[t] + 1.3 y2[t] == 0; sol = NDSolve[{eq1, eq2, y1'[ti] == 1, y1[ti] == 0, y2'[ti] == 1, y2[ti] == 0}, {y1, y2, A}, {t, ti, tf}][[1]]; Plot[Evaluate[{y1[t], y2[t]}], {t, ti, tf}, PlotRange -> All, PlotLegends -> Automatic] – Soumyajit Roy May 18 '15 at 13:38
• @Guesswhoitis. But, defining A[t] as piecewise function prior to NDSolve is not solving the issue, because there is no y1[t] and y2[t] values to check before solving the eqns. So, I think, the checking is to be done in every time step (t) within NDSolve. – Soumyajit Roy May 18 '15 at 13:46
• Welcome to Mathematica.SE! I suggest that: 1) You take the introductory Tour now! 2) When you see good questions and answers, vote them up by clicking the gray triangles, because the credibility of the system is based on the reputation gained by users sharing their knowledge. Also, please remember to accept the answer, if any, that solves your problem, by clicking the checkmark sign! 3) As you receive help, try to give it too, by answering questions in your area of expertise. – bbgodfrey May 18 '15 at 14:09

Here is one way to use Piecewise, which might well be what Guess who it is. had in mind.

ti = 0; tf = 30;
eq1 = y1''[t] + 0.5 y2'[t] + 0.7 y1[t] + A[t] == 0;
eq2 = y2''[t] + 0.3 y1'[t] + 1.3 y2[t] == 0;

{sol} = NDSolve[{
eq1, eq2, y1'[ti] == 1, y1[ti] == 0, y2'[ti] == 1, y2[ti] == 0,
A[t] == Piecewise[{{Sin[t], y1[t] > y2[t]}}, 0]},
{y1, y2, A}, {t, ti, tf}];

Plot @@ {Through[{y1, y2, 10 A[#] &}[t]] /. sol, Flatten[{t, y1["Domain"] /. sol}]}

• That's exactly it, yes. :D (Please settle for an upvote IOU for the time being…) – J. M. is away May 18 '15 at 16:22


text:
# At a certain time in a reaction, substance a is disappearing at a rate of 4.0×10−2 m/s, substance

2 answers
###### Question:

At a certain time in a reaction, substance a is disappearing at a rate of 4.0×10−2 m/s, substance b is appearing at a rate of 2.0×10−2 m/s, and substance c is appearing at a rate of 6.0×10−2 m/s. which of the following could be the stoichiometry for the reaction being studied?
(a) 2 a → b + 3 c
(b) a → 2 b + 3 c
(c) 2 a + b → 3 c
(d) a + 2 b → 3 c
(e) 4 a → 2 b + 3 c

## Answers

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