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Show that something cannot be a ring
February 24th 2009, 12:55 PM #1
Feb 2009
Ring question
I have been asked to show that the set of n x n real matrices with determinant 1 is not a ring.
I've been fine with this topic so far, but this question stumped me. I'm not sure how I can go about manipulating examples of n x n matrices with the only piece of information that I have being
that det(M) = 1. The only thing I can think of is linking it in some way with invertible matrices.
Last edited by FractalMath; February 24th 2009 at 11:57 PM.
Well, in order for it to be a ring, it would have to be an abelian group under addition right? In particular, it would have to contain the additive identity, but what is the additive identity?
What is its determinant?
February 24th 2009, 02:23 PM #2
Jul 2008
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Pembroke Pines SAT Math Tutor
Find a Pembroke Pines SAT Math Tutor
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I have been tutoring for over 20 years at the high school level (mainly private tutoring and 6 years at the University level). I am extremely passionate about my students success and will go the
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14 Subjects: including SAT math, chemistry, calculus, geometry
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Foundation Maths supports the UK School Curriculum Early Years Foundation Stage and Key Stage 1. Suitable for children aged 3 to 6 and voiced with a lovely soft English accent giving positive verbal
encouragement throughout.
Foundation Maths includes 500 maths questions across 4 skill levels in 8 topic areas
- Sums can be read out loud with a tap of the speaker button
- Help in each section is given with a tap of the help button
Eight Topic Areas to Learn:
- Learn the Numbers: learn the operator symbols and the numbers 1 to 30 with fruit to count
- Learn to Count: Random fruit are shown and the child counts and picks the right number
- Number Patterns: Children find the missing number in a sequence of 5 numbers
- Write the Numbers: Learn to correctly form the numbers 1 - 9 by tracing dotted number forms
- Addition: Learn to add two numbers together
- Subtraction: Children learn to take one number from another
- Multiplication: One number multiplied by another
- Division: One number divided by another
Four Levels to Progress Through:
- Beginner: Simple questions with answers up to 10 and illustrations to help young children
- Intermediate: Questions with answers up to 20
- Advanced: Questions with answers up to 30
- Genius: Algebra questions, the answer is given and the child finds the missing part of the question.
Foundation Maths supports the following Early Years learning goals:
- Counting and using numbers to at least 10 in familiar contexts
- Recognising numerals 1 to 9
- Beginning to understand addition as 'combining' two groups of objects
- Beginning to understand subtraction as 'taking away'
- Using early mathematical ideas to solve practical problems.
Foundation Maths also supports the following Key Stage 1 learning outcomes:
- Use the correct language, symbols and vocabulary associated with number and data
- Count reliably up to 20 objects at first and recognise, be familiar with the numbers 11 to 20; gradually extend counting.
- Create and describe number patterns; explore and record patterns related to addition and subtraction, and then patterns of multiples of 2, 5 and 10 explaining the patterns and using them to make
predictions; recognise sequences, including odd and even numbers to 30 then beyond; recognise the relationship between halving and doubling.
- Use the symbol '=' to represent equality; solve simple missing number problems [for example, 2 + ? = 6 ]
We hope you enjoy Foundation Maths and find it useful in teaching your children some basic maths skills, if you have any suggestions or queries we would love to hear from you!
Email contact@readwritephonics.co.uk or tweet @readwritephonic if you have any questions or problems - remember we can't reply to comments!
Level up your mathematics skills and become King of Math!
King of Math is a fast-paced mathematics game with lots of fun and diverse problems in different areas. Starting as a male or female farmer, you level up your character by answering math questions
and improving your total score. New character design and music for each of the ten levels. Collect stars, get achievements and compare your scores against your friends and players all over the world!
Playing King of Math is a great way to improve or refresh you mathematical skills and you will have a lot of fun doing it! The mathematics level is about Middle School/Junior High School.
The game includes:
- Addition
- Subtraction
- Mixed 1
- Multiplication
- Division
- Arithmetic
- Geometry
- Fractions
- Powers
- Statistics
- Equations
- Mixed 2
Math BINGO
The object of Math BINGO is to practice math facts while playing BINGO!
-Choose from 5 games: Addition, Subtraction, Multiplication, Division and Mixed
-Choose from 3 different levels of difficulty: Easy, Medium and Hard
-Create up to 5 player profiles
-Choose from 8 different fun cartoon avatars
-Keep track of number of games played by player profile
-The Scoreboard keeps track of scores for each game and level
-Collect and play with BINGO Bugs when you earn a high score!
-Fun bonus game: BINGO Bug Bungee
How to Play and Win Math Bingo
The object of Math Bingo is to get a pattern of five Bingo Bugs in a row by correctly answering math problems. Math problems are presented at the top of the game screen. Feedback is presented at the
bottom of the game screen. Correct solutions to problems answered incorrectly will be displayed.
The score in Math Bingo is determined by the time to complete a game plus a two second penalty for each problem answered incorrectly.
Time to complete game 0:45 seconds
Incorrect answers: 3
Score: 45+6=51
Player Profiles
Math Bingo saves player information with custom player profiles. Each player profile consists of a unique name (up to 20 characters) and a fun cartoon avatar. The player profiles display total number
of games played, total number of high scores achieved, and total number of Bingo Bugs collected. The guest profile does not store any information and therefore cannot achieve high scores or collect
Bingo Bugs. The guest profile is useful for adults who want to play Math Bingo without their scores going into the scoreboard.
Selecting a Game
There are five different games in Math Bingo: Addition, Subtraction, Multiplication, Division and Mixed. Players will be prompted to select a game type before game play.
Selecting a Level
There are three levels of game play in Math Bingo: easy, medium, and hard. Players will be prompted to select a game level before game play.
Score Board
Math Bingo displays high scores in the Score Board. There are four different Score Boards: addition, subtraction, multiplication and division. Each Score Board displays the top three scores for each
level: easy, medium, and hard.
Bingo Bugs
Bingo Bugs are rewards for achieving a high score. A player who earns a high score will collect a Bingo Bug! Players can collect a variety of Bingo Bugs and interact with them by tapping the Bingo
Bug button in the player profile or by tapping the My Bingo Bugs button after playing a game. HINT: Tilt your device to make the Bingo Bugs move around, tap on a Bingo Bug to make them giggle and
The settings menu can be accessed by clicking the information button on the lower right hand side the introduction screen. The settings menu will allow you to reset all high scores and to delete
player profiles.
Math BINGO:
- Does not contain third-party ads.
- Does not contain in-app purchases.
- Does not contain integration with social networks.
- Does not use analytics/data collection tools.
- Does include links to www.abcya.com
For more information on our privacy policy please visit:
Browse over 1,400 formulas, figures, and examples to help you with math, physics, chemistry and more. Use an expanding list of helpful tools such as a unit converter, quadratic solver, and triangle
solver to perform common calculations.
* 2nd Place Best Young Adults App - 2011 Best App Ever Awards *
* 3rd Place Best Education App - 2012 Best App Ever Awards *
Current Main Categories:
- Algebra
- Geometry
- Trigonometry
- Linear Algebra
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- Derivatives
- Integration
- Table of Integrals
- Vector Calculus
- Differential Equations
- Discrete
- Probability and Statistics
- Physics
- Chemistry (includes Periodic Table)
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- Financial (includes Real Estate)
- Prime Numbers
- Greek Alphabet
- Tools ranging from Algebra to Physics
Anything you'd like to see, email us at: info@happymaau.com
See our site www.happymaau.com for more information.
*Network access only needed for sending feedback.
"We were amazed by the large variety of exercises" - Top Kids Apps
"A great help to children who are trying to sharpen their math skills" - The iPhone Mom
"I like the variety of ways that problems are presented. . . . It gives the kids different concepts related to the arithmetic they’re doing." - GeekDad
King of Math Junior is a mathematics game in a medieval environment where you climb the social ladder by answering math questions and solving puzzles. Collect stars, get medals and compete against
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King of Math Junior is suitable for age 6 and up and introduces mathematics in an accessible and inspiring manner. Its educational strength lies in awakening curiosity and putting mathematics in a
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Kids Maths is a great app perfect for preschoolers to learn about numbers and counting. The kid can have hands on following activities:
* Learn Numbers- Choose from Play mode, previous and
next buttons
* Count and tell
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* Find bigger number
* Find Smaller number
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* Draw and Learn- Practice to draw the numbers within the stencil with color of your choice.
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KS1 (Key Stage 1) Maths presents maths exercises for kids aged 5-7 years. Thousands of maths exercise are available to practice which kids will like and will enjoys learning. There are various of
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Basic Counting
Counting in Multiples
Numbers up to 100
Comparing Numbers
Numbers in Words
Additions with pictures
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Subtraction - Numbers upto 10
Additions - Numbers upto 20
Subtraction - Numbers upto 20
Multiplication with pictures
Geometry - 2D Shapes
Geometry - 3D Shapes
Money - Recongise the coins
Money - count the money with 2 coins
Money - count the money with 2+ coins
Time - Read the Clock (hour and half past the hour...
Time - Match the clock and time
Year 2:
Skip Counting with pictures
Skip Counting - Forward/Backward
Place Values (tens, units)
Number Sequence/Line
Comparing Numbers
Numbers in words - upto 100
Addition with Objects
Additions - A two- digit number and ones
Additions - A two- digit number and tens
Additions - Two two-digit numbers
Additions - Adding three one-digit numbers
Additions - Making Tens
Additions - Word Problems
Subtraction - With Objects
Subtraction - A one-digit number from two-digit nu...
Subtraction - A two-digit number from two-digit nu...
Subtract tens
Subtraction - Word Problems
Multiplication sentences
Multiplication tables (2, 5 and 10)
Multiplication tables (up to 10)
Division (by 2, 5, and 10)
Division sentences
Fractions - recognise fractions
Fractions - name the fraction
Fractions - match the fraction
Measurement - Units
Measurement - Units for Length
Measurement - Units for Weight/Mass
Geometry - Identify 2D Shapes
Geometry - Identify 3D Shapes
Geometry - Properties of 2D Shapes
Geometry - Properties of 3D Shapes
Geometry - Shapes of everyday objects
Money - count the money
Time - Read the Clock (hour,half past the hour, quarter past hour)
Time - Match the clock and time
The principal focus of mathematics teaching in key stage 1 is to ensure that pupils develop confidence and mental fluency with whole numbers, counting and place value. This should involve working
with numerals, words and the four operations, including with practical resources [for example, concrete objects and measuring tools ].
At this stage, pupils should develop their ability to recognise, describe, draw, compare and sort different shapes and use the related vocabulary. Teaching should also involve using a range of
measures to describe and compare different quantities such as length, mass, capacity/volume, time and money.
By the end of year 2, pupils should know the number bonds to 20 and be precise in using and understanding place value. An emphasis on practice at this early stage will aid fluency.
Pupils should read and spell mathematical vocabulary, at a level consistent with their increasing word reading and spelling knowledge at key stage 1
★ Math for kids is either a big task or a milestone. Teaching your child math isn't something that happens overnight.
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★ Kids Learn Math ( Kids Math ) is an interactive app which makes learning maths easy, simple and fun which helps kids learn faster.( Kids 1-20 Kids Math )
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- Tap on the largest number from the numbers shown.
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- Tap on a tile to turn it over and reveal a number. Then tap on the second tile, trying to find the matching number. If the images match, the tiles are removed from the board.
★ It is said that Humans are only fully Human when they Play; well, then kids will be only fully kids when they play. Even better if they learn while playing! Research also shows that kids, who enjoy
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Maths Formulae lists some of the useful maths formulas that helps you as a quick reference (cheat sheet).Using this App you can access the formulas anywhere and anytime you want.It covers formulas
for College Grade/Higher Grade Students/School Students.It lists out all the important formulas/topics in Algebra, Geometry, Trigonometry and Statistics.Review these formulas/concepts regularly and
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1.) ALGEBRA
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3.) TRIGONOMETRY
- Angles
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4.) STATISTICS
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The application "math expert" is a collection of formulas out of mathematics and physics. The special feature is that the application can calculate the formulas. The calculation is based on the motto
"Tell me what you know, and I will check which calculations are possible."
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Solve Math problems and plot functions.
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Basic Algebra
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What's 51 x 51?
Can your child do this in less than five seconds?
Well, he really can do! This Mental Maths app helps improve your mental calculation skills through introduction to various methods & exercises. This app has the most exhaustive content around mental
maths. With this app, our goal is to provide you easy to read material which you can read and follow at your own pace. Chapters are written in conversational language with lot of pictorial
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Please send in your feedback and comments to exploreinandroid2010@gmail.com
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This is a algebra calculator that solves algebra for you, and gives you answers teachers will accept! Unlike other similar apps this app is built directly for students. The answers that appear will
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and is a calculator that calculates square roots, cubed roots and any other root. We have added the foil method and exponents to the list of problems that can be solved. Reduce, add, subtract, divide
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Math Helper Free solves math problems and shows step-by-step solution.
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MathHelper is a universal assistant for anyone who has to deal with higher mathematics, calculus, algebra. You can be a student of a college or graduate, but if you suddenly need emergency assistance
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Train your brain, learn math tricks and amaze others!
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11 * 86 = 946
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Check out the screenshots and see for yourself!
*** Issue with our app? - Please send us an email: support@jmtapps.com ***
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Follow us on Twitter: https://www.twitter.com/whitesof
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Use Horizontal swipe on individual formulas if content not fully displayed
Mathematical Formulas aggregated in most useful way.
Please email us at " contactus@whitesof.com" to add any new formulas or suggestions or topics.
The App covers topics like
9)Analytical Geometry
10)Boolean Algebra
11) Series
The app is continuously updated with latest details and added with new topics frequently.
Math BINGO is a fun way for children to practice math facts on your phone/tablet. Choose from addition, subtraction, multiplication or division BINGO, then select a level of difficulty.
More from developer
Teach your child to read and write the easy way!
Read Write Phonics teaches the 44 phonic sounds of the English language, how to blend the phonic sounds together to form words, and how to write the letters of the alphabet by tracing dotted letter
- Read: learn the correct 44 phonic sounds of the English language!
- Write: learn to write the letters of the English alphabet, with star rewards!
- Phonics: learn how to blend the phonic sounds to form words!
Learn by doing: tap and swipe through the App listening to the correct sounds and blending the sounds into words.
Fun encouragement and reward stars are given in the Write section.
Clear and simple flash cards system with simple intuitive navigation.
Clear Sassoon font developed especially for children’s first books and traditional learning aids.
Clean design, there are no distracting images or sounds in the Write or Phonic sections.
Focus on learning the sounds and learning how to form letters, without distraction.
The example word cards in the Read section each use one simple bright word image.
Move through the App using the arrow keys, the speaker button replays sounds and the redo button restarts animation. The home key takes you up a menu level and the envelope key will open an email to
us at Read Write Phonics.
The Read section contains a flash card for each of the 44 sounds, showing the letter or letter combination which represent them. Once the sound has played wait for the letter/s to animate into an
example word, tapping the speaker button plays the sound or word again.
The Write section shows an example of how letters are written, then allows the child to try tracing the dotted letter, with a prompt where to begin and a prompt if they forget to dot the i or cross
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stars are achieved.
The Phonics section includes small, two and three letter words and big, four letter and above example words. Tapping the speaker button plays the full word, tapping each letter or letter combination
pays the phonic sound. Swiping or stroking through the word plays the sounds in order blending or synthesising the word.
(Note: the ck sound has been included three times with cards, c, ck, and k all making the same phonic sound. qu and x also included for completeness but are actually consonant blends (k + w) and (k +
s). The letters and letter combinations chosen for read write phonics are the most common ones and the ones taught first for example “ee” in “bee” instead of “y” in “many”)
Email contact@readwritephonics.co.uk or tweet @readwritephonic if you have any questions or problems - remember we can't reply to comments!
This is a free *demonstration* version of Read Write Phonics and includes 10 sounds, 8 letters to try tracing and 4 words to blend.
The full version of Read Write Phonics is £1.49 and teaches the 44 phonic sounds of the English language, how to blend the phonic sounds together to form words, and how to write the letters of the
alphabet by tracing dotted letter forms!
- Read: learn the correct 44 phonic sounds of the English language!
- Write: learn to write the letters of the English alphabet, with star rewards!
- Phonics: learn how to blend the phonic sounds to form words!
Learn by doing: tap and swipe through the App listening to the correct sounds and blending the sounds into words.
Fun encouragement and reward stars are given in the Write section.
Clear and simple flash cards system with simple intuitive navigation.
Clear Sassoon font developed especially for children’s first books and traditional learning aids.
Clean design, there are no distracting images or sounds in the Write or Phonic sections.
Focus on learning the sounds and learning how to form letters, without distraction.
The example word cards in the Read section each use one simple bright word image.
Move through the App using the arrow keys, the speaker button replays sounds and the redo button restarts animation. The home key takes you up a menu level and the envelope key will open an email to
us at Read Write Phonics.
The Read section contains a flash card for each of the 44 sounds (in the full version), showing the letter or letter combination which represent them. Once the sound has played wait for the letter/s
to animate into an example word, tapping the speaker button plays the sound or word again.
The Write section shows an example of how letters are written, then allows the child to try tracing the dotted letter, with a prompt where to begin and a prompt if they forget to dot the i or cross
the ts! Then wait for your mark, 0 to 3 stars! The App marks the child's effort based on the smoothness of their line and the distance from the ideal line. Encouragement is always given even if no
stars are achieved.
The Phonics section includes small, two and three letter words and big, four letter and above example words. Tapping the speaker button plays the full word, tapping each letter or letter combination
pays the phonic sound. Swiping or stroking through the word plays the sounds in order blending or synthesising the word.
Email contact@readwritephonics.co.uk or tweet @readwritephonic if you have any questions or problems - remember we can't reply to comments!
This is a free *demonstration* version of Foundation Maths and includes 60 questions and demonstrations of each section. The full version includes 500 questions.
Foundation Maths supports the UK School Curriculum Early Years Foundation Stage and Key Stage 1. Suitable for children aged 3 to 6 and voiced with a lovely soft English accent giving positive verbal
encouragement throughout.
Foundation Maths includes 500 maths questions across 4 skill levels in 8 topic areas
- Sums can be read out loud with a tap of the speaker button
- Help in each section is given with a tap of the help button
Eight Topic Areas to Learn:
- Learn the Numbers: learn the operator symbols and the numbers 1 to 30 with fruit to count
- Learn to Count: Random fruit are shown and the child counts and picks the right number
- Number Patterns: Children find the missing number in a sequence of 5 numbers
- Write the Numbers: Learn to correctly form the numbers 1 - 9 by tracing dotted number forms
- Addition: Learn to add two numbers together
- Subtraction: Children learn to take one number from another
- Multiplication: One number multiplied by another
- Division: One number divided by another
Four Levels to Progress Through:
- Beginner: Simple questions with answers up to 10 and illustrations to help young children
- Intermediate: Questions with answers up to 20
- Advanced: Questions with answers up to 30
- Genius: Algebra questions, the answer is given and the child finds the missing part of the question.
Foundation Maths supports the following Early Years learning goals:
- Counting and using numbers to at least 10 in familiar contexts
- Recognising numerals 1 to 9
- Beginning to understand addition as 'combining' two groups of objects
- Beginning to understand subtraction as 'taking away'
- Using early mathematical ideas to solve practical problems.
Foundation Maths also supports the following Key Stage 1 learning outcomes:
- Use the correct language, symbols and vocabulary associated with number and data
- Count reliably up to 20 objects at first and recognise, be familiar with the numbers 11 to 20; gradually extend counting.
- Create and describe number patterns; explore and record patterns related to addition and subtraction, and then patterns of multiples of 2, 5 and 10 explaining the patterns and using them to make
predictions; recognise sequences, including odd and even numbers to 30 then beyond; recognise the relationship between halving and doubling.
- Use the symbol '=' to represent equality; solve simple missing number problems [for example, 2 + ? = 6 ]
We hope you enjoy the Foundation Maths DEMO and find it useful in teaching your children some basic maths skills, if you have any suggestions or queries we would love to hear from you!
Email contact@readwritephonics.co.uk or tweet @readwritephonic if you have any questions or problems - remember we can't reply to comments!
300 High Frequency Words is a great interactive way to teach your child to recognise and read the first 300 high frequency words of the English language. These are the common words such as: the, and,
said, he, I, be, they, go, etc etc. Some of these words are decodable using phonics but some have to be learnt as "Sight" or Red" words. Giving your child a way to quickly learn these will enable
them to read more challenging books more quickly. Tapping the screen voices the word in a soft English accent.
300 High Frequency Words is split into 8 manageable sets of words starting at the most common words moving through to the least common.
Within each set the words are shown in a different order each time the app is used, this ensures that your child cannot simply learn the order of the words.
300 High Frequency words can be used by a child alone tapping each word to hear it and then tapping the arrow to move through to the next one learning the words at their own pace.
Parents can use the app in a similar way to traditional flash cards to support learning and test their child's knowledge, asking them to read the word, then tap the word to confirm or correct, then
move to the next.
Once a child grows in confidence you can see how quickly they can move through each set of words timing and offering rewards for shorter and shorter times. The quicker they can move through the set
the sooner they will be able to use the knowledge in reading and enjoying books.
The 300 High Frequency Words are the same words found in Appendix 7 of the Letters and Sounds: Principles and Practice of High Quality Phonics document produced by the UK Department for Education and
commonly used in UK schools.
300 High Frequency Words DEMO is a *demonstration* version including 40 of the 300 words in the full App. This is a great interactive way to teach your child to recognise and read the first 300 high
frequency words of the English language. These are the common words such as: the, and, said, he, I, be, they, go, etc etc. Some of these words are decodable using phonics but some have to be learnt
as "Sight" or Red" words. Giving your child a way to quickly learn these will enable them to read more challenging books more quickly. Tapping the screen voices the word in a soft English accent.
300 High Frequency Words is split into 8 manageable sets of words starting at the most common words moving through to the least common.
This DEMO version includes 40 words, 5 sample words in each set.
Within each set the words are shown in a different order each time the app is used, this ensures that your child cannot simply learn the order of the words.
300 High Frequency words DEMO can be used by a child alone tapping each word to hear it and then tapping the arrow to move through to the next one learning the words at their own pace.
Parents can use the app in a similar way to traditional flash cards to support learning and test their child's knowledge, asking them to read the word, then tap the word to confirm or correct, then
move to the next.
Once a child grows in confidence you can see how quickly they can move through each set of words timing and offering rewards for shorter and shorter times. The quicker they can move through the set
the sooner they will be able to use the knowledge in reading and enjoying books.
The 300 High Frequency Words are the same words found in Appendix 7 of the Letters and Sounds: Principles and Practice of High Quality Phonics document produced by the UK Department for Education and
commonly used in UK schools.
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David G. Bonagura Jr. - Common Core’s Newer Math
The following sentences from the New York Times could have been written today in homage to the Common Core Standards Initiative, the recently adopted national standards for the teaching of
mathematics and English-language arts in grades K–12.
“Instead of this old method, the educators would stress from the earliest grades the new concept of the unity of mathematics and an understanding of its structure, using techniques that have been
developed since the turn of the century. . . . The new concepts must be taught in high school to prepare the students for the type of mathematics that they will find when they reach college.”
But the century in question here is the 20th, not the 21st. This article, written in 1961, is not about today’s Common Core, but about New Math, the program that was supposed to transform mathematics
education by emphasizing concepts and theories rather than traditional computation. Instead, after a few short years of propagating ignorance of all things mathematical, New Math became the butt of
jokes nationwide (the Peanuts comic strip took aim more than once) before it was unceremoniously abandoned.
Flash forward 50 years, and Common Core is today making the same
: “The standards are designed to be robust and relevant to the real world, reflecting the knowledge and skills that our young people need for success in college and careers. With American students
fully prepared for the future, our communities will be best positioned to compete successfully in the global economy.”
But what makes us think Common Core will live up to its hype? And how is it substantially different from New Math, as well as subsequent math programs such as Sequential Math, Math A/B, and the
National Council of Teachers of Mathematics Standards? These have all failed America’s children — even though each program promised to transform them into young Einsteins and Aristotles.
The problem with Common Core is not that it provides standards, but that, despite its claims, there is a particular pedagogy that accompanies the standards. And this pedagogy is flawed, for, just as
in New Math, from the youngest ages Common Core buries students in concepts at the expense of content.
Take, for example, my first-grade son’s Common Core math lesson in basic subtraction. Six- and seven-year-olds do not yet possess the ability to think abstractly; their mathematics instruction,
therefore, must employ concrete methodologies, explanations, and examples. But rather than, say, count on a number line or use objects, Common Core’s standards mandate teaching first-graders to
“decompose” two-digit numbers in an effort to emphasize the concept of place value. Thus 13 – 4 is warped into 13 – 3 = 10 – 1 = 9. Decomposition is a useful skill for older children, but my
first-grade son has no clue what it is about or how to do it. He can, however, memorize the answer to 13 – 4. But Common Core does not advocate that tried-and-true technique.
Common Core’s elevation of concept over computation continues in its place-value method for multiplying two-digit numbers, which is taught in fourth grade. Rather than multiply each digit of the
number from right to left, Common Core requires students to multiply each place value so that they have to add four numbers, rather than two, as the final step in finding the product.
Common Core’s most distinctive feature is its insistence that “mathematically proficient students” express understanding of the underlying concepts behind math problems through verbal and written
expression. No longer is it sufficient to solve a word problem or algebraic equation and “show your work”; now the work is to be explained by way of written sentences.
I have seen this “writing imperative” first-hand in my sons’ first- and third-grade Common Core math classes. There is certainly space in their respective books for traditional computation, but the
books devote enormous space to word problems that have to be answered verbally as well as numerically, some in sections called Write Math. The reason, we are told, is that the Common Core–driven
state assessments will contain large numbers of word problems and spaces for students to explain their answers verbally. This prescription immediately dooms grammar-school students who have reading
difficulties or are not fluent in English: The mathematical numbers that they could have grasped are now locked into sentences they cannot understand.
The most egregious manifestation of the “writing imperative” is the Four Corners and a Diamond graphic organizer that my sons’ school has implemented to help prepare for the writing portion of the
state assessments. The “fourth corner” requires students to explain the problem and solution in multiple sentences. How all this writing helps them with math is yet to be demonstrated.
Hence Common Core looks terribly similar to the failed New Math program, which also emphasized “the why rather than the how, the fundamental concepts that unify the various specialties, from
arithmetic to the calculus and beyond, rather than the mechanical manipulations and rule memorizations.” Common Core may not completely eschew the “how,” and it may not be obsessed with binary sets
and matrices as New Math was, but it is likely to lose the “how” — the content — in its efforts to move the “why” — the concepts — into the foreground.
The problem is not that students, including those in the primary grades, should not be presented the basic concepts of mathematics — they should be. But there is a difference between learning basic
concepts and expressing the intricacies of true mathematical proofs that Common Core desires. Mathematical concepts require a high aptitude for abstract thinking — a skill not possessed by young
children and never attained by many. What will happen to students who already struggle with math when they not only are forced to explain what they do not understand, but are presented new material
in abstract conceptual formats?
All students must learn to perform the basic mathematical operations of addition, subtraction, multiplication, and division in order to function well in society. Knowing why these operations work as
they do is a great benefit, but it is not essential. And in mathematics, concepts are often grasped long after students have mastered content — not before.
In trying to learn both the “why” and the “how” in order to prepare for the state assessments, students will not fully grasp either: They will not receive the instructional time needed to learn how
to do the operations because teachers will be forced to devote their precious few classroom minutes to explaining concepts, as the assessments require. The “how” of the basic operations, which need
to be memorized and practiced over and over, will be insufficiently learned, since Common Core orders teachers to serve two masters.
The result is simple arithmetic: Instead of developing college- and career-ready students, we will have another generation of students who cannot even make change from a $5 bill, all courtesy of the
latest set of bureaucrat-promoted standards that promise to save American education.
By giving concept priority over content, Common Core has failed to learn the history lesson from New Math. Students instructed according to Common Core standards will ultimately know neither the
“why” nor the “how,” and we will eventually consign these standards to the ever-expanding dustbin of failed educational initiatives, until the next messianic program is unveiled.
And, of course, this doomed educational experiment, like its predecessors, has a high cost: our children’s ability to do math.
— David G. Bonagura Jr. is a teacher and writer in New York. He has written about education for Crisis, The Catholic Thing, The University Bookman, and the Wall Street Journal.
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Billerica Statistics Tutor
Find a Billerica Statistics Tutor
...I also used excel to show that the area and parameter of a regular polygon approaches the area and circumference of a circle as n (number of sides) approaches infinity. I've built on these
results to show that the volume of a pyramid approaches a cone as n approaches infinity. I've used tables to show the relation between numbers of sides to the sum of interior and exterior angles.
13 Subjects: including statistics, physics, ASVAB, algebra 1
...I am a second year graduate student at MIT, and bilingual in French and English. I earned my high school diploma from a French high school, as well as a bachelor of science in Computer Science
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...A full Precalculus contains a great deal of material and is often a daunting task for the student. I have taught at the University level as the professor for Precalculus. I take the topics in
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...Consulting me on study skill techniques will make your study time efficient and effective. I was a Special Education Teacher in ELA and Math for students with moderate special needs in grades
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Psychology 100a > Mcauliffe > Flashcards > Psych 100A Midterm #1 | StudyBlue
There is a clear order, but the size of the differences between one value and the next is not the same. (A ranking) --> Not often used in research studies.
i.e. Agree, Strongly Agree, Disagree, strongly disagree
i.e. top 5 fave foods
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A000241 - OEIS
Verified for n=11, 12 by Shengjun Pan and R. Bruce Richter, in "The Crossing Number of K_11 is 100", J. Graph Theory 56 (2) (2007) 128-134.
The values for n >= 13 are probably only conjectural.
Also the sum of the dimensions of the irreducible representations of su(3) that first occur in the [n-5]th tensor power of the tautological representation. - james dolan (jdolan(AT)math.ucr.edu), Jun
It appears that a(n)=C(floor(n/2),2)*C(floor((n-1)/2),2). [From Paul Barry, Oct 02 2008]
From Paul Barry, Oct 02 2008: (Start)
We conjecture that this sequence is given by one half of the third coefficient of the denominator polynomial of the n-th convergent to the g.f. of n!, in which case the next numbers are
784,1008,1296,1620, 2025, 2475,...
Essentially sum{k=0..n, (-1)^(n-k) floor(k/2)ceiling(k/2)floor((k-1)/2)ceiling((k-1)/2)/2}. (End)
One of the most basic questions in knot theory remains unresolved: is crossing number additive under connected sum? In other words, does the equality c(K1#K2) = c(K1) + c(K2) always hold, where c(K)
denotes the crossing number of a knot K and K1#K2 is the connected sum of two (oriented) knots K1 and K2? Theorem 1.1. Let K1, . . .,Kn be oriented knots in the 3-sphere. Then (c(K1) + . . . + c(Kn))
/ 152 <= c(K1# . . . #Kn) <= c(K1) + . . . + c(Kn). [From Jonathan Vos Post, Aug 26 2009]
Ábrego, Bernardo M.; Aichholzer, Oswin; Fernández-Merchant, Silvia; Ramos, Pedro; Salazar, Gelasio. The 2-Page Crossing Number of K_n. Discrete Comput. Geom. 49 (2013), no. 4, 747--777. MR3068573
Jean-Paul Delahaye, in Pour La Science, Feb. 2013, #424, Logique et Calcul. Le problème de la fabrique de briques. (The problem of the brick factory), in French.
P. Erdos and R. K. Guy, Crossing number problems, Amer. Math. Monthly, 80 (1973), 52-58.
R. K. Guy, The crossing number of the complete graph, Bull. Malayan Math. Soc., Vol. 7, pp. 68-72, 1960.
Kainen, Paul C. On a problem of P. Erdos. J. Combinatorial Theory 51968 374--377. MR0231744 (38 #72)
D. McQuillan and R. B. Richter, A parity theorem for drawings of complete ... graphs, Amer. Math. Monthly, 117 (2010), 267-273.
A. Owens, On the biplanar crossing number, IEEE Trans. Circuit Theory, 18 (1971), 277-280.
Shengjun Pan and R. Bruce Richter, "The Crossing Number of K_11 is 100", J. Graph Theory 56 (2) (2007) 128-134.
Saaty, Thomas L., On polynomials and crossing numbers of complete graphs. J. Combinatorial Theory Ser. A 10 (1971), 183--184. MR0291013 (45 #107)
T. L. Saaty, The number of intersections in complete graphs, Engrg. Cybernetics 9 (1971), no. 6, 1102-1104 (1972).; translated from Izv. Akad. Nauk SSSR Tehn. Kibernet. 1971, no. 6, 151-154
(Russian). Math. Rev. 58 #21749.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
C. Thomassen, Embeddings and minors, pp. 301-349 of R. L. Graham et al., eds., Handbook of Combinatorics, MIT Press.
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West Boxford Statistics Tutors
...Many computer applications involve having to solve these types of problems, and most use iterative methods of estimation. We studied these methods and then compared them to the methods used by
the Mathematica program (Wolfram Research) for accuracy and benchmark performance. We then set out to write our own programming functions to solve numerous classic Differential and Integral
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Skillman Math Tutor
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Complex Coefficient Polynomial Roots
October 5th 2010, 06:25 AM #1
Oct 2010
I am looking for an algorithm which can be used to approximate the complex roots of a complex coefficient polynomial. Such as:
(3-4j)x^4 + (1+2j)x + (-4+1)
The polynomial is not of any specific order, and could be quite large.
Any help would be appreciated as i am completely stuck at the moment.
I am looking for an algorithm which can be used to approximate the complex roots of a complex coefficient polynomial. Such as:
(3-4j)x^4 + (1+2j)x + (-4+1)
The polynomial is not of any specific order, and could be quite large.
Any help would be appreciated as i am completely stuck at the moment.
First you need to localise the roots, so that you know where to look. The Cauchy bound and its veriants will do this for you. Newton-Raphson will find complex roots if you start close enough.
October 6th 2010, 12:14 AM #2
Grand Panjandrum
Nov 2005
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Quadratic equation
November 25th 2008, 07:26 AM
Quadratic equation
12/x+1 for length and 6/4-x for width
The rectangle given has a permieter of 14 units. Find the values of x factoring a quadratic equation.
So basically i really have no idea, i drew it, but from there Im not sure, help me out please
November 25th 2008, 08:01 AM
next time, use parentheses (as corrected above) next time to make your problem more clear to those that read it.
you should know that P = 2(L+W) ...
that means L+W = half the perimeter, correct?
so, solve the equation ...
$\frac{12}{x+1} + \frac{6}{4-x} = 7$
first step is to get a common denominator.
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The Long Straddle and Gamma Scalping
The Long Straddle
The long straddle is a limited risk strategy used when the tactical option investor is forecasting a large move in the underlying or an increase in the implied volatility or both.
I don’t believe that traders should have a strategy bias. Traders should study the market and apply the strategy that best applies to their market forecast. Traders with a strategy bias tend to limit
their opportunities.
However, I do think that it’s ok to have a few favorite strategies that you employ when you think that the time is right. One of my preferred strategies is the long straddle. For those of you
unfamiliar with the strategy, the long straddle is the simultaneous purchase of at the money call and put. For example if XYZ stock is trading at $25, an investor would purchase the 25 strike call
and put at the same time. The long straddle is a limited risk, theoretically unlimited profit potential strategy.
Why purchase a straddle? A straddle should be purchased when the investor forecasts a large price move in the underlying or an increase in implied volatility, or both. It is easier to forecast a move
in the implied volatility than to attempt to predict price movement. One of the best times to put on a long straddle is in the weeks preceding a quarterly earnings report. Implied volatilities can
have a tendency to rise in anticipation of the earnings numbers and peak out just prior to the announcement.
A straddle purchased before the volatility increase can be profitable. Generally they should be put on 3-4 weeks before the announcement, so they can be purchased when the implied volatility is low
and appreciate in value as the implied volatility increases as the earnings announcement date is approached.
What are the risks associated with the long straddle? Well the implied volatility may not increase and the stock price may remain very stable. In that case, the enemy of the option purchaser, time
decay or theta will take its toll on the position. You may also want to consider the volatility of the broad market before purchasing a straddle. If the broad market has a very high volatility level
due to some recent event, it may not be the best time for a straddle. If the VIX is at high levels you might want to consider another strategy. If the VIX is at normal levels or has declined and the
investor is forecasting a rise in the VIX and a rise in the implied volatility of an individual equity due to an impending earnings announcement, a long straddle may be an appropriate strategy.
Is there any way to offset the effects of the time decay as measured by the theta? One tool that can be employed by aggressive traders is known as gamma scalping. When you purchase a straddle you
have the right to buy or sell the underlying at the strike price. So, if you are long or short the stock in the same amount of shares as your equivalent number of straddle contracts, you have
protection against an adverse move in your stock position regardless of whether you are long or short.
I like to do some scanning to find stocks with a history of implied volatility increase as the earnings date approaches. Then I use a 20-day window and try to locate issues that are trading near a
strike price and at a 20 day moving average. I use soft numbers, so the entry can be plus or minus a few cents from either parameter. Then I calculate the daily standard deviation by dividing the
annual standard deviation by sixteen.
Why use the number 16? The square root of time is used to calculate standard deviations across multiple time frames. There are 256 trading days in a year. The square root of 256 is 15.87, so that is
rounded to sixteen.
So I have now entered my straddle when the price of the underlying is at a strike and near a moving average. My position is close to being delta neutral. The at the money calls and puts should have
roughly the same delta. Again, I use soft numbers, so I consider a delta of -50 to +50 as being delta neutral. Because I have a long position it will be gamma positive, so that means that the delta
can change rapidly with movement in the underlying and that I will profit from large price swings.
Once the option position is on, if the stock moves up by one standard deviation, I’ll short enough shares to make my position delta neutral again. If the stock moves down by one standard deviation,
I’ll take a long position in the stock. Using round lots, I’ll buy enough shares to become delta neutral once again. When the stock returns to its mean, I’ll close the position for a small gain.
Remember if the stock moves one standard deviation from its mean, there is a 68% probability that it will return to the mean. If it makes a two standard deviation move, there is a 95% chance that it
will return to the mean.
By gamma scalping this way, the investor can attempt to earn day trading profits sufficient enough to offset the time decay of the position. When things go right, I have been able to earn enough day
trading profits from this method to completely cover the cost of the straddle before it is liquidated around the time of the earnings announcement. I don’t have a hard rule for exiting the straddle.
If profits are adequate from the implied volatility increase I may liquidate the entire position just before the announcement. On the other hand I may decide to sell part of the position and keep
some through the announcement and try to profit from a large move in the stock. If the position has been paid for by gamma scalping on a daily basis along the way, the investor has a lot of
flexibility with their money management strategy at the exit.
About sellacalloption
Portfolio manager and chief option strategist for Fusion Asset Management. Professional option trader with VTrader Pro, LLC.
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Calculate a Ratio in Excel
In Excel, if you divide 2 by 8, the result is 0.25. if you format the cell as a fraction, the cell might show 1/4 as the result.
What if you want to show the result as a ratio? How can you get the cell to show 1:4 instead of 1/4? There may be other ways, but here's the formula that I used.
=B2/GCD(B2,C2) & ":" & C2/GCD(B2,C2)
This formula requires that the Analysis ToolPak be installed, in Excel 2003 and earlier versions. It divides each cell by the greatest common divisor (GCD), and puts a colon between the two numbers.
Would you use a different formula? __________________
Share and Enjoy
Thank! I have an immediate use for this.
Works well if numbers have a common denominator or divisor.
Without a GCD the actuall numbers display for example
17 and 20 will display as 17:20
Any suggestions on how to resolve this?
Thanks for the info Debra. IIt was good to be able to follow your example. Some of my numbers haven't divided easily either but it helped to have the ratio formula as a starter and benchmark.
17:20 (17/20) can't be reduced anymore. What needs to be resolved?
Elizabeth – when the result is irreducible, just estimate – if you can get away with it : 17:20 = 17/20 = 0.85 = 1:1.85. Not good enough for precision engineering but adequate for other applications.
If you require the value to be precisely as what excel calculates, just replace the GCD function with MIN stastical function. The only thing you need to take care is that the numbers should be in
ascending order, if you want it in the format "1:XX". Otherwise, the result would be "XX:1? if the numbers are in descending order.
And totbean, just for your information 1:1.85 is not the same as 0.85. "0.85 = 1:1.1764?
I am not so good at maths
@ Debra: can u pls give any other formula...the formula u gave here is not working
Great timp Stya. How do you control the number of decimal places the MIN statistical function calculates to? Something in the cell format?
I meant 'tip' :[ Sorry.
John, Cell format is something that is used to control our view what excel does. so there is no way that you can control that using cell format. Even you change the format, the result does not vary.
what you can do is to use the "Round" function with either "up" or "down" or as it is to control the result. One another way to keep control over excel calculation is to change it from excel options
itself, the precision calculation. I am not sure about this, but did come across elsewhere.
Hi Debra, I tried your formula and it did not work. It is "#NAME?" instead of a figure. And no, my quotes are not the curley ones, they are the straight ones just like yours.
Please advise. Thank You:)
Hi Debra,
Thanks a million. It works now and this is great:)
If you know the largest number will never have more than 7 digits in it (seems to be an Excel limit for this method), I think you can use the following formula (which does not require the Analysis
ToolPak add-in) to calculate your ratio...
A **quick** test seems to show that I don't need all those # signs in the numerator for the formula I posted earlier, this seems to work the same way...
Rick- can you make it a 1:XXX with the above formula?
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Another random problem
Post reply
Another random problem
a point is randomly selected with a rectangle whose vertices are (0,0), (2,0), (2,3) and (0,3). What is the probability that the x-coordinate of the point is less than the y-coordinate?
Re: Another random problem
I think it is 1/3
The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
Re: Another random problem
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
Re: Another random problem
Hi bobbym
how did you get that?
The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
Re: Another random problem
x ∈ [0,2], y ∈ [0,3] both are uniform distributions. There are 3 y's for every 2 x's.
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
Re: Another random problem
Hi bobbym
i think i switched x and y coordinates.
The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
Re: Another random problem
Or the answer is the area enclosed by the rectangle on top and the line y = x.
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
Re: Another random problem
Hi bobbym
yes that's how i did it,i just flipped the rectangle around the y=x line.
The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
Re: Another random problem
Hi anonimnystefy;
It is late and time for you to rest. I will see you tomorrow.
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
Re: Another random problem
The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
Re: Another random problem
Is it not 2:20 AM?
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
Re: Another random problem
Yes it is.
The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
Re: Another random problem
Yawning, sleepy eyelids, a general buildup of toxins and a lowering of body temperature all signalling the need for sleep...
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
Re: Another random problem
And yet math is awaiting.
The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
Re: Another random problem
Let the problems of the day be sufficient for the day.
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
Re: Another random problem
Don't understand.
The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
Re: Another random problem
Math will wait.
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
Full Member
Re: Another random problem
actually, bobbym
The point (x,y) satisfies x<y if and only if it belongs to the shaded triangle bounded by the lines x=y,y=2 , and , x=0 the area of which is 2. The rectangle has area 6, so the probability in
question is 1/3
I see you have graph paper.
You must be plotting something
Re: Another random problem
Hi cooljackiec
Bobbym's answer is correct. Take a closer look and try plotting a few values fir (x,y) to see which area the points will belong.
And, welcome to the forum!
The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
Re: Another random problem
Hi cooljackiec;
The area above the triangle that the line y = x makes is twice as large as the area below the triangle. y is a 2 to 1 favorite. There is only one probability that is twice its complement. That is 2 /
3. The probability the P(y > x ) = 2 / 3 so P(x < y ) = 2 / 3.
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
Post reply
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Arbitrary Gridlines
Don't you wish you had more control over Excel's chart gridlines? Wouldn't you like Excel to draw gridlines where you want them? There isn't a lot of built in support for this, but over the years
people have learned a few tricks to generate charts like this:
Here is how to get your own custom gridlines, with arbitrary spacing (and arbitrary labels). You can use this technique for many purposes:
• Show target or cut off levels for the plotted values.
• Show statistics, such as mean, control limits, etc.
• Generate log or probability gridline scales with your own scale parameters.
See "Flexible Log Axes" at http://www.tushar-mehta.com
This technique is closely related to that in Arbitrary Axis Scale, which demonstrates how to put axis ticks and labels at arbitrary positions along an axis. I use a slightly different approach in Add
a Horizontal Line to a Column Chart.
Chart 1 is a scatter chart showing 40 randomly generated data points.
The data I will use to generate my gridlines (and labels, while I'm at it) is shown below:
Axis X Axis Y Label
0 0.7 alpha
0 1.2 beta
0 1.9 gamma
0 2.2 delta
0 3.1 epsilon
0 3.6 eta
0 4.2 theta
To make gridlines parallel to the X axis, all my X values are equal to the Y axis minimum (zero in this case). The Axis Y values correspond to the places I want my axis ticks. (To generate gridlines
parallel to the Y axis, put the values under X and zeros--or Y axis minimums--under Y.)
Chart 2 shows the dummy gridline data plotted, as a series of magenta squares along the Y axis.
In Chart 3, I have added positive X Error Bars to my dummy series, long enough to stretch across the plot area. I also manually set the X axis maximum; adding the error bars causes Excel to extend
the axis scale. (The error bars are magenta for clarity).
In Chart 4, I have added dummy labels to the gridlines, using the technique described in Arbitrary Axis Scale. The easiest way to get the grid labels is to apply data labels to each point in the
dummy series, then change the label to display the desired text. To do this easily, you need Rob Bovey's XY Chart Labeler, a free and absolutely must have add-in available from http://www.appspro.com
. It's compatible with every version of Excel since 97 (and I think there's also an earlier version, but even I use 97). Use the Labeler to add the 'Labels' to the dummy series added above, aligned
to the left of the points. See the magenta labels in Chart 4.
In Chart 5 I have cleaned things up for presentation. I removed the original (default) horizontal gridlines. I formatted my custom gridlines so they were hairline width (like the default ones) and I
removed the crossbar (the "T") at their ends. In the patterns tab of the Format Y Axis dialog, I set the major ticks, minor ticks, and tick labels to "none". I adjusted the size of the plot and the X
axis scales. I changed all the magenta stuff to black, and formatted the dummy series ("Grid Y") to have no marker. I removed the "Axis Y" entry from the legend. (Single click twice--don't double
click--on the legend entry until only the entry is selected, then press delete.)
Now wasn't that easy!? DummySeries.zip is a zipped Excel file with an easy demonstration of the steps to create arbitrary gridlines, or an arbitrary axis (see Arbitrary Axis Scale).
My Simulated Probability Chart Example (below) is an application of the Arbitrary Axis Scale and Arbitrary Gridlines techniques.
The Reciprocal Axis Chart is another implementation of these techniques.
This zipped Excel file has yet another example of a chart using an arbitrary axis scale and gridlines. The data is sparse for early times and dense later, so neither a Line chart nor an XY Scatter
chart will do.
Arbitrary Gridlines and Axis Labels shows how to improve the appearance of an axis and its related gridlines using this technique.
Peltier Technical Services, Inc., Copyright © 2013. All rights reserved.
You may link to this article or portions of it on your site, but copying is prohibited without permission of Peltier Technical Services. Microsoft Most Valuable Professional
My MVP Profile
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C++ Operator: How to deal with operator overloading?
September 29th, 2006, 04:28 PM #1
Elite Member Power Poster
Join Date
Feb 2005
"The Capital"
C++ Operator: How to deal with operator overloading?
Q: How to deal with operator overloading?
A: Operator overloading looks like a high hill for starters but the key point in their understanding is not considering operator overloading anything other/different from a simple function. This
helps a lot especially for beginners!
Consider them normal functions with pre-defined prototypes (for syntactical reasons/benefits while putting them to use so that they just look the way they are used for the fundamental types).
The arithmetic operators work on two operands, for example +, -, /, *, etc. Since there are two things that they work on the function needs to know what those two things(operands) are. The
operands are passed to them as arguments to the function. For member functions we show only one argument because of the implicit "this" argument. So, member overloads work on "this" and the
second argument commonly named as "rhs" from right-hand-side (of course, you are free to name it differently). As for non-member overloads, there is no "this" involved, hence you give two
arguments. The operators would work for your types as the same way they work with the pre-defined types (like say any integral type). They will work for the same way for your class' objects as
well. That's it!
(Note that there are also operators that work on just one operand ++, --. Please refer to this FAQ - C++ Operator: How to overload postfix increment and decrement operators?)
That's all there is in operator overloading 101 - the basics. Operator overloading is not something more than mere functions.
Q: How to better overload arithmetic operators?
A: It is recommended to write operator 'op' in terms of their operator 'op='. That is, operator+ in terms of operator+=, operator- in terms of operator-= and similarly operator* and operator/.
There are basically two advantages of doing it that way:
1. You would need to only maintain 'op=' version for your classes.
2. There could be a gain in performance owing to Return Value Optimization or RVO (see references).
Take a look at the following sample:
// just need to maintain this function...
template<typename T>
T& operator+=(const T& rhs)
// do stuff..
return *this;
// no maintenance required... once += is maintained well
template<typename T>
const T operator+(const T& lhs, const T& rhs)
return T(lhs)+=rhs; //RVO
// first argument by value
template<typename T>
const T operator+(T lhs, const T& rhs)
// copy constructor called as well
return lhs+=rhs; //no RVO as the return object is named
There is no problem with having overloads independently but why ignore the above mentioned advantages? 'operator+=' and 'operator+' can be completely independently defined as well. But don't you
think that is a kind of rework/duplicacy (writing same logic twice) when there is an alternative way that may give you an extra bit of performance and also keep your worries away while
maintaining the code that you only need to modify 'operator+=' when you need to and leave operator+ untouched?
Another point to note above are two versions of 'operator+'. You only need 1 of those. Even if 'operator+' is or is not defined in terms of 'operator+=' what happens is the object gets "named"
and hence all possibilities of "un-named RVO" are lost. There could be gains on certain compilers due to "named RVO". In that case (named RVO), both would be equivalent. And the conclusion that
we can derive is that however there (while comparing first argument passed as 'T' or 'const T&') may be no performance boost on such compilers (with both named and unnamed RVO) still you get the
advantage related to code maintainance. Doing it this way also makes it unnecessary to make the overloads 'op' as friends.
Q: Why do we need operator overloads as friends?
A: There are basically 2 options - member overloads or non-member overloads. Non-static member functions have this implicit "this" pointer as an argument. Refer to this FAQ - C++ General: What is
the 'this' pointer?. This makes functions that require the first argument to be something different that the same object's type to be written as members. For example, consider 'operator<<'. It is
usually overloaded for 'ostream' operations. The prototype is similar to this:
template <typename T>
std::ostream& operator<<(std::ostream& ostr, const Vector<T>& vect);
It could also be written the following way because there is no "special" restriction about keeping 'std::ostream&' as the first operand/argument:
template<typename T>
std::ostream& operator<<(const Vector<T>& vect, std::ostream& ostr);
But now let us see how that affects us. For the first case, you can use the operator the following way:
std::cout << Vector<int>(); //considering class template Vector<T> has a default constructor
For the second definition of 'operator<<', you would not be able to use the above syntax. The primary objective of operator overloading is to keep the semantics as close to their counter-parts
for fundamental types as possible. For the second version, we would need to call the operator as follows:
operator<<(Vector<int>(), std::cout);
which is not we want. Otherwise, what's the use of operator overloading when we can even write a function 'void DisplayVectorContents()' to do that?
Hence, 'operator<< cannot be a member - it has to be a non-member for us to be able to specify the first argument as 'std::ostream'. Take a look at the following working sample for
// forward declaration
template <typename T>
class Vector;
template <typename T> std::ostream& operator<<(std::ostream&, const Vector<T>&);
template <typename T>
class Vector
std::complex<T> mycomplex;
Vector() {}
Vector(T i_component, T j_component) : mycomplex(i_component, j_component) {}
// member operators
Vector& operator+=(const Vector& rhs)
return *this;
Vector& operator-=(const Vector& rhs)
return *this;
Vector& operator*=(const Vector& rhs)
return *this;
Vector& operator/=(const Vector& rhs)
return *this;
friend std::ostream& operator<< <T>(std::ostream&, const Vector<T>&);
template <typename T>
const Vector<T> operator+(const Vector<T>& lhs, const Vector<T>& rhs)
return Vector<T>(lhs)+=rhs;
template <typename T>
const Vector<T> operator-(const Vector<T>& lhs, const Vector<T>& rhs)
return Vector<T>(lhs)-=rhs;
template <typename T>
const Vector<T> operator*(const Vector<T>& lhs, const Vector<T>& rhs)
return Vector<T>(lhs)*=rhs;
template <typename T>
const Vector<T> operator/(const Vector<T>& lhs, const Vector<T>& rhs)
return Vector<T>(lhs)/=rhs;
template <typename T>
std::ostream& operator<<(std::ostream& ostr, const Vector<T>& vect)
ostr << "i component : " << vect.mycomplex.real() << "\n";
ostr << "j component : " << vect.mycomplex.imag() << "\n";
return ostr;
int main()
Vector<double> first_vector(10, 100);
Vector<double> second_vector(10, 100);
std::cout << first_vector;
std::cout << second_vector;
std::cout << "Sum : \n" << (first_vector+second_vector);
std::cout << "Diff : \n" << (first_vector-second_vector);
std::cout << "Prod : \n" << (first_vector*second_vector);
std::cout << "Div : \n" << (first_vector/second_vector);
return 0;
You can remove the templatization of the code if it bothers you and write 'class Vector' for a type where 'T' is double.
1. Return value optimization at comp.lang.c++.moderated
2. Named return value optimization at msdn.microsoft.com
3. Operator overloading
Last edited by cilu; April 16th, 2007 at 02:48 AM.
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Question about the graph of averages
September 7th 2012, 07:18 AM #1
Sep 2012
Question about the graph of averages
So I was studying the graph of averages for a scatterplot (when you plot the average of all y-values corresponding to a single x-value) and my book said that if the graph of averages falls on a
straight line, then that line is the regression line. However, I'm not sure when the graph of averages would fall on a straight line. Would this only occur if the scatterplot itself had a
correlation coefficient of 1 or -1?
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[FOM] Formalization Thesis
T.Forster@dpmms.cam.ac.uk T.Forster at dpmms.cam.ac.uk
Sat Dec 29 06:45:27 EST 2007
Tim, Thank you for your interesting post. It has unleashed a storm of
discussion, as i knew it would - none of which have i had time to read -
tho' i am looking forward to it. I am risking contributing this post
without having read the rest of the correspondence beco's i suspect that
the point i am about to make will be made by no-one else except possibly
Randall Holmes, and he seems to be off-line.
I am thinking about the attitude to the paradoxes taken by ZF and its
congenors. The basic message it brings is that there is a simple uniform
explanation of the paradoxes, and that is the error of thinking that the
problematic objects are sets. (As you know, i am a student of a set theory
that doesn't take this point of view, so of course i would be saying all
this wouldn't i!) We should remember that altho' the non-sethood of some of
these problematic collections is a theorem of pure logic (LPC), the
non-sethood of - for example - the universe - is not. Perhaps this
difference matters..?
Chow's principle could be false if there are genuine mathematical
facts about some of these large dodgy collections (V for example) that do
not lend themselves to representation as facts about wellfounded sets. I do
not have any compelling examples of such mathematical assertions, but i am
alive to the possibility that there might be some. By setting its face
against taking these entities seriously ZF & co are betting on there *not*
being many such mathematics. There may, indeed, not be any such
mathematics, but it's by no means an open-and-shut case. It would be if the
nonexistence of all the naughty objects were provable in predicate calculus
but it isn't.
Happy new Year
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Trigonometric functions
Posted by Jon on Sunday, February 10, 2008 at 1:53pm.
Find the value of Sec A. and I have a right triangle BC=30 and BA=34
I chose D
• Trigonometric functions - Guido, Sunday, February 10, 2008 at 2:04pm
There is missing data here. Is 30 the hypotenuse or one of the legs of the right triangle? The same for 34. There is no way for me to know if BC is one the legs or is BA is the hypotenuse or
Once I know that, I can proceed to help you.
• Trigonometric functions - Jon, Sunday, February 10, 2008 at 2:13pm
BA is the hypotenuse
BC is the opposite leg
□ Trigonometric functions - Guido, Sunday, February 10, 2008 at 2:28pm
We use the Pythagorean Theorem to find length AC as step 1. In other words, we MUST find the ADJACENT LEG.
AC^2 + BC^2 = BA^2
AC^2 + (30)^2 = (34)^2
AC^2 900 = 1156
AC^2 = 1156 - 900
AC^2 = 256
We now take the square root of both sides of the equation to find AC.
sqrt{AC^2} = sqrt{256}
AC = 16
We now have all three sides.
We need secant.
What is secant?
Secant = hypotenuse/adjacent
Well, 34 is our hypotenuse and we just found AC (our adjacent). See it?
Secant = 34/16
The improper fraction 34/16 = 2_1/8 as a mixed number when reduced to the lowest term.
However, 2_1/8 written as an improper fraction is 17/8.
Final Answer: Choice (A)
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MathsNet: Postgraduate Projects
You may contact a Proposer directly about a specific project or contact the Postgraduate Admissions Secretary with general enquiries.
Title Network performance subject to agent-based dynamical processes
Group(s) Industrial and Applied Mathematics, Statistics and Probability
Proposer(s) Dr Keith Hopcraft, Dr Simon Preston
Networks – systems of interconnected elements – form structures through which information or matter is conveyed from one part of an entity to another, and between autonomous units. The
form, function and evolution of such systems are affected by interactions between their constituent parts, and perturbations from an external environment. The challenge in all
application areas is to model effectively these interactions which occur on different spatial- and time-scales, and to discover how
i) the micro-dynamics of the components influence the evolutionary structure of the network, and
ii) the network is affected by the external environment(s) in which it is embedded.
Description Activity in non-evolving networks is well characterized as having diffusive properties if the network is isolated from the outside world, or ballistic qualities if influenced by the
external environment. However, the robustness of these characteristics in evolving networks is not as well understood. The projects will investigate the circumstances in which memory can
affect the structural evolution of a network and its consequent ability to function.
Agents in a network will be assigned an adaptive profile of goal- and cost-related criteria that govern their response to ambitions and stimuli. An agent then has a memory of its past
behaviour and can thereby form a strategy for future actions and reactions. This presents an ability to generate ‘lumpiness’ or granularity in a network’s spatial structure and
‘burstiness’ in its time evolution, and these will affect its ability to react effectively to external shocks to the system. The ability of externally introduced activists to change a
network’s structure and function - or agonists to test its resilience to attack - will be investigated using the models. The project will use data of real agent’s behaviour.
Title Fluctuation Driven Network Evolution
Group(s) Industrial and Applied Mathematics, Statistics and Probability
Proposer(s) Dr Keith Hopcraft, Dr Simon Preston
A network’s growth and reorganisation affects its functioning and is contingent upon the relative time-scales of the dynamics that occur on it. Dynamical time-scales that are short
compared with those characterizing the network’s evolution enable collectives to form since each element remains connected with others in spite of external or internally generated
Description ‘shocks’ or fluctuations. This can lead to manifestations such as synchronicity or epidemics. When the network topology and dynamics evolve on similar time-scales, a ‘plastic’ state can
emerge where form and function become entwined. The interplay between fluctuation, form and function will be investigated with an aim to disentangle the effects of structural change from
other dynamics and identify robust characteristics.
Title Numerical methods for stochastic partial differential equations
Group(s) Scientific Computation, Statistics and Probability
Proposer(s) Prof Michael Tretyakov
Numerics for stochastic partial differential equations (SPDEs) is one of the central topics in modern numerical analysis. It is motivated both by applications and theoretical study.
Description SPDEs essentially originated from the filtering theory and now they are also widely used in modelling spatially distributed systems from physics, chemistry, biology and finance acting in
the presence of fluctuations. The primary objectives of this project include construction, analysis and testing of new numerical methods for SPDEs.
Other Web-page http://www.maths.nott.ac.uk/personal/pmzmt
Title Uncertainty quantification for evolutionary PDEs
Group(s) Scientific Computation, Statistics and Probability
Proposer(s) Dr Kris van der Zee
Uncertainty quantification for evolutionary PDEs
(Or- Can we rely on our computational simulations, no matter how accurate they are?)
The field of uncertainty quantification (UQ), as applied to partial differential equations (PDEs), provides a means for understanding the effect of uncertainty in the parameters on
output quantities of interest. This is, for example, extremely important in evolutionary models that describe the degradation of nuclear waste containment systems: A small uncertainty in
the parameters may have a large influence on the degradation, and thus may result in unexpected damaged systems that leak nuclear waste. Fortunately, observational data can help reduce
uncertainty in models. The models can then become predictive and allow for an assessment of its true future behavior!
Description Challenges for students:
* How can one quantify the uncertainty in the complex evolutionary models?
* Can one determine suitable observation scenarios using Bayesian experimental design principles?
* How complex does a model need to be for a predictive simulation?
* Can we employ UQ methodologies to reliably predict degradation of nuclear waste containment systems?
Depending on the interest of the student, several of these issues (or others) can be addressed.
Also, the student is encouraged to suggest a second supervisor, possibly from another group!
Relevant • I. Babuska, F. Nobile, and R. Tempone, A systematic approach to model validation based on Bayesian updates and prediction related rejection criteria, Comput. Meth. Appl. Mech. Engrg.
Publications 197 (2008), pp. 2517-2539
Other • This work connects to a large interdisciplinary multi-university EPSRC project.
Title Index policies for stochastic optimal control
Group(s) Statistics and Probability
Proposer(s) Dr David Hodge
Since the discovery of Gittins indices in the 1970s for solving multi-armed bandit processes the pursuit of optimal policies for this very wide class of stochastic decision processes has
been seen in a new light. Particular interest exists in the study of multi-armed bandits as problems of optimal allocation of resources (e.g. trucks, manpower, money) to be shared
between competing projects. Another area of interest would be the theoretical analysis of computational methods (for example, approximative dynamic programming) which are coming to the
Description fore with ever advancing computer power.
Potential project topics could include optimal decision making in the areas of queueing theory, inventory management, machine maintenance and communication networks.
Other Keywords: multi-armed bandits, dynamic programming, Markov decision processes
Title Uncertainty quantification in palaeo-climate reconstruction
Group(s) Statistics and Probability
Proposer(s) Dr Richard Wilkinson
The climate evolves slowly. Even if we stopped emitting green-houses gases today, we wouldn't see the full effect of the damage already done for at least another 100 years. The
instrumental record of past climate and weather goes back at most 300 years, and before then we have to rely on indirect (and inaccurate) data sources. Because of the slow evolution of
the climate, this is like only having a very small number of acccurate observations, and so consquently we have very little information that can be used to assess the accuracy of climate
simulators, which are the key tool used for predicting how the climate will behave in the future.
An important source of information on what the climate was like in the past comes from proxy data sources such as pollen taken from lake deposits, or measurements of the air-content
Description (specifically the ratio of oxygen-18 to oxygen-16) stored in glaciers hundreds of thousands of years ago. Reconstructing past climate from these data sources is a difficult task as the
measurements are noisy, correlated, and don't have accurate dates attached to them, yet the task is important if we are to understand how the climate evolves and hence be able to predict
the future.
In this project, we will look at statistical methods for accurate palaeo-climate reconstruction, and aim to provide believeable uncertainty quantifications that accurately represent our
degree of confidence/ignorance about what the climate was like in the past. The complex nature of the problem means that it is likely that state-of-the-art Monte Carlo methods will be
needed, as well as potentially developing new methods in order to do the inference.
Title Semi-Parametric Time Series Modelling Using Latent Branching Trees
Group(s) Statistics and Probability
Proposer(s) Dr Theodore Kypraios
A class of semi-parametric discrete time series models of infinite order where we are be able to specify the marginal distribution of the observations in advance and then build their
dependence structure around them can be constructed via an artificial process, termed as Latent Branching Tree (LBT). Such a class of models can be very useful in cases where data are
Description collected over long period and it might be relatively easy to indicate their marginal distribution but much harder to infer about their correlation structure. The project is concerned
with the development of such models in continuous-time as well as developing efficient methods for making Bayesian inference for the latent structure as well as the model parameters.
Moreover, the application of such models to real data would be also of great interest.
Title Bayesian methods for analysing computer experiments
Group(s) Statistics and Probability
Proposer(s) Dr Richard Wilkinson
Computer experiments (ie simulators) are used in nearly all areas of science and engineering. The statistical analysis of computer experiments is an exciting and rapidly growing area of
statistics which looks at the question of how best to learn from computer models. Examples of the types of challenges faced, and possible areas for a Ph.D, are given below.
(i) Computer models are often process models where the likelihood function is intractable (as is common in genetics, ecology, epidemiology etc) and so to do inference we have to use
likelihood-free techniques. Approximate Bayesian computation (ABC) methods are a new class of Monte Carlo methods that are becoming increasingly popular with practitioners, but which are
largely unstudied by statisticians, and there remains many open questions about their performance. Application areas which use ABC methods are mostly biological (genetics and ecology in
particular), but their use is growing across a wide range of fields.
(ii) Expensive simulators which take a considerable amount of time to run (eg, climate models), present the challenge of how to learn about the model (its parameters, validity, or its
Description predictions etc.) when we only have a limited number of model evaluations available for use. A statistical tool developed in the last decade is the idea of building statistical emulators
of the simulator. Emulators are cheap statistical models of the simulator (models of the model) which can be used in place of the simulator to make inferences, and are now regularly used
in complex modelling situations such as in climate science. However, there are still many questions to be answered about how best to build and then use emulators. Possible application
areas for these methods include climate science, and engineering (such as ground-water flow problems for radio-active waste), as well as many others.
(iii) "All models are wrong, but some are useful" - In order to move from making statements about a model to making statements about the system the model was designed to represent, we
must carefully quantify the model error - interest lies in what will actually happen, rather than in what your model says will happen! Failure to account for model errors can mean that
different models of the same system can give different predictions (see for example the controversy regarding the differing predictions of the large climate models - none of which
account for model error!). Assessing and incorporating model error is a new and rapidly growing idea in statistics, and is done by a combination of subjective judgement and statistical
learning from data. The range of potential application areas is very wide, but in particular meterology and mechanical engineering are areas where these methods are needed.
• Wilkinson, Approximate Bayesian computation (ABC) gives exact results under the assumption of model error, in submission. Available as arXiv:0811.3355.
• Wilkinson, Bayesian calibration of expensive multivariate computer experiments. In ‘Large-scale inverse problems and quantification of uncertainty’, 2010, John Wiley and Sons,
Relevant Series in Computational Statistics.
Publications • Quantifying simulator discrepancy in discrete-time dynamical simulators, Wilkinson, M. Vrettas, D. Cornford, J. E. Oakley. Journal of Agricultural, Biological, and Environmental
Statistics: Special issue on Computer models and spatial statistics for environmental science, 16(4), 554-570, 2011
Other See http://www.maths.nottingham.ac.uk/personal/pmzrdw/ for more information.
Title Ion channel modelling
Group(s) Statistics and Probability
Proposer(s) Prof Frank Ball
The 1991 Nobel Prize for Medicine was awarded to Sakmann and Neher for developing a method of recording the current flowing across a single ion channel. Ion channels are protein
molecules that span cell membranes. In certain conformations they form pores allowing current to pass across the membrane. They are a fundamental part of the nervous system.
Mathematically, a single channel is usually modelled by a continuous time Markov chain. The complete process is unobservable but rather the state space is partitioned into two classes,
Description corresponding to the receptor channel being open or closed, and it is only possible to observe which class of state the process is in. The aim of single channel analysis is to draw
inferences about the underlying process from the observed aggregated process. Further complications include (a) the failure to detect brief events and (b) the presence of (possibly
interacting) multiple channels. Possible projects include the development and implementation of Markov chain Monte Carlo methods for inferences for ion channel data, Laplace transform
based inference for ion channel data and the development and analysis of models for interacting multiple channels.
Title Optimal control in yield management
Group(s) Statistics and Probability
Proposer(s) Dr David Hodge
Serious mathematics studying the maximization of revenue from the control of price and availability of products has been a lucrative area in the airline industry since the 1960s. It is
particularly visible nowadays in the seemingly incomprehensible price fluctuations of airline tickets. Many multinational companies selling perishable assets to mass markets now have
Description large Operations Research departments in-house for this very purpose. This project would be working studying possible innovations and existing practices in areas such as: customer
acceptance control, dynamic pricing control and choice-based revenue management. Applications to social welfare maximization, away from pure monetary objectives, and the resulting game
theoretic problems are also topical in home energy consumption and mass online interactions.
Title Stochastic Processes on Manifolds
Group(s) Statistics and Probability
Proposer(s) Dr Huiling Le
As well as having a wide range of direct applications to physics, economics, etc, diffusion theory is a valuable tool for the study of the existence and characterisation of solutions of
partial differential equations and for some major theoretical results in differential geometry, such as the 'Index Theorem', previously proved by totally different means. The problems
Description which arise in all these subjects require the study of processes not only on flat spaces but also on curved spaces or manifolds. This project will investigate the interaction between the
geometric structure of manifolds and the behaviour of stochastic processes, such as diffusions and martingales, upon them.
Title Analytic methods in probability theory
Group(s) Statistics and Probability
Proposer(s) Notice: Undefined index: pmzsu in /maths/www/html/postgraduate/projects/index.php on line 477 Notice: Undefined index: pmzsu in /maths/www/html/postgraduate/projects/index.php on
line 477
My research focuses on interactions between probability theory, combinatorics and topology. Topics of particular interest:
• dependence, limit theorems for dependent variables with applications to dynamical systems, examples and counterexamples to limit theorems;
Description • modern analytic methods in probability theory such as stochastic orderings, Stein's type operator, contraction, Poincare - Hardy - Sobolev type inequalities and their
• stochastic analysis of rare events such as Poisson approximation and large deviations;
• Random groups, Hausdorff dimension and related topics
Other information
Title Statistical Theory of Shape
Group(s) Statistics and Probability
Proposer(s) Dr Huiling Le
Devising a natural measure between any two fossil specimens of a particular genus, assessing the significance of observed 'collinearities' of standing stones and matching the observed
systems of cosmic 'voids' with the cells of given tessellations of 3-spaces are all questions about shape.
Description It is not appropriate however to think of 'shapes' as points on a line or even in a euclidean space. They lie in their own particular spaces, most of which have not arisen before in any
context. PhD projects in this area will study these spaces and related probabilistic issues and develop for them a revised version of multidimensional statistics which takes into account
their peculiar properties. This is a multi-disciplinary area of research which has only become very active recently. Nottingham is one of only a handful of departments at which it is
Title Automated tracking and behaviour analysis
Group(s) Statistics and Probability
Proposer(s) Dr Christopher Brignell
In collaboration with the Schools of Computer and Veterinary Science we are developing an automated visual surveillance system capable of identifying, tracking and recording the exact
movements of multiple animals or people. The resulting data can be analysed and used as an early warning system in order to detect illness or abnormal behaviour. The three-dimensional
Description targets are, however, viewed in a two dimensional image and statistical shape analysis techniques need to be adapted to improve the identification of an individual's location and
orientation and to develop automatic tests for detecting specific events or individuals not following normal behaviour patterns.
Title Asymptotic techniques in Statistics
Group(s) Statistics and Probability
Proposer(s) Prof Andrew Wood
Asymptotic approximations are very widely used in statistical practice. For example, the large-sample likelihood ratio test is an asymptotic approximation based on the central limit
theorem. In general, asymptotic techniques play two main roles in statistics: (i) to improve understanding of the practical performance of statistics procedures, and to provide insight
into why some proceedures perform better than others; and (ii) to motive new and improved approximations. Some possible topics for a Ph.D. are
Description • Saddlepoint and related approximations
• Relative error analysis
• Approximate conditional inference
• Asymptotic methods in parametric and nonparametric Bayesian Inference
Title Statistical Inference for Ordinary Differential Equations
Group(s) Statistics and Probability
Proposer(s) Dr Theodore Kypraios, Dr Simon Preston, Prof Andrew Wood
Ordinary differential equations (ODE) models are widely used in a variety of scientific fields, such as physics, chemistry and biology. For ODE models, an important question is how best
to estimate the model parameters given experimental data. The common (non-linear least squares) approach is to search parameter space for parameter values that minimise the sum of
Description squared differences between the model solution and the experimental data. However, this requires repeated numerical solution of the ODEs and thus is computationally expensive;
furthermore, the optimisation's objective function is often highly multi-modal making it difficult to find the global optimum. In this project we will develop computationally less
demanding likelihood-based methods, specifically by using spline regression techniques that will reduce (or eliminate entirely) the need to solve numerically the ODEs.
Title Bayesian approaches in palaeontology
Group(s) Statistics and Probability
Proposer(s) Dr Richard Wilkinson
Palaeontology provides a challenging source of problems for statisticians, as fossil data are usually sparse and noisy. Methods from statistics can be used to help answer scientific
questions such as when did species originate or become extinct, and how diverse was a particular taxonomic group. Some of these questions are of great scientific interest - for example -
did primates coexist with dinosaurs in the Cretaceous? There is no hard evidence either way, but statistical methods can be used to assess the probability that they did coexist.
This project will involve building a stochastic forwards model of an evolutionary scenario, and then fitting this model to fossil data. Quantifying different sources of uncertainty is
likely to play a key part in the analysis.
Relevant • Dating primate divergences through an integrated analysis of palaeontological and molecular data, Wilkinson, M. Steiper, C. Soligo, R.D. Martin, Z. Yang, and S. Tavare, Systematic
Publications Biology, 60(1): 16-31, 2011.
Other See http://www.maths.nottingham.ac.uk/personal/pmzrdw/ for more information
Title Statistical shape analysis with applications in structural bioinformatics
Group(s) Statistics and Probability
Proposer(s) Dr Christopher Fallaize
In statistical shape analysis, objects are often represented by a configuration of landmarks, and in order to compare the shapes of objects, their configurations must first be aligned as
closely as possible. When the landmarks are unlabelled (that is, the correspondence between landmarks on different objects is unknown) the problem becomes much more challenging, since
both the correspondence and alignment parameters need to be inferred simultaneously.
An example of the unlabelled problem comes from the area of structural bioinformatics, when we wish to compare the 3-d shapes of protein molecules. This is important, since the shape of
Description a protein is vital to its biological function. The landmarks could be, for example, the locations of particular atoms, and the correspondence between atoms on different proteins is
unknown. This project will explore methods for unlabelled shape alignment, motivated by the problem of protein structure alignment. Possible topics include development of:
i) efficient MCMC methods to explore complicated, high-dimensional distributions, which may be highly multimodal when considering large proteins;
ii) fast methods for pairwise alignment, needed when a large database of structures is to be searched for matches to a query structure;
iii) methods for the alignment of multiple structures simultaneously, which greatly exacerbates the difficult problems faced in pairwise alignment.
Relevant • Green, P.J. and Mardia, K.V. (2006) Bayesian alignment using hierarchical models, with applications in protein bioinformatics. Biometrika, 93(2), 235-254.
Publications • Mardia, K.V., Nyirongo, V.B., Fallaize, C.J., Barber, S. and Jackson, R.M. (2011). Hierarchical Bayesian modeling of pharmacophores in bioinformatics. Biometrics, 67(2), 611-619.
Title High-dimensional molecular shape analysis
Group(s) Statistics and Probability
Proposer(s) Prof Ian Dryden
In many application areas it is of interest to compare objects
and to describe the variability in shape as an object evolves over time.
For example in molecular shape analysis it is common to have several thousand
atoms and a million time points. It is of great interest to reduce the
dimension to a relatively small number of dimensions, and to describe
the variability in shape and coverage properties over time. Techniques
Description from manifold learning will be explored, to investigate if the variability can
be effectively described by a low dimensional manifold. A recent method for
shapes and planar shapes called principal nested spheres will be adapted for
3D shape and surfaces. Also, other non-linear dimension reduction techniques such as
multidimensional scaling will be explored, which approximate the geometry
of the higher dimensional manifold. The project will involve collaboration
with Dr Charlie Laughton of the School of Pharmacy.
Relevant Publications • Jung, S., Dryden, I.L. and Marron, J.S. (2012). Analysis of principal nested spheres. Biometrika, 99, 551–568.
Other information
Title Uncertainty quantifcation for models with bifurcations
Group(s) Statistics and Probability, Industrial and Applied Mathematics
Proposer(s) Prof Ian Dryden Notice: Undefined index: pmzkac in /maths/www/html/postgraduate/projects/index.php on line 479 Notice: Undefined index: pmzkac in /maths/www/html/postgraduate/projects/
index.php on line 479 ,
The project will consider Uncertainty Quantification (UQ) when there are bifurcations or discontinuities in the models. Gaussian Process Emulators (GPE) and Generalised Polynomial Chaos
(gPC) techniques will be used to construct fast approximations to high-cost deterministic models. Also, an important component of Bayesian UQ is the difficult task of elicitation of the
prior distributions of the parameters of interest, which will be investigated. We will exploit the flexibility in the choice of GPE covariance function to deal with cases where the
Description dependence on the inputs is not smooth. Lack of smoothness can be handled by dividing the parameter space into elements and using of gPC or GPE on each element, but this is difficult to
do automatically. We propose to compute the hypersurfaces at which discontinuities occur, using techniques from numerical bifurcation theory, as preparation for discretising with gPC or
GPE methods. Bifrucations arise in carbon sequestration applications, and radioactive waste disposal is another area where elicitation and Bayesian emulation are useful.
Title Statistical analysis of neuroimaging data
Group(s) Statistics and Probability, Mathematical Medicine and Biology
Proposer(s) Dr Christopher Brignell
The activity of neurons within the brain can be detected by function magnetic resonance imaging (fMRI) and magnetoencephalography (MEG). The techniques record observations up to 1000
times a second on a 3D grid of points separated by 1-10 millimetres. The data is therefore high-dimensional and highly correlated in space and time. The challenge is to infer the
Description location, direction and strength of significant underlying brain activity amongst confounding effects from movement and background noise levels. Further, we need to identify neural
activity that are statistically significant across individuals which is problematic because the number of subjects tested in neuroimaging studies is typically quite small and the
inter-subject variability in anatomical and functional brain structures is quite large.
Title Identifying fibrosis in lung images
Group(s) Statistics and Probability, Mathematical Medicine and Biology
Proposer(s) Dr Christopher Brignell
Many forms of lung disease are characterised by excess fibrous tissue developing in the lungs. Fibrosis is currently diagnosed by human inspection of CT scans of the affected lung
regions. This project will develop statistical techniques for objectively assessing the presence and extent of lung fibrosis, with the aim of identifying key factors which determine
Description long-term prognosis. The project will involve developing statistical models of lung shape, to perform object recognition, and lung texture, to classify healthy and abnormal tissue.
Clinical support and data for this project will be provided by the School of Community Health Sciences.
Title Modelling hospital superbugs
Group(s) Statistics and Probability, Mathematical Medicine and Biology
Proposer(s) Prof Philip O'Neill
The spread of so-called superbugs such as MRSA within healthcare settings provides one of the major challenges to patient welfare within the UK. However, many basic questions regarding
Description the transmission and control of such pathogens remain unanswered. This project involves stochastic modelling and data analysis using highly detailed data sets from studies carried out in
hospital, addressing issues such as the effectiveness of patient isolation, the impact of different antibiotics, and the way in which different strains interact with each other.
Title Modelling of Emerging Diseases
Group(s) Statistics and Probability, Mathematical Medicine and Biology
Proposer(s) Prof Frank Ball
When new infections emerge in populations (e.g. SARS; new strains of influenza), no vaccine is available and other control measures must be adopted. This project is concerned with
Description addressing questions of interest in this context, e.g. What are the most effective control measures? How can they be assessed? The project involves the development and analysis of new
classes of stochastic models, including intervention models, appropriate for the early stages of an emerging disease.
Title Structured-Population Epidemic Models
Group(s) Statistics and Probability, Mathematical Medicine and Biology
Proposer(s) Prof Frank Ball
The structure of the underlying population usually has a considerable impact on the spread of the disease in question. In recent years the Nottingham group has given particular attention
to this issue by developing, analysing and using various models appropriate for certain kinds of diseases. For example, considerable progress has been made in the understanding of
epidemics that are propogated among populations made up of households, in which individuals are typcially more likely to pass on a disease to those in their household than those
Description elsewhere. Other examples of structured populations include those with spatial features (e.g. farm animals placed in pens; school children in classrooms; trees planted in certain
configurations), and those with random social structure (e.g. using random graphs to describe an individual's contacts). Projects in this area are concerned with novel advances in the
area, including developing and analysing appropriate new models, and methods for statistical inference (e.g. using pseudo-likelihood and Markov chain Monte Carlo methods).
Title Bayesian Inference for Complex Epidemic Models
Group(s) Statistics and Probability, Mathematical Medicine and Biology
Proposer(s) Prof Philip O'Neill
Data-analysis for real-life epidemics offers many challenges; one of the key issues is that infectious disease data are usually only partially observed. For example, although numbers of
Description cases of a disease may be available, the actual pattern of spread between individuals is rarely known. This project is concerned with the development and application of methods for
dealing with these problems, and involves using Markov Chain Monte Carlo (MCMC) techniques.
Title Bayesian model choice assessment for epidemic models
Group(s) Statistics and Probability, Mathematical Medicine and Biology
Proposer(s) Prof Philip O'Neill
During the last decade there has been a significant progress in the area of parameter estimation for stochastic epidemic models. However, far less attention has been given to the issue
Description of model adequacy and assessment, i.e. the question of how well a model fits the data. This project is concerned with the development of methods to assess the goodness-of-fit of epidemic
models to data.
Title Epidemics on random networks
Group(s) Statistics and Probability, Mathematical Medicine and Biology
Proposer(s) Prof Frank Ball
There has been considerable interest recently in models for epidemics on networks describing social contacts. In these models one first constructs an undirected random graph, which gives
the network of possible contacts, and then spreads a stochastic epidemic on that network. Topics of interest include: modelling clustering and degree correlation in the network and
Description analysing their effect on disease dynamics; development and analysis of vaccination strategies, including contact tracing; and the effect of also allowing for casual contacts, i.e.
between individuals unconnected in the network. Projects in this area will address some or all of these issues.
• Ball F G and Neal P J (2008) Network epidemic models with two levels of mixing. Math Biosci 212, 69-87.
Relevant • Ball F G, Sirl D and Trapman P (2009) Threshold behaviour and final outcome of an epidemic on a random network with household structure. Adv Appl Prob 41, 765-796.
Publications • Ball F G, Sirl D and Trapman P (2010) Analysis of a stochastic SIR epidemic on a random network incorporating household structure. Math Biosci 224, 53-73.
Title Robustness-performance optimisation for automated composites manufacture
Group(s) Statistics and Probability, Scientific Computation
Proposer(s) Prof Frank Ball Notice: Undefined index: pmzkac in /maths/www/html/postgraduate/projects/index.php on line 479 Notice: Undefined index: pmzkac in /maths/www/html/postgraduate/projects/
index.php on line 479 , , Prof Michael Tretyakov
Multidisciplinary collaborations are a critical feature of material science research enabling integration of data collection with computational and/or mathematical modelling. This PhD
study provides an exciting opportunity for an individual to participate in a project spanning research into composite manufacturing, stochastic modelling and statistical analysis, and
scientific computing. The project is integrated into the EPSRC Centre for Innovative Manufacturing in Composites, which is led by the University of Nottingham and delivers a co-ordinated
programme of research at four of the leading universities in composites manufacturing, the Universities of Nottingham, Bristol, Cranfield and Manchester.
This project focuses on the development of a manufacturing route for composite materials capable of producing complex components in a single process chain based on advancements in the
knowledge, measurement and prediction of uncertainty in processing. The necessary developments comprise major manufacturing challenges. These are accompanied by significant mathematical
problems, such as numerical solution of coupled non-linear partial differential equations with randomness, the inverse estimation of composite properties and their probability
Description distributions based on real-time measurements and the formulation and solution of a stochastic model of the variability in fibre arrangements. The outcome of this work will enable a step
change in the capabilities of composite manufacturing technologies to be made, overcoming limitations related to part thickness, component robustness and manufacturability as part of a
single process chain, whilst yielding significant developments in mathematics with generic application in the fields of stochastic modelling and inverse problems.
The specific aims of this project are: (i) Stochastic simulation of multi-dimensional non-linear stochastic problems; (ii) Stochastic and statistical modelling of fibre variability in
Automated Fibre Placement to permit the predictive simulation of range of potential outcomes conditional on monitoring observations made during the process; (iii) Solution of the
anisotropic conductivity inverse problem under uncertainty to translate monitoring and simulation of observable parameters to uncertainty quantification of critical unobservable
The PhD programme contains a training element, which includes research work as well as traditional taught material. The exact nature of the training will be mutually agreed by the
student and their supervisors and will have a minimum of 30 credits (approximately ¼ of a Master course/taught component of an MSc course) of assessed training. The graduate programmes
at the School of Mathematical Sciences and the EPSRC Centre for Innovative Manufacturing in Composites provide a variety of appropriate training courses.
information We require an enthusiastic graduate with a 1^st class degree in Mathematics (in exceptional circumstances a 2(i) class degree can be considered), preferably of the MMath/MSc level, with
good programming skills and willing to work as a part of an interdisciplinary team. A candidate with a solid background in statistics and stochastic processes will have an advantage.
The studentship is available for a period of three and a half years and provides an annual stipend of £13,726 and full payment of Home/EU Tuition Fees. Students must meet the EPSRC
eligibility criteria.
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Topic: Help with a school project, a statistical survey on education
Replies: 15 Last Post: Jul 13, 2013 4:14 PM
Messages: [ Previous | Next ]
loom91 Re: Help with a school project, a statistical survey on education
Posted: Jul 22, 2007 2:36 PM
Posts: 17
Registered: 4/25/06 On Jul 21, 4:36 pm, The Qurqirish Dragon <qurqiri...@aol.com> wrote:
> On Jul 20, 10:55 am, loom91 <loo...@gmail.com> wrote:
> > I'm also looking for specific help on the following topics:
> > i)What is a suitable measure of whether girls are stronger at some
> > subjects while boys at other subjects?
> Assuming you have the individual subject-test scores for each person,
> then let mu_bi be the popolation mean for boys in subject i, and mu_gi
> be the population mean for girls in subject i. You want to test the
> hypotheses (1) mu_bi > mu_gi and (2) mu_gi > mu_bi. performing this
> sort of hypothesis test is one of the first things that most
> statistics texts cover when considering two related populations.
> If exactly one test fails, then your evidence supports a gender bias
> in the subject. If both tests fail, then the evidence indicates no
> bias (or, more correctly stated, fails to show there is a bias. If
> this is a beginning statistics class, then you should be certain to
> tell them the difference between those two phrasings)
Thanks for your comments. But we are not aiming to measure the
absolute difference between boys and girls. I had tried to give a
picture of what we are looking for above:
"i)What is a suitable measure of whether girls are stronger at some
subjects while boys at other subjects? I'm thinking of comparing the
percentage of total marks obtained in one subject, standardised
against the whole population. For example, consider the variable X =
percentage of total marks earned in History+Geography. Next, we
the standardised (wrt the entire population) variate corresponding to
X, let it be Z. Now we compute the mean of Z over the girls schools
([itex]E_1(Z)[/itex]) and the mean over the boys schools
If the first value is larger than the second value (it seems one will
have to be positive and the other negative), then we may say that
girls prefer humanities more over other subjects than boys. Next we
can do the same analysis on the boys vs girls population in coed
schools and see if the difference is less. By using the absolute
instead of expressing it as percentage of total marks, we can also
compare the relative performance (as opposed to preference) of boys
and girls in humanities. The same can be done for languages and
sciences. Is this a statistically sound measure (unlikely, since I
just made it up)? What are the alternatives?"
As nyou see, we are not comparing whether boys score more marks than
girls in math. We are aiming to measure whether girls are weaker in
math than boys *relative to the other subjects*. For example, if a boy
scores very low marks in all subjects, but scores comparatively better
in Biology, you would say he was strong in Biology, even though mny
boys scored more in Biology than him. This is why I was expressing
marks in the subject as parcentage of total marks obtained, to judge
the relative contribution of the subject irrespective of whether the
student is good or bad overall. Then I standardise wrt to the whole
population to see whether the subject contributes more or less to a
students score than the population average. By taking the mean over
the boys population and seeing whether it is more than the mean over
the girls population, we can judge whether the subject contributes
more to the totals of one sex than the other. Does this make sense?
Will it work well? Will something else work better?
> > iii)Is there some easily available (preferably free) software that
> > will let me do all this analysis (brownie points for fitting
> > probability distributions and graphing)? It would be a nightmare to do
> > this by hand since we usually work with less than 50 data points
> > instead of several hundred.
> Off hand, I don't know of free software, but it is likely that your
> school has one or more of them already on the school's computers. For
> that matter, at this level even Excel will have sufficient tools
> (although you may need to install the statistical measurements pack.)
Actually, our proposed stat computer lab is stalled because there are
not enough plugpoints to put up two more computers :-)
> > iv)As it stand right now, we will sample two boys schools, two girls
> > schools and one coed school. Is this enough to be statistically
> > significant? How many data points should we sample from each school?
> > Should this be a constant or proportional to the total number of
> > students?
> To compare the individual schools, of course, this is fine (as long as
> you have a decent sized sample from each). To compare TYPES of
> schools, then no, it is insufficient, as you only have have a few data
> points. As for the sample size (from each school), I would suggest a
> minimum of the larger of 30 or 5% of the student population (these
> numbers are the same at 600 students) This way you can likely use a
> normal approximation to score distributions, even if the scores are
> not normally distributed. Many statistical tests have simpler forms
> for normally distributed data. This may allow you to have the class do
> the analysis by hand. If you use a software package, then this need is
> not important, obviously. In any event, large sample sizes will
> improve your confidence levels in the hypothesis tests.
We plan on taking 50 data points from each school, about 20-25% of the
class size. So you say that we should take the same number of points
even if one school has more students than the other? Also, do you mean
that the sample size is insufficient to draw reliable conclusions
about whether coed schools really lessen the gender differences?
> > v)Finally, is the whole proposition so glaringly ridiculous that all
> > serious statisticians will simply laugh at it? I hope not :redface:
> Not at all. It is great if you can use an example like this (as
> opposed to textbook work). This should be a very good problem for a
> first-year statistics class. Of course, if your results show a
> significant difference between the schools in your district, there may
> be some bruised egos in the administration(s), but that is problem
> outside the scope of statistics ;-)- Hide quoted text -
> - Show quoted text -
Just to clear up something, you don't think I'm the teacher, do you?
I'm just a 11th grade student. Our syllabus covers very little
estimation, mostly descriptive stat, so it'll be helpful if you gave
me a few pointers about common problems encountered by statisticians
when doing this type of study.
Also, the question about which I've abolutely no idea at all is the
ii) What is a good way of identifying whether the population in a
school indeed consists of discreet stratas? This could be good
students/bad students (there is indication from previous results that
this may be the case) or in coed schools boys/girls (very likely the
case). In case of coed schools, there may even be four stratas: good
boys, good girls, bad boys, bad girls. It will be interesting to
whether bad boys vs girls show more difference than good boys vs
girls. All this sounds very pretty, but I don't know how to separate
the population into stratas.
Can you help me? Thanks.
Date Subject Author
7/20/07 Help with a school project, a statistical survey on education loom91
7/20/07 Re: Help with a school project, a statistical survey on education richardstartz@comcast.net
7/22/07 Re: Help with a school project, a statistical survey on education loom91
7/21/07 Re: Help with a school project, a statistical survey on education The Qurqirish Dragon
7/22/07 Re: Help with a school project, a statistical survey on education loom91
7/22/07 Re: Help with a school project, a statistical survey on education Richard Ulrich
7/23/07 Re: Help with a school project, a statistical survey on education loom91
7/22/07 Re: Help with a school project, a statistical survey on education loom91
7/22/07 Re: Help with a school project, a statistical survey on education loom91
7/22/07 Re: Help with a school project, a statistical survey on education Bob
7/23/07 Re: Help with a school project, a statistical survey on education loom91
7/24/07 Re: Help with a school project, a statistical survey on education Nick
7/25/07 Re: Help with a school project, a statistical survey on education John Kane
7/25/07 Re: Help with a school project, a statistical survey on education loom91
7/13/13 statistical survey to be done on which topics? divya.nair421@gmail.com
7/26/07 Re: Help with a school project, a statistical survey on education loom91
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box shape question [Archive] - Car Audio Forum - CarAudio.com
06-04-2004, 10:04 AM
I am planning on building a sealed box for a 10" Alpine Type R (SWR 1021d) sub. I have looked and read a bit on the whole process, but I have seen some differing points regarding shape and wanted to
see what you guys think. I am clear on the whole size (volume) issue - in my case, my sub suggests between a .5 and a .8 cubic volume box. I want to make a box with around .66 cubic feet internal
volume. Here is where my confusion lies.........Does the shape matter? I have heard that a perfect cube is a bad idea. I would like to stay away from slanted sides (different bottom and top sizes)
because I don't want to have to make angular/mitre cuts. Basically, something resembling a rectangle is easier to make. :) Many people say that as long as it isn't an exact cube you are good to go,
but I have seen several places (for example)
that talk about the "golden ratio". Is this ratio of side length that important? Will there be a noticable difference in sound? I calculated my side length with this ratio and I don't really like the
shape. Also, it appears that the pre-made boxes (Q logic for example) do not follow this ratio. Anyone have any insights?
Also, what gauge speaker wire do you suggest? I am pushing this sub with a RF 250a2.
By the way, the shape I want to use (outside edge to outside edge) is 12 x 12 x 12 (with 3/4" MDF) (internally 10.5 x 10.5 x 10.5) which gives me an internal volume of .6699. What do you guys think
of that?
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Here's the question you clicked on:
if an aroma of vanillin fills a room, having the dimensions of 12ftx10.5ftx10.0ft. if the density is 2.4x10^-11g/L how many kilograms of vanillin are in the room?
• one year ago
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Best Response
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Density formula: \[\rho=\frac{m}{V}\] \[m=\rho V\] ------------------------------------------------------- Volume= 12x10.5x10.0=1260ft^3= 35679 liters \[m=\rho V\] \[m=( 2.4x10^{-11}g/L )(35679L)
\] \[m=8.56\times 10^{-7}g=8.56 \times 10^{-10}kg\]
Best Response
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hey thanks- can you explain how you get from cubic feet to liters?
Best Response
You've already chosen the best response.
You either remembering it or from a table of conversions, 1 Cubic Feet = 28.32 Liters You can arrange it so that the cubic feet is at the bottom when you multiply, to cancel the cubic feet. You
want liters here, so: \[1280~ ft^3 \times \frac{28.32~liters}{1~Cubic~feet}=35679~liters\]
Best Response
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got it. thank you so much!
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Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.
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Fundamental theories are gauge theories
It has long puzzled me that such unwanted degrees of freedom of the photon field (arising from the so-called "continuous spin" operators in the Poincare unirrep) should automatically hook up with
local phase changes of the fermion field.
The thing is however, that the extra degrees of freedom are not just unwanted, but unphysical. The photon is simply not an object suited to being described by fields, since it is a rep of the little
group ##ISO(2)##, but fields can only naturally carry reps of the little group ##SO(3)##. Only constrained fields can described the photon. So, when one uses an unconstrained field like ##A_{\mu}##,
then as you know, we get two extra nonphysical polarisations that cause unitarity to break down. This extra polarisations can then also interact with the electron, either through scattering, but more
importantly, they form part of its Coloumb field.
In the field formalism, these extra polarisations manifest as gauge degrees of freedom, so the electron also picks up a guage variance. Since the photons form the electrons Coloumb field or electric
charge, which is transformed by global phase changes, these non-physical photons induce a gauged phase.
But I doubt that nature "cares" whether we can/can't solve our full interacting theory by perturbation around a free theory.
It's nothing to do with solving the theory. Theories with divergences do not exist. So perhaps a clearer way to say it is that there doesn't exist a field theory in four dimensions that doesn't
contain a massless spin-1 particle and hence has gauge symmetry in the field formalism.
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On best simultaneous approximation
S.V.R. Naidu
Department of Applied Mathematics, A.U.P.G. Centre Nuzvid 521201, India
Abstract: For nonempty subsets $F$ and $K$ of a nonempty set $V$ and a real valued function $f$ on $X\times X$ the notion of $f$-best simultaneous approximation to $F$ from $K$ is introduced as an
extension of the known notion of best simultaneous approximation in normed linear spaces. The concept of uniformly quasi-convex function on a vector space is also introduced. Sufficient conditions
for the existence and uniqueness of $f$-best simultaneous approximation are obtained.
Classification (MSC2000): 41A28, 41A50, 41A52; 54H99, 46A99
Full text of the article:
Electronic fulltext finalized on: 2 Nov 2001. This page was last modified: 16 Nov 2001.
© 2001 Mathematical Institute of the Serbian Academy of Science and Arts
© 2001 ELibM for the EMIS Electronic Edition
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How tall is 67 inches in feet?
A system of measurement is a set of units which can be used to specify anything which can be measured and were historically important, regulated and defined because of trade and internal
commerce. In modern systems of measurement, some quantities are designated as fundamental units, meaning all other needed units can be derived from them, whereas in the early and most historic
eras, the units were given by fiat (see statutory law) by the ruling entities and were not necessarily well inter-related or self-consistent.
Zeng Jinlian (simplified Chinese: 曾金莲; traditional Chinese: 曾金蓮; pinyin: Zēng Jīnlián; June 25, 1964 – February 13, 1982) was the tallest woman ever verified in medical history, taking over
Jane Bunford's record. Zeng is also the only woman counted among the 14 people who have reached verified heights of eight feet or more.
At the time of her death at the age of 17 in Hunan, China, Zeng, who would have been 8 feet, 1.75 inches tall (she could not stand straight because of a severely deformed spine), was the tallest
person in the world. She had also broken Robert Wadlow's previous record of tallest 17 year-old ever. In the year between Don Koehler's death and her own, she surpassed fellow "eight-footers" Gabriel
Estêvão Monjane and Suleiman Ali Nashnush. That Zeng's growth patterns mirrored those of Robert Wadlow is shown in the table below.
Many different units of length have been used around the world. The main units in modern use are U.S. customary units in the United States and the Metric system elsewhere. British Imperial units
are still used for some purposes in the United Kingdom and some other countries. The metric system is sub-divided into SI and non-SI units.
The base unit in the International System of Units (SI) is the metre, defined as "the length of the path travelled by light in vacuum during a time interval of 1/299,792,458 of a second." It is
approximately equal to 1.0936 yards. Other units are derived from the metre by adding prefixes from the table below:
A rack unit, U or RU is a unit of measure that describes the height of equipment designed to mount in a 19-inch rack or a 23-inch rack. The 19-inch (482.6 mm) or 23-inch (584.2 mm) dimension
refers to the width of the equipment mounting frame in the rack including the frame; the width of the equipment that can be mounted inside the rack is less). One rack unit is 1.75 inches (44.45
mm) high.
The size of a piece of rack-mounted equipment is frequently described as a number in "U". For example, one rack unit is often referred to as "1U", 2 rack units as "2U" and so on.
Law Crime
The term crime does not, in modern times, have any simple and universally accepted definition, but one definition is that a crime, also called an offence or a criminal offence, is an act harmful
not only to some individual, but also to the community or the state (a public wrong). Such acts are forbidden and punishable by law.
Related Websites:
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(ROC) L
Receiver Operating Characteristic (ROC) Literature Research
PI: Kelly H. Zou, Ph.D.
Grant Acknowledgement: This research has been supported by NIH R01-LM007861 "Improved Tumor Resection in Image-Guided Neurosurgery."
A. BACKGROUND (EARLY PAPERS AND TEXTBOOKS).
• J.P. Egan. Signal Detection Theory and ROC Analysis, Academic Press,New York, 1975.
• D.M. Green and J.A. Swets. Signal Detection Theory and Psychophysics. Robert E. Krieger Publishing Co., Huntington,New York, 1974.
• D. Jalihal and L.W. Nolte. Signal detection theory and reconstruction algorithms-performance for images in noise. IEEE Transactions on Biomedical Engineering, 41:501-4, 1994.
• L.B. Lusted. Signal detectability and medical decision-making. Science, 171:1217-1219, 1971.
• N.A. Macmillan and C.D. Creelman. Detection Theory: A User's Guide, Cambridge University Press, Cambridge, 1991.
• B.J. McNeil and S.J. Adelstein. Determining the value of diagnostic and screening tests. Journal of Nuclear Medicine, 17:439-448, 1976.
• B.J. McNeil, E.Keeler, and S.J. Adelstein. Primer on certain elements of medical decision making. New England Journal of Medicine, 293:211-215, 1975.
• W.W. Peterson, T.G. Birdsall, and W.C. Fox. The theory of signal detectibility. Transactions of the IRE Professional Group in Information Theory, PGIT, 2-4:171-212, 1954.
• J.A. Swets and R.M. Pickett. Evaluation of Diagnostic Systems: Methods from Signal Detection Theory. Academic Press,New York, 1982.
• J.A. Swets. Signal Detection Theory and ROC Analysis in Psychology and Diagnostics: Collected Papers. Lawrence Erlbaum Associates, Publishers, Mahwah, New Jersy, 1995.
• M. Treisman and A. Faulkner. The effect of signal probability on the slope of the receiver operating characteristic given by the rating procedure British J. Math. Statist. Psych, 37:199,215,1984.
B. OVERVIEWS / REVIEWS.
• C.B. Begg. Statistical methods in medical diagnosis. CRC Critical Reviews in Medical Informatics, 1:1-22, 1986.
• C.B. Begg. Advances in statistical methodology for diagnostic medicine in the 1980's. Statistics in Medicine, 10:1887-1895, 1991.
• G. Champbell. Advances in statistical methodology for the evaluation of diagnostic and laboratory tests. Statistics in Medicine, 13 :499-508,1994.
• R.M. Centor. Signal detectability: The use of ROC curves and their analyses. Medical Decision Making, 11:102-6, 1991.
• L.S. Erdreich and E.T. Lee. Use of relative operating characteristic analysis in epidemiology: A method for dealing with subjective judgement. American Journal of Epidemiology, 114:649-62, 1981.
• D.J. Goodenough, K.Rossmann, and L.B. Lusted. Radiographic applications of receiver operating characteristic (ROC) curves. Radiology, 110:89-95, 1974.
• J.A. Hanley. Receiver operating characteristic (ROC) methodology: The state of the art. Critical Reviews in Diagnostic Imaging, 29:307-35, 1989.
• A.R. Henderson. Assessing test accuracy and its clinical consequences: a primer for receiver operating characteristic curve analysis. [Review] Annals of Clinical Biochemistry,30:521-39, 1993.
• J.K. Hsiao, J.J. Bartko, and W.Z. Potter. Diagnosing diagnoses: Receiver operating characteristic methods and psychiatry. Archives of General Psychiatry, 46:664-7, 1989.
• C.E. Metz. Basic principles of ROC analysis. Seminars In Nuclear Medicine, 8:283-98, 1978.
• C.E. Metz. ROC methodology in radiologic imaging. Investigative Radiology, 21:720-733, 1986.
• C.E. Phelps. The methodologic foundations of studies of the appropriateness of medical care [see comments]. New England Journal of Medicine, 329(17):1241-5, 1993.
• Y.T. van der Schouw, A.L. Verbeek, and S.H.Ruijs. Guidelines for the assessment of new diagnostic tests. Invest Radiol, 30:334-40,1995.
• M. Schulzer. Diagnostic tests: a statistical review. [Review] Muscle and Nerve, 17(7):815-9, 1994.
• J.A. Swets. ROC analysis applied to the evaluation of medical imaging techniques. Investigative Radiology, 14:109-21, 1979.
• J.A. Swets. Measuring the accuracy of diagnostic systems. Science, 240:1285-93, 1988.
• D.A. Turner. An intuitive approach to receiver operating characteristic curve analysis. Journal of Nuclear Medicine, 19:213-20, 1978.
C. DESIGN OF ROC STUDIES / BIAS.
• C.B. Begg and R.A. Greenes. Assessment of diagnostic tests when disease verification is subject to selection bias. Biometrics, 39:207-215, 1983.
• C.B. Begg. Biases in the assessment of diagnostic tests. Statistics in Medicine, 6:411-423, 1987.
• C.B. Begg. Experimental design of medical imaging trials: Issues and options. Investigative Radiology, 24:934-936, 1989.
• C.B. Begg and B.J. McNeil. Assessment of radiologic tests: Control of bias and other design considerations. Radiology, 167:565-9, 1988.
• K.S. Berbaum, D.D. Dorfman, and E.A. Franken, Jr. Measuring observer performance by ROC analysis: Indications and complications. Investigative Radiology, 24:228-33, 1989.
• G.A. Diamond. ROC steady: A receiver operating characteristic curve that is invariant relative to selection bias. Medical Decision Making, 7:238-43, 1987.
• G.A. Diamond. ROCky III. Medical Decision Making, 7:247-9, 1987.
• G.A. Diamond. Can the discriminant accuracy of a test be determined in the face of selection bias? Medical Decision Making,11:48-56, 1991.
• G.A. Diamond. Scotched on the ROCs. Medical Decision Making, 11:198-200, 1991.
• G.A. Diamond. What is the effect of sampling error on ROC analysis in the face of verification bias? Medical Decision Making, 12:155-6, 1992.
• C.A. Gatsonis and X.H. Zhou. Group sequential designs for comparative ROC studies in diagnostic medicine, Technical report, Department of Health Care Policy, Harvard Medical School, Boston,
Massachusetts, 1993.
• W.F. Good, D.Gur, W.H. Straub, and J.H. Feist. Comparing imaging systems by ROC studies: Detection versus interpretation. Investigative Radiology, 24:932-3, 1989.
• R.Gray, C.B. Begg, and R.A. Greenes. Construction of receiver operating characteristic curves when disease verification is subject to selection bias. Medical Decision Making, 4:151-64, 1984.
• R.A. Greenes and C.B. Begg. Assessment of diagnostic technologies: Methodology for unbiased estimation from samples of selectively-verified patients. Investigative Radiology, 20:751-756, 1985.
• D.Gur, J.L. King, H.E. Rockette, C.A. Britton, F.L. Thaete, and R.J. Hoy. Practical issues of experimental ROC analysis: Selection of controls. Investigative Radiology, 25:583-6, 1990.
• D.Gur, H.E. Rockette, W.F. Good, B.S. Slasky, L.A. Cooperstein, W.H. Straub, N.A. Obuchowski, and C.Metz. Effect of observer instruction on ROC study of chest images. Investigative Radiology,
25:230-4, 1990.
• J.A. Hanley and C.B. Begg. Response to ROC steady. Medical Decision Making, 7:244-6, 1987.
• J.A. Hanley. Verification bias and the one-parameter logistic ROC curve-some clarifications. [Comment on Diamond 1991]. Medical Decision Making, 11:203-7, 1991.
• M.G. Hunink and C.B. Begg. Diamond's correction method-a real gem or just cubic zirconium? [Comment on Diamond 1991]. Medical Decision Making, 11:201-3, 1991.
• J.L. King, C.A. Britton, G.D., H.E. Rockette, and P.L. Davis. On the validity of the continuous and discrete confidence rating scales in receiver operating characteristic studies. Investigative
Radiology,28:962-963, 1993.
• C.E. Metz. Some practical issues of experimental design and data analysis in radiological ROC studies. Investigative Radiology, 24:234-45, 1989.
• C.E. Metz and J.H. Shen. Gains in accuracy from replicated readings of diagnostic images: Prediction and assessment in terms of ROC analysis. Medical Decision Making, 12:60-75, 1992.
• N.A. Obuchowski. Computing sample size for receiver operating characteristic studies. Investigative Radiology, 29:238-243, 1994.
• D.F. Ransohoff and A.R. Feinstein. Problems of spectrum and bias in evaluating the efficacy of diagnostic tests. New England Journal of Medicine, 299:926-930, 1978.
• H.E. Rockette, D.Gur, L.A. Cooperstein, N.A. Obuchowski, J.L. King, C.R. Fuhrman, E.K. Tabor, and C.E. Metz. Effect of two rating formats in multi-disease ROC study of chest images. Investigative
Radiology, 25:225-9, 1990.
• H.E. Rockette, D.Gur, and C.E. Metz. The use of continuous and discrete confidence judgments in receiver operating characteristic studies of diagnostic imaging techniques. Investigative
Radiology, 27:169-172, 1992.
D. CURVE-FITTING.
• I.G. Abrahamson and H.Levitt. Statistical analysis of data from experiments in human signal detection. [Maximum likelihood ROC curve - general location-scale family]. Journal of Mathematical
Psychology, 6:391-417, 1969.
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• D.D. Dorfman and E.Alf, Jr. Maximum likelihood estimation of parameters of signal detection theory and determination of confidence intervals-rating method data. [Binormal ROC curve-ordinal data].
Journal of Mathematical Psychology, 6:487-496, 1969.
• D.R. Grey and B.J.T. Morgan. Some aspects of ROC curve-fitting: Normal and logistic models. [Corrections and improvements to Dorfman-Alf; minimum chi-square bilogistic]. Journal of Mathematical
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• J.A. Hanley. The robustness of the ``binormal'' assumptions used in fitting ROC curves. Medical Decision Making, 8:197-203, 1988.
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• H.L. Kundel, D.D. Dorfman and K.S. Berbaum. Degeneracy and discrete receiver operating characteristic rating data, 2 :907-15,1995.
• C.E. Metz and X. Pan. "Proper" binormal ROC curves: theory and maximum-likelihood estimation, unpublished manuscript, University of Chicago.
• D. Mossman. Resampling techniques in the analysis of non-binormal ROC data. Med Decis Making , 15:358-66,1995.
• J.C. Ogilvie and C.D. Creelman. Maximum-likelihood estimation of receiver operating characteristic curve parameters. Journal of Mathematical Psychology, 5:377-391, 1968.
• E. Somoza. Eccentric Diagnostic Tests. Medical Decision Making, 16:15-23, 1996.
• J.A. Swets. Form of empirical ROCs in discrimination and diagnostic tasks: Implications for theory and measurement of performance. Psychological Bulletin, 99:181-198, 1986.
• A.N. Tosteson and C.B. Begg. A general regression methodology for ROC curve estimation. Medical Decision Making, 8:204-15, 1988.
• A.N. Tosteson, J. Wittenberg, and C.B. Begg. ROC curve regression analysis: the use of ordinal regression models for diagnostic test assessment. Environmental Health Perspectives, 8:73-8, 1994.
• X.H. Zhou. Testing an underlying assumption on a ROC curve based on rating data. Med Decis Making, 15:276-82,1995.
• R.M. Centor and J.S. Schwartz. An evaluation of methods for estimating the area under the receiver operating characteristic (ROC) curve. Medical Decision Making, 5:149-56, 1985.
• C. Cox. Location-scale cumulative odds models for ordinal data: a generalized non-linear model approach. Statistics in Medicine,14:1191-203, 1995.
• J.A. Hanley and B.J. McNeil. The meaning and use of the area under a receiver operating characteristic (ROC) curve. Radiology, 143:29-36, 1982.
• R.D. Hays. ROC: Estimation of the area under a receiver operating characteristic curve. Appl. Psych. Meas, 14 :208,1990.
• J.Hilden. The area under the ROC curve and its competitors. Medical Decision Making, 11:95-101, 1991.
• D.Katz and B.Foxman. How well do prediction equations predict? Using receiver operating characteristic curves and accuracy curves to compare validity and generalizability. Epidemiology,4
(4):319-26, 1993.
• D.K. McClish. Analyzing a portion of the ROC curve. Medical Decision Making, 9:190-5, 1989.
• T.O. Nelson. ROC curves and measures of discrimination accuracy: A reply to Swets. [Comment on Swets 1986a]. Psychological Bulletin, 100:128-132, 1986.
• J.J. Riera-Diaz. The estimation of event related potentials affected by random shifts and scalings. International Journal of Bio-Medical Computing,38:109-20,1995.
• A.J. Simpson and M.J. Fitter. What is the best index of detectability? [Recommends the area under the binormal curve]. Psychological Bulletin, 80:481-488, 1973.
• J.A. Swets. Indices of discrimination or diagnostic accuracy: Their ROCs and implied models. Psychological Bulletin, 99:100-117, 1986.
• A.Agresti. A survey of models for repeated ordered categorical response data. Statistics in Medicine, 8:1209-1224, 1989.
• D.Bamber. The area above the ordinal dominance graph and the area below the receiver operating graph. Journal of Mathematical Psychology, 12:387-415, 1975.
• C.A. Beam. Random-effects models in the receiver operating characteristic curve-based assessment of the effectiveness of diagnostic imaging technology: concepts, approaches, and issues. Academic
Radiology, 2: supple: 4-13, 1995.
• C.A. Beam and H.S. Wieand. A statistical method for the comparison of a discrete diagnostic test with several continuous diagnostic tests. Biometrics, 47:907-919, 1991.
• D.A. Block. Comparing two diagnostic tests against the same "gold standard" in the same sample. Biometrics, 53: 73-85, 1997.
• C. Cox. Location-scale cumulative odds models for ordinal data: a generalized non-linear model approach. Statistics in Medicine, 15 :1191-203, 1995.
• E.R. DeLong, W.B. Vernon, and R.R. Bollinger. Sensitivity and specificity of a monitoring test. Biometrics, 41:947-958, 1985.
• E.R. DeLong, D.M. DeLong, and D.L. Clarke-Pearson. Comparing the areas under two or more correlated receiver operating characteristic curves: A nonparametric approach. Biometrics, 44:837-45,
• D.D. Dorfman, K.S. Berbaum, and C.E. Metz. Receiver operating characteristic rating analysis: Generalization to the population of readers and patients with the jackknife method. Investigative
Radiology, 27:723-31, 1992.
• B. Emir, S.Wieand, J.Q. Su, and S. Cha. Analysis of repeated markers used to predict progression of cancer. Statistics in Medicine, 17: 2563-78, 1998.
• B. Emir, S. Wieand, S.-H. Jung, and Z. Ying. Comparison of diagnostic markers with repeated measurements: A non-parametric ROC curve approach. Staistics in Medicine, 19: 511-23, 2000.
• C.A. Gatsonis. Random-effects models for diagnostic accuracy data. Academic Radiology, 2:suppl:14-21, 1995.
• J.A. Hanley. Alternative approaches to receiver operating characteristic analyses. Radiology, 168:568-70, 1988.
• J.A. Hanley and B.J. McNeil. A method of comparing the areas under receiver operating characteristic curves derived from the same cases. Radiology, 148:839-43, 1983.
• R.A. Hilgers. Distribution-free confidence bounds for ROC curves. Methods of Information In Medicine, 30:96-101, 1991.
• F. Hsieh and B.W. Turnbull. Nonparametric and semiparametric estimation of the receiver operating characteristic curve. Annals of Statistics, 24: 25-40, 1996.
• K. Jensen, H.-H. Muller and H. Schafer. Regional confidence bands for ROC curves. Statistics in Medicine, 19: 493-509, 2000.
• V. Kairisto and A. Poola. Software for illustrative presentation of basic clinical characteristics of laboratory tests--GraphROC for Windows. Scandinavian Journal of Clinical & Laboratory
Investigation, 222(suppl):43-60, 1995.
• K. Kim. A bivariate cumulative probit regression model for ordered categorical data. Statistics in Medicine, 14 :1341-1352, 1995.
• G.Ma and W.J. Hall. Confidence bands for receiver operating characteristic curves. Medical Decision Making, 13:191-197, 1993.
• D.K. McClish. Comparing the areas under more than two independent ROC curves. Medical Decision Making, 7:149-55, 1987.
• D.K. McClish. Determining a range of false-positive rates for which ROC curves differ. Medical Decision Making, 10:283-7, 1990.
• B.J. McNeil and J.A. Hanley. Statistical approaches to the analysis of receiver operating characteristic (ROC) curves. Medical Decision Making, 4:137-50, 1984.
• A. Moise, B. Clement, M. Raissis and P. Manopoulos. A test for crossing receiver operating characteristic (ROC) curves. Comm. Statist. A -- Theory Methods, 17:1985-2004,1988.
• C.E. Metz. Statistical analysis of ROC data in evaluating diagnostic performance. In D.E. Herbert and R.H. Myers, editors, Multiple regression analysis: Applications in the health sciences, pages
365-384,American Institute of Physics, New York,1986.
• C.E. Metz. Quantification of failure to demonstrate statistical significance: The usefulness of confidence intervals. Investigative Radiology, 28:59-63, 1993.
• C.E. Metz and H.B. Kronman. Statistical significance tests for binormal ROC curves. Journal of Mathematical Psychology, 22:218-243, 1980.
• C.E. Metz, J.H. Shen, and B.A. Herman. New methods for estimating a binormal ROC curve from continuously-distributed test results. Unfinished manuscript; presented at the 1990 Joint Statistical
Meetings in Anaheim, California.
• C.E. Metz, P.-L. Wang, and H.B. Kronman. A new approach for testing the significance of differences between ROC curves measured from correlated data. In F.Deconinck, editor, Information
Processing in Medical Imaging: Proceedings of the Eighth Conference, 432-445.Martinus Nijhoff Publishers, The Hague, 1984.
• A.Moise, B.Clement, and M.Raissis. A test for crossing receiver operating characteristic (ROC) curves. Communications in Statistics-Theory and Methods, 17:1985-2003, 1988.
• P.A. Murtaugh. ROC curves with multiple marker measurements. Biometrics, 51 :1514-22, 1995.
• N.A. Obuchowski. Multireader, multimodality receiver operating characteristic curve studies: hypothesis testing and sample size estimation using an analysis of variance approach with dependent
observations. Academic Radiology, 2: supple: 22-29,1995.
• N.A. Obuchowski Nonparametric analysis of clustered ROC curve data. Unpublished manuscript, 1996 ENAR meeting.
• H.E. Rockette, N.A. Obuchowski, and D.Gur. Nonparametric estimation of degenerate ROC data sets used for comparison of imaging systems. Investigative Radiology, 25:835-7, 1990.
• H.E. Rockette, N.Obuchowski, C.E. Metz, and D.Gur. Statistical issues in ROC curve analysis. Proceedings of the SPIE, 1234:111-119, 1990.
• D.E. Shapiro. The interpretation of diagnostic tests. Statistical Methods in Medical Research, 8: 113-134, 1999.
• E.K. Shultz. Multivariate receiver-operating characteristic curve analysis: prostate cancer screening as an example.[Review] Clinical Chemistry, 41:1248-55, 1995.
• P.J. Smith, T.J. Thompson, M.M. Engelgau and W.H. Herman. A generalized linear model for analysing receiver operating characteristic curves. Statistics in Medicine, 15 :323-33, 1996.
• E. Svensson, and S. Holm. Separation of systematic and random differences in ordinal rating scales. Statistics in Medicine,13:2437-53, 1994.
• M. Swaving, H. Van Houwelingen, F.P. Ottes, T. Steerneman. Statistical comparison of ROC curves from multiple readers. Medical Decision Making, 16 : 143-153, 1996.
• M.L. Thompson and W.Zucchini. On the statistical analysis of ROC curves. Statistics In Medicine, 8:1277-90, 1989.
• A.L. Toledano and C.A. Gatsonis. Regression analysis of correlated receiver operating characteristic data. Academic Radiology,2:supple:30--36, 1995.
• A.Y. Toledano and C.A. Gatsonis. Ordinal regression methodology for roc curves derived from correlated data. Statistics in Medicine,15:1807-26,1996.
• S. Wieand, M.H. Gail, B.R. James, and K.L. James. A family of nonparametric statistics for comparing diagnostic markers with paired or unpaired data. Biometrika , 76:585-592, 1989.
• X.H. Zhou. Testing an underlying assumption on a ROC curve based on rating data. Medical Decision Making,15:276-282,1995.
• X.H. Zhou Testing an underlying assumption on a ROC curve based on rating data. Medical Decision Making, 15 :276-82, 1995.
• X.H. Zhou and C.A. Gatsonis. A smple method for comparing correlated roc curves using incomplete data. Statistics in Medicine, 15 :1687-1693, 1996.
• K.H. Zou, C. M Tempany, J. R. Fielding, and S. G. Silverman. Original smooth receiver operating characteristic curve estimation from continous data: Statistical methods for analyzing the
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• G. Campbell and J.M. Deleo. Fundamentals of fuzzy receiver operating characteristic (ROC) functions. Comput. Sci. Statist.: Proc. 21st Symp. Interface (Kenneth Berk and Linda Malone, eds.), Amer.
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• P.Deneef. Evaluating rapid test for streptococcal pharyngitis: The apparent accuracy of a diagnostic test when there are errors in the standard of comparison. Medical Decision Making, 7:92-96,
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• J.J. Gart and A.A. Buck. Comparison of a screening test and a reference test in epidemiological studies: II. a probabilistic model for the comparison of diagnostic tests. American Journal of
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• D.A. Grayson. Statistical diagnosis and the influence of diagnostic error. Biometrics, 43:975-984, 1987.
• W.J. Hall & D.E. Shapiro A receiver operating characteristic curve for subjective probability of disease assessments in the absence of a gold standard (abstract only). Medical Decision Making,
14:430, 1994.
• R.M. Henkelman, I.Kay, and M.J. Bronskill. Receiver operator characteristic (ROC) analysis without truth. Medical Decision Making, 10:24-9, 1990.
• S.L. Hui and S.D. Walter. Estimating the error rates of diagnostic tests. Biometrics, 36:167-171, 1980.
• S.L. Hui and X.H. Zhou. Evaluation of diagnostic tests without gold standards. Statistical methods in medical research, 7: 354-370, 1998.
• H.C. Kraemer. The robustness of common measures of 2X2 association to bias due to misclassifications. American Statistician, 39:286-290, 1985.
• N.J.D. Nagelkerke, V.Fidler, and M.Buwalda. Instrumental variables in the evaluation of diagnostic test procedures when the true disease state is unknown. Statistics in Medicine, 7:739-744, 1988.
• C.E. Phelps and A.Hutson. Estimating diagnostic test accuracy using a `fuzzy' gold standard. Medical Decision Making, 15:44-57, 1995.
• R.M. Poses, R.D. Cebul, and R.M. Centor. Evaluating physicians' probabilistic judgements. Medical Decision Making, 8:233-240, 1988.
• R.M. Poses, C.B. Bekes, R.L. Winkler, W.E. Scott, and F.J. Copare. Are two (inexperienced) heads better than one (experienced) head? Archives of Internal Medicine, 150:1874-1878, 1990.
• M.Staquet, M.Rozencweig, Y.J. Lee, and F.M. Muggia. Methodology for the assessment of new dichotomous diagnostic tests. Journal of Chronic Disease, 34:599-610, 1981.
• V. Torrance-Rynard and S. Walter. Effects of dependent errors in the assessment of diagnostic test performance. Statistics in Medicine, 16: 2157-2175, 1997.
• J.S. Uebersax. Statistical modeling of expert ratings on medical treatment appropriateness. Journal of the American Statistical Association, 88:421-427, 1993.
• J.S. Uebersax and W.M. Grove. Latent class analysis of diagnostic agreement. Statistics in Medicine, 9:559-572, 1990.
• J.S. Uebersax and W.M. Grove. A latent trait finite mixture model for the analysis of rating agreement. Biometrics, 49:823-835, 1993.
• P.M. Vacek. The effect of conditional dependence on the evaluation of diagnostic tests. Biometrics, 41:959-968, 1985.
• P.N. Valenstein. Evaluating diagnostic tests with imperfect standards. American Journal of Clinical Pathology, 93:252-258, 1990.
• S.D. Walter and L.M. Irwig. Estimation of test error rates, disease prevalence, and relative risk from misclassified data. Journal of Clinical Epidemiology, 41:923-38, 1988.
• X.H. Zhou and R. Higgs. Assessing the relative accuracies of two screening tests in the presence of verification bias. Statistics in Medicine, 19: 1697-1705, 2000.
• X.H. Zhou and R. Higgs. COMPROC and CHECKNORM: computer programs for comparing accuracies of diagnostic tests using ROC curves in the presence of verification bias. Computer Methods and Programs
in Biomedicine, 57: 179-186,1998.
• X.H. Zhou and C.A. Rodenberg. Estimating a ROC curve in the presence of verification bias. Communication in Statistics, Ther., 27: 635-657.
• X.H. Zhou. Maximum likelihood estimators of sensitivity and specificity corrected for verification bias. Commun. Statist. Theor. Meth, 22: 3177-3198, 1993.
• X.H. Zhou. Effect of verification bias on positive and negative predictive values. Statistics in Medicine, 13: 1737-1745, 1994.
• X.H. Zhou. Nonparametric ML estimate of a ROC area corrected for verification bias. Biometrics, 52: 310-316, 1996.
• X.H. Zhou. Comparing the correlated areas under the ROC curves of two diagnostic tests in the presence of verification bias. Biometrics, 54, 453-470, 1998.
• X.H. Zhou. Comparing accuracies of two screening tests in the presence of verication bias. JRSS, Ser C, 47: 135-147, 1998.
• X.H. Zhou. Correction for verificatin bias in studies of a diagnostic test's accuracy. Statistics Methods in Medical Research, 7: 337-353, 1998.
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• B. Sahiner, H.P. Chan, N. Petrick, et al. Feature selection and classifier performance in computer-aided diagnosis: The effect of finite sample size. Medical Physics, 27(7):1509-1522, 2000.
• M.S. Scher, B.L. Jones, D.A. Steppe, et al. Functional brain maturation in neonates as measured by EEG-sleep analyses. Clinical Neurophysiology, 114(5):875-882, 2003.
• E.F. Schisterman, D. Faraggi, B.Reiser. Adjusting the generalized ROC curve for covariates. Statistics in Medicine, 23(21):3319-3331, 2004.
• P.M. Shankar, V.A. Dumane, T. George, et al. Classification of breast masses in ultrasonic B scans using Nakagami and K distributions. Physics in Medicine and Biology, 48(14):2229-2240, 2003.
• P.M. Shankar, V.A. Dumane, C.W. Piccoli, et al. Computer-aided classification of breast masses in ultrasonic B-scans using a multiparameter approach. IEEE Transactions on Ultrasonics
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• H.C. Sox, Jr. Probability theory in the use of diagnostic tests. Annals of Internal Medicine, 104:60-66, 1986.
• C. Stephan, S. Wesseling, T. Schink, et al. Comparison of eight computer programs for receiver-operating characteristic analysis. Clinical Chemistry, 49(3):433-439, 2003.
• R.G. Swensson, J.L. King, D. Gur. A constrained formulation for the receiver operating characteristic (ROC) curve based on probability summation. Medical Physics, 28(8):1597-1609, 2001.
• Y. Takahashi, K. Murase, H. Higashino, et al. Receiver operating characteristic (ROC) analysis of images reconstructed with iterative expectation maximization algorithms. Annals of Nuclear
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• L.A. Thibodeau. Evaluating diagnostic tests. Biometrics, 37:801-804, 1981.
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• M.L. Thompson. Assessing the diagnostic accuracy of a sequence of tests. Biostatistics, 4(3):341-351, 2003.
• M.Treisman and A.Faulkner. The effect of signal probability on the slope of the receiver operating characteristic given by the rating procedure. British Journal of Mathematical and Statistical
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• P.M. Vacek. The effect of correlation between two diagnostic tests on maximum likelihood estimates of test error rates and disease prevalences. In ASA Proceedings of the Statistical Computing
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• P.M. Vacek, M.Kresnow, and R.M. Potter. The sampling distributions of estimators for the error rates of diagnostic tests. ASA Proceedings of the Statistical Computing Section, 438-440, 1985.
• R. Wagner. Multiple-reader, multiple-case (MRMC) ROC analysis in diagnostic imaging, computer-aided diagnosis, and statistical pattern recognition. Medical Physics, 29(6):1331-1331, 2002.
• R.F. Wagner, S.V. Beiden, G. Campbell, et al. Assessment of medical imaging and computer-assist systems: Lessons from recent experience. Academic Radiology, 9(11):1264-1277, 2002.
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• T. Wigren, S. Remle, B. Wahlberg. Analysis of a low-complexity change detection scheme. International Journal of Adaptive Control and SIgnal Processing, 14(5):481-503, 2000.
• A.A. White and J.R. Landis. A general categorical data methodology for evaluating medical diagnostic tests. Communication in Statistics, Part A-Theory and Methods, 117:567-605, 1982.
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Each tutorial is in one or more of the following four formats:
cdf: computable document format
Interactive presentation that can be explored in your browser with the free CDF Player plugin. Can also be viewed and modified using Mathematica 8+.
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These are interactive presentations which can be run in Mathematica 5+. Also viewable with the CDF Player plugin, but you won't be able to run them. Try the html versions instead.
pdf: portable document format
Download Acrobat Reader to view pdfs.
Can be viewed right here in your browser.
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html nb
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More tutorials:
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• Nanoscopy with nitrogen-vacancy centers in diamond: the movie (m4v)
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Ordinary Generating Function for Mobius
up vote 8 down vote favorite
Is there any information known for the Ordinary Generating Function for Mobius? $$ \sum_{n=1}^{\infty} {\mu(n)}x^n $$ I know that
1. It has radius of convergence 1.
2. Does not have limit as $x\rightarrow 1$.
My question is
1. Does it have limit as $x\rightarrow \textrm{exp}(i\theta)\neq 1$?
2. Is there a similar result for $\textrm{exp}(i\theta)\neq 1$ with $$ \sum_{n\leq x}\mu(n)=o(x)$$ i.e. is there a result of this type? $$ \sum_{n\leq x}\mu(n)\textrm{exp}(in\theta) = o(x)$$
EDIT1: Aside from my questions, there are some known results for the OGF for Mobius.
EDIT2: Found a reference for (III)
Reference: JASON P. BELL, NILS BRUIN, AND MICHAEL COONS "TRANSCENDENCE OF GENERATING FUNCTIONS WHOSE COEFFICIENTS ARE MULTIPLICATIVE" available at arXiv:1003.2221v2
P. Borwein, T. Erd´elyi, and F. Littman, Polynomials with coefficients from a finite set available at http://www.cecm.sfu.ca/personal/pborwein/PAPERS/P195.pdf
(I) This function has the unit circle as natural boundary.
(II) This function is a transcendental function.
(III) It is not bounded on any open sector($\{z:|z|<1, z=re^{i\theta}, \alpha<\theta<\beta\}$)of unit disc.
This result (III) together with Baire Category answers my question 1 with a dense set of $\theta$'s, but as Prof Tao mentioned, there can be a possibility that it is bounded for some values of $\
nt.number-theory analytic-number-theory
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2 Answers
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Mobius randomness heuristics suggest that $\sum_n \mu(n) (r e^{i\theta})^n$ does not converge to a limit as $r \to 1^-$ for any (or at least almost any) $\theta$. If it did converge for some $
\theta$, then we would have
$$\sum_n \mu(n) e^{in\theta} \psi_k(n) \to 0$$ as $k \to \infty$, where
$$ \psi_k(n) := (1-2^{-k-1})^n - (1-2^{-k})^n.$$
The cutoff function $\psi_k$ is basically a smoothed out version of the indicator function $1_{[2^k,2^{k+1}]}$. The Mobius randomness heuristic (discussed for instance in this article of
Sarnak) then suggests that the sum $\sum_n \mu(n) e^{in\theta} \psi_k(n)$ should have a typical size of $2^{k/2}$, and so should not decay to zero as $k \to \infty$.
vote Note from Plancherel's theorem that the $L^2_\theta$ mean of $\sum_n \mu(n) e^{in\theta} \psi_k(n)$ is indeed comparable to $2^{k/2}$. This does not directly preclude the (very unlikely)
12 scenario that this exponential sum is very small for many $\theta$ and only large for a small portion of the $\theta$, but if one optimistically applies various versions of the Mobius
down randomness heuristic (with square root gains in exponential sums) to guess higher moments of $\sum_n \mu(n) e^{in\theta} \psi_k(n)$ in $\theta$, one is led to conjecture a central limit
vote theorem type behaviour for the distribution of this quantity (as $\theta$ ranges uniformly from $0$ to $2\pi$), i.e. it should behave like a complex gaussian with mean zero and variance $\sim
2^k$, and in particular it should only be $O(1)$ about $O(2^{-k})$ of the time, and Borel-Cantelli then suggests divergence for almost every $\theta$ at least. Unfortunately this is all very
heuristic, and it seems difficult with current technology to unconditionally rule out a strange conspiracy that makes $\sum_n \mu_n (re^{i\theta})^n$ bounded as $r \to 1$ for some specific
value of $\theta$ (say $\theta = \sqrt{2} \pi$), though this looks incredibly unlikely to me. (One can use bilinear sums methods, e.g. Vaughan identity, to get some nontrivial pointwise upper
bounds on these exponential sums when $\theta$ is highly irrational, but I see no way to get pointwise lower bounds, since $L$-function methods will not be available in this setting, and there
may well be some occasional values of $\theta$ and $k$ for which these sums are actually small. But it is likely at least that the $\theta=0$ theory can be extended to rational values of $\
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For (2), the answer is yes - see for example this paper of Baker and Harman.
up vote 9 down
Thank you for this. Actually their paper is conditional. Indeed, unconditional result by H. Davenport is qjmath.oxfordjournals.org/content/os-8/1/313.full.pdf. Anyway, that
answers 2. – i707107 Apr 3 '13 at 21:01
1 Fair enough, although on the first page it cites unconditional results as well. – Greg Martin Apr 4 '13 at 5:57
Yeah. They cited Davenport's result. I enjoyed reading it. – i707107 Apr 4 '13 at 18:27
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Kaggle Digit Recognizer: K-means optimisation attempt
I recently wrote a blog post explaining how Jen and I used the K-means algorithm to classify digits in Kaggle’s Digit Recognizer problem and one of the things we’d read was that with this algorithm
you often end up with situations where it’s difficult to classify a new item because if falls between two labels.
We decided to have a look at the output of our classifier function to see whether or not that was the case.
As a refresher, this was our original function to determine the label for a new test data row…
(defn which-am-i [unranked-value]
(let [all-the-gaps (map #(find-gap %1 unranked-value) all-the-averages)]
[(ffirst (sort-by second all-the-gaps)) all-the-gaps]))
…and that would return a result like this:
user> (which-am-i (first test-data))
[0 ([0 1763.5688862988827] [1 2768.1143197890624] [2 2393.9091578180937]
[3 2598.4629450761286] [4 2615.1233720558307] [5 2287.1791665580586]
[6 2470.096959417967] [7 2406.0132574502527] [8 2489.3635108564304] [9 2558.0054056506265])]
We first changed the function to return another entry in the vector representing the top two picks:
(defn which-am-i [unranked-value]
(let [all-the-gaps (map #(find-gap %1 unranked-value) all-the-averages)
top-two (take 2 (sort-by second all-the-gaps))]
[(ffirst (sort-by second all-the-gaps)) top-two all-the-gaps]))
If we run that:
user> (which-am-i (first test-data))
[2 ([2 1855.3605185002546] [0 2233.654619238101])
([0 2233.654619238101] [1 2661.7148686603714] [2 1855.3605185002546]
[3 2512.429018687221] [4 2357.637631775974] [5 2457.9850052966344]
[6 2243.724487123002] [7 2554.1158473740174] [8 2317.567588716217] [9 2520.667565741239])]
In this case the difference between the top 2 labels is about 400 but we next changed the function to output the actual difference so we could see how close the top 2 were over the whole test set:
(defn which-am-i [unranked-value]
(let [all-the-gaps (map #(find-gap %1 unranked-value) all-the-averages)
top-two (take 2 (sort-by second all-the-gaps))
difference-between-top-two (Math/abs (apply - (map second top-two)))]
[(ffirst (sort-by second all-the-gaps)) top-two difference-between-top-two all-the-gaps]))
We then ran it like this, taking just the first 10 differences on this occasion:
user> (take 10 (->> test-data (map which-am-i) (map #(nth % 2))))
(378.2941007378465 523.6102802591759 73.57510749262792 3.8557350283749656 5.806672422475231
183.90928740097775 713.1626629833258 335.38646365464047 538.6191727330108 161.68429111785372)
From visually looking at this over a larger subset we noticed that there seemed to be a significant number of top twos within a distance of 50.
We therefore changed the function to use shuffle to randomly pick between those two labels when the distance was that small.
(defn which-am-i [unranked-value]
(let [all-the-gaps (map #(find-gap %1 unranked-value) all-the-averages)
top-two (take 2 (sort-by second all-the-gaps))
difference-between-top-two (Math/abs (apply - (map second top-two)))
very-close (< difference-between-top-two 50)
best-one (if very-close (ffirst (shuffle top-two)) (ffirst top-two))]
[best-one top-two all-the-gaps]))
Our assumption was that this might marginally improve the performance of the algorithm but it actually made it marginally worse, with a new accuracy of 79.3%
At this point we weren’t really sure what we should be doing to try and improve the algorithm’s performance so we moved onto random forests but it’s interesting that you can get a reasonable accuracy
even with such a simple approach!
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Source: http://www.sing365.com
Reebok Commercial Part 2
Beware, if E equals MC squared
Than A.I. equals ankles in intensive care
The answer keeps you glued to the ground suspended in air
The answer to every prayer for every player everywhere
Try to compare
Try to block the shot if you dare
You get the best seat in the house, it’s called the wheelchair
The math ?umopis?, you should be a reebok owner
Killer crossovers split opponent’s in half like moses
All contenders are left defenseless
They end up in hospital beds with swollen tendens, takin’ anit-depressants
They can’t seem to deal with them pressure
I came to give you this message
I got the ammo for your weapons, and the answer for your question /
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Geometric Quantization
John Baez
August 11, 2000
Geometric quantization is a marvelous tool for understanding the relation between classical physics and quantum physics. However, it's a bit like a power tool - you have to be an expert to operate it
without running the risk of seriously injuring your brain. Here's a brief sketch of how it goes. This is pretty terse; for the details you'll have to read the series of articles on geometric
quantization on the sci.physics.research archive.
1. We start with a CLASSICAL PHASE SPACE: mathematically, this is a manifold X with a symplectic structure ω.
2. Then we do PREQUANTIZATION: this gives us a Hermitian line bundle L over X, equipped with a U(1) connection D whose curvature equals iω. L is called the PREQUANTUM LINE BUNDLE.
Warning: we can only do this step if ω satisfies the BOHR-SOMMERFELD CONDITION, which says that ω/2π defines an integral cohomology class. If this condition holds, L and D are determined to
isomorphism, but not canonically.
3. The Hilbert space H[0] of square-integrable sections of L is called the PREQUANTUM HILBERT SPACE. This is not yet the Hilbert space of our quantized theory - it's too big. But it's a good step in
the right direction. In particular, we can PREQUANTIZE CLASSICAL OBSERVABLES: there's a map sending any smooth function on X to an operator on H[0]. This map takes Poisson brackets to
commutators, just as one would hope. The formula for this map involves the connection D.
4. To cut down the prequantum Hilbert space, we need to choose a POLARIZATION, say P. What's this? Well, for each point x in X, a polarization picks out a certain subspace P[x] of the complexified
tangent space at x. We define the QUANTUM HILBERT SPACE, H, to be the space of all square-integrable sections of L that give zero when we take their covariant derivative at any point x in the
direction of any vector in P[x]. The quantum Hilbert space is a subspace of the prequantum Hilbert space.
Warning: for P to be a polarization, there are some crucial technical conditions we impose on the subspaces P[x]. First, they must be ISOTROPIC: the complexified symplectic form ω must vanish on
them. Second, they must be LAGRANGIAN: they must be maximal isotropic subspaces. Third, they must vary smoothly with x. And fourth, they must be INTEGRABLE.
5. The easiest sort of polarization to understand is a REAL POLARIZATION. This is where the subspaces P[x] come from subspaces of the tangent space by complexification. It boils down to this: a real
polarization is an integrable distribution P on the classical phase space where each space P[x] is Lagrangian subspace of the tangent space T[x]X.
6. To understand this rigamarole, one must study examples! First, it's good to understand how good old SCHRÖDINGER QUANTIZATION fits into this framework. Remember, in Schrödinger quantization we
take our classical phase space X to be the cotangent bundle T^*M of a manifold M called the CLASSICAL CONFIGURATION SPACE. We then let our quantum Hilbert space be the space of all
square-integrable functions on M.
Modulo some technical trickery, we get this example when we run the above machinery and use a certain god-given real polarization on X = T^*M, namely the one given by the vertical vectors.
7. It's also good to study the BARGMANN-SEGAL REPRESENTATION, which we get by taking X = C^n with its god-given symplectic structure (the imaginary part of the inner product) and using the god-given
KÄHLER POLARIZATION. When we do this, our quantum Hilbert space consists of analytic functions on C^n which are square-integrable with respect to a Gaussian measure centered at the origin.
8. The next step is to QUANTIZE CLASSICAL OBSERVABLES, turning them into linear operators on the quantum Hilbert space H. Unfortunately, we can't quantize all such observables while still sending
Poisson brackets to commutators, as we did at the prequantum level. So at this point things get trickier and my brief outline will stop. Ultimately, the reason for this problem is that
quantization is not a functor from the category of symplectic manifolds to the category of Hilbert spaces - but for that one needs to learn a bit about category theory.
Basic Jargon
Here are some definitions of important terms. Unfortunately they are defined using other terms that you might not understand. If you are really mystified, you need to read some books on differential
geometry and the math of classical mechanics before proceeding.
• complexification - We can tensor a real vector space with the complex numbers and get a complex vector space; this process is called complexification. For example, we can complexify the tangent
space at some point of a manifold, which amounts to forming the space of complex linear combinations of tangent vectors at that point.
• distribution - The word "distribution" means many different things in mathematics, but here's one: a "distribution" V on a manifold X is a choice of a subspace V[x] of each tangent space T[p](X),
where the choice depends smoothly on x.
• Hamiltonian vector field - Given a manifold X with a symplectic structure ω, any smooth function f: X -> R can be thought of as a "Hamiltonian", meaning physically that we think of it as the
energy function and let it give rise to a flow on X describing the time evolution of states. Mathematically speaking, this flow is generated by a vector field v(f) called the "Hamiltonian vector
field" associated to f. It is the unique vector field such that
ω(.,v(f)) = df
In other words, for any vector field u on X we have
ω(u,v(f)) = df(u) = u f
The vector field v(f) is guaranteed to exist by the fact that ω is nondegenerate.
• integrable distribution - A distribution on a manifold X is "integrable" if at least locally, there is a foliation of X by submanifolds such that V[x] is the tangent space of the submanifold
containing the point x.
• integral cohomology class - Any closed p-form on a manifold M defines an element of the pth deRham cohomology of M. This is a finite-dimensional vector space, and it contains a lattice called the
pth integral cohomology group of M. We say a cohomology class is integral if it lies in this lattice. Most notably, if you take any U(1) connection on any Hermitian line bundle over M, its
curvature 2-form will define an integral cohomology class once you divide it by 2 π i. This cohomology class is called the first Chern class, and it serves to determine the line bundle up to
• Poisson brackets - Given a symplectic structure on a manifold M and given two smooth functions on that manifold, say f and g, there's a trick for getting a new smooth function {f,g} on the
manifold, called the Poisson bracket of f and g.
This trick works as follows: given any smooth function f we can take its differential df, which is a 1-form. Then there is a unique vector field v(f), the Hamiltonian vector field associated to
f, such that
ω(-,v(f)) = df
Using this we define
{f,g} = ω(v(f),v(g))
It's easy to check that we also have {f,g} = dg(v(f)) = v(f) g
so {f,g} says how much g changes as we differentiate it in the direction of the Hamiltonian vector field generated by f.
In the familiar case where M is R^2n with momentum and position coordinates p[i], q[i], the Poisson brackets of f and g work out to be
{f,g} = sum[i] df/dp[i] dg/dq[i] - df/dq[i] dg/dp[i]
• square-integrable sections - We can define an inner product on the sections of a Hermitian line bundle over a manifold X with a symplectic structure. The symplectic structure defines a volume
form which lets us do the necessary integral. A section whose inner product with itself is finite is said to be square-integrable. Such sections form a Hilbert space H[0] called the "prequantum
Hilbert space". It is a kind of preliminary version of the Hilbert space we get when we quantize the classical system whose phase space is X.
• symplectic structure - A symplectic structure on a manifold M is a closed 2-form ω which is nondegenerate in the sense that for any nonzero tangent vector u at any point of M, there is a tangent
vector u at that point for which w(u,v) is nonzero.
• U(1) connection - The group U(1) is the group of unit complex numbers. Given a complex line bundle L with an inner product on each fiber L[x], a U(1) connection on L is a connection such that
parallel translation preserves the inner product.
• vertical vectors - Given a bundle E over a manifold M, we say a tangent vector to some point of E is vertical if it projects to zero down on M.
The only way to learn the rules of this Game of games is to take the usual prescribed course, which requires many years, and none of the initiates could ever possibly have any interest in making
these rules easier to learn. - Hermann Hesse, The Glass Bead Game
© 2000 John Baez
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A fifth of alcohol is equal to how many pints of alcohol?
United States customary units are a system of measurements commonly used in the United States. The U.S. customary system developed from English units which were in use in the British Empire
before American independence. Consequently most U.S. units are virtually identical to the British imperial units. However, the British system was overhauled in 1824, changing the definitions of
some units used there, so several differences exist between the two systems.
The majority of U.S. customary units were redefined in terms of the meter and the kilogram with the Mendenhall Order of 1893, and in practice, for many years before. These definitions were
refined by the international yard and pound agreement of 1959. The U.S. primarily uses customary units in its commercial activities, while science, medicine, government, and many sectors of
industry use metric units. The SI metric system, or International System of Units is preferred for many uses by NIST
An alcoholic beverage is a drink that contains ethanol. Alcoholic beverages are divided into three general classes for taxation and regulation of production: beers, wines, and spirits (distilled
beverages). They are legally consumed in most countries around the world. More than 100 countries have laws regulating their production, sale, and consumption. Beer is the third most popular
drink in the world, after water and tea.
Alcoholic beverages have been consumed by humans since the Neolithic era; the earliest evidence of alcohol was discovered in Jiahu, dating from 7000–6600 BC. The production and consumption of
alcohol occurs in most cultures of the world, from hunter-gatherer peoples to nation-states.
Two Pints of Lager and a Packet of Crisps is a British comedy television series sitcom that ran from 26 February 2001 to 24 May 2011 and starring Sheridan Smith, Will Mellor, Ralf Little, Natalie
Casey, Kathryn Drysdale and Luke Gell. Created and written by Susan Nickson, it is set in the town of Runcorn in Cheshire, England, and originally revolves around the lives of five twenty-somethings.
In September 2007 Little confirmed in an interview on This Morning that he had left the series. The seventh series aired following a two year hiatus in 2008 with Little's character being killed off
in the first episode. Smith and Drysdale did not return for the ninth and final series with their departures being mentioned briefly in the first episode, Freddie Hogan and Georgia Henshaw were
brought in as their replacements making Mellor and Casey the only original characters left in the series. Writer & Creator, Susan Nickson also left before the ninth series.
The core cast have been augmented by various recurring characters throughout the series, including Beverly Callard, Lee Oakes, Hayley Bishop, Alison Mac, Thomas Nelstrop and Jonathon Dutton. The show
was first broadcast in 2001 on BBC Two. The title was inspired by the 1980 hit single "Two Pints of Lager and a Packet of Crisps Please" by Splodgenessabounds.
In recipes, quantities of ingredients may be specified by mass (commonly called weight), by volume, or by count.
For most of history, most cookbooks did not specify quantities precisely, instead talking of "a nice leg of spring lamb", a "cupful" of lentils, a piece of butter "the size of a walnut", and
"sufficient" salt. Informal measurements such as a "pinch", a "drop", or a "hint" (soupçon) continue to be used from time to time. In the US, Fannie Farmer introduced the more exact specification of
quantities by volume in her 1896 Boston Cooking-School Cook Book.
The system of imperial units or the imperial system (also known as British Imperial) is the system of units first defined in the British Weights and Measures Act of 1824, which was later refined
and reduced. The system came into official use across the British Empire. By the late 20th century, most nations of the former empire had officially adopted the metric system as their main system
of measurement, but some Imperial units are still used in the United Kingdom and Canada.
Scotland had a distinct system of measures and weights until at least the late 18th century, based on the ell as a unit of length, the stone as a unit of mass and the boll and the firlot as units
of dry measure. This official system coexisted with local variants, especially for the measurement of land.
The system is said to have been introduced by David I of Scotland (1124–53), although there are no surviving records until the 15th century when the system was already in normal use. Standard
measures and weights were kept in each burgh, and these were periodically compared against one another at "assizes of measures", often during the early years of the reign of a new monarch.
Nevertheless, there was considerable local variation in many of the units, and the units of dry measure steadily increased in size from 1400 to 1700.
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A multibody factorization method for independently moving objects
Results 1 - 10 of 130
, 1999
"... Scene flow is the three-dimensional motion field of points in the world, just as optical flow is the twodimensional motion field of points in an image. Any optical flow is simply the projection
of the scene flow onto the image plane of a camera. In this paper, we present a framework for the computat ..."
Cited by 127 (9 self)
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Scene flow is the three-dimensional motion field of points in the world, just as optical flow is the twodimensional motion field of points in an image. Any optical flow is simply the projection of
the scene flow onto the image plane of a camera. In this paper, we present a framework for the computation of dense, non-rigid scene flow from optical flow. Our approach leads to straightforward
linear algorithms and a classification of the task into three major scenarios: (1) complete instantaneous knowledge of the scene structure, (2) knowledge only of correspondence information, and (3)
no knowledge of the scene structure. We also show that multiple estimates of the normal flow cannot be used to estimate dense scene flow directly without some form of smoothing or regularization. 1
- IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE , 2003
"... This paper presents an algebro-geometric solution to the problem of segmenting an unknown number of subspaces of unknown and varying dimensions from sample data points. We represent the
subspaces with a set of homogeneous polynomials whose degree is the number of subspaces and whose derivatives at a ..."
Cited by 117 (29 self)
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This paper presents an algebro-geometric solution to the problem of segmenting an unknown number of subspaces of unknown and varying dimensions from sample data points. We represent the subspaces
with a set of homogeneous polynomials whose degree is the number of subspaces and whose derivatives at a data point give normal vectors to the subspace passing through the point. When the number of
subspaces is known, we show that these polynomials can be estimated linearly from data; hence, subspace segmentation is reduced to classifying one point per subspace. We select these points optimally
from the data set by minimizing certain distance function, thus dealing automatically with moderate noise in the data. A basis for the complement of each subspace is then recovered by applying
standard PCA to the collection of derivatives (normal vectors). Extensions of GPCA that deal with data in a highdimensional space and with an unknown number of subspaces are also presented. Our
experiments on low-dimensional data show that GPCA outperforms existing algebraic algorithms based on polynomial factorization and provides a good initialization to iterative techniques such as
K-subspaces and Expectation Maximization. We also present applications of GPCA to computer vision problems such as face clustering, temporal video segmentation, and 3D motion segmentation from point
correspondences in multiple affine views.
- In European Conference on Computer Vision , 2004
"... Recovery of three diensWXzm (3D) sD) e and otion of non-sN;m[ s cenes fro a onocular videosdeomWW is i portant forapplications like robot navigation and hu an co puter interaction. If every
point in thes cene rando ly oves it is i - posW=J= to recover the non-rigids-r es In practice, any non-rigid o ..."
Cited by 72 (10 self)
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Recovery of three diensWXzm (3D) sD) e and otion of non-sN;m[ s cenes fro a onocular videosdeomWW is i portant forapplications like robot navigation and hu an co puter interaction. If every point in
thes cene rando ly oves it is i - posW=J= to recover the non-rigids-r es In practice, any non-rigid objects e.g. the hu an face under various expres[XFX] defor with certains tructures Theirs hapes
can be regarded as a weighted co bination of certains hapebasXJ Shape and otion recovery unders uchs ituations has attracted uch interesX Previous work onthis proble [6, 4, 13] utilized only orthonor
ality consWJNm ts on the ca era rotations (ro- tation constraints).This paper proves that usJ] only the rotation cons]N]m ts res]N] in a biguous and invalid smWWX];m[ The a biguity arisX fro the fact
that thesmX e bas+ are not unique becaus their linear transJW ation is a news et of eligiblebasib To eli inate the a biguity, we propos as et of novel consNXNm ts basis constraints, which uniquely
deter ine thesmW e bas;F We prove that, under the weak-p ers ective projection odel, enforcing both the bas= and the rotation consW+;m ts leads to a closNm[JF slosNm to the proble of non-rigids hape
and otion recovery. The accuracy and robus;Wm[ of ourclos=;m[J slos=; is evaluated quantitatively on sm thetic data and qualitatively on real videoseomWN;JN 1
- In CVPR , 2009
"... We propose a method based on sparse representation (SR) to cluster data drawn from multiple low-dimensional linear or affine subspaces embedded in a high-dimensional space. Our method is based
on the fact that each point in a union of subspaces has a SR with respect to a dictionary formed by all oth ..."
Cited by 71 (6 self)
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We propose a method based on sparse representation (SR) to cluster data drawn from multiple low-dimensional linear or affine subspaces embedded in a high-dimensional space. Our method is based on the
fact that each point in a union of subspaces has a SR with respect to a dictionary formed by all other data points. In general, finding such a SR is NP hard. Our key contribution is to show that,
under mild assumptions, the SR can be obtained ’exactly ’ by using ℓ1 optimization. The segmentation of the data is obtained by applying spectral clustering to a similarity matrix built from this SR.
Our method can handle noise, outliers as well as missing data. We apply our subspace clustering algorithm to the problem of segmenting multiple motions in video. Experiments on 167 video sequences
show that our approach significantly outperforms state-of-the-art methods. 1.
- In ECCV , 2006
"... Abstract. We cast the problem of motion segmentation of feature trajectories as linear manifold finding problems and propose a general framework for motion segmentation under affine projections
which utilizes two properties of trajectory data: geometric constraint and locality. The geometric constra ..."
Cited by 69 (0 self)
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Abstract. We cast the problem of motion segmentation of feature trajectories as linear manifold finding problems and propose a general framework for motion segmentation under affine projections which
utilizes two properties of trajectory data: geometric constraint and locality. The geometric constraint states that the trajectories of the same motion lie in a low dimensional linear manifold and
different motions result in different linear manifolds; locality, by which we mean in a transformed space a data and its neighbors tend to lie in the same linear manifold, provides a cue for
efficient estimation of these manifolds. Our algorithm estimates a number of linear manifolds, whose dimensions are unknown beforehand, and segment the trajectories accordingly. It first transforms
and normalizes the trajectories; secondly, for each trajectory it estimates a local linear manifold through local sampling; then it derives the affinity matrix based on principal subspace angles
between these estimated linear manifolds; at last, spectral clustering is applied to the matrix and gives the segmentation result. Our algorithm is general without restriction on the number of linear
manifolds and without prior knowledge of the dimensions of the linear manifolds. We demonstrate in our experiments that it can segment a wide range of motions including independent, articulated,
rigid, non-rigid, degenerate, non-degenerate or any combination of them. In some highly challenging cases where other state-of-the-art motion segmentation algorithms may fail, our algorithm gives
expected results. 2 1
- In CVPR , 2007
"... Over the past few years, several methods for segmenting a scene containing multiple rigidly moving objects have been proposed. However, most existing methods have been tested on a handful of
sequences only, and each method has been often tested on a different set of sequences. Therefore, the compari ..."
Cited by 64 (5 self)
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Over the past few years, several methods for segmenting a scene containing multiple rigidly moving objects have been proposed. However, most existing methods have been tested on a handful of
sequences only, and each method has been often tested on a different set of sequences. Therefore, the comparison of different methods has been fairly limited. In this paper, we compare four 3-D
motion segmentation algorithms for affine cameras on a benchmark of 155 motion sequences of checkerboard, traffic, and articulated scenes. 1.
, 2007
"... This paper describes methods for recovering time-varying shape and motion of non-rigid 3D objects from uncalibrated 2D point tracks. For example, given a video recording of a talking person, we
would like to estimate the 3D shape of the face at each instant, and learn a model of facial deformation. ..."
Cited by 50 (1 self)
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This paper describes methods for recovering time-varying shape and motion of non-rigid 3D objects from uncalibrated 2D point tracks. For example, given a video recording of a talking person, we would
like to estimate the 3D shape of the face at each instant, and learn a model of facial deformation. Time-varying shape is modeled as a rigid transformation combined with a non-rigid deformation.
Reconstruction is ill-posed if arbitrary deformations are allowed, and thus additional assumptions about deformations are required. We first suggest restricting shapes to lie within a lowdimensional
subspace, and describe estimation algorithms. However, this restriction alone is insufficient to constrain reconstruction. To address these problems, we propose a reconstruction method using a
Probabilistic Principal Components Analysis (PPCA) shape model, and an estimation algorithm that simultaneously estimates 3D shape and motion for each instant, learns the PPCA model parameters, and
robustly fills-in missing data points. We then extend the model to model temporal dynamics in object shape, allowing the algorithm to robustly handle severe cases of missing data.
- In European Conference on Computer Vision , 2000
"... . This paper extends the recovery of structure and motion to image sequences with several independently moving objects. The motion, structure, and camera calibration are all a-priori unknown.
The fundamental constraint that we introduce is that multiple motions must share the same camera paramete ..."
Cited by 47 (0 self)
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. This paper extends the recovery of structure and motion to image sequences with several independently moving objects. The motion, structure, and camera calibration are all a-priori unknown. The
fundamental constraint that we introduce is that multiple motions must share the same camera parameters. Existing work on independent motions has not employed this constraint, and therefore has not
gained over independent static-scene reconstructions. We show how this constraint leads to several new results in structure and motion recovery, where Euclidean reconstruction becomes possible in the
multibody case, when it was underconstrained for a static scene. We show how to combine motions of high-relief, low-relief and planar objects. Additionally we show that structure and motion can be
recovered from just 4 points in the uncalibrated, fixed camera, case. Experiments on real and synthetic imagery demonstrate the validity of the theory and the improvement in accuracy obtained usin...
, 2006
"... We present an algebraic geometric approach to 3-D motion estimation and segmentation of multiple rigid-body motions from noise-free point correspondences in two perspective views. Our approach
exploits the algebraic and geometric properties of the so-called multibody epipolar constraint and its asso ..."
Cited by 42 (15 self)
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We present an algebraic geometric approach to 3-D motion estimation and segmentation of multiple rigid-body motions from noise-free point correspondences in two perspective views. Our approach
exploits the algebraic and geometric properties of the so-called multibody epipolar constraint and its associated multibody fundamental matrix, which are natural generalizations of the epipolar
constraint and of the fundamental matrix to multiple motions. We derive a rank constraint on a polynomial embedding of the correspondences, from which one can estimate the number of independent
motions as well as linearly solve for the multibody fundamental matrix. We then show how to compute the epipolar lines from the first-order derivatives of the multibody epipolar constraint and the
epipoles by solving a plane clustering problem using Generalized PCA (GPCA). Given the epipoles and epipolar lines, the estimation of individual fundamental matrices becomes a linear problem. The
clustering of the feature points is then automatically obtained from either the epipoles and epipolar lines or from the individual fundamental matrices. Although our approach is mostly designed for
noise-free correspondences, we also test its performance on synthetic and real data with moderate levels of noise.
, 2006
"... In its full generality, motion analysis of crowded objects necessitates recognition and segmentation of each moving entity. The difficulty of these tasks increases considerably with occlusions
and therefore with crowding. When the objects are constrained to be of the same kind, however, partitioning ..."
Cited by 42 (1 self)
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In its full generality, motion analysis of crowded objects necessitates recognition and segmentation of each moving entity. The difficulty of these tasks increases considerably with occlusions and
therefore with crowding. When the objects are constrained to be of the same kind, however, partitioning of densely crowded semi-rigid objects can be accomplished by means of clustering tracked
feature points. We base our approach on a highly parallelized version of the KLT tracker in order to process the video into a set of feature trajectories. While such a set of trajectories provides a
substrate for motion analysis, their unequal lengths and fragmented nature present difficulties for subsequent processing. To address this, we propose a simple means of spatially and temporally
conditioning the trajectories. Given this representation, we integrate it with a learned object descriptor to achieve a segmentation of the constituent motions. We present experimental results for
the problem of estimating the number of moving objects in a dense crowd as a function of time. 1
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Implicit Differentiation
October 24th 2008, 11:29 AM #1
Oct 2008
Implicit Differentiation
When air expands adiabatically (without gaining or losing heat), its pressure P and volume V are related by the equation PV^1.4=C where C is a constant. Suppose that at a certain instant the
volume is 510 cubic centimeters and the pressure is 81kPa and is decreasing at a rate of 7 kPa/minute. At what rate in cubic centimeters per minute is the volume increasing at this instant?
how do approach this problem?
Differentiate your equation wrt time.
$P\cdot V^{1.4}=C$
$P\cdot 1.4V^{0.4}\cdot \frac{dV}{dt}+V^{1.4}\cdot \frac{dP}{dt}=0$
Enter in all your knowns and solve for dV/dt
October 24th 2008, 12:20 PM #2
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What is the smallest diameter ring a non-convex polyhedron can pass through in 3-space?
up vote 4 down vote favorite
The question is mostly in the title.
Imagine I have some non-convex polyhedron $P$, and I would like to find the smallest diameter ring that it can pass through in 3-space, undergoing any necessary rotations as it does so. Is there an
efficient way to calculate $D_{ring}$? Pressing my luck, can I find the set of rotations for $P$ as it passes through the ring?
geometry polyhedra computer-science
add comment
2 Answers
active oldest votes
This is the "piano movers problem", also known as the motion planning problem, which has an enormous literature. Check out http://en.wikipedia.org/wiki/Motion_planning
up vote 6 down vote accepted
Very cool, thanks! – UltraBlue06 Nov 30 '11 at 21:50
add comment
Just a side remark on convex polyhedra: Each of the regular polyhedra except the cube can pass through a circle of radius smaller than the smallest-radius cylinder circumscribing the
polyhedron. This is proved in Tudor Zamfirescu's delightful paper, "Convex polytopes passing through circles" (PDF link). There is quite a nice (non-algorithmic) literature on this
up vote 5 problem.
down vote
1 Thanks, I enjoyed reading the paper. =) – UltraBlue06 Dec 2 '11 at 19:19
Zamfirescu has a light touch despite deep thoughts. :-) – Joseph O'Rourke Dec 2 '11 at 20:33
add comment
Not the answer you're looking for? Browse other questions tagged geometry polyhedra computer-science or ask your own question.
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Crum Lynne Math Tutor
Find a Crum Lynne Math Tutor
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Conservation of mechanical energy
I have a problem with my train of thought.
I have narrowed it down to several premises, and I think that one of them must be wrong.
Premise 1
Δ total energy (E) = Δ kinetic energy (K) + Δ potential energy (U)
(assuming that there is no change in any of the other forms of energy in the system)
Premise 2
Δ E = q + w
Consider the system of a block.
The block is initially traveling upwards (+y direction), away from the earth's surface.
At t=0 it has a kinetic energy of 1/2mv^2 (v is non zero). It initially also has gravitational potential energy of 0.
The velocity eventually decreases to 0 (at t[1]). The whole time (from t=0 to t[1]), the earth's gravitational attraction is exerting a force of mg on the block.
No other force is acting on the block, since it is already in motion at t=0.
It appears that the work done by the gravitational force on the block is -mgh.
But, ΔE = 0, because the kinetic energy has simply been transformed into potential energy.
So, how can work be non zero (-mgh)?
It seems that (looking at the block as the system)
ΔE = 0 = W= -mgh
(no heat transfer, so q=0, no friction, no other energy transfer)
Since this is impossible, I know that I did something wrong.
Thanks for your help!
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MathGroup Archive: April 2010 [00565]
[Date Index] [Thread Index] [Author Index]
Re: problems with NMinimize
• To: mathgroup at smc.vnet.net
• Subject: [mg109363] Re: problems with NMinimize
• From: Bill Rowe <readnews at sbcglobal.net>
• Date: Fri, 23 Apr 2010 03:49:45 -0400 (EDT)
On 4/22/10 at 6:43 AM, ngb at ecs.soton.ac.uk (Neil Broderick) wrote:
>Hi, I am trying to use NMinimize to find the solutions to various
>numerical equations and I keep getting error messages concerning
>non-numerical values. For example consider the following:
>In[2]:= NMinimize[Abs[NIntegrate[Sin[x], {x, -a, b}]]^2, {a, b}]
<error messages snipped>
Even if this did work as you expected with no error messages,
this would be a very inefficient way to solve the problem.
NMinimize works by repeatedly evaluating the expression to be
minimized. That means the numerical integration problem is
repeatedly solved. In this case, there is a symbolic result for
the integral. So doing:
In[4]:= int = Integrate[Sin[x], {x, -a, b}]
Out[4]= cos(a)-cos(b)
And this clearly is zero for Abs[a]==Abs[b]. So, any pair of
numbers satisfying this last will be a minimum for the square of
the absolute value for the integral.
But if you didn't want to find the minimum by inspection, then doing
In[5]:= NMinimize[Abs[int], {a, b}]
Out[5]= {4.80727*10^-14,{a->0.0961182,b->0.0961182}}
returns one of many possible solutions. Note, I didn't bother
with squaring the absolute value since minimizing the absolute
value is the same as minimizing the square of the absolute value.
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Translating Words into Math
Date: 8 Mar 1995 22:51:36 -0500
From: Anonymous
Subject: Algebra.
Dear Dr. Math,
I need help with my Algebra. We are doing story problems.
I DON'T want you to do them for me. I just need help on
understanding them so that I can solve them by myself. I am
stuck on one particular problem.
The number is decreased by 1/3 of the number. The result is 62.
Find the number. Please help me.
Brittany I
Date: Thu, 09 Mar 1995 12:57:09 +0000
From: Dr. Math
Subject: Re: Algebra.
Hello there!
I think the hardest thing about doing these problems is translating
sentences into mathematical equations. Basically, you have to pick
them apart, piece by piece, translating one part at a time until what
you have left is an equation that you can solve using Algebra. So
let's dig in.
Actually, I'll make up a new problem that's similar to yours, and I'll
lead you through the translation process on that one. Here it is:
A number is increased by 3/4 of the number, and the result is 28.
What is the number?
One of the first things we want to do is to replace "the number" by
a variable. You can call it anything you want, and in this case I'll
call it x. So we go through the sentence and wherever we see the
phrase "the number" or "itself" or something that obviously refers
to the number we're talking about, we replace that by an x. So we get:
x is increased by 3/4 of x, and the result is 28. What is x?
Also, one of the first things you want to do in these sentences is
find out where the equal sign goes. Typically, you'll look for the
verb "is," or "the result is," or if they're really throwing it out at
your face, "is equal to." In our problem, we find our equal sign in
"and the result is":
x increased by 3/4 of x = 28.
(I changed the grammar just a teency bit)
Now what does 3/4 of x mean? It means we multiply 3/4 times x.
So now we have
x increased by 3/4 * x = 28.
Now there's only one thing left. We need to figure out what
"increased by" means. What would I mean if we were baking
cookies, and I said to you "increase the amount of chocolate
chips by two cups"? I'd mean that you would add two extra
cups of chocolate chips to the batter, i.e. ADD two cups, and
we'd both have a stomach ache. So the phrase "increased by" is
probably going to mean "plus."
x + 3/4 * x = 28.
And now we have a bona fide Algebraic sentence. Notice the
period at the end. Usually people don't write those, but I
wanted to show you that even when we write down something
as weird as "x + 3/4 x = 28", it can always be read as a real
English sentence, with punctuation at the end and everything.
So see if you can apply the same techniques to solve your
problem. If you still have trouble, write back!
-Ken "Dr." Math
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OpenMx - Advanced Structural Equation Modeling
Tue, 03/16/2010 - 15:28
Your objective function needs
Your objective function needs to be changed.
The code you have:
mxAlgebra(-2*sum(log(pclass1%x%Gompertz1.objective + pclass2%x%Gompertz1.objective)), name="mixtureObj")
adds the Gompertz1 objective to itself. Change the second one to Gompertz2, and it should work.
Tue, 03/16/2010 - 15:44
PEBKAC error. Everything
PEBKAC error. Everything comes up roses when I try to mix two different classes instead of two of the same class. Pending my permission to post the data, I'll wrap this up into something for the
models/passing directory and the documentation.
Mon, 07/26/2010 - 13:10
Hi all, I am following Ryne's
Hi all,
I am following Ryne's code on fitting GMM, but ran into a few error messages. My model is simpler: I fit a Latent Basis 2-class GMM with 9 repeated assessments, constraining time loadings at times 1
& 4. I am starting with a model that has no within-class intercept & slope variances (so, my 'phi' matrix is constrained to 0).
I get several error messages.
First, when I am creating Model Expectations for Mixture (Parent) Model by specifying the sum of class proportions equal to 1, I get the following error:
> classP <- mxMatrix ("Full", 2, 1, free = TRUE, lbound = 0,
+ labels = c ("pclass1", "pclass2"), name = "classProbs")
> classS <- mxAlgebra (sum(classProbs), name = "classSum")
> constant <- mxMatrix ("Iden", 1, name = "con")
> classC <- mxConstraint ("classSum", "=", "con", name = "classCon")
Error in mxConstraint("classSum", "=", "con", name = "classCon") :
unused argument(s) ("=", "con")
Of course, subsequently, I can't specify the MxModel, since I don't have the classC object, and I get an error when running the full mixture model.
I am attaching the complete R syntax file and would greatly appreciate any leads on how to modify it.
Attachment Size
Grouped-based GMM.R 4.21 KB
Mon, 07/26/2010 - 13:21
The mxConstraint() syntax has
The mxConstraint() syntax has changed. Try using:
mxConstraint(classSum == con, name = "classCon")
Note the absence of quotes in the first argument. MxConstraints and now specified much like MxAlgebras are specified.
Mon, 07/26/2010 - 15:06
Thank you! That solved the
Thank you! That solved the issure of constrains.
However, when I run the final model with
> mixedResults <- mxRun (mixedModel)
I am getting the following error message:
Error: The job for model 'Group Based GMM 2cl' exited abnormally with the error message: Expected covariance matrix is not positive-definite in data row 30 at iteration 266.
In addition: Warning message:
In model 'Group Based GMM 2cl' NPSOL returned a non-zero status code 6. The model does not satisfy the first-order optimality conditions to the required accuracy, and no improved point for the merit
function could be found during the final linesearch (Mx status RED)
Mon, 07/26/2010 - 15:39
I would be tempted to keep
I would be tempted to keep theta elements positive, thusly:
manCov <- mxMatrix ("Diag", 9, 9, free = TRUE,
values = rep (25, times = 9),
labels = 'residual',
name = 'theta')
manCov <- mxMatrix ("Diag", 9, 9, free = TRUE,
values = rep (25, times = 9),
labels = 'residual',
lbound = 0,
name = 'theta')
And I wonder if this has anything to do with a "mancave" :)
Tue, 07/27/2010 - 13:43
Thank you,
Thank you, Michael.
specifying lower bounds on the theta matrix did help the estimation process. What I realized, however, is that factor loadings for the slope parameter were equal across latent classes, thus,
representation for one of classes was 0% (makes sense, since trajectories were specified as being the same).
When specifying class-specific loadings for the slope (so that trajectories would differ across classes), I run into a new problem of R complaining:
Error: Unknown reference 'lambdaS1' detected in the entity 'lambda1' in model 'groupGMM1'
Here is the modified syntax I'm using to which, I think, the error message is referring:
## factor loadings for slope cl1
time1 <- mxMatrix ("Full", 9, 1, free = c(FALSE, TRUE, TRUE, FALSE, TRUE, TRUE, TRUE, TRUE, TRUE),
values = c(0, .5, .8, 1, 1, 1, 1, 1, 1),
label = paste ('t', 1:9, 'cl1', sep = ""),
name = 'lambdaS1')
## factor loadings for slope cl2
time2 <- mxMatrix ("Full", 9, 1, free = c(FALSE, TRUE, TRUE, FALSE, TRUE, TRUE, TRUE, TRUE, TRUE),
values = c(0, .5, .8, 1, .8, .7, .6, .5, .4),
label = paste ('t', 1:9, 'cl2', sep = ""),
name = 'lambdaS2')
## factor loading matrix lambda
loadings1 <- mxAlgebra (cbind (lambdaInt, lambdaS1), name = 'lambda1')
loadings2 <- mxAlgebra (cbind (lambdaInt, lambdaS2), name = 'lambda2')
## model expectations:
meanAlg1 <- mxAlgebra (alpha1 %*% t(lambda1), dimnames = list(NULL, names),
name = 'mean1')
covAlg1 <- mxAlgebra (lambda1 %*% phi %*% t(lambda1) + theta,
dimnames = list(names, names),
name = 'cov1')
meanAlg2 <- mxAlgebra (alpha2 %*% t(lambda2), dimnames = list(NULL, names),
name = 'mean2')
covAlg2 <- mxAlgebra (lambda2 %*% phi %*% t(lambda2) + theta,
dimnames = list(names, names),
name = 'cov2')
model1 <- mxModel ("groupGMM1",
factorMeans1, factorCov, manCov,
unit, time, loadings1,
meanAlg1, covAlg1,
mxFIMLObjective ('cov1', 'mean1'))
Tue, 08/03/2010 - 17:10
time1 is not included in your
time1 is not included in your model. That should do the trick.
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The Secret History of Mathematicians
The Secret History of Mathematicians
Arthur Cayley: Mathematician Laureate of the Victorian Age. Tony Crilly. xxiv + 609 pp. Johns Hopkins University Press, 2006. $69.96.
James Joseph Sylvester: Jewish Mathematician in a Victorian World. xvi + 461 pp. Johns Hopkins University Press, 2006. $69.96.
Historian George Sarton often said that science advances in darkness, invisible to the majority of people, who are more interested in battles and other noisier activities. In his 1957 book The Study
of the History of Mathematics, Sarton went on to say that if the history of science is secret, then the history of mathematics is doubly so, "for the growth of mathematics is unknown not only to the
general public, but even to scientific workers."
Sarton's words help us understand why few have ever heard of Arthur Cayley (1821-95) or James Joseph Sylvester (1814-97), two of the most profound and prolific mathematicians of the Victorian era.
Cayley's seminal investigations of matrix algebra, which constituted only a tiny portion of his 967 papers, were crucial for the development of linear algebra. The terms matrix, determinant and
Jacobian, familiar to most science students, were invented by Sylvester, an enthusiastic poet who called himself the "mathematical Adam."
It is not clear when Cayley and Sylvester first met, but by 1847 they were corresponding to share thoughts about mathematics. They rallied and inspired each other for nearly four decades, drawn
together by their common circumstances. Each had triumphed on the University of Cambridge's fearsome Tripos examinations: Sylvester placed second in 1837 and Cayley first in 1842. These
accomplishments qualified them for mathematics professorships, but such positions were few and openings rare. Cayley was a Trinity College fellow at Cambridge for a few years and could have remained
so, teaching routine courses and living a communal life. But had he married, he would have been compelled to resign his fellowship and then perhaps scratch out a clergyman's living at one of the
parishes belonging to Trinity, far from the academic community. Sylvester's options were fewer. As a Jew, he could not even collect his Cambridge degree, since in order to do so he would have had to
subscribe to the Thirty-Nine Articles of the Church of England.
Cayley and Sylvester would travel other avenues. By 1846 both were studying law at London's Inns of Court, and each stayed on there for several years, Cayley drafting legal documents as a
barrister-at-law and Sylvester working as an actuary and barrister. The first had escaped the stone, parental grip of the Church of England. The second was its unloved bastard.
Turning away legal and actuarial work, hoarding every spare minute for mathematics, Cayley and Sylvester communicated with each other about their research almost daily, building a new theory of
algebraic invariants. So intertwined were their lives as mathematicians, both during and after their careers at the Inns of Court, that when E. T. Bell wrote his famous (and flawed) history, Men of
Mathematics, he christened them the "Invariant Twins." It is appropriate that Johns Hopkins University Press should publish their biographies simultaneously.
Arthur Cayley: Mathematician Laureate of the Victorian Age, by well-known historian of mathematics Tony Crilly, is the first full-length account of Cayley's life. The book draws on many primary
sources. Crilly begins every chapter with a brief abstract and divides the material into short sections, with subheads that help keep the reader oriented. He includes surprisingly few mathematical
formulas and integrates his explanations of these smoothly with the biographical material.
Cayley was born into a comfortable family. His father, Henry, was a "Russia merchant," and Arthur spent the first seven years of his life in St. Petersburg, where his family enjoyed gracious living
and high culture among British expatriates, at the price of harsh winters and isolation. The family returned to England in 1828.
Young Cayley attended King's College London, where his abilities were quickly recognized, and then entered Trinity College, Cambridge, in 1838—the year of Queen Victoria's coronation.
When a biographical subject has led a calm, pleasant life, it is the biographer who must suffer, searching for engaging matter. Fortunately for Crilly, the Victorian society in which Cayley moved was
close-knit and lively. George Boole, Charles Dickens, Francis Galton, James Clerk Maxwell and William Thomson (later Lord Kelvin) are but a few Cayley intimates found in these pages. (Crilly includes
such details as that Cayley disliked Dickens's novels, preferring instead those of Jane Austen.)
Anyone interested in the emerging role of the research mathematician in England will find Crilly's book particularly rewarding. Cayley finally attained a professorship at the age of 41, when he
became the first person elected to the Sadlerian chair of mathematics at Cambridge. In accepting the position, he happily traded income for leisure.
Scientific research had by that time gained acceptance at Cambridge. However, pure mathematics was regarded as mere mental gymnastics, or at best a scientific tool; "mixed mathematics" (what today we
call applied mathematics) was valued more. Shortly after Cayley was appointed, William Thomson expressed that popular bias to Hermann von Helmholtz: "Oh! that the CAYLEYS would devote that skill that
they have to such things [Kirchoff's work on electrical conducting plates] instead of to pieces of algebra which possibly interest four people in the world, certainly not more, and possibly also only
the one person who works."
Cayley was elected president of the British Association for the Advancement of Science in 1883, an honor that no pure mathematician had enjoyed in nearly 40 years. The association's annual meeting in
Southport attracted scientists, politicians, preachers and the working class. Fear of mathematics being nothing new, an anxious rumor spread there that the new president would require a blackboard
and a supply of chalk! Cayley's portrait was painted, and his address was reported in the Times.
A look back at the meeting's proceedings confirms that there were very few presentations in pure mathematics. The reader is left with the sad impression that despite England's pride in its
"mathematician laureate," Cayley might have been better appreciated in France or Germany.
Cayley's lifelong friend Sylvester was loquacious and quick-tempered, Cayley's emotional antipode. Sylvester has been a focus of research for historian and mathematician Karen Parshall for more than
two decades. In her 1998 book James Joseph Sylvester: Life and Work in Letters, she examined his mathematics with skill and insight. Sylvester sprinkled all of his writing with witty reflections,
philosophical digressions and even lines of verse. Readers of his letters have developed a thirst to know more about him, and with the arrival of Parshall's masterful biography, James Joseph
Sylvester: Jewish Mathematician in a Victorian World, they can imbibe at last.
Parshall turns over the political and religious soils that alternately starved and nourished Sylvester's work. Mathematics fertilizes her account, but she distributes it sparingly. What grows is a
rich story of a stubborn genius willing to fight the world so that he might bestow his gifts.
We are introduced to Sylvester as a troubled child who is becoming increasingly aware of the anti-Semitism that will impede him. We later watch his hopes soar and then plummet as he struggled with
the reality that in the 19th century few academic positions, particularly those at Oxford and Cambridge, could be achieved by an English Jew. After college, he had an unhappy stint for a few years as
a professor of natural philosophy at secular University College London. In his mid-20s, he resigned the position to take another, across the Atlantic at the University of Virginia.
By offering an invitation to one of England's rising scholars, the University of Virginia hoped to elevate its own prestige. By accepting, Sylvester hoped to fashion there the sort of research
professorship that he had longed to find at home. The two daydreams combined into a nightmare. When he arrived in 1841, the students, who were sons of the Southern aristocracy, cheered him. They
welcomed the great man with an "illumination" in his honor, burning candles between stolen barrels of tar and, according to a contemporary account, "raising an unholy racket." When the students'
fantasy of an Oxbridge don was spoiled by "a little, bluff, beef-fed English cockney, perfectly insignificant in his appearance, and raw and awkward in his manners," their admiration soon gave way to
insubordination and threats of violence. That Sylvester acquired a "sword cane" for protection, and that he was later attacked by a pair of brothers, are very likely, Parshall concludes. That
Sylvester drew it and struck a nonfatal blow, as a student reported, remains an unconfirmed legend.
Back in England, Sylvester anticipated a bleak future. He had no job and had accomplished little mathematically in the preceding two years. In December 1844 he accepted an offer of a position as an
actuary at the newly created Equity and Law Life Assurance Society at Lincoln's Inn Fields in London. To do so must have been painful for someone who wanted so much to do research as a mathematician.
A couple of years later he began studying for the bar.
Sylvester's business efforts and his intense mathematical collaboration with Cayley during this phase of his life were successful but exhausting. When a professorship at the Royal Military Academy in
Woolrich became available in 1855, one that did not require taking an oath "on the true faith of a Christian," he applied, and with the help of an influential friend, Lord Brougham, he got the
position. The Academy was more concerned with training future officers than with producing pure mathematics. Nonetheless, for 15 years Sylvester persisted in struggling to mold his position to suit
his aspirations, until in 1869 he was forced into mandatory retirement.
Seeking solace, he turned to poetry, publishing Laws of Verse (1870), an unreadable study of versification that prompted one reviewer to advise the author to concentrate instead on mathematics. Over
the next six years, Sylvester kept busy in London with a variety of activities that included music and public speaking. It seemed impossible for him to attain another academic position.
In 1876, salvation came from America in the form of an offer from the new Johns Hopkins University to lead a graduate program in mathematics and launch the American Journal of Mathematics, a research
journal that continues today. For the next seven years, Sylvester would be surrounded by enthusiastic students and appreciative colleagues. While he was there, the political and social scenery back
in England was being transformed as new rights were won by working men, the emerging middle class and Jews.
At age 70, Sylvester was elected to the Savilian professorship of geometry at Oxford. He wiped away tears as he left behind his grateful students and colleagues in Baltimore. It is difficult for us
to appreciate fully the emotions that he felt as he was finally embraced by Oxford. "She is a good dear mother our University here," he wrote to Cayley, "and stretches out her arms with impartial
fondness to take all her children to her bosom even those she has not reared at her breast."
Cayley and Sylvester differed wildly in personality and background. But mathematically they were often of a single mind. Together they were pioneers, breaking free of the powerful attraction exerted
by Newton, who was at the center of the solar system that was English mathematics. The abstract algebra that Cayley and Sylvester developed, so useless in the opinion of Thomson, would prove
essential to Werner Heisenberg, Albert Einstein and other future scientists. Both of these biographies will be appreciated by anyone who is amused by this cosmic irony.
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Does probability come from quantum physics?
(Phys.org)—Ever since Austrian scientist Erwin Schrodinger put his unfortunate cat in a box, his fellow physicists have been using something called quantum theory to explain and understand the nature
of waves and particles.
But a new paper by physics professor Andreas Albrecht and graduate student Dan Phillips at the University of California, Davis, makes the case that these quantum fluctuations actually are responsible
for the probability of all actions, with far-reaching implications for theories of the universe.
Quantum theory is a branch of theoretical physics that strives to understand and predict the properties and behavior of atoms and particles. Without it, we would not be able to build transistors and
computers, for example. One aspect of the theory is that the precise properties of a particle are not determined until you observe them and "collapse the wave function" in physics parlance.
Schrodinger's famous thought experiment extends this idea to our scale. A cat is trapped in a box with a vial of poison that is released when a radioactive atom randomly decays. You cannot tell if
the cat is alive or dead without opening the box. Schrodinger argued that until you open the box and look inside, the cat is neither alive nor dead but in an indeterminate state.
For many people, that is a tough concept to accept. But Albrecht says that, as a theoretical physicist, he concluded some years ago that this is how probability works at all scales, although until
recently, he did not see it as something with a crucial impact on research. That changed with a 2009 paper by Don Page at the University of Alberta, Canada.
"I realized that how we think about quantum fluctuations and probability affects how we think about our theories of the universe," said Albrecht, a theoretical cosmologist.
One of the consequences of quantum fluctuations is that every collapsing wave function spits out different realities: one where the cat lives and one where it dies, for example. Reality as we
experience it picks its way through this near-infinity of possible alternatives. Multiple universes could be embedded in a vast "multiverse" like so many pockets on a pool table.
There are basically two ways theorists have tried to approach the problem of adapting quantum physics to the "real world," Albrecht said: You can accept it and the reality of many worlds or multiple
universes, or you can assume that there is something wrong or missing from the theory.
Albrecht falls firmly in the first camp.
"Our theories of cosmology say that quantum physics works across the universe," he said. For example, quantum fluctuations in the early universe explain why galaxies form as they did—a prediction
that can be confirmed with direct observations.
The problem with multiple universes, Albrecht said, is that it if there are a huge number of different pocket universes, it becomes very hard to get simple answers to questions from quantum physics,
such as the mass of a neutrino, an electrically neutral subatomic particle.
"Don Page showed that the quantum rules of probability simply cannot answer key questions in a large multiverse where we are not sure in which pocket universe we actually reside," Albrecht said.
One answer to this problem has been to add a new ingredient to the theory: a set of numbers that tells us the probability that we are in each pocket universe. This information can be combined with
the quantum theory, and you can get your math (and your calculation of the mass of a neutrino) back on track.
Not so fast, say Albrecht and Phillips. While the probabilities assigned to each pocket universe may seem like just more of the usual thing, they are in fact a radical departure from everyday uses of
probabilities because, unlike any other application of probability, these have already been shown to have no basis in the quantum theory.
"If all probability is really quantum theory, then it can't be done," Albrecht said. "Pocket universes are much, much more of a departure from current theory than people had assumed."
The paper is currently posted on the ArXiv.org preprint server and submitted for publication and has already stimulated considerable discussion, Albrecht said.
"It forces us to think about the different kinds of probability, which often get confused, and perhaps can help draw a line between them," he said.
More information: albrecht.ucdavis.edu/
1.9 / 5 (18) Feb 05, 2013
The whole point of the schroedinger cat experiment is the absurdity of extrapolating from quantum to macro or even universe scales. The cat knows if its alive, even if we only know a probability
figure for its demise. An autopsy would clearly show when the cat died, not when it entered some superimposed state.
3.4 / 5 (5) Feb 05, 2013
The paper is pretty neat. An example of what is being talked about here is coin flipping. How can the probability of the outcome of a coin flip be traced back to quantum mechanical uncertainty?
As it turns out, it all starts with water and polypeptides in your nervous system. Quantum uncertainty at that level introduces enough probability in molecular transport to evoke less-than-perfect
physical prowess, which ultimately leads imprecise control of the energy required to repeatedly produce a perfect flip and snatch.
1.2 / 5 (18) Feb 05, 2013
Quantum computing is neither faster, neither more secure than the classical computing (but it's still good grant and salary generator for physicists, who manage to pretend the opposite): Why quantum
computing is hard - and quantum cryptography is not provably secure, No quantum trick – no matter how complex or exotic – can improve the capacity of quantum optical communication
1.7 / 5 (12) Feb 05, 2013
BTW The article title "does probability come from quantum physics" is logically orthogonal to the preprint abstract "..we claim there is no physically verified fully classical theory of
4.6 / 5 (9) Feb 05, 2013
Just another non-falsifiable speculation.
Whydening Gyre
1.4 / 5 (11) Feb 05, 2013
"God does not play with Dice." - Albert Einstein.
He has, however, found a coin or two to flip...
2 / 5 (8) Feb 05, 2013
@EyeNStein The cat
There are at least two cats. Your feeble attempt to square QM with classical physics employing a hidden English predication failed
Just another non-falsifiable speculation.
You're wrong. Perhaps SOME experiments are beyond present abilities, but others are doable by brighter minds than you
1.4 / 5 (9) Feb 05, 2013
two cats.
The invention of a second non-verifiable cat doesn't add to our understanding of the universe it only adds confusion. If you had mentioned that micro scale vibrators can be simultaneously
vibrating and not: I would have agreed. But concious cats and the cyanide release sensors constitute a measurement-improbability collapses-clasical reality appears. No second cat is required.
4.5 / 5 (15) Feb 05, 2013
You can accept it and the reality of many worlds or multiple universes, or you can assume that there is something wrong or missing from the theory.
WOW! What a ridiculous statement! Many, many physicists accept the veracity of quantum theory but reject the idea of multiple universes. Multiple universe are NOT required by the theory, and are
completely speculative! Whether or not other universes exist is a matter of philosophical OPINION. Moreover, the issue is unscientific in nature, since there's not even any way to test it!
You'd expect to hear this sort of thing on TV and among the general public, but from a physicist! Unbelievable!
1.6 / 5 (14) Feb 05, 2013
Whether or not other universes exist is a matter of philosophical OPINION
Well, not quite. For me it's mostly a question of physically and logically rigorous definition of multiverse.
1.9 / 5 (9) Feb 05, 2013
Whether or not other universes exist is a matter of philosophical OPINION.
So is the question of existence itself. Existence is completely dependent upon what David Bohm described as "implicit order," which is only peripherally understood
1.3 / 5 (16) Feb 05, 2013
One of the consequences of quantum fluctuations is that every collapsing wave function…..
It is interesting to note that up to now we still could not understand 'wave function of what, or what is wavy? Conventionally, it is just a mathematical abstraction; this is one real problem in
quantum mechanics. Understanding the mechanism of quantum mechanics as below, could answer the problem.
3.9 / 5 (11) Feb 05, 2013
Quantum computing is neither faster, neither more secure than the classical computing . . .
Unfortunately, this article is about probability and its origins, not quantum computing.
1 / 5 (2) Feb 05, 2013
Studies reported on this site have shown that "collapse" of a wave function is an "overshadowing" by another wave function and that the overshadowed function(s)still exist. In addition, the
"observer" is not the research staff but a hidden, ordering "field."
1.6 / 5 (13) Feb 05, 2013
The whole point of the schroedinger cat experiment is the absurdity of extrapolating from quantum to macro or even universe scales. The cat knows if its alive, even if we only know a probability
figure for its demise. An autopsy would clearly show when the cat died, not when it entered some superimposed state.
Cat knowing, forensics, the ASPCA, and T gondii are some of the many parameters not included in the thought experiment.
3.4 / 5 (14) Feb 05, 2013
The human sensors are a magical waveform collapsing mechanism... um... no.
The observer is as part of the system as anything else... there IS NO OBSERVER.
Yes, think about it... the human eye looking at something to observe is no different than photons interacting with a rock.
Interacting with matter... be it a rock, a magnetic field, a human eye... they are all the same, they all change the properties of the thing they are interacting with.
Why the continual, somewhat arrogant, fixation that the human body is a magical device that collapses waveforms into a reality? Is this because you think there is a 'soul'... some part of the human
body that is outside the bounds of physics?
2.3 / 5 (9) Feb 05, 2013
The cat knows if its alive, even if we only know a probability figure for its demise. An autopsy would clearly show when the cat died, not when it entered some superimposed state.
Collapsing the wavefunction also decides when the cat died.
Consider that at the moment of observation the cat exists in all possible states it can take while it is in the box, which includes all the possible histories, and all possible times of death.
The laws of quantum mechanism apply equally to what and where and when. The isotope that relases the poison by decaying is not in superposition only for decaying or not decaying, but when it decays
if it does.
The cat knowing that it is alive is irrelevant, because nobody outside the box can read its mind telepathically.
1 / 5 (8) Feb 05, 2013
One crucial factor missed out is the human mind, not measurable and yet it could be the main factor responsible for materializing which probability or pocket of universes it desires. At one time, the
earth was flat, but now the mind now "knows" that the earth is actually a ball in space.
1 / 5 (5) Feb 05, 2013
One other explanation for probability is that the lightcone of any event is not complete prior to the event, so there is no way to verify total input prior to an event. The laws may be deterministic,
but input is not.
As for multiworlds, it is a question of whether we view time as a vector from a determined past into a probabilistic future, or the changing configuration of what is, that turns future probability
into actuality. Consider this image; http://en.wikiped...film.svg
Do we view it as the present moving from left to right, or the frames moving right to left?
Before a race, there are many possible winners, but after it, only one.
2.9 / 5 (7) Feb 05, 2013
Hello ValeriaT & Whydening Gyre it's that crazy 'language' guy again.
I am more curious about what make US think in terms of probability? Probability is OUR problem, is it not, as we are unable to probe the quantum region with our own senses. We can't determine many
things for certain so we have to use a 'best (mathematical) guess'. I don't see anything wrong in that, as long as we remember it's WE who are 'guessing' and NOT the system under examination.
Take a macro situation for example. Police may use Rossmo's formula, 'Dragnet' computer program or other stats to predict where some criminal will strike again. They know the past events criminal so
look for a pattern. Sometimes it works sometimes it doesn't. But the reality is that only the criminal knows (doesn't know) where to strike next.
I've always had a 'realistic' problem with the poor old cat; the cat will try to escape before dying and evidence of that will be on the walls of the box. The cat certainly knows what's coming!
1.5 / 5 (8) Feb 06, 2013
seems it at least partially depends on if you consider that the cat is in BOTH states or in NEITHER state, if its in NEITHER then there is no requirement for multiverse obviously.
But this whole 'collapsing the waveform' or 'quantum uncertainty' seems to be mistaking ignorance for epiphany to me.
Ugh, if a phenomena requires an observer to be in a collapsed state, how did dinosaurs exist, or perhaps they didnt? Copenhagen taken to its extreme form seems to be sort of creationism to me, with
the implicit assumption that before humans there was some sort of 'universal observer'.
3.2 / 5 (5) Feb 06, 2013
The state of the cat is entirely irrelevant to the issue of probability.
The probability that the cat is dead or alive is identical - by the design of the experiment - as the probability that the radioactive source decays.
The state of the cat is only a macroscopic indicator of that decay.
The problem with the question of course, is not having a precise definition of probability that is devoid from connection with the real world.
The words "choice" or "selection" are part of the definition of probability and therefore make the definition dependent upon physical existence.
There can be no choice or selection without physical existence.
On the other hand defining probability without any physical connection seems to leave the definition simply a matter of creating a weighted set, and comparing the weight of one element with the
weight of the set.
This certainly has no connection to quantum mechanics.
1 / 5 (5) Feb 06, 2013
The only hope to reach a convincing interpretations of quantum mechanics is by modelling the dynamics of measurements. This has been done in a rich enough model, which clarifies the cat issue: the
cat dies because of its interaction with the apparatus. This approach leads to a minimal interpretation: the statistical interpretation, about which Einstein wrote even in his last year. Trying to
solve our problems with QM in other universes, as the authors do, is a non-minimal approach; it can and should be neglected.
2.6 / 5 (5) Feb 06, 2013
StarGazer2011, yes I can your point but then you are making a distinction about who is valid observer? While the dinosaurs were around so were other creatures that had sensory organs so are you
saying that they are not valid observers in QM context? The same goes for the sound of a falling tree. On a planet where life is prolific we can hardly claim the absence of an observer so it is only
'experiments' like the cat about which we can argue. Put it another way, has anyone actually seen a subatomic particle? No, of couse not, we don't have the physical apparatus with which to do so yet
here we all are (me included)discussing 'prob. in QM'. Yet again there are those that have a 'biocentric' view.
It seems that I can no longer claim Cogito ergo sum and if that assertion is wrong, I haven't written anything and you aren't reading it.
One thing for sure; If nobody opens the box the cat will die anyway but then maybe hpothetical cats live forever!
2.3 / 5 (12) Feb 06, 2013
Copenhagen taken to its extreme form seems to be sort of creationism to me, with the implicit assumption that before humans there was some sort of 'universal observer'.
The problem many people seem to be having with QM is that they interpret the word "observer" to mean a literal concious and cognitive observer.
What the word in QM means is simply the interchange of information. The superposition can only be in all states if no other thing depends on it being in a certain state. The classical world arises
from these dependencies where A depends on B depends on C depends on... where each limits the possible states of the other according to the rules of physics until you're left with no wiggle room for
the wavefunctions.
For that purpose, a rock can be the observer, and that is also the reason why Shrödingers cat has to be in a box so nothing else can interact with it until the box is opened. Quantum systems only
behave in "weird" ways when you isolate them.
1.6 / 5 (7) Feb 06, 2013
Okay natello so you're an AWT and battling against hard to get you're point across...man, you're in for a rough ride. But I've been in (and still am) that situation so hang in there.
Y'know what is really confusing is this:[actual converstation between me and an astrophysicist] LD;'...Black hole is what?...' AsPh; '...in laymans terms a superdense object causing distortion.' LD;
'Er, I'm sorry, since the Aether theory has been rejected... a distortion in what?' AsPh; '...in the fabric of space-time'. LD; 'Fabric...what is that then?'
Instances like this tell me that terms like 'a sea of virtual particles slipping in and out of existence...' is no more valid than an Aether.
But then I have a theory of my own that is different to most of the others which is partly expressed in the the public domain. The only trouble is that I don't fully understand it myself, but I am
working on it. When I'm ready it will be in print.
As I said, hang in there.
vlaaing peerd
2 / 5 (1) Feb 06, 2013
I don't think flipping a coin on macro scale has anything to do with probability or in the least so little it has no influence on the outcome.
When you are able to count in all influencing factors one should be able to exactly predict the outcome of the flipping coin. Probably (no pun intended) the bigger the scale the less likely quantum
scale probability would have influence on it.
1 / 5 (7) Feb 06, 2013
Had the professor and his student done their homework, they would have found it to be the other way around. Quantum theory began with Max Planck who first proposed the existence of fixed energy
states. Planck's famous radiation equation was based on it and the laws of probability that were first applied to thermodynamics. Planck noteded the similarity between gases and electron movements
and called random electron currents as a "gas". His energy equation was definitely based highly on probability equations, resulting in his energy state equation, which led to his radiation equation.
See "Planck's Columbia Lectures".
5 / 5 (3) Feb 06, 2013
The whole point of the schroedinger cat experiment is the absurdity of extrapolating from quantum to macro or even universe scales.
I didn't see anyone really give you a full response. The whole point of the schroedinger cat is not to point out the absurdity of scaling up quantum effects to macro-scale objects. It is to provide a
reference that is based in reality that surrounds us. For a LOT of people, it is far easier to understand "the cat is in an undetermined state" than it is to understand any explanation using waves
and particles. The cat "experiment" was never meant to be taken seriously. When I explain it, I always end it with, "But in this case we know the cat is DEAD. Nobody poked air holes in the box..."
You'd be surprised how often someone says, "Yeah if they poked air holes in we could see the cat..." And ta-da, a light goes off. :)
1.5 / 5 (8) Feb 06, 2013
Axe and Tigger, well said. But I like LarryD's line:
"as long as we remember it's WE who are 'guessing' and NOT the system under examination."
Assume humans don't exist. Everything else in the universe is guided by physical laws that govern energy/magnetism, period. We make choices and attempts to measure, therefore we are the only reason
for "probability". The universe outside the sphere of influence of any being capable of choice has absolutely no random occurences....just physics. Probability as in the precise point position of a
quantum of energy such an electron or photon is again, only because we can't measure it with enough precision to make a definitive statement...the universe knows precisely where every quantum of
energy present within it's boundaries is.
We have to remember that we can only perceive/alter one local reality and that, at the end of the day we are still governed by the same physical laws that the universe is.
3.4 / 5 (5) Feb 06, 2013
To quote Paul Steinhardt:
"An infinite number of universes with an infinite variety of physical properties implies that everything that can happen does happen in some universe. A theory that predicts everything predicts
1.7 / 5 (13) Feb 06, 2013
I do not believe in QT. First of all "random" does not exist. Period. We use the word random the same way we use the words "dark matter". We know there is some forces determining the result of an
event. We just don't know what they all are, and we don't understand how they work entirely. If we did have complete understanding of any given system, we would be able to 100% predict any behavior.
So to answer the articles title question. NO probability does not come from quantum physics. In fact quantum physics comes from our inability to understand all the possible factors. Probability
exists only as a factor of our uncertainty. AKA lack of understanding. The word "random" is an insult to science.
If we were able to copy all of the information contained in every atom of the "radioactive element" into a computer simulation. And we knew at the sub atomic lvl what causes one atom to decay, which
is not "random". We could determine the exact moment of the cats death. Unwitnessed.
3 / 5 (6) Feb 06, 2013
For an intersting thought experiment on how the conscious observer is (or isn't) a crucial part of QM one may take a look at the "Wiegner's friend" extension of the Schrödinger's cat experiment.
A rather more dramatic form can be achieved by placing a conscious observer (again Wiegner's friend) inside the box.
1.4 / 5 (9) Feb 06, 2013
It's kind of funny and ironic to me if you really think the whole thing through, the thing we try to predict most often is most likely the most "random"/unpredictable thing in the universe. A
conscious decision. It is otherwise obvious to me that a multiverse does not exist. There is no subatomic "fly-in-the-ointment" where everthing else in the universe happened exactly the same way as
in another univers until in one universe this "fly" being akin to a plinko chip went left instead of right, thus branching off and making another universe. This is just stupid. It is fun to imagine
but so obviously not the case. Only the most simplistic of minds could believe that. This is why Einstein said "God does not play dice with the universe." Now having said that, is it possible that
there could be a universe where I made the conscious choice to stop and have coffee this morning on the way to work. Thus changing the causality of every other event branching out from my tree of
1.4 / 5 (9) Feb 06, 2013
Well yes I believe so, but it would not mean an infinite number of universes. There would simply be one universe for every conscious decision ever made by every being that ever existed. Far less than
5 / 5 (2) Feb 06, 2013
I do not believe in QT. First of all "random" does not exist. Period. We know there is some forces determining the result of an event. We just don't know what they all are, and we don't
understand how they work entirely. If we did have complete understanding of any given system, we would be able to 100% predict any behavior.
I don't know how you can say with certainty that randomness does not exist when you openly admit we do not know everything there is to know about physics. Simply put, what if the laws of physics
include randomness? We have plenty of evidence of random events occuring. It seems more likely that random behaviour DOES exist, and we simply don't know why it exists. That question would then fall
into a huge category of very interesting unknowns we can confirm, but do not fully understand yet.
1.5 / 5 (10) Feb 06, 2013
I don't know how you can say with certainty that randomness does not exist when you openly admit we do not know everything there is to know about physics. Simply put, what if the laws of physics
include randomness? We have plenty of evidence of random events occuring. It seems more likely that random behaviour DOES exist, and we simply don't know why it exists. That question would then
fall into a huge category of very interesting unknowns we can confirm, but do not fully understand yet.
I know because you cannot show me a single event in nature that is random. In fact many things we can predict with current technology would be considered whichcraft 100 years ago. The weather per se.
The Schrodingers cat thought experiment the article sites refers specifcally to the "random" decay of radioactive isotope, yet this event is so "random" we know exactly how long it takes and in fact
use it in our most acurate clocks. Making it one of the least "random" things we know of.
3.9 / 5 (11) Feb 06, 2013
I know because you cannot show me a single event in nature that is random.
Radioactive decay would be one (of many) examples.
While you can fit curves to random processes (distributions) you cannot predict when an individual event will occur. Randomness is constrained - but it can be shown that it cannot be the result of
smaller, fully deterministic processes. (such smaller processes would constitute 'hidden variables').
And by the Bell tests we have a strong argument that hidden variables do not exist. You could only save that with non-local hidden variables as in the deBrogli-Bohm interpretation of QM. But that
opens up an entirely different box of problems.
And there is a strong indication that complete determinism isn't in the cards. (e.g. if you do double slit experiments with single electrons in the apparatus you still get interference patterns)
1.3 / 5 (12) Feb 06, 2013
"Radioactive decay would be one (of many) examples."
I already dealth with this specific arguement in my previous post. If radioactive decay is random, then how is it the most precise event we can use to determine the passage of time with.
If we can predict exactly how long it will take to happen, than how can it possibly be a random event? There is a very specific set of rules governing radioactive decay. We simply do not understand.
I tell you again. Your quantum theory is "witchcraft" and very very unscientific.
3.3 / 5 (7) Feb 06, 2013
"Radioactive decay would be one (of many) examples."
I already dealth with this specific arguement in my previous post. If radioactive decay is random, then how is it the most precise event we can use to determine the passage of time with.
If we can predict exactly how long it will take to happen, than how can it possibly be a random event? There is a very specific set of rules governing radioactive decay. We simply do not
understand. I tell you again. Your quantum theory is "witchcraft" and very very unscientific.
You don't know how an atomic clock works.
It has nothing to do with radioactive decay, which is random.
1.5 / 5 (10) Feb 06, 2013
I wasn't talking about atomic clocks. I was talking about half-lifes which is a probabalistic event. (You know, related to what we're discussing). Yet is predictable enough that we can use carbon-14
dating to determine the "probabalist" yet very accurate age of many things. In fact can you name a method used more often to determine the age of something? Again the mere fact that we can come up
with a formula to accurately predict the nature of something "random" implies that it is in fact not random at all.
3.8 / 5 (10) Feb 06, 2013
If radioactive decay is random, then how is it the most precise event we can use to determine the passage of time with.
Over many decay events you can average out - that's how we can use decay for measuring time (if there were few events then the dating via radiooactivity wouldn't work nearly as well).
It's like flipping a coin - you can't predict any one specific outcome, but over a million coin tosses you can map all the possible paths and see that the paths that lead to a near 50/50 split are
VASTLY in the majority over those that are all heads - so betting on teh average outcome is better than betting on the extreme one.
I think you need to look up the difference between a probabilistic event and a probability distribution.
1.6 / 5 (13) Feb 06, 2013
First of all "random" does not exist. Period.
I don't think so. Sometimes the information is simply lost. For example, the surface of water droplet cannot reflect the movements of all water molecules which are bouncing and colliding inside of
it. Some of these movements are simply lost for the observer from outside for ever. How we can be sure, that the randomness of quantum mechanics doesn't work in the similar way?
1 / 5 (1) Feb 06, 2013
The paper is pretty neat. An example of what is being talked about here is coin flipping. How can the probability of the outcome of a coin flip be traced back to quantum mechanical uncertainty?
As it turns out, it all starts with water and polypeptides in your nervous system. Quantum uncertainty at that level introduces enough probability in molecular transport to evoke
less-than-perfect physical prowess, which ultimately leads imprecise control of the energy required to repeatedly produce a perfect flip and snatch.
I just have to say Tektrix, that this almost a goddamn PERFECT comment. I've reused it (with a reference) on Facebook, and about to add it to Google .
1.7 / 5 (12) Feb 06, 2013
Nothing comes from nothing.
1.4 / 5 (10) Feb 06, 2013
Guys, "random" is a human construct. Every immediate reality (what we term as "the present") has a set of variables that came together to create it, this happens everywhere in the universe.
If 2 supposed identical samples containing C14 display differing rates of decay, there is a reason...a mechanism causing the differential, whether we know what it is or not. If you knew EVERY
variable, there is no probability.
This is why we need it. We don't know and likely never will.
QT is a wonderful thought exercise, but as long as we experience time as linear it will always be just that.
"It's like flipping a coin - you can't predict any one specific outcome.." - AP
See Tetrix's 1st post. Control every variable down to the quantum level and you can make the coin land the same way every time.
1.4 / 5 (10) Feb 06, 2013
Antialias. You are making a moot point. The simple fact that we can detect that different elements have different decay rates at all tells you it isn't random. The fact that these rates are
measurable and can be generalized with a formula tells you that they are not random. It't obvious if you think things through to their logical conclusion. We simply cannot observe all of the data and
information of the system to understand it completely. At the risk of being obtuse through repitition I say again. I like to use a computer simulation analogy. It's also akin to a star trek
transporter. If we could map all of the information contained in an atom, (or group of atoms) and we knew completely how all the subatomic actions worked, we would be able to simulate with 100%
certainty how the system would evolve by running the simulation faster than real time. Or teleport said object by remapping the data onto remotely located atoms.
1.5 / 5 (8) Feb 06, 2013
Not only randomness is an evident essential aspect of QM, claiming determinism as a universal fact to predict everything is not serious today: already deterministic gravitationally bound systems
exceeding 3 bodies have been shown for more than a century to totally lose determinable properties beyond a time dependant predictability horizon. Beyond that horizon, only probalilistic approaches
allow to do statisical predictions for those chaotic systems ruled by resonance. Determinism, although very useful, is only a practical simplification, just as linearity and so on, applicable in some
limited artificial cases, which are idealising reality, which is evidently independant of us humans. It is our perception and interpretation to understanding, which is totally subjective.
BTW, the theory of scale relativity still gives the most elegant interpretation, but this is another sensitive argument.
3.9 / 5 (7) Feb 06, 2013
Control every variable down to the quantum level and you can make the coin land the same way every time.
Since you can't control it to the quantum level that's a nonsensical approach. At the quantum level values aren't discrete but a probability distribution.
1 / 5 (10) Feb 06, 2013
The problem with multiple universes, Albrecht said, is that it if there are a huge number of different pocket universes
This article got into exactly the opposite conclusion. In this article The Multiverse Created Probability To Explain The Multiverse (synopsis)
Whydening Gyre
1 / 5 (11) Feb 06, 2013
Does probability come from quantum physics?
I would say -
Whydening Gyre
1.4 / 5 (11) Feb 06, 2013
One crucial factor missed out is the human mind, not measurable and yet it could be the main factor responsible for materializing which probability or pocket of universes it desires. At one time,
the earth was flat, but now the mind now "knows" that the earth is actually a ball in space.
Humans knew that LONG before they thought it was flat...
3.7 / 5 (3) Feb 06, 2013
Your faith is not supported by experiment.
"We could determine the exact moment of the cats death. Unwitnessed." - Jalmy
4.2 / 5 (5) Feb 06, 2013
You are confusing random with unpredictable.
A coin toss can produce completely random results, and yet the statistical character of those tosses are absolutely predictable.
"The fact that these rates are measurable and can be generalized with a formula tells you that they are not random." - Jalmy
If Statistics wasn't predictable, then it would have no value.
Since it does, it is.
Since it is, you are wrong.
1.4 / 5 (10) Feb 07, 2013
That is my whole point fool. Something being predictable at any lvl means it is not random. If there is a patern to it then it isn't "random" that's the whole point. Way to entirely miss the big
picture though. Try to read the entirety of my post, then make a stupid comment after please. TY.
1.6 / 5 (13) Feb 07, 2013
Your faith is not supported by experiment.
Really? So the literally uncountable amount of information we are processing in computer simulations is all wrong? We might as well stop doing it. Since we cannot support the results with experiment.
Why do we spend billions of dollars devolping these huge supercomputers to run science simulations? Because the enormous amount of math and the complexity and vastness of the data sets make it
impossible to test with experiment. Your uncertainty principle is wrong. Your so called "experiments" that you think prove it are just misinterpreted. Einstein thought it was BS and so do I. If you
actually spent some time thinking about things maybe you would too.
1.4 / 5 (9) Feb 07, 2013
"The cat knowing its alive" is not irrelevant. The cat would be very aware of being locked in a confined space. Its only human vanity that is prepared to create whole universes rather than admit the
obvious classical nature of a cat collapsing its own wave function even if it is not fully sentient.
1 / 5 (5) Feb 07, 2013
Control every variable down to the quantum level and you can make the coin land the same way every time.
Since you can't control it to the quantum level that's a nonsensical approach. At the quantum level values aren't discrete but a probability distribution.
No, they are discrete. Just not by OUR measuring capabilities, which is why I conceded we need "probability" in the same post you drew the above quote from.
IF you had the ability to measure the orbit of the electron around a the nucleus of Hydrogen atom you could apply a vector coordinate to every position it occupies. These would be discrete
coordinates, not probabilistic ones.
3.1 / 5 (14) Feb 07, 2013
At the quantum level values aren't discrete but a probability distribution.
In the scope of CI QM, yes.
But what about out of it? What if QM does not = reality? Is it still all "random"? Because you said so? Because the Copenhagenists said so? Or just because the sky is blue?
What makes you so certain (that nothing is certain)? Isn't that allready a flawed logic construct to begin with?
The Bell experiments? There were no loophole-free Bell tests done yet. Double-slit? Without a truckload of fundamental assumptions, essentially inconclusive.. What else?
Nuclear decay? As random as a humans lifespan. While the average might converge to a certain value, it doesn't strictly dictate the path of an individual (which cannot be predicted without fully
understanding the underlying principles and having sufficient data).
At this point, the only reasonable and honest conclusion that can be drawn is, that we can NOT be certain of either..
Ergo, WE SIMPLY DO NOT KNOW (yet).
3.7 / 5 (6) Feb 07, 2013
Something being predictable at any lvl means it is not random. If there is a patern to it then it isn't "random" that's the whole point.
Your whole point is wrong, then.
You can measure the probability of something happening with great accuracy. That does not mean that an individual event of that class is not random.
Since you say half-life is not random, consider carefully what half-life means. For a given sample of element X, 50% of the atoms in that sample will decay within the half-life. So if element X has a
half-life of 10 minutes, some atoms will decay in 1 second. Some in 2 minutes. Some in 9 minutes and four seconds. Some in 15 minutes. Some in an hour...
You can say with great certainty that the half-life is 10 minutes. You can not say with any certainty when exactly a single atom (a single event) will decay. This is not because there are hidden
variables we do not know about. It is because the decay is random.
2.8 / 5 (11) Feb 07, 2013
IF you had the ability to measure the orbit of the electron around a the nucleus of Hydrogen atom you could apply a vector coordinate to every position it occupies. These would be discrete
coordinates, not probabilistic ones.
I'm affraid it might be a bit more complicated..
There is no such thing as an absolute discrete coordinate either, as that would also require an absolute discrete point in time. And in the light of last several years of data, Planck time doesn't
seem to fit that bill.
Everything is in constant motion, so every "discrete" coordinate is in essence just an average over a chosen time interval. No matter how sensitive/precise the measurements are, it will allways be
like that. Sort of like the HUP in praxis.
In a certain sense, one could almost say that there is no such thing as a tangible NOW - just past, and future..
This actually makes all the arguments about discrete or probabilistic reality quite moot, as the ultimate answer seems to be neither, nor.
1.7 / 5 (12) Feb 07, 2013
Sorry rkolter you missed my point again. Perhaps you did not thorougly read. The important part of it is that the probability amounts of each element is different and repeatable to a certain degree
of uncertainty. If it was truely random. All elements would have the same probability of decay. But they dont. They have different decay rates. And repeatable. Even though you have to go to a great
many years to see the patern. There IS A PATERN. Maybe your problem is you do not understand the word "random". Here is the definition as given by Webster's.
1 a: lacking a definite plan, purpose, or pattern
See the word pattern is key.
Another corroborating piece of evidence is the fairly recent observation that the sun is influencing these decay rates. In times of solar flares for example decay rates increase. Again this is
showing there are things influencing your completely not random "random" decay rates. I guess Erwin Schrödinger needs to come up with a new trigger device.
Whydening Gyre
1.4 / 5 (11) Feb 07, 2013
You are confusing random with unpredictable.
A coin toss can produce completely random results, and yet the statistical character of those tosses are absolutely predictable.
Aren't statistics and probability flip sides of the same coin?
"The fact that these rates are measurable and can be generalized with a formula tells you that they are not random." - Jalmy
If Statistics wasn't predictable, then it would have no value.
Since it does, it is.
Since it is, you are wrong.
Aren't statistics and probability flip sides of the same coin?
1 / 5 (6) Feb 07, 2013
"I'm affraid it might be a bit more complicated.."
Daywalk - I'll add the words "in a specific inertial reference frame" to what I said regarding the vector coodinates. Relative to the universe, everything is in motion. Magnetically trapped in a
vaccuum chamber here on earth, we simply lack the ability to measure the coordinates the electron occupies in our inertial reference frame.
4.2 / 5 (5) Feb 07, 2013
...The important part of it is that the probability amounts of each element is different and repeatable... If it was truely random. All elements would have the same probability of decay...
1 a: lacking a definite plan, purpose, or pattern
See the word pattern is key.
I removed your personal attacks and snide comments to try to pull the core of your argument out of your rant.
At the core of your arguement is the assumption that "no pattern" means necessarily "no solution is more likely than any other". This is not the case. It is perfectly reasonable to have a process
that is random, with some options more or less likely.
Also, you excluded the second definition from Webster:
(2) Relating to, having, or being elements or events with definite probability of occurrence
This is what we've been talking about.
1.4 / 5 (9) Feb 07, 2013
Exactly rkolter, your version of random is the same "random" that we use when talking about computers generating random numbers. Which again is not at all actually random. It is simply beyond our
conscious ability to descern, or for the sake of most uses it is good enough. But for the specifics of this topic and our argument that type of random is not good enough, and it is most definitly not
what I have been talking about. Yes "no solution is more likely than any other" is true randomness. Which my whole arguement that you keep helping me prove. True randomness DOES NOT EXIST. The word
is a tool to explore with, just as is the random generator in a computer. It is a tool, nothing more nothing less. Just as quantum mechanics is a tool. I think we got a bit side tracked from the
original article though. Just to clairfy what my stance is. I think the asumption that true randomness exists is the flaw that leads to the creation of multiverse which the existance of is obsurd.
2.1 / 5 (7) Feb 07, 2013
I don't think flipping a coin on macro scale has anything to do with probability or in the least so little it has no influence on the outcome.
Regardless of what you think, the authors of the article use this very example to illustrate the premise. And AP'org is right- the intrinsic nature of QM makes the levels of control you are
suggesting, impossible. The idea that perfect knowledge lends perfect control is centuries old and has been firmly refuted by the huge body of work in QM.
1 / 5 (3) Feb 07, 2013
LarryD...your comment about a "biocentric" view is well received. I tend to believe that anything, both matter and wave can be considered an observer...anything that can interact Consciousness is not
required. The cat is either dead or alive, not both. A silly take on this would be, at what point in the classical sense of this mind experiment is the cat actually dead? Or is it alive? Is it when
the first photon enters the observer's eye and an image of the cat forms? Is it when the observer examines the cat at rest to see if it is alive? Is it when the vet declares the poor beast deceased
or in good health? I know it's silly, but my point is, existence is observation to some extent, perhaps entirely. The cat is observed by virtue of the fact that it exists. Obviously, such a claim has
far reaching implications and really isn't done any justice in my flip response.
3.2 / 5 (13) Feb 07, 2013
Aren't statistics and probability flip sides of the same coin?
Yes, as long as you keep in mind that "statistics" only deals with things that have happened,,,, useful assessing causality.
Probability is the assessment of what might happen, once IT happens, it's no longer a probably event, it is an actual event with 100% probability (it happened.)
It is an important distinction.
3.5 / 5 (8) Feb 07, 2013
when talking about computers generating random numbers.
That's why no one talks about computer generating random numbers but only about computers generating pseudo-random numbers.
They are numbers that are generated via an algorithm (and hence absolutely predictable since the next number depends on the previous one).
The difference with truly random events in real life is that they aren't dependent on each other (and again there are tests that can be used to distinguish whether events are indpenedent or not)
1.5 / 5 (8) Feb 07, 2013
Yes. Independent events. That search continues...
...and language evolves to give meaning to that.
1.8 / 5 (8) Feb 07, 2013
Probability comes from the language used.
Lite never comments. No probability there. lol
Whydening Gyre
1.4 / 5 (11) Feb 07, 2013
Probability comes from the language used.{/q]
SOrry, Tau. gotta disagree with ya there - probability was here first...:)
Lite never comments. No probability there. lol
And I'm waiting for the day he/she/it does not 1 every comment I make. I'll have to go out and by a lottery ticket if THAT ever happens.
3.2 / 5 (14) Feb 08, 2013
Probability comes from the language used. Lite never comments. No probability there. lol
Lite is a bot.
It has a blacklist of usernames, and continuously scours ALL articles in a predefined time interval. If it finds a comment by a blacklisted user, it automatically rates it with an assigned value
(mostly 1).
As the bot is running on a server/computer, which runs off the electricity grid, and is connected to the internet through a multitude of devices (etc.), a fair bit of probability to its function
still applies..
The admins who (should) manage this site seemingly don't give a damn about the comments sections, which in turn often gets pestered by spam, etc. So the next best thing in case the admins don't give
a f***, is to give a s***.
Yea, I know, very on topic.. NOT! But who cares, eh? :-)
Whydening Gyre
1.4 / 5 (11) Feb 08, 2013
That might be right except for a couple small quirks. I've noticed lite does not rate certain users or if it does, it rates them a 5.
Interesting coincidence, that...
If I didn't know better, I'd say it's my very competitive wife. However, she has to ask me how to use Word, so...
3 / 5 (12) Feb 08, 2013
Daywalk - I'll add the words "in a specific inertial reference frame" to what I said regarding the vector coodinates.
Doesn't matter. There is no such thing as 'perfect vacuum', neither is it possible to perfectly isolate any volume from the 'outside world' - no matter what you do, it will allways remain part of the
Even passing an event horizon can not be strictly considered a causal disconnect (as often believed). What falls inside a BH does not dissapear, it becomes a contributing part of a bigger whole,
which still affects the 'outside', be it either by gravity, resulting magnetic field(s), etc..
The only way how to achieve such 'disconnect' (on paper) is to define a threshold bellow which everything is considered negligible, or essentially non-existent. Well hello, I present to thee the
magnifficient theory of QM..
When it comes to motion (or 'change') in a fundamental context, it can neither be considered absolutely discrete, nor truely random.
1 / 5 (6) Feb 08, 2013
...uncountable amount of information...
is another label for what is called the Continuum.
That introduces randomness.
Yes, all computer simulations are wrong.
Spending money is fun. You are taught that.
1 / 5 (5) Feb 08, 2013
DW- Well said. Especially your last line. It is essentially the point I was trying to make. If no sentient life populated the universe, everything would occur as "the universe's" laws of physics
dictate they should...nothing can be truly random.
1 / 5 (1) Feb 08, 2013
@ValeriaT - Quantum computers would not be faster regular ones when calculating something like 3x5 or 2 4. However when calculating probabilities it clearly should be much much faster regular
computers (when it would start working at all). And grants would be one of the ways to speed up that development.
1 / 5 (6) Feb 08, 2013
The answer is the theory is wrong or God DOES throw dice.
1.8 / 5 (10) Feb 08, 2013
No 'theory' is wrong. Just obsolete or superseded.
See Heilocentrism vs. Geocentrism.
Luckily humans weary of complexity.
The less math (or assumption) involved, the greater chance(!) of appeal among humans. (Deists and/or Theists love this added advantage of simplicity until belief and/or faith becomes obsolete or
Self referential-ism is frowned upon in science. You know this.
1.4 / 5 (9) Feb 09, 2013
I'm not even convinced that the single radioactive atom which is imperiling Schroedingers cat is in a superposition state. Its more similar to a point on the screen of the two slits experiment. That
point is either illuminated or not by impact of a single photon. The only superposition in that experiment exists while the photon is in flight through both of the slits simultaneously. The
scattering of dots on the screen is a classical probability distribution not a superimposed fuzzy pattern. A geiger counter never registers half a count, even when sealed in a box with a cat. A
'classical' autopsy still reveals when the cat died once the box is observed.
2.6 / 5 (5) Feb 09, 2013
The scattering of dots on the screen is a classical probability distribution
No it is not. If it were you'd see as a result the combined results of the same experiment done twice, but with one slit occluded each time.
But instead you see an interference pattern.
1.4 / 5 (9) Feb 10, 2013
antialias_physorg: You misunderstand me. I'm just saying that the photon dots are either there or not: Rather than the photons which travel through both slits and are superimposed in flight. The dots
distribution follows a classical calculatable wave interference, but like decaying atoms you never know where the next one will be till it has happened.
1 / 5 (7) Feb 10, 2013
The existence of photons dots on the target screen is clear tangible evidence, that these photons do behave like the localized particles. The wave-interference-like distribution of photons indicates,
these photons are surrounded with wave, which affects their path. Because photons itself do behave like the particles (and they're usually much smaller, than the slit itself), it's evident, that the
wave behavior of photons doesn't come from photons itself, but from vacuum which surrounds them.
It's a simple clean logics, which is inaccessible for people, who are using the complex math. It's simply another hidden layer of reality for them. But the physicists and high-school teachers don't
want you to use such a clean transparent logics, because they would lose their informational monopoly and social influence, if not job. Most of people want to understand the things as cleanly as
possible and they resort to complex formal description of reality only when they don't see any other option.
1.4 / 5 (10) Feb 10, 2013
I admit, for fooled sheep who are downvoting me obstinately is difficult to admit, they're fooled sheep - but this is simply, how the reality currently is. This is the power of new ideas, the power
of new vision of reality. The problem with its understanding isn't in complexity of math involved - it is in your head and your prejudices only - as it always was. The situation well known from
medieval times just repeats again. You didn't expect, you may appear in the position of hypocritical reactionary and ignorant opponents of Galielo or Pasteur?
Well, it just happens now. The world is changing - but the thinking of average people remains.
1 / 5 (2) Feb 10, 2013
'Oh wise oracle my head is full of pain. Free me of this confusion and make the answer plain.'
'What troubles you, my son? Your question does not seem a simple one'
'Oh wise and wonderful Oracle I'm on nerve, what say you about the Normal D Curve.'
'Probability is the game, I roll the dice after careful aim'
'But Oracle, if on probability you depend surely means you know not the end!'
'I know all without a doubt, of QM and what it's about!'
'Oracle, why play if you can see what every outcome is to be?'
'How typical and not very bright, I toss the die to see if I'm right.'
'Oh come now Oracle with me you jest, surely numbers on a die have equal probability at best.'
'For a human that's pretty neat, insinuating that I'm a cheat'
'Wave or particle, where is it, Oracle?'
'Neither and both, not here yet everywhere you see, that makes 100% probability'
Don't forget that Bohr said to Einstein ' Don't tell God what to do'.
Hmmm. what kind of probability density function describes God?
2 / 5 (4) Feb 10, 2013
Not a theory, just a "what if" idea I have been contemplating...
When we look out at all the galaxies, we are actually looking out past the boundry of our universe, like inside a marble looking out and space time bends in on itself, and so we are looking out at
every possible Universe, every possible shape and fluctuation the big bang could have taken since the begining of time? Like waves of photons interacting, these island universes can also interact
within the greater multiverse.
This might explain a lot.. Thoughts?
not rated yet Feb 12, 2013
Well, here's a clue. The "observer" is only a simplification for what is really happening. In reality, the waveform is colapsed by decoherence from interaction with the environment. So, if my pet
yellow dog is trained to bark at a live cat, and he doesn't bark when the box is opened, he becomes the observer and by hearing his bark or lack of it, I transfer that information to the environment
and colapse the waveform. If however, I am out to pick up the paper and not there to hear the bark or lack of it, the cat is still in an undetermined state, with equal probability of being both alive
and dead. Yes, you can perform a necropsy (on an animal, not an autopsy) and determine if the cat died earlier. This is simular to quantum erasure. It doesn't change the outcome of the thought
experiment. Of course, that trick is easily demonstrated with electrons and the right setup of slits and detectors. This stuff really isn't that hard. Really, it isn't.
not rated yet Feb 12, 2013
Some may have missed my point. The outcome of the thought experiment doesn't depend on a human observer, or a dog or an insect. It only depends on the colapse of the probability wave by decoherence
through interaction with the surrounding environment. Of course, if the outcome of the experiment has absolutely no interaction with the surrounding environment, none whatsoever, then the wave has
not colapsed and the cat lives forever in an indeterminate state. More likely though, the cat would start to smell, the smell would attract flies, and so on until the interaction with the environment
was great enough to colapse the waveform. Just accept it. Quantum mechanics is here to stay. As the Borg would say, resistance is futile.
1 / 5 (1) Feb 13, 2013
GPhillip, yes you are probably right QM is here to stay but then so are Newtons Laws, Classical music and Hot Dogs. Every generation thinks it has the final answer and history has proved that to be
wrong (Hot Dogs excluded). Each generation provides another step up a very long ladder and QM is such. A couple of hundred years hence they will be claiming their version of QM is 'it' with maybe a
new 'state' other than wave or particle (a 'wavicle'???). But that's progress;accept it and move on. The theories we have now answer some questions but a good theory not only makes predictions it
asks the all important and vital next question. Forget the cat and try something that really needs an answer. For example,an experiement on whether a ghost is a collapsed wave from the person
claiming to see one. You may or not believe in such entities but there many millions of people (maybe billions) both layman and scientists who do. Is QM up to the task?
3 / 5 (2) Feb 13, 2013
Rather than the photons which travel through both slits and are superimposed in flight. The dots distribution follows a classical calculatable wave interference, but like decaying atoms you never
know where the next one will be till it has happened.
If that were the case they would fly through one slit or the other - because you could as easily put the screen at the slit (in which case you DON'T get an interference pattern)
But we DO see an interference pattern in the 2-slit experiment and hence they don't fly through either one slit or the other. So the 'dot' is not present until measured.
The photon is superimposed with itself because you get an interference pattern even if you space the photons so far apart that there is on average only one in the apparatus at any one time. (You can
also do it with single electrons and get the same result)
not rated yet Feb 13, 2013
Thinking that Quantum Mechanics is just a temporary best solution and that something will come along later that makes more intuitive sense is just another way of trying to reject the science that has
been proven to be absolutely accurate in 10's of thousands of experiments. Quantum Mechanics is no more a temporary best solution than the math of 2 and 2 = 4. It's correct and will always be correct
and will not be proven incorrect at any point in the future. Of course we may find simpler expressions of the same concept as it is studied further. Einstein's original presentation of GR took grad
level math abilities to understand, but after studying it for decades, simpler explanations of the same concept, without the complex math were possible.
As far as trying to explain paranormal phenomena with quantum effects, I'm afraid that is beyond the capabilities of the science. The scientific method requires experiments to be performed that can
disprove the validity of a hypothesis.
not rated yet Feb 13, 2013
The very reason that some phenomena is classified as paranormal is that is cannot be subjected to falsification by experiment. So, some things like ghosts, UFO's, big foot, etc., are likely to always
remain outside the capabilities of scientific investigation. Simply, science can not and does not provide an answer to everything. It has it's limitations.
1 / 5 (5) Feb 13, 2013
The very reason that some phenomena is classified as paranormal is that is cannot be subjected to falsification by experiment
Unfortunately in contemporary physics many phenomena are ignored if not denied just because they depend on higher number of parameters, than the contemporary physicists can recognize. Typically the
cold fusion and various psychic phenomena fall into this category: despite they're manifesting with quite tangible effects often, the general lack of reproducibility refuses the physicists who are
motivated with safety and reliability of research, not with desire to reveal something new. Illustratively speaking, for modern physicists even the lightning is not real, until it cannot be reliably
predicted and reproduced.
1 / 5 (5) Feb 13, 2013
"refuses the physicists" should be "repels the physicists"
1 / 5 (4) Feb 13, 2013
GPhillip, nonsense! In no way would or could I reject QM. Just because I can point to a certain type of 'pseudo failure' doesn't mean that I reject QM. I think your remark indicated that you didn't
get my point but ValeriaT did! I really do think that we have reached the point, or will be soon, where we can tackle such questions as the 'paranormal'; and therein lies the problem. As long as we
give that label to some event scientists won't touch it. Some such events have been shown to be electromagnetic but as far as I know no one has yet tried to reproduce alleged events not because it is
outside our understanding but because of location. Setting up a fully equipped lab in someone's house is, I admit, hardly feasible but come on fellas, surely there's some bright individual(s) who can
overcome this. It seems that I have more faith in our science.
5 / 5 (2) Feb 15, 2013
I probably didn't say this well at all. Richard Hamming, a founder of modern computer science, said it much better:
Mathematics addresses only a part of human experience. Much of human experience does not fall under science or mathematics but under the philosophy of value, including ethics, aesthetics, and
political philosophy. To assert that the world can be explained via mathematics amounts to an act of faith.
1 / 5 (3) Feb 16, 2013
GPhillip have you ever read the complete works of Plato? There are numerous remarks there that would also apply...but then those guys had a lot of time to sit and think etc. So I do agree with you
that if we think maths will provide all the answers then we are misguided. But then again, the topic here is about probability and maths can only supply us with possibilities. The 'numbers game' with
regard to criminals has many times failed (The Yorkshire Ripper being a famous one)and where human intuition, philosophy etc become the way forward.
But having said that, maths and science are 'good inventions' so we ought to use them.
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Teaching to the Math Standards with
Volume 3, Issue C ::: September 1999
Teaching to the Math Standards with Adult Learners
by Esther D. Leonelli
For the last 10 years, I have been an advocate for standards-based teaching of mathematics and numeracy to adult basic education (ABE), General Educational Development (GED), and adult English for
Speakers of Other Languages (ESOL) students. It has been quite a journey, a learning experience, and the most fulfilling part of my adult education career since I returned to teaching adults in 1985.
By "standards-based," I mean a set of values and important ideas used to judge methods of instruction and assessment. With respect to math instruction, I mean both content and methodology based upon
the National Council of Teachers of Mathematics (NCTM) Curriculum and Evaluation Standards for School Mathematics (1989), which was adapted for ABE instruction by Massachusetts teachers.
My Conversion
I was trained as a secondary math education major in the late 1960s. When I first taught adults, from 1971 to 1973, I tried to incorporate the methods I learned in college. These methods were based
upon the "new math movement" and Piaget's work with children, used manipulatives, and included a deductive, but very directed, approach. These did not translate well into my adult education work at
that time. I found that my students, who were in an individualized math lab in a Boston program that prepared them for medical training and professions, did not want to be led through these
"discovery" lessons. They wanted to be shown the "rule" so they could apply it to the problems in the book and to the math tasks needed for the particular job they were planning to pursue.
I could do this very well. And so I taught math skills one on one, in a linear way, using pencil and paper and remedial arithmetic skills textbooks. I taught basic computation by rote, using
decontexualized situations. Once the students mastered computation skill using only numbers, then I showed them how the skills were applied to word problems, which were chosen for the particular
skill to be practiced and mastered.
I continued to teach this way when I returned to the adult education classroom in 1985, teaching ABE and GED level students. I was reluctant to use the manipulatives the Cuisenaire rods, the
base-10 blocks that were a part of my math education training. I gave up trying to have my students discover the math concepts they were trying to master, although I wished that my students could
rely on their own reasoning powers to reconstruct the theory or rule if forgotten. I reverted to teaching math the way I was taught.
My methods worked okay. I relied on textbooks with many practice exercises and the answers in the back of the back. I could get my students to pass the competency-based math tests my center used for
the alternative adult diploma credential we granted, one test at a time. That made my students feel confident and good. But I felt something was lacking when they couldn't remember how to divide
fractions once we moved on to another math topic. I was disconcerted when they had to return to my class to prepare for the Licensed Practical Nurse (LPN) test or the college entry test. Something
wasn't sticking. The math was "learned" for the test and then forgotten. Students depended on me and on the textbook for answers and rules. And, my methods did not work well for the students who were
non-native English speakers. Needless to say, I felt that the approach I was taking needed changing.
By 1989, the GED test had changed to include more emphasis on problem-solving and higher-order thinking skills. It allowed more use of estimation skills and required that fewer complicated
calculations be done without the use of calculator. A team of GED teachers in Massachusetts took a good look at the test and came to the conclusion that how we were teaching math should change. I
joined that team. Around the same time, I attended a multisession workshop on teaching basic mathematics at the Adult Literacy Resource Institute in Boston. This was a mathematical re-awakening for
me and an invitation to reconsider my own practice. The workshop introduced me to new national developments in the area of curriculum, methodology, evaluation, and teacher training in school
mathematics that moved math teaching beyond the "back to basics" movement of the last two decades. These ideas, along with research and practice in classrooms and constructivist theory, were
incorporated into NCTM Standards, the seminal document for the math-standards movement. I read the Standards, became a "believer," and in the process, also became a lifelong math learner.
Through attendance at NCTM conferences I got to see first-hand the exciting changes in pedagogy and assessment advocated by proponents of the NCTM standards. I saw less direct instruction and more
modeling of mathematical behavior by teachers. I saw less "drill and kill" practice and more interesting problems for investigation by students. I saw less individual seatwork and more cooperative
lessons and conversation in the classroom around math, and fewer answers from the teacher and more sharing by and among students of individual strategies for solving problems. The workshops were
intended to engage me in learning more math, which they did. And they showed me how the activities could be engaging for my learners.
In one workshop I was introduced to international developments in math education, the "realistic maths" curriculum from the Freudenthal Institute, the Netherlands. Their approach asks students to
make mathematical sense from graphical images of the real world. One of the "geometry" problems that I like to pose to my students came from that workshop (see below). It requires not only
visualization, but also the physical handling of concrete materials and group discussion to come up with an optimal solution.
Here are two views of a building:
Front View
Side View
What is the least number of blocks you can use to build the building?
What is the maximum number of blocks you can use to build the building?
This activity let me view students at work alone and together, solving a concrete problem. When they build their structures I see what they saw. What I learned is that many of my students have never
had the opportunity to build and play and visualize. I also realized that I was making a lot of assumptions when I "lectured" or "demonstrated." I had assumed that my students could "read" a picture
and could learn to interpret word problems by my teaching of "key" words and formulas.
Standards and Frameworks
The NCTM Standards were based upon the assumption that, in the late twentieth century, American society has four new social goals for school education: (1) mathematically literate workers, (2)
lifelong learning, (3) opportunity for all, and (4) an informed electorate. To meet these (1989) societal goals, the Standards state further, that: Educational goals for students must reflect the
importance of mathematical literacy. Toward this end, the K-12 standards articulate five general goals for all students: (1) that they learn to value mathematics, (2) that they become confident in
their ability to do mathematics, (3) that they become mathematical problem solvers, (4) that they learn to communicate mathematically, and (5) that they learn to reason mathematically. Points 2
through 3 appear in the first three "process standards" of the document: Math as communications, math as problem solving, and math as reasoning. The fourth process standard mathematical connections
relates to the inter-relatedness of math topics and the connection of math to other disciplines.
Several instructional themes permeate the NCTM Standards: Concrete and problem-centered approaches to teaching math concepts; emphasis on estimation and visualization in realistic contexts; and using
cooperative learning techniques. The Massachusetts ABE Math team found that these practices coupled with the four process standards are completely in harmony with notions of good adult education
practice and so they included these in Massachusetts ABE Math Standards.
The "how" of teaching math is followed by the "what to teach." The NCTM Standards content strands are described for three groups of K-12 learners: K-5; middle grades 6-8; high school level 9-12. The
ABE math team found that the content of much of ABE and GED mathematics fell within the middle-grade math range and so focussed their content standards on those standards. My own view is that today's
GED test covers school mathematics content up to 8th grade. The Massachusetts Numeracy Framework roughly parallels these content stands with seven Numeracy strands.
I find that a standards-based approach to teaching adult basic math fits well with good adult education practice. The approach is learner-centered, involves a solid theory of learning for
understanding, and addresses the wide diversity of cultural background, learning styles, and abilities of the learners whom I teach. And, it addresses math content and skills that are relevant for
the new millenium.
What I take personally from the NCTM Standards is this:
- Learning (and doing) mathematics empowers adult learners;
- Math (and number sense) comes from real life;
- Mathematics is more than arithmetic competency and a set of rules to be memorized;
- Mathematics is investigation, communication, and a way of thinking about the world.
In terms of content, how and what math I teach, the math must be meaningful and connected to adults but also must stretch them beyond where they are. It must be more than teaching computation. And,
since algebra is a "gatekeeper" to entry into and success in further education, my commitment to civil rights and equal opportunity compels me to ensure that adult math instruction includes some
algebra (Moses, 1997).
In My Classroom
The learning of math as well as the doing of math includes moving along a continuum from and among the concrete to the representational to the abstract. "Digits" were born to represent fingers
and toes; to "calculate" originally meant to use stones to count. In actual practice, mathematicians, scientists, technicians, draftsmen, engineers people who use math everyday often use graphic
and concrete models to do math work. So I try to incorporate the use of a "hands-on" curriculum. It starts, as in real life, with concrete models, incorporates graphics and representational
activities, includes, as well, games, writing, and the use of mathematical language and symbolism, and finally, integrates technology.
For example, I use a range of visual models to help learners conceptualize fractions, decimals, and percents. They construct number lines using folded paper, to demonstrate halves, quarters, eighths.
Pie graphs of a day's activities are drawn with colored pencils or developed on computer from spread sheets. I try to teach decimals and percentages at the same time, so that the students can relate
these two concepts. Thus, students use the folded paper which represent fractions to analyze a candy sale's bar graphs, which are calibrated in percentages. They describe the pie graphs in
fractions as well as in percentages. Building on students' experiences with percents in everyday life, we construct the meaning of percents in more complex situations.
But I try to do more with manipulatives than just use them to develop and demonstrate concepts. The blocks or tiles or other concrete things are often themselves part of the problem. I recently
conducted a bean-bag race in the hallway of the learning center where I work. Two students walked along a track, dropping bean-bags every two seconds, while a third student kept time. The rest of the
class, who hadn't viewed it directly, had to look at the bean-bag drops and tell which student walked faster and how they knew. One student spent a number of minutes animatedly explaining to me how
the walks differed and how he had analyzed the situation. In explaining his reasoning, he pointed out the differences in the "proportional" distances between the two sets of bean-bags and how that
translated into different speeds. As he talked, he got very excited with his own understanding and explanation, and exclaimed at the end of his analysis "and that's math!"
From Real Life
Students from other countries use different procedures than found in many adult education texts, particularly for several of the common computation operations such as subtraction and long division
(Schmitt, 1991). Despite this, and although the operations did not make sense as taught to many American-born students the first time around, adult education texts teach only the US algorithm (a rule
or recipe for a mathematical procedure or operation). Standards-based math teaching respects students' thinking, background knowledge, and development of their own algorithms for computation. I try
to teach my students by listening to their explanations of their own thinking and ways of doing math.
One of my GED students, Leo, was a "street smart" learner. He could apply his own experience in playing the numbers to solving the combinatorial problems I posed in class. (For example, "how many
different outfits can you make with three shirts and four pairs of pants?") But he couldn't do a two-digit division problem the "long way," and he felt that would hamper his passing the GED. We spend
about 15 minutes after class one day, talking through a long-division problem. In drawing out his thinking, I found he understood the concept of division as repeated subtraction and urged him to use
that strategy. In the process, he came up with a method of division that made sense to him and which he could articulate and repeat successfully. Although he claims that he now used a method I
"showed" him, it was his own algorithm that he was able to apply confidently in his work, not one that you could find in any GED book.
With my more basic students, those still working on addition and subtraction, I use an investigation of the concepts of carrying and borrowing. Several useful card games, such as Close to 100 and
Close to 0, build on the learner's sense and experience with numbers using 100 and 1000 as benchmarks (see box).
"Closed to 100"
• Using only single-digit cards, deal players hands of six cards.
• Players choose, from their hands, four cards, forming two two-digit
numbers that add up to a number as close to 100 as possible.
• Players keep their own score. Points are the difference between the
sum of the two two-digit numbers as 100.
• Deal seven hands. The player with the lowest total score wins.
(Russell, et al., 1998)
The games gives learners a chance to use numbers in the context of a real-life social situation: a card game. Results can be discussed, strategies shared, and which simulates mental math activities
that adults need for daily life, such as calculating change from a dollar, adding or subtracting percents, making purchases. Besides being fun, it is learning in a social context.
One of my formerly homeless students graciously shared with me many of his strategies for estimation. He often practiced his multiplication skills by estimating the bricks in a building wall, then
counting one-by-one or multiplying row by height to check his number reasoning. Today I give "Elliot's Walk" a true story as a problem to my students to assess their proportional reasoning and
communication skills in using math.
Here it is: Elliot took a walk from his apartment to Harvard Square one day and counted his paces as he walked. He figures that his pace is approximately 2 feet long. He counted as high as 3,000
paces and then stopped counting just as he got to the Square. Approximately how far did he walk?
Investigation, Communication
I try to teach for understanding, using a problem-posing, questioning approach that connects the areas in which learners have strengths. For example, instead of directly teaching my learners to do
these problems:
1. 3x5 + 6
2. 102 + 10x5
3. 25/5 + 35/7
using the PEMDAS or "Please Excuse My Dear Aunt Sally," (parenthesis, exponents, multiplication, division, addition, subtraction ) rule for order of operations, I ask them to generate their own
expressions in my Number of the Day activity. This also gives me a daily assessment of the depth and breadth of my students' grasp of computation, order of operations, and use of symbolic notation. I
write "Number of the Day" on the board, and a box with a number next to it. I ask the students to write as many numerical expressions as they can that equal the number of the day. Student responses
the day the number was 350 included: 300 + 50; 200 + 100 + 25 + 25; 70 x 5; 50 x 6 + 50; 182 + 26; 150x2 + 50.
My adult learners have access in the classroom to appropriate technology: calculators and computers. Rather than being a crutch, calculators are an invaluable aid in teaching basic mathematics
because of their speed, accuracy, decimal display, and memory. Many of my learners already rely on calculators in every day situations as workers and consumers. I use them as instructional tools to
assist in the development of concepts, to help reinforce skills, to promote higher level thinking, and to enhance problem-solving instruction. By freeing them from the routine, long, or complex
calculations, more time can be spent on conjecturing and reasoning.
In Conclusion
It is challenging to change one's practice to the values and practices of the NCTM and ABE Math Standards, particularly when the ideal is not readily shared by other teachers and is not in the
experience of our learners. I find many learners look to me for answers when I'm trying to have them develop that capacity themselves. One of my students complained to his counselor that "at the end
of the day, we're tired from working, and she expects us to think." It is easy to fall back into old methods of direct instruction and worksheets and workbook pages. Also, teaching using the math
standards means covering less material while taking time for discovery. That's hard when students need their GEDs by June. So sometimes this way of teaching brings its own discomforts, to me and to
my students.
The only evidence I have for the success of my approach that "less is more" is that I rarely cover all the material in the GED textbooks in the 14 to 15 weeks of my course. Yet most of my students
seem to have the confidence to take and pass the test. My experience with teaching a range of learners, from literacy students to GED, has convinced me that teaching for learning in the vision of the
math standards is good adult education practice.
Curry, D., Schmitt, M.J., & Waldron, S. (1996). A Framework for Adult Numeracy Standards: The Mathematical Skills and Abilities Adults Need to be Equipped for the Future. Boston, MA: World Education.
Leonelli, E.& Schwendeman, R. (eds.) (1994). The ABE Math Standards Project, Vol 1: The Massachusetts Adult Basic Education Math Standards. Holyoke, MA: Holyoke Community College/SABES Regional
Massachusetts Department of Education (1996) Massachusetts Curriculum Frameworks for K-12 and Adults. Malden, MA: MA DOE.
Moses, R. (1997) "Mathematics, Literacy, Citizenship, and Freedom," presentation at NCTM 75th Annual Meeting, April 19, Minneapolis, MN.
National Council of Teachers of Mathematics (1989). Curriculum and Evaluation Standards for School Mathematics. Reston VA: NCTM.
Russell, S.,J., Tierney, C., Mokros, J., Economopoulos, K., et al. (1998). Investigations in Number, Data, and Space (TM). White Plains, NY: Cuisenaire/Dale Seymour Publications.
Schmitt, M.J., (1991). The Answer is Still the Same It Doesn't Matter How You Got It! A Booklet for Math Teachers and Math Students Who Come from Multicultural Backgrounds. Boston, MA: Author.
About the Author
Esther D. Leonelli is technology coordinator and a math instructor at the Community Learning Center, Cambridge, MA. She is a co-founder and past president of the Adult Numeracy Network, an affiliate
of the National Council of Teachers of Mathematics. Esther moderates the numeracy list, an electronic discussion list for adult education practitioners (numeracy @worldstd.com). During 1998-99 she
was an NIFL Leadership Fellow.
Updated 7/27/07 :: Copyright © 2005 NCSALL
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Hotel Occupancy
Hello! I am trying to build a hotel occupancy program Write a program that calculates the occupancy rate for each floor of a hotel. The program should start by asking for the number of floors that
the hotel has. A loop should then iterate once for each floor. During each iteration, the loop should ask the user for the number of rooms on the floor and the number of them that are occupied. After
all the iterations, the program should display the number of rooms the hotel has, the number of them that are occupied, the number that are vacant, and the occupancy rate for the hotel.
Input Validation: Do not accept a number less than one for the number of floors. Do not accept a number less than 10 for the number of rooms on a floor. The equation is: Occupancy rate = number of
rooms occupied / total number of rooms. As far as the code is concerned, I think it is good except I don't know why my occupied rooms is not display /calculating incorrectly. Here's my code:
import javax.swing.JOptionPane; //needed for GUI
import java.text.DecimalFormat; //needed to format the Output
public class DTHotelOccupancy
{//Begin class
public static void main(String[] args)
{//Begin main method
String input; //To hold the user's input
final int MIN_FLOORS = 1; //Minimum amount of floors
final int MIN_ROOMS = 10; //Minimum amount of rooms per floor
int floors, //Number of floors in hotel
rooms = 0, //Number of available rooms in each floor
occrooms = 0, //Number of rooms occupied
totalrooms = 0, //Number of total rooms
totalroomsocc = 0, //Total Number of rooms occupied
vacant = 0; //Number of vacant rooms
double occrate = 0; //Occupancy rate
DecimalFormat formatter = new DecimalFormat("#,##0.0"); //format the scores
//Get the number of floors in the hotel
input =JOptionPane.showInputDialog("How many Floors are in the Hotel?\n"+
"\tThe number of floors must be at least " + MIN_FLOORS);
//Convert floors into integer
floors = Integer.parseInt(input);
//Validate the number of floors entered.
while (floors < MIN_FLOORS)
input =JOptionPane.showInputDialog("You have Entered a value less than the 1"+
"How many Floors are in the Hotel? "+
"The number of floors must be at least " + MIN_FLOORS);
//Convert floors into integer
floors = Integer.parseInt(input);
//For Loop
for (int i = 1; i <= floors;i++)
input = JOptionPane.showInputDialog("\tHow many rooms are on floor " + i + "? ");
//Convert rooms into integer
rooms = Integer.parseInt(input);
//Validate the number of rooms per floor
while (rooms < 10)
input = JOptionPane.showInputDialog("You have entered the number of rooms less than 10 per floor!"+
"\t\nHow many rooms are on floor " + i + "? ");
//Convert rooms into integer
rooms = Integer.parseInt(input);
input = JOptionPane.showInputDialog("How many rooms on floor " + i + " are occupied? ");
//Convert occupied rooms into integer
occrooms = Integer.parseInt(input);
while (occrooms > rooms)
input = JOptionPane.showInputDialog("You have entered the number of occupied rooms greater than the nummber of rooms per floor!"+
"\t\nHow many rooms are on floor " + i + "? ");
//Convert occupiedrooms into integer
occrooms = Integer.parseInt(input);
totalrooms = (floors * rooms); //Calculate the Number of rooms
totalroomsocc = (totalrooms - occrooms); //Calculate the total number of rooms occupied
vacant = (totalrooms - totalroomsocc); //Calculate the Number of rooms Vacant
occrate = ((double)occrooms / totalrooms); //Calculate OccupancyRate
//Display the results
JOptionPane.showMessageDialog(null, "\tTotal Rooms:" + totalrooms +
"\t\nRooms occupied: " +totalroomsocc+
"\t\nRooms vacant: " +vacant +
"\t\nOccupancy Rate: " + formatter.format(occrate),
//End the program.
}//End main method
}//End class
Please help me out with the calculating for the total rooms occupied
Thank you
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% tutorial.tex - a short descriptive example of a LaTeX document % For additional information see Tim Love's ``Text Processing using LaTeX'' % http://www-h.eng.cam.ac.uk/help/tpl/textprocessing/ %
You may also post questions to the newsgroup comp.text.tex % draft, flee, opening, 12pt % insert PostScript figures % controllabel line spacing %\usepackage[nofiglist,notablist]{endfloat} %\
usepackage{endfloat} %\usepackage{natbib} %\usepackage{epstopdf} %\renewcommand{\efloatseparator}{\mbox{}} % the following produces 1 inch margins all around with no header or footer % beyond 25.mm %
beyond 25.mm % beyond 25.mm %\bibpunct{(}{)}{;}{a}{,}{,} % SOME USEFUL OPTIONS: % indent paragraph by this much % space between paragraphs % \mathindent 20.mm % indent math equations by this much %
post-script figures here or in /. % Helps LaTeX put figures where YOU want % 90% of page top can be a float % 90% of page bottom can be a float % only 10% of page must to be text % make title
footnotes alpha-numeric %An Empirical Model of Spatial Competition with an Application to %Cement % --------------------- end of the preamble --------------------------- \documentclass[12pt]{article}
\usepackage{chicago} \usepackage{graphicx} \usepackage{setspace} \usepackage{amsmath,amsthm,amssymb,amstext} \usepackage{rotating} \usepackage{soul} \usepackage{epsfig} \usepackage{lscape} \
setcounter{MaxMatrixCols}{10} %TCIDATA{OutputFilter=LATEX.DLL} %TCIDATA{Version=5.50.0.2953} %TCIDATA{} %TCIDATA{BibliographyScheme=BibTeX} %TCIDATA{LastRevised=Thursday, April 08, 2010 13:15:05}
%TCIDATA{} \topmargin =15.mm \oddsidemargin =0.mm \evensidemargin =0.mm \headheight =0.mm \headsep =0.mm \textheight =220.mm \textwidth =165.mm \parindent 10.mm \parskip 0.mm \newcommand{\MyTabs}{ \
hspace*{25.mm} \= \hspace*{25.mm} \= \hspace*{25.mm} \= \hspace*{25.mm} \= \hspace*{25.mm} \= \hspace*{25.mm} \kill } \graphicspath{{../Figures/}{../data/:}} \renewcommand{\topfraction}{0.9} \
renewcommand{\bottomfraction}{0.9} \renewcommand{\textfraction}{0.1} \alph{footnote} \input{tcilatex} \begin{document} \begin{abstract} % beginning of the abstract The theoretical literature of
industrial organization shows that the distances between consumers and firms have first-order implications for competitive outcomes whenever transportation costs are large. To assess these effects
empirically, we develop a structural model of competition among spatially differentiated firms and introduce a GMM estimator that recovers the structural parameters with only regional-level data. We
apply the model and estimator to the portland cement industry. The estimation fits, both in-sample and out-of-sample, demonstrate that the framework explains well the salient features of competition.
We estimate transportation costs to be \$0.30 per tonne-mile, given diesel prices at the 2000 level, and show that these costs constrain shipping distances and provide firms with localized market
power. To demonstrate policy-relevance, we conduct counter-factual simulations that quantify competitive harm from a hypothetical merger. We are able to map the distribution of harm over geographic
space and identify the divestiture that best mitigates harm. \end{abstract} % REQUIRED \thispagestyle{empty} \begin{center} ECONOMIC ANALYSIS GROUP DISCUSSION PAPER \vspace{2.8cm} Competition among
Spatially Differentiated Firms: An Empirical Model with an Application to Cement \vspace{0.25cm} By \vspace{0.25cm} Nathan H. Miller* and Matthew Osborne** \\[0pt] EAG 10-2 $\quad$ March 2010 \end
{center} \vspace{0.45cm} \noindent EAG Discussion Papers are the primary vehicle used to disseminate research from economists in the Economic Analysis Group (EAG) of the Antitrust Division. These
papers are intended to inform interested individuals and institutions of EAG's research program and to stimulate comment and criticism on economic issues related to antitrust policy and regulation.
The Antitrust Division encourages independent research by its economists. The views expressed herein are entirely those of the authors and are not purported to reflect those of the United States
Department of Justice or the Bureau of Economic Analysis. \vspace{0.25cm} \noindent Information on the EAG research program and discussion paper series may be obtained from Russell Pittman, Director
of Economic Research, Economic Analysis Group, Antitrust Division, U.S. Department of Justice, LSB 9446, Washington DC 20530, or by e-mail at russell.pittman@usdoj.gov. Comments on specific papers
may be addressed directly to the authors at the same mailing address or at their email addresses. \vspace{0.25cm} \noindent Recent EAG Discussion Paper and EAG Competition Advocacy Paper titles are
listed at the end of this paper. To obtain a complete list of titles or to request single copies of individual papers, please write to Janet Ficco at the above mailing address or at
janet.ficco@usdoj.gov or call (202) 307-3779. In addition, recent papers are now available on the Department of Justice website at http://www.usdoj.gov/atr/public/eag/discussion\_papers.htm.
Beginning with papers issued in 1999, copies of individual papers are also available from the Social Science Research Network at www.ssrn.com. \vspace{0.25cm} \noindent\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\
_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_% \_\_\_\_\_\_\_\_\_ \noindent * \ Economic Analysis Group, Antitrust Division, U.S. Department of Justice. Email: nathan.miller@usdoj.gov. \noindent ** \ Bureau of
Economic Analysis. Email: matthew.osborne@bea.gov. We thank Robert Johnson, Ashley Langer, Russell Pittman, Chuck Romeo, Jim Schmitz, Charles Taragin, Raphael Thomadsen, and seminar participants at
the Bureau of Economic Analysis, Georgetown University, and the U.S. Department of Justice for valuable comments. Sarolta Lee, Parker Sheppard, and Vera Zavin provided research assistance. \newpage \
thispagestyle{empty} % end of the abstract \newpage \thispagestyle{empty} \setcounter{page}{1} \onehalfspacing \section{Introduction} Geography is understudied in the empirical literature of
industrial organization. Although the theoretical literature has established that the physical distances between firms and consumers have first-order implications for competitive outcomes whenever
transportation costs are large (e.g., % \citeN{hotelling29}, \citeN{agt79}, \citeN{salop79}, \citeN{thissevives88}, % \citeN{econ89}, \citeN{vogel08}), the complexities associated with modeling
spatial differentiation have made it difficult to translate theoretical insights into workable empirical models.\footnote{% Surely it would be too strong to claim that, in empirical industrial
organization, space is the final frontier.} Standard empirical methodologies simply sidestep spatial differentiation through the delineation of distinct geographic markets. This simplifies estimation
but requires the dual assumptions that (1) transportation costs are sufficiently large to preclude substantive competition across market boundaries, and (2) transportation costs are sufficiently
small that spatial differentiation is negligible within markets. It can be difficult to meet both conditions.\footnote{\citeN{syverson04} discusses how this tension can compel researchers to seek
compromise between markets that are ``too small'' and markets that are ``too large''. It is sometimes argued that markets that are too large may overstate the intensity of competition while markets
that are too small may understate competition.} In practice, markets are often based on political borders of questionable economic significance (e.g., state or county lines). Nonetheless, market
delineation is employed routinely in studies of industries characterized by high transportation costs, including ready-mix concrete (e.g., \citeN{syverson04}, % \citeN{syversonhortacsu07}, \citeN
{cw09}), portland cement (e.g., % \citeN{salvo08}, \citeN{ryan06}), and paper (e.g., \citeN{pes03}).\footnote{% These valuable contributions focus on wide range of topics, including the competitive
impacts of horizontal and vertical mergers, heterogeneity in plant productivity and its implications for competition, the inference of market power, and dynamic investment decisions.} Our purpose is
to introduce an alternative empirical framework. To that end, we develop and estimate a structural model of competition among spatially differentiated firms that accounts for transportation costs in
a realistic and tractable manner. We focus on production and consumption within a two dimensional Euclidean space, which we refer to informally as a geographic space. Competition involves a discrete
number of plants, each endowed with a physical location, and a continuum of consumers that spans the space. Each plant sets a distinct price to each consumer, taking into consideration its proximity
to the consumer and the proximities of its competitors. Thus, plants discriminate between elastic and inelastic consumers based on the pre-determined plant and consumer locations, and the model
resembles the theoretical work of \citeN{thissevives88}. Competitive outcomes depend on the magnitude of transportation costs and the degree of spatial differentiation within the geographic space. We
discretize the geographic space into small ``consumer areas'' to operationalize the model. Since each plant may ship to each consumer area, the model diverges starkly from more conventional
approaches in which plants do not compete across market boundaries. We employ standard differentiated-product methods to model competition within consumer areas. On the supply side, domestic plants
compete in prices given capacity constraints and the existence of a competitive fringe of foreign importers. On the demand side, consumers select the plant that maximizes utility, taking into
consideration their proximity to the plants, the plant prices, and a nested logit error term. To be clear, the plants are differentiated primarily by price and location -- we assume that the product
itself is homogeneous. We derive an equation that characterizes the equilibrium price and market share for each plant-area pair as a function of data and parameters.\footnote{% We refer to the
fraction of potential demand in a consumer area that is captured by a given plant as the ``market share'' of the plant. We select the term purely for expositional convenience. We do not argue that
consumer areas reflect antitrust markets in any sense; indeed, a defining characteristic of the model is that it avoids market delineation entirely.} The central challenge for estimation is that
prices and market shares are unobserved in the data, at least at the plant-area level. We develop a generalized method of moments (GMM) estimator that exploits variation in data that are more often
observed: regional level consumption, production, and prices. The key insight is that each candidate parameter vector corresponds to an equilibrium set of plant-area prices and market shares. We
compute numerical equilibrium for each candidate parameter vector using a large-scale nonlinear equation solver developed in \citeN{dfsane06}. We then aggregate the predictions of the model to the
regional level and evaluate the distance between the data and the predictions. The estimator can be interpreted as having inner and outer loops: the outer loop minimizes an objective function over
the parameter space while the inner loop computes numerical equilibrium for each candidate parameter vector. We show that the estimator consistently recovers the structural parameters of the data
generating process in an artificial data experiment. %\footnote{We conduct an artificial data %experiment to evaluate the accuracy of the estimator. %The results suggest that the estimator
consistently recovers the structural parameters %of the data generating process.} We apply the model and the estimator to the portland cement industry in the U.S. Southwest over the period 1983-2003.
The choice of industry conveys at least three substantive advantages to the analysis. First, transportation costs contribute substantially to overall consumer costs because portland cement is
inexpensive relative to its weight. Second, it may be reasonable to treat portland cement as a homogenous product because strict industry standards govern the production process.\footnote{% Many
plants produce a number of different types of portland cement, each with slightly different specifications and characteristics (e.g., superior early strength or higher sulfate resistance). These
products are close substitutes for most construction projects.} This conformity matches the simplicity of the demand system, in which spatial considerations (e.g., plant and consumer locations) are
the main source of plant heterogeneity. Third, high quality data on the industry are publicly available. We obtain information on regional consumption, production, and average prices, as well as
limited information on cross-region shipments, from annual publications of the United States Geological Survey. We pair these regional-level metrics with information on the location and
characteristics of portland cement plants from publications of the Portland Cement Association. We exploit variation in these data to estimate the model. The results of estimation suggest that
consumers pay roughly \$0.30 per tonne-mile, given diesel prices at the 2000 level.\footnote{% Strictly speaking, the model identifies consumer willingness-to-pay for proximity to the plant, which
incorporates transportation costs as well as any other distance-related costs (e.g., reduced reliability). We refer to this willingness-to-pay as the transportation cost, although the concepts may
not be precisely equivalent.} Given the shipping distances that arise in numerical equilibrium, this translates into an average transportation cost of \$24.61 per metric tonne over the sample period
-- sufficient to account for 22 percent of total consumer expenditure. Costs of this magnitude have real effects on competition in the industry. We focus on two such effects: First, transportation
costs constrain the distance that cement can be shipped economically. The results indicate that cement is shipped only 92 miles on average between the plant and the consumer; by contrast, a
counter-factual simulation suggests that the average shipment would be 276 miles absent transportation costs. Second, transportation costs insulate firms from competition and provide localized market
power. For instance, the prices that characterize numerical equilibrium decrease systematically in the distance between the plant and the consumer, as do the corresponding market shares.\footnote{%
These patterns are precisely what economic theory would predict given that consumers pay the costs of transportation in the portland cement industry.} The estimation procedure produces impressive
in-sample and out-of-sample fits despite parsimonious demand and marginal cost specifications. The model predictions explain 93 percent of the variation in regional consumption, 94 percent of the
variation in regional production, and 82 percent of the variation in regional prices. The model predictions also explain 98 percent of the variation in cross-region shipments, even though we withhold
the bulk of these data from estimation. As we detail in an appendix, the quality of these fits is underscored by the rich time-series variation in these data. Together, the regression fits suggest
that a small quantity of exogenous data, properly utilized, may be sufficient to explain some of the most salient features of competition in the portland cement industry. We interpret this as
substantial support for the power of the analytical framework. We suspect that our method may prove useful for future research and policy endeavors relating to international trade, environmental
economics, and industrial organization. One such application is merger simulation, an important tool for competition policy. We use counter-factual simulations to evaluate the effects of a
hypothetical merger between two multi-plant cement firms in 1986. We find that the merger reduces consumer surplus by \$1.4 million if no divestitures are made, and we are able to map the
distribution of this harm over the U.S. Southwest. The overall magnitude of the effect is modest relative to the amount of commerce -- by way of comparison, we calculate total pre-merger consumer
surplus to be more than \$239 million. We then consider the six possible single-plant divestitures, and find that the most powerful reduces consumer harm by 56 percent.\footnote{% We refer to the
divesture plan that offsets the greatest amount of consumer harm among the set of single-plant divestitures as the most powerful. We do not attempt to characterize, in any way, the appropriate course
of action for an antitrust authority.} At least two caveats are important. First, the estimation procedure rests on the uniqueness of equilibrium at each candidate parameter vector, but there is no
theoretical reason to expect this condition to hold generally. To assess the issue, we conduct a Monte Carlo experiment that computes numerical equilibrium using several different starting points for
each of 6,300 randomly drawn candidate parameter vectors. The results are strongly supportive of uniqueness, at least in our application (see the appendices for details). Second, the promise of
spatial differentiation creates incentives for firms to locate optimally in order to secure a base of profitable customers, provide separation from an efficient competitor, and/or deter nearby entry.
We abstract from these considerations entirely and instead assume that firm location is pre-determined and exogenous. Nonetheless, our framework could help define stage-game payoffs in more dynamic
models that endogenize firm location choices (e.g., as in % \citeN{seim06} and \citeN{aguirregabiria06}). Our work builds on recent contributions to the industrial organization literature that model
competition among spatially differentiated retail firms (e.g., \citeN{thomadsen051}, \citeN{davis06}, \citeN{mcmanus09}). These papers employ a framework in which each firm sets a single price to all
consumers, consistent with practice in most retail settings, and recover the structural parameters with firm-level data and more standard estimation techniques (e.g., \citeN{blp95}).\footnote{%
Thomadsen (2005) may be the closest antecedent to our work. Thomadsen shows that a supply-side equilibrium condition can be substituted for firm-level market shares in estimation. We develop the
potency of equilibrium conditions more fully: given an equilibrium condition and aggregate data, estimation is feasible with neither firm-level prices nor firm-level market shares. \citeN
{aguirregabiria06} develop a sophisticated model of spatial differentiation but do not take the model to data.} We make two distinct contributions to this literature. First, our model extends the
existing framework to incorporate firms that price discriminate among consumers. In such a setting, firm-level data are no longer sufficient to support standard estimation techniques. Thus, our
second contribution is the demonstration that estimation is feasible with relatively aggregated data. Of course, the regional-level data we use in our application would also support estimation in
simpler retail settings. More generally, our results indicate that firm-level heterogeneity can affect competitive outcomes even when the product is relatively homogenous. Refinements to the standard
toolbox available for structural research in these settings have been outstripped in empirical industrial organization by models of observed and (especially) unobserved product heterogeneity. We
suspect that models of competition for homogenous-product industries are currently of heightened value, not only because of the imbalance in the literature, but also because these industries provide
fertile testing grounds for methodological innovations regarding industry dynamics (e.g., as in \citeN{ryan06}). We hope that the framework we introduce helps, incrementally, to redirect the
attention of researchers to these settings. The paper proceeds as follows. In Section \ref{sec:ind}, we sketch the relevant institutional details of the portland cement industry, focusing on
transportation costs, production technology, and trends in production and consumption. We develop the empirical model of Bertrand-Nash competition in Section \ref{sec:model} and discuss our data
sources in Section \ref% {sec:data}. Then, in Section \ref{sec:est}, we develop the estimator and provide identification arguments. We present the estimation results in Section \ref{sec:res}, discuss
the merger simulations in Section \ref% {sec:sim}, and then conclude. \section{The Portland Cement Industry \label{sec:ind}} Portland cement is a finely ground dust that forms concrete when mixed
with water and coarse aggregates such as sand and stone. Concrete is an essential input to many construction and transportation projects, either as pourable fill material or as pre-formed concrete
blocks, because its local availability and lower maintenance costs make it more economical than substitutes such as steel, asphalt, and lumber (e.g., \citeN{vanoss02}).% \footnote{% We draw heavily
from the publicly available documents and publications of the United States Geological Survey and the Portland Cement Association to support the analysis in this section. We defer detailed discussion
of these sources for expositional convenience.} Most portland cement is shipped by truck to ready-mix concrete plants or construction sites, in accordance with contracts negotiated between individual
purchasers and plants.\footnote{% A smaller portion is shipped by train or barge to terminals, and only then distributed to consumers by truck. Shipment via terminals reduces transportation costs for
more distant consumers. Roughly 23 percent of portland cement produced in the United States was shipped through terminals in 2003.} Transportation costs contribute substantially to overall consumer
expenditures because portland cement is inexpensive relative to its weight, a fact that is well understood in the academic literature. For example, Scherer et al (1975) estimates that transportation
would have accounted for roughly one-third of total consumer expenditures on a hypothetical 350-mile route between Chicago and Cleveland, and a 1977 Census Bureau study determines that most portland
cement is consumed locally -- for example, more than 80 percent is transported within 200 miles.\footnote{% Scherer et al (1975) examined more than 100 commodities and determined that the
transportation costs of portland cement were second only to those of industrial gases. Other commodities identified as having particularly high transportation costs include concrete, petroleum
refining, alkalies/chlorine, and gypsum.} More recently, \citeN{salvo08b} presents evidence consistent with the importance of transportation costs in the Brazilian portland cement industry. A recent
report prepared for the Environmental Protection Agency identifies five main variable input costs of production: raw materials, coal, electricity, labor, and kiln maintenance (\citeN{epa09}). In the
production process itself, a feed mix composed of limestone and supplementary materials is fed into large rotary kilns that reach temperatures of 1400-1450$^\circ$ Celsius. The combustion of coal is
the most efficient way to generate this extreme heat. Kilns generally operate at peak capacity with the exception of an annual shutdown period for maintenance. It is possible to adjust output by
extending or shortening the maintenance period -- for example, plants may simply forego maintenance at the risk of kiln damage and/or breakdowns. The feed mix exits the kiln as semi-fused clinker
modules. Once cooled, the clinker is mixed with a small amount of gypsum, placed into a grinding mill, and ground into tiny particles averaging ten micrometers in diameter. This product -- portland
cement -- is shipped to purchasers either in bulk or packaged in smaller bags. We focus on the production and consumption of portland cement in the U.S. Southwest -- by which we mean California,
Arizona and Nevada -- over the period 1983-2003. This region accounts for roughly 15 percent of domestic portland cement production and consumption during the sample period. Figure % \ref{fig:map1}
provides a map of the region based on the plant locations in the final year of the sample. As shown, most plants are located along an interstate highway and nearby one or more population centers.
Although some firms own more than one plant, production capacity in the area is not particularly concentrated -- the capacity-based Herfindahl-Hirschman Index (HHI) of 1260 is well below the
threshold level that defines highly concentrated markets in the 1992 Merger Guidelines. The plants also face competition from foreign importers that ship portland cement through the customs offices
of San Francisco, Los Angeles, San Diego, and Nogales. Still, transportation costs may insulate some plants from both foreign and domestic competition.\footnote{% We observe little entry and exit
over the sample period. The sole entrant (Royal Cement) began operations in 1994 and the only two exits occurred in 1988. This stability is consistent with substantial sunk costs of plant
construction, as documented in \citeN{ryan06}. Plant ownership is somewhat more fluid; we observe fourteen changes in plant ownership, spread among nine of the sixteen plants.} \begin{figure}[t] \
centering \includegraphics[height=4in]{map-col-hwy-2.eps} \caption{{\protect\footnotesize {Portland Cement Production Capacity in the U.S. Southwest circa 2003. }}} \label{fig:map1} \end{figure} %
The Figure is a map of California, Arizona, and Nevada. Cement plants are marked with circles, with % higher capacity plants getting bigger circles. The four import points -- San Francisco, Los
Angeles, % San Diego, and Nogales -- are marked with black squares. See text for details. In Figure \ref{fig:imports}, we plot total consumption and production in the U.S. Southwest over the sample
period, together with two measures of foreign imports. Several patterns are apparent. Both consumption and production increase over 1983-2003, and both metrics are highly cyclical. However,
consumption is more cyclical than production, so that the gap between consumption and production increases in overall activity; foreign importers provide additional supply whenever domestic demand
outstrips domestic capacity. Strikingly, observed foreign imports are nearly identical to ``apparent imports,'' which we define as consumption minus production, consistent with negligible net trade
between the U.S. Southwest and other domestic areas. Finally, we note that the average free-on-board price charged by domestic plants in the region (not shown) falls over the sample period from \$107
per metric tonne to \$74 per metric tonne, primarily due to lower coal and electricity costs.\footnote{% Prices are in real 2000 dollars.} \begin{figure}[t] \centering \includegraphics[height=3in]
{imports.eps} \caption{{\protect\footnotesize {Consumption, Production, and Imports of Portland Cement. Apparent imports are defined as consumption minus production. Observed imports are total
foreign imports shipped into San Francisco, Los Angeles, San Diego, and Nogales.}}} \label{fig:imports} \end{figure} %The figure plots consumption, production, and imports over the sample period. It
also plots the apparent imports, %defined as consumption minus production. The text describes the relevant data patterns in detail. \section{Empirical Model \label{sec:model}} \subsection{Overview}
We develop a structural model of competition that accounts for transportation costs in a realistic and tractable manner. We focus on production and consumption within a single geographic space.
Competition involves a discrete number of plants, each endowed with a physical location, and a continuum of consumers that spans the space. Each plant sets a distinct price to each consumer, taking
into consideration its proximity to the consumer and the proximities of its competitors. To operationalize the model, we discretize the geographic space into small consumer areas. We employ standard
techniques to model competition within consumer areas: plants charge area-specific prices subject to exogenous capacity constraints, and consumer demand is nested logit. We derive an equation that
characterizes the equilibrium price and market share for each plant-area pair. \subsection{Supply} We take as given that there are $F$ firms, each of which operates some subset $\Im_f$ of the $J$
plants. Each plant is endowed with a set of attributes, including a physical location. Consumers exist in $N$ different areas that span the geographic space. Every plant can ship into every area.
Firms set prices at the plant-area level to maximize variable profits: $$\pi_{f} = \underbrace{\sum_{j \in \Im_{f}} \sum_{n} P_{jn}Q_{jn}(\boldsymbol{% \theta_d})}_{\text{variable revenues}} - \
underbrace{\sum_{j \in \Im_{f}} \int_0^{Q_j (\boldsymbol{\theta_d})}MC_j(Q; \boldsymbol{\theta_c}) dQ}_{% \text{variable costs}}, \label{eq:vprofs}$$ where $Q_{jn}$ and $P_{jn}$ are the quantity and
price, respectively, of plant $j$ in area $n$, $MC_j$ is the marginal cost of production, and $Q_j$ is total plant production (i.e., $Q_j=\sum_n Q_{jn}$). The vectors $% \boldsymbol{\theta_c}$ and $\
boldsymbol{\theta_d}$ include the supply and demand parameters, respectively, and together form the joint parameter vector $\boldsymbol{\theta} = (\boldsymbol{\theta_c}, \boldsymbol{\theta_d})$% .
The marginal cost function can accommodate most forms of scale economies or diseconomies. We choose a functional form that captures the plant-level capacity constraints that are important for
portland cement production. In particular, we let marginal costs increase flexibly in production whenever production exceeds some threshold level. The marginal cost function is: $$MC(\boldsymbol{w}_
{j},Q_{j}(\boldsymbol{\theta_d}); \boldsymbol{\theta_c}) = \boldsymbol{w}_{j}^\prime \boldsymbol{\alpha} + \gamma \; 1 \left\{ \frac{% Q_j(\boldsymbol{\theta_d})}{CAP_j}>u \right\} \left(\frac{Q_{j}
(% \boldsymbol{\theta_d})}{CAP_{j}} - u \right) ^\phi , \label{eq:mc}$$ where $\boldsymbol{w_j}$ is a vector that includes the relevant marginal cost shifters and $CAP_j$ is the maximum quantity that
plant $j$ can produce. The parameter $\nu$ is the utilization threshold above which marginal costs increase in production, the parameter $\phi$ determines the curvature of the marginal cost function
when utilization exceeds $\nu$, and the combination $\gamma(1-\nu)^\phi$ represents the marginal cost penalty associated with production at capacity. The marginal cost function is continuously
differentiable in production for $\phi>1$. The vector of cost parameters is $\boldsymbol{\theta_c}= (\boldsymbol{\alpha}, \gamma, \nu, \phi)$. We let the domestic firms compete against a competitive
fringe of foreign importers, which we denote as plant $J+1$. We assume that the fringe is a non-strategic actor and that import prices are exogenously set based on some marginal cost common to all
importers. The fringe is endowed with one or more geographic locations and ships into every consumer area. It sets a single price across all consumer areas (i.e., the fringe does not price
discriminate), consistent with perfect competition among importers. \subsection{Demand} We specify a nested logit demand system that captures the two most important characteristics that differentiate
portland cement plants: price and location. To that end, we assume that each area features many potential consumers. Each consumer either purchases cement from one of the domestic plants or the
importer, or foregoes a purchase of portland cement altogether. We refer to the domestic plants and the importer as the inside goods, and refer to the option to forego a purchase as the outside good.
We place the inside goods in a separate nest from the outside good. We express the indirect utility that consumer $i$ receives from plant $j$ as a function of the relevant plant and location
observables: $$u_{ij} = \beta_0 + \boldsymbol{x}_{jn}^\prime \boldsymbol{\beta} + \zeta_i + \lambda \epsilon_{ij}, \label{eq:util}$$ where the vector $\boldsymbol{x}_{jn}$ includes the price of
cement and the distance between the plant and the area, as well as other plant-specific demand shifters. The idiosyncratic portion of the indirect utility function is composed of consumer-specific
shocks to the desirability of the inside good ($\zeta_i$) and the desirability of each plant ($\epsilon_{it}$). We assume that the combination $\zeta_i + \lambda \epsilon_{ij}$ has an extreme value
distribution in which the parameter $\lambda$ characterizes the extent to which valuations of the inside good are correlated across consumers.% \footnote{\citeN{cardell97} derives the conditions
under which the specified taste shocks produce the extreme value distribution. Tastes are perfectly correlated if $\lambda=0$ and tastes are uncorrelated if $\lambda=1$. In the latter case the model
collapses to a standard logit.} We normalize the mean utility of the outside good to zero, so that the indirect utility associated with foregoing cement purchase is $u_{i0}=\epsilon_{i0}$, with $\
epsilon_{i0} $ also having the extreme value distribution. The vector of demand parameters is $\boldsymbol{\theta_d}=(\beta_0, \boldsymbol{\beta}, \lambda)$.% \footnote{% We exclude product
characteristics from the specification because portland cement is a homogenous good, at least to a first approximation. It may be desirable to control for observed product characteristics in other
applications. (Though the presence of unobserved characteristics would pose a challenge to the estimation procedure.) Also, we assume that consumers pay the cost of transportation, consistent with
practice in the cement industry. One could alternatively incorporate distance into the marginal cost function.% } The nested logit structure yields an analytical expression for the market shares
captured by each plant. The market shares are specific to each plant-area pair because the relative desirability of each plant varies across areas: $$S_{jn}(\boldsymbol{P}_{n}; \boldsymbol{\theta_d})
= \frac{\exp(\beta_0+ \lambda I_{n})}{1+\exp(\beta_0+\lambda I_{n})} * \frac{\exp(\boldsymbol{x}% _{jn}^\prime \boldsymbol{\beta})}{\sum_{k=1}^{J+1} \exp(\boldsymbol{x}% _{kn}^\prime \boldsymbol{\
beta})}, \label{eq:shares}$$ where $\boldsymbol{P}_n$ is a vector of area-specific prices and $I_{n}= \ln \left( \sum_{k=1}^{J+1} \exp(\boldsymbol{x}_{kn}^\prime \boldsymbol{\beta}) \right)$ is the
inclusive value of the inside goods. The first factor in this expression is the marginal probability that a consumer in area $n$ selects an inside good, and the second factor is the conditional
probability that the consumer purchases from plant $j$ given selection of an inside good. Both take familiar logit forms due to the distributional assumptions on idiosyncratic consumer tastes. The
demand system maps cleanly into supply: the quantity sold by plant $j$ to area $n$ is $Q_{jn} = S_{jn}M_n$, where $M_n$ is the potential demand of area $n$.\footnote{% The substitution patterns
between cement plants are characterized by the independence of irrelevant alternatives (IIA) within the inside good nest. We argue that IIA is a reasonable approximation for our application. Portland
cement is purchased nearly exclusively by ready-mix concrete plants and other construction companies. These firms employ similar production technologies and compete under comparable demand
conditions. We are therefore skeptical that meaningful heterogeneity exists in consumer preferences for plant observables (e.g., price and distance). Without such heterogeneity, the IIA property
arises quite naturally -- for example, the random coefficient logit demand model collapses to standard logit when the distribution of consumer preferences is degenerate.} \subsection{Equilibrium} The
profit function yields the first-order conditions that characterize equilibrium prices for each plant-area pair: $$Q_{jn} + \sum_{n} \sum_{k \in \Im_{f}} (P_{kn}-MC_{kn})\frac{\partial Q_{kn}% }{\
partial P_{jn}}=0.$$ Since each plant competes in every consumer area and may price differently across areas, there are $J\times N$ first-order conditions. For notational convenience, we define the
block-diagonal matrix $\boldsymbol{\Omega(P)}$ as the combination of $n=1,\dots,N$ sub-matrices, each of dimension $J \times J$% . The elements of the sub-matrices are defined as follows: $$\Omega^n_
{jk}(\boldsymbol{P}) = \left\{ \begin{array}{ll} \frac{\partial Q_{jn}}{\partial P_{kn}} & \text{if j and k have the same owner} \\ 0 & \text{otherwise},% \end{array} \right. \label{eq:omega}$$ where
the demand derivatives take the nested logit forms. Thus, the elements of each sub-matrix $\Omega^n_{jk}(\boldsymbol{P})$ characterize substitution patterns within area $n$, and the matrix $\
boldsymbol{\Omega(P)}$ has a block diagonal structure because prices in one area do not affect demand in other areas. We can now stack the first-order conditions: $$\boldsymbol{P} = \boldsymbol{MC
(P)} - \boldsymbol{\Omega(P)}^{-1}\boldsymbol{% Q(P)}, \label{eq:equil}$$ where $\boldsymbol{P}$, $\boldsymbol{MC(P)}$, and $\boldsymbol{Q(P)}$ are vectors of prices, marginal costs and quantities,
respectively. Provided that marginal cost parameter $\phi$ exceeds one, the mappings $\boldsymbol{% MC(P)}$, $\boldsymbol{\Omega(P)}$, and $\boldsymbol{Q(P)}$ are each continuously differentiable.
Further, the price vector belongs to a compact set in which each price is (1) greater or equal to the corresponding marginal cost, and (2) smaller than or equal to the price a monopolist would charge
to the relevant area. Therefore, by Brower's fixed-point theorem, Bertrand-Nash equilibrium exists and is characterized by the price vector $% \boldsymbol{P^*}$ that solves Equation \ref{eq:equil}.\
footnote{% The existence proof follows \citeN{aguirregabiria06}. Multiple equilibria may exist.} Spatial price discrimination is at the core of the firm's pricing problem: firms maximize profits by
charging higher prices to nearby consumers and consumers without a close alternative. Aside from price discrimination, the firm's pricing problem follows standard intuition. For example, a firm that
contemplates a higher price for cement from plant $j$ to area $n$ must evaluate a number of effects: (1) the tradeoff between lost sales to marginal consumers and greater revenue from inframarginal
consumers; (2) whether the firm would recapture lost sales with its other plants; and (3) whether the lost sales would ease capacity constraints and make the plant more competitive in other consumer
areas. \section{Data Sources and Summary Statistics \label{sec:data}} \subsection{Data sources} We cull the bulk of our data from the Minerals Yearbook, an annual publication of the U.S. Geological
Survey (USGS). The Minerals Yearbook is based on an annual census of portland cement plant and contains regional-level information on portland cement consumption, production, and free-on-board
prices.\footnote{% The census response rate is typically well over 90 percent (e.g., 95 percent in 2003), and USGS staff imputes missing values for the few non-respondents based on historical and
cross-sectional information. Other academic studies that feature these data include \citeN{mcbride83}, \citeN{rosenbaumreading88}% , \citeN{rosenbaum94}, \citeN{jansrosenbaum96}, \citeN{ryan06}, and
% \citeN{syversonhortacsu07}. The Minerals Yearbook provides the average free-on-board price charged the plants located in each region, rather than the price paid by the consumers in each region.}
The four relevant regions include Northern California, Southern California, Arizona, and Nevada. We observe annual consumption in each region over the period 1983-2003. The USGS combines the Arizona
and Nevada regions when reporting production and prices over 1983-1991, and no usable production or price information is available for Nevada over 1992-2003.\footnote{% The USGS combines Nevada with
Idaho, Montana and Utah starting in 1992. We adjust the Arizona data to remove the influence of a single plant located in New Mexico whose production is aggregated into the region. We detail this
adjustment in an appendix. Also, it is worth noting that the USGS does not intend for its regions to approximate geographic markets. Rather, the regions are delineated such that plant-level
information cannot be backward-engineered from the Minerals Yearbook.} The Minerals Yearbook also includes information on the price and quantity of portland cement that is imported into the U.S.
Southwest via the customs offices in San Francisco, Los Angeles, San Diego, and Nogales. We also make use of more limited data on cross-region shipments from the California Letter, a second annual
publication of the USGS. The California Letter provides information on the quantity of portland cement that is shipped from plants in California to consumers in Northern California, Southern
California, Arizona, and Nevada. However, the level of aggregation varies over the sample period, some data are redacted to protect sensitive information, and no information is available before 1990.
In total, we observe 96 data points: \begin{tabular}{p{2.7in}p{2.7in}} & \end{tabular} % CA to N. CA over 1990-2003 S. CA to N. CA over 1990-1999 % CA to S. CA over 2000-2003 S. CA to S. CA over
1990-1999 % CA to AZ over 1990-2003 S. CA to AZ over 1990-1999 % CA to NV over 2000-2003 S. CA to NV over 1990-1999 % N. CA to N. CA over 1990-1999 N. CA to AZ over 1990-1999 \noindent We withhold
the bulk of these data from the estimation procedure and instead use the data to conduct out-of-sample checks on the model predictions.\footnote{% The California Letter is based on a monthly survey
rather than on the annual USGS census. As a result, the data are not always consistent with the Minerals Yearbook. We normalize the data prior to estimation so that total shipments equal total
production in each year.} We supplement the USGS data with basic plant-level information from the Plant Information Survey (PIS), an annual publication of the Portland Cement Association. The PIS
provides the location of each portland cement plant in the United States, together with its owner, the annual kiln capacity, and various other kiln characteristics.\footnote{% We multiply kiln
capacity by 1.05 to approximate cement capacity, consistent with the industry practice of mixing clinker with a small amount of gypsum (typically 3 to 7 percent) in the grinding mills.} We
approximate consumer areas using counties, which meet the criterion of being small relative to the overall geographic space -- there are 90 counties in the U.S. Southwest. We collect county-level
data from the Census Bureau on construction employment and residential construction permits to account for county-level heterogeneity in potential demand. Finally, we collect data on diesel, coal,
and electricity prices from the Energy Information Agency, data on average wages of durable good manufacturing employees from the BEA, and data on crushed stone prices from the USGS; we exploit
state-level variation for all but the diesel data. \subsection{Summary statistics} We provide summary statistics on consumption, production, and prices for each of the regions in Table \ref
{tab:swData}. Some patterns stand out: First, substantial variation in each metric is available, both inter-temporally and across regions, to support estimation. Second, Southern California is larger
than the other regions, whether measured by consumption or production. Third, consumption exceeds production in Northern California, Arizona, and Nevada; these shortfalls must be countered by
cross-region shipments and/or imports. The observation that plants in these regions charge higher prices is consistent with transportation costs providing some degree of local market power.\footnote
{% The data on cross-region shipments are also suggestive of large transportation costs. For example, more than 90 percent of portland cement produced in Northern California was shipped to consumers
in Northern California over 1990-1999.} Finally, imports are less expensive than domestically produced portland cement. This discrepancy may exist in part because the reported prices exclude duties;
more speculatively, domestic producers may be more reliable or may maintain relationships with consumers that support higher prices. \begin{table}[tbp] \caption{Consumption, Production, and Prices} \
begin{center} \begin{tabular}[h]{lcccc} \hline\hline \rule[0mm]{0mm}{5.5mm}Description & Mean & Std & Min & Max \\ \hline \rule[0mm]{0mm}{5.5mm}\textit{Consumption} & & & & \\ $\;$ Northern
California & 3,513 & 718 & 2,366 & 4,706 \\ $\;$ Southern California & 6,464 & 1,324 & 4,016 & 8,574 \\ $\;$ Arizona & 2,353 & 650 & 1,492 & 3,608 \\ $\;$ Nevada & 1,289 & 563 & 416 & 2,206 \\ \rule
[0mm]{0mm}{5.5mm}\textit{Production} & & & & \\ $\;$ Northern California & 2,548 & 230 & 1,927 & 2,894 \\ $\;$ Southern California & 6,316 & 860 & 4,886 & 8,437 \\ $\;$ Arizona-Nevada & 1,669 & 287 &
1050 & 2,337 \\ \rule[0mm]{0mm}{5.5mm}\textit{Domestic Prices} & & & & \\ $\;$ Northern California & 85.81 & 11.71 & 67.43 & 108.68 \\ $\;$ Southern California & 82.81 & 16.39 & 62.21 & 114.64 \\ $\;
$ Arizona-Nevada & 92.92 & 14.24 & 75.06 & 124.60 \\ \rule[0mm]{0mm}{5.5mm}\textit{Import Prices} & & & & \\ $\;$ U.S. Southwest & 50.78 & 9.30 & 39.39 & 79.32 \\ \hline \multicolumn{5}{p{3.8in}}{{\
footnotesize {\rule[0mm]{0mm}{0mm}Statistics are based on observations at the region-year level over the period 1983-2003. Production and consumption are in thousands of metric tonnes. Prices are per
metric tonne, in real 2000 dollars. Import prices exclude duties. The region labeled ``Arizona-Nevada" incorporates information from Nevada plants only over 1983-1991.}}}% \end{tabular}% \end{center}
\end{table} In Table \ref{tab:swDistData}, we explore the spatial characteristics of the regions in more depth, based on the plant locations of 2003. First, the average county in Northern California
is 65 miles from the nearest domestic cement plant. Since the comparable statistics for Southern California, Arizona, and Nevada are 72 miles, 92 miles, and 100 miles, respectively, one may infer
that average transportation costs may differ substantively across the four regions. Second, the average \textit{additional} distance to the second closest domestic plant is 44 miles in Northern
California, 11 miles in Southern California, 82 miles in Arizona, and 77 miles in Nevada, suggestive that plants in Arizona and Nevada may hold more local market power than plants elsewhere. Finally,
the average distance to the nearest customs office is 123 miles in Northern California, 110 miles in Southern California, 181 miles in Arizona, and 281 miles in Nevada, suggestive that imports may
constrain domestic prices less severely in Arizona and Nevada. The latter two empirical patterns are consistent with the higher cement prices observed in Arizona and Nevada. \begin{table}[tbp] \
caption{Distances between Counties and Plants} \begin{center} \begin{tabular}[h]{lcccc} \hline\hline \rule[0mm]{0mm}{5.5mm}Description & Mean & Std & Min & Max \\ \hline \multicolumn{5}{l}{\rule[0mm]
{0mm}{5.5mm}\textit{Miles to the closest plant}} \\ $\;$ Northern California & 64.65 & 30.00 & 7.36 & 115.39 \\ $\;$ Southern California & 72.28 & 39.74 & 18.58 & 127.46 \\ $\;$ Arizona & 91.62 &
40.01 & 29.13 & 163.99 \\ $\;$ Nevada & 100.04 & 61.97 & 17.38 & 232.03 \\ \multicolumn{5}{l}{\rule[0mm]{0mm}{5.5mm}\textit{Additional miles to the second closest plant}} \\ $\;$ Northern California
& 43.95 & 49.84 & 0.49 & 176.07 \\ $\;$ Southern California & 11.22 & 8.83 & 0.49 & 31.08 \\ $\;$ Arizona & 81.96 & 55.65 & 0.69 & 172.38 \\ $\;$ Nevada & 77.38 & 43.28 & 12.82 & 177.20 \\ \
multicolumn{5}{l}{\rule[0mm]{0mm}{5.5mm}\textit{Miles to the closest import point}} \\ $\;$ Northern California & 122.65 & 67.42 & 4.33 & 283.00 \\ $\;$ Southern California & 110.17 & 67.43 & 30.29 &
221.28 \\ $\;$ Arizona & 180.80 & 90.26 & 14.07 & 314.91 \\ $\;$ Nevada & 281.45 & 81.93 & 170.09 & 442.02 \\ \hline \multicolumn{5}{p{3.8in}}{{\footnotesize {\rule[0mm]{0mm}{0mm}Distances are
calculated based on plant locations in 2003. There are 46 counties in Northern California, 12 counties in Southern California, 15 counties in Arizona, and 17 counties in Nevada.}}}% \end{tabular}% \
end{center} \end{table} \section{Estimation \label{sec:est}} \subsection{Overview} The challenge for estimation is that prices and market shares are unobserved in the data, at least at the plant-area
level. We develop a GMM estimator that exploits variation in data that are more often observed: regional level consumption, production, and prices. The key insight is that each candidate parameter
vector corresponds to an equilibrium set of plant-area prices and market shares. We compute equilibrium numerically for each candidate parameter vector, aggregate predictions of the model to the
regional-level, and then evaluate the ``distance'' between the data and the aggregate predictions. The estimation routine is an iterative procedure defined by the following steps: \begin{enumerate} \
item Select a candidate parameter vector $\boldsymbol{\widetilde{\theta}}$. \item Compute the equilibrium price and market share vectors. \item Calculate regional-level metrics based on the
equilibrium vectors. \item Evaluate the regional-level metrics against the data. \item Update $\boldsymbol{\widetilde{\theta}}$ and repeat steps 1-5 to convergence. \end{enumerate} The estimation
procedure can be interpreted as having both an inner loop and an outer loop: the inner loop computes equilibrium for each candidate parameter vector and the outer loop minimizes an objective function
over the parameter space. We discuss the inner loop and the outer loop in turn, and then address some additional details. \subsection{Computation of numerical equilibrium} We use a large-scale
nonlinear equation solver developed in \citeN{dfsane06} to compute equilibrium. The equation solver employs a quasi-Newton method and exploits simple derivative-free approximations to the Jacobian
matrix; it converges more quickly than other algorithms and does not sacrifice precision. We define a numerical Bertrand-Nash equilibrium as a price vector for which $\parallel \boldsymbol{\iota(P)}
\parallel / \; \text{dim}(% \boldsymbol{\iota(P)}) < \delta$, where $\parallel \cdot \parallel$ denotes the Euclidean norm operator and $$\boldsymbol{\iota(P)} = \boldsymbol{\Omega(P)}(\boldsymbol{P}
-\boldsymbol{% MC(P)}) - \boldsymbol{Q(P)}.$$ We denote the equilibrium prices that correspond to the candidate parameter vector $\boldsymbol{\widetilde{\theta}}$ as $\widetilde{P}_{jnt}(\boldsymbol
{% \widetilde{\theta}},\boldsymbol{\chi})$, where $\boldsymbol{\chi}$ includes the exogenous data. We denote the corresponding equilibrium market shares as $\widetilde{S}_{jnt}(\boldsymbol{\widetilde
{\theta}},\boldsymbol{\chi})$. From a computational standpoint, our construction of $\boldsymbol{\iota(P)}$ avoids the burden of inverting $\boldsymbol{\Omega(P)}$ that would be required by the
straight application of Equation \ref{eq:equil}. Further, the structure of the problem permits us to compute equilibrium separately for each period. The price vector that characterizes the
equilibrium of a given period has length $J_t \times N$ so that, for example, the equilibrium price vector for 2003 has $14 \times 90 =1,260$ elements.\footnote{% We set $\delta=$1e-13. Numerical
error can propagate into the outer loop when the inner loop tolerance is substantially looser (e.g., 1e-7), which slows overall estimation time and can produce poor estimates. The inner loop
tolerance is not unit free and must be evaluated relative to the price level. We also note that, in settings characterized by constant marginal costs, one could ease the computational burden of the
inner loop by solving for equilibrium prices in each consumer area separately.} The empirical analogs to the computed plant-area prices and market shares are not observed in the data, so we aggregate
the prices and market shares to construct more useful regional metrics. For notational precision, we define the sets $\aleph_r$ and $\jmath_r$ as the counties and plants, respectively, located in
region $r$. The aggregated region-period metrics take the form: \begin{eqnarray} \widetilde{C}_{rt}(\boldsymbol{\widetilde{\theta}},\boldsymbol{\chi}% )&=&\sum_{n\in\aleph_r}\left(1-\widetilde{S}_
{0nt}(\boldsymbol{\widetilde{% \theta}},\boldsymbol{\chi})\right)M_{nt} \notag \\ \widetilde{Q}_{rt}(\boldsymbol{\widetilde{\theta}},\boldsymbol{\chi}% )&=&\sum_{j\in\jmath_r}\widetilde{S}_{jnt}(\
boldsymbol{\widetilde{\theta}},% \boldsymbol{\chi})M_{nt} \\ \widetilde{P}_{rt}(\boldsymbol{\widetilde{\theta}},\boldsymbol{\chi}% )&=&\sum_{j\in\jmath_r}\sum_n\frac{\widetilde{S}_{jnt}(\boldsymbol{%
\widetilde{\theta}},\boldsymbol{\chi})M_{nt}}{\sum_{j\in\jmath_r}\sum_n% \widetilde{S}_{jnt}(\boldsymbol{\widetilde{\theta}},\boldsymbol{\chi})M_{nt}}% \widetilde{P}_{jnt}(\boldsymbol{\widetilde{\
theta}},\boldsymbol{\chi}), \notag \end{eqnarray} where $\widetilde{C}_{rt}(\boldsymbol{\widetilde{\theta}},\boldsymbol{\chi})$% , $\widetilde{Q}_{rt}(\boldsymbol{\widetilde{\theta}},\boldsymbol{\
chi})$, and $\widetilde{P}_{rt}(\boldsymbol{\widetilde{\theta}},\boldsymbol{\chi})$ are total consumption, total production, and weighted-average price, respectively. We calculate regional
consumption for each of the four regions in the data -- Northern California, Southern California, Arizona, and Nevada. We calculate production and prices for Northern California, Southern California,
and a combined Arizona-Nevada region. We denote the empirical analogs $C_{rt}$, $Q_{rt}$, and $P_{rt}$. We find that, in practice, we can better identify some of the model parameters by exploiting
information on aggregated cross-region shipments. We denote the quantity of shipments from region $r$ to region $s$ as $% \widetilde{Q}^s_{rt}(\boldsymbol{\widetilde{\theta}},\boldsymbol{\chi})$. The
shipments take the form: \begin{eqnarray} \widetilde{Q}^s_{rt}(\boldsymbol{\widetilde{\theta}},\boldsymbol{\chi}% )&=&\sum_{j\in\jmath_r}\sum_{n\in\aleph_s}\widetilde{S}_{jnt}(\boldsymbol{% \
widetilde{\theta}},\boldsymbol{\chi})M_{nt}, \end{eqnarray} We calculate the quantity of portland cement produced by plants in California (both Northern and Southern) that is shipped to consumers in
Northern California. The empirical analog, which we denote $Q^s_{rt}$, is available over the period 1990-2003. We withhold data on other cross-region shipments from estimation because the number of
parameters that we estimate exceeds the lengths of the available time-series. The withheld data, however, provide natural out-of-sample tests on the model predictions. \subsection{Objective function}
We estimate the parameters using the standard GMM framework for systems of nonlinear regression equations (e.g., \citeN{greene03}, page 369). The equations we use compare the metrics computed in the
inner loop to their empirical analogs: \begin{eqnarray} \boldsymbol{C}_r&=&\boldsymbol{\widetilde{C}}_r(\boldsymbol{\theta},% \boldsymbol{\chi})+\boldsymbol{e_r^1} \notag \\ \boldsymbol{Q}_r&=&\
boldsymbol{\widetilde{Q}}_r(\boldsymbol{\theta},% \boldsymbol{\chi})+\boldsymbol{e_r^2} \\ \boldsymbol{P}_r&=&\boldsymbol{\widetilde{P}}_r(\boldsymbol{\theta},% \boldsymbol{\chi})+\boldsymbol{e_r^3}
\notag \\ \boldsymbol{Q}^s_{r}&=&\boldsymbol{\widetilde{Q}}^s_r(\boldsymbol{\theta},% \boldsymbol{\chi})+\boldsymbol{e_{rs}^4}. \notag \label{eq:moments} \end{eqnarray} We write the equations in
vector form with one element per period. There are four consumption moments, three production moments, three price moments, and one cross-region shipments moment. We have 21 observations on each of
the consumption, production and price moments, and 14 observations on the cross-region shipments moment. We interpret the disturbances as measurement error, and assume the disturbances to have
expectation zero and contemporaneous covariance matrix $\boldsymbol{\Sigma}$. The GMM estimator is: $$\boldsymbol{\widehat{\theta}} = \arg \min_{\boldsymbol{\Theta}} \boldsymbol{e% }(\boldsymbol{\
widetilde{\theta}},\boldsymbol{\chi})^\prime \boldsymbol{A}% ^{-1} \boldsymbol{e}(\boldsymbol{\widetilde{\theta}},\boldsymbol{\chi}),$$ where $\boldsymbol{e}(\boldsymbol{\widetilde{\theta}},\
boldsymbol{\chi})$ is a vector of empirical disturbances obtained by stacking the nonlinear regression equations and $\boldsymbol{A}$ is a positive definite weighting matrix. We employ the usual
two-step procedure to obtain consistent and efficient estimates (\citeN{hansen82}). We first minimize the objective function using $\boldsymbol{A}=\boldsymbol{I}$. We then estimate the
contemporaneous covariance matrix and minimize the objective function a second time using the weighting matrix $\boldsymbol{A} = \boldsymbol{% \widehat{\Sigma}} \otimes\boldsymbol{I}.$ We compute
standard errors that are robust to both heteroscedasticity and arbitrary correlations among the error terms of each period, using the methods of \citeN{hansen82} and % \citeN{neweymcfadden94}.\
footnote{% Measurement error that is zero in expectation is sufficient for consistency. Estimation of the contemporaneous covariance matrix $\boldsymbol{\Sigma}$ is complicated by the fact that we
observe consumption, production, and prices over 1983-2003 but cross-region shipments over 1990-2003. We use methods developed in \citeN{sz73} and \citeN{hhs90} to account for the unequal numbers of
observations.} \subsection{Potential demand} We normalize the potential demand of each county using two exogenous demand predictors that we observe at the county level: the number of construction
employees and the number of new residential building permits. We regress regional portland cement consumption on the demand predictors (aggregated to the regional level), impute predicted consumption
at the county level based on the estimated relationships, and then scale predicted consumption by a constant of proportionality to obtain potential demand.\footnote{% The regression of regional
portland cement consumption on the demand predictors yields an $R^2$ of 0.9786, which foreshadows an inelastic estimate of aggregate demand. Additional predictors, such as land area, population, and
percent change in gross domestic product, contribute little additional explanatory power. We use a constant of proportionality of 1.4, which is sufficient to ensure that potential demand exceeds
observed consumption in each region-year observation.} The results indicate that potential demand is concentrated in a small number of counties. In 2003, the largest 20 counties account for 90
percent of potential demand, the largest 10 counties account for 65 percent of potential demand, and the largest two counties -- Maricopa County and Los Angeles County -- together account for nearly
25 percent of potential demand.\footnote{% The largest five counties are Maricopa County (3,259 thousand metric tonnes), Los Angeles County (3,128 thousand metric tonnes), Clark County (1,962
thousand metric tonnes), Riverside County (1,803 thousand metric tonnes) and San Diego County (1,733 thousand metric tonnes).} In the time-series, potential demand more than doubles over 1983-2003,
due to greater activity in the construction sector and the onset of the housing bubble. \subsection{The geographic space \label{sec:ussw}} Our restricted geographic focus eases the computational
burden of the estimation routine. For instance, a national sample would require the computation of more than 300 thousand equilibrium plant-county prices in each time period, for every outer loop
iteration. The geographical restriction is valid provided that gross domestic inflows/outflows are insubstantial. The data provide some support. Most directly, the California Letter indicates that
more than 98 percent of cement produced in Southern California was shipped within the U.S. Southwest over the period 1990-1999, and more than 99 percent of cement produced in California (both
Northern and Southern) was shipped within the U.S. Southwest over the period 2000-2003.% \footnote{% Analogous statistics for Northern California over 1990-1999 are unavailable due to data
redaction.} We consider outflows from Arizona unlikely because the Minerals Yearbook indicates that consumption routinely exceeds production in that state, and we consider outflows from Nevada
unlikely because production capacity is low relative to potential demand. Since \textit{net} domestic inflows/outflows are insubstantial (see Figure \ref% {fig:imports}), these data patterns suggest
that gross inflows are also insubstantial. \subsection{Identification \label{sec:ID}} We use an artificial data experiment to test identification. We draw 40 data sets, each with 21 time periods,
using our model and a vector of ``true'' parameters as the data generating process. We then seek to recover the parameter values with the GMM estimation procedure. The exogenous data includes the
plant capacities, the potential demand of counties, the diesel price, the import price, and two cost shifters. We randomly draw capacity and potential demand from the data (with replacement), and we
draw the remaining data from normal distributions.\footnote{% Specifically, we use the following distributions: diesel price $\sim N(1,0.28)$, import price $\sim N(50,9)$, cost shifter 1 $\sim N
(60,15)$, and cost shifter 2 $\sim N(9,2).$ We redraw data that are below zero. We also redraw data that lead the estimator to nonsensical areas of parameter space.} We hold plant and county
locations fixed to maintain tractability, and rely on the random draws on capacity, potential demand, and diesel prices to create variation in the distances between production capacity and consumers.
Table \ref{tab:fakedata} shows the results of the experiment. Interpretation is complicated somewhat because we use non-linear transformations to constrain the price and distance coefficients below
zero, constrain the coefficients on the cost shifters and the over-utilization cost above zero, and constrain the inclusive value and utilization threshold coefficients between zero and one. We defer
details on these transformations to Appendix % \ref{app:estdet}. As shown, the means of the estimated coefficients are close to (transformed) true parameters. The means of the price and distance
coefficients, which are of particular interest, are within 6 percent and 11 percent of the truth, respectively. The root mean-squared errors tend to be between 0.45 and 0.66 -- the two exceptions
that generate higher mean-squared errors are the import dummy and the over-utilization cost, which appear to be less cleanly identified. \begin{table}[tbp] \caption{Artificial Data Test for
Identification} \begin{center} \begin{tabular}[h]{lccccc} \hline\hline \rule[0mm]{0mm}{6.5mm}Variable & Parameter & Truth ($\theta$) & Transformed (% $\widetilde{\theta}$) & Mean Est & RMSE \\ \hline
\multicolumn{3}{l}{\rule[0mm]{0mm}{4.5mm}\textit{Demand}} & & & \\ \; Cement Price & $\beta_1$ & -0.07 & -2.66 & -2.51 & 0.66 \\ \; Miles$\times$Diesel Price & $\beta_2$ & -25.00 & 3.22 & 2.86 & 0.59
\\ \; Import Dummy & $\beta_3$ & -4.00 & -4.00 & -6.07 & 1.23 \\ \; Intercept & $\beta_0$ & 2.00 & 2.00 & 1.11 & 0.51 \\ \; Inclusive Value & $\lambda$ & 0.09 & -2.31 & -1.73 & 0.54 \\ \multicolumn
{3}{l}{\rule[0mm]{0mm}{4.5mm}\textit{Marginal Costs}} & & & \\ \; Cost Shifter 1 & $\alpha_1$ & 0.70 & -0.36 & -0.88 & 0.51 \\ \; Cost Shifter 2 & $\alpha_2$ & 3.00 & 1.10 & 0.54 & 0.45 \\ \;
Utilization Threshold & $\nu$ & 0.90 & 2.19 & 1.71 & 0.59 \\ \; Over-Utilization Cost & $\gamma$ & 300.00 & 5.70 & 6.14 & 1.05 \\ \hline \multicolumn{6}{p{6.3in}}{{\footnotesize {\rule[0mm]{0mm}{0mm}
Results of GMM estimation on 40 data sets that are randomly drawn based on the ``true'' parameters listed. The parameters are transformed prior to estimation to place constraints on the parameter
signs/magnitudes (see Appendix \ref% {app:estdet}). Mean Est and RMSE are the mean of the estimated (transformed) parameters and the root mean-squared error, respectively. }}}% \end{tabular}% \end
{center} \end{table} To further build intuition, we explore some of the empirical relationships (graphed in Figure \ref{fig:ID}) that drive parameter estimates in our application. On the demand side,
the price coefficient is primarily identified by the relationship between the consumption and price moments. In panel A, we plot cement prices and the ratio of consumption to potential demand
(``market coverage'') over the sample period. The two metrics have a weak negative correlation, consistent with downward-sloping but inelastic aggregate demand. The distance coefficient is primarily
identified by (1) the cross-region shipments moment, and (2) the relationship between the consumption and production moments. To explore the second source of identification, we plot the gap between
production and consumption (``excess production'') for each region over the sample period. In many years, excess production is positive in Southern California and negative elsewhere, consistent with
inter-regional trade flows. The magnitude of these implied trade flows helps drive the distance coefficient. Interestingly, the implied trade flows are higher later in the sample, when the diesel
fuel is less expensive. \begin{figure}[t] \centering \includegraphics[height=4in]{stats_SWsample.eps} \caption{{\protect\footnotesize {Empirical Relationships in the U.S. Southwest. Panel A plots
average cement prices and market coverage. Prices are in dollars per metric tonne and market coverage is defined as the ratio of consumption to potential demand (times 100). Panel B plots excess
production in each region, which we define as the gap between between production and consumption. Excess production is in millions of metric tonnes. Panel C plots average coal prices, electricity
prices, durable-goods manufacturing wages, and crushed stone prices in California. For comparability, each time-series is converted to an index that equals one in 2000. Panel D plots the average
cement price and industry-wide utilization (times 100). } }} \label{fig:ID} \end{figure} On the supply side, the parameters on the marginal cost shifters are primarily identified by the price
moments. In panel C, we plot the coal price, the electricity price, the durable-goods manufacturing wage, and the crushed stone price for California. Coal and electricity prices are highly correlated
with the cement price (e.g., see panel A), consistent with a strong influence on marginal costs; inter-regional variation in input prices helps disentangle the two effects. It is less clear that
wages and crushed stone prices are positively correlated with cement prices. The utilization parameters are primarily identified by the relationship between the production moments (which determine
utilization) and the price moments. In panel D, we plot cement prices and industry-wide utilization over the sample period. The two metrics are negatively correlated over 1983-1987 and positively
correlated over 1988-2003. \section{Estimation Results \label{sec:res}} \subsection{Specification and fits} We estimate the model with parsimonious specifications of the utility and marginal cost
functions. Specifically, the utility specification includes the plant-county price, the ``distance'' between the plant and county, a dummy for the import option, and an intercept. We proxy distance
using a diesel price index interacted with the miles between the plant and the center of the county (in thousands). The marginal cost specification incorporates the five variable inputs identified by
\citeN{epa09}. The constant portion of marginal costs includes shifters for the price of coal, the price of electricity, the average wages of durable-goods manufacturing employees, and the price of
crushed stone. We let marginal costs increase in production once utilization exceeds some (estimated) threshold value, as written in Equation \ref{eq:mc}, and we normalize $\phi=1.5$ to ensure the
theoretical existence of equilibrium. Before turning to the parameter estimates, we note that this specification produces impressive in-sample and out-of-sample fits. In Figure \ref% {fig:fitsregX},
we plot observed consumption against predicted consumption (panel A), observed production against predicted production (panel B), and observed prices against predicted prices (panel C). The model
explains 93 percent of the variation in regional consumption, 94 percent of the variation in regional production, and 82 percent of the variation in regional prices. The model also generates accurate
out-of-sample predictions. In panel D, we plot observations on cross-region shipments against the corresponding model predictions. We use only 14 of these observations in the estimation routine --
the remaining 82 data points are withheld from the estimation procedure and do not directly influence the estimated parameters. Even so, the model explains 98 percent of the variation in these data.\
footnote{% We provide additional information on the estimation fits in Appendix \ref% {app:fits}.} \begin{figure}[t] \centering \includegraphics[height=4in]{fits-regX.eps} \caption{{\protect\
footnotesize {GMM Estimation Fits for Regional Metrics. Consumption, production, and cross-region shipments are in millions of metric tonnes. Prices are constructed as a weighted-average of plants in
the region, and are reported as dollars per metric tonne. The lines of best fit and the reported $R^2$ values are based on univariate OLS regressions.}}} \label{fig:fitsregX} \end{figure} %The figure
shows four scatter-plots. The first plots predicted consumption against observed consumption (panel A). The second %plots predicted production against observed consumption (panel B). The third plots
predicted prices against observed prices %(panel C). The fourth plots predicted cross-region shipments against observed shipments. Each plot also shows the line of best fit. %and the resulting
r-squared value. The text discusses the fits in detail. \subsection{Demand estimates and transportation costs \label{sec:demres}} Table \ref{tab:GMM} presents the parameter estimates of the GMM
procedure. The price and distance coefficients are the two primary objects of interest on the demand side; both are negative and precisely estimated.\footnote{% The other demand parameters take
reasonable values and are precisely identified. The negative coefficient on the import dummy may be due to consumer preferences for domestic plants or to the fact that observed import prices do not
reflect the full price of imported cement (e.g., the data exclude duties). The inclusive value coefficient suggests that consumer tastes for the different cement providers are highly correlated,
inconsistent with the standard (non-nested) logit model.} The ratio of these coefficients identifies consumers' willingness-to-pay for proximity, incorporating transportation costs and any other
distance-related costs (e.g., reduced reliability). In the following discussion, we refer to the willingness-to-pay as the transportation cost, although the two concepts may not be strictly
equivalent. \begin{table}[tbp] \caption{Estimation Results } \begin{center} \begin{tabular}[h]{lccc} \hline\hline \rule[0mm]{0mm}{6.5mm}Variable & Parameter & Estimate & St. Error \\ \hline \
multicolumn{3}{l}{\rule[0mm]{0mm}{4.5mm}\textit{Demand}} & \\ \; Cement Price & $\beta_1$ & -0.087 & 0.002 \\ \; Miles$\times$Diesel Price & $\beta_2$ & -26.42 & 1.78 \\ \; Import Dummy & $\beta_3$ &
-3.80 & 0.06 \\ \; Intercept & $\beta_0$ & 1.88 & 0.08 \\ \; Inclusive Value & $\lambda$ & 0.10 & 0.004 \\ \multicolumn{3}{l}{\rule[0mm]{0mm}{4.5mm}\textit{Marginal Costs}} & \\ \; Coal Price & $\
alpha_1$ & 0.64 & 0.05 \\ \; Electricity Price & $\alpha_2$ & 2.28 & 0.47 \\ \; Hourly Wages & $\alpha_3$ & 0.01 & 0.04 \\ \; Crushed Stone Price & $\alpha_4$ & 0.29 & 0.31 \\ \; Utilization
Threshold & $\nu$ & 0.86 & 0.01 \\ \; Over-Utilization Cost & $\gamma$ & 233.91 & 38.16 \\ \hline \multicolumn{4}{p{4.3in}}{{\footnotesize {\rule[0mm]{0mm}{0mm}GMM estimation results. Estimation
exploits variation in regional consumption, production, and average prices over the period 1983-2003, as well as variation in shipments from California to Northern California over the period
1990-2003. The prices of cement, coal, and crushed stone are in dollars per metric tonne. Miles are in thousands. The diesel price is an index that equals one in 2000. The price of electricity is in
cents per kilowatt-hour, and hourly wages are in dollars per hour. The marginal cost parameter $\phi$ is normalized to 1.5, which ensures the theoretical existence of equilibrium. Standard errors are
robust to heteroscedasticity and contemporaneous correlations between moments.}}}% \end{tabular}% \end{center} \end{table} First, however, we briefly summarize the implied price elasticities. We
estimate that the aggregate elasticity is -0.16 in the median year. This inelasticity is precisely what one should expect based on economic theory and the fact that portland cement composes only a
small fraction of total construction expenses. Indeed, \citeN{syverson04} makes a similar argument for ready-mix concrete, which accounts for only two percent of total construction expenses according
to the 1987 Benchmark Input-Output Tables. The cost share of portland cement (an input to ready-mix concrete) is surely even lower. By contrast, we estimate that the median firm-level elasticity is
-5.70, consistent with substantive competition between firms. Finally, we estimate that the domestic elasticity -- which captures the responsiveness of domestic demand to domestic prices, holding
import prices constant -- is -1.11 in the median year. The discrepancy between the aggregate and domestic elasticities is suggestive that imports have a disciplining effect on domestic prices.
Returning to transportation costs, we estimate that consumers pay roughly \$0.30 per tonne mile, given diesel prices at the 2000 level.\footnote{% The calculation is simply $\frac{26.42}{0.087}\frac
{index}{1000}=0.3037$, where $index=1$ in 2000.} Given the shipping distances that arise in numerical equilibrium, this translates into an average transportation cost of \$24.61 per metric tonne over
the sample period -- sufficient to account for 22 percent of total consumer expenditure. Transportation costs of this magnitude have real effects on the industry. We develop two such effects here:
(1) transportation costs constrain the distance that cement can be shipped economically; and (2) transportation costs insulate firms from competition and provide some degree of localized market
power. In Figure \ref{fig:miles4}, we plot the estimated distribution of the shipping distances over 1983-2003. We calculate that portland cement is shipped an average of 92 miles, that 75 percent of
portland cement is shipped under 110 miles, and that 90 percent is shipped under 175 miles.% \footnote{% The average shipping distance fluctuates between a minimum of 72 miles in 1983 and a maximum
of 114 miles in 1998, and is highly correlated with the diesel price index.} To better place these numbers in context, we ask the question: ``How far would portland cement be shipped if
transportation costs were negligible?'' We perform a counter-factual simulation in which we normalize the distance coefficient to zero, keeping the other coefficients at their estimated values. The
results suggest that, in an average year, portland cement would have been shipped on average 276 miles absent transportation costs. Intriguingly, the ratio of actual miles shipped to this simulated
measure provides a unit-free measure that could enable cross-industry comparisons. The ratio in our application is 0.33. \begin{figure}[t] \centering \includegraphics[height=3in]{miles4.eps} \caption
{{\protect\footnotesize {The Estimated Distribution of Miles Shipped over 1983-2003.}}} \label{fig:miles4} \end{figure} % The figure is a non-parametric bar chart that characterizes the distances
that cement is shipped. % See text for summary statistics and more details. We now develop the empirical evidence regarding localized market power. We start with an illustrative example. Figure \ref
{fig:priceshare} shows the prices (Map A) and market shares (Map B) that characterize numerical equilibrium for the Clarksdale plant in 2003, evaluated the optimized coefficient vector. We mark the
location of the Clarksdale plant with a star, and mark other plants with circles. The Clarksdale plant captures more than 40 percent of the market in the central and northeastern counties of Arizona.
It charges consumers in these counties its highest prices, typically \$80 per metric tonne or more. Both market shares and prices are lower in more distant counties, and in many counties the plant
captures less than one percent of demand despite substantial discounts. The locations of competitors may also influence market share and prices, though these effects are difficult to discern on the
map. \begin{figure}[t] \centering \includegraphics[height=4in]{priceshare-erika.eps} \caption{{\protect\footnotesize {Equilibrium Prices and Market Shares for the Clarksdale Plant in 2003. The
Clarksdale plant is marked with a star, and other plants are marked with circles.}}} \label{fig:priceshare} \end{figure} %The figure is a map of California, Arizona, and Nevada. In Map A, counties
for which the Clarksdale plant has higher prices %are shaded darker. In Map B, counties for which the plant has higher market shares are shaded darker. The location of the %Clarksdale plant is marked
with a star, and the locations of competitors are marked with circles. The text provides a description % of the spatial price and share patterns that emerge. We explore these relationships more
rigorously with regression analysis, based on the prices and market shares that characterize equilibrium at the optimized coefficient vector. We regress price and market share on three independent
variables: the distance between the plant and the county, the distance between the county and the nearest other domestic plant, and the estimated marginal cost of the plant. We define distance as
miles times the diesel index, and use a log-log specification to ease interpretation. We focus on three data samples, composed of the plant-county pairs with distances of 0-100, 100-200, and 200-300,
respectively. Our objective is purely descriptive and the regression coefficients should not be interpreted as consistent estimates of any underlying structural parameters. Table \ref{tab:disteffect}
presents the results, which are consistent with the illustrative example and demonstrate that (1) plants have higher prices and market shares in counties that are closer; and (2) plants have lower
prices and market shares in counties that have nearby alternatives. For the closest plants and counties, a 10 percent reduction in distance is associated with prices and market shares that are 0.9
percent and 14 percent higher, respectively. For the same sample, a 10 percent reduction in the distance separating the county from its the closest alternative is associated with prices and market
shares that are 0.7 percent and 11 percent lower, respectively. Interestingly, these price effects attenuate for plants and counties that are somewhat more distant, whereas the market share effects
amplify. \begin{table}[tbp] \caption{Plant Prices and Market Shares} \begin{center} \begin{tabular}[h]{lcccccc} \hline\hline \rule[0mm]{0mm}{5.5mm}{\footnotesize {Dependent Variable:}} & {\
footnotesize {ln(Price)}} & {\footnotesize {ln(Price)}} & {\footnotesize {ln(Price)}} & {\footnotesize {ln(Share)}} & {\footnotesize {ln(Share)}} & {\footnotesize {% ln(Share)}} \\ \rule[0mm]{0mm}
{5.5mm}{\footnotesize {Distance from Plant:}} & {\footnotesize {0-100}} & {\footnotesize {100-200}} & {\footnotesize {200-300% }} & {\footnotesize {0-100}} & {\footnotesize {100-200}} & {\
footnotesize {% 200-300}} \\ \hline \rule[0mm]{0mm}{6.5mm}{\footnotesize {ln(Distance from Plant)}} & {\footnotesize {-0.098*}} & {\footnotesize {-0.038*}} & {\footnotesize {% -0.003}} & {\
footnotesize {-1.369*}} & {\footnotesize {-2.655*}} & {\footnotesize {-5.902*}} \\ & {\footnotesize {(0.026)}} & {\footnotesize {(0.009)}} & {\footnotesize {% (0.009)}} & {\footnotesize {(0.102)}} &
{\footnotesize {(0.186)}} & {\footnotesize {(0.195)}} \\ \rule[0mm]{0mm}{6.5mm}{\footnotesize {ln(Distance to Nearest}} & {\footnotesize {0.071*}} & {\footnotesize {0.018*}} & {\footnotesize {0.007*}
% } & {\footnotesize {1.073*}} & {\footnotesize {0.813*}} & {\footnotesize {% 1.279*}} \\ $\; \;$ {\footnotesize {Alternative)}} & {\footnotesize {(0.019)}} & {\footnotesize {(0.004)}} & {\
footnotesize {(0.002)}} & {\footnotesize {% (0.117)}} & {\footnotesize {(0.081)}} & {\footnotesize {(0.098)}} \\ \rule[0mm]{0mm}{6.5mm}{\footnotesize {ln(Marginal Cost)}} & {\footnotesize {% 0.723*}}
& {\footnotesize {0.835*}} & {\footnotesize {0.841*}} & {\footnotesize {-1.126*}} & {\footnotesize {-2.057*}} & {\footnotesize {% -3.212*}} \\ & {\footnotesize {(0.054)}} & {\footnotesize {(0.014)}}
& {\footnotesize {% (0.013)}} & {\footnotesize {(0.304)}} & {\footnotesize {(0.645)}} & {\footnotesize {(0.874)}} \\ & & & & & & \\ {\footnotesize {R}$^2$} & {\footnotesize {0.7186}} & {\footnotesize
{0.9209}} & {\footnotesize {0.9499}} & {\footnotesize {0.5278}} & {\footnotesize {% 0.4735}} & {\footnotesize {0.5407}} \\ {\footnotesize {\# of Obs.}} & {\footnotesize {2,840}} & {\footnotesize {%
5,460}} & {\footnotesize {6,088}} & {\footnotesize {2,840}} & {\footnotesize {5,460}} & {\footnotesize {6,088}} \\ \hline \multicolumn{7}{p{5.8in}}{{\footnotesize {\rule[0mm]{0mm}{0mm}Results of OLS
regression. The units of observation are at the plant-county-year level. The dependent variables are the natural logs of the plant-county specific prices and market shares that characterize numerical
equilibrium at the GMM estimates. Distance from Plant is the miles between the plant and county, times a diesel price index that equals one in 2000. Distance to Nearest Alternative is the miles
between the county and the nearest other domestic plant, times the diesel price index. Marginal Cost is the marginal cost of the plant implied by the GMM estimates. All regressions include an
intercept. Standard errors are robust to heteroscedasticity and correlations among observations from the same plant. Statistical significance at the one percent level is denoted by *.}}}% \end
{tabular}% \end{center} \end{table} \subsection{Marginal cost estimates} The marginal costs estimates shown in Table \ref{tab:GMM} correspond to a marginal cost of \$69.40 in the mean plant-year
(weighted by production). Of these marginal costs, \$60.50 is attributable to costs related to coal, electricity, labor and raw materials, and the remaining \$8.90 is attributable to high utilization
rates. In Figure \ref{fig:costs}, we plot the three metrics over 1983-2003, together with the average prices that arise in numerical equilibrium at the optimized parameter vector. The constant
portion of marginal costs declines through the sample period, due primarily to cheaper coal and electricity, whereas the utilization portion increases. We estimate that utilization-related expenses
account for roughly 25 percent of overall marginal costs over 1997-2003, during the onset of the housing bubble. Finally, we note that the average markup (i.e., price minus marginal cost) is quite
stable through the sample period around its mean of \$17.20. \begin{figure}[t] \centering \includegraphics[height=3in]{costs.eps} \caption{{\protect\footnotesize {Estimated Marginal Costs and Average
Prices.% }}} \label{fig:costs} \end{figure} % The figure plots estimated marginal costs, estimated constant marginal costs, estimated marginal costs due to high % utilization, and average prices,
over the sample period. The text discusses the relationships between these % metrics in detail. We calculate that the average plant-year observation has variable costs of \$51 million by integrating
the marginal cost function over the production levels that arise in numerical equilibrium. Virtually all of these variable costs -- 98.5 percent -- are due to coal, electricity, labor and raw
materials, rather than due to high utilization. Thus, although capacity constraints may have substantial affects on marginal costs, the results suggest that their cumulative contribution to plant
costs can be minimal. Taking the accounting statistics further, we calculate that the average plant-year has variable revenues of \$73 million and that the average gross margin (variable profits over
variable revenues) is 0.32. As argued in % \citeN{ryan06}, margins of this magnitude may be needed to rationalize entry given the sunk costs associated with plant construction.\footnote{% These gross
margins are consistent with publicly-available accounting data. For instance, Lafarge North America -- one of the largest domestic producers -- reports an average gross margin of 0.33 over
2002-2004.}$^,$\footnote{% Fixed costs are well understood to be important for production, as well. The trade journal \textit{Rock Products} reports that high capacity portland cement plants incurred
averaged \$6.96 in maintenance costs per production tonne in 1993 (\citeN{rock94}). Evaluated at the production levels that correspond to numerical equilibrium in 1993, this number implies that the
average plant would have incurred \$5.7 million in maintenance costs relative to variable profits of \$17.7 million. The GMM estimation results suggest that the bulk of these maintenance costs are
best considered fixed rather than due to high utilization rates. Of course, the static nature of the model precludes more direct inferences about fixed costs.} Finally, we discuss the individual
parameter estimates shown in Table \ref% {tab:GMM}, each of which deviates somewhat from production data available from the Minerals Yearbooks and \citeN{epa09}. To start, the coal parameter implies
that plants burn 0.64 tonnes of coal to produce one tonne of cement, whereas in fact plants burn roughly 0.09 tonnes of coal to produce each tonne of cement. The electricity parameter implies that
plants use 228 kilowatt-hours per tonne of cement, whereas the true number is closer to 150. Each tonne of cement requires approximately 0.34 employee-hours yet the parameter on wages is essentially
zero. Lastly, the crushed stone coefficient of 0.29 is too small, given that roughly 1.67 tonnes of raw materials (mostly limestone) are used per tonne of cement. We suspect that these discrepancies
are due to measurement error in the data.\footnote{% In particular, the coal prices in the data are free-on-board and do not reflect any transportation costs paid by cement plants; cement plants may
negotiate individual contracts with electrical utilities that are not reflected in the data; the wages of cement workers need not track the average wages of durable-goods manufacturing employees; and
cement plants typically use limestone from a quarry adjacent to the plant, so the crushed stone price may not proxy the cost of limestone acquisition (i.e., the quarry production costs).} \subsection
{A comparison to standard methods} The standard method of structural analysis for homogenous product industries assumes independent markets and Cournot competition. In this section, we contrast some
of our results to those generated by the standard method in % \citeN{ryan06}, a recent paper that estimates a structural model of the portland cement industry based on data from the Minerals Yearbook
and the Plant Information Summary. We focus on two economic concepts -- the aggregate elasticity of demand and the consequences of high utilization -- for which our model generates distinctly
different estimates than the standard method. These discrepancies do not diminish the substantial contribution of \citeN{ryan06}, which embeds the standard method within an innovative dynamic
discrete choice game and focuses primarily on the dynamic parameters. Rather, the discrepancies suggest two reasons that our model may sometimes provide more reasonable results than conventional
approaches. First, we estimate the aggregate elasticity of demand to be -0.16 in the median sample year whereas Ryan works with an aggregate elasticity of -2.96, obtained from a constant elasticity
demand system. The difference is due to specific specification choices -- the constant elasticity demand system produces an aggregate elasticity of -0.15 once housing permits are included as a
control.\footnote{% See Table 3 in \citeN{ryan06}. We consider the inelastic estimate more plausible because portland cement is a minor cost for most construction projects (see Section \ref
{sec:demres}).} Ryan cannot use the inelastic estimate because, within the context of Cournot competition, it implies firm elasticities that are small and inconsistent with profit maximization. This
occurs because the Cournot model restricts each firm elasticity to be linearly related to the aggregate elasticity according the relationship $e_j = e/s_j$, where $e_j$, $e$, and $s_j$ denote the
firm elasticity, the aggregate elasticity, and the firm market shares, respectively. This critique is fundamental: the standard method can be inappropriate for intermediate goods, such as portland
cement, that account for only a fraction for the total production costs of the final good. By contrast, the nested logit demand system divorces the firm elasticities from the aggregate elasticity
and, in our case, produces inelastic aggregate demand and elastic firm demand. Second, the two methods produce vastly different estimates of the marginal cost curve once utilization reaches the
threshold level (which both methods place just above 0.85). We estimate that marginal costs increase gradually so that full utilization increases marginal costs by a total of \$12.25 relative to
utilization below the threshold. By contrast, Ryan estimates that the slope of the marginal cost curve past the threshold is nearly infinite.\footnote{% Our coefficient $\gamma$ is roughly analogous
to Ryan's $\delta_2$ coefficient. We estimate $\gamma$ to be 233.91, while Ryan estimates $% \delta_2$ to be $1157\times 10^7$. See Table 4 in \citeN{ryan06}.} We suspect that the difference is data
driven. The standard method requires data on firm-level utilization. However, firm production is not available from the publicly-available data, and Ryan imputes utilization as annual capacity
divided by annualized daily capacity. In Figure \ref{fig:ryan}, we plot total production, total annual capacity, and total annualized daily capacity in the U.S. Southwest over 1983-2003, together
with total consumption. Both production and consumption are pro-cyclical, and actual utilization (i.e., production over annual capacity) varies substantively and predictably with demand conditions.
By contrast, annual capacity simply tracks annualized daily capacity so that Ryan's utilization proxy is uncorrelated with demand conditions. The strength of the relationship between utilization and
demand is precisely what identifies the magnitude of utilization costs. Thus, we suspect that the lower data requirements of our model -- estimation is feasible when some variables of interest (e.g.,
firm-level production) are unobserved -- may improve economic estimates. \begin{figure}[t] \centering \includegraphics[height=3in]{ryan.eps} \caption{{\protect\footnotesize {Consumption, Production,
and Two Capacity Measures.}}} \label{fig:ryan} \end{figure} % The figure plots production, consumption, annual utilization, and annualized daily utilization, over the sample % period. The text
discusses the relationships between these metrics in detail. \section{An application to competition policy \label{sec:sim}} The model and estimator may prove useful for a variety of policy endeavors.
One potential application is merger simulation, an important tool for competition policy. In this subsection, we use counter-factual simulations to evaluate a hypothetical merger between Calmat and
Gifford-Hill in 1986. During that year, Calmat and Gifford-Hill operated six plants and accounted for 43 percent of industry capacity in the U.S. Southwest. We calculate the loss of consumer surplus
due to the unilateral effects of the merger, map the distribution of harm over the U.S. Southwest, and evaluate six alternative divestiture plans.\footnote{% We follow standard practice to perform
the counterfactuals. For each merger simulation, we define a matrix $\boldsymbol{\Omega^{post}(P)}$ using Equation \ref{eq:omega} and the post-merger structure of the industry. We compute the
equilibrium post-merger price vector as the solution to Equation % \ref{eq:equil}, substituting $\boldsymbol{\Omega^{post}(P)}$ for $% \boldsymbol{\Omega(P)}.$ Following \citeN{mcfadden81} and % \
citeN{smallrosen81}, the change in consumer surplus due to the merger is: $$\Delta CS =\sum_{n=1}^N \frac{\ln (1+\exp(\beta_0+ \lambda I_{nt}^{pre}))-\ln (1+\exp(\beta_0+ \lambda I_{nt}^{post}))}{\
beta_1} M_n, otag$$ where $I_{n}^{pre}$ is the inclusive value of the inside goods calculated using equilibrium pre-merger prices, $I_n^{post}$ is the inclusive value calculated using equilibrium
post-merger prices, and $\beta_1$ is the price coefficient.} Table \ref{tab:mergsim} shows that the merger reduces consumer surplus by \$1.40 million in 1986, absent a divestiture. The magnitude of
the effect is modest relative to the amount of commerce; by way of comparison, we calculate total pre-merger consumer surplus to be more than \$239 million. We refer to the six plants available for
divestiture as Calmat1, Calmat2, Calmat3, Gifford-Hill1, Gifford-Hill2, and Gifford-Hill3, respectively. The single-plant divestitures mitigate between 31 percent and 56 percent of the harm. The
``optimal'' divestiture -- that of Gifford-Hill2 -- results in consumer harm of only \$614 thousand. \begin{table}[tbp] \caption{Divestitures and Consumer Surplus} \begin{center} \begin{tabular}[h]
{lccccccc} \hline\hline & \multicolumn{7}{c}{\rule[0mm]{0mm}{5.5mm}\underline{Required Divestiture}} \\ \rule[0mm]{0mm}{4.5mm} & {\footnotesize {None}} & {\footnotesize {Calmat1}} & {\footnotesize
{Calmat2}} & {\footnotesize {Calmat3}} & {\footnotesize {% Gifford-Hill1}} & {\footnotesize {Gifford-Hill2}} & {\footnotesize {% Gifford-Hill3}} \\ \hline \rule[0mm]{0mm}{5.5mm} {\footnotesize {$\
Delta$ Surplus}} & -1,397 & -964 & -618 & -797 & -827 & -614 & -891 \\ \rule[0mm]{0mm}{7.5mm} {\footnotesize {\% Mitigated}} & $\cdot$ & 31\% & 56\% & 43\% & 41\% & 56\% & 36\% \\ \hline \multicolumn
{8}{p{6.1in}}{{\footnotesize {\rule[0mm]{0mm}{0mm}Results of counterfactual simulations. $\Delta$ Surplus is the change in consumer surplus due to a hypothetical merger between Calmat and
Gifford-Hill in 1986, and is reported in thousands of 2000 dollars. \% Mitigated is calculated relative to the change in consumer surplus that occurs when no plant is divested. We consider six
single-plant divestiture plans, and refer to the different plants as Calmat1, Calmat2, Calmat3, Gifford-Hill1, Gifford-Hill2, and Gifford-Hill3. }}}% \end{tabular}% \end{center} \end{table} We map
the distribution of consumer harm over the U.S. Southwest in Figure % \ref{fig:mapdivest}, both for the merger without divestiture (panel A) and under the optimal divestiture plan (panel B). As shown
in panel A, the unilateral effects of the merger are concentrated in Southern California and Arizona. Together, Maricopa County and Los Angeles County account for 60 percent of consumer harm, and
more than 90 percent of consumer harm occurs only 10 counties. The best single-plant divestiture mitigates consumer harm in Southern California but does little to reduce harm in Maricopa County (see
panel B). The results of an additional counterfactual exercises, in which we also divest one of the Arizona plants, suggest that a two-plant divestiture can mitigate this harm as well (results not
shown). \begin{figure}[t] \centering \includegraphics[height=4in]{Consumer-Harm2.eps} \caption{{\protect\footnotesize {Loss of Consumer Surplus Due to a Hypothetical Merger between Calmat and
Gifford-Hill}}} \label{fig:mapdivest} \end{figure} \section{Conclusion \label{sec:conc}} We develop a structural model of competition among spatially differentiated firms. The model accounts for
transportation costs in a realistic and tractable manner. We estimate the model with relatively disaggregated data and recover the underlying structural parameters. We argue that the model and
estimator together provide an appealing framework with which to evaluate competition in industries characterized by transportation costs and relatively homogenous products. We apply the model and
estimator to the portland cement industry and demonstrate that (1) the framework explains the salient features of competition, (2) the framework provides novel insights regarding transportation costs
and spatial differentiation, and (3) the framework could inform merger analysis and other competition policy endeavors. Although the model is static, it could be utilized to define payoffs in more
dynamic settings. Such extensions could examine a number of research topics -- such as entry deterrence and product differentiation -- that have been emphasized in the theoretical literature of
industrial organization since at least \citeN{hotelling29}. \newpage \bibliographystyle{chicago} \bibliography{cement} \clearpage \section{References} Aguirregabiria, V. and G. Vicentini (2006).
Dynamic spatial competition between multistore firms. Mimeo. Berry, S., J. Levinsohn, and A. Pakes (1995, July). Automobile prices in market equilibrium. Econometrica, 841–890. Cardell, S. N. (1997).
Variance components structures for the extreme value and logistic distributions with applications to models of heterogeneity. Journal of Economic Theory 13, 185–213. Collard-Wexler, A. (2009).
Productivity dispersion and plant selection in the ready-mix concrete industry. Mimeo. d’Aspremont, C., J. Gabszewicz, and J.-F. Thisse (1979). On Hotelling’s “Stability in Competion”. Econometrica
47, 1145–1150. Davis, P. (2006). Spatial competition in retail markets: Movie theaters. The RAND Journal of Economics 37, 964–982. Dunne, T., S. Klimek, and J. A. Schmitz (2009). Does foreign
competition spur productivity? Evidence from post-WWII U.S. cement manufacturing. Federal Reserve Bank of Minneapolis Staff Report. Economides, N. (1989). Stability in competition. Journal of
Economic Theory 47, 178–194. EPA (2009). Regulatory Impact Analysis: National Emission Standards for Hazardous Air Pollutants from the Portland Cement Manufacturing Industry. Prepared by RTI
International. Greene, W. H. (2003). Econometric Analysis 5th Ed. Prentice-Hall Upper Saddle River NJ. Hansen, L. (1982). Large sample properties of generalized method of moments estimators.
Econometrica 50, 1029–1054. Hotelling, H. (1929). Stability in competition. Economic Journal 39, 41–57. Hwang, H.-s. (1990). Estimation of a linear SUR model with unequal numbers of observations.
Review of Economics and Statistics 72. Jans, I. and D. Rosenbaum (1997, May). Multimarket contact and pricing: Evidence from the U.S. cement industry. International Journal of Industrial Organization
15, 391–412. La Cruz, W., J. Mart´ınez, and M. Raydan (2006). Spectral residual method without gradient information for solving large-scale nonlinear systems of equations. Mathematics of Computation
75, 1429–1448. Levenberg, K. (1944). A method for the solution of certain non-linear problems in least squares. Quarterly Journal of Mathematics 2, 164–168. Marquardt, D. (1963). An algorithm for
least-squares estimation of nonlinear parameters. SIAM Journal of Applied Mathematics 11, 431–331. McBride, M. (1983, December). Spatial competition and vertical integration: Cement - concrete
revisited. American Economic Review 73, 1011–1022. McFadden, D. (1981). Econometric Models of Probabilistic Choice. MIT Press. Structural Analysis of Discrete Data, C.F. Manski and D. McFadden
(eds.). McManus, B. (Forthcoming 2009). Nonlinear pricing in an oligopoly market: the case of specialty coffee. RAND Journal of Economics. Newey, W. K. and D. McFadden (1994). Large sample estimation
and hypothesis testing. Handbook of Econometrics 4. Pesendorfer, M. (2003). Horizontal mergers in the paper industry. RAND Journal of Economics 34, 495–515. Rock-Products (1994). Cement plant
operating cost study. Overland Park, KS: Intertic Publishing. Rosenbaum, D. and S. Reading (1988). Market structure and import share: A regional market analysis. Southern Economic Journal 54 (3),
694–700. Rosenbaum, D. I. (1994). Efficiency v. collusion: evidence cast in cement. Review of Industrial Organization 9, 379–392. Ryan, S. (2009). The costs of environmental regulation in a
concentrated industry. Mimeo. Salop, S. (1979). Monopolistic competition with outside goods. Bell Journal of Economics 10, 141–156. Salvo, A. (2008). Inferring market power under the threat of entry:
The case of the Brazilian cement industry. Mimeo. Salvo, A. (2010). Trade flows in a spatial oligolopy: Gravity fits well, but what does it explain? Canadian Journal of Economics 43, 63–96. Seim, K.
(2006). An empirical model of firm entry with endogenous product-type choices. The RAND Journal of Economics 37, 619–640. Small, K. and H. Rosen (1981). Applied welfare economics with discrete choice
methods. Econometrica 49, 105–130. Srivastava, J. and M. K. Zaatar (1973). A Monte Carlo comparison of four estimators of the dispersion matrix of a bivariate normal population, using incomplete
data. Journal of the American Statistical Association 68, 180–183. Syverson, C. (2004, December). Market structure and productivity: A concrete example. Journal of Political Economy 112, 1181–1222.
Syverson, C. and A. Horta¸csu (2007). Cementing relationships: Vertical integration, foreclosure, productivity, and prices. Journal of Political Economy 115, 250–301. Thisse, J.-F. and X. Vives
(1988). On the strategic choice of spatial price policy. American Economic Review 78, 122–137. Thomadsen, R. (2005). The effect of ownership structure on prices in geographically differentiated
industries. RAND Journal of Economics 36, 908–929. Van Oss, H. G. and A. Padovani (2002). Cement manufacture and the environment, Part I: Chemistry and technology. Journal of Industrial Ecology 6,
89–105. Vogel, J. (2008, June). Spatial competition with heterogeneous firms. Journal of Political Economy 116 (3), 423–466. \appendix %\singlespace \section{Regression fits \label{app:fits}} \
renewcommand{\theequation}{A-\arabic{equation}} \setcounter{equation}{0} %\renewcommand{\thefigure}{A-\arabic{figure}} %\setcounter{figure}{0} In this appendix, we develop further the quality of the
estimation fits. We focus on the ability of the model to predict the inter-temporal variation that exist in the data. Figure \ref{fig:fits1} aggregates the data and the model predictions across
regions, and plots the resulting time-series. Panel A shows consumption, panel B shows production, panel C shows imports, and panel D shows average prices (imports are defined as production minus
consumption). In each case, the model predictions mimic the inter-temporal patterns observed in the data. Univariate regressions of the data on the prediction explain 96 percent of the variation
total consumption, 75 percent of the variation in total production, 76 percent of the variation in imports, and 91 percent of the variation in average prices.\footnote{% The model does not fully
capture the fall in average prices over the 1980s and early 1990s. One possible explanation is that the model, as specified, does not incorporate potential changes to total factor productivity. % \
citeN{schmitz09} review the evidence regarding productivity and argue that the gradual elimination of onerous clauses from labor contracts improved productivity in the 1980s.} \begin{figure}[t] \
centering \includegraphics[height=4in]{fits-insample.eps} \caption{{\protect\footnotesize {GMM Estimation Fits for Aggregate Metrics. The solid lines plot data and the dashed lines plot predictions.
Consumption, production, and imports are in millions of metric tonnes. Imports are defined as production minus consumption. Prices are constructed as a weighted-average of the plant-county prices and
are reported in dollars per metric tonne. The $R^2$ values are calculated from univariate regressions of the observed metric on the predicted metric. }}} \label{fig:fits1} \end{figure} %The figure
shows four graphs that characterize the estimation fits. Panel A is for total consumption, panel B is for %total production, panel C is for apparent imports, and panel D is for average prices. Each
graph plots the predicted % and observed values over the sample period. The graphs also show the R-squared from a regression of the observed value % on the predicted value. See text for details.
Figure \ref{fig:outfits2} provides analogous time-series fits for eight of the time-series for cross-region shipments. The data in panel A pertain to shipments from California (both Northern and
Southern) to Northern California, and are used in estimation. The data in the remaining panels are excluded from the estimation procedure, so the corresponding fits are out-of-sample. As shown, the
model predictions are close to the data in each panel and tend to track the variation well (when variation exists). \begin{figure}[t] \centering \includegraphics[height=4in]{fits-outsample.eps} \
caption{{\protect\footnotesize {GMM Estimation Fits for Cross-Region Shipments. The solid lines plot data and the dashed lines plot predictions. Shipments are expressed as a percentage of production
in California (panels A-D) or Southern California (panels E-H).}}} \label{fig:outfits2} \end{figure} %The figure shows eight graphs that characterize the regression fits for the cross-region
shipments. %Each graph plots both predicted and observed shipments for one particular region-to-region %combination, over the sample period. The region-to-region combination are, in order: %CA to
N.CA, CA to S.CA, CA to AZ, CA to NV, S.CA to N.CA, S.CA to S.CA, S.CA to AZ, and S.CA to NV. %See text for details. \section{The uniqueness of equilibrium} The estimation procedure rests on
uniqueness of equilibrium at each candidate parameter vector. The results of a Monte Carlo experiment suggest the assumption holds, at least in our application. We consider 300 parameter vectors for
each of the 21 years in the sample, for a total of 6,300 candidate parameter vectors. For each $\theta_i \in \boldsymbol{\theta}$, we draw from the distribution $N(\widehat{\mu}_i,\widehat{\sigma}_i^
2)$, where $% \widehat{\mu}_i$ and $\widehat{\sigma}_i$ are the coefficient and standard error, respectively, reported in Table \ref{tab:GMM}. We then compute the numerical equilibrium for each
parameter vector, using eleven different starting vectors. We define the elements of the starting vectors to be $% P_{jnt}=\phi \overline{P_t},$ where $\overline{P_t}$ is the average price of
portland cement and $\phi = 0.5, 0.6, \dots, 1.4, 1.5$. Thus, we start the equation solver at initial prices that sometimes understate and sometimes overstate the average prices in the data. The
experiment produces eleven equilibrium price vectors for each parameter vector. We calculate the standard deviation of each price element across the eleven observations. Thus, we would calculate
1,260 standard deviations for a typical equilibrium price vector with 1,260 plant-county elements. The experiment provides support for the uniqueness condition if these standard deviations are
small.% \footnote{% The equal-solver computes numerical equilibria for 90.3 percent of the candidate vectors. See Appendix \ref{app:estdet} for a discussion of non-convergence in the inner-loop.}
This proves to be the case. In fact, the maximum maximum standard deviation is \textit{zero}, considering all prices and draws, so the experiment finds no evidence of multiple equilibria. \section
{Estimation details \label{app:estdet}} We minimize the objective function using the Levenberg-Marquardt algorithm (% \citeN{levenberg44}, \citeN{marquardt63}), which interpolates between the
Gauss-Newton algorithm and the method of gradient descent. We find that the Levenberg-Marquardt algorithm outperforms simplex methods such as simulated annealing and the Nelder-Mead algorithm, as
well as quasi-Newton methods such as BFGS. We implement the minimization procedure using the nls.lm function in R, which is downloadable as part of the minpack.lm package. We compute numerical
equilibrium using Fortran code that builds on the source code of the dfsane function in R. The dfsane function implements the nonlinear equation solver developed in \citeN{dfsane06} and is
downloadable as part of the BB package. We find that Fortran reduces the computational time of the inner loop by a factor of 30 or more, relative to the dfsane function in R. The computation of
equilibrium for each time period can be parallelized, which further speeds the inner loop calculations. The numerical computation of equilibrium takes between 2 and 12 seconds for most candidate
parameter vectors when run on a 2.40GHz dual core processor with 4.00GB of RAM. We use observed prices to form the basis of the initial vector in the inner loop computations, which limits the
distance that the nonlinear equation solver must ``walk'' to compute numerical equilibrium. In practice, the equation solver occasionally fails to compute a numerical equilibrium at the specified
tolerance level (1e-13) within the specified maximum number of iterations (600). The candidate parameter vectors that generate non-convergence in the inner loop tend to be less economically
reasonable, and may be consistent with equilibria that are simply too distant from observed prices. When this occurs, we construct regional-level metrics based on the price vector that comes closest
to satisfying our definition of numerical equilibrium. We constrain the signs and/or magnitudes of some parameters, based on our understanding of economic theory and the economics of the portland
cement industry, because some parameter vectors hinder the computation of numerical equilibrium in the inner loop. For instance, a positive price coefficient would preclude the existence of
Bertrand-Nash equilibrium. We use the following constraints: the price and distance coefficients ($\beta_1$ and $% \beta_2$) must be negative; the coefficients on the marginal cost shifters ($% \
boldsymbol{\alpha}$) and the over-utilization cost ($\gamma$) must be positive; and the coefficients on the inclusive value ($\lambda$) and the utilization threshold ($\nu$) must be between zero and
one. We use nonlinear transformations to implement the constraints. As examples, we estimate the price coefficient using $\widetilde{\beta_1}=\log(-\beta_1)$ in the GMM procedure, and we estimate the
inclusive value coefficient using $\widetilde{% \lambda} = \log\left(\frac{\lambda}{1-\lambda}\right)$. We calculate standard errors with the delta method. \section{Data details} We make various
adjustments to the data in order to improve consistency over time and across different sources. We discuss some of these adjustments here, in an attempt to build transparency and aid replication. The
Minerals Yearbook reports the total production and average price of plants in the ``Nevada-Arizona-New Mexico'' region over 1983-1991, and in the ``Arizona-New Mexico'' region over 1992-2003. We
scale the USGS production data downward, proportional to plant capacity, to remove for the influence of the single New Mexico plant. Since the two plants in Arizona account for 89 percent of kiln
capacity in Arizona and New Mexico in 2003, we scale production by 0.89. The portland cement plant in Riverside closed its kiln permanently in 1988 but continued operating its grinding mill with
purchased clinker. We include the plant in the analysis over 1983-1987, and we adjust the USGS production data to remove the influence of the plant over 1988-2003 by scaling the data downward,
proportional to plant grinding capacities. Since the Riverside plant accounts for 7 percent of grinding capacity in Southern California in 1988, so we scale the production data for that region by
0.93. We exclude one plant in Riverside that produces white portland cement. White cement takes the color of dyes and is used for decorative structures. Production requires kiln temperatures that are
roughly 50$^\circ$C hotter than would be needed for the production of grey cement. The resulting cost differential makes white cement a poor substitute for grey cement. The PCA reports that the
California Cement Company idled one of two kilns at its Colton plant over 1992-1993 and three of four kilns at its Rillito plant over 1992-1995, and that the Calaveras Cement Company idled all kilns
at the San Andreas plant following the plant's acquisition from Genstar Cement in 1986. We adjust plant capacity accordingly. The data on coal and electricity prices from the Energy Information
Agency are available at the state level starting in 1990. Only national-level data are available in earlier years. We impute state-level data over 1983-1989 by (1) calculating the average discrepancy
between each state's price and the national price over 1990-2000, and (2) adjusting the national-level data upward or downward, in line with the relevant average discrepancy. %%% Appendix Figures %%%
%\renewcommand{\thefigure}{A-\arabic{figure}} %\setcounter{figure}{0} \end{document}
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Vermont, CA ACT Tutor
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Week 4 Ilab by Dvv402
Statistics – Lab Week 4
Statistical Concepts:
* Probability
* Binomial Probability Distribution
Calculating Binomial Probabilities
* Open a new MINITAB worksheet.
* We are interested in a binomial experiment with 10 trials. First, we will make the probability of a success ¼. Use MINITAB to calculate the probabilities for this distribution. In column C1 enter
the word ‘success’ as the variable name (in the shaded cell above row 1. Now in that same column, enter the numbers zero through ten to represent all possibilities for the number of successes. These
numbers will end up in rows 1 through 11 in that first column. In column C2 enter the words ‘one fourth’ as the variable name. Pull up Calc > Probability Distributions > Binomial and select the radio
button that corresponds to Probability. Enter 10 for the Number of trials: and enter 0.25 for the Event probability:. For the Input column: select ‘success’ and for the Optional storage: select ‘one
fourth’. Click the button OK and the probabilities will be displayed in the Worksheet.
* Now we will change the probability of a success to ½. In column C3 enter the words ‘one half’ as the variable name. Use similar steps to that given above in order to calculate the probabilities for
this column. The only difference is in Event probability: use 0.5.
* Finally, we will change the probability of a success to ¾. In column C4 enter the words ‘three fourths’ as the variable name. Again, use similar steps to that given above in order to calculate the
probabilities for this column. The only difference is in Event probability: use 0.75.
Plotting the Binomial Probabilities
1. Create plots for the three binomial distributions above. Select Graph > Scatter Plot and Simple then for graph 1 set Y equal to ‘one fourth’ and X to ‘success’ by clicking on the variable name and
using the “select” button below the list of variables. Do this two more times and for...
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Stochastic Theory of a Data-Handling System with Multiple Sources
Results 1 - 10 of 262
- IEEE/ACM Transactions on Networking , 1993
"... Absfruct- The emerging high-speed networks, notably the ATM-based Broadband ISDN, are expected to integrate through statistical multiplexing large numbers of traffic sources having a broad range
of burstiness characteristics. A prime instrument for controlling congestion in the network is admission ..."
Cited by 267 (5 self)
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Absfruct- The emerging high-speed networks, notably the ATM-based Broadband ISDN, are expected to integrate through statistical multiplexing large numbers of traffic sources having a broad range of
burstiness characteristics. A prime instrument for controlling congestion in the network is admission control, which limits calls and guarantees a grade of service determined by delay and loss
probability in the multiplexer. We show, for general Markovian traffic sources, that it is possible to assign a notional effective bandwidth to each source which is an explicitly identi-fied, simply
computed quantity with provably correct properties in the natural asymptotic regime of small loss probabilities. It is the maximal real eigenvalue of a matrix which is directly obtained from the
source characteristics and the admission criterion, and for several sources it is simply additive. We consider both fluid and point process models and obtain parallel results. Numerical results show
that the acceptance set for heterogeneous classes of sources is closely approximated and conservatively bounded by the set obtained from the effective bandwidth approximation. Also, the
bandwidth-reducing properties of the Leaky Bucket regulator are exhibited numerically. For a source model of video teleconferencing due to Heyman et al. with a large number of states, the effective
bandwidth is easily computed. The equivalent bandwidth is bounded by the peak and mean source rates, and is monotonic and concave with respect to a parameter of the admission criterion. Coupling of
state transitions of two related asynchronous sources always increases their effective bandwidth. 1.
, 2002
"... A fundamental problem in text data mining is to extract meaningful structure from document streams that arrive continuously over time. E-mail and news articles are two natural examples of such
streams, each characterized by topics that appear, grow in intensity for a period of time, and then fade aw ..."
Cited by 260 (2 self)
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A fundamental problem in text data mining is to extract meaningful structure from document streams that arrive continuously over time. E-mail and news articles are two natural examples of such
streams, each characterized by topics that appear, grow in intensity for a period of time, and then fade away. The published literature in a particular research field can be seen to exhibit similar
phenomena over a much longer time scale. Underlying much of the text mining work in this area is the following intuitive premise --- that the appearance of a topic in a document stream is signaled by
a "burst of activity," with certain features rising sharply in frequency as the topic emerges.
, 1993
"... We show the existence of effective bandwidths for multiclass Markov fluids and other types of sources that are used to model ATM traffic. More precisely,we show that when such sources share a
buffer with deterministic service rate, a constraint on the tail of the buffer occupancy distribution is a l ..."
Cited by 187 (14 self)
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We show the existence of effective bandwidths for multiclass Markov fluids and other types of sources that are used to model ATM traffic. More precisely,we show that when such sources share a buffer
with deterministic service rate, a constraint on the tail of the buffer occupancy distribution is a linear constraint on the number of sources. That is, for a small loss probability one can assume
that each source transmits at a fixed rate called its effective bandwidth. When traffic parameters are known, effective bandwidths can be calculated and may be used to obtain a circuit-switched style
call acceptance and routing algorithm for ATM networks. The important feature of the effective bandwidth of a source is that it is a characteristic of that source and the acceptable loss probability
only.Thus, the effective bandwidth of a source does not depend on the number of sources sharing the buffer nor on the model parameters of other types of sources sharing the buffer.
- IEEE Transactions on Automatic Control , 1994
"... Motivated by recent development in high speed networks, in this paper we study two types of stability problems: (i) conditions for queueing networks that render bounded queue lengths and bounded
delay for customers, and (ii) conditions for queueing networks in which the queue length distribution of ..."
Cited by 171 (20 self)
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Motivated by recent development in high speed networks, in this paper we study two types of stability problems: (i) conditions for queueing networks that render bounded queue lengths and bounded
delay for customers, and (ii) conditions for queueing networks in which the queue length distribution of a queue has an exponential tail with rate `. To answer these two types of stability problems,
we introduce two new notions of traffic characterization: minimum envelope rate (MER) and minimum envelope rate with respect to `. Based on these two new notions of traffic characterization, we
develop a set of rules for network operations such as superposition, input-output relation of a single queue, and routing. Specifically, we show that (i) the MER of a superposition process is less
than or equal to the sum of the MER of each process, (ii) a queue is stable in the sense of bounded queue length if the MER of the input traffic is smaller than the capacity, (iii) the MER of a
departure process from a stable queue is less than or equal to that of the input process (iv) the MER of a routed process from a departure process is less than or equal to the MER of the departure
process multiplied by the MER of the routing process. Similar results hold for MER with respect to ` under a further assumption of independence. These rules provide a natural way to analyze
feedforward networks with multiple classes of customers. For single class networks with nonfeedforward routing, we provide a new method to show that similar stability results hold for such networks
under the FCFS policy. Moreover, when restricting to the family of two-state Markov modulated arrival processes, the notion of MER with respect to ` is shown to be
- IEEE Journal on Selected Areas in Communications , 1995
"... Abstract-A new approach to determining the admissibility of variable bit rate (VBR) traffic in buffered digital networks is developed. In this approach all traffic presented to the network is
assumed to have been subjected to leaky-bucket regulation, and extremal, periodic, on-off regulated traffic ..."
Cited by 150 (9 self)
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Abstract-A new approach to determining the admissibility of variable bit rate (VBR) traffic in buffered digital networks is developed. In this approach all traffic presented to the network is assumed
to have been subjected to leaky-bucket regulation, and extremal, periodic, on-off regulated traffic is considered; the analysis is based on fluid models. Each regulated traffic stream is allocated
bandwidth and buffer resources which are independent of other traffic. Bandwidth and buffer allocations are traded off in a manner optimal for an adversarial situation involving minimal knowledge of
other traffic. This leads to a single-resource statistical-multiplexing problem which is solved using techniques previously used for unbuffered traffic. VBR traffic is found to be divisible into two
classes, one for which statistical multiplexing is effective and one for which statistical multiplexing is ineffective in the sense that accepting small losses provides no advantage over requiring
lossless performance. The boundary of the set of admissible traffic sources is examined, and is found to be sufficiently linear that an effective bandwidth can be meaningfully assigned to each VBR
source, so long as only statistically-multiplexable sources are considered, or only nonstatistically-multiplexable sources are considered. If these two types of sources are intermixed, then nonlinear
interactions occur and fewer sources can be admitted than a linear theory would predict. A qualitative characterization of the nonlinearities is presented. The complete analysis involves conservative
approx-imations; however, admission decisions based on this work are expected to be less overly conservative than decisions based on alternative approaches. I.
, 1995
"... The algorithms described in this thesis are designed to schedule cells in a very high-speed, parallel, input-queued crossbar switch. We present several novel scheduling algorithms that we have
devised, each aims to match the set of inputs of an input-queued switch to the set of outputs more effici ..."
Cited by 138 (4 self)
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The algorithms described in this thesis are designed to schedule cells in a very high-speed, parallel, input-queued crossbar switch. We present several novel scheduling algorithms that we have
devised, each aims to match the set of inputs of an input-queued switch to the set of outputs more efficiently, fairly and quickly than existing techniques. In Chapter 2 we present the simplest and
fastest of these algorithms: SLIP --- a parallel algorithm that uses rotating priority ("round-robin") arbitration. SLIP is simple: it is readily implemented in hardware and can operate at high
speed. SLIP has high performance: for uniform i.i.d. Bernoulli arrivals, SLIP is stable for any admissible load, because the arbiters tend to desynchronize. We present analytical results to model
this behavior. However, SLIP is not always stable and is not always monotonic: adding more traffic can actually make the algorithm operate more efficiently. We present an approximate analytical model
of this behavior. SLIP prevents starvation: all contending inputs are eventually served. We present simulation results, indicating SLIP's performance. We argue that SLIP can be readily implemented
for a 32x32 switch on a single chip. In Chapter 3 we present i-SLIP, an iterative algorithm that improves upon SLIP by converging on a maximal size match. The performance of i-SLIP improves with up
to log 2 N iterations. We show that although it has a longer running time than SLIP, an i-SLIP scheduler is little more complex to implement. In Chapter 4 we describe maximum or maximal weight
matching algorithms based on the occupancy of queues, or waiting times of cells. These algorithms are stabl...
, 1996
"... This paper presents a personal view of work to date on effective bandwidths, emphasising the unifying role of the concept: as a summary of the statistical characteristics of sources over
different time and space scales; in bounds, limits and approximations for various models of multiplexing unde ..."
Cited by 132 (4 self)
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This paper presents a personal view of work to date on effective bandwidths, emphasising the unifying role of the concept: as a summary of the statistical characteristics of sources over different
time and space scales; in bounds, limits and approximations for various models of multiplexing under quality of service constraints; and as the basis for simple and robust tariffing and connection
acceptance control mechanisms for poorly characterized traffic. The framework assumes only stationarity of sources, and illustrative examples include periodic streams, fractional Brownian input,
policed and shaped sources, and deterministic multiplexing.
- IEEE/ACM Transactions on Networking , 1993
"... We propose a new way for evaluating the performance of packet switching communication networks under a fixed (session based) routing strategy. Our approach is based on properly bounding the
probability distribution functions of the system input processes. The bounds we suggest, which are decaying ex ..."
Cited by 122 (3 self)
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We propose a new way for evaluating the performance of packet switching communication networks under a fixed (session based) routing strategy. Our approach is based on properly bounding the
probability distribution functions of the system input processes. The bounds we suggest, which are decaying exponentials, possess three convenient properties: When the inputs to an isolated network
element are all bounded, they result in bounded outputs, and assure that the delays and queues in this element have exponentially decaying distributions; In some network settings bounded inputs
result in bounded outputs; Natural traffic processes can be shown to satisfy such bounds. Consequently, our method enables the analysis of various previously intractable setups. We provide sufficient
conditions for the stability of such networks, and derive upper bounds for the interesting parameters of network performance. 1 Introduction In this paper we consider data communication networks, and
the problem of ev...
, 1995
"... We analyse the queue Q L at a multiplexer with L inputs. We obtain a large deviation result, namely that under very general conditions lim L!1 L \Gamma1 log P[Q L ? Lb] = \GammaI (b) provided
the offered load is held constant, where the shape function I is expressed in terms of the cumulant ..."
Cited by 114 (11 self)
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We analyse the queue Q L at a multiplexer with L inputs. We obtain a large deviation result, namely that under very general conditions lim L!1 L \Gamma1 log P[Q L ? Lb] = \GammaI (b) provided the
offered load is held constant, where the shape function I is expressed in terms of the cumulant generating functions of the input traffic. This provides an improvement on the usual effective
bandwidth approximation P[Q L ? b] e \Gammaffib , replacing it with P[Q L ? b] e \GammaLI(b=L) . The difference I(b) \Gamma ffi b determines the economies of scale which are to be obtained in large
multiplexers. If the limit = \Gamma lim t!1 t t (ffi) exists (here t is the finite time cumulant of the workload process) then lim b!1 (I(b) \Gamma ffi b) = . We apply this idea to a number of
examples of arrivals processes: heterogeneous superpositions, Gaussian processes, Markovian additive processes and Poisson processes. We obtain expressions for in these cases. is zero for independent
arrivals, but positive for arrivals with positive correlations. Thus economies of scale are obtainable for highly bursty traffic expected in ATM multiplexing.
- IEEE/ACM Trans. Networking , 1993
"... This paper, together with [1] and [2], opens a new window for the study of queueing performance in a richer, heterogeneous input environment. It offers a unique way to understand the effect of
second- and higher-order input statistics on queues, and develops new concepts of traffic measurement, netw ..."
Cited by 114 (28 self)
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This paper, together with [1] and [2], opens a new window for the study of queueing performance in a richer, heterogeneous input environment. It offers a unique way to understand the effect of
second- and higher-order input statistics on queues, and develops new concepts of traffic measurement, network control and resource allocation for high speed networks in the frequency domain. The
technique developed in this paper applies to the analysis of queue response to the individual effects of input power spectrum, bispectrum, trispectrum, and input rate steady state distribution. Our
study provides clear evidence that of the four input statistics, the input power spectrum is most essential to queueing analysis. Furthermore, input power in the lowfrequency band has a dominant
impact on queueing performance, whereas high-frequency power to a large extent can be neglected. The research reported here was supported by NSF under grant NCR-9015757 and by Texas Advanced Research
Program under gr...
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Nonlinear Polarization <<< | >>> | Feedback
In the RP Photonics Buyer's Guide you can find plenty of suppliers for hundreds of photonics products.
Ask RP Photonics how calculations and simulations of nonlinear processes may benefit your product development.
Definition: the part of the light-induced electric polarization which depends nonlinearly on the electric field of the light
German: nichtlineare Polarisation
When light propagates in a transparent medium, its electric field causes some amount of electric polarization in the medium, i.e. some density of electric dipole moment. (This must not be confused
with the polarization of the light field, which is the direction of its electric field.) That polarization propagates together with the electromagnetic field in the form of a polarization wave.
Whereas at low light intensities the electric polarization is proportional to the electric field strength, nonlinear contributions become important at high optical intensities, as they can e.g. be
produced with lasers.
Second-order Nonlinear Polarization
The second (lowest) order of nonlinear polarization can arise from a χ^(2) nonlinearity which can occur only in crystal materials with a non-centrosymmetric crystal structure. (Nonlinear effects at
crystal surfaces are an exception.) The nonlinear polarization then has a component which depends quadratically on the electric field of an incident light wave. More precisely, the tensor nature of
the nonlinear susceptibility needs to be considered:
where P[i] is the i-th Cartesian coordinate of the polarization, χ^(2) is the nonlinear susceptibility, and E(t) is the optical electric field. More commonly, this is written as
with the nonlinear tensor d. Many tensor components can actually be zero for symmetry reasons, depending on the crystal class.
The nonlinear polarization contains frequency components which are not present in the exciting beam(s). Light with such frequencies can then be generated in the medium (→ nonlinear frequency
conversion). For example, if the input field is monochromatic, the nonlinear polarization also has a component with twice the input frequency (→ frequency doubling). As the polarization has the form
of a nonlinear polarization wave, the frequency-doubled light is also radiated in the direction of the input beam. Other examples are sum and difference frequency generation, optical rectification,
parametric amplification and oscillation.
Third-order Nonlinear Polarization
The next higher order of nonlinear polarization can arise from a χ^(3) nonlinearity, as it occurs in basically all media. This can give rise to various phenomena:
Phase Matching
In many cases, the nonlinear mixing products can be efficiently accumulated over a greater length of crystal only if phase matching is achieved. Otherwise, the field amplitudes at the exit face,
generated at different locations in the crystal, essentially cancel each other, and the apparent nonlinearity is weak. Some nonlinear effects, however, are either automatically phase-matched (e.g.
self-phase modulation) or do not need phase matching (e.g. Raman scattering).
[1] D. A. Kleinman, “Nonlinear dielectric polarization in optical media”, Phys. Rev. 126 (6), 1977 (1962)
See also: nonlinearities, nonlinear index, polarization waves, nonlinear crystal materials, nonlinear frequency conversion, phase matching
Categories: nonlinear optics, physical foundations
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Understanding Nonlinear Vector Analysis
October 10, 2011
Today's R&D engineers and scientists tasked with researching and designing high-performance, active RF devices face many challenges. One key challenge involves characterizing the devices' nonlinear
behavior so that it can be reduced to provide linear high-power solutions. This task is especially vital in the telecommunications industry where nonlinear device behavior contributes to information
interference and the reduction in effective bandwidth.
In this industry, amplifiers are considered an indispensable component and yet due to their nonlinear behavior, are often the cause of wasted frequency spectrum. Designing a power amplifier to
operate only within its region of linear operation also results in an inefficient use of the available power. Amplifiers are often driven into the nonlinear region of operation and then linearized
around that point. Consequently, it has become increasingly important to understand the nonlinear behaviors of active RF components such as power amplifiers and frequency doublers which, in turn,
makes accurate measurement of a device's nonlinear behavior all the more crucial.
All active devices exhibit nonlinear behaviors to varying degrees and therefore, the list of possible devices that could benefit from this measurement insight can be quite large. Examples of this are
LNAs, T/R modules and converters, as well as passive devices such as inductors with magnetic cores or filters driven at high powers.
Unfortunately, making nonlinear measurements is not an easy task, especially considering that currently available tools and models for accomplishing this goal are generally difficult to use and do
not provide the information required. What's needed is a solution specifically designed to accurately measure the nonlinear effects of active RF devices. With this information, designers will then be
better equipped to control and minimize nonlinear device behavior and more accurately, easily and quickly create linear high-power solutions.
Agilent has introduced two primary methods for measuring the nonlinear effects of a device under test (DUT): nonlinear component characterization and X-parameters. Nonlinear component
characterization provides calibrated, vector-corrected waveforms of the incident, transmitted and reflected waves from the DUT. Vector calibration, power calibration and the use of a poly-harmonic
phase reference and calibration removes the systematic error terms.With this measurement, all receivers must be measured simultaneously for each frequency point. The information derived from
nonlinear component characterization measurements enables the engineer to better understand and more deterministically control the nonlinear behavior of the DUT.
X-parameters are the logical, mathematically-correct extension of S-parameters into a nonlinear, large-signal operating environment (see the sidebar, "A Closer Look at X-Parameters"). X-parameter
measurements require an additional source which is used to drive the DUT with both a large and small signal tone at the appropriate frequencies and phases, at the same time. Careful control of the
phase and amplitude of these signals is therefore critical. Measuring the amplitudes and phases of the scattered waves under these conditions allows for the identification of X-parameters. These
parameters provide the engineer with information on such things as device gain and match, while the device is operating in either a linear or nonlinear state. The X-parameters can be extracted into
Agilent's Advanced Design System (ADS) or displayed like S-parameters.
A Closer Look at X-Parameters
X-parameters are to nonlinear measurements what S-parameters are to linear measurements. Consider, for example, that S-parameters were developed as a method to analyze and model the linear behavior
of RF components and play a key role in analyzing, modeling and designing complex systems which cascade multiple individual components. They are related to familiar measurements such as S11 input
match, S22 output match, S21 gain/loss, and S12 isolation, and can be easily imported into electronic simulation tools. While extremely useful and powerful, S-parameters do have limitations and are
defined only for small-signal linear systems.
S-parameters are measured using traditional network analyzers. To a smaller extent, network analyzers can also provide insight into the nonlinear behavior of devices via approximation techniques such
as using un-ratioed receiver measurements and offsetting the measurement receivers' frequency from the source stimulus frequency. By employing these techniques, simple gain compression, harmonic
amplitude only, frequency converter match, conversion loss/gain, and group delay can be measured.
In contrast to S-parameters, X-parameters were developed to represent and analyze the nonlinear and linear behavior of RF components in a much more robust and complete manner. As an extension of
S-parameters under large-signal operating conditions, the devices are driven into saturation (the real-word operating environment for many components) and then the X-parameters are measured under
these conditions. When making this measurement, no knowledge is used or required concerning the internal circuitry of the DUT. Rather, the measurement is a stimulus response model of the voltage
waves. In other words, the absolute amplitude and cross frequency relative phase of the fundamental, and all related harmonics, are accurately measured and represented by X-parameters.
Because the X-parameters relate cross-frequency dependencies, there are usually many more X-parameters than S-parameters, such as in the case of the gain of the output fundamental frequency to the
input third harmonic. Here, there are eight X-parameters for this simple case with only one harmonic and no power dependency. In contrast, there can never be more than four S-parameters.X-parameters
also depend explicitly on the large signal state of the device, making input power a variable. In contrast, S-parameters are assumed to be power independent.
The accurate and robust nature of X-parameters makes them extremely useful for engineers and scientists trying to better understand the nonlinear behavior of their active components. As an example,
consider the design of a power amplifier in which the designer drives the amplifier into the nonlinear region to get the maximum output power and to extract the maximum efficiency. A feedback circuit
is then used to compensate for the nonlinear effects, causing the output to behave like a high-power linear device.
In this case, the typical approach to suppressing the power amplifier's harmonic outputs is through the use of filters and other components. But, if the filtering component's input match does not
match the output match of the specific harmonic of interest (generated by the amplifier), then the harmonic's attenuation level could be significantly different from what the designer anticipates.
This situation may cause the designer to brut force a solution by 'trial and error' -- a very tedious and time consuming experience at best. One way to avoid this dilemma is by obtaining accurate
phase and amplitude information from the X-parameters and then employing appropriate simulation tools. With this approach, designers can design the most robust systems possible in the shortest amount
of time and with the highest degree of accuracy.
Utilizing a solution that employs both nonlinear component characterization and X-parameters provides critical insight, essential to accurately measuring a device's nonlinear behavior. Agilent
Technologies' Nonlinear Vector Network Analyzer (NVNA) supports both methods in a highly integrated, powerful and simple to use instrument. With a minimum amount of external hardware, this solution
effectively converts a 4-port PNA-X microwave network analyzer into a high-performance nonlinear analyzer from 10 MHz to 67 GHz (Figure 1). Because the NVNA is based on a standard PNA-X microwave
network analyzer, it provides all the power, flexibility and measurement capability of the PNA-X for linear measurements. It can then easily switch into the NVNA mode for nonlinear measurement.
Figure 1: Agilent’s NVNA software, for use with the PNA-X microwave network analyzer, establishes a new industry standard in RF nonlinear network analysis from 10 MHz to 67 GHz.
With the NVNA's nonlinear component characterization, all of the DUT's input and output spectra are measured. Both the amplitude and phase of the full spectra -- fundamental, harmonics, mixing terms
and cross-frequency products -- are then displayed in the PNA-X network analyzer. The relative phase and absolute amplitude of any of the frequencies of interest can also be displayed. With this
information, the designer can design a matching circuit to remove the nonlinear effects of, for example, the third harmonic as the amplitude and phase of that spectral component is explicitly known.
This data can be displayed in frequency, time or power domains, as well as in terms of user-defined ratios such as I/V to display dynamic loadlines.
With its ability to display data in various domains, the NVNA provides the designer with greater insight into nonlinear component behavior. For example, if the DUT's output is distorted in the time
domain, the designer can change to the frequency domain display and observe the individual frequency components' amplitude and phase. Next, the power can be varied to observe the sensitivity and
level of significance which that spectral component has for given power levels, relative to the fundamental frequency of interest. The designer might also want to measure the group delay through a
frequency doubler. This is a relatively easy task with the NVNA as it can measure the input and output stimulus phase -- relative to the calibrated phase reference -- as well as the signals
amplitude. As an additional benefit, all measurement-based data collected with the NVNA can be exported to design models of the designer's choice.
The NVNA's other method of measuring nonlinear component behavior is through the use of nonlinear scattering parameters or X-parameters. Such functionality provides an accurate portrayal of both
nonlinear device and cascaded nonlinear device behavior using measurement-based data. Additionally, the X-parameters can be accurately cascaded from individual devices using Agilent's ADS to simulate
and design more complex modules and systems. These parameters separate out a critical term from the measurement, XT, which enables accurate nonlinear design and simulation by taking cross-frequency
mismatch properly into account for nonlinear components.
When a single-tone stimulus is applied to the DUT, the X-parameters capture the device behavior at the fundamental frequencies and harmonics. Using a two-tone stimulus captures the fundamentals,
harmonics and mixing products behavior. This is particularly useful in analyzing the bandwidth dependent characteristics and gaining additional insight into memory effects. X-parameters can also be
extracted from 3-port mixers and converters. One large tone is applied to the RF and one large tone is applied to the LO. All three ports will be fully characterized in a similar fashion to the
previously stated two port amplifier case. Combinations of amplifiers and mixers/converters can then be cascaded to predict and optimize module and system design and performance.
Accurately measuring and reducing a device's nonlinear behavior is crucial to creating linear high-power solutions for use in a range of applications in aerospace and defense and telecommunications,
just to name a few. While conventional solutions to this challenge fail to provide the information and accuracy required, Agilent's NVNA software uses component characterization and X-parameters to
quickly, easily and with the highest degree of accuracy measure nonlinear behavior in a DUT. With its full match correction and accurate amplitude, as well as cross-frequency relative phase
information, the NVNA is today providing a new standard in accuracy and insight into the behaviors of nonlinear components.
About Agilent Technologies
Agilent Technologies Inc. (NYSE: A) is the world's premier measurement company and a technology leader in communications, electronics, life sciences and chemical analysis. The company's 19,000
employees serve customers in more than 110 countries. Agilent had net revenues of $5.4 billion in fiscal 2007. Information about Agilent is available on the Web at www.agilent.com.
Press Release: Agilent Technologies Introduces World’s Highest Performing 67 GHz PNA-X Vector Network Analyzer
Agilent Technologies Expands Industry’s Most Flexible PNA-X Network Analyzer for Active Device Test with 13.5, 43.5, 50 GHz Models
Agilent Technologies Introduces Breakthrough Technology to Analyze Nonlinear Behaviors of Active Components
Backgrounder: Understanding Component Characterization Using the PNA-X Series Microwave Network Analyzers
Agilent Documents: For more information, go to www.agilent.com/find/nvna
Contacts: Janet Smith, Agilent
+1 970 679 5397
> More Press Releases
> More Backgrounders
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Kepler's 3rd Law and the Doppler Effect
I'm assuming that "c" is the speed of light.
What is the M? Mass of the planet?
Sorry, I forgot to define my variables: M is the planet's mass, v is the velocity relative to your line of sight, v
is the circular velocity of the orbit, [itex]\Delta \lambda[/itex] is the shift in wavelength relative to its rest frame value, [itex]\lambda[/itex] is the rest frame value of the wavelength, c is
the speed of light, a is the semimajor axis, and G is the gravitational constant.
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CSR PS-308 Split Packages
Split Bundles of Flat-Size and Irregular Parcel Mailpieces
Updated September 2005
PS-308 (705.8.9.4)
This CSR discusses balancing (or leveling) of bundles for those limited situations in which the pieces that would ordinarily constitute the “last bundle” for a presort destination are less than the
minimum amount required for the destination.
In addition, this CSR clarifies rate eligibility for the preparation of bundles of flat-size and irregular parcel mailpieces prepared under the following circumstances:
1. When the pieces that would ordinarily be in the “last bundle” or “balanced bundle” for a presort destination are fewer than the minimum number required for the destination.
2. When the weight of the last bundle or balanced bundle is less than the required minimum weight.
Mailers use software that balances (or levels) the number of pieces or the weight of their bundles. This may result in multiple bundles with fewer than the required minimum number of pieces or less
than the required minimum weight for a presort destination. The number of bundles created because of balancing may not exceed the number of bundles that would have resulted if only the last bundle
did not meet the minimum requirements.
We allow mailers to prepare a last bundle or a balanced bundle for a presort destination when the bundle is less than the minimum weight or it contains fewer than the minimum number of pieces without
loss of rate eligibility providing the “logical bundle” to a presort destination meets the minimum quantity for the rate claimed.
For example, consider two mailings of bundles of flat-size Standard Mail pieces that mailers might bundle and place on pallets:
● Mailing A has 17 pieces to a presort destination; 12 pieces in one bundle reach the maximum height for one bundle to maintain its integrity, leaving just 5 pieces for the “last bundle."
The mailer could make one bundle of 12 pieces and one “last bundle” of 5 pieces (a total of two bundles) as allowed under DMM 705.8.9.4*. Alternatively, the mailer could balance the bundles by
placing 9 pieces in one bundle and 8 pieces in the other bundle (still just two bundles).
● Mailing B has 18 pounds of pieces to a presort destination; 10 pounds in one bundle reach the maximum height for one bundle to maintain its integrity.
This mailer could make one bundle of 10 pounds and one “last bundle” of 8 pounds (a total of two bundles); or the mailer could balance the bundles by making two bundles of 9 pounds each (still just 2
In both cases, the mailers are creating the same number of bundles, but are balancing them to create more stable bundles. Bundle balancing may result in bundles to one presort destination having less
than the minimum number of pieces or being of less than the minimum weight. However, in all cases, this must NOT create additional bundles for the Postal Service to handle.
Similar balancing (or leveling) also is acceptable for bundles of flat-size Periodicals and Package Services mailpieces placed in sacks or on pallets, bundles of Standard Mail flat-size pieces placed
in sacks, bundles of Standard Mail, Periodicals, and Package Services irregular parcels placed in sacks or on pallets, and for bundles of First-Class Mail flat-size mailpieces placed in flat trays.
*See also DMM 705.8.9.3, 705.8.9.5, 335.2.7, 345.2.10, 365.2.8, 375.2.7, 385.2.7, 445.2.8, 465.2.6, 475.2.6, and 485.2.6.
Sherry Suggs
Mailing Standards
United States Postal Service
Washington DC 20260-3436
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Great Neck Estates, NY
Find a Great Neck Estates, NY Calculus Tutor
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Local ring
December 5th 2010, 11:17 AM
Local ring
Could You help me with this task?
"Proof that a commutative ring a is a local ring iff for any $a, b \in A$, from this that $a+b=1$ we have $a$ is an invertible element or $b$ is an invertible element."
Proof. $(\Rightarrow)$$a,b\in A$ - any element which satysfied equality $a+b=1$ and suppose that $a,botin U(A)$, where $U(A)$ is a set of invertible elements of $A$. We have then that $a,b\in \
mathfrak{m}$. Since $\mathfrak{m}$ is an ideal, then also $1=a+b\in \mathfrak{m}$, and so $\mathfrak{m}=A$. This is contradiction, so $a \in U(A)$ or $b \in U(A)$.
$(\Leftarrow)$ - I'm not sure. Could You help me with this implication?
December 5th 2010, 12:20 PM
Could You help me with this task?
"Proof that a commutative ring a is a local ring iff for any $a, b \in A$, from this that $a+b=1$ we have $a$ is an invertible element or $b$ is an invertible element."
Proof. $(\Rightarrow)$$a,b\in A$ - any element which satysfied equality $a+b=1$ and suppose that $a,botin U(A)$, where $U(A)$ is a set of invertible elements of $A$. We have then that $a,b\in \
mathfrak{m}$. Since $\mathfrak{m}$ is an ideal, then also $1=a+b\in \mathfrak{m}$, and so $\mathfrak{m}=A$. This is contradiction, so $a \in U(A)$ or $b \in U(A)$.
$(\Leftarrow)$ - I'm not sure. Could You help me with this implication?
Let $\displaystyle{S:=A-U(A)$ , and take $x,y\in S$ . If $x+yotin S$ then
$x+y\in U(A)\Longrightarrow \exists c\in A\,\,s.t.\,\,1=c(x+y)=cx+cy\Longrightarrow cx\in U(A)\,\,or\,\,cy\in U(A)$ ,
but in both cases we get a straightforward contradiction (can you see why?) , so it must be
that $x+y\in S$ and from here $S$ is an ideal, and we're thus done.
December 5th 2010, 12:53 PM
I found longer solution to this implication. Your is shorter and easier ;)
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WebSCHEDULE - Course Details
MATH 695 - Independent Study at Cañada College for Spring 2013 (CRN : 43851) Most Mathematics courses have prerequisites that are listed as part of the course description in the Schedule of Classes.
Before registering for a Mathematics course, be sure you have completed the stated prerequisite.
Note to all Algebra students:
The Math Department uses a single textbook for the Elementary and Intermediate Algebra sequence. This allows students to complete the Algebra sequence in three different ways: a four semester
sequence of MATH 111, 112, 122, and 123 each covering one fourth of the book; a two semester sequence of MATH 110 and MATH 120 each covering half of the book; or a combination of the above. Please
see your counselor to be sure you take the correct course.
MATH 695 INDEPENDENT STUDY
Units (Grade Option) 0.5-6 (No more than 3 units per semester); Class Hours: By Arrangement; Recommended: Eligibility for READ 836, and ENGL 836 or ESL 400; Prerequisite(s): None.
Description: Self-paced, individualized instruction is provided in selected areas to be arranged with an instructor and student and approved by the dean. Varying modes of instruction can be used --
lecture, laboratory, research, skill development, etc. May be repeated for credit up to 6 units. Transfer: CSU.
Cañada College
MAP Department:
Number of Units:
Textbook: View textbook in Bookstore
Section Information as of Sunday, April 20th 2014 - 09:41:58 pm
│Instructor│Meeting Date│Meeting Time│Days│Building│Room│Section│ Session │
│Enriquez,A│01/14-05/17 │TBA │ │16 │0108│AA │Other Independent Study │
┃ Critical Dates for this Course ┃
┃Last day to add class: │28-JAN-2013 ┃
┃Last day to drop with a refund: │28-JAN-2013 ┃
┃Last day to drop without a "W": │03-FEB-2013 ┃
┃Last day to drop with a "W": │25-APR-2013 ┃
┃Transferable: │CSU ┃
View all sections for MATH 695 - Independent Study
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iterative deepening
iterative deepening
<algorithm> A graph search algorithm that will find the shortest path with some given property, even when the graph contains cycles. When searching for a path through a graph, starting at a given
initial node, where the path (or its end node) has some desired property, a depth-first search may never find a solution if it enters a cycle in the graph. Rather than avoiding cycles (i.e. never
extend a path with a node it already contains), iterative deepening explores all paths up to length (or "depth") N, starting from N=0 and increasing N until a solution is found.
Last updated: 2004-01-26
Try this search on Wikipedia, OneLook, Google
Nearby terms: ITAR « Iterated Function System « iteration « iterative deepening » iterator » Iternet » IT governance
Copyright Denis Howe 1985
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markov process and classification of states
April 18th 2011, 06:56 PM #1
Mar 2010
markov process and classification of states
a)classify each of the states for the following markov chain (ie transient, recurent absorbing) and state how many classes there are
0.3 0.7 0
0.3 0 0.7
i have that each state is recurent and there is a single recurent class.
b)show that this markov chain is aperiodic without computing P^n (ie use the fact that if we can transition to a state in integer values>1 then the chain has no specific period.
i am still trying to make sense of the periodicity concept. Does it mean that we can get back to a certain state, having started there after transitioning to all other possible states in only
certain integer valus or multiples of this. Or does it mean that we can get back to a certain state having started there in any number of steps, but it is not actually necassary to transition
through all possible states.
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Posted by Jasmine20 on Wednesday, January 24, 2007 at 10:42pm.
Okay now i need explanation how to do these types of problems.
Directions: Divide
Realize that you have 7x^3y^3 on the bottom. Now in order to get rid of that, you want to have the same thing in the top, multiplied by something else. That way, the 7x^3y^3 on the top and bottom
will cross out, because any number over itself is 1. So, take out 7x^3y^3 from the numerator:
Now once you cross out the (7x^3y^3) from the fraction, you are left with x^2y^2-3xy+2. So, to factor this, you need the beginnings of the parentheses to be xy, so when you multiply xy times xy you
will get x^2y^2.
(xy )(xy )
Now, you know that the ends of the parentheses must multiply to 2, so they will be 2 and 1, but we don't know if they are positive and negative yet.
(xy 2)(xy 1)
Now, you want the outer terms multiplied plus the inner terms multiplied to equal negative 3. So, if the 2 and 1 were positive, we'd end up with +3 and we want -3, so we know the signs inside must be
where did you get:
the -3xy+2
I took out 7x^3y^3 from everything in the numerator. So, from the 7x^5y^5, I took out 7x^3y^3 and separated it from x^2y^2, because if you multiply those 2, you will end up with the original 7x^5y^5.
I did the same for -21x^4y^4. If you remove from that 7x^3y^3 (basically, what will you multiply by 7x^3y^3 to get back to -21x^4y^4?), you will get -3xy. For the last part, 7x^3y^3 times 2 is the
original, 14x^3y^3. Wow, that sounds way more confusing than it is. If you ask your teacher to show you it is much easier written and pointed out than me trying to explain it on here.
• math,algebra,help - Tina, Monday, April 25, 2011 at 6:45pm
3y divided by 7
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Summary: An Invariant-Sum Characterization of Benford's Law
Pieter C. Allaart
Vrije Universiteit Amsterdam1
The accountant Nigrini remarked that in tables of data distributed according to
Benford's Law, the sum of all elements with rst digit d (d = 1 2 :: 9) is approxi-
mately constant. In this note, a mathematical formulation of Nigrini's observation
is given and it is shown that Benford's Law is the unique probability distribution
such that the expected sum of all elements with rst digits d1 :: dk is constant for
every xed k.
Keywords and phrases: First signi cant digit, Benford's law, mantissa function,
AMS 1990 subject classi cation: 60A10.
1 Introduction
The main goal of this article is to give a mathematical proof of an empirical observation
of the accountant M. Nigrini. In his Ph.D. thesis (1992), Nigrini observed that tables of
unmanipulated accounting data closely follow Benford's Law (see x2 below), and that
in su ciently long lists of data for which Benford's Law holds,
the sum of all entries with leading digit d is constant for various d.
(cf. Nigrini, 1992, pp. 70/71).
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Process Standards
Posted by: degarcia | November 24, 2008
Process Standards
The five fundamental processes that characterize “doing” mathematics are problem solving, communication, reasoning and proof, representation, and connections.
(Elementary Mathematics is Anything but Elementary p. 5)
Problem Solving
Problem solving in math means “becoming involved in a task for which the solution method is not known in advance. To find a solution, students must use previously acquired knowledge and, through this
process, gain new mathematical understandings” (Bahr & Garcia, 2010). Problem solving should be an integral part of daily math. Problems can be drawn from real life experiences and applications. The
problems selected should be carefully analyzed so the teacher can predict the mathematical ideas that will underscore students’ problem solving efforts.
To promote problem solving, students must feel safe in their learning environment. They need to know they are free to explore, take risks, share, and argue strategies with one another. This type of
environment will also allow them to confidently work through problems and challenges, and view themselves as capable mathematical thinkers. Specific problem solving strategies, or heuristics, is also
a focus of the teacher. This includes things like simplifying the complexity of a problem, drawing a diagram or picture of a problem, seeing patterns, guessing and checking, and working backwards.
Problem solving also has a metacognitive component. Teachers can promote this level of thoughtfulness by asking questions:
• What do we know from the information in this problem?
• What information does it ask for?
• Is there any missing information that might make solving the problem easier?
• What should we do first, second, etc.?
• Are we getting closer to the solution?
• Should we try something else?
• Why is that true?
These types of questions places the responsibility for success on the students.
Mathematics communication is both a means of transmission and a component of what it means to “do” mathematics. Teacher have to provide an environment in which students can risk expressing their
beginning efforts to communicate their thinking. Teachers must be patient while students begin to do this, because communicating in math doesn’t come naturally to students.
The NCTM standards provide a complete list of standards in communication and the benefits of communicating in math. To view them, click here.
When students share, they should genuinely listen to one another, compare it to their own ideas, evaluate it, then share their own opinions. Teachers can use probing and prompting questions during
discussions as scaffolding. In older grades, students should be encouraged to elaborate more.
Writing in math is also beneficial to deepening mathematical understanding. In the primary grades, they rely more on pictures and as they get older will be able to form more complete sentences and
thoughts. Writing in math also allows them to practice using mathematical vocabulary and symbols. Their writing skills are consequently enhanced as they practice justifying and writing in this
expository form. Just like writing in any other content area, the teacher will have to model how this should be done effectively.
Reasoning and Proof
Reasoning is a habit and should provide a context for developing important mathematical ideas. Questioning is the key! Ask WHY? Mathematics involves discovery, so invite students to make conjectures
and create, refine, and evaluate them. Also, allow students to explore and explain their own reasoning. It’s often best to start students off with what they know, and then build from there.
This is where students can take advantage of manipulatives and using technology to solve problems and explore their conjectures. Several virtual manipulative websites exist to get the same practice
with manipulatives while utilizing technology, especially if the manipulatives in the classroom are limited. One such website is found here. Using manipulatives will help reinforce concepts to all
students, especially students with learning disabilities and English language learners.
Sometimes younger children need a discrepant or contradictory event to verify their reasoning. They tend to overgeneralize an idea, which means they may apply reasoning from one context to a context
where the same reasoning does not really apply. Through enough exploration and discovery the students will be able to accommodate and assimilate the new mathematical reasoning into the correct
Also encourage students to look for patterns. These patterns can be spatial, temporal, logical, and sequential.
This is asking students to show a mathematical idea in more than one way. There are five ways to represent thinking:
1) manipulative models
2) static pictures
3) written symbols
4) spoken/written language
5) real-world situations or contexts
Real- life situations are very valuable to the students because it gives them something more concrete to work with, and they begin to see the real purpose and meaning behind using the mathematical
concept. Because adults think in symbols and children do not, children support their thinking with examples they have seen in the real world.
Representations are used by children first to display the problem, then to find a solution, and finally use tools to solve similar problems. This will especially be useful to special needs children
and English language learners to use situations they are familiar with.
The following are examples of solving problems using models, pictures, numbers and words.
“Connecting is the experience of mentally relating one object to another” (Bahr & Garcia, 2010). Elementary Mathematics is Anything but Elementary (2010) identifies six types of connections
distinguished by what types of thoughts are being connected:
1) representations
2) problem solving strategies or conjectures
3) prior and current math learning
4) mathematical topics
5) mathematics and other subjects
6) mathematics and real-life situations
If you encourage these connections, it will increase your students’ mathematical reasoning abilities. One of the roles of teachers is to compare strategies students share and help students see the
connections between those strategies. It is essential to make these connections with prior math learning, so the learning is logical and builds from what students know. Just when you build a
structure, you need to have a firm foundation before you can start building the structure. The same is true in math. When students make connections between math concepts it’s like they have formed a
neighborhood of strong buildings and can see how the neighborhood works and functions together, instead of each building functioning or existing on its own. Children learn about the world in
connected ways, so balanced math instruction will help children do this.
Integrating other subjects with math has to be meaningful. Therefore, if you simply read a book with a math concept in it, that is not successful integration. Here is also a website that has
successful ways to integrate math with literature, history, geography, health, art, and music.
“When mathematics is consistently used to solve problems in other subject area contexts, connections, to real life occur consistently” (Bahr & Garcia, 2010).
Bahr, Damon L. & Lisa Ann de Garcia. (2010). Elementary mathematics is anything but elementary. USA: Wadsworth, Cengage Learning.
Lesh, R. A., T. R. Post, and M. J. Behr. 1987. Representations and translations among representations in mathematics learning and problem solving. In Problems of representation in the teaching and
learning of mathematics, ed. C. Janvier, 33-40. Hillsdale, NJ: Lawrence Erlbaum.
Posted in Pedagogy | Tags: Process Standards in Math
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Lawrence Leff, M.S. - All books by this author
Barron’s E-Z Series - All books in this series
Known for many years as Barron’s Easy Way Series, the new editions of these popular self-teaching titles are now Barron’s E-Z Series. Brand-new cover designs reflect all new page layouts, which
feature extensive two-color treatment, a fresh, modern typeface, and more graphic material than ever— charts, graphs, diagrams, instructive line illustrations, and where appropriate, amusing
cartoons. Meanwhile, the quality of the books’ contents remains at least as high as ever. Barron’s E-Z books are self-help manuals focused to improve students’ grades in a wide variety of academic
and practical subjects. For most subjects, the level of difficulty ranges between high school and college-101 standards. Although primarily designed as self-teaching manuals, these books are also
preferred by many teachers as classroom supplements—and for some courses, as main textbooks. E-Z books review their subjects in detail, and feature both short quizzes and longer tests with answers to
help students gauge their learning progress. Subject heads and key phrases are set in a second color as an easy reference aid.
An experienced math teacher breaks down Barron’s E-Z Precalculus into easy-to-follow lessons for self-teaching and rapid learning. The book features a generous number of step-by-step demonstration
examples as well as tables, graphs, and graphing-calculator-based approach. Major topics include: algebraic methods; functions and their graphs; complex numbers; polynomial and rational functions;
exponential and logarithmic functions; trigonometry and polar coordinates; counting and probability; binomial theorem; calculus preview; and much more. Exercises at the end of each chapter.
About The Author:
Lawrence Leff is Assistant Principal and Chairman of the Mathematics Department at Franklin D. Roosevelt High School, New York City.
Table of Contents:
STUDY UNIT I: ALGEBRA AND GRAPHING METHODS
● Basic Algebraic Methods
● Rational and Irrational Expressions
● Graphing and Systems of Equations
● Functions and Quadratic Equations
● Complex Numbers and the Quadratic Formula
STUDY UNIT II: FUNCTIONS AND THEIR GRAPHS
● Special Functions and Equations
● Polynomial and Rational Functions
● Exponential and Logarithmic Functions
STUDY UNIT III: TRIGONOMETRIC ANALYSIS
● Trigonometry
● Graphing Trigonometric Functions
● Trigonometric Identities and Equations
● Solving Triangles
STUDY UNIT IV: POLAR COORDINATES AND CONIC SECTIONS
● Polar Coordinates and Parametric Equations
● Conic Sections and Their Equations
STUDY UNIT V: NUMBER PATTERNS AND CALCULUS PREVIEW
● Sequences, Series, and Counting
● Calculus Preview
Answers to Checkup Exercises
Paperback / 480 Pages / 7 13/16 x 10 / 2010
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Kids.Net.Au - Encyclopedia > Coulomb's Law
In physics, Coulomb's Law is an inverse-square law indicating the magnitude and direction of electrical force that one stationary, electrically charged substance of small volume exerts on another.
When one is interested only in the magnitude of the force (and not in its direction), it may be easiest to consider a simplified, scalar version of the Law
F = \frac{\left|q_1 q_2\right|}{4 \pi \epsilon_0 r^2} </math>
where q[1] is the charge on one substance, q[2] is the charge on the other, r is the distance between them, and ε[0] is a universal constant, the permittivity of vacuum. (See physical constants for
more information. Note that 1/(μ[0]ε[0]) = c^2 × 10^-7, where μ[0] is the permeability of vacuum and c is the speed of light.)
Among other things, this formula says that the magnitude of the force is directly proportional to the magnitude of the charges of each substance and inversely proportional to the square of the
distance between them.
The force F acts on the line connecting the two charged objects.
For calculating the direction and magnitude of the force simultaneously, one will wish to consult the full-blown vector version of the Law
\mathbf{F} = \frac{q_1 q_2 \mathbf{r}}{4 \pi \epsilon_0 \left|\mathbf{r}\right|^3} </math>
where the vector r connects the two substances, and the other symbols are as before.
(The r vector in the numerator indicates that the force should be along the vector connecting the two substances. |r| has been raised to the third power instead of the second in the denominator in
order to normalize the length of the r in the numerator to 1.)
In either formulation, Coulomb's Law is fully accurate only when the substances are static (i.e. stationary), and remain approximately correct only for slow movement. When movement takes places,
magnetic fields are produced that alter the force on the two substances. Especially when rapid movement takes place, the electric field will also undergo a transformation described by Einstein's
theory of relativity.
All Wikipedia text is available under the terms of the GNU Free Documentation License
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How to Divide Double Digits
Edit Article
Edited by N'cha!, Trigtchr, FlaminPunkMonkey, Flickety and 23 others
Dividing double digits is very similar to long division with a single-digit divisor, but it does require some extra multiplication and thinking.
1. 1
Set your problem up in long division format:
☆ Example: 236 & divide; 28 becomes
2. 2
Guess how many times the divisor (the 28, in our example) can go into the dividend (the 236, in our example), then write down that number.
3. 3
Multiply your guess and the divisor (which in our example is 28) and write the result under the original dividend.
4. 4
Subtract the dividend and the multiplication result from step 3.
5. 5
Continue the guess-multiply-subtract process until you reach zero OR a subtotal which is smaller than the divisor.
6. 6
Decide how to deal with the remainder. When the result of the subtraction is smaller than the divisor, you have a remainder. You can write the remainder as a fraction, using the divisor as the
□ In our example, the answer would be 8 12/28, which would reduce in lowest terms to 8 3/7.
7. 7
If you want to produce a decimal rather than a fraction, you need to add a ".0" to the end of your original dividend. (In our example, the 236 becomes 236.0)
8. 8
Bring down the zero and stick it on the end of your latest subtraction result.
9. 9
Estimate how many times your divisor can go into this new subtotal and write that down.
10. 10
Multiply again...
11. 11
...then subtract again.
12. 12
Keep repeating the "stick on a zero/estimate/multiply/subtract" process until you have enough decimal places OR until it subtracts to zero, whichever comes first.
• In this example, we were working with 28. Keep in mind that 10 x 28 = 280, which means that 5 x 28 is half of that, or 140. Since 236 is between 280 and 140, your first guess should be between 5
and 10. That's one reason why 8 is a good number.
• If, at any point, your subtraction results in a negative number, your guess was too high. Erase that entire step and try a smaller guess.
• If, at any point, your subtraction results in a number larger than your divisor, your guess wasn't high enough. Erase that entire step and try a larger guess.
Things You'll Need
• Pencil
• Paper
• Calculator (the quickest way)
Article Info
Thanks to all authors for creating a page that has been read 83,371 times.
Was this article accurate?
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What Constraints Entail: Part 2
Thu 3 Nov 2011
Last time we derived an entailment relation for constraints, now let's get some use out of it.
Reflecting Classes and Instances
Most of the implications we use on a day to day basis come from our class and instance declarations, but last time we only really dealt with constraint products.
For example given:
#if 0
class Eq a => Ord a
instance Eq a => Eq [a]
we could provide the following witnesses
ordEq :: Ord a :- Eq a
ordEq = Sub Dict
eqList :: Eq a :- Eq [a]
eqList = Sub Dict
But this would require a lot of names and become remarkably tedious.
So lets define classes to reflect the entailment provided by class definitions and instance declarations and then use them to reflect themselves.
class Class b h | h -> b where
cls :: h :- b
infixr 9 :=>
class b :=> h | h -> b where
ins :: b :- h
instance Class () (Class b a) where cls = Sub Dict
instance Class () (b :=> a) where cls = Sub Dict
Now we can reflect classes and instances as instances of Class and (:=>) respectively with:
-- class Eq a => Ord a where ...
instance Class (Eq a) (Ord a) where cls = Sub Dict
-- instance Eq a => Eq [a] where ...
instance Eq a :=> Eq [a] where ins = Sub Dict
That said, instances of Class and Instance should never require a context themselves, because the modules that the class and instance declarations live in can't taken one, so we can define the
following instances which bootstrap the instances of (:=>) for Class and (:=>) once and for all.
#ifdef UNDECIDABLE
instance Class b a => () :=> Class b a where ins = Sub Dict
instance (b :=> a) => () :=> b :=> a where ins = Sub Dict
These two instances are both decidable, and following a recent bug fix, the current version of GHC HEAD supports them, but my local version isn't that recent, hence the #ifdef.
We can also give admissable-if-not-ever-stated instances of Class and (:=>) for () as well.
instance Class () () where cls = Sub Dict
instance () :=> () where ins = Sub Dict
Reflecting the Prelude
So now that we've written a handful of instances, lets take the plunge and just reflect the entire Prelude, and (most of) the instances for the other modules we've loaded.
instance Class () (Eq a) where cls = Sub Dict
instance () :=> Eq () where ins = Sub Dict
instance () :=> Eq Int where ins = Sub Dict
instance () :=> Eq Bool where ins = Sub Dict
instance () :=> Eq Integer where ins = Sub Dict
instance () :=> Eq Float where ins = Sub Dict
instance () :=> Eq Double where ins = Sub Dict
instance Eq a :=> Eq [a] where ins = Sub Dict
instance Eq a :=> Eq (Maybe a) where ins = Sub Dict
instance Eq a :=> Eq (Complex a) where ins = Sub Dict
instance Eq a :=> Eq (Ratio a) where ins = Sub Dict
instance (Eq a, Eq b) :=> Eq (a, b) where ins = Sub Dict
instance (Eq a, Eq b) :=> Eq (Either a b) where ins = Sub Dict
instance () :=> Eq (Dict a) where ins = Sub Dict
instance () :=> Eq (a :- b) where ins = Sub Dict
instance Class (Eq a) (Ord a) where cls = Sub Dict
instance () :=> Ord () where ins = Sub Dict
instance () :=> Ord Bool where ins = Sub Dict
instance () :=> Ord Int where ins = Sub Dict
instance ():=> Ord Integer where ins = Sub Dict
instance () :=> Ord Float where ins = Sub Dict
instance ():=> Ord Double where ins = Sub Dict
instance () :=> Ord Char where ins = Sub Dict
instance Ord a :=> Ord (Maybe a) where ins = Sub Dict
instance Ord a :=> Ord [a] where ins = Sub Dict
instance (Ord a, Ord b) :=> Ord (a, b) where ins = Sub Dict
instance (Ord a, Ord b) :=> Ord (Either a b) where ins = Sub Dict
instance Integral a :=> Ord (Ratio a) where ins = Sub Dict
instance () :=> Ord (Dict a) where ins = Sub Dict
instance () :=> Ord (a :- b) where ins = Sub Dict
instance Class () (Show a) where cls = Sub Dict
instance () :=> Show () where ins = Sub Dict
instance () :=> Show Bool where ins = Sub Dict
instance () :=> Show Ordering where ins = Sub Dict
instance () :=> Show Char where ins = Sub Dict
instance Show a :=> Show (Complex a) where ins = Sub Dict
instance Show a :=> Show [a] where ins = Sub Dict
instance Show a :=> Show (Maybe a) where ins = Sub Dict
instance (Show a, Show b) :=> Show (a, b) where ins = Sub Dict
instance (Show a, Show b) :=> Show (Either a b) where ins = Sub Dict
instance (Integral a, Show a) :=> Show (Ratio a) where ins = Sub Dict
instance () :=> Show (Dict a) where ins = Sub Dict
instance () :=> Show (a :- b) where ins = Sub Dict
instance Class () (Read a) where cls = Sub Dict
instance () :=> Read () where ins = Sub Dict
instance () :=> Read Bool where ins = Sub Dict
instance () :=> Read Ordering where ins = Sub Dict
instance () :=> Read Char where ins = Sub Dict
instance Read a :=> Read (Complex a) where ins = Sub Dict
instance Read a :=> Read [a] where ins = Sub Dict
instance Read a :=> Read (Maybe a) where ins = Sub Dict
instance (Read a, Read b) :=> Read (a, b) where ins = Sub Dict
instance (Read a, Read b) :=> Read (Either a b) where ins = Sub Dict
instance (Integral a, Read a) :=> Read (Ratio a) where ins = Sub Dict
instance Class () (Enum a) where cls = Sub Dict
instance () :=> Enum () where ins = Sub Dict
instance () :=> Enum Bool where ins = Sub Dict
instance () :=> Enum Ordering where ins = Sub Dict
instance () :=> Enum Char where ins = Sub Dict
instance () :=> Enum Int where ins = Sub Dict
instance () :=> Enum Integer where ins = Sub Dict
instance () :=> Enum Float where ins = Sub Dict
instance () :=> Enum Double where ins = Sub Dict
instance Integral a :=> Enum (Ratio a) where ins = Sub Dict
instance Class () (Bounded a) where cls = Sub Dict
instance () :=> Bounded () where ins = Sub Dict
instance () :=> Bounded Ordering where ins = Sub Dict
instance () :=> Bounded Bool where ins = Sub Dict
instance () :=> Bounded Int where ins = Sub Dict
instance () :=> Bounded Char where ins = Sub Dict
instance (Bounded a, Bounded b) :=> Bounded (a,b) where ins = Sub Dict
instance Class () (Num a) where cls = Sub Dict
instance () :=> Num Int where ins = Sub Dict
instance () :=> Num Integer where ins = Sub Dict
instance () :=> Num Float where ins = Sub Dict
instance () :=> Num Double where ins = Sub Dict
instance RealFloat a :=> Num (Complex a) where ins = Sub Dict
instance Integral a :=> Num (Ratio a) where ins = Sub Dict
instance Class (Num a, Ord a) (Real a) where cls = Sub Dict
instance () :=> Real Int where ins = Sub Dict
instance () :=> Real Integer where ins = Sub Dict
instance () :=> Real Float where ins = Sub Dict
instance () :=> Real Double where ins = Sub Dict
instance Integral a :=> Real (Ratio a) where ins = Sub Dict
instance Class (Real a, Enum a) (Integral a) where cls = Sub Dict
instance () :=> Integral Int where ins = Sub Dict
instance () :=> Integral Integer where ins = Sub Dict
instance Class (Num a) (Fractional a) where cls = Sub Dict
instance () :=> Fractional Float where ins = Sub Dict
instance () :=> Fractional Double where ins = Sub Dict
instance RealFloat a :=> Fractional (Complex a) where ins = Sub Dict
instance Integral a :=> Fractional (Ratio a) where ins = Sub Dict
instance Class (Fractional a) (Floating a) where cls = Sub Dict
instance () :=> Floating Float where ins = Sub Dict
instance () :=> Floating Double where ins = Sub Dict
instance RealFloat a :=> Floating (Complex a) where ins = Sub Dict
instance Class (Real a, Fractional a) (RealFrac a) where cls = Sub Dict
instance () :=> RealFrac Float where ins = Sub Dict
instance () :=> RealFrac Double where ins = Sub Dict
instance Integral a :=> RealFrac (Ratio a) where ins = Sub Dict
instance Class (RealFrac a, Floating a) (RealFloat a) where cls = Sub Dict
instance () :=> RealFloat Float where ins = Sub Dict
instance () :=> RealFloat Double where ins = Sub Dict
instance Class () (Monoid a) where cls = Sub Dict
instance () :=> Monoid () where ins = Sub Dict
instance () :=> Monoid Ordering where ins = Sub Dict
instance () :=> Monoid [a] where ins = Sub Dict
instance Monoid a :=> Monoid (Maybe a) where ins = Sub Dict
instance (Monoid a, Monoid b) :=> Monoid (a, b) where ins = Sub Dict
instance Class () (Functor f) where cls = Sub Dict
instance () :=> Functor [] where ins = Sub Dict
instance () :=> Functor Maybe where ins = Sub Dict
instance () :=> Functor (Either a) where ins = Sub Dict
instance () :=> Functor ((->) a) where ins = Sub Dict
instance () :=> Functor ((,) a) where ins = Sub Dict
instance () :=> Functor IO where ins = Sub Dict
instance Class (Functor f) (Applicative f) where cls = Sub Dict
instance () :=> Applicative [] where ins = Sub Dict
instance () :=> Applicative Maybe where ins = Sub Dict
instance () :=> Applicative (Either a) where ins = Sub Dict
instance () :=> Applicative ((->)a) where ins = Sub Dict
instance () :=> Applicative IO where ins = Sub Dict
instance Monoid a :=> Applicative ((,)a) where ins = Sub Dict
instance Class (Applicative f) (Alternative f) where cls = Sub Dict
instance () :=> Alternative [] where ins = Sub Dict
instance () :=> Alternative Maybe where ins = Sub Dict
instance Class () (Monad f) where cls = Sub Dict
instance () :=> Monad [] where ins = Sub Dict
instance () :=> Monad ((->) a) where ins = Sub Dict
instance () :=> Monad (Either a) where ins = Sub Dict
instance () :=> Monad IO where ins = Sub Dict
instance Class (Monad f) (MonadPlus f) where cls = Sub Dict
instance () :=> MonadPlus [] where ins = Sub Dict
instance () :=> MonadPlus Maybe where ins = Sub Dict
Of course, the structure of these definitions is extremely formulaic, so when template-haskell builds against HEAD again, they should be able to be generated automatically using splicing and reify,
which would reduce this from a wall of text to a handful of lines with better coverage!
An alternative using Default Signatures and Type Families
Many of the above definitions could have been streamlined by using default definitions. However, MPTCs do not currently support default signatures. We can however, define Class and (:=>) using type
families rather than functional dependencies. This enables us to use defaulting, whenever the superclass or context was ().
#if 0
class Class h where
type Sup h :: Constraint
type Sup h = ()
cls :: h :- Sup h
default cls :: h :- ()
cls = Sub Dict
class Instance h where
type Ctx h :: Constraint
type Ctx h = ()
ins :: Ctx h :- h
default ins :: h => Ctx h :- h
ins = Sub Dict
instance Class (Class a)
instance Class (Instance a)
#ifdef UNDECIDABLE
instance Class a => Instance (Class a)
instance Instance a => Instance (Instance a)
instance Class ()
instance Instance ()
This seems at first to be a promising approach. Many instances are quite small:
#if 0
instance Class (Eq a)
instance Instance (Eq ())
instance Instance (Eq Int)
instance Instance (Eq Bool)
instance Instance (Eq Integer)
instance Instance (Eq Float)
instance Instance (Eq Double)
But those that aren't are considerably more verbose and are much harder to read off than the definitions using the MPTC based Class and (:=>).
#if 0
instance Instance (Eq [a]) where
type Ctx (Eq [a]) = Eq a
ins = Sub Dict
instance Instance (Eq (Maybe a)) where
type Ctx (Eq (Maybe a)) = Eq a
ins = Sub Dict
instance Instance (Eq (Complex a)) where
type Ctx (Eq (Complex a)) = Eq a
ins = Sub Dict
instance Instance (Eq (Ratio a)) where
type Ctx (Eq (Ratio a)) = Eq a
ins = Sub Dict
instance Instance (Eq (a, b)) where
type Ctx (Eq (a,b)) = (Eq a, Eq b)
ins = Sub Dict
instance Instance (Eq (Either a b)) where
type Ctx (Eq (Either a b)) = (Eq a, Eq b)
ins = Sub Dict
Having tested both approaches, the type family approach led to a ~10% larger file size, and was harder to read, so I remained with MPTCs even though it meant repeating "where ins = Sub Dict" over and
In a perfect world, we'd gain the ability to use default signatures with multiparameter type classes, and the result would be considerably shorter and easier to read!
Fake Superclasses
Now, that we have all this machinery, it'd be nice to get something useful out of it. Even if we could derive it by other means, it'd let us know we weren't completely wasting our time.
Let's define a rather horrid helper, which we'll only use where a and b are the same constraint being applied to a newtype wrapper of the same type, so we can rely on the fact that the dictionaries
have the same representation.
evil :: a :- b
evil = unsafeCoerce refl
We often bemoan the fact that we can't use Applicative sugar given just a Monad, since Applicative wasn't made a superclass of Monad due to the inability of the Haskell 98 report to foresee the
future invention of Applicative.
There are rather verbose options to get Applicative sugar for your Monad, or to pass it to something that expects an Applicative. For instance you can use WrappedMonad from Applicative. We reflect
the relevant instance here.
instance Monad m :=> Applicative (WrappedMonad m) where ins = Sub Dict
Using that instance and the combinators defined previously, we can obtain the following
applicative :: forall m a. Monad m => (Applicative m => m a) -> m a
applicative m =
m \\ trans (evil :: Applicative (WrappedMonad m) :- Applicative m) ins
Here ins is instantiated to the instance of (:=>) above, so we use trans to compose ins :: Monad m :- Applicative (WrappedMonad m) with evil :: Applicative (WrappedMonad m) :- Applicative m to obtain
an entailment of type Monad m :- Applicative m in local scope, and then apply that transformation to discharge the Applicative obligation on m.
Now, we can use this to write definitions. [Note: Frustratingly, my blog software inserts spaces after <'s in code]
(< &>) :: Monad m => m a -> m b -> m (a, b)
m < &> n = applicative $ (,) < $> m < *> n
Which compares rather favorably to the more correct
(< &>) :: Monad m => m a -> m b -> m (a, b)
m < &> n = unwrapMonad $ (,) < $> WrapMonad m < *> WrapMonad n
especially considering you still have access to any other instances on m you might want to bring into scope without having to use deriving to lift them onto the newtype!
Similarly you can borrow < |> and empty locally for use by your MonadPlus with:
instance MonadPlus m :=> Alternative (WrappedMonad m) where ins = Sub Dict
alternative :: forall m a. MonadPlus m => (Alternative m => m a) -> m a
alternative m =
m \\ trans (evil :: Alternative (WrappedMonad m) :- Alternative m) ins
The correctness of this of course relies upon the convention that any Applicative and Alternative your Monad may have should agree with its Monad instance, so even if you use Alternative or
Applicative in a context where the actual Applicative or Alternative instance for your particular type m is in scope, it shouldn't matter beyond a little bit of efficiency which instance the compiler
picks to discharge the Applicative or Alternative obligation.
Note: It isn't that the Constraint kind is invalid, but rather that using unsafeCoerce judiciously we can bring into scope instances that don't exist for a given type by substituting those from a
different type which have the right representation.
13 Responses to “What Constraints Entail: Part 2”
1. Doug McClean Says:
November 16th, 2011 at 3:17 pm
Working on the problem of expressing the pattern behind the relationship between sort/sortBy, maximum/maximumBy, etc, I have a few observations.
First, you’ve shown the existence of a type Dict :: Constraint -> * (which I suggest might more naturally be named Evidence in keeping with most of the work I have seen on qualified types, but
that’s not important). A little bit of magic (the TH approach that you mention would do it) should suffice to make this a surjective type function. There’s a plausible argument that it’s
injective as well (since we take a nominal typing approach to constraints and ignore ordering of constraint conjunctions), but I don’t think this accomplishes much so let’s ignore it.
Second, I postulate the existence of another (surjective, but surely not injective) type function Qualifier :: * -> Constraint which extracts the context of its argument and returns the empty
constraint if its argument is unqualified.
Third, twisting the same idea, I postulate the existence of another surjective type function (for which I don’t have a good name…) Unqualified :: * -> * which discards the context if its argument
is qualified and is otherwise the identity function.
Taking 1 and assuming 2 and 3, then shouldn’t we be able to have a function:
by :: t -> Evidence (Qualifier t) -> Unqualified t?
Working through the translation for an example, and ignoring the term level for a bit, we’d get:
sort :: (Ord a) => [a] -> [a]
Qualifier ((Ord a) => [a] -> [a]) ~ Ord a
Evidence (Ord a) ~ OrdDict a
Unqualified ((Ord a) => [a] -> [a]) ~ [a] -> [a]
(sort `by`) :: OrdDict a -> [a] -> [a]
… which is what we were looking for.
Sadly/oddly this is applicable at types which are not (possibly-qualified) arrow types, but it seems fairly harmless:
(True `by`) :: () -> Bool (assuming () is the type of evidence for the empty constraint?)
Did I skip any steps? Is there a reason in principle why we couldn’t write either Qualifier or Unqualified (I could see how we might need a language extension…).
Stepping back to the term level, I’m a bit stuck. I think I am stuck on understanding your concrete syntax more than on any fundamental reason why this is impossible, but I’m not sure.
2. Doug McClean Says:
November 16th, 2011 at 3:30 pm
One more note:
The existence of GHC is a proof for the possibility of doing this at the term level, right, because it erases qualified types during the translation to the core language by doing this same
evidence passing scheme?
3. Doug McClean Says:
November 16th, 2011 at 3:32 pm
Oops, one more:
If you really were concerned about expressions like (True `by`) you could strengthen the type of by from
by :: t -> Evidence (Qualifier t) -> Unqualified t
by :: (Unqualified t ~ a -> b) => t -> Evidence (Qualifier t) -> Unqualified t
4. Edward Kmett Says:
November 16th, 2011 at 5:28 pm
The problem with Qualifier and Unqualified is that unification of types involving => isn’t done by simple unification, but instead by bi-implication for constraints, so while you can write some
types with them, you typically can’t build inhabitants.
5. Doug McClean Says:
November 16th, 2011 at 5:51 pm
That explains why you can’t define them in Haskell, but I don’t think it bars you from adding them as primitives, does it? And the type checker already has the implementation that would back the
I don’t have access to a machine with bleeding edge GHC on it at the moment, but is (c => t) -> Dict c -> t a syntactically valid type? If not, why not?
6. Edward Kmett Says:
November 16th, 2011 at 7:20 pm
discharge :: (c => t) -> Dict c -> t
discharge t Dict = t
works fine.
as for whether the type checker has the machinery, that is somewhat harder to answer, becuase it would depend on the details of the largely underspecified constraint simplifier.
7. Edward Kmett Says:
November 16th, 2011 at 7:23 pm
Dan Peebles has some nice examples for things like a more polymorphic ‘on’ that could benefit, but in the meantime, one can use explicit proxies to work around the inability to name the
constraint implicitly.
8. Doug McClean Says:
November 17th, 2011 at 11:39 am
Ah, ok, I’m on the right page now. This can’t work because context reduction/constraint simplification is such a mess.
Probably a good thing, too, because for something like this you wouldn’t want any context reduction in the first place, you’d want the naively inferred context that is based solely on what class
functions are actually mentioned in a definition.
This is because you might want to occasionally use an ordering on, say, lists other than the lexical one in the prelude (say, by sum or whatever). If the context gets reduced from Ord ([Salary])
to Ord Salary then you’ve already lost your chance.
Proxies it is.
9. Edward Kmett Says:
November 17th, 2011 at 2:13 pm
As an aside, you can _write_
type family Constraints t :: Constraints
type instance Constraints (p => q) = p
you just can’t use it anywhere. =/
The compiler complains when you go to use it that you need LiberalTypeSynonyms even if you have LiberalTypeSynonyms turned on. ;)
10. Doug McClean Says:
November 17th, 2011 at 2:43 pm
Yeah, I was thinking of that definition and the corresponding
type family Unqualified t :: *
type instance Unqualified (p => q) = q
Alas there is a logical reason why it doesn’t make sense.
{-# LANGUAGE CommunistTypeSynonyms #-}?
11. Edward Kmett Says:
November 17th, 2011 at 2:57 pm
I raised the issue to Max. Not sure there is a good way for it to be defined in user land, because
() => q and q are interchangeable, so it would require compiler support.
The most compelling use case is to make code like
on :: (p a, p b) => (c -> c -> d) -> (forall x. p x => x -> c) -> a -> b -> d
inhabitable for
((++) `on` show)
but it impacts all sorts of unrelated areas in the compiler.
12. Philip J-F Says:
March 1st, 2012 at 2:04 am
Inspired by this discussion I wrote a post showing how to get multiple, passable, effectively first-class, instances. I borrowed your applicative example: http://joyoftypes.blogspot.com/2012/02/
13. zzo38 Says:
June 26th, 2012 at 2:08 pm
I like this! Nevertheless they are not perfect, not like real superclass, it is why I wanted to define my own programming language to correct these things, but at least Haskell has been improved
by these thing so that much works OK.
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Math Practices Video Series: Mathematical Practice #5
We are continuing with Mathematical Practice #5 in our video series on the Standards for Mathematical Practice. If you’ve missed any of this video series, you can catch up with our previous posts:
• Mathematical Practice #1: Make sense of problems and persevere in solving them.
• Mathematical Practice #2: Reason abstractly and quantitatively.
• Mathematical Practice #3: Construct viable arguments and critique the reasoning of others.
• Mathematical Practice #4: Model with mathematics.
There are multiple parts to each practice. The parts help students develop the habit of mind that is the main practice. Remember that the practices are defined as ways to help students become
mathematically proficient. As we look at each practice, think of ways we can help students to take ownership of these practices.
In the fifth video, students are learning how to determine the sum of integers. The Essential Question asks: “Are the sum of two integers positive, negative or zero and how can you tell?”
Observe how the teacher immediately makes a real life connection for the students giving meaning to the mathematics. What questions does she ask? Students are using multiple representations to
display the mathematics. When they begin to work on problems in their groups, they will be able to use these strategies, thereby building their proficiency.
Mathematical Practice #5. Use appropriate tools strategically.
• Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a
calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software.
• Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight
to be gained and their limitations.
• Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve
problems. They are able to use technological tools to explore and deepen their understanding of concepts.
As you look at your classroom, you probably see students with varying degrees of expertise in this practice. Our job, as educators, is to help students develop a habit of mind that helps them
naturally think before they begin, make sense of what they are doing and persevere in their work. Ask yourself:
Do your students use manipulatives to assist them in making sense of a problem?
Are students aware that there may be more than one way to find a solution?
Are students given time to discover the rules of integers rather than just being told?
As students take ownership of their learning and develop expertise using the mathematical practices, the content standards (knowledge, skills and understandings, procedural skills and fluency, and
application and problem solving) will make sense, allowing students to achieve success in mathematics.
Comments are closed for this entry.
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We prove a variant of a theorem of Corrádi and Hajnal (1963) [4] which says that if a graph G has at least 3k vertices and its minimum degree is at least 2k, then G contains k vertex-disjoint
March 2014
Robert J. Marsh | Sibylle Schroll
We study a circular order on labelled, m-edge-coloured trees with k vertices, and show that the set of such trees with a fixed circular order is in bijection with the set of RNA m-diagrams of
March 2014
Axel Hultman
Given a permutation statistic s:Sn→R, define the mean statistic s¯ as the class function giving the mean of s over conjugacy classes. We describe a way to calculate the expected value of s on a
February 2014
Miguel A. Méndez | Jean Carlos Liendo
A symmetric set operad is a monoid in the category of combinatorial species with respect to the operation of substitution. From a symmetric set operad, we give here a simple construction of a
February 2014
Susanna Dann
The lower dimensional Busemann–Petty problem asks whether origin-symmetric convex bodies in Rn with smaller volume of all k-dimensional sections necessarily have smaller volume. The answer is
February 2014
Sean McGuinness
In 1971, Nash-Williams proved that if G is a simple 2-connected graph on n vertices having minimum degree at least 13(n+2), then any longest cycle C in G is also edge-dominating; that is, each
February 2014
Alexey Ovchinnikov
This paper is devoted to integrability conditions for systems of linear difference and differential equations with difference parameters. It is shown that such a system is difference isomonodromic
February 2014
Zhicong Lin | Jiang Zeng
Generalizing recent results of Egge and Mongelli, we show that each diagonal sequence of the Jacobi–Stirling numbers Jc(n,k;z) and JS(n,k;z) is a Pólya frequency sequence if and only if z∈[−1,1]
February 2014
Fumihiko Nakano | Taizo Sadahiro
We study a generalization of Holteʼs amazing matrix, the transition probability matrix of the Markov chains of the ‘carries’ in a non-standard numeration system. The stationary distributions are
February 2014
Miklós Bóna
We show that for every k, the probability that a randomly selected vertex of a random binary search tree on n nodes is at distance k−1 from the closest leaf converges to a rational constant ck as
January 2014
William Y.C. Chen | Jeremy J.F. Guo | Larry X.W. Wang
We introduce the notion of infinitely log-monotonic sequences. By establishing a connection between completely monotonic functions and infinitely log-monotonic sequences, we show that the
January 2014
Hassane Kone
In A. Tsang (2010) [21], a representation theorem was established for continuous valuations on Lp-spaces whose underlying measure is non-atomic. In this paper, we generalize the results of A.
January 2014
Carolyn Chun | Dillon Mayhew | James Oxley
Let M be an internally 4-connected binary matroid and N be an internally 4-connected proper minor of M. In our search for a splitter theorem for internally 4-connected binary matroids, we proved
January 2014
Carolyn Chun | Dillon Mayhew | James Oxley
In our quest to find a splitter theorem for internally 4-connected binary matroids, we proved in the preceding paper in this series that, except when M or its dual is a cubic Möbius or planar
October 2013
Masao Ishikawa | Masahiko Ito | Soichi Okada
A compound determinant identity for minors of rectangular matrices is established. Given an (s+n−1)×sn matrix A with s blocks of n columns, we consider minors of A by picking up in each block the
October 2013
Antonio Giambruno | Mikhail Zaicev
If L is a special Lie algebra over a field of characteristic zero, its sequence of codimensions is exponentially bounded. The PI-exponent measures the exponential rate of growth of such sequence
October 2013
Bridget Eileen Tenner
Mesh patterns are a generalization of classical permutation patterns that encompass classical, bivincular, Bruhat-restricted patterns, and some barred patterns. In this paper, we describe all mesh
October 2013
Thorsten Holm | Peter Jørgensen | Martin Rubey
We give a complete classification of torsion pairs in the cluster category of Dynkin type Dn, via a bijection to new combinatorial objects called Ptolemy diagrams of type D. For the latter we give
October 2013
William Y.C. Chen | Oliver X.Q. Gao | Peter L. Guo
A signed labeled forest is defined as a (plane) forest labeled by 1,2,…,n along with minus signs associated with some vertices. Signed labeled forests can be viewed as an extension of signed
October 2013
Marc Chamberland | Armin Straub
Convergent infinite products, indexed by all natural numbers, in which each factor is a rational function of the index, can always be evaluated in terms of finite products of gamma functions. This
October 2013
Matthias Lenz
We show that the f-vector of the matroid complex of a representable matroid is log-concave. This proves the representable case of a conjecture made by Mason in 1972....
September 2013
Patrick Devlin | Edinah K. Gnang
We investigate using Sage [5] the special class of formulas made up of arbitrary but finite combinations of addition, multiplication, and exponentiation gates. The inputs to these formulas are
September 2013
Zhi-Wei Sun
The Franel numbers given by fn=∑k=0n(nk)3 (n=0,1,2,…) play important roles in both combinatorics and number theory. In this paper we initiate the systematic investigation of fundamental
September 2013
Su-Ping Cui | Nancy S.S. Gu
For a given prime p, by studying p-dissection identities for Ramanujanʼs theta functions ψ(q) and f(−q), we derive infinite families of congruences modulo 2 for some ℓ-regular partition functions,
September 2013
Jayant Shah
Analyzing shape manifolds as Riemannian manifolds has been shown to be an effective technique for understanding their geometry. Riemannian metrics of the types H0 and H1 on the space of planar
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Quantum physicists simulate Dirac equation, one of the cornerstones in physics
The latest news from academia, regulators research labs and other things of interest
Posted: January 6, 2010
Quantum physicists simulate Dirac equation, one of the cornerstones in physics
(Nanowerk News) Researchers of the Institute for Quantum Optics and Quantum Information (IQOQI) in Innsbruck, Austria, used a calcium ion to simulate a relativistic quantum particle, demonstrating a
phenomenon that has not been directly observable so far: the Zitterbewegung. They have published their findings in the current issue of the journal Nature.
In the 1920s quantum mechanics was already established and in 1928 the British physicist Paul Dirac showed that this theory can be merged with special relativity postulated by Albert Einstein.
Dirac's work made quantum physics applicable to relativistic particles, which move at a speed that is comparable to the speed of light. The Dirac equation forms the basis for groundbreaking new
insights, e.g. it provides a natural description of the electron spin and predicts that each particle also has its antiparticle (anti matter). In 1930, as a result of the analysis of the Dirac
equation, the Austrian Nobel laureate Erwin Schrödinger first postulated the existence of a so called Zitterbewegung (quivering motion), a kind of fluctuation of the motion of a relativistic
Rainer Blatt, Gerhard Kirchmair, Rene Gerritsma, Florian Zähringer and Christian Roos used a calcium ion to simulate a relativistic quantum particle, demonstrating a phenomenon that has not been
directly observable so far: the Zitterbewegung.
"According to the Dirac equation such a particle does not move in a linear fashion in a vacuum but 'jitters' in all three dimensions," Christian Roos from the Institute for Quantum Optics and Quantum
Information (IQOQI) of the Austrian Academy of Sciences (ÖAW) explains. "It is not clear whether this Zitterbewegung can be observed in real systems in nature."
Quantum simulation of a particle
Physical phenomena are often described by equations, which may be too complicated to solve. In this case, researchers use computer simulations to answer open questions. However, even for small
quantum systems, classical computers have not enough power to manage the processing of the data; thus, scientists, such as Richard Feynman, proposed to simulate these phenomena in other quantum
systems experimentally. The preconditions for doing this – detailed knowledge about the physics of these systems and an excellent control over the technology and set-up - have been set by the
research group headed by Rainer Blatt by conducting experiments with quantum computers over the last few years; they are now able to carry out quantum simulations experimentally.
"The challenges with these experiments are to recreate the equations in the quantum system well, to have a high level of control over the various parameters and to measure the results," Christian
Roos says.
The experimental physicists of the IQOQI trapped and cooled a calcium ion and in this well-defined state, a laser coupled the state of the particle and the state of the relativistic particle to be
simulated. "Our quantum system was now set to behave like a free relativistic quantum particle that follows the laws of the Dirac equation," Rene Gerritsma explains, a Dutch Postdoc working at the
IQOQI and first author of the work published in Nature. Measurements revealed the features of the simulated particle.
"Thereby, we were able to demonstrate Zitterbewegung in the experimental simulation and we were also able to determine the probability of the distribution of a particle," Gerritsma says. In this very
small quantum system the physicist simulated the Dirac equation only in one spatial dimension. "This simulation was a proof-of-principle experiment," Roos says, "which, in principle, can also be
applied to three-dimensional dynamics if the technological set-up is adjusted accordingly."
Simulation of antiparticles
Due to the extremely high level of control over the physical regime of the simulated particle, the scientists were able to modify the mass of the object and to simulate antiparticles. "In the end,
our approach was very simple but you have to come up with the idea first," says Christian Roos, whose team of scientists was inspired by a theoretical proposal of a Spanish group of researchers. The
work was supported by the Austrian Science Funds (FWF) and the European Commission.
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Undergraduate Course Descriptions
Unless otherwise noted, all courses are 3 credits.
MATH 1060 Basic Mathematics with Algebra
This course covers the arithmetic of whole numbers, signed numbers, fractions, decimals and percents, its primary coverage is polynomial arithmetic, algebraic expressions, factoring, solving
equations (linear and quadratic) with applications and graphing.
Note: Credits for this Basic Skills course are not applicable toward degree requirements. Prerequisite: None
Successful completion of Math Basic Skills requirement is necessary for all the following courses. Successful completion of Math Basic Skills Requirements means obtaining a score of 20 on Basic
Skills Math test or grade P in Math 1060.
MATH 1090 Mathematical Concepts (This is a UCC - Area 3E course.)
This course is intended to provide a wide ranging exposure to mathematical ideas expected of a liberal arts undergraduate. Topics include: Voting, Fair division, Apportionment, Graphs and Networks,
Consumer Finance, Statistics and Probability. The course is designed for students not majoring in business, the sciences or math.
MATH 1100 Contemporary Mathematics (This is a UCC - Area 3E course.)
This course is intended to provide a wide ranging exposure to mathematical ideas expected of a liberal arts undergraduate. Topics include Sets, Logic, Statistics, Probability, Number Systems and
Problem Solving. The course is designed for students not majoring in business, the sciences or math.
MATH 1110 Algebra and Geometry with Applications
This is a course with emphasis on studying practical problems with mathematical models. Topics include: Problem solving, number theory, introduction to functions and modeling, systems of equations
and matrices, exponential and logarithmic functions, linear inequalities in two variables and geometry. Prerequisite: Math 1100
MATH 1150 College Algebra
A comprehensive study of algebraic functions and their properties. Topics include the real numbers system, exponents and radicals, solving equations and inequalities, functions and their graphs,
polynomial functions and rational functions.
MATH 1160 Precalculus (This is a UCC - Area 3E course.)
A comprehensive study of exponential, logarithmic and trigonometric functions. Topics include function properties, exponential, logarithmic and trigonometric functions (their properties and graphs),
solving exponential and logarithmic equations, trigonometric functions (their properties and graphs), trigonometric identities and solving trigonometric equations. Prerequisite: MATH 1150 or by
placement with permission from the Department Chairperson.
MATH 1170 Business Math (This is a UCC - Area 3E course.)
A study of algebraic and transcendental functions, including their properties and graphs with a focus on applications to business. Topics include algebraic fundamentals, equations and inequalities,
polynomial functions and graphs, exponential and logarithmic functions and mathematics of finance.
MATH 1300 Elementary Statistics (This is a UCC - Area 3E course.)
This course studies the development of statistical concepts with applications to various disciplines. Topics include descriptive and inferential statistical techniques. The latter are explained in
terms of concepts from probability theory such as normal distribution, t-distribution, sampling theory, estimation, confidence intervals, hypothesis testing, t-test, Chi square test, analysis of
variance and regression and correlation. The software package SPSS is used to perform statistical analysis. Not open to science or mathematics majors.
MATH 1350 Algebra, Trigonometry, and Functions (4 credits) (This is a UCC - Area 3E course.)
A comprehensive study of algebraic and elementary transcendental functions. Topics include: the real number system, solving equations and inequalities, function properties, algebraic functions and
their graphs, exponential and logarithmic functions (their properties and graphs), solving exponential and logarithmic equations, trigonometric functions (their properties and graphs) trigonometric
identities and solving trigonometric equations. Students may be admitted into the course based on the results of a placement test.
MATH 1400 Quantitative Mathematics I
An introduction to functions, equations, matrix algebra, linear programming, and mathematics of finance. Topics include Equations and Inequalities, Functions and Graphs, Matrix Algebra, Linear
Programming: Graphical Analysis as well as the Simplex Method, and Mathematics of Finance, Markov Chains (optional)
MATH 1450 Quantitative Mathematics II
This course covers essential ideas of the calculus: functions, limits, continuity, differentiation and applications, antiderivatives and definite integrals. Business applications are stressed.
Trigonometry is not required.
Prerequisite: MATH 1400
MATH 1600 Calculus I (4 credits) (This is a UCC - Area 3E course.)
Limit and continuity of functions, the Intermediate Value Theorem, derivatives, differentiation rules, Rolle's theorem and the Mean Value Theorem, applications of differentiation, antiderivatives,
definite integrals and the Fundamental Theorem of Calculus. Prerequisite: Math 1160 or Math 1350 or by placement.
MATH 1610 Calculus II (4 credits)
Indefinite and definite integrals and their estimation, techniques of integration., improper integrals, L'Hospital's Rule, applications of integration, infinite series, power series and introduction
to taylor polynomials and approximations. Prerequisite: MATH 1600
MATH 2000 Logic and Methods of Higher Mathematics
An introduction to rigorous reasoning through logical and intuitive thinking. The course will provide logical and rigorous mathematical background for study of advanced math courses. Students will be
introduced to investigating, developing, conjecturing, proving and disproving mathematical results. Topics include formal logic, set theory, proofs, mathematical induction, functions, partial
ordering, relations and the integers.
Prerequisite: MATH 1600
MATH 2010 Calculus III (4 credits)
Study of vectors and the Geometry of Space; vector valued functions, differentiation and integration of vector-valued functions; calculus of functions of several variables including partial
differentiation and multiple integrals; higher order derivatives and their applications; Vector Fields; Line and Surface integrals.
Prerequisite: MATH 1610
MATH 2020 Linear Algebra
An introductory course in the theory of linear transformations and vector spaces. Topics include: systems of equations, matrices, determinants, general vector spaces, inner product spaces,
eigenvalues and eigenvectors.
Prerequisite: MATH 1610
MATH 2120 Survey of Mathematics
This course surveys number theory, graph theory and combinatorics, and the history of mathematics.
Prerequisite: MATH 1610
MATH 2300 Statistics (4 credits)
A rigorous course for math and science majors covering: measures of central tendency, measures of variation, graphical techniques for univariate and bivariate data, correlation and regression,
probability, binomial and normal distributions, estimation, confidence interval, testing of hypothesis, contingency tables, analysis of variance, nonparametric methods; use of packages such as SAS,
Minitab , etc. is strongly emphasized.
Prerequisite: None
MATH 3010 Modern Algebra
An introduction to groups, isomorphisms, rings, integral domains, fields and polynomial rings. Emphasis is placed on the development of theorems and techniques of proofs using definitions and
Prerequisite: MATH 2000 or CS 2600
MATH 3110 Number Theory
This is an introductory course in Number Theory for students interested in mathematics and the teaching of mathematics. The course covers basic notions of integers and sequences, divisibility, and
mathematical induction. It also covers standard topics such as Prime Numbers; the Fundamental Theorem of Arithmetic; Euclidean Algorithm; the Diophantine Equations; Congruence Equations and their
Applications (e.g. Fermat’s Little Theorem); Multiplicative Functions (e.g. Euler’s Phi Function); Application to Encryption and Decryption of Text; The Law of Quadratic Reciprocity.
Prerequisite: MATH 2000
MATH 3220 Differential Equations
A study of the methods of solution and applications of ordinary differential equations. Topics include: first and second order equations, existence and uniqueness of solutions, separation of
variables, exact equations, integrating factors, linear equations, undetermined coefficients, variation of parameters, transform methods, series solutions, systems of equations and elementary
numerical methods.
Prerequisite: MATH 1610
MATH 3230 Foundations of Geometry
Foundations of Geometry presents the different axiomatic approaches to the study of geometry with specific applications to finite, Euclidean, and non-Euclidean geometries with extensive use of
constructions to explore ideas, properties, and relationships. Technology will be used throughout the course to encourage these open-ended explorations. The role of different types of proofs will be
developed throughout the course.
Prerequisites: MATH 1610 and (MATH 2000 or CS 2600)
MATH 3240 Probability and Statistics (4 credits)
A mathematical treatment of probability as well as statistics. Topics include probability axioms, discrete and continuous sample spaces, random variables, mathematical expectation, probability
functions; basic discrete and continuous distribution functions; multivariate random variables. Also covered is Central Limit Theorem, confidence intervals, hypotheses testing and Linear regression.
Software such as SAS or Minitab may be used for hypotheses testing and regression problems.
Prerequisite: MATH 1610
MATH 3260 Mathematical Models in Finance and Interest Theory
A course on the formulation, analysis, and interpretation of advanced mathematical models in finance and interest theory. Computers and technology will be used to give students a hands-on experience
in developing and solving their own models. Applications to “real-world” problems in interest theory, including the development of complex annuity models, will be emphasized. The course will cover
the fundamentals needed for the second actuarial exam. The primary focus will be on the financial models.
Prerequisite: MATH 1610
MATH 3320 Statistical Computing
Students solve statistical problems on the computer with the help of statistical packages, such as SAS, BMD, Mystat, etc., and learn to interpret the outputs and draw inferences. Topics include
analysis of variance with and without interactions, correlation and regression analysis, general linear models, multiple comparisons and analysis of contingency tables.
Prerequisite: MATH 3240
MATH 3340 Applied Regression Analysis
This is a comprehensive treatment of regression analysis course, statistical topics including: simple linear regression, least square estimates, ANOVA table, F-test, R-square, multiple regression,
using dummy variables, selections of the “best subset” of predictor variables, checking model assumptions and Logistic regression. The computer package, SAS, will be used through out the course and
applications to real life data will be an integral part of the course.
Prerequisite: MATH 3240
MATH 3350 Introduction to Numerical Analysis
Treatment of numerical methods including numerical integration, numerical solution of equations and systems of equations, approximation of functions, numerical solution of differential equations,
applications and computer implementation of numerical methods.
Prerequisite: MATH 2020
MATH 3800 Linear and Nonlinear Optimization
Iterative Algorithms, Optimization process and Linear Programming (LP), including the Graphical method and Simplex method. Duality and Sensitivity analysis, LP applications in business and health.
Nonlinear Unconstrained problems and various Descent methods. Nonlinear Constrained optimization, including Primal, Penalty, and Barrier methods.
Prerequisite: MATH 2020
MATH 3990 Selected Topics (1- 3 credits)
A topic not covered by an existing course is offered as recommended by the department and approved by the dean. The number of credits for MATH 3990 may vary from 1 to 3 for a selected topic. MATH
3990 can not be credited more than twice, each on a different topic, towards degree requirements.
Prerequisite: Department Chairperson's permission
MATH 4010 Applied Algebra
A course covering applications of Modern Algebra. Topics include Boolean algebras and applications to switching theory; symmetry groups in three dimensions, monoids and machines, applications of
rings, fields, and Galois theory including error-correcting codes.
Prerequisite: MATH 3010
MATH 4110 Advanced Discrete Mathematics
This is an advanced course in discrete mathematics primarily dealing with discrete dynamical systems, algorithms, combinatorics and Graph Theory. Emphasis is placed on complexity of algorithms, on
existence and optimization problems in Graph Theory and on associated algorithms.
Prerequisite: MATH 2020 or CS 2600
MATH 4120 Time Series Analysis
This is an applied statistical methods course in time series modeling of empirical data observed over time.
Prerequisite: MATH 3340 or (MATH 3240 and instructor's permission)
MATH 4130 Experimental Design for Statistics
For processes of any kind that have measurable inputs and outputs, Design of Experiments (DOE) methods guide you in the optimum selection of inputs for experiments, and in the analysis of results.
Full factorial as well as fractional factorial designs are covered – see the course outline for additional details. Software such as SAS or S-Plus will be used for testing and regression problems.
Prerequisite: MATH 3240 with at least a C- or the approval by the chairperson of Math Dept
MATH 4150 Topics from Applied Mathematics
The objective of this course is to give the student an understanding and appreciation of applied mathematics. This will be accomplished by working through a variety of problems from the physical
sciences. Emphasis will be on modeling scientific phenomena rather than developing mathematical methods. Basic ideas and concepts will first be illustrated on simple problems, and they will
eventually be extended to more complicated systems.
Prerequisite: MATH 3220
MATH 4210 Mathematical Statistics
A theoretical treatment of statistical topics including: distribution theory, sampling, point and interval estimation, methods of estimation such as maximum likelihood estimation, properties of
estimators, Neyman-Pearson Lemma, hypothesis testing, power of a test, and linear models.
Prerequisite: MATH 3240
MATH 4220 Complex Analysis
Elements of complex analysis. Topics include: complex numbers, analytic functions, Cauchy integral theorem, Cauchy integral formula, power series and conformal mapping.
Prerequisite: MATH 2010
MATH 4230 Real Analysis
A rigorous approach to the theory of functions of real variables. Topics include: metric spaces, sets, limits, sequences, continuity, uniform continuity, differentiation, Riemann integration,
sequences and series of functions, and Riemann-Stieltjes integral.
Prerequisite: MATH 2010
MATH 4250 Introduction to Topology
This course uses the concepts of set theory, analysis and group theory in a rigorous manner to explore the fundamental mathematical definitions of continuity, homeomorphism, metrics, product
topology, compactness, connectedness, the fundamental group and homotopy theory.
Prerequisite: MATH 3010
MATH 4900 Mathematics Seminar (2 credits) (This is a Writing Intensive-WI course.)
This is a required course for all mathematics majors and should be taken, if possible, in the junior year. The course will be led by a faculty member and conducted in an inquiry based fashion, with
coverage of topics determined by the interests of the student and faculty. Each student will complete a project of study in an area of mathematics or it applications, culminating in a presentation to
the faculty and students, and a final paper submitted to the faculty advisor.
Prerequisite: Must be registered in a 4000-level mathematics course, or must have successfully completed a 4000-level course. Math 4900 registration form may be downloaded from here.
MATH 4990 Independent Study (1- 3 credits)
An individual research project under the direction of a faculty member and with the approval of the chairperson. The number of credits for each independent study may vary from 1 to 3 per semester, up
to a limit of 6 credits.
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January 2012
A surprising amount of what we do in computer security relies upon the use of random numbers and yet not many of us actually take the time to think about how these numbers are being generated. We
blithely assume that our computer, when required, can generate a truly random number. But wait! How can a machine that works in a deterministic manner generate something that is truly random. The
simple answer is that it can't. The best we can do is to generate a number that is
The fact that we rely upon pseudorandom numbers is a potential problem for IT security. After all, if a machine is using a known algorithm to generate a number that your system then treats as random,
what is to stop an attacker from calculating that same number if he knows your algorithm. It is a fundamental truth of any strong computer security that you must assume that an attacker knows the
algorithms that you have used. So, perhaps one should spend a little more time understanding how pseudorandom numbers are generated in order that they are not relied upon inappropriately.
The measure of random in a number is known as
. Not the entropy as physicists use it, but entropy as cryptographers use it. Entropy is, in essence, how uncertain you are about a number. This means that the entropy is not necessarily related to
the number of bits in a number but instead to the number of possible values that number could have taken. Imagine being able to discard the number of bits in a number about which you were certain.
The number of bits that remain are the
number of bits of entropy
. Suppose, for example, a 100 bit number could take on two possible values, then you only need one bit to differentiate those two values. Hence, such a number would be described as having 1bit of
entropy. Obviously this is an extreme case, and I have not complicated the issue by factoring in the different types of probability distribution.
Most mathematicians tend to define the entropy of a number X as:
H(X) ≔ − ∑P(X=x) log₂P(X=x)
where the summation is from 0 to a number x, and P(X=x) is the probability of X taking the value x.
So, how do you generate a number that has to greatest possible entropy. There has been much research on using activities such as mouse movements or keystrokes to generate a number that can be
But so far it appears that the activities picked to generate the numbers have proved to be not quite as random as first thought: some typists are remarkably accurate and consistent! Other work has
focused on incorporating some form of physical device into a computer that can generate truly random numbers based upon provably
behaviour. This has two drawbacks: generating large random numbers can take an unacceptably long time, and the device might fail rendering the numbers predictable, without you knowing.
Which brings us back to the generation of pseudorandom numbers. Most accept that this is the best method of generating numbers with sufficient entropy, whilst making the mechanism efficient and
accessible enough for widespread use in computing. There are many algorithms in the literature but I would recommend an approach such as described in Fortuna which was an improvement by Ferguson,
Schneier and Koho on a previous method called Yarrow. Fortuna, like most approaches, relies upon a
which is a relatively small but truly random number. But, it allows for the use of modern encryption techniques such as AES which mean that although the algorithm is known it is highly unlikely that
the resulting pseudorandom number sequence can be reproduced. What Fortuna shows is that rather than rely upon standard libraries on their own, you construct a routine that, yes, uses standard
libraries, but recognises the potential attacks against those and seeks to minimise the threat by incorporating elements such as AES and SHA thereby reserving some secret information to the local
machine, decreasing the chances that a third party could reproduce your number(s).
So, in short, the best way to use random numbers is by generating pseudrandom numbers but in such a way that "unreliable" sources of randomness are replaced by effectively creating your own store of
entropy upon which you can draw and keeping the means by which that store is generated secret by virtue of a secret key.
For a superb, free book that goes into great depth about random number generation, see
Luc Devroye
's book
(originally published with Springer-Verlag, New York, 1986).
One of the perennial problems with symmetric-key encryption has been establishing a secure channel over which to pass the secret key. It has always been much easier to compromise the transmission of
the keys than to try to find some weakness in the encryption algorithm. Military and government organisations have put in place elaborate methods of passing secret keys: they pass secrets more
generally so using similar channels to pass an encryption key is not a great leap.
However, as everyone has become more connected, and especially with the commercialisation of the Internet, encryption has become a requirement for the vast majority of networked users. Thus, the
traditional methods of passing secret keys is impractical, if only because you might not actually know who you want to communicate with in an encrypted fashion.
Diffie-Helman Key Exchange
It was realised in the 1970’s that encryption needed to be more accessible, so a great deal of work was done on algorithms that could ensure that a key had been passed over relatively insecure
channels was not compromised. Leaders in the field were Whifield Diffie and Martin Helman. In doing this work, it was realised that there was another way of tackling the problem. In the late 1970’s
Diffie and Helman published their seminal paper entitled “New Directions in Cryptography” and public-key encryption was born.
When the implementation of the early public-key encryption methods was compared to symmetric-key encryption it was found that the public-key encryption was significantly slower and hence
communication using public-key encryption alone was, at the very least, going to require far more processing power than would otherwise have been the case. But wait! The original problem being
studied was secure key transmission so why not use public-key encryption to securely transmit the symmetric-key, and then use the faster, more efficient symmetric-key encryption algorithms. Iin
essence, that is how most so called public-key encryption works today.
The majority of public-key encryption algorithms rely upon a mathematical subtlety. One of the earliest was based upon prime numbers. If one has a number (N) that is derived from multiplying two
prime numbers, and N is very large, it is practically impossible to calculate what the two constituent prime numbers were (known as factorisation). It’s not that you can’t calculate the two prime
numbers. There are many algorithms for doing so. It is that it takes such a long time to successfully compute the algorithm (sometimes thousands of years) that by the time you finish, the information
you can recover is worthless. Even with the huge budgets available to governments, it is possible to reduce these timescales only marginally.
Public-key algorithms therefore are able to use one of the prime number constituents of N as part of the public-key element of the encryption process without fearing that the other might be
discovered. It’s a fact of life that even with the massive increases in computing power over recent years, traditional algorithms for factorising these large numbers means public-key encryption is
still quite secure.
Of course, “secure” is a relative word and there are many ways of recovering the private element that was used to derive N. These do not involve some fiendishly clever mathematical method (although a
small army of mathematicians are seeking to do this) but rather simple methods such as accessing the PC that holds the private element! As ever, the weakest link determines the strength of the chain.
Having said that, there is now an emerging threat the public-key encryption which does not rely upon such trivial methods: Quantum Computing, a field first introduced by Richard Feynman in 1982. To
understand why Quantum Computing poses a threat one first needs to understand a little about how it works.
In the computers that we are familiar with, processing is done using bits. A bit has two possible values: 0 and 1. A quantum computer uses qubits. A qubit can also have the values 0 and 1, but also
any combination of the two simultaneously. In quantum physics this is called superposition. So, if you have 2 qubits you can have 4 possible states, 3 qubits gives 8 possible states; and all
simultaneously. In a bit-based computer you have the same number of possible states but only one exists at any one time. It is the fact that these states can exist simultaneously in a quantum
computer which is both counter-intuitive and extraordinarily powerful.
These qubits are manipulated using quantum logic gates in the same way that conventional computation is done by manipulating bits using logic gates. The detailed maths is not important here. Suffice
to say that you can operate on multiple simultaneous states, thereby increasing the amount of computation you can undertake over what you could otherwise do in a conventional computer. So, if you had
a 2 qubit quantum computer you could theoretically compute at 4 times the speed of a 2 bit conventional computer. There are a few problems though, in translating theory into practice.
First, the algorithms developed over many years for conventional computers have been optimised for their architecture, and different algorithms are needed to run a quantum computer. Hence, trying to
compare speeds of conventional and quantum computers is like comparing apples with bananas. However, one of the earliest algorithms developed for quantum computing was by Peter Shor. Shor’s 1994
algorithm was specifically designed to factorise numbers into their prime number components. Shor’s algorithm runs on a quantum computer in polynomial time, which means that for a number N it takes
only logN to successfully complete the algorithm. Even with the largest numbers in use today in public-key encryption, that means that it is perfectly feasible to factorise the number in meaningful
Since Peter Shor developed his quantum computing algorithm many others have been developed, and it is worth noting that a significant number of these algorithms are aimed at breaking the underlying
mathematics that supports more recent public-key encryption. The fact that methods other than the use of prime numbers have been developed is being very quickly followed by their quantum computing
algorithm counterpart.
A Young Heisenberg
Second, there is the much quoted, but less well understood, Heisenberg’s Uncertainty Principle. The bottom line is that this principle tells us that if you observe something then you affect it.
Hence, measuring a qubit causes it to adopt one state but that state might not have been the state resulting from the computation but could have been altered by you observing it. So, the moment you
try to measure the answer calculated by your superfast quantum computer it loses its ability to give you the correct answer. That would seem to render quantum computing rather pointless. But, there
is another quantum effect that can be employed: quantum entanglement. This is where, if two objects have previously interacted, they stay “connected” such that by studying the state of one object you
can determine the state of the other. Hence, you can determine the state of the qubit with your answer by studying another object with which it is entangled. Again, this is counter-intuitive, but has
been proven, so all one really needs to know is that there is a way of getting your answer out of your quantum computer.
Quantum Computer On An Optical Bench
Lastly, there is a small matter of implementation. Until recently this was a show stopper. Typically quantum computers are based on light, as photons are relatively easy to work with and measure.
However, anyone who has seen an optical bench will know that they are enormous. They require great weight to prevent even the slightest vibration and they are prone to all manner of environmental
factors. Hardly the small “chips” we are used to! Also, typically these implementations are aimed at running one algorithm: the algorithm is built into the design.
Having said that it wasn’t that long ago that conventional computers with far less processing power than the average PC required air conditioned luxury and small army of attendants to keep them
functioning. It did not take very long for the size to shrink dramatically and for the environmental requirements to be quite relaxed. Not surprising then that there is a company in Canada (D Wave
Systems Inc) who already offer access to quantum based computing. It’s expensive and the size of a room, but we all know that won’t be the case for long.
D-Wave's Bold Statement On Their Website
2011 bought about some major developments which might well make 2012 the year quantum computing comes of age. Most significant of these was by a team at Bristol University, who developed a small
chip which housed a quantum computer. It had only 2 qubits but it was of a size that could be used for the mass market, and, crucially, it was programmable.
We are now entering a new era where we have programmable, relatively inexpensive, relatively small, quantum computers visible on the horizon, and we know that such computers have the potential to
undermine the mathematics upon which current public-key encryption depends. Given all of that, maybe it’s time to reconsider the role of public-key encryption and where it might be more sensible to
rely wholly on symmetric-key encryption.
Here is a brief video we produced at the University of Surrey to introduce cyber security, and the work we are doing there:
If, like me, you use Facebook, Twitter, LinkedIn (and a whole host of other social media sites) then, again like me, you might have wondered how people actually read your pages. I've long wondered
how much of the page visitors read. Especially so after I read the statistics about how little time visitors spend on each page on the web. In a matter of a few seconds, what information is the
reader really scanning?
I've seen a lot of very impressive research being done on the subject and many erudite papers being published. All of this is backed by a great deal of experimentation which involves complex
equipment and analysis. Hence, when I saw the product being offered by Eyetrack I thought that it represented a fun, accessible way that everyone can answer the question about their own sites.
Using a webcam, plus a bit of calibartion, the software will track the eyes of a reader to a specific page. The data collected can then be turned into a report. Not some great long winded piece of
statistics but an easy to understand visual map of how the visitor uses your page.
You might not be surprised to learn that the item most Facebook visitors look are the faces. In addition to how much time the user is spending on each part of the page and the corollary, what they
don't read, you can also track how a visitor visually moves around the page.
And, it's not the same for each type of social networking site. Take, for example, LinkedIn (below) and compare it with Facebook (above):
It would appear that visitors do actually read the text on LinkedIn, but very much at the "headline" level. All those extra bits of infromation added by LinkedIn to the right hand pane about who esle
was viewed, etc seem to go to waste.
And then, of course, there's Twitter:
Are viewers reading the first one or two words of a Tweet only? That's all the time you have to capture their attention! But compare Twitter with Facebook: it does appear to be text not photos being
scanned in Twitter.
One very interesting area that is worth studying is the deviation in viewing patterns caused by the client that is being used to view the page. With so many apps now available for viewing each of the
social networking sites, it might be interesting to see if the viewing pattern is in any way governed by the way it is presented. Twitter in particular has a large variety of clients, so how does
that affect the viewing:
So, thanks to Eyetrack for enabling us all to do a bit of research that is both fun and produces useful results.
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Methods for designing submerged breakwaters are still being developed, particularly in respect of the 3D nature of wave-breakwater interaction. Many of the available design tools are inefficient as
they are not able to provide any information on the spatial distribution of the wave field around breakwaters, and cannot therefore guarantee reliability and accuracy for the engineer. There is thus
a need for an engineering design tool with the ability to model spatial variation of wave height. This paper proposes a method based on machine learning algorithms for predicting the nearshore wave
field behind a submerged breakwater that includes both 2D and 3D effects. The proposed numerical model has been validated by various scales of laboratory data. Comparisons reveal the ability of the
proposed model to predict the wave field around submerged breakwater.
Submerged Breakwaters; Wave field; 3D; Wave transmission; Artificial Neural Networks; Numerical modeling
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question in pilot wave theory
1. Velocity v is continuous.
2. v is the velocity of the particle. (Note, however, that group velocity is a velocity of a wave.)
3. Particle.
4. Wave guides the particle in space.
5. Pilot wave theory is the original de Broglie matter wave theory.
I would also suggest you to read some basics, such as
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Is the set of primes "translation-finite"?
up vote 29 down vote favorite
The definition in the title probably needs explaining. I should say that the question itself was an idea I had for someone else's undergraduate research project, but we decided early on it would be
better for him to try adjacent and less technical questions. So it's not of importance for my own work per se, but I'd be interested to know if it easily reduces to a known conjecture/fact/
counterexample in number theory.
Apologies if the question is too technical/localized/unappealing/bereft of schemes.
Given a subset $X$ of the natural numbers $N$, and given $n \in N$, we write $X-n$ for the backward translate of $X$, i.e. the set {$\{x-n : x\in X\}$}.
We say that $X$ is translation-finite if it has the following property: for every strictly increasing sequence n[1] < n[2] < in $N$, there exists k (possibly depending on the sequence) such that
$(X-n_1) \cap (X-n_2) \cap \dots\cap (X-n_k)$
is finite or empty.
Thus every finite set is trivially translation-finite: and if the elements of $X$ form a sequence in which the difference between successive terms tends to infinity, then $X$ is translation-finite
and we can always take k=2. Moreover:
• if $X$ contains an infinite arithmetic progression, or if it has positive (upper) Banach density, then it is NOT translation finite;
• there exist translation-finite sets which, when enumerated as strictly increasing sequences, grow more slowly than any faster-than-liner function.
• there exist translation-finite sets containing arbitrarily long arithmetic progressions.
These resultlets suggest the question in the title, but I don't know enough about number theory to know if it's a reasonable question. Note that if, in the definition, we were to fix k first (i.e.
there exists k such that for any sequence (n[j])...) then we would get something related to Hardy-Littlewood conjectures; but I was hoping that this might not be necessary to resolve the present
EDIT (2nd Nov) It's been pointed out below that the question reduces in some sense to a pair of known, hard, open problems. More precisely: if the answer to the question is yes, then we disprove the
Hardy-Littlewood k-tuples conjecture; if the answer is no, then there are infinitely many prime gaps bounded by some absolute constant, and this is thought to be beyond current techniques unless one
assumes the Eliott-Halberstam conjecture.
Added in 2013: Stefan Kohl points out that the latter is Yitang Zhang's famous recent result. However, as Will Sawin points out in comments, a negative answer to the main question would imply there
are 3-tuple configurations occurring infinitely often in the primes, and (see thelink in Will's comment) this is thought to be out of reach even if we assume the EH conjecture holds.
1 What do you mean by "the elements of X form a sequence tending to infinity"? Does that imply something other than X being an infinite set? – Darsh Ranjan Oct 30 '09 at 0:07
aargh, typo/brain spasm - I missed out some important words. I meant that the gap size has to tend to infinity. I've edited the original post to correct this. – Yemon Choi Oct 30 '09 at 2:00
5 Yay for non-scheme questions! – Theo Johnson-Freyd Oct 30 '09 at 2:12
1 @Scott: makes sense to me. @Yemon: thanks for nicely summarizing th results in the question body! – Ilya Nikokoshev Nov 27 '09 at 23:52
1 Because the existence of any $3$-tuples which occurs infinitely often in primes is still out of reach, this is still open. See: mathoverflow.net/questions/132731/… – Will Sawin Jul 4 '13 at 16:26
show 1 more comment
1 Answer
active oldest votes
As you mention, this is related to the Hardy-Littlewood k-tuple conjecture. In particular, if their conjecture is true, then the primes are not translation-finite. Indeed, it is
possible to find an increasing sequence n[1] < n[2] < n[3] < ⋯ so that for every k, the first k n[i]s form an admissible k-tuple. (For example, I think n[i] = (i+1)! works.) Then, by
the k-tuple conjecture, infinitely many such prime constellations exist and so for all k, (X-n[1]) ∩ (X-n[2]) ∩ ⋯ ∩ (X-n[k]) is infinite. (Here and below, X is the set of primes.)
However, maybe we can prove that the primes are not translation finite by some other means. Unfortunately, the technology is not quite good enough to do that. Proving that the primes
up vote 19 are not translation finite would, in particular, prove that there exist n[1] < n[2] such that (X-n[1]) ∩ (X-n[2]) is infinite. In particular, this implies that the gap n[2]-n[1] occurs
down vote infinitely often in primes, and so p[n+1]-p[n] is constant infinitely often. (The standard notation p[n] indicates the n^th prime.)
The best known upper bound for the size of small gaps in primes is that liminf[n→∞] (p[n+1]-p[n])/log p[n] = 0. This was established by Goldston and Yildirim around 2003 and the proof
was later simplified. To the best of my knowledge, the best conditional result is by the same authors; they show that given the Elliott-Halberstam conjecture, the prime gap is
infinitely often at most 20 or so.
Thanks! That's very clear. So, if I understand you correctly: a proof of HL's k-tuple conjecture would imply the primes are not translation finite; while a proof that the primes are
not translation finite would imply a bound on prime gaps that is beyond current knowledge. So my question essentially reduces to known open problems I guess. – Yemon Choi Nov 1 '09
at 23:52
Which probably means it deserves a CW status + open-problems tag. – Ilya Nikokoshev Nov 2 '09 at 0:04
As well as rewriting the question so the two open problems become clear. – Ilya Nikokoshev Nov 2 '09 at 0:04
1 @Yemon Choi: That's correct. Proving the primes are translation finite would disprove the k-tuples conjecture, while proving the primes aren't translation finite would prove bounds
on small prime gaps. – aorq Nov 2 '09 at 3:27
1 Why does the infinitude of $(X-n_1)\cap(X-n_2)$ imply that the gap $n_2-n_1$ occurs infinitely often? Couldn't some smaller gaps occur infinitely often instead? – François G. Dorais♦
Dec 27 '12 at 4:59
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GRONDMAN et al. : A SURVEY OF ACTOR-CRITIC REINFORCEMENT LEARNING: STANDARD AND N ATURAL POLICY GRADIENTS 3 with u drawn from the probability distribution function (x; )
3 Reinforcement Learning zForm of unsupervised learning – Machine is never told what the correction action is. – Negative / positive rewards given to the machine based
A Brief Survey of Operant Behavior ... process is not trial-and-error learning. ... Operant reinforcement not only shapes the topography of behavior, it maintains it in strength long after an operant
has been formed. Schedules of reinforcement are
Reinforcement Learning Assume the world is a Markov Decision Process - transition and rewards unknown; states and actions known. Two objectives:
Reinforcement Learning in Online Stock Trading Systems Abstract Applications of Machine Learning (ML) to stock market analysis include Portfolio ... with a survey of earlier approaches outlines in
[13]. However, a plethora of alternatives exist, some
A (Revised) Survey of Approximate Methods for Solving Partially Observable Markov Decision Processes Douglas Aberdeen National ICT Australia, Canberra, Australia.
Multi-Instance Learning: A Survey Zhi-Hua Zhou National Laboratory for Novel Software Technology, Nanjing University, ... reinforcement learning where the labels of the training instances are
delayed, in multi-instance learning there is no any delay.
Reinforcement learning: A survey. Journal of Artificial Intelligence Research, 4:237–285, 1996. [136] S. Kapetanakis and D. Kudenko. Improvingon the reinforcementlearning of coordinationin
cooperativemulti-agent systems.
Reinforcement Learning is considerably more difficult for continuous-time systems than for discrete-time systems, and its development has lagged. ... Invited survey paper. [9] A. G. Barto, R. S.
Sutton, and C. Anderson, “Neuron-like adaptive ele-
Reinforcement Learning and Automated Planning: A Survey . Ioannis Partalas, Dimitris Vrakas and Ioannis Vlahavas . Department of Informatics . Aristotle University of Thessaloniki
Reinforcement Learning Sampler CS 536: Machine Learning Littman (Wu, TA) ... “Reinforcement Learning: A survey” in Journal of Artificial Intelligence Research. Bertsekas & Tsitsiklis (1996).
Neuro-Dynamic Programming. Tesauro (1992). “Practical Issues in Temporal Difference
Recommended Reading Sutton & Barto (1998). Reinforcement Learning: An Introduction. Kaelbling, Littman and Moore (1996). “Reinforcement Learning: A survey” in Journal of Artificial Intelligence
In this paper we briefly survey reinforcement learning, a machine learning paradigm that is especially well-suited to learning control policies for mobile robots. We discuss some of its shortcomings,
and introduce a framework for effectively
A Brief Survey of Parametric Value Function Approximation ... Reinforcement learning aims at estimating the optimal policy without knowing the model and from interactions with the system. Value
functions can no longer be computed,
Reinforcement Learning: A Survey. Journal of Artificial Intelligence Research. Volume 4, 1996. • G. Tesauro. TD-Gammon, a self-teaching backgammon program, achieves master-level play. Neural
Computation 6(2), 1995. • http://ai.stanford.edu/~ang/
Reinforcement Learning II: Q-learning Hal Daumé III Computer Science University of Maryland [email protected] CS 421: Introduction to Artificial Intelligence 28 Feb 2012 Many slides courtesy of Dan
Klein, Stuart Russell, ... Midcourse survey, qualitative
Transfer in Reinforcement Learning: a Framework and a Survey Alessandro Lazaric Abstract Transfer in reinforcement learning is a novel research area that focuses
Multi-Agent Reinforcement Learning: A Survey Lucian Bus¸oniu Robert Babuˇska Bart De Schutter Delft Center for Systems and Control Delft University of Technology
survey of how reinforcement learning methods react in general to canine training techniques. The SARSA Algorithm: The SARSA algorithm is an on-policy temporal difference reinforcement learning
method. The algorithm builds a value table using the SARSA update rule, which updates
which is based on reinforcement learning from spectrum survey data introduced in [3]. The rein-forcement learning algorithm returns the vector of the scored channels based on possible interference in
particular channel.
A survey of machine learning First edition by Carl Burch for the Pennsylvania Governor’s School for the Sciences c 2001, Carl Burch. ... Reinforcement learning is much more challenging than
supervised learning, and researchers still don’t
Reinforcement learning: A survey. Journal of Artificial Intelligence Research, 4:237–285, 1996. [136]S. Kapetanakis and D. Kudenko. Improving on the reinforcement learning of coordination in
cooperative multi-agent systems. In Proceedings
Reinforcement learning in robotics, an application using Lego Mindstorms ... Reinforcement Learning: A Survey. Journal of Artificial Intelligence Research, 4:237-285, 1996. Title: Microsoft
PowerPoint - Robotics 1 - Cioffi.ppt Author: Administrator
Reinforcement learning: a survey. Journal of Artificial Intelligence Research , 4:237–285, 1996. Andrew Y. Ng, Adam Coates, Mark Diel, Varun Ganapathi, Jamie Schulte, Ben Tse, Eric Berger, and Eric
Liang. Autonomous inverted helicopter flight via reinforcement lea rning.
ing developed in machine learning and data mining areas. There has been a large amount of work on transfer learning for reinforcement learning inthe machine learning literature
A survey of both the statistical and ethical problems may be found in [Ros96]. ... Robin Pemantle/Random processes with reinforcement 35 4.4. Learning A problem of longstanding interest to
psychologists is how behavior is learned.
BUS¸ONIU et al.: A COMPREHENSIVE SURVEY OF MULTIAGENT REINFORCEMENT LEARNING 157 A. Contribution and Related Work This paper provides a detailed discussion of the MARL
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General Structure Functions
Montero de Juan, Francisco Javier and Tejada Cazorla, Juan Antonio and Yáñez Gestoso, Francisco Javier (1994) General Structure Functions. Kybernetes, 23 (3). pp. 10-19. ISSN 0368-492X
Restricted to Repository staff only until 2020.
Official URL: http://www.emeraldinsight.com/journals.htm?articleid=875592&show=abstract
A general concept of a structure function is proposed by considering a general order topology, where possible degrees of performance for the system and its components are going to be represented.
Finite multistate structure functions and continuum structures can therefore be viewed as particular cases. Gives general definitions of minimal path and minimal cut, allowing general reliability
bounds based on them. These are applied to some multivalued structures.
Item Type: Article
Uncontrolled Continuum structures
Subjects: Sciences > Mathematics > Operations research
ID Code: 15961
References: Barlow, R.E. and Proschan, B., Statistical Theory of Reliability and Life Testing, To Begin With, Silver Spring, MD, 1975.
Baxter, L.A., “Continuum Structures”, International Journal of Applied Probability, Vol. 21 No. 4, 1984, pp. 802-15.
Baxter, L.A., Continuum Structures II, Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 99 No. 2, 1986, pp. 331-8.
Block, H.W. and Savits, T.W., “Continuous Multistate Structure Functions”, Operations Research, Vol. 32, 1984, pp. 703-14.
Montero, J., “Coherent Fuzzy Systems”, Kybernetes, Vol. 17 No. 4, 1988, pp. 28-33.
Montero, J. and Tejada, J., “Some Results on Fuzzy Systems”, in Rose, J. (Ed.), Cybernetics and Systems: The Way Ahead, Thales, Lytham St Annes, 1987, pp. 591-4.
Goguen, L., “L-fuzzy Sets”, Journal of Mathematical Analysis and Applications, Vol. 18 No. 1, 1967, pp. 145-74.
French, S., Decision Theory, Ellis Horwood, Chichester, 1986.
Vansnick, J.C., “Intensity of Preferences”, in Sawaragi, Y., Inoue, K. and Nakayama, H. (Eds), Toward Interactive and Intelligent Decision Support Systems II, Springer-Verlag,
Berlin, 1987, pp. 220-9.
Willard, S., General Topology, Addison-Wesley, Reading, MA, 1970.
Montero, J., “Observable Structure Functions”, Kybernetes, Vol. 22 No. 2, 1993, pp. 31-9.
Montero, J., Tejada, J. and Yañez, J., ”Structural Properties of Continuum Systems”, European Journal of Operational Research, Vol. 45 Nos 2-3, 1990, pp. 231-40.
Deposited On: 16 Jul 2012 11:05
Last Modified: 06 Feb 2014 10:35
Repository Staff Only: item control page
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(idea) by Woundweavr Sat Nov 13 1999 at 14:22:36
Level of extremeness. As something increases in size or the amount, magnitude increases.
(idea) by Adalgeirr Thu Jul 20 2000 at 18:08:40
The magnitude of a
can be found by the following
Assume V= ai+bj+ck
Then: ||V|| = (a^2+b^2+c^2)^1/2
(thing) by Grzcyrgba Fri Mar 23 2001 at 13:38:18
In astronomy, magnitude is a logarithmic measurement of observed brightness. The logarithmic nature of magnitudes comes from the fact that the response of the human eye to light is logarithmic. The
magnitude units devised by ancient Greek astronomers correspond to equal flux ratios, or
m(a) - m(b) = const * log ------ (1)
implying a logarithmic scale. Five magnitudes corresponds to a factor of 100 difference in flux, which makes the constant in the equation above equal to -2.5.
Magnitudes are typically measured with reference to an astronomical standard, usually the star Vega which has a magnitude of 0.0. Increasing magnitudes denote fainter objects, thus a star with a
magnitude of -1.0 is brighter than a star with a magnitude of +1.0. The unaided human eye can detect objects with magnitudes as faint as approximately 6.0. The deepest telescopic image -- the Hubble
Deep Field -- detected objects as faint as 28.0 to 29.0, more than a thousand million times fainter.
The apparent magnitude of an object, usually denoted by a lower case m, is the perceived brightness of an object. Thus The Sun has an apparent magnitude of -26.7, but only because it is nearby,
cosmically speaking.
The absolute magnitude of an object, denoted by an upper case M is the apparent magnitude an object would have if we were observing it from a distance of ten parsecs. The absolute magnitude of the
Sun is around +4.75, while the apparent magnitude of an average galaxy would be about -19.5.
The apparent and absolute magnitudes of an object (if both are known) can be used to determine the distance to an object, using the formula
m - M = ( 5 * log(D) ) - 5 + A (2)
where D is the distance in parsecs, and A is a fudge factor which accounts for extinction by dust and gas between us and the object. m is usually straightforward to measure, but M requires that you
know something about the object in advance.
Magnitudes are often measured in specific wavelengths or filters, and many filter sets have been defined for specific uses. Often, objects are described in terms of their color, which is the
difference in observed magnitudes in different filters. For example,
(B - V) = m(B) - m(V) (3)
are filters in the
filter set, measuring
light, respectively. By equation (1), this corresponds to a ratio of fluxes observed in the different filters. The color can, for example, tell you the
of an object, or tell you the amount of
interstellar reddening
. You may also see
bolometric magnitude
, which is the sum of all emitted radiation from
radio waves
gamma rays
, used to measure the total
of an object.
(definition) by Webster 1913 Wed Dec 22 1999 at 1:00:39
Mag"ni*tude (?), n. [L. magnitudo, from magnus great. See Master, and cf. Maxim.]
Extent of dimensions; size; -- applied to things that have length, breath, and thickness.
Conceive those particles of bodies to be so disposed amongst themselves, that the intervals of empty spaces between them may be equal in magnitude to them all. Sir I. Newton.
2. Geom.
That which has one or more of the three dimensions, length, breadth, and thickness.
Anything of which greater or less can be predicated, as time, weight, force, and the like.
Greatness; grandeur.
"With plain, heroic
of mind."
Greatness, in reference to influence or effect; importance; as, an affair of magnitude.
The magnitude of his designs. Bp. Horsley.
Apparent magnitude Opt., the angular breadth of an object viewed as measured by the angle which it subtends at the eye of the observer; -- called also apparent diameter. -- Magnitude of a star
Astron., the rank of a star with respect to brightness. About twenty very bright stars are said to be of first magnitude, the stars of the sixth magnitude being just visible to the naked eye.
Telescopic stars are classified down to the twelfth magnitude or lower. The scale of the magnitudes is quite arbitrary, but by means of photometers, the classification has been made to tenths of a
magnitude. <-- the difference in actual brightness between magnitudes is now specified as a factor of 2.512, i.e. the difference in brightness is 100 for stars differing by five magnitudes. -->
© Webster 1913.
The 25 brightest stars of the night sky atom, molecule, nucleus, proton, neutron, electron scientific notation Exodus 8:2
Proxima Centauri color Vanity of vanities Stellar Description
Ladyfest Brazilian flag Adaptive Differential Pulse Code Modulation dwarf nova
Greek Architecture fixed stars Response.Write October 4, 2001
light curve Apalachicola River sign bit how to locate an earthquake's epicenter
vector component Mirfak resurgent dome
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Steam Tables Pdf
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First Quantization is a mystery... but de-quantizing perhaps not
up vote 5 down vote favorite
There is an well-known infamous DICTUM:
-Second Quantization is a functor, First Quantization is a mystery-.
Indeed, second quantization is the "Fock functor", which builds the Fock space in a canonical way out of the Hilbert space of a single particle.
But, what about first quantization? There is probably no hope to canonically associate an Hilbert space to the manifold of states of a classical particle (mathematically, there seem to be an inherent
element of choice as far as turning functions into operators).
However, there is (I suspect) some functorial description for going the other way around, FROM the quantum scenario INTO the classical one (corresponding to the limit $h\rightarrow 0$). If this is
true, there maybe a "fiber" of candidate quantum descriptions, all collapsing into the same classical one.
Any place where this has been worked out clearly?
quantum-mechanics ct.category-theory higher-category-theory
Possibly relevant question: mathoverflow.net/questions/10678 – José Figueroa-O'Farrill Jun 10 '11 at 18:49
add comment
1 Answer
active oldest votes
You may be interested in the answers to a question on physics.SE, "Quantum Mechanics on Manifold".
The gist of it is that there is a plethora of different quantization schemes (canonical quantization, geometric quantization, ...). As you note, they all have the same classical theory as a
limit. Overview:
Quantization Methods: A Guide for Physicists and Analysts
up vote 4 That said, speaking as a physicist (caveat emptor), I think the paper
down vote
pretty much covers everything that is of practical relevance. Unsurprisingly, the Schrödinger equation on an embedded manifold $M \subseteq \mathbb{R}^3$ depends not only on intrinsic data,
like the metric or curvature, but also on extrinsic data about the embedding of the manifold inside 3D space. That's to be expected because an electron can tunnel through the full 3D space
and hence take "extrinsic short-cuts". I think that's also the reason why there is so much ambiguity in quantization schemes: it's simply not something that you can do completely
add comment
Not the answer you're looking for? Browse other questions tagged quantum-mechanics ct.category-theory higher-category-theory or ask your own question.
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Marcus Hook Math Tutor
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More Meter Madness
Okay, after having read WB8ICN’s postings on QRPedia, I decided to try to measure the impedence of both of my meters. I dug a 1.5M ohm resistor out of the pile, and hooked some jumper leads to it. I
then did the following:
1. Measure the resistance of the resistor, using the DVM as an ohm meter (R)
2. Measure the voltage of a 9 volt battery, using the DVM as a volt meter. (V)
3. Hook the resistor in series with the battery, and measure the resulting voltage. (V[m])
What we are looking for is the resistance of the meter, which we will call R[m]. A little math should show that R[m] = R * V[m] / (V – V[m]). Okay, that heavy lifting aside, let’s see what we
discovered about my two meters:
For my better meter, a nice Radio Shack PC interfaceable multimeter, I got the following readings:
1. R = 1.496M ohms
2. V = 9.24 volts
3. V[m] = 7.94 volts
Plugging these values into the formula, we get a value for R[m] of 9.18M ohms, considerably off from the 10M ohms that we used as the “nominal” value. Later, I’ll go back and work up the correction
factor for the previous nights example, but for now, let’s move onto my second meter, a cheapy $20 one that I’ve used mostly to check my car battery and continuity. I recorded the following values:
1. R = 1.480M ohms
2. V = 9.14 volts
3. V[m] = 3.68 volts
Plugging these values in, I get about .997M ohms, or about 1Mohm! No wonder my previous readings were so crazy, with the 4.4M ohm resistance of my series resistors, this thing was dividing the peak
value down considerably.
Okay, let’s go back to my dummy load experiment. First, on meter number one, we find that R[m] of 9.18M ohms. If we go through my various calculations and correct for the measured values, we find
that the output power should be around 4.71 watts. Doing the same for measurement number two, we’d get an output power of around 4.67 watts, both in excellent agreement!
(Note, I didn’t measure the voltage drop of the diode, I’m assuming 0.7 v. The data sheet says the maximum drop would be 1v).
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• m0n5t3r on Visual Cryptography
• Barrie Gilbert on Crystal Sets to Sideband, by Frank W. Harris
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Centripetal Force vs Centrifugal Force
Neither are forces in the sense that neither have the tendency to cause masses to accelerate.
Careful here.
Yes the centripetal
is real and measurable with real forcemeters.
Yes the
centrifugal force
is a convenient fiction to make the maths easy.
How come?
Velocity is a vector quantity and change of velocity may be effected by change of magnitude or change of direction.
Either constitute an acceleration.
Now for centripetal
, this is always provided by the normal suppliers or agent of
So for instance consider a weight whirling round on a string.
The centripetal
is provided by the tension in the string - they are one and the same.
However the weight is also in constant acceleration.
This acceleration is directed towards the centre of of rotation (inwards).
This is in accordance with Newtons second law
= mass times acceleration.
So the weight is not in equilibrium.
This is so fundamental and important I will repeat it
So the weight is not in equilibrium.
Shazam ! * # ! Enter the
centrifugal force
This ficticious
is exactly equal in magnitude to but opposite in direction to the centripetal
Let us pretend that such a
is acting on the weight.
Now the weight is in equilibrium Yay!
The weight is said to be in equilibrium under the action of the tension in the string and the opposing
centrifugal force
This trick or transformation was first proposed by D'Alembert.
It enables us to reduce a problem in dynamics ( using Newtons Laws of motion) to a simpler one in statics (using the laws of equilibrium).
Note that you never see both the centripetal
and the
centrifugal force
appear in the same analysis you see either one or the other.
When the
centrifugal force
is used the acceleration is zero, since the system is in equilibrium.
When the centripetal
is invoked the system is not in equilibrium but accelerates. This acceleration is provided by the centripetal
So as regards gravity and acceleration, the acceleration due to gravity = the centripetal acceleration or the
of gravity = the centripetal
. Here '= ' means the same as.
Gravity and the centripetal
(acceleration) are not additional, they are one and the same.
Does this help?
Oh and welcome to Physics Forums
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Creative Mathematical Genius
Born: Erlangen, Germany, March 23, 1882
Died: Bryn Mawr, Pennsylvania, April 14, 1935
Creative Mathematical Genius
It might be that Emmy Noether was designed for mathematical greatness. Her father Max was a math professor at the University of Erlangen. Scholarship was in her family; two of her three brothers
became scientists as well. Emmy would surpass them all. Ultimately Max would become best known as Emmy Noether's father.
Amalie Emmy Noether spent an average childhood learning the arts that were expected of upper middle class girls. Girls were not allowed to attend the college preparatory schools. Instead, she went to
a general "finishing school," and in 1900 was certified to teach English and French. But rather than teaching, she pursued a university education in mathematics
She audited classes at Erlangen as one of two women among thousands of men, then took the entrance exam. She entered the University of Göttingen in 1903, again as an auditor, and transferred back to
Erlangen in 1904 when the university finally let women enroll. She received her mathematics Ph.D. in 1907.
Noether worked at the Mathematical Institute of Erlangen, without pay or title, from 1908 to 1915. It was during this time that she collaborated with the algebraist Ernst Otto Fischer and started
work on the more general, theoretical algebra for which she would later be recognized. She also worked with the prominent mathematicians Hermann Minkowski, Felix Klein, and David Hilbert, whom she
had met at Göttingen. In 1915 she joined the Mathematical Institute in Göttingen and started working with Klein and Hilbert on Einstein's general relativity theory. In 1918 she proved two theorems
that were basic for both general relativity and elementary particle physics. One is still known as "Noether's Theorem."
But she still could not join the faculty at Göttingen University because of her gender. Noether was only allowed to lecture under Hilbert's name, as his assistant. Hilbert and Albert Einstein
interceded for her, and in 1919 she obtained her permission to lecture, although still without a salary. In 1922 she became an "associate professor without tenure" and began to receive a small
salary. Her status did not change while she remained at Göttingen, owing not only to prejudices against women, but also because she was a Jew, a Social Democrat, and a pacifist.*
During the 1920s Noether did foundational work on abstract algebra, working in group theory, ring theory, group representations, and number theory. Her mathematics would be very useful for physicists
and crystallographers, but it was controversial then. There was debate whether mathematics should be conceptual and abstract (intuitionist) or more physically based and applied (constructionist).
Noether's conceptual approach to algebra led to a body of principles unifying algebra, geometry, linear algebra, topology, and logic.
In 1928-29 she was a visiting professor at the University of Moscow. In 1930, she taught at Frankfurt. The International Mathematical Congress in Zurich asked her to give a plenary lecture in 1932,
and in the same year she was awarded the prestigious Ackermann-Teubner Memorial Prize in mathematics.
Nevertheless, in April 1933 she was denied permission to teach by the Nazi government. It was too dangerous for her to stay in Germany, and in September she accepted a guest professorship at Bryn
Mawr College. She also lectured at the Institute for Advanced Study in Princeton. The guest position was extended, but in April 1935 she had surgery to remove a uterine tumor and died from a
postoperative infection.
* Gottfried E. Noether, "Emmy Noether (1882-1935)," in Louise S. Grinstein and Paul J. Campbell: Women of Mathematics: A Bibliographic Sourcebook (New York, Greenwood Press), 1987, pp. 165-170.
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Apache Junction Algebra 1 Tutor
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If you can dream it, you can do it.
I'm sensing a pattern here..
Stefan wants to be with Katherine.
Damon wants to be with Katherine.
Stefan wants to be with Elena.
Damon wants to be with Elena.
Stefan leaves town.
Damon leaves town.
Stefan breaks up with Elena.
Damon breaks up with Elena.
Stefan becomes a ripper.
Damon becomes a ripper.
Seriously Damon..
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Answer to Random Drug Testing Question 12/24/11
Moderators: mvs_staff, forum_admin, Marilyn
In today's Parade there was a question about random drug testing. Every 3 months one quarter of a fixed population is randomly selected for testing. The question was what is the probability that a
person would be selected over the course of one year. Marilyn's answer was 25%, but the actual probability is 68%. The probability of not being selected is 0.75x0.75x0.75x0.75 = 0.316, so the
probability of being selected at least once is 1 - 0.316 = 0.684.
The binomial probability function defines the probabilities of being selected n times out of N possible times with p being the probability of being selected each time, and q = 1-p:
P(n/N) = [N!/n!(N-n)!] p^n q^(N-n)
The probability of being selected exactly once over the course of the year is:
P(1/4) = [4!/3!1!] [0.25^1][0.75^3] = 0.422
P(2/4) = [4!/2!2!] [0.25^2][0.75^2] = 0.211
Three times:
P(3/4) = [4!/1!3!] [0.25^3][0.75^1] = 0.047
Four times:
P(4/4) = [4!/4!0!] [0.25^4][0.75^0] = 0.004
The most likely outcome is being selected once (42.2%). A probability of 25% is nowhere to be found in any of the outcomes.
Posts: 3
Joined: Sat Sep 06, 2008 4:16 pm
On 12/25/11, Marilyn wrote:
I manage a drug-testing program for an organization with 400 employees. Every three months, a random-number generator selects 100 names for testing. Afterward, these names go back into the
selection pool. Obviously, the probability of an employee being chosen in one quarter is 25 percent. But what is the likelihood of being chosen over the course of a year?
-- Jerry Haskins, Vicksburg, Miss.
The probability remains 25 percent, despite the repeated testing. One might think that as the number of tests grows, the likelihood of being chosen increases, but as long as the size of the pool
remains the same, so does the probability. Goes against your intuition, doesn't it?
Juan Largo wrote:In today's Parade there was a question about random drug testing. Every 3 months one quarter of a fixed population is randomly selected for testing. The question was what is the
probability that a person would be selected over the course of one year. Marilyn's answer was 25%, but the actual probability is 68%. The probability of not being selected is 0.75x0.75x0.75x0.75
= 0.316, so the probability of being selected at least once is 1 - 0.316 = 0.684.
The binomial probability function defines the probabilities of being selected n times out of N possible times with p being the probability of being selected each time, and q = 1-p:
P(n/N) = [N!/n!(N-n)!] p^n q^(N-n)
The probability of being selected exactly once over the course of the year is:
P(1/4) = [4!/3!1!] [0.25^1][0.75^3] = 0.422
P(2/4) = [4!/2!2!] [0.25^2][0.75^2] = 0.211
Three times:
P(3/4) = [4!/1!3!] [0.25^3][0.75^1] = 0.047
Four times:
P(4/4) = [4!/4!0!] [0.25^4][0.75^0] = 0.004
The most likely outcome is being selected once (42.2%). A probability of 25% is nowhere to be found in any of the outcomes.
Marilyn frequently answers a diferent question then what was seems, to others, to be what was actually asked. This isn't an unpredictable situation, since the questions are short and English can be
ambiguous. But it would be helpeful if she could try to indicate what it is she is answering.
It could be that she was answering "Over the course of a year, does the probability that an employee will be chosen
in any given quarter
change?" Many people intuitively feel that if you aren't chosen in the spring test, the chances you will be selected in the summer will go up. And if you aren't chosen at all in the first three tests
of a year, it is almost certain you will be chosen for the fourth since each test picks 1/4 of the employees. She could have easily indicated that by saying "the likelihood of being chosen
for any given test
We can't know for certain what she was thinking when she gave this blatantly incorrect answer. But I don't think she will understand your math;
she has ignored
examples in the past where similar math has proven her incorrect. Her application of probability is usually based more on intuition than math, as evidenced by here repeated incorrect (or at least,
correct only for a question that isn't what most others see in the text) replies about sequences of dice rolls. She does have a better intuition than most, but she places too much confidence in it.
Posts: 1814
Joined: Tue Mar 10, 2009 11:01 am
I'd be willing to go along with what you said, but Marilyn did not explain her answer at all. She just threw out a number without offering any reasoning or rationale behind it, which is very unlike
her. Maybe she was having a bad day.
Posts: 3
Joined: Sat Sep 06, 2008 4:16 pm
Juan Largo wrote:I'd be willing to go along with what you said, but Marilyn did not explain her answer at all. She just threw out a number without offering any reasoning or rationale behind it,
which is very unlike her. Maybe she was having a bad day.
On the contrary, it is very unusual for her to offer any support for her answers. She usually just states it and expects everybody to see how obvious it is. The best recent example is the question
about dice sequences. She answered it three times teh exact same way, by stating that it should seem obvious which sequence was more likely.
Posts: 1814
Joined: Tue Mar 10, 2009 11:01 am
Once again Marilyn presents her question in an ambiguous manner. The phrase "being chosen over the course of a year" has several different meanings. It could mean at any time during the year a test
occurs, what's the probability of being selected. In that case, Marilyn is correct. But it could also mean the probability of being selected sometime during the year; i.e., there exists a date among
the 365/6 in the year such that a drug test is held on that date and you were selected. That is 68%, as our first commenter just computed. (In fact, it is close to 1-1/e).
Posts: 6
Joined: Sat Nov 29, 2008 8:43 pm
jimvb wrote:Once again Marilyn presents her question in an ambiguous manner. The phrase "being chosen over the course of a year" has several different meanings. It could mean at any time during
the year a test occurs, what's the probability of being selected.
Only by misinterpretation. It's an easy mistake to make, but clearly in error. A direct comparison was made between "being chosen in one quarter" and "being chosen over the course of a year." That
is, comparing one period to another. Your interpretation above requires the comparison to be between "being chosen in one quarter" and "being chosen in another quarter," in which case the word "year"
is superfluous.
Posts: 1814
Joined: Tue Mar 10, 2009 11:01 am
Well done, Juan!
In 17 words and a one line of math you have succinctly explained the
answer to the problem posed in this week's Parade column.
This is a simple problem from freshman statistics, so it's odd that Marilyn got it wrong. One can only hope that readers will be motivated to ponder this problem and seek to understand for themselves
how to solve it, rather than accepting the incorrect answer given in the column.
Juan Largo wrote:In today's Parade there was a question about random drug testing. Every 3 months one quarter of a fixed population is randomly selected for testing. The question was what is the
probability that a person would be selected over the course of one year. Marilyn's answer was 25%, but the actual probability is 68%. The probability of not being selected is 0.75x0.75x0.75x0.75
= 0.316, so the probability of being selected at least once is 1 - 0.316 = 0.684.
The binomial probability function defines the probabilities of being selected n times out of N possible times with p being the probability of being selected each time, and q = 1-p:
P(n/N) = [N!/n!(N-n)!] p^n q^(N-n)
The probability of being selected exactly once over the course of the year is:
P(1/4) = [4!/3!1!] [0.25^1][0.75^3] = 0.422
P(2/4) = [4!/2!2!] [0.25^2][0.75^2] = 0.211
Three times:
P(3/4) = [4!/1!3!] [0.25^3][0.75^1] = 0.047
Four times:
P(4/4) = [4!/4!0!] [0.25^4][0.75^0] = 0.004
The most likely outcome is being selected once (42.2%). A probability of 25% is nowhere to be found in any of the outcomes.
Posts: 1
Joined: Sun Oct 23, 2011 12:34 pm
Parade 12/25/11 wrote:
I manage a drug-testing program for an organization with 400 employees. Every three months, a random-number generator selects 100 names for testing. Afterward, these names go back into the
selection pool. Obviously, the probability of an employee being chosen in one quarter is 25 percent. But what is the likelihood of being chosen over the course of a year?
-- Jerry Haskins, Vicksburg, Miss.
Marilyn responds:
The probability remains 25 percent, despite the repeated testing. One might think that as the number of tests grows, the likelihood of being chosen increases, but as long as the size of the pool
remains the same, so does the probability. Goes against your intuition, doesn't it?
I think Marilyn did not answer the question actually asked which is "what is the likelihood of being chosen over the course of a year?"
This would be one minus the product of not being chosen in any quarter for four consecutive quarters:
Code: Select all
cumulative quarters, cumulative probability of being chosen
1, 1-0.75 =0.25
2, 1-(0.75)^2=0.4375
3, 1-(0.75)^3=0.5781
4, 1-(0.75)^4=0.6836
So, over the course of four consecutive quarters any employee has a 0.6836 probability of being chosen for testing.
Posts: 2076
Joined: Mon Jun 18, 2007 9:21 am
The probability remains 25 percent, despite the repeated testing. One might think that as the number of tests grows, the likelihood of being chosen increases, but as long as the size of the pool
remains the same, so does the probability. Goes against your intuition, doesn't it?
After thinking about this some more, what Marilyn undoubtedly meant was that the likelihood of being selected over the course of a year is unchanged by being selected or not being selected in any
previous quarters. It's a common fallacy that if you weren't selected beforehand you are "due" to be selected next time. Maybe Marilyn inferred that the person was committing this fallacy from the
way the question was stated and she was just trying to dispel it.
If that's the case, my apologies to Marilyn.
Posts: 3
Joined: Sat Sep 06, 2008 4:16 pm
I don't think you need to apologize to Marilyn. Marilyn didn't answer the question that was asked. Someone could have asked a question about what your odds are of being picked in the 4th drawing of
the year if you weren't picked in the first 3 (or if you were picked in the first 3). Either way, the answer would be 25%. But that wasn't the question asked.
According to the earlier post, here is what the question said, in part:
"Obviously, the probability of an employee being chosen in one quarter is 25 percent. But what is the likelihood of being chosen over the course of a year?"
I just don't see any reasonable way to interpret the question the way that Marilyn did. The question writer is asking what are the chances of being chosen sometime in the year (presumably in at least
1 of the 4 drawings), not the chances of being chose in any one quarter. It is right there in the question "Obviously, the probability of an employee being chosen in one quarter is 25%." Then Marilyn
goes on to explain that the chance of being chosen in any one quarter is 25%.
Marilyn definitely blew this one. I hope she writes a correction.
Posts: 2
Joined: Sat Jan 07, 2012 9:47 am
svalancius wrote:Marilyn definitely blew this one. I hope she writes a correction.
She has already admitted the answer was incorrect, but not a mistake. Instead, she made a tongue-in-cheek comment about being influenced by eggnog, and said she will address it in two weeks (probably
the delay caused by the publishing process).
Posts: 1814
Joined: Tue Mar 10, 2009 11:01 am
I'm new to the forum - where did she say she would address this in two weeks?
What Marilyn wrote was true; it just didn't answer the question. I'm glad to hear she is going to address it in a future column.
Posts: 2
Joined: Sat Jan 07, 2012 9:47 am
svalancius wrote:I'm new to the forum - where did she say she would address this in two weeks?
She answers daily questions on the parade website. Follow the links from the "home" page of this forum, or use
this link
What Marilyn wrote was true; it just didn't answer the question. I'm glad to hear she is going to address it in a future column.
It's also true that frogs have short stumpy legs, but that has no bearing on the question she was asked. An answer can still be wrong, even if true, if it doesn't address the question.
But Marilyn has a very poor record of admitting when she is wrong - as in, almost never, and only when she can find an acceptable excuse. There were at least two other answers in the past couple of
months that were outright wrong, and she ignored them. Probably because she can't argue that her answers were right for a different question. There was even one a couple of years ago where she broke
the answer down into three separate cases, and each one was wrong for different reasons. I mention that because the proof that one was wrong follows the same reasoning as the drug-test answer, that
the chance of a thing happening in at least one of two attempts is less than the twice the chance of it happening in one.
Posts: 1814
Joined: Tue Mar 10, 2009 11:01 am
I enjoy Marilyn's column, but I was so taken aback at her answer to the question on the morning of Dec. 25 that I registered to participate in this site so that I could comment, only to run into the
(reasonable) delays in establishing eligibility to participate. In the meantime, the points I wished to make have been ably and thoroughly made by other commenters.
I will make one observation that simply rephrases the point others have made. I focus on intuition; that is, beyond the conclusive mathematical demonstration presented earlier in the comments,
intuition shows the error of Marilyn's answer to the question that was -- unambiguously -- asked. Marilyn's answer was: "One might think that as the number of tests grows, the likelihood of being
chosen increases, but as long as the size of the pool remains the same, so does the probability. Goes against your intuition, doesn't it?" Sorry, but as the number of tests grows with a constant
pool, the probability of the event occurring at least once in the course of the tests increases. To wit: The odds of rolling a "6" when a six-sided die is tossed once is 1/6. So I throw the die 100
times (an increased number of tests) as the size of the pool remains the same (the die has six sides in each test); do the odds of getting at least one "6" across the 100 throws remain 1 in 6? Of
course not. That's why Russian Roulette played 100 times in sequence -- if our intuition means anything --is nearly certain to be fatal.
Looking forward to Marilyn's correction.
Posts: 1
Joined: Sun Dec 25, 2011 9:21 am
Location: Gainesville FL
jerrygf wrote:Sorry, but as the number of tests grows with a constant pool, the probability of the event occurring at least once in the course of the tests increases.
We can only speculate on how and/or why Marilyn made this mistake, but the following sequence of inferences does seem plausible:
• Many people do feel - intuitively - that random chance has a memory. That if heads has come up on three consecutive coin flips, somehow tails is more likely on the next one so as to "even things
out." A person on this list, who feels he understands probabilty but does not, has incorrectly called this "regression towards the mean" and claimed it was true in a more complex situation.
• I call it "correction toward the mean."
• Marilyn knows of this tendency, and has probably answered many questions involving "correction toward the mean." Her first mistake was, without reading the question carefully, she assumed this
was another.
• The only way it made sense to be a question about "correction toward the mean," was if the probability asked about was the probability in one quarter.
• So Marilyn's second mistake was that she answered the wrong question, and ...
• Her third was when she admonished the reader about thinking "correction toward the mean" was true.
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: Greek Pupils Excel at the 53rd International Mathematical Olympiad
Greek Pupils Excel at the 53rd International Mathematical Olympiad
By Areti Kotseli on July 20, 2012 in News
Greek pupils shined during the 53rd International Mathematical Olympiad receiving a Gold, a Silver and three Bronze medals as well as an honorable mention in Argentina on Monday July 16, 2012.
During the contest, Greek pupils proved to be the brightest minds of the Eurozone, leaving Germans, Italians, Dutchmen, Spaniards and others behind.
Every single pupil representing Greece received a distinction, bringing a Gold Medal to our country for the second consecutive year.
Our talented youth continued the long-year tradition of distinctions in International Mathematical Olympiads proving right the volunteer work of the Hellenic Mathematical Society.
Greek pupils were escorted by Professor at the National Technical University of Athens Anargyros Fellouris and Mathematician Evaggelos Zotos.
A total of 548 participants from 100 countries competed July 4-16 in the annual math competition designed for pre-collegiate students.
South Korea ranked first overall in the 2012 competition, with China as runner-up. Next years event will take place in Santa Marta, Colombia.
The Greek students who received distinctions are:
1.Lolas Panagiotis, Trikala, Gold Medal
2.Dimakis Panagiotis, Athina, Silver Medal
3.Mousatov Alexandros, Athens, Bronze Medal
4.Skiadopoulos Athinagoras, Rhodes, Bronze Medal
5.Tsinas Konstantinos, Trikala, Bronze Medal
6.Tsampasidis Zacharias, Katerini Honorable Mention
Power through Knowledge
"I hope for nothing. I fear nothing. I am free."
Nikos Kazantzakis
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How to replace failure by a list of successes
Results 11 - 20 of 61
- International Conference on Logic Programming , 1999
"... At present, the field of declarative programming is split into two main areas based on different formalisms; namely, functional programming, which is based on lambda calculus, and logic
programming, which is based on firstorder logic. There are currently several language proposals for integrating th ..."
Cited by 20 (3 self)
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At present, the field of declarative programming is split into two main areas based on different formalisms; namely, functional programming, which is based on lambda calculus, and logic programming,
which is based on firstorder logic. There are currently several language proposals for integrating the expressiveness of these two models of computation. In this thesis we work towards an integration
of the methodology from the two research areas. To this end, we propose an algebraic approach to reasoning about logic programs, corresponding to the approach taken in functional programming. In the
first half of the thesis we develop and discuss a framework which forms the basis for our algebraic analysis and transformation methods. The framework is based on an embedding of definite logic
programs into lazy functional programs in Haskell, such that both the declarative and the operational semantics of the logic programs are preserved. In spite of its conciseness and apparent
simplicity, the embedding proves to have many interesting properties and it gives rise to an algebraic semantics of logic programming. It also allows us to reason about logic programs in a simple
calculational style, using rewriting and the algebraic laws of combinators. In the embedding, the meaning of a logic program arises compositionally from the meaning of its constituent subprograms and
the combinators that connect them. In the second half of the thesis we explore applications of the embedding to the algebraic transformation of logic programs. A series of examples covers simple
program derivations, where our techniques simplify some of the current techniques. Another set of examples explores applications of the more advanced program development techniques from the Algebra
of Programming by Bird and de Moor [18], where we expand the techniques currently available for logic program derivation and optimisation. To my parents, Sandor and Erzsebet. And the end of all our
exploring Will be to arrive where we started And know the place for the first time.
- In 2nd International Workshop on Practial Aspects of Declarative Languages, volume 1753 of LNCS , 2000
"... . Pattern matching is a great convenience in programming. However, pattern matching has its problems: it conicts with data abstraction; it is complex (at least in Haskell, which has pattern
guards, irrefutable patterns, n+k patterns, as patterns, etc.); it is a source of runtime errors; and lastly, ..."
Cited by 17 (1 self)
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. Pattern matching is a great convenience in programming. However, pattern matching has its problems: it conicts with data abstraction; it is complex (at least in Haskell, which has pattern guards,
irrefutable patterns, n+k patterns, as patterns, etc.); it is a source of runtime errors; and lastly, one cannot abstract over patterns as they are not a rst class language construct. This paper
proposes a simplication of pattern matching that makes patterns rst class. The key idea is to treat patterns as functions of type a!Maybe bi.e., a!(Nothing|Just b); thus, patterns and pattern
combinators can be written as functions in the language. 1 Introduction A hotly debated issue in the language Haskell [HJW92] has been patterns. What are their semantics? Do we want n+1 patterns? Do
we need @-patterns? When do we match lazily and when do we match strictly? Do we need to extend patterns with pattern guards? And etc. In this paper I will propose, not another extension, but a
- Department of Computer Science, University of Utrecht , 1999
"... The distinctive merit of the declarative reading of logic programs is the validity ofallthelaws of reasoning supplied by the predicate calculus with equality. Surprisingly many of these laws are
still valid for the procedural reading � they can therefore be used safely for algebraic manipulation, pr ..."
Cited by 16 (4 self)
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The distinctive merit of the declarative reading of logic programs is the validity ofallthelaws of reasoning supplied by the predicate calculus with equality. Surprisingly many of these laws are
still valid for the procedural reading � they can therefore be used safely for algebraic manipulation, program transformation and optimisation of executable logic programs. This paper lists a number
of common laws, and proves their validity for the standard (depth- rst search) procedural reading of Prolog. They also hold for alternative search strategies, e.g. breadth- rst search. Our proofs of
the laws are based on the standard algebra of functional programming, after the strategies have been given a rather simple implementation in Haskell. 1
- In Proc. 2011 ACM Conference on Object-Oriented Programming Systems, Languages, and Applications , 2011
"... In many projects, lexical preprocessors are used to manage different variants of the project (using conditional compilation) and to define compile-time code transformations (using macros).
Unfortunately, while being a simple way to implement variability, conditional compilation and lexical macros hi ..."
Cited by 12 (4 self)
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In many projects, lexical preprocessors are used to manage different variants of the project (using conditional compilation) and to define compile-time code transformations (using macros).
Unfortunately, while being a simple way to implement variability, conditional compilation and lexical macros hinder automatic analysis, even though such analysis is urgently needed to combat
variability-induced complexity. To analyze code with its variability, we need to parse it without preprocessing it. However, current parsing solutions use unsound heuristics, support only a subset of
the language, or suffer from exponential explosion. As part of the TypeChef project, we contribute a novel variability-aware parser that can parse almost all unpreprocessed code without heuristics in
practicable time. Beyond the obvious task of detecting syntax errors, our parser paves the road for further analysis, such as variability-aware type checking. We implement variability-aware parsers
for Java and GNU C and demonstrate practicability by parsing the product line MobileMedia and the entire X86 architecture of the Linux kernel with 6065 variable features.
- The Journal of Logic Programming , 1994
"... This paper presents a new program analysis framework to approximate call patterns and their results in functional logic computations. We consider programs containing non-strict, nondeterministic
operations in order to make the analysis applicable to modern functional logic languages like Curry or TO ..."
Cited by 10 (0 self)
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This paper presents a new program analysis framework to approximate call patterns and their results in functional logic computations. We consider programs containing non-strict, nondeterministic
operations in order to make the analysis applicable to modern functional logic languages like Curry or TOY. For this purpose, we present a new fixpoint characterization of functional logic
computations w.r.t. a set of initial calls. We show how programs can be analyzed by approximating this fixpoint. The results of such an approximation have various applications, e.g., program
optimization as well as verifying safety properties of programs. 1
- In ????, pages , 2007
"... Abstract. Parser combinators are higher-order functions used to build parsers as executable specifications of grammars. Some existing implementations are only able to handle limited ambiguity,
some have exponential time and/or space complexity for ambiguous input, most cannot accommodate left-recurs ..."
Cited by 10 (0 self)
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Abstract. Parser combinators are higher-order functions used to build parsers as executable specifications of grammars. Some existing implementations are only able to handle limited ambiguity, some
have exponential time and/or space complexity for ambiguous input, most cannot accommodate left-recursive grammars. This paper describes combinators, implemented in Haskell, which overcome all of
these limitations.
- In Proc. Haskell workshop
"... In this paper, we show how to manipulate syntax with binding using a mixed representation of names for free variables (with respect to the task in hand) and de Bruijn indices [5] for bound
variables. By doing so, we retain the advantages of both representations: naming supports easy, arithmetic-free ..."
Cited by 9 (2 self)
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In this paper, we show how to manipulate syntax with binding using a mixed representation of names for free variables (with respect to the task in hand) and de Bruijn indices [5] for bound variables.
By doing so, we retain the advantages of both representations: naming supports easy, arithmetic-free manipulation of terms; de Bruijn indices eliminate the need for α-conversion. Further, we have
ensured that not only the user but also the implementation need never deal with de Bruijn indices, except within key basic operations. Moreover, we give a hierarchical representation for names which
naturally reflects the structure of the operations we implement. Name choice is safe and straightforward. Our technology combines easily with an approach to syntax manipulation inspired by Huet’s
‘zippers’[10]. Without the ideas in this paper, we would have struggled to implement EPIGRAM [19]. Our example—constructing inductive elimination operators for datatype families—is but one of many
where it proves invaluable.
- IN PROC. OF THE 20TH INTERNATIONAL WORKSHOP ON FUNCTIONAL AND (CONSTRAINT) LOGIC PROGRAMMING (WFLP 2011 , 2011
"... In this paper we present our first steps towards a new system to compile functional logic programs of the source language Curry into purely functional Haskell programs. Our implementation is
based on the idea to represent non-deterministic results as values of the data types corresponding to the res ..."
Cited by 9 (4 self)
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In this paper we present our first steps towards a new system to compile functional logic programs of the source language Curry into purely functional Haskell programs. Our implementation is based on
the idea to represent non-deterministic results as values of the data types corresponding to the results. This enables the application of various search strategies to extract values from the search
space. We show by several benchmarks that our implementation can compete with or outperform other existing implementations of Curry.
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Utilizing the Concept of the Golden Proportion
Written by Dino S. Javaheri, DMD, and Sara Shahnavaz, DDS Friday, 31 May 2002 19:00
Adhering to smile design principles is essential when initiating the restoration of teeth in the anterior region of the mouth. When the dentist does not use smile design guidelines, there is no
blueprint for tooth preparation and restoration. In the absence of a plan, teeth may be underprepared or overprepared, leading to a compromise of the aesthetic and functional results. In addition,
the laboratory technician must be provided with an adequate amount of information to allow fabrication of ideal restorations. Providing the technician with only a stone model communicates minimal
aesthetic information.
In order to achieve optimal aesthetic results, the dentist must provide the technician with a plan specifically designed for each patient. This must include the proper position of the incisal edges,
incisal plane, midline position, the length and width of the teeth, tissue height, axial inclination, embrasure form, line angles, and surface texture.^1-3 A mathematical concept called the Golden
Proportion has provided guidelines for tooth preparation and fabrication of the restorations. Application of this principle will, therefore, help to improve design, functional results, laboratory
communication, and patient satisfaction. This article discusses the Golden Proportion and its role in dental aesthetics.
The Golden Proportion is based on a very specific mathematical proportion that is acquired from mathematics and nature.^4-6 This proportion, also referred to as the Divine Proportion or Magic ratio,
describes a number often encountered when taking the ratio of distance in simple geometric figures such as the pentagram, decagon, and dodecagon.^7 To understand the Golden Proportion from a
mathematical standpoint, consider a line segment with a length of one. Divide this segment into two segments, a shorter segment, x, and a longer segment, 1-x. The Golden Proportion is one where the
ratio of the shorter segment to the longer segment (x:1-x) is equal to the ratio of the longer segment to the whole segment(1-x:1).^8 The mathematical equation that needs to be solved for x is x/
(1-x)=(1-x)/1. Solving this equation results in a shorter segment length of 0.382 and the longer segment length at 0.618.^7,9 This is the only ratio that solves the equation.
This proportion can be found throughout the universe; from the spirals of galaxies to the spiral of a Nautilus seashell; from the harmony of music to the beauty in art.8,9 A botanist will find it in
the growth patterns of flowers and plants, while a zoologist sees it in the breeding of rabbits.^10
The ancient Greeks have often been credited with defining the Golden Proportion.^11 Pythagorus and Phidius are believed to have identified it.^11 The Golden Proportion was closely studied by the
Greek sculptor Phidias, hence the designated mathematical symbol for the Golden Proportion ratio is phi.^9 The Greeks studied phi through their mathematics and used it in their architecture. The
Parthenon at Athens is a classic example of the use of the Golden Proportion. The initial discovery of the Golden Proportion by the Greeks has been challenged by architects and archeologists.8 Badawy
^12 claims that the use of proportions was initiated by the ancient Egyptians, based on his study of Egyptian buildings, sculptures, and paintings. The ancient Egyptians used it in the construction
of the great pyramids and in the design of hieroglyphs found on tomb walls.^13 Separately, the ancients of Mexico applied the Golden Proportion while building the Sun pyramid at Teotihuacan.^8
Although debate remains as to the origin of this proportion, it is clearly evident in Egyptian, Greek, Roman, Romanesque, Gothic, and Renaissance culture.^9,14
The Golden Proportion is also evident in the arts.^15 Renowned artists such as Michelangelo, Raphael, and Leonardo da Vinci made use of this concept.^16 Leonardo da Vinci drew the "ideal man" using
the Golden Proportion, and the head of Mona Lisa was drawn using this relationship. Evidence suggests that classical music composed by Mozart, Beethoven, and Bach incorporated the Golden Proportion.^
17 Whether the use of the Golden Proportion by these artists and musicians was by design, intuition, or accident is not known.
In the 19th century, the French mathematician and scientist, Edouard Lucas, documented the Golden Proportion as it occurs in nature. The French architect Le Corbusier applied his concepts to modern
architecture. Today, windows, furniture, automobiles, and smiles are designed using the concept of the Golden Proportion.
In dentistry, the Golden Proportion has been suggested as one possible mathematical approach to development of ideal size and shape relationships for maxillary teeth. These principles can be used to
determine the width of the teeth as they relate to each other. There are many important design considerations that accompany application of the Golden Proportion. Only after the incisal edge
position, incisal plane, gingival plane, and central incisor length have been determined, can the Golden Proportion be applied. As applied to the maxillary teeth, the Golden Proportion requires a 62%
reduction in the viewing width of each tooth, beginning with the central incisor, and proceeding posteriorly.
It is important to distinguish between the viewing width and the clinical width of a tooth. Viewing width is measured on a straight-on frontal photograph of the patient's smile. Anatomical
measurements taken in a patient's mouth or on a model will not provide this relationship. To apply the concept of the Golden Proportion, a lateral incisor can be arbitrarily assigned a viewing width
of 1, in which case the central incisor width should be 1.6, the cuspid 0.62, the first premolar 0.38, and the second premolar 0.23.^4,18 This allows for a smile dominated by the central incisors,
with the other teeth becoming progressively smaller.
A variation of the Golden Proportion was suggested by Snow.^19 He proposed that the Golden Proportion could be used to develop symmetry, dominance, and proportion for aesthetically pleasing smiles.
To simplify these proportions, the assigned values for the six anterior teeth were added (assigning 1.6 to the central, 1 to the lateral, and 0.62 to the cuspid). Then the assigned width of each
tooth was divided by the total to determine the relative percentage. The resulting percentages were: central incisor 25%, lateral incisor 16%, and cuspid 9%. These percentages illustrate the
dominance of the central incisors, which are 50% of the cuspid-to-cuspid width.
Following is a clinical case in which the concept of the Golden Proportion is applied.
Case presentation
Figure 1. Preoperative view of patient presenting with aesthetic concerns.
The patient, a 42-year-old female, was unhappy with the appearance of her maxillary anterior teeth (Figure 1). She had four veneers that had been placed approximately 15 years ago. One had debonded
and the other three had marginal leakage. She was unhappy with her smile because of the color and shape/size discrepancies of the maxillary anterior teeth. The patient did not display the mandibular
teeth at full smile or during conversation, and therefore was not concerned about the appearance of these teeth.
Clinical Exam
The clinical examination revealed leaking porcelain veneers on the central incisors and the left lateral incisor. The veneer on the maxillary right cuspid had debonded. The tissue heights were
slightly uneven, resulting in disproportional sizes of the teeth. The general periodontal health, however, was good. The patient's occlusion exhibited loss of anterior guidance, which created group
function. The patient did not report any tooth pain, and there was no history of TMJ symptoms.
The patient sought a smile of uniform symmetry, appropriate tooth shape, and lighter color. Tissue recontouring was discussed to allow for development of proper proportions of the new restorations.
To evaluate the position of the incisal edge of the teeth, a direct mock-up was prepared. Although this can be accomplished with a wax-up, a direct mock-up not only allows the patient to preview and
approve the length of the veneers, but serves as a preparation guide for the clinician.
Determining the location of the incisal edge of the central incisors is one of the most important factors for a successful reconstruction. The three determinants to be considered are occlusion,
phonetics, and aesthetics. The occlusion should be characterized by anterior guidance, cuspid disclusion, lack of slide between centric relation and centric occlusion, absence of balancing or working
side interferences, and equal and simultaneous force on each tooth when the teeth are in contact, with no deflection when additional force is exerted. Phonetics is also used as a guide to determine
the position of the incisal edge. During the production of "F" or "V" sounds, the maxillary anterior teeth should lightly touch the vermilion border of the lower lip. Finally, both the clinician and
the patient should evaluate the aesthetics of the proposed incisal edge position.
Figure 2. Tracing from preoperative photograph to apply smile design principles. Figure 3. Proposed incisal edge position and gingival heights are marked.
Figure 4. New central incisor length is measured using a ruler. Figure 5. A 75% width-to-height ratio is determined and marked.
Figure 6. The shapes of the central incisors are drawn using the length and width measurements.
After determining the incisal edge position, the length and tissue heights of the central incisors are established. From this point a trace drawing can be used to apply design concepts. It is
important to remember that the Golden Proportion is applied to the viewing width of teeth, using a frontal photograph. Using tracing paper, the patient's teeth are outlined (Figure 2). The new
incisal edge position and the gingival position are placed on the working outline (Figure 3). To give the central incisors a distinct rectangular shape, the width-to-height ratio should be
approximately 75%. The length of the central incisors is measured on the photograph and multiplied by 0.75 to give the correct photographic width for these teeth (Figures 4 and 5). After marking the
width measurement, the proposed shape of the new central incisors can be drawn (Figure 6). At this time, the width of the other anterior teeth can be established using the Golden Proportion.
Figure 7. The width of the central incisors is reduced by 62% to give the width of the lateral incisors. Figure 8. The shapes of the lateral incisors are drawn.
Figure 9. A 62% reduction in width of the lateral incisors gives the profile width of the cuspids.
The width of the central incisors is multiplied by 0.62, thereby establishing the width of the lateral incisors. This new width of the lateral incisors is marked, and the shapes are drawn (Figures 7
and 8). The same technique is utilized for the profile view of the cuspids and premolars (Figure 9). It is important to note that the Golden Proportion only approximates the width of each tooth, not
the length.
This tracing helps to define the treatment plan. The tracing shows where tissue recontouring is required and helps in preparation design. The tracing demonstrates whether the widths of the teeth must
be altered in order to design a proportional and symmetrical smile. To do this may require aggressive preparation of the teeth to open the contact areas, so the laboratory technician can establish
the new proportions.
Prior to prescribing the final veneers, the provisional restorations are fabricated to the guidelines set forth in the smile design process. This gives the practitioner and patient an opportunity to
evaluate the proposed aesthetics and occlusion. If necessary, changes can be made in the provisional restorations and relayed to the laboratory technician.
Figure 10. The final smile designed using the concept of the Golden Proportion.
After patient approval, the final pressed ceramic restorations were placed using a resin cement. The patient's occlusion and aesthetics were re-evaluated 1 week after cementation. The patient
reported no sensitivity or complications. Occlusion and gingival health were excellent. The patient was given a mandibular bleaching tray to achieve a shade match between the mandibular and maxillary
teeth. The completed case is shown in Figure 10.
It is not necessary to achieve the width of the anterior teeth exactly as described by the Golden Proportion when designing an anterior reconstruction. In fact, these exact proportions rarely occur
in the natural dentition. The Golden Proportion is just one of many factors involved in smile design. The value of the Golden Proportion is as a diagnostic tool in evaluating a smile, and as a guide
to veneer preparation and fabrication.
In fact, several studies have indicated that wide variation exists for patients and dentists regarding ideal anterior tooth proportions.^20-22 In a study by Rosenstiel et al,^21 549 dentists
evaluated computer images of the same six maxillary anterior teeth. Dentists preferred 80% proportions when viewing short or very short teeth, and the Golden Proportion (62%) for very tall teeth.
There was no identifiable preference for teeth of normal length or tall teeth, and choices could not be predicted based on gender, specialist training, experience, or patient volume. The results of a
similar study by Kokich et al^22 demonstrated that orthodontists, general dentists, and lay people detect specific aesthetic discrepancies at different levels of change. In the case of the Golden
Proportion, lay people did not discern a lateral incisor narrowing until the deviation reached 4 mm.
The patient who presents for cosmetic rehabilitation will probably not be comfortable with some of the wide deviations identified in the previous studies. Nevertheless, these studies demonstrate that
no single rule or formula can be used to generalize across a population.
The design elements presented here do not represent a complete discussion of the available principles and techniques of smile design. In addition, patients and dentists will vary in their
preferences. The principles described here offer one set of guidelines for aesthetic dentistry. In the case of the Golden Proportion, the exact proportions are not as important as are the concepts of
symmetry and the use of a logical approach to aesthetic restoration of the maxillary anterior teeth.F
The authors would like to thank Frontier Dental Laboratory for fabricating the restorations presented in the article.
1. Moskowitz ME, Nayyar A. Determinants of dental esthetics: a rational for smile analysis and treatment. Compend Contin Educ Dent. 1995;16:1164-1166.
2. Eubank J. Macroesthetic elements of smile design. J Am Dent Assoc. 2001;132:39-45.
3. Morely J. Smile design: specific considerations. J Calif Dent Assoc. 1997;25:633-637.
4. Preston JD. The Golden Proportion revisited. J Esthet Dent. 1993;5:247-251.
5. Gillen RJ, Schwartz RS, Hilton TJ, et al. An analysis of selected normative tooth proportions. Int J Prosthodont. 1994;7:410-417.
6. Herz-Fischler R. A Mathematical History of the Golden Number. New York, NY: Dover Publications Inc; 1998.
7. Ghyka MC. The Geometry of Art and Life. New York, NY: Dover Publications Inc; 1977.
8. Williams R. The Golden Proportion. The Geometrical Foundation of Natural Structure: Source Book of Design. New York, NY: Dover Publications Inc; 1979.
9. Huntley HE. The Divine Proportion: A Study In Mathematical Beauty. New York, NY: Dover Publications Inc; 1970.
10. Fischler R. How to find the golden number without really trying. Fibonacci Quart. 1981:19:406-410.
11. Boyer CB. History of Mathematics. New York, NY: Wiley & Sons; 1968.
12. Badawy A. A History of Egyptian Architecture. 3 volumes. New York, NY: University of California Press; 1966-68.
13. Robins G, Shute C. Mathematical bases of ancient Egyptian architecture and graphic art. Historia Math. 1985;12:107-122.
14. Fields M. Practical mathematics of roman times. Mathematics Teacher. 1933;26:77-84.
15. Kappraff J. Connections: The Geometric Bridge Between Art and Science. New York, NY: McGraw-Hill, Inc; 1991.
16. Kemp, Martin. Spirals of life: D'Arcy Thompson and Theodore Cook, with Leonardo and Durer in retrospect. Physis Riv Internaz Storia Sci. 1995;32:37-54.
17. Evans B. Number and Form and Content: A Composer's Path Of Inquiry. The Visual Mind. Cambridge, Mass: Leonardo Book Series, MIT Press; 1993.
18. Gillen RJ, Schwartz RS, Hilton TJ, et al. An analysis of selected normative tooth proportions. Int J Prosthodont. 1994;7:410-417.
19. Snow SR. Esthetic smile analysis of maxillary anterior tooth width: the golden percentage. J Esthet Dent. 1999;11:177-184.
20. Wagner IV, Carlsson GE, Ekstrand K, et al. A comparative study of assessment of dental appearance by dentists, dental technicians, and laymen using computer-aided image manipulation. J Esthet
Dent. 1996;8:199-205.
21. Rosenstiel SF, Ward DH, Rashid RG. Dentists' preferences of anterior tooth proportion: a web-based study. J Prosthodont. 2000;9:123-136.
22. Kokich VO Jr, Kiyak HA, Shapiro PA. Comparing the perception of dentists and lay people to altered dental esthetics. J Esthet Dent. 1999;11:311-324.
Dr. Javaheri is an assistant professor in the Advanced Education in General Dentistry residency program at University of the Pacific (UOP). He is the course director for the UOP continuing education
program "Setting New Standards in Cosmetic Dentistry." Dr. Javaheri has published numerous articles in various publications. He maintains an aesthetic/restorative practice in Alamo, Calif. Dr.
Javaheri can be reached by e-mail at This e-mail address is being protected from spambots. You need JavaScript enabled to view it .
Dr. Shahnavaz maintains a practice in Alamo, Calif.
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POP Seminar, 02-07-03
Patricia Johann
Rutgers University
Parametric Polymorphism, Observational Equivalence, and Monadic Computation
In this talk we propose a calculus, called PolyComp, supporting polymorphism, type- and term-level recursion, and computation (i.e., monadic) types. We give PolyComp an operational semantics, which
we use to construct parametric models for it. We show that equivalence in these models coincides with corresponding notions of observational equivalence and that these, in turn, depend on the types
at which observations can be made. Finally, we use our models to derive operational properties of PolyComp which are independent of the choice of monad, and also discuss how monads which model
specific computational effects can be incorporated into our framework. Successfully incorporating new language features into parametric models of observational equivalence is nontrivial because
operational techniques are often not modular. In particular, term-level recursion typically does not interact well with other features. To our knowledge, ours is the first parametric model of
observational equivalence for a calculus supporting all the features of PolyComp.
This is joint work with Neil Ghani.
Principles of Programming Seminars
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C# .NET
It is bit difficult to give exact algorithm for this problem but Just for reference you can refer following solution-
float[] used = new float[n + 1];
//used[j] is the amount of space in bin j already used up.
int i, j;
//Initialize all used entries to 0.0
//Sort S into descending(nonincreasing)order, giving the sequence S1 >= S2 >= ... >= Sn.
for(i = 1; i <= n; i++)
//Look for a bin in which s[i] fits.
for(j = 1; j <= n; j++)
if (used[j] + S[i] <= 1.0) {
bin[i] = j;
used[j] += S[i];
break; //exit for(j)
except this refer these links for information-
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Example of multi dimensional array in turbo c?
i dont know how to manipulate array in turbo c
Example of multi dimensional array in turbo c?
An array is a variable that has space to hold 1 or more instances of only ONE data type.
unsigned int fuzz[256];
This allocates enough memory for 256 unsigned integers, referenced by the variable name fuzz.
You can store up to 256 unsigned integer numbers in that array. Example: 4,5,7,10,etc.
If you want to write a number to the array, you would write something like: fuzz[2]=4;
In this case, the unsigned integer 4 is written to the 3rd location of array fuzz (array numbers start at 0, not 1!).
A multi-dimensional array is like a matrice.
Using my previous example, you can think of a single array as 1 row of numbers. A multi-dimensional array can be thought as having multiple rows of data.
unsigned int foo[256][256];
This is a two dimensional array.
Think of it as 1 row of numbers, with another row of numbers beneath it.
To write to the first location in the SECOND row, you could do this: foo[1,0]=4.
The first number in the bracket represents the row number (0-1 in this case) and the second number represents the location on that particular row. The number 4 is written to the second row, 1st
column on that row.
No comments:
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Gödel and the end of physics
Stephen Hawking
"Gödel and the end of physics"
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This lecture is the intellectual property of Professor S.W.Hawking. You may not reproduce, edit, translate, distribute, publish or host this document in any way with out the permission of Professor
In this talk, I want to ask how far can we go, in our search for understanding and knowledge.wIll we ever find a complete form of the laws of nature. bY a complete form, I mean a set of rules, that
in principle at least, enable us to predict the future to an arbitrary accuracy, knowing the state of the universe at one time. A qualitative understanding of the laws, has been the aim of
philosophers and scientists, from Ahristottal onwards. But it was Newton's Principia Mathematica in 1687, containing his theory of universal gravitation, that made the laws quantitative and precise.
This led to the idea of scientific determinism, which seems first to have been expressed by Le-plass. iF at one time, one knew the positions and velocities of all the particles in the universe, the
laws of science should enable us to calculate their positions and velocities, at any other time, past or future. The laws may or may not have been ordained by God, but scientific determinism asserts
that he does not intervene, to break them.
At first, it seemed that these hopes for a complete determinism would be dashed, by the discovery early in the 20th century, that events like the decay of radio active atoms, seemed to take place at
random. It was as if God was playing dice, in Einstein's phrase. But science snatched victory from the jaws of defeat, by moving the goal posts, and redefining what is meant by a complete knowledge
of the universe. It was a stroke of brilliance, whose philosophical implications have still not been fully appreciated.mUch of the credit belongs to Paul Dirac, my pree-decessor but one in the
Lucasian chair, though it wasn't motorized in his time. Dirac showed how the work of Erwin Schroedinger and Werner Heisenberg, could be combined in new picture of reality, called quantum theory. In
quantum theory, a particle is not characterized by two quantities, its position and its velocity, as in classical Newtonian theory.iNstead it is described by a single quantity, the wave function. tHe
size of the wave function at a point, gives the probability that the particle will be found at that point, and the rate at which the wave function changes from point to point, gives the probability
of different velocities. One can have a wave function that is sharply peaked at a point.tHis corresponds to a state in which there is little uncertainty in the position of the particle.hOwever, the
wave function varies rapidly, so there is a lot of uncertainty in the velocity. Similarly, a long chain of waves has a large uncertainty in position, but a small uncertainty in velocity.oNe can have
a well defined position, or a well defined velocity, but not both.
This would seem to make complete determinism impossible.iF one can't accurately define both the positions, and the velocities, of particles at one time, how can one predict what they will be in the
future. It is like weather forecasting.tHe forecasters don't have an accurate knowledge of the atmosphere at one time.jUst a few measurements at ground level, and what can be learnt from satellite
photographs.tHats why weather forecasts are so unreliable. However, in quantum theory, it turns out one doesn't need to know both the positions, and the velocities. If one knew the laws of physics,
and the wave function at one time, then something called the Schroedinger equation, would tell one how fast the wave function was changing with time. This would allow one to calculate the wave
function at any other time. One can therefore claim that there is still determinism, but it is a determinism on a reduced level.iNstead of being able accurately to predict two quantities, position
and velocity, one can predict only a single quantity, the wave function. We have re-defined determinism, to be just half of what Le-plass thought it was. sOme people have tried to connect the
unpredictability of the other half, with con'shusness, or the intervention of supernatural beings.bUt it is difficult to make either case, for something that is completely random.
In order to calculate how the wave function develops in time, one needs the quantum laws that govern the universe.sO how well do we know these laws. As Dirac remarked, Maxwell's equations of light,
and the relativistic wave equation, which he was too modest to call the Dirac equation, govern most of physics, and all of chemistry and biology. So in principle, we ought to be able to predict human
behavior, though I can't say I have had much success myself. The trouble is that the human brain contains far too many particles, for us to be able to solve the equations.bUt it is comforting to
think we might be able to predict the nematode worm, even if we can't quite figure out humans. Quantum theory, and the Maxwell and Dirac equations, indeed govern much of our life, but there are two
important areas beyond their scope.oNe is the nuclear forces.tHe other is gravity. The nuclear forces are responsible for the Sun shining, and the formation of the elements, including the carbon and
oxygen of which we are made. And gravity caused the formation of stars and planets, and indeed, of the universe itself.sO it is important to bring them into the scheme.
The so called weak nuclear forces, have been unified with the Maxwell equations, by Abdus Salahm and Stephen Weinberg, in what is known as, the Electro weak theory. The predictions of this theory
have been confirmed by experiment, and the authors rewarded with Nobel prizes. The remaining nuclear forces, the so called strong forces, have not yet been successfully unified with the electro weak
forces, in an observationally tested scheme. Instead, they seem to be described by a similar but separate theory, called QCD. It is not clear who, if anyone, should get a Nobel prize for QCD, but
David Gross and Gerard teh Hooft, share credit for showing the theory gets simpler at high energies. I had quite a job to get my speech synthesizer to pronounce Gerrard's surname.iT wasn't familiar
with ap-osstrophee t. The electro weak theory, and QCD, together constitute the so called Standard Model of particle physics, which aims to describe everything except gravity.
The standard model seems to be adequate for all practical purposes, at least for the next hundred years.bUt practical or economic reasons, have never been the driving force in our search for a
complete theory of the universe. No one working on the basic theory, from Galeelaeo onward, has carried out their research to make money, though Dirac would have made a fortune if he had patented the
Dirac equation. He would have had a royalty on every television, walkman, video game and computer.
The real reason we are seeking a complete theory, is that we want to understand the universe, and feel we are not just the victims of dark and mysterious forces.iF we understand the universe, then we
control it, in a sense. The standard model is clearly unsatisfactory in this respect. fIrst of all, it is ugly and ad hoc.tHe particles are grouped in an apparently arbitrary way, and the standard
model depends on 24 numbers, whose values can not be deduced from first principles, but which have to be chosen to fit the observations. What understanding is there in that?cAn it be Nature's last
word. The second failing of the standard model, is that it does not include gravity.iNstead, gravity has to be described by Einstein's General Theory of Relativity. General relativity, is not a
quantum theory, unlike the laws that govern everything else in the universe. aLthough it is not consistent to use the non quantum general relativity, with the quantum standard model, this has no
practical significance at the present stage of the universe, because gravitational fields are so weak. However, in the very early universe, gravitational fields would have been much stronger, and
quantum gravity would have been significant. Indeed, we have evidence that quantum uncertainty in the early universe, made some regions slightly more or less dense, than the otherwise uniform
background. We can see this in small differences in the background of microwave radiation from different directions. The hotter, denser regions will condense out of the expansion as galaxies, stars
and planets. All the structures in the universe, including ourselves, can be traced back to quantum effects in the very early stages.iT is therefore essential to have a fully consistent quantum
theory of gravity, if we are to understand the universe.
Constructing a quantum theory of gravity, has been the outstanding problem in theoretical physics, for the last 30 years. It is much, much more difficult than the quantum theories of the strong and
electro weak forces.tHese propagate in a fixed background of space and time.oNe can define the wave function, and use the Schroedinger equation to evolve it in time. But according to general
relativity, gravity is space and time.sO how can the wave function for gravity, evolve in time.aNd anyway, what does one mean by the wave function for gravity. It turns out that, in a formal sense,
one can define a wave function, and a Schroedinger like equation for gravity, but that they are of little use in actual calculations.
Instead, the usual approach is to regard the quantum spacetime, as a small perturbation of some background spacetime, generally flat space. The perturbations can then be treated as quantum fields,
like the electro weak and QCD fields, propagating through the background spacetime. In calculations of perturbations, there is generally some quantity, called the effective coupling, which measures
how much of an extra perturbation, a given perturbation generates. If the coupling is small, a small perturbation, creates a smaller correction, which gives an even smaller second correction, and so
on.pErturbation theory works, and can be used to calculate to any degree of accuracy. An example is your bank account.tHe interest on the account, is a small perturbation.a very small perturbation if
you are with one of the big banks). The interest is compound.tHat is, there is interest on the interest, and interest on the interest on the interest.hOwever, the amounts are tiny.tO a good
approximation, the money in your account, is what you put there. On the other hand, if the coupling is high, a perturbation generates a larger perturbation, which then generates an even larger
perturbation. An example would be borrowing money from loan sharks.tHe interest can be more than you borrowed, and then you pay interest on that.iT is disastrous.
With gravity, the effective coupling is the energy or mass of the perturbation, because this determines how much it warps spacetime, and so creates a further perturbation. However, in quantum theory,
quantities like the electric field, or the geometry of spacetime, don't have definite values, but have what are called, quantum fluctuations. These fluctuations have energy.iN fact, they have an
infinite amount of energy, because there are fluctuations on all length scales, no matter how small. Thus treating quantum gravity as a perturbation of flat space, doesn't work well, because the
perturbations are strongly coupled.
Supergravity was invented in 1976 to solve, or at least improve, the energy problem.It is a combination of general relativity with other fields, such that that each species of particle, has a super
partner species. tHe energy of the quantum fluctuations of one partner is positive, and the other negative, so they tend to cancel.It was hoped the infinite positive and negative energies would
cancel completely, leaving only a finite remainder. In this case, a perturbation treatment would work, because the effective coupling would be weak. However in 1985, people suddenly lost confidence
that the infinities would cancel. tHis was not because anyone had shown that they definitely didn't cancel.iT was reckonned it would take a good graduate student, 300 years to do the calculation, and
how would one know they hadn't made a mistake on page two. Rather it was because Ed Witten declared that string theory, was the true quantum theory of gravity, and supergravity was just an
approximation, valid when particle energies are low, which in practice, they always are. In string theory, gravity is not thought of as the warping of spacetime.iNstead, it is given by string
diagrams, networks of pipes that represent little loops of string, propagating through flat spacetime. The effective coupling, that gives the strength of the junctions where three pipes meet, is not
the energy, as it is in supergravity. Instead it is given by what is called, the dilaton, a field that has ~not been observed.if the dilaton had a low value, the effective coupling would be weak, and
string theory, would be a good quantum theory.bUt it is no earthly use for practical purposes.
In the years since 1985, we have realized that both supergravity and string theory, belong to a larger structure, known as M theory.why it should be called M Theory, is completely obscure. M theory,
is not a theory in the usual sense.Rather it is a collection of theories, that look very different, but which describe the same physical situation. These theories are related by mappings, or
correspondences, called dualities, which imply that they are all reflections of the same underlying theory. Each theory in the collection, works well in the limit, like low energy, or low dilaton, in
which its effective coupling is small, but breaks down when the coupling is large. This means that none of the theories, can predict the future of the universe, to arbitrary accuracy.for that, one
would need a single formulation of M-theory, that would work in all situations.
Up to now, most people have implicitly assumed that there is an ultimate theory, that we will eventually discover.Indeed, I myself have suggested we might find it quite soon. However, M-theory has
made me wonder if this is true.Maybe it is not possible to formulate the theory of the universe in a finite number of statements. This is very reminiscent of Goedel's theorem.This says that any
finite system of axyoms, is not sufficient to prove every result in mathematics.
Goedel's theorem is proved using statements that refer to themselves.sUch statements can lead to paradoxes.aN example is, this statement is false. If the statement is true, it is false.and if the
statement is false, it is true. Another example is, the barber of Corfoo shaves every man who does not shave himself. Who shaves the barber?if he shaves himself, then he doesn't, and if he doesn't,
then he does. Goedel went to great lengths to avoid such paradoxes, by carefully distinguishing between mathematics, like 2+2 =4,and meta mathematics, or statements about mathematics, such as
mathematics is cool, or mathematics is consistent. that is why his paper is so difficult to read.but the idea is quite simple. First Goedel showed that each mathematical formula, like 2+2=4, can be
given a unique number, the Goedel number.the Goedel number of 2+2=4, is *. Second, the meta mathematical statement, the sequence of formulas A, is a proof of the formula B, can be expressed as an
arithmetical relation between the Goedel numbers for A- and B. Thus meta mathematics can be mapped into arithmetic, though I'm not sure how you translate the meta mathematical statement, 'mathematics
is cool'. Third and last, consider the self referring Goedel statement, G.tHis is, the statement G can not be demonstrated from the axyoms of mathematics. Suppose that G could be demonstrated.tHen
the axyoms must be inconsistent, because one could both demonstrate G, and show that it can not be demonstrated. On the other hand, if G can't be demonstrated, then G is true.bY the mapping into
numbers, it corresponds to a true relation between numbers, but one which can not be deduced from the axyoms. Thus mathematics is either inconsistent, or imcomplete.tHe smart money, is on imcomplete.
What is the relation between Goedels theorem, and whether we can formulate the theory of the universe, in terms of a finite number of principles. One connection is obvious.aCcording to the positivist
philosophy of science, a physical theory, is a mathematical model.sO if there are mathematical results that can not be proved, there are physical problems that can not be predicted. One example might
be the Golbach conjecture.gIven an even number of wood blocks, can you always divide them into two piles, each of which can not be arranged in a rectangle.tHat is, it contains a prime number of
Although this is incompleteness of sort, it is not the kind of unpredictability I mean.gIven a specific number of blocks, one can determine with a finite number of trials, whether they can be divided
into two primes. But I think that quantum theory and gravity together, introduces a new element into the discussion, that wasn't present with classical Newtonian theory. In the standard positivist
approach to the philosophy of science, physical theories live rent free in a Platonic heaven of ideal mathematical models. That is, a model can be arbitrarily detailed, and can contain an arbitrary
amount of information, without affecting the universes they describe. But we are not angels, who view the universe from the outside.iNstead, we and our models, are both part of the universe we are
describing. Thus a physical theory, is self referencing, like in Goedels theorem.oNe might therefore expect it to be either inconsistent, or imcomplete.tHe theories we have so far, are ~both
inconsistent, and imcomplete.
Quantum gravity is essential to the argument..THe information in the model, can be represented by an arrangement of particles.aCcording to quantum theory, a particle in a region of a given size, has
a certain minimum amount of energy. Thus, as I said earlier, models don't live rent free.tHey cost energy.By Einsteins famous equation, E = mc squared, energy is equivalent to mass.aNd mass causes
systems to collapse under gravity. It is like getting too many books together in a library.tHe floor would give way, and create a black hole that would swallow the information. Remarkably enough,
Jacob Bekenstein and I, found that the amount of information in a black hole, is proportional to the area of the boundary of the hole, rather than the volume of the hole, as one might have expected.
The black hole limit on the concentration of information, is fundamental, but it has not been properly incorporated into any of the formulations of M theory that we have so far. They all assume that
one can define the wave function at each point of space.bUt that would be an infinite density of information, which is not allowed. On the other hand, if one can't define the wave function point
wise, one can't predict the future to arbitrary accuracy, even in the reduced determinism of quantum theory. What we need, is a formulation of M theory, that takes account of the black hole
information limit.bUt then our experience with supergravity and string theory, and the analogy of Goedels theorem, suggest that even this formulation, will be imcomplete.
Some people will be very disappointed if there is not an ultimate theory, that can be formulated as a finite number of principles.I used to belong to that camp, but I have changed my mind. I'm now
glad that our search for understanding will never come to an end, and that we will always have the challenge of new discovery.wIthout it, we would stagnate. Goedels theorem ensured there would always
be a job for mathematicians.I think M theory will do the same for physicists. I'm sure Dirac would have approved.
Thank you for listening.
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search results
Expand all Collapse all Results 1 - 4 of 4
1. CJM 2010 (vol 62 pp. 961)
Multiplicative Isometries and Isometric Zero-Divisors
For some Banach spaces of analytic functions in the unit disk (weighted Bergman spaces, Bloch space, Dirichlet-type spaces), the isometric pointwise multipliers are found to be unimodular
constants. As a consequence, it is shown that none of those spaces have isometric zero-divisors. Isometric coefficient multipliers are also investigated.
Keywords:Banach spaces of analytic functions, Hardy spaces, Bergman spaces, Bloch space, Dirichlet space, Dirichlet-type spaces, pointwise multipliers, coefficient multipliers, isometries,
isometric zero-divisors
Categories:30H05, 46E15
2. CJM 2009 (vol 61 pp. 124)
Characterizing Complete Erd\H os Space
The space now known as {\em complete Erd\H os space\/} $\cerdos$ was introduced by Paul Erd\H os in 1940 as the closed subspace of the Hilbert space $\ell^2$ consisting of all vectors such that
every coordinate is in the convergent sequence $\{0\}\cup\{1/n:n\in\N\}$. In a solution to a problem posed by Lex G. Oversteegen we present simple and useful topological characterizations of $\
cerdos$. As an application we determine the class of factors of $\cerdos$. In another application we determine precisely which of the spaces that can be constructed in the Banach spaces $\ell^p$
according to the `Erd\H os method' are homeomorphic to $\cerdos$. A novel application states that if $I$ is a Polishable $F_\sigma$-ideal on $\omega$, then $I$ with the Polish topology is
homeomorphic to either $\Z$, the Cantor set $2^\omega$, $\Z\times2^\omega$, or $\cerdos$. This last result answers a question that was asked by Stevo Todor{\v{c}}evi{\'c}.
Keywords:Complete Erd\H os space, Lelek fan, almost zero-dimensional, nowhere zero-dimensional, Polishable ideals, submeasures on $\omega$, $\R$-trees, line-free groups in Banach spaces
Categories:28C10, 46B20, 54F65
3. CJM 2004 (vol 56 pp. 225)
Complex Uniform Convexity and Riesz Measure
The norm on a Banach space gives rise to a subharmonic function on the complex plane for which the distributional Laplacian gives a Riesz measure. This measure is calculated explicitly here for
Lebesgue $L^p$ spaces and the von~Neumann-Schatten trace ideals. Banach spaces that are $q$-uniformly $\PL$-convex in the sense of Davis, Garling and Tomczak-Jaegermann are characterized in terms
of the mass distribution of this measure. This gives a new proof that the trace ideals $c^p$ are $2$-uniformly $\PL$-convex for $1\leq p\leq 2$.
Keywords:subharmonic functions, Banach spaces, Schatten trace ideals
Categories:46B20, 46L52
4. CJM 1999 (vol 51 pp. 26)
Separable Reduction and Supporting Properties of Fréchet-Like Normals in Banach Spaces
We develop a method of separable reduction for Fr\'{e}chet-like normals and $\epsilon$-normals to arbitrary sets in general Banach spaces. This method allows us to reduce certain problems involving
such normals in nonseparable spaces to the separable case. It is particularly helpful in Asplund spaces where every separable subspace admits a Fr\'{e}chet smooth renorm. As an applicaton of the
separable reduction method in Asplund spaces, we provide a new direct proof of a nonconvex extension of the celebrated Bishop-Phelps density theorem. Moreover, in this way we establish new
characterizations of Asplund spaces in terms of $\epsilon$-normals.
Keywords:nonsmooth analysis, Banach spaces, separable reduction, Fréchet-like normals and subdifferentials, supporting properties, Asplund spaces
Categories:49J52, 58C20, 46B20
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When does it work?
I have been sharing my experience teaching a group of fifth-graders how to problem-solve around fraction computation and using it as an opportunity to demonstrate the Teaching-Learning Cycle in
action. Previously, I wrote about how I planned for a problem-solving lesson and then described my instruction during the following lesson. In this post, I want to discuss how I used assessment and
evaluation to monitor the learners' progress and inform future planning and instruction.
We were using the clock model as a context for adding fractions and I wanted to gather data about whether or not the learners could determine when this model was an effective approach. I used an
existing set of textbook items and asked the fifth-graders to: "Look at the expressions shown below - circle the ones that you think you could use the clock model to solve and place an 'X' through
those you could not."
from Scott Foresman – Addison Wesley Math [5^th Grade]
Once the kids had completed this task, I asked them to solve one of the problems they had circled. As they worked, I gathered data on whether or not they were able to determine when the clock model
could work.
In analyzing my observations, I noticed what the fifth-graders could do and what they were trying to do. First, they all recognized that fractions involving ninths and sevenths were poor candidates
for the clock model. Those who chose to solve #2, #9, and #10 were also fluent in applying prior experiences with the time context. Some learners thought eighths could work (circling #4) and others
struggled to see that fifths could work ('X'ing out #3, #5, and #8). These last two areas of approximation gave me some ideas about what to focus on next.
The last assessment I gave was intended to gather data about how the fifth-graders might apply the idea of context to a problem that could not be easily solved using the clock model. As a ticket out
the door, I asked, "Now what could you do to solve a problem you put an 'X' through?" Based on my evaluation of these assessments, I was prepared to plan for future lessons.
What would you do next?
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