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Bolingbrook Math Tutor
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Topic: Whole, Natural, & Counting Numbers?
Replies: 5 Last Post: May 5, 2008 11:27 AM
Messages: [ Previous | Next ]
Re: Whole, Natural, & Counting Numbers?
Posted: Oct 21, 2001 7:37 PM
> Zero is not a Natural Number because it is not a counting number, and infact
> was developed at a later time. (Or so they say.) You don't count to 3
> saying 0, 1, 2, 3.
Zero is a natural number according to some people such
as Eric Hehner from the University of Toronto. The following
is what he said in one of his books:
"Your life begins at year 0, a highway begins at mile 0,
and so on."
If we treat the number zero differently, things can get
unnecessarily complicated. For example, if people
used the number zero like the way they use other numbers,
we wouldn't have something like "unchanged" or "<= =>"
in the stock market report.
As I understand, the definition as to whether or not
zero is a natural number is not universal. Personally,
I consider zero as a natural number.
submissions: post to k12.ed.math or e-mail to k12math@sd28.bc.ca
private e-mail to the k12.ed.math moderator: kem-moderator@thinkspot.net
newsgroup website: http://www.thinkspot.net/k12math/
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Date Subject Author
9/30/01 Whole, Natural, & Counting Numbers? Cassie Weed
10/1/01 Re: Whole, Natural, & Counting Numbers? C. McGinnis
10/1/01 Re: Whole, Natural, & Counting Numbers? Alexander Bogomolny
5/5/08 Re: Whole, Natural, & Counting Numbers? Guest
11/4/01 Re: Whole, Natural, & Counting Numbers? C. McGinnis
10/10/01 Re: Natural/Counting Numbers vs. Whole Numbers Monique
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Course Content and Outcome
Course Content and Outcome Guide for CS 251
Course Number:
CS 251
Course Title:
Discrete Structures II
Credit Hours:
Lecture Hours:
Lecture/Lab Hours:
Lab Hours:
Special Fee:
Course Description
Introduces discrete structures and computational techniques in the areas of functions, relations, probability, graph theory, algorithm analysis, and finite state automata. Prerequisite: CS 250. Audit
Addendum to Course Description
Intended Outcomes for the course
Upon the successful completion of this course students will be able to:
? Formulate, interpret, and apply properties of propositional and first-order predicate calculus in real world
? Use analytical problem solving strategies to solve problems using multiple approaches and to interpret the
results in practical terms.
? Utilize those techniques in discrete mathematics and logic that are used in the study and practice of computer
? Be successful in subsequent coursework in the mathematical foundation of Computer Science.
Course Activities and Design
Outcome Assessment Strategies
Assessment must include:
1. At least two in-class proctored examinations, one of which may be the final exam, and
2. At least two of the following additional measures, where at least one includes writing:
a) Take-home examinations. (Group and/or individual)
b) Projects. (Group and/or individual)
c) Quizzes. (Group and/or individual)
d) Graded homework/worksheets.
e) In-class activities.
f) Attendance.
Course Content (Themes, Concepts, Issues and Skills)
Major Topics:
· Apply the properties of propositional calculus to: determine whether a wff is a tautology, a contradiction, or a contingency by truth tables and by Quine's method; construct equivalence
proofs; and transform truth functions and wffs into conjunctive or disjunctive normal form.
· Describe the basic inference rules and use them to write formal proofs in propositional calculus.
· Apply the properties of first-order predicate calculus to: determine whether a wff is valid, invalid, satisfiable, or unsatisfiable; construct equivalence proofs; and transform first-order
wffs into prenex conjunctive or disjunctive normal form.
· Describe the rules of inference for quantifiers and use them along with the basic inference rules to write formal proofs in first-order predicate calculus.
· Write formal proofs in first-order predicate calculus with equality.
· Construct partial correctness proofs of simple imperative programs and construct termination proofs for simple loops.
· Transform first-order wffs into clausal form; and unify atoms from a set of clauses.
· Describe the resolution inference rule; use it to write formal proofs in first-order logic; and describe how resolution is used to execute a logic program.
· Transform simple English sentences into formal logic (propositional, first-order, or higher-order).
· Apply appropriate algebraic properties to: simplify Boolean expressions; simplify regular expressions; write recursive definitions for simple functions in terms of operations for abstract
data types; write expressions to represent relations constructed in terms of operations for relational databases; and work with congruences.
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Book Record
Accession # 11681 Title: The Cost of Her Pride Author: Alexander, _ Mrs Original Source: View Page
+Transcribed Ledger Data
+Book Record
Author Alexander, Mrs., 1825-1902. Pen Name: Contributors: Title: The cost of her pride. Uniform Title: Publisher: J.B. Lippincott Co. Where Published: Philadelphia, PA Subject(s): Additional Info:
+Transaction History (101)
Transaction Date Original Acc. Title Borrower Name Pat Listed Patron Transaction Comments
Document # # Name
Saturday, November View 11681 The cost of her Beulah Smith 5477 Beulah Smith
22nd, 1902 Page pride.
Friday, September 5th, View 11681 The cost of her Mrs. M. S. 79 Lizzie Burson
1902 Page pride. Claypool
Tuesday, August 26th, View 11681 The cost of her Bessie Meyers 6005 Bessie Meyers
1902 Page pride.
Monday, August 18th, View 11681 The cost of her Agnes Bodicut 6121 Agnes Bodicut
1902 Page pride.
Monday, July 21st, 1902 View 11681 The cost of her Thos Richards 2653 T. Y. Richards
Page pride.
Wednesday, July 16th, View 11681 The cost of her Elizabeth Helms 6042 Elizabeth Helm
1902 Page pride.
Monday, July 14th, 1902 View 11681 The cost of her Ida Spradling 5371 Ida Spaulding
Page pride.
Saturday, July 5th, View 11681 The cost of her H.L. Kinert 3876 H. L. Kinert
1902 Page pride.
Tuesday, June 24th, View 11681 The cost of her Mrs. Kate 4940 Mrs. Kate A
1902 Page pride. Souder Souder
Monday, June 16th, 1902 View 11681 The cost of her Louis Hargrave 5971 Louis Hargraves
Page pride.
Saturday, June 7th, View 11681 The cost of her Linnie Coffeen 4025 Linnie Coffeen
1902 Page pride.
Thursday, May 22nd, View 11681 The cost of her Lulu Over 5967 Lulu Over
1902 Page pride.
Saturday, May 17th, View 11681 The cost of her Margaret 5920 Margaret
1902 Page pride. McBreen McBreen
Saturday, May 10th, View 11681 The cost of her Katharine Garst 4827 Katharine S.
1902 Page pride. Garst
Monday, May 5th, 1902 View 11681 The cost of her Mrs. Wm Gill 3096 Mrs. Wm. F.
Page pride. Gill
Monday, April 28th, View 11681 The cost of her Carrie Paden 3700 Carrie Paden
1902 Page pride.
Wednesday, April 2nd, View 11681 The cost of her Nora Spurgeon 5810 Nora Spurgeon
1902 Page pride.
Monday, March 24th, View 11681 The cost of her Mrs. Millie 6080 Mrs. Millie
1902 Page pride. Hendricks Kendricks
Saturday, March 15th, View 11681 The cost of her Louise Phinney 3240 Louise Phinney
1902 Page pride.
Saturday, March 1st, View 11681 The cost of her H.A. Morrow 5386 H. A. Morrow
1902 Page pride.
Saturday, February View 11681 The cost of her Mrs. John 3118 Mrs. John
22nd, 1902 Page pride. Shanahan Shanahan
Tuesday, February 18th, View 11681 The cost of her Hazel Sherritt 3971 Hazel Sheritt
1902 Page pride.
Tuesday, February 11th, View 11681 The cost of her Mrs. S.P. 2080 Mrs. S. P.
1902 Page pride. Wildman Wildman
Saturday, February 8th, View 11681 The cost of her Eva Braden 5991 Eva Braden
1902 Page pride.
Wednesday, February View 11681 The cost of her Harry Morgan 3326 Harry Leach
5th, 1902 Page pride.
Tuesday, January 21st, View 11681 The cost of her J.H. Fenton 4697 J. H. Fenton
1902 Page pride.
Thursday, January 16th, View 11681 The cost of her Mrs. Kate 2814 Bobbie Knowlton
1902 Page pride. Knowlton
Friday, January 3rd, View 11681 The cost of her Carolyn Cohn 3742 Carrie Cohn
1902 Page pride.
Saturday, December 7th, View 11681 The cost of her Mrs. Herman 3820 Mrs. Herman
1901 Page pride. Cohen Cohen
Saturday, November View 11681 The cost of her Elliot Brady 2860 Ethel Brady
30th, 1901 Page pride.
Friday, November 22nd, View 11681 The cost of her Hattie Karn 3842 Hattie Karn
1901 Page pride.
Friday, November 15th, View 11681 The cost of her Maggie B. 5442 Maggie Carvel
1901 Page pride. Carvel
Saturday, November 9th, View 11681 The cost of her Walter Howe 3014 Walter J. Howe
1901 Page pride.
Monday, November 4th, View 11681 The cost of her Lena Pierce 5417 Lena Pierce
1901 Page pride.
Thursday, October 24th, View 11681 The cost of her Mrs. Fannie 5918 Mrs. Fannie
1901 Page pride. Wilbur Wilber
Monday, October 14th, View 11681 The cost of her Clare E. 5898 Miss. Clare E
1901 Page pride. Johnson Johnson
Monday, October 7th, View 11681 The cost of her Mrs. Frances 5517 Mrs. Frances
1901 Page pride. Paden Paden
Tuesday, September View 11681 The cost of her Mrs. Maude 3925 Mrs. Maud
24th, 1901 Page pride. Albright Albright
Monday, September 9th, View 11681 The cost of her Maude Harris 3276 Maude Harris
1901 Page pride.
Monday, August 26th, View 11681 The cost of her Margaret Mann 5063 Margaret Mann
1901 Page pride.
Friday, August 16th, View 11681 The cost of her Elijah Ward 5100 Elijah Ward
1901 Page pride.
Friday, July 19th, 1901 View 11681 The cost of her Mrs. Millie 4801 Mrs. Milt Gray
Page pride. Gray
Monday, July 15th, 1901 View 11681 The cost of her E. A. Edwards 4565 E. A. Edwards
Page pride.
Tuesday, July 2nd, 1901 View 11681 The cost of her Bessie Brownell 5719 Anna Stuckey
Page pride.
Tuesday, June 18th, View 11681 The cost of her Arlie Tieche 3360 Arlie Tieche
1901 Page pride.
Friday, June 14th, 1901 View 11681 The cost of her Mrs Herman 3820 Mrs. Herman
Page pride. Cohen Cohen
Monday, June 10th, 1901 View 11681 The cost of her Mable Meeker 3265 Mabel Mecker
Page pride.
Monday, June 3rd, 1901 View 11681 The cost of her George Postma 5183 George Postma
Page pride.
Tuesday, May 28th, 1901 View 11681 The cost of her Eva Davis 5431 Eva Davis
Page pride.
Monday, May 13th, 1901 View 11681 The cost of her Mrs Kate 2814 Bobbie Knowlton
Page pride. Knowlton
Saturday, April 27th, View 11681 The cost of her Mrs T J Price 5631 Martha Robbins
1901 Page pride.
Saturday, April 20th, View 11681 The cost of her Chester Wardlow 5315 Chester Wardlow
1901 Page pride.
Friday, April 5th, 1901 View 11681 The cost of her Mrs Blanche 4502 Mrs. Blanche fixed patron number (orig. 4501)
Page pride. Turner Turner
Friday, March 29th, View 11681 The cost of her Margaret Scott 3838 Margaret Leoth
1901 Page pride.
Friday, March 22nd, View 11681 The cost of her E. L. Chalfant 3218 E. L. Chalfant
1901 Page pride.
Friday, March 15th, View 11681 The cost of her Mrs R. W. 5629 Mrs. R. W.
1901 Page pride. Thompson Thompson
Saturday, March 9th, View 11681 The cost of her Thos Larmore 4863 Lucy Larmore
1901 Page pride.
Saturday, March 2nd, View 11681 The cost of her O B Barmister 5094 O. B. Bannister
1901 Page pride.
Saturday, February View 11681 The cost of her George Burt 2957 George Burt
23rd, 1901 Page pride.
Thursday, February View 11681 The cost of her Mrs W C Denney 5081 Mrs. M. B.
21st, 1901 Page pride. Street
Tuesday, February 19th, View 11681 The cost of her Mrs Cora E. 5619 Mrs. Cora E.
1901 Page pride. Nold Nobel
Tuesday, February 5th, View 11681 The cost of her Mrs Sarah 5574 Mrs. Sarah
1901 Page pride. Sommers Summers
Monday, February 4th, View 11681 The cost of her Mrs Kate 2814 Bobbie Knowlton
1901 Page pride. Knowlton
Saturday, February 2nd, View 11681 The cost of her Mrs Ellen 383 Ellen E. Smith
1901 Page pride. Meeker
Friday, January 18th, View 11681 The cost of her R W Monroe 1217 R. W. Monroe
1901 Page pride.
Tuesday, January 15th, View 11681 The cost of her I. B. Saxon 3325 I. B. Saxon
1901 Page pride.
Monday, January 7th, View 11681 The cost of her Mrs Minnie 4308 Minnie Thomas
1901 Page pride. Thomas
Saturday, December View 11681 The cost of her Mrs Eva Cole 4850 Mrs. Eva I.
29th, 1900 Page pride. Cole
Tuesday, December 18th, View 11681 The cost of her John Clancy 4997 John Clancy
1900 Page pride.
Saturday, December 8th, View 11681 The cost of her Elijah Ward 5100 Elijah Ward
1900 Page pride.
Saturday, November View 11681 The cost of her Frank Burt 4055 Frank E. Burt
17th, 1900 Page pride.
Tuesday, November 13th, View 11681 The cost of her Rob Knowlton 715 Addie Knowlton
1900 Page pride.
Monday, October 29th, View 11681 The cost of her Jeannette 5153 Jeannette
1900 Page pride. Englebach Engelbach
Friday, October 19th, View 11681 The cost of her Sue M. Neely 69 Sue H. Maddy
1900 Page pride.
Wednesday, October View 11681 The cost of her Mrs. Kate 2814 Bobbie Knowlton
17th, 1900 Page pride. Knowlton
Tuesday, October 9th, View 11681 The cost of her Katherine Garst 4827 Katharine S.
1900 Page pride. Garst
Wednesday, October 3rd, View 11681 The cost of her Annie McDonald 5068 Annie McDonald
1900 Page pride.
Wednesday, September View 11681 The cost of her D. W. Leonard 3950 D Leonard
26th, 1900 Page pride.
Friday, September 21st, View 11681 The cost of her Allie Emerson 367 Allie Emerson
1900 Page pride.
Saturday, September View 11681 The cost of her Mary Brady 1621 M M Brady
15th, 1900 Page pride.
Wednesday, September View 11681 The cost of her Mrs. Fred Klein 5398 Mrs. Fred Klein
12th, 1900 Page pride.
Saturday, September View 11681 The cost of her Mrs. O. A. 5516 Mrs. O. A.
8th, 1900 Page pride. Johnson Johnson
Wednesday, September View 11681 The cost of her Mrs. A. M. 3870 Mrs. A M Klein
5th, 1900 Page pride. Klein
Saturday, August 25th, View 11681 The cost of her Vida Cassaday 2344 Vida Cassady
1900 Page pride.
Tuesday, August 21st, View 11681 The cost of her Mrs. D. W. 2649 Helen Hickman "52" is written to the right of borrower name
1900 Page pride. Stewart
Friday, August 17th, View 11681 The cost of her Thos. H. Ruby 4269 Mr. T. H. Kirby fixed patron number (orig. 4262)
1900 Page pride.
Thursday, August 16th, View 11681 The cost of her Mrs. Fannie 3743 Mrs. Fannie "44" written to the right of Book ID number
1900 Page pride. Cohn Cohn
Thursday, August 2nd, View 11681 The cost of her Nora Hawk 4192 Nora Hawk
1900 Page pride.
Monday, July 30th, 1900 View 11681 The cost of her Howard Coffeen 2278 Howard Coffeen Original transaction lists patron # as 2878, but this was likely an error and has been changed to reflect
Page pride. Howard Coffeen's actual patron # 2278.
Thursday, July 19th, View 11681 The cost of her Mrs. Emma 2874 Anthony
1900 Page pride. Turicchi Turicchi
Thursday, July 5th, View 11681 The cost of her Claude McElwee 3750 Claude McElwee
1900 Page pride.
Saturday, June 30th, View 11681 The cost of her Carolyn Cohen 3742 Carrie Cohn
1900 Page pride.
Friday, June 22nd, 1900 View 11681 The cost of her Mrs. Lola 2801 Lola Harrington
Page pride. Needham
Thursday, June 7th, View 11681 The cost of her Mrs. Lois E. 5047 Lois Evans
1900 Page pride. Evans
Wednesday, May 23rd, View 11681 The cost of her Mrs. Chas 5252 Mrs. Chas G.
1900 Page pride. Foresman Foresman
Friday, May 18th, 1900 View 11681 The cost of her Samuel Sutton 3804 S. R. Sutton
Page pride.
Thursday, May 10th, View 11681 The cost of her Mrs. L.L. 4411 Mrs. L. L.
1900 Page pride. Turner Turner
Saturday, May 5th, 1900 View 11681 The cost of her Mrs. Geo 4797 Gertie
Page pride. Nicholson Nicholson
Thursday, May 3rd, 1900 View 11681 The cost of her Josie Jones 4429 Vida Stacy
Page pride.
Wednesday, April 18th, View 11681 The cost of her Ada Carmack 3794 Mrs. W. F.
1900 Page pride. Warner
Tuesday, April 10th, View 11681 The cost of her Karl G. Sample 3130 Karl G. Sample
1900 Page pride.
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Space-time is not the same for everyone
Astronomy, Astrophysics, Breakingnews, Universe 11:00 AM
Before the Big Bang, space-time as we know it did not exist. So how was it born? The process of creating normal space-time from an earlier state dominated by quantum gravity has been studied for
years by theorists at the Faculty of Physics, University of Warsaw. Recent analyses suggest a surprising conclusion: not all elementary particles are subject to the same space-time.
In the mathematical model constructed by theorists at the Department of Physics, University of Warsaw, classical space-time is created by the interaction of matter with quantum gravity. The process
resembles how an ice crystal lattice (symbolizing classical space-time) is formed by freezing liquid water (quantum gravity). Recent studies on the model suggest that different elementary particles
generate different classical space-times [Credit: Copyright Faculty of Physics, University of Warsaw]
Several billion years ago, in the era soon after the Big Bang, the Universe was so dense and so hot that elementary particles felt the existence of gravity strongly. For decades, physicists around
the world have been attempting to discover the laws of quantum gravity describing this phase of the evolution of the Universe. Recently Professor Jerzy Lewandowski's group at the Faculty of Physics,
University of Warsaw (FUW) proposed its own model of the quantum Universe. Recent studies of its properties, discussed during the 20th International Conference on General Relativity and Gravitation
(GR20), being held in Warsaw in conjunction with the 10th Edoardo Amaldi Conference on Gravitational Waves (Amaldi10), have surprised researchers. The analyses performed by Prof. Lewandowski and his
PhD student Andrea Dapor show that different elementary particles "experience" the existence of different space-times. One of the attempts to describe quantum gravity is called loop quantum gravity
(LQG). This theory assumes that space-time is structurally somewhat similar to a fabric: It consists of a large number of very small fibres entangled in loops. A field with an area of one square
centimetre might hold a million trillion trillion trillion trillion trillion (10^66) such fibres. Three years ago, Prof. Lewandowski's group developed a consistent mathematical model of LQG that
combines quantum mechanics with general relativity. The model assumes the existence of two interacting fields. One is a gravitational field, which can be identified with a space (since, according to
the general theory of relativity, gravity warps space-time, and this curved space-time gives rise to gravitational effects). The second field in the model is a (scalar) field that assigns a number to
each point in space. This field is interpreted as the simplest type of matter. The image of reality in the model put forward by the Warsaw University physicists is quantum, and so has characteristics
extremely different from those of the world we deal with every day. "In this situation, it seemed natural to ask: How does the space-time known to all of us emerge from the primary states of quantum
gravity? And since normal space-time would be born as a result of the interaction between matter and quantum gravity, can we be certain that each type of matter definitely interacts with a space-time
that has the same properties?," says Prof. Lewandowski. To find answers to these questions, theorists first derived patterns of interaction between quantum gravity effects and matter for the two
mathematically simplest cases: for zero rest mass particles and for simple (scalar) non-zero rest mass particles. In the Standard Model, which in modern physics describes the elementary particles and
their interactions, the relevant massless particles would be photons, and scalar non-zero rest mass particles with mass -- the famous Higgs boson, responsible for the mass of the other particles:
quarks and electrons, muons, taus and their associated neutrinos. After deriving the equations representing the behaviour of particles in accordance with the laws of the quantum gravity model, FUW
physicists started to check whether similar equations could be obtained with the use of ordinary space-time with different symmetries. For massless particles this turned out to be possible. The
sought-for space-time was isotropic, i.e. it had the same properties in all directions. "According to the simplified model we researched, regardless of whether the photon has greater momentum or
less, more energy or less, space-time appears to it to be the same in all directions," explains Prof. Lewandowski. For particles with mass, the situation was different. The existence of mass imposes
a specific additional condition on the theory. The FUW physicists showed that a classical space-time, which would simultaneously meet the mass condition and have the same properties in all
directions, cannot be constructed. The appropriate space-time could be found only among anisotropic space-times. The preferred direction of these space-times was the particle's direction of motion.
"Particles with mass not only experience different space-times than photons do, but each sees its own private version of space-time depending on the direction it moves in. This finding really took us
by surprise," says PhD student Andrea Dapor. Does this latest discovery mean that the Universe of particles with mass is not isotropic? Such an assertion would be of huge experimental and
observational importance. However, the answer is no, the Universe does not have a preferred direction. As observers studying the behaviour of elementary particles, we are classical, rather than
quantum, systems and in a sense we are "outside" the particles' world. It is not then important what each particle "experiences" of its space-time. Regardless of the direction of flight, all
particles recorded in the laboratory will have exactly the same characteristics. For this reason, experimentally confirming the theoretical predictions of the FUW team will be no trivial task. The
work of Professor Lewandowski's team was funded by grants from the Polish Ministry of Science and Higher Education and the Polish National Science Centre. Source: Faculty of Physics, University of
Warsaw [July 09, 2013]
. You can follow any responses to this entry through the
RSS 2.0
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Oral puzzles
Re: Oral puzzles
Hi bobbym,
The solution #1552 is correct. Excellent!
#1553. Two right circular cylinders of equal volumes have their heights in the ratio 1:2. What is the ratio of their radii?
Character is who you are when no one is looking.
Re: Oral puzzles
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: Oral puzzles
Hi bobbym and Denominator,
The solution #1553 is perfect. Brilliant!
#1554. A well with 14 meters inside diameter is dug 10 meters deep. Earth taken out of it has been evenly spread all around it to a width of 21 meters to form an embankment. What is the height of the
embankment? (Use pi = 22/7)
Character is who you are when no one is looking.
Re: Oral puzzles
Hi ganesh;
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: Oral puzzles
Hi ganesh;
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: Oral puzzles
Hi bobbym,
The .
#1555. If (a + b) : (b + c) : (c + a) = 6:7:8 and (a + b + c) = 14, then find the value of 'c'.
Character is who you are when no one is looking.
Re: Oral puzzles
Hi ganesh;
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: Oral puzzles
Hi bobbym,
The solution #1555 is correct. Neat job!
#1556. $1210 is divided among A, B, and C so that A : B = 5 : 4 and B : C = 9 : 10. Then, how much does C gets?
Character is who you are when no one is looking.
Re: Oral puzzles
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: Oral puzzles
Hi bobbym and Denominator,
The solution #1556 is perfect. Good work!
#1557. The salaries of Roger and Steve are in the ratio 2 : 3. If the salary of each is increased by $4000, the new ratio becomes 40 : 57. What is Steve's present salary?
Character is who you are when no one is looking.
Re: Oral puzzles
Hi ganesh;
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: Oral puzzles
Hi bobbym and Denominator,
The solution #1557 is correct. Good work!
#1558. The third proportional to 0.36 and 0.48 is ____________
Character is who you are when no one is looking.
Re: Oral puzzles
Hi ganesh;
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: Oral puzzles
Hi bobbym,
#1559. In a ratio, which is equal to 3 : 4, if the antecedent is 12, then the consequent is ______________
Character is who you are when no one is looking.
Re: Oral puzzles
Hi ganesh;
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: Oral puzzles
Hi bobbym and Denominator,
The solution #1559 is correct. Neat work!
#1560. If 2A = 3B = 4C, then A : B : C is ______________
Character is who you are when no one is looking.
Re: Oral puzzles
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: Oral puzzles
1. 45
2. 7
3. 120
4. 7 and 10
Re: Oral puzzles
Hi bobbym and Denominator,
The solution #1560
Good work, Denominator!
#1561. The average age of three boys is 25 years and their ages are in the proportion 3 : 5 : 7. Find the age of the youngest boy.
Character is who you are when no one is looking.
Re: Oral puzzles
Hi ganesh;
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.
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Subject:University level mathematics books
From Wikibooks, open books for an open world
< Mathematics
University level mathematics books
This category contains books which are typically appropriate for a University setting, whether at an undergraduate level or beyond. For books that are intended for an audience that is before the
University level please see
K-12 mathematics
Completed books
In subsections:
Books nearing completion
In subsections:
Half-finished books
In subsections:
Partly developed books Featured Books
In subsections:
Freshly started books
In subsections: • Formal Logic
Unknown completion In subsections:
In subsections:
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Homework Help
Posted by 6th grader on Wednesday, July 29, 2009 at 4:24pm.
How do you find out the square root of a decimal?
ex. 2.31
• Math - Ms. Sue, Wednesday, July 29, 2009 at 4:29pm
You could use a calculator.
• Math - 6th grader, Wednesday, July 29, 2009 at 4:35pm
I need to do it without a calculator
• Math - MathMate, Wednesday, July 29, 2009 at 4:38pm
This used to be taught to 6th graders back in the 1910's. With the advent of calculators and computers, the method got lost from the system. Some home schooling programmes still do it.
1. By modern calculator
Assuming you have a calculator, press 2.31 followed by the √ key.
2. By simple calculator
If your calculator does not have a √ key, you can do it by trial and error. Try to square 1.5 (=2.25), too small. Try 1.6² (=2.56), too big. So the square root is somewhere in between Keep trying
and halving the gap, for example, try 1.55 in this case.
3. By simple calculator using Newton's method.
From a given estimate x, calculate a better estimate using
x1=x + (2.31-x²)/(2x)
Keep repeating the same procedure until the desired accuracy is achieved.
which is already accurate to 6 figures after the decimal point.
4. Manual method.
It is rather difficult to describe the method without having an easy way to align numbers in the response. So I will refer you to a web-site that describes the method:
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Infinite Descent (idea)
Infinite Descent is a proof technique of use in solving problems from number theory, ranging from elementary problems to vital theorems of modern algebraic number theory. Infinite Descent was first
described by name by the 17th century mathematician Fermat in the paper La méthode de la 'descente infinie ou indéfinie.
Proof by Infinite Descent is a special case of proof by contradiction (reductio ad absurdum), although from a mathematical logic point of view it can be seen as equivalent to the axiom of induction,
or the well-order property of the natural numbers. The technique is usually employed to demonstrate non-existence of collections of numbers with a particular property. To do so, it is shown that if
one set exists, then another exists which is in some sense 'smaller'. In this way, infinitely many successively smaller collections can be found, yet the problem is posed in such a way that there
cannot be infinitely many such collections- usually because of the existence of a bound. This is a contradiction, so the assumption of existence of a solution isn't tenable and we thus conclude no
solutions exist.
This is probably best understood by way of an example. So consider the following problem:
Show that there is no quadruple (x,y,z,u) of positive integers satisfying
x^2 + y^2 = 3(z^2 + u^2)
The important thing here is that we are looking for positive integers -the counting numbers 1,2,3,...- and thus an infinite descent cannot exist: a sequence of such values cannot keep strictly
decreasing forever. By contrast, infinite strictly decreasing sequences of rational numbers or real numbers are not inherently paradoxical- the sequence obtained by successive halving (1, 1/2, 1/4, 1
/8,...) being an example.
Descent Proof
We proceed therefore to give a proof by infinite descent. Suppose we have four positive integer values x,y,z,u satisfying the required condition. Then, since 3(z^2 +u^2) is divisible by three, and x^
2 + y^2 is equal to that expression, it must be that x^2+y^2 is divisible by 3. Formally, we say that x^2 + y^2 is congruent to 0 mod 3, which is a fancy way of saying we are left with remainder 0
after division by 3.
An aside on working mod 3. Here a number is essentially either 0, 1 or 2: 3 is 0 mod 3 (as are 6,9,12 etc.) , 4 is 1 mod 3 (as are 7,10,13,etc) and so on. Addition works as usual except you discard
multiples of 3, so, for example, 2+2 = 4 = 3+1 so 2+2 =1.
Knowing that x^2+y^2=0 mod 3, we conclude that one of three cases holds:
• Both x^2 and y^2 are 0 mod 3
• x^2 = 1 mod 3 and y^2 = 2 mod 3
• x^2 = 2 mod 3 and y^2 = 1 mod 3
However, you can easily verify (test all cases!) that the square of any number mod 3 is 0 or 1. That is, there are no numbers which square to 2 mod 3, so the second two cases above are impossible. We
know therefore that x^2 and y^2 are both 0 mod 3, meaning they are divisible by 3.
But if the square of an integer can be divided by 3, then the integer itself can be divided by 3. So we can write x=3x' and y=3y' for some x' and y'. Since x and y were strictly positive, x' and y'
are definitely smaller (x' being a third of x, y' a third of y). We can plug these new versions into the equation to get (3x')^2 + (3y')^2 = 3(z^2 + u^2).
That simplifies to 3.3(x'^2+y'^2) = 3(z^2+u^2), and cancelling a 3 from each side we get
So we can deduce from the presence of a 3 on the left hand side that z^2+u^2 =0 mod 3, and by the same reasoning as for x and y earlier can find ourselves z' and u' such that z=3z' and u=3u'.
Plugging into the equation again, we find
3(x'^2 +y'^2)= (3z')^2 + (3u')^2
So 3(x'^2 +y'^2)= 9(z'^2 + u'^2)
i.e x'^2+y'^2=3(z'^2 + u'^2)
Hence we're right back where we started, except all the numbers are a third of those we started with. There's nothing to stop us applying the same thought process to find a set x'',y'',z'',u''. In
fact, we can do this infinitely many times, with the terms strictly decreasing in size each time.
Thus admitting one solution gives rise to an infinite descent, so there can be no solutions.
Direct proof by contradiction
To confirm proof by descent is a contradiction argument in disguise (even though we needn't explicitly appeal to that contradiction to complete a proof; as in the example above the observation of an
infinite descent of integers is sufficient), here's a proof of the same result by slightly different means.
Suppose for contradiction there exist positive integer solutions to the problem. Let x,y,z,u be a solution such that V=x^2 + y^2 = 3(z^2 + u^2) is minimal. Then, by the earlier reasoning, we can find
x',y,z',u' satisfying the conditions. But they give rise to V'=x'^2 + y'^2 = 3(z'^2 + u'^2) = V/9. So V'<V, contradicting the minimality of V. So positive integer solutions do not exist.
Constructive proof
One last proof can be given, with a more constructivist feel- it demonstrates that any non-negative integer solution must be the solution 0,0,0,0. Thus, there cannot be a positive integer solution,
since x,y,z,u a solution implies x=y=z=u=0 and x,y,z,u arbitrary positive integers implies x,y,z,u non-zero; the method of proof by contraposition then ensures x,y,z,u aren't a solution and their
arbitrary nature proves there are no such solutions.
To reason in this way, let X,Y,Z,U be a non-negative integer solution (so zero is allowed). Then we can construct a new solution X',Y',Z',U' with X=3X'. Repeating, we get X'' st X=3X' = 3.3X'' and so
on; concluding that 3^n divides X for every n. But the only number which is infinitely divisible by 3 is zero, so X=0. Further, the same reasoning gives each of Y,Z and U =0. We are done.
Some notable results by Infinite Descent
Fermat's theorem on sums of two squares (the Christmas Theorem)
An odd prime p can be written as p=x^2 + y^2 iff p is congruent to 1 mod 4
Typically for Fermat, he didn't prove this. Euler gave the first proof, by means of infinite descent.
No solution to x^4+y^4=z^2
This was known to Fermat, and allows an elementary proof of his last theorem in the case n=4 (clearly, if the sum cannot be a square it cannot be a fourth power).
Irrationality of the square root of 2
That the square root of 2 cannot be rational was known to the Greeks. Euclid gave a proof by infinite descent, so the technique actually predates Fermat by nearly 2000 years! The result generalises
to the square root of any prime.
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Patente US7698312 - Performing recursive database operations
This application claims priority under 35 U.S.C. §120 as a continuation of U.S. patent application Ser. No. 10/867,923, filed on Jun. 14, 2004, now U.S. Pat. No. 7,155,446, which application lists as
inventors Thierry Cruanes, Wei Li, Ari Mozes and Benoit Dagville, which application is titled PERFORMING RECURSIVE DATABASE OPERATIONS, and which application claims domestic priority to provisional
U.S. Patent Application Ser. No. 60/571,441, entitled PERFORMING RECURSIVE DATABASE OPERATIONS, filed May 14, 2004; the contents of both of which applications are hereby incorporated by reference in
their entirety for all purposes. This application is also related to the following U.S. patent applications: Ser. No. 10/643,629, entitled FREQUENT ITEMSET COUNTING USING CLUSTERED PREFIXES AND INDEX
SUPPORT, filed on Aug. 18, 2003; Ser. No. 10/643,563, entitled DYNAMIC SELECTION OF FREQUENT ITEMSET COUNTING TECHNIQUE, filed on Aug. 18, 2003; and Ser. No. 10/643,628, entitled EXPRESSING FREQUENT
ITEMSET COUNTING OPERATIONS, filed on Aug. 18, 2003; the contents of each of which are hereby incorporated by reference in their entirety for all purposes.
The present invention relates to databases, and in particular, to performing recursive database operations.
Some computational tasks are especially suitable for recursive processing. One example of an operational task that lends itself especially well to recursive processing is “frequent itemset”
determination. An “itemset” is a set of items. For example, one itemset might include the items (apple, banana), while another itemset might include the items (apple, orange), while yet another
itemset might include the items (banana, orange). An itemset is “frequent”, relative to a set of data structures, if the number of the data structures that contain all of the items in the itemset is
at least a specified fraction of the total number of the data structures in the set.
For example, in a set of three data structures, each data structure might represent a different customer's transaction at a supermarket. A first data structure might contain the items (apple, banana,
milk), while a second data structure might contain the items (apple, banana, milk, orange), while a third data structure might contain the item (orange). Assuming that the specified fraction is ⅔,
the itemset (apple, banana) is a frequent itemset because “apple” occurs with “banana” in two of the three data structures, but the itemsets (apple, orange) and (banana, orange) are not frequent
itemsets because “apple” occurs with “orange” in only one of the three data structures and “banana” occurs with “orange” in only one of the three data structures. As the number of data structures in
a set of data structures increases, the determination of whether a particular itemset is frequent relative to that set of data structures becomes more computationally intensive.
Frequent itemset determination lends itself especially well to recursive processing due at least in part to the observation that an N-element itemset cannot be a frequent itemset relative to a set of
data structures unless all of the (N−1)-element subsets of the N-element itemset are also frequent itemsets relative to that set of data structures. For example, the 3-element itemset (apple, banana,
milk) cannot be a frequent itemset relative to the set of data structures in the above example unless all of the 2-element subsets of that 3-element itemset, namely, (apple, banana), (apple, milk),
and (banana, milk), are also frequent itemsets relative to the set of data structures in the above example.
This observation allows the computationally intensive determination of whether itemsets are frequent to be performed for fewer itemsets. The determination of whether a particular N-element itemset is
frequent needs to be performed only if all of the (N−1)-element subsets of the particular N-element itemset are also frequent. Thus, for each successive value of N, the group of N-element itemsets
for which this determination needs to be performed can be based on the determinations already performed for the (N−1)-element itemsets. Frequent itemset counting is, therefore, a task that can be
performed more efficiently using a recursive approach.
According to one theoretical approach, frequent itemsets might be determined in the following manner. An application that is external to a database server might send a query to the database server.
When executed, the query would cause the database server to select, from a set of data structures, each data structure that contains all of the items in a specified itemset. The database server would
execute the query and return the selected data structures to the application. The application might count the selected data structures and determine whether the number of selected data structures
meets a specified threshold. If the number of selected data structures met the specified threshold, then the application might place the specified itemset in a set of frequent itemsets. The
application might perform the above steps for each 1-element itemset that is a subset of an M-element itemset, one 1-element itemset at a time, and one 1-element itemset after another.
Once the application had performed the above steps for each such 1-element itemset, the application might determine, for each particular 2-element subset of the M-element itemset, whether all of the
1-element subsets of that particular 2-element subset are contained in the set of frequent itemsets. If all of the 1-element subsets of the particular 2-element subset were contained in the group of
frequent itemsets, then the application might send, to the database server, a query that would cause the database server to select, from the set of data structures, each data structure that contains
all of the items in the particular 2-element itemset. The database server would execute the query and return the selected data structures to the application. The application might count the selected
data structures and determine whether the number of selected data structures meets the specified threshold. If the number of selected data structures met the specified threshold, then the application
might place the particular 2-element itemset in the set of frequent itemsets. The application might perform the above steps for each 2-element itemset that is a subset of the M-element itemset, one
2-element itemset at a time, and one 2-element itemset after another.
For each successive value of N, the application might perform the above steps for the N-element itemsets that are subsets of the M-element itemset until N was greater than M or there were no (N−1)
-element itemsets in the set of frequent itemsets, whichever came first. Thus, by sending a multitude of queries to a database server in serial manner and counting the results of such queries, the
application might determine frequent itemsets that are subsets of the M-element itemset.
Unfortunately, considerable overheard would be involved in the above approach. It would take significant time for the application to send the many queries to the database server and for the database
server to send the results of the many queries back to the application.
Furthermore, because most of the operations performed in the above approach would be performed by the application (the database server would just execute queries and return the results), application
programmers would be burdened with implementing the functionality required to perform most of the operations involved in the above approach.
These are some of the problems that would attend the above approach. A technique that overcomes these problems is needed.
The approaches described in this section are approaches that could be pursued, but not necessarily approaches that have been previously conceived or pursued. Therefore, unless otherwise indicated, it
should not be assumed that any of the approaches described in this section qualify as prior art merely by virtue of their inclusion in this section.
The present invention is illustrated by way of example, and not by way of limitation, in the figures of the accompanying drawings and in which like reference numerals refer to similar elements and in
FIG. 1 is a block diagram that illustrates a system in which recursive database operations may be performed in a parallelized manner, according to an embodiment of the present invention;
FIG. 2 is a flow diagram that illustrates a technique for performing a recursive database operation using two stages of concurrently executing slaves, according to an embodiment of the present
invention; and
FIG. 3 is a block diagram that illustrates a computer system upon which an embodiment of the invention may be implemented.
A method and apparatus is described for performing recursive database operations. In the following description, for the purposes of explanation, numerous specific details are set forth in order to
provide a thorough understanding of the present invention. It will be apparent, however, that the present invention may be practiced without these specific details. In other instances, well-known
structures and devices are shown in block diagram form in order to avoid unnecessarily obscuring the present invention.
In order to perform recursive database operations more efficiently, according to one embodiment of the invention, a plurality of first-stage slaves and a plurality of second-stage slaves are
established in a database server. In a first iteration of a recursive database operation, the first-stage slaves concurrently process data items and send the results of the first-stage slaves'
processing to the second-stage slaves. The second-stage slaves receive the first results of the first-stages slaves' processing and concurrently process those first results. The second-stage slaves
store the first results of the second-stage slaves' processing in a data repository.
In a second iteration of the recursive database operation, the first-stage slaves obtain the first results of the second-stage slaves' processing from the data repository, concurrently process those
first results, and send the second results of the first-stage slaves' processing to the second-stage slaves. The second-stage slaves receive the second results of the first-stages slaves' processing
and concurrently process those second results. The second-stage slaves store the second results of the second-stage slaves' processing in the data repository.
Subsequent iterations of the recursive database operation proceed in this manner until the recursive database operation has been completed. In each iteration, the first-stage slaves consume the
product of the second-stage slaves' processing during the previous iteration, and the second-stage slaves consume the product of the first-stage slaves' processing during the current iteration.
Because the first-stage slaves and second-stage slaves are implemented in the database server, the above embodiment makes it unnecessary for an application to send multiple queries to and receive
multiple results from the database server during the performance of the recursive database operation. Additionally, application programmers are spared the burden of programming applications to
perform the processing that is performed by the first-stage slaves and second-stage slaves.
Multi-Slave Database Server
FIG. 1 is a block diagram that illustrates a system 100 in which recursive database operations may be performed in a parallelized manner, according to an embodiment of the present invention. System
100 comprises a database server 102 and a database 104. Database server 102 is communicatively coupled to database 104.
Database server 102 comprises a plurality of first-stage slaves 106A-N and a plurality of second-stage slaves 108A-N. Each of first-stage slaves 106A-N may be a separate thread of a first process,
and each of second-stage slaves 108A-N may be a separate thread of a second process. Alternatively, each of first-stage slaves 106A-N and second-stage slaves 108A-N may be a separate thread of the
same process. Alternatively, each of first-stage slaves 106A-N and each of second-stage slaves 108A-N may be a separate process. First-stage slaves 106A-N execute concurrently with each other and
each of second-stage slaves 108A-N.
Database server 102 also comprises a query coordinator 110. Query coordinator 110 is a process that executes concurrently with first-stage slaves 106A-N and second-stage slaves 108A-N. Query
coordinator 110 sends messages to, and receives messages from, first-stage slaves 106A-N and second-stage slaves 108A-N. Such messages may be sent and received using inter-process communication
mechanisms, for example.
Database 104 comprises data structures 112A-N. Each of data structures 112A-N may be, for example, a separate row of a database table. In one embodiment, each of data structures 112A-N contains one
or more items. For example, one of data structures 112A-N might contain items such as “apple,” “banana,” and “orange.” Database 104 also comprises a data repository 114. Data repository 114 may be,
for example, a temporary database table.
In one embodiment, data repository 114 comprises two separate segments 116A and 116B. One of segments 116A-B is designated as a “read” segment, and one of segments 116A-B is designated as a “write”
segment. In one embodiment, query coordinator 110 maintains state information that indicates which of segments 116A-B is currently designated as the “read” segment, and which of segments 116A-B is
currently designated as the “write” segment. The designations of segments 116A-B can be swapped.
Dual-Stage Parallelized Performance of Recursive Database Operations
FIG. 2 is a block diagram that illustrates a technique 200 for performing a recursive database operation using two stages of concurrently executing slaves, according to an embodiment of the present
invention. In block 202, a plurality of first-stage slaves is established in a database server. For example, first-stage slaves 106A-N may be established in database server 102. In block 204, a
plurality of second-stage slaves in established in the database server. For example, second-stage slaves 108A-N may be established in database server 102.
An iteration of the recursive database operation is represented in blocks 206-212. In block 206, data items in a data repository are processed by causing the first-stage slaves to process
concurrently the data items. For example, database server 102 may process a plurality of data items that are stored in data repository 114 by causing first-stage slaves 106A-N to process concurrently
the data items. In one embodiment, each data item is a separate itemset.
In block 208, the products of the first-stage slaves' processing are processed by causing the second-stage slaves to process concurrently those products. For example, database server 102 may process
the products of the first-stage slaves' earlier processing by causing second-stage slaves 108A-N to process concurrently the first-stage slaves' products.
In block 210, the products of the second-stage slaves' processing are stored in the data repository. For example, each of second-stage slaves 108A-N may store, in data repository 114, data items that
are the products of the second-stage slaves' processing.
In block 212, it is determined whether the recursive database operation is complete. For example, query coordinator 110 may determine whether the recursive database operation is complete. If the
recursive database operation is complete, then technique 200 ends. Alternative, if the recursive database operation is not complete, then control passes back to block 206.
Thus, if the recursive database operation is not complete, then, during a next iteration of the recursive database operation, the first-stage slaves will, in block 206, process concurrently the
products produced by the second-stage slaves during the then-previous iteration of the recursive database operation. These products may be, for example, data items that second-stage slaves 108A-N
stored in data repository 114.
Selecting Criteria-Satisfying Data Items
In one embodiment of the invention, at least some of the processing performed by first-stage slaves 106A-N, second-stage slaves 108A-N, or both, involves determining whether data items satisfy
specified criteria, and producing only those of the data items that satisfy the specified criteria.
In one embodiment, second-stage slaves 108A-N concurrently process data items produced by the processing of first-stage slaves 106A-N by determining, for each particular data item, whether that
particular data item satisfies specified criteria. In one embodiment, second-stage slaves 108A-N select only the data items that satisfy the specified criteria, and store in data repository 114 only
the selected data items. In one embodiment, second-stage slaves 108A-N return, to an application or other set of processes or threads, only the selected data items. In one embodiment, because only
the selected data items are stored in data repository 114, first-stage slaves 106A-N process only the selected data items during the next iteration of the recursive database operation.
Determining Candidate Itemsets
In one embodiment, the data items processed and produced by first-stage slaves 106A-N and second-stage slaves 108A-N are itemsets. In one embodiment, the processing performed by first-stage slaves
106A-N involves determining candidate itemsets and counting how many of data structures 112A-N contain all of the elements of each such candidate itemset.
As is discussed above, the observation that an N-element itemset cannot be a frequent itemset relative to a set of data structures unless all of the (N−1)-element subsets of the N-element itemset are
also frequent itemsets relative to that set of data structures allows the computationally intensive determination of whether itemsets are frequent to be performed for fewer itemsets during each
successive iteration of a recursive frequent itemset-determining operation.
Accordingly, in one embodiment, during an iteration of the recursive frequent itemset-determining operation, at least one of first-stage slaves 106A-N does the following. At least one of first stage
slaves 106A-N reads (N−1)-element itemsets from data repository 114. The (N−1)-element itemsets are, in one embodiment, frequent (N−1)-element itemsets that second-stage slaves 108A-N stored in data
repository 114 during a previous iteration of the operation. At least one of first-stage slaves 106A-N determines all of the different possible N-element itemsets that are subsets of an M-element
itemset, where the M-element itemset includes one instance of every item that occurs in data structures 112A-N.
For example, if items (apple, banana, milk, orange) are the only items that occur once or more in data structures 112A-N, then the possible 2-element itemsets of that 4-element itemset are (apple,
banana), (apple, milk), (apple, orange), (banana, milk), (banana, orange), and (milk, orange).
In one embodiment, for each such possible N-element subset, at least one of first-stage slaves 106A-N determines whether all of the (N−1)-element subsets of that possible N-element subset are
frequent (N−1)-element itemsets that were read from data repository 114. If all of the (N−1)-element subsets of a particular N-element itemset are frequent (N−1) element itemsets, then at least one
of first-stage slaves 106A-N places the particular N-element itemset in a group of candidate N-element itemsets. Otherwise, the particular N-element itemset is not placed in the group of candidate
N-element itemsets.
For example, if (apple), (banana), and (orange) are the only frequent 1-element itemsets relative to data structures 112A-N, then (apple, banana), (apple, orange), and (banana, orange) might be
frequent 2-element itemsets relative to data structures 112A-N, and are therefore candidate 2-element itemsets. However, in this example, (apple, milk), (banana, milk), and (milk, orange) cannot be
frequent 2-element itemsets relative to data structures 112A-N because (milk) is not a frequent 1-element itemset relative to data structures 112A-N. Consequently, the determinations of whether
(apple, milk), (banana, milk), and (milk, orange) actually are frequent 2-element itemsets relative to data structures 112A-N do not need to be performed.
In one embodiment, during the initial iteration of the operation, first-stage slaves 106A-N do not determine candidate 1-element itemsets based on the contents of data repository 114, because at that
point, data repository 114 is empty. Instead, first-stage slaves 106A-N assume that all of the 1-element subsets of the M-element itemset referenced above are candidate 1-element itemsets.
In one embodiment, only one of first-stage slaves 106A-N determines the group of candidate N-element itemsets as discussed above. In one embodiment, after making this determination, that first-stage
slave sends, to the others of first-stage slaves 106A-N, the candidate N-element itemsets.
Concurrently Counting Occurrences of Candidate Itemsets
As is discussed above, in one embodiment, a group of candidate N-element itemsets is generated based on the contents of data repository 114, and the contents of that group are obtained by each of
first-stage slaves 106A-N. Once the group of candidate N-element itemsets is known, the occurrences of those candidate N-element itemsets in data structures 112A-N can be counted. Occurrences of
non-candidate N-element itemsets do not need to be counted.
In order to count these occurrences more efficiently, in one embodiment, the counting is performed concurrently, or in parallel, by first-stage slaves 106A-N. Each of first-stage slaves 106A-N may be
assigned a separate subset of data structures 112A-N. For example, if there were 10 first-stage slaves 106A-N and 100 data structures 112A-N, then each of first-stage slaves 106A-N might be assigned
10 separate data structures of data structures 112A-N. The divvying of data structures 112A-N among first-stage slaves 106A-N might be done according to range, for example, or according to a hash
mapping, for another example.
Each of data structures 112A-N may indicate an identifier that is unique to that data structure. A particular data structure's identifier may be used to determine the range in which the particular
data structure belongs. Alternatively, a particular data structure's identifier may be used as input to a hash function that produces a hash value to which the particular data structure corresponds.
For example, if there were 10 first-stage slaves 106A-N and 100 data structures 112A-N, then the hash function might, for each data structure, divide that data structure's identifier by 10 and take
the whole remainder resulting from that division to be the hash value associated with that data structure.
In one embodiment, each of first-stage slaves 106A-N counts, concurrently with each other of first-stage slaves 106A-N, the occurrences of the candidate N-element itemsets in data structures in the
subset of data structures 112A-N that has been assigned to that first-stage slave. In one embodiment, because multiple slaves count occurrences of the candidate N-element itemsets in parallel, the
time taken to count all such occurrences is reduced.
For example, the contents of data structures 112A-C assigned to first-stage slave 106A might be (apple, banana, milk), (apple, banana, milk, orange), and (orange), respectively, and the contents of
data structures 112D-F assigned to first-stage slave 106B might be (banana, milk, orange), (apple, milk, orange), and (apple, banana, orange), respectively. In this example, assuming that the
candidate 2-element itemsets are (apple, banana), (apple, milk), (apple, orange), (banana, milk), (banana, orange), and (milk, orange), first-stage slave 106A would count 2 occurrences of (apple,
banana), 2 occurrences of (apple, milk), 1 occurrence of (apple, orange), 2 occurrences of (banana, milk), 1 occurrence of (banana, orange), and 1 occurrence of (milk, orange). In this example,
assuming the same candidate 2-element itemsets, second-stage slave 106B would count 1 occurrence of (apple, banana), 1 occurrence of (apple, milk), 2 occurrences of (apple, orange), 1 occurrence of
(banana, milk), 2 occurrences of (banana, orange), and 2 occurrences of (milk, orange).
In one embodiment, the counting of occurrences of candidate N-element itemsets is performed using either a “bitmap intersection” technique or a “prefix tree counting” technique. Both of these
techniques are specifically described in co-pending U.S. patent application Ser. No. 10/643,563, titled “DYNAMIC SELECTION OF FREQUENT ITEMSET COUNTING TECHNIQUE.” In one embodiment, the technique
that is used to count occurrences of candidate N-element itemsets is dynamically determined at each iteration of the operation. Thus, the counting technique used during one iteration of the operation
may differ from the counting technique used during another iteration of the operation. Techniques for dynamically selecting counting techniques also are described in U.S. patent application Ser. No.
Distributing Preliminary Counts Among Second-Stage Slaves
In one embodiment, each of first-stage slaves 106A-N counts the total number of data structures in the subset of data structures 112A-N assigned to that first-stage slave. In one embodiment, each
particular first-stage slave of first-stage slaves 106A-N sends, to one or more of second-stage slaves 108A-N, at least the following type of information: (a) one or more of the candidate N-element
itemsets, (b) a count of occurrences of those candidate N-element itemsets in those of data structures 112A-N assigned to the particular first-stage slave, and (c) a count of the total number of
those of data structures 112A-N assigned to the particular first-stage slave.
In one embodiment, each of first-stage slaves 106A-N sends the above type of information to multiple second-stage slaves of second-stage slaves 108A-N. In one embodiment, all of the counts associated
with a particular candidate N-element itemset are sent to the same second-stage slave, regardless of which first stage-slave determined the counts. However, counts associated with different N-element
itemsets may be sent to different second-stage slaves.
For example, assuming the candidate 2-element itemsets of the example above, each of first-stage slaves 106A-N may send counts to various ones of second-stage slaves 108A-N in the following manner:
counts associated with (apple, banana) may be sent to second-stage slave 108A, counts associated with (apple, milk) may be sent to second-stage slave 108B, counts associated with (apple, orange) may
be sent to second-stage slave 108C, counts associated with (banana, milk) may be sent to second-stage slave 108D, counts associated with (banana, orange) may be sent to second-stage slave 108E, and
counts associated with (milk, orange) may be sent to second-stage slave 108F.
For another example, if there are fewer second-stage slaves 108A-N, then counts associated with (apple, banana) and counts associated with (apple, milk) may be sent to second-stage slave 108A, counts
associated with (apple, orange) and counts associated with (banana, milk) may be sent to second-stage slave 108B, and counts associated with (banana, orange) and counts associated with (milk, orange)
may be sent to second-stage slave 108C.
In one embodiment, the candidate itemsets are divvied among second-stage slaves 108A-N in as balanced a manner as possible, so that each of second-stage slaves 108A-N receives counts associated with
approximately the same number of candidate itemsets during a particular iteration of the operation. Candidate itemsets may be divvied among second-stage slaves 108A-N through a hash-mapping
technique, for example. For example, the elements of a particular candidate itemset may be enumerated, combined, and input into a hash function. Counts associated with the particular candidate
itemset may be sent to the second-stage slave that is associated with the hash value produced by the hash function.
Concurrently Aggregating Preliminary Counts
In one embodiment, the processing performed by second-stage slaves 108A-N involves aggregating preliminary counts received from first-stage slaves 106A-N, and selecting one or more candidate
N-element itemsets based on whether aggregate counts for those itemsets meet a specified threshold.
In one embodiment, due to the techniques described above, each of second-stage slaves 108A-N receives, from first-stage slaves 106A-N, one or more preliminary occurrence counts respectively
associated with one or more separate subsets of the candidate N-element itemsets. In one embodiment, each of second-stage slaves 108A-N also receives, from each of first-stage slaves 106A-N that
sends a preliminary occurrence count to that second-stage slave, a count of the total number of data structures 112A-N that were assigned to that first-stage slave.
For example, each of first-stage slaves 106A-N might determine, for 10 separate database structures of database structures 112A-N, how many occurrences of each of the candidate 2-element itemsets are
within those database structures. Continuing the example, second-stage slave 108A might be associated with candidate 2-element itemsets (apple, banana) and (apple, milk). Therefore, in this example,
each particular first-stage slave of first-stage slaves 106A-N sends, to second-stage slave 108A, information that indicates: the number of database structures that the particular first-stage slave
evaluated (in this case, 10), the number of occurrences of (apple, banana) in the database structures that the particular first-stage slave evaluated, and the number of occurrences of (apple, milk)
in the database structures that the particular first-stage slave evaluated.
In one embodiment, each particular second-stage slave of second-stage slaves 108A-N separately aggregates (i.e., adds up), for each of the candidate N-element itemsets with which the particular
second-stage slave is associated, the preliminary counts that the particular second-stage slave receives from first-stage slaves 106A-N. For example, second-stage slave 108A might receive, from
first-stage slave 106A, a count of 2 for (apple, banana) and a count of 2 for (apple, milk). Continuing the example, second-stage slave 108A might receive, from first-stage slave 106B, a count of 1
for (apple, banana) and a count of 1 for (apple, milk). Therefore, in this example, second-stage slave 108A would determine an aggregate count of 3 for (apple, banana) and an aggregate count of 3 for
(apple, milk).
In one embodiment, second-stage slaves 108A-N concurrently determine aggregate counts for candidate N-element itemsets. For example, second-stage slave 108A may aggregate counts for (apple, banana)
and (apple, milk) at the same time that second-stage slave 108B aggregates counts for (apple, orange) and (banana, milk), and at the same time that second-stage slave 108C aggregates counts for
(banana, orange) and (milk, orange).
In one embodiment, second-stage slaves 108A-N aggregate counts concurrently with first-stage slaves 106A-N determining the counts and sending the counts to second-stage slaves 108A-N.
Selecting Frequent Itemsets from Among Candidate Itemsets
As is described above, each of second-stage slaves 108A-N receive, from each of first-stage slaves 106A-N that sends a preliminary occurrence count to that second-stage slave, a count of the total
number of data structures 112A-N that were assigned to that first-stage slave. In one embodiment, each of second-stage slaves aggregates each such total number of data structures to determine an
aggregate total number of data structures 112A-N.
In one embodiment, once a particular second-stage slave of second-stage slaves 108A-N has determined an aggregate count for a particular candidate N-element itemset, as described above, the
particular second-stage slave determines whether the aggregate count for the particular candidate N-element itemset is at least as great as a specified fraction of the aggregate total number of data
structures 112A-N. The specified fraction is referred to herein as the “threshold.” If the aggregate count for the particular candidate N-element itemset is at least as great as the number derived
from the specified threshold, then the particular second-stage slave determines that the particular candidate N-element is a frequent N-element itemset relative to data structures 112A-N.
For example, second-stage slave 108A might receive, from first-stage slave 106A, an indication that first-stage slave 106A evaluated 3 of database structures 112A-N. Continuing the example,
second-stage slave 108A might receive, from first-stage slave 106B, an indication that first-stage slave 106B evaluated 3 of database structures 112A-N. Assuming for purposes of example that these
were the only “database structure totals” received from first-stage slaves 106A-B, second-stage slave 108A determines that the aggregate total number of data structures 112A-N is 6 (i.e., 3+3).
Continuing the example, if the aggregate count for (apple, banana) is 3, then second-stage slave 108A determines that the fraction of data structures 112A-N that contain (apple, banana) is ½ (i.e., 3
/6). Continuing the example, if the aggregate count for (apple, milk) is 3, then second-stage slave 108A determines that the fraction of data structures 112A-N that contain (apple, milk) is also ½
(i.e., 3/6). Assuming for purposes of the example that the specified threshold is ⅓, second-stage slave 108A determines that both (apple, banana) and (apple, milk) are frequent 2-element itemsets
relative to data structures 112A-N.
In one embodiment, each of second-stage slaves 108A-N determines, for each of the candidate N-element itemsets with which that second-stage slave is associated, whether that candidate N-element
itemset is a frequent N-element itemset relative to data structures 112A-N. In one embodiment, each of second-stage slaves 108A-N selects, from among the candidate N-element itemsets with which that
second-stage slave is associated, only the frequent N-element itemsets.
In one embodiment, for a particular iteration of the operation, each of second-stage slaves 108A-N stores, in data repository 114, only the selected frequent N-element itemsets. In one embodiment,
each of second-stage slaves 108A-N sends, to one or more other entities that may include an application external to database server 102, the selected frequent N-element itemsets. In one embodiment,
each of second-stage slaves 108A-N performs the above determination, selection, storage, and sending concurrently with each of the others of second-stage slaves 108A-N.
In one embodiment, one or more of first-stage slaves 106A-N uses the frequent N-element itemsets stored in data repository 114 to determine candidate (N+1)-element itemsets during a next iteration of
the operation. Thus, the determinations of each successive iteration may be based upon the determinations of previous iterations.
Flow Control
In one embodiment, each particular first-stage slave of first-stage slaves 106A-N sends, to second-stage slaves 108A-N, a message that indicates that the particular first-stage slave has finished
evaluating the subset of database structures 112A-N to which the particular first-stage slave was assigned. A particular first-stage slave sends the message only after the particular first-stage
slave has evaluated all of the database structures in the particular first-stage slave's assigned subset of data structures. In one embodiment, each of first-stage slaves 106A-N waits, after sending
such a message, to receive a signal from query coordinator 110 before proceeding to the next iteration of the operation.
In one embodiment, when a particular second-stage slave of second-stage slaves 108A-N has received such a message from each of first-stage slaves 106A-N, the particular second-stage slave begins to
determine, based on the aggregated counts and database structure total, which of the candidate N-element itemsets are frequent N-element itemsets. In one embodiment, second-stage slaves 108A-N do not
begin to perform this determination until such a message has been received from each of first-stage slaves 106A-N.
In one embodiment, for each particular second-stage slave of second-stage slaves 108A-N, when the particular second-stage slave has finished storing frequent N-element itemsets in data repository 114
, the particular second-stage slave sends a message to query coordinator 110. The message indicates to query coordinator 110 that the particular second-stage slave has finished. In one embodiment,
second-stage slaves 108A-N store frequent N-element itemsets in a particular segment of segments 116A-B that has been designated as the “write” segment.
In one embodiment, after query coordinator 110 has received a message from each of second-stage slaves 108A-N indicating that those second-stage slaves have finished, query coordinator 110 swaps the
designations of the “read” and “write” segments 116A-B, so that the former “read” segment becomes the new “write” segment for the next iteration of the operation, and the former “write” segment
becomes the new “read” segment for the next iteration of the operation. After the designations have been swapped, the newly designated “write” segment may be emptied.
In one embodiment, after query coordinator 110 swaps the designations of the “read” and “write” segments, query coordinator 110 sends, to second-stage slaves 108A-N, a reference to the newly
designated “write” segment. In one embodiment, when second-stage slaves 108A-N store frequent N-element itemsets in the “write” segment, second-stage slaves 108A-N do so by writing the frequent
N-element itemsets to a location based on the reference to the “write” segment.
In one embodiment, after query coordinator 110 swaps the designations of the “read” and “write” segments, query coordinator 110 sends, to first-stage slaves 106A-N, a reference to the newly
designated “read” segment, which contains the frequent itemsets stored during the previous iteration of the operation. In one embodiment, one or more of first-stage slaves 106A-N generates candidate
(N+1)-element itemsets based on frequent N-element itemsets read from a location based on the reference to the “read” segment.
In one embodiment, after sending the newly designated “read” segment reference to each of first-stage slaves 106A-N as described above, query coordinator 110 sends, to each of first-stage slaves 106
A-N, a signal that informs first-stage slaves 106A-N that first-stage slaves 106A-N may proceed with the next iteration of the operation. In one embodiment, query coordinator 110 then waits to
receive “finished” messages from each of second-stage slaves 108A-N as described above.
In one embodiment, after at least one of first-stage slaves 106A-N has received a signal from query coordinator 110 that first-stage slaves 106A-N may proceed with the next iteration of the
operation, that first-stage slave (or slaves) determines (a) whether the currently designated “write” segment of segments 116A-B is empty and (b) whether “N” is greater than “M” as used in the above
context of N-element itemsets and the M-element itemset. In one embodiment, if the currently designated “write” segment is empty (meaning that there are no candidate (N+1)-element itemsets for the
next iteration of the operation), or if “N” is greater than “M,” then the recursive operation is ended. Otherwise, “N” is incremented and the next iteration of the recursive is performed as described
Hardware Overview
FIG. 3 is a block diagram that illustrates a computer system 300 upon which an embodiment of the invention may be implemented. Computer system 300 includes a bus 302 or other communication mechanism
for communicating information, and a processor 304 coupled with bus 302 for processing information. Computer system 300 also includes a main memory 306, such as a random access memory (RAM) or other
dynamic storage device, coupled to bus 302 for storing information and instructions to be executed by processor 304. Main memory 306 also may be used for storing temporary variables or other
intermediate information during execution of instructions to be executed by processor 304. Computer system 300 further includes a read only memory (ROM) 308 or other static storage device coupled to
bus 302 for storing static information and instructions for processor 304. A storage device 310, such as a magnetic disk or optical disk, is provided and coupled to bus 302 for storing information
and instructions.
Computer system 300 may be coupled via bus 302 to a display 312, such as a cathode ray tube (CRT), for displaying information to a computer user. An input device 314, including alphanumeric and other
keys, is coupled to bus 302 for communicating information and command selections to processor 304. Another type of user input device is cursor control 316, such as a mouse, a trackball, or cursor
direction keys for communicating direction information and command selections to processor 304 and for controlling cursor movement on display 312. This input device typically has two degrees of
freedom in two axes, a first axis (e.g., x) and a second axis (e.g., y), that allows the device to specify positions in a plane.
The invention is related to the use of computer system 300 for implementing the techniques described herein. According to one embodiment of the invention, those techniques are performed by computer
system 300 in response to processor 304 executing one or more sequences of one or more instructions contained in main memory 306. Such instructions may be read into main memory 306 from another
computer-readable medium, such as storage device 310. Execution of the sequences of instructions contained in main memory 306 causes processor 304 to perform the process steps described herein. In
alternative embodiments, hard-wired circuitry may be used in place of or in combination with software instructions to implement the invention. Thus, embodiments of the invention are not limited to
any specific combination of hardware circuitry and software.
The term “computer-readable medium” as used herein refers to any medium that participates in providing instructions to processor 304 for execution. Such a medium may take many forms, including but
not limited to, non-volatile media, volatile media, and transmission media. Non-volatile media includes, for example, optical or magnetic disks, such as storage device 310. Volatile media includes
dynamic memory, such as main memory 306. Transmission media includes coaxial cables, copper wire and fiber optics, including the wires that comprise bus 302. Transmission media can also take the form
of acoustic or light waves, such as those generated during radio-wave and infra-red data communications.
Common forms of computer-readable media include, for example, a floppy disk, a flexible disk, hard disk, magnetic tape, or any other magnetic medium, a CD-ROM, any other optical medium, punchcards,
papertape, any other physical medium with patterns of holes, a RAM, a PROM, and EPROM, a FLASH-EPROM, any other memory chip or cartridge, a carrier wave as described hereinafter, or any other medium
from which a computer can read.
Various forms of computer readable media may be involved in carrying one or more sequences of one or more instructions to processor 304 for execution. For example, the instructions may initially be
carried on a magnetic disk of a remote computer. The remote computer can load the instructions into its dynamic memory and send the instructions over a telephone line using a modem. A modem local to
computer system 300 can receive the data on the telephone line and use an infra-red transmitter to convert the data to an infra-red signal. An infra-red detector can receive the data carried in the
infra-red signal and appropriate circuitry can place the data on bus 302. Bus 302 carries the data to main memory 306, from which processor 304 retrieves and executes the instructions. The
instructions received by main memory 306 may optionally be stored on storage device 310 either before or after execution by processor 304.
Computer system 300 also includes a communication interface 318 coupled to bus 302. Communication interface 318 provides a two-way data communication coupling to a network link 320 that is connected
to a local network 322. For example, communication interface 318 may be an integrated services digital network (ISDN) card or a modem to provide a data communication connection to a corresponding
type of telephone line. As another example, communication interface 318 may be a local area network (LAN) card to provide a data communication connection to a compatible LAN. Wireless links may also
be implemented. In any such implementation, communication interface 318 sends and receives electrical, electromagnetic or optical signals that carry digital data streams representing various types of
Network link 320 typically provides data communication through one or more networks to other data devices. For example, network link 320 may provide a connection through local network 322 to a host
computer 324 or to data equipment operated by an Internet Service Provider (ISP) 326. ISP 326 in turn provides data communication services through the world wide packet data communication network now
commonly referred to as the “Internet” 328. Local network 322 and Internet 328 both use electrical, electromagnetic or optical signals that carry digital data streams. The signals through the various
networks and the signals on network link 320 and through communication interface 318, which carry the digital data to and from computer system 300, are exemplary forms of carrier waves transporting
the information.
Computer system 300 can send messages and receive data, including program code, through the network(s), network link 320 and communication interface 318. In the Internet example, a server 330 might
transmit a requested code for an application program through Internet 328, ISP 326, local network 322 and communication interface 318.
The received code may be executed by processor 304 as it is received, and/or stored in storage device 310, or other non-volatile storage for later execution. In this manner, computer system 300 may
obtain application code in the form of a carrier wave.
In the foregoing specification, embodiments of the invention have been described with reference to numerous specific details that may vary from implementation to implementation. Thus, the sole and
exclusive indicator of what is the invention, and is intended by the applicants to be the invention, is the set of claims that issue from this application, in the specific form in which such claims
issue, including any subsequent correction. Any definitions expressly set forth herein for terms contained in such claims shall govern the meaning of such terms as used in the claims. Hence, no
limitation, element, property, feature, advantage or attribute that is not expressly recited in a claim should limit the scope of such claim in any way. The specification and drawings are,
accordingly, to be regarded in an illustrative rather than a restrictive sense.
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Plandome, NY Algebra 1 Tutor
Find a Plandome, NY Algebra 1 Tutor
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elliptic movement?
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same. If you can help I would appreciate it. "Kimberley MacEwan" <mixedmicelles@hotmail.com> writes: > what would be the code to have movement along the same vector? Right > now I have gaussian
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Bushism, Bushist movement is not a clown movement.
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elliptic integrals
Matlab function ellipke(m) is limited to the input domain 0<m<1. I have to compute complete elliptic integrals with a negative argument. Help! Thanks. ...
no mouse movement
Hi, This is a good one. I have been getting no mouse movement/response on my pc for about 4 days now. Its as if there is no mouse. First, I noticed that the mouse would work but from time to time it
would seem to freeze then gradually the freezes got longer and then it stopped completely. It would behave normally upon reboot, but gradually got completely frozen. It now does not respond at all.
(If I shutdown and unplug then plug it back in it works for a while.) Here is the funny part. I got no response in Windows XP, until I logged in as another user. When I logged in it worked & then
Elliptical ROI
Hello, I would like to create an elliptical ROI on an image (I'm using a GUI - so this is user selected) that performs a calculation on the pixels in the image within the elliptical ROI. I would also
like to be able to move the ellipsoid around once it has been created. Is there a way to do this in Matlab? Thanks so much for your help! maybe u could use rect = getrect; to select a rectangle and
then use rectangle('position',rect,'Curvature',[1,1]) to create an ellipse (check help rectangle) another option is to use roipoly hope this helps, eitan JMM wrote: > >...
NoSQL Movement?
recently i wrote a blog article on The NoSQL Movement at http://xahlee.org/comp/nosql.html i'd like to post it somewhere public to solicit opinions, but in the 20 min or so, i couldn't find a proper
newsgroup, nor private list that my somewhat anti-NoSQL Movement article is fitting. So, i thought i'd post here to solicit some opinins from the programer community i know. Here's the plain text
version ----------------------------- The NoSQL Movement Xah Lee, 2010-01-26 In the past few years, there's new fashionable thinking about anti relational database, now blessed wi...
Movement describer.
I was working on a voice video description program. With this program movement in the webcam would be stored and described. (Upper body / Lower body) I am using a basic language and readpixel
programming method on a scaled down webcam image. That would store the amount of moves beeing registered. Is there a programming jargon word that can be named and explained for Jesting? (I want to
play (Mortal combat with the webcam. Shadow boxing)? (Usually people send solution in the natural way within millisecond) (Nature) (He is 'ing') (If someone has solution (scooter...
More precise movement
Does anyone know how i can have more precise movement of objects that are on there own layers? When moving small images you can only move an objects a certain amount and that sometimes is to much, i
think that it snaps to grid which i do not want it to do. Does anyone know what i am on about and know a solution to stop this? Thanks in advance "Pauline" <tattrill03@aol.com> wrote in message
news:4097dccd$0$98379$c397aba@news.newsgroups.ws... > Does anyone know how i can have more precise movement of objects that are on > there own layers? > > When moving small...
Camera movement.
With the camera you have many different ways of moving. Examples are: * Point of origine, point where I look at and a normal indicating the up camera. * You also have rotation around the camera axis.
* You have rotation around the object * You have an orbit view to rotate arond an object pointing to the horizon of that object. My question, is there any good way, to create code in such a way that
all moving options are synchronized? The problem is that I have too many choices, of rotating, and moving, and it is very hard to choose the best solution. Would storing the position, lookat
Elliptic filters
Hi, I have a speech signal that has noise introduced at various frequencies. The FFt magnitude of the speech signal is of the order of 10 power 6 at these frequencies. The whole point is I have to
remove these components. When I used an elliptic filter with a passband attenuation of 3dB and a stopband attenuation of 50dB, the magnitude at these points becomes of the order of 10 power 5. how do
i reduce the magnitudes at these points to zero? According to me, if there is a stopband attenuation of 50db, it means it is of the order of 10 power 5 and the magnitudes should be of the order of
How how i create shaped windows in clarion. I can create elliptical window, I need more shapes..Help In article <8bd6d7cc9d071c6f9d312810f098e05b@localhost.talkaboutprogramming.com>,
sylvester@esponline.co.za says... > How how i create shaped windows in clarion. > I can create elliptical window, I need more shapes..Help You might want to look at these two products. Keep in mind
the iAlchemy site is NOT FireFox browser friendly, so you need to view that site in IE. Product Description - Skin, MFG - iAlchemy, LTD Internet Link - http://www.ialchemy.com/_skin.html Product
Cursor Movement
Hi, Is there exists some function for controlling the cursor movement in C++. With control of cursor movement I mean, is there exists any function that let me to relocate the cursor as per coordinate
values given by the user. Amar Prakash Tripathi Amar Prakash Tripaithi wrote: > Hi, > Is there exists some function for controlling the cursor movement in > C++. With control of cursor movement I
mean, is there exists any > function that let me to relocate the cursor as per coordinate values > given by the user. http://www.parashift.com/c++-faq-lite/input-output.html#faq-...
Elliptic integral
Hello, does anybody know of a Mathematica program which converts an elliptic integral of the form int z^n/sqrt(P(z)) dz, where P is a fourth-order polynomial, into one of the standard forms? The
reason that I would like this is to try and evaluate Integrate[ Exp[q*I*x]/(Sqrt(-A*Exp[4*I*x] - I*B*Exp[3*I*x] + (C+I*D)*Exp[2*I*x] + I*B*Exp[I*x] - A)), {x, 0, 2*Pi}, Assumptions -> {Element[q,
Integers], A > 0, B > 0, C > 0, D > 0] but Mathematica won't do it in one go, even if I choose specific values for A, B, C, D and q. I believe that an analytic solutio...
elliptic integrals
Are there really no in-built functions to compute incomplete elliptic integrals? Or does someone have a pre-built routine already (save myself the effort of typing out and translating some NR
code...)? -Jeremy. To answer my own question, it looks like there is with the IMSL license. Oh well. -Jeremy. On Wed, 9 Feb 2011, Jeremy Bailin wrote: > Are there really no in-built functions to
compute incomplete elliptic integrals? Or does someone have a pre-built routine already (save myself the effort of typing out and translating some NR code...)? > > -Jeremy. > Have ...
more movement in fsolve
Hi, i'm running an fsolve routine but the independent variables are not moving far at all from their original values. How can I allow the vector x to move further from its original point? ...
Movement limits
Hi, Does anybody know how to configure a certain fixed movement distance in an assembly? Maybe I am doing something wrong, but what I want to achieve is: Example: - I mate a cylindric rod (shaft) and
with a let us say cube with a shaft dia hole. - concentric - ok, when I make a free drag I can go with a shaft up and down in infinity... - well I want to have limited drag in a defined distance The
point is that I want to have limits in movement of certain parts in assembly, how do I accomplish that? Thanks. Oz Distance Limit mates Read help or get training. matt "yozotrinity...
Elliptical path ?
A girl i know, who is going to do her degree thesis, asked me to animate an atom for her powerpoint dissertation. "Atomic" realism is not required, but i was wondering about the elliptical motion
path of the electrons. Doing a rotation is easy, but i don't have a clue how to realize the elliptical motion. I know that a real electron should probably follow a circulator motion, but an
elliptical one is more "cinematic" :) Suggestions ? -- Gareth Jax Ihgger #7*3*2 Indegno Neurone della Mente alveare Big Bad of Con-Fu Fighting Gareth Jax wrote: | A girl i know, w...
Elliptical Arc
Hi, I have an elliptical arc in a graphic system that is defined based on the following: Start point P1(x1,y1,1.0) End point P2(x2,y2,1.0) Center Point C(Cx,Xy, w) Direction Clock Wise/Counter
Clockwise I can get additional points on the arc (if the arc length is greather than 2) Given this information I need obtain obtain the additional properties of the ellipse Start point P1(x1,y1,1.0)
(same) End point P2(x2,y2,1.0) (same) Center Point C(Cx,Xy, w)(same) Semi Major Axis a Semi Minor Axis b Orientation Theta direction Clock Wise/Counter Clockwise The following image...
cursor movement
Hello, On a dynamically updating XY plot, I have two cursors so that the user can intergrate between the two x points. The trouble is the cursor keeps moving to the right with the next point being
plotted. ( I am using "Snap to Point" on the cursor. How do I stop the cursor from auto-updating (moving to the right?). The only movement I want on the cursor is the one the user makes using the
mouse or the cursor legend. Thanks, KB Hi, Try using "free" lock style for the cursor. This will fix the X and Y co-ordinates of the cursor until the user moves the...
Curved movement
I am working on a 2D game in which I would like to move a sprite from one location on the screen to another but instead of moving in a straight line I would like the path to be curved. The starting
point and ending points can be anywhere on the screen. Ideally adjusting the amount of curvature and which side of the set of points the curve occurs would be great. Any help would be appreciated,
thanks. On Wed, 29 Oct 2008 10:19:06 -0400, Bill <Bill@hotmail.com> wrote: >I am working on a 2D game in which I would like to move a sprite from >one location on the screen ...
Elliptic Code
Hi Does anyone have/know of a python implementation of the elliptic curve factoring algorithm (lenstra) which is both: simply and cleanly coded functional I'm aware of William Stein's code (from
elementary number theory book) but I don't understand his coding style and the algorithm doesn't seem to work efficiently. For that matter has anyone come across any useable math/number theory
packages apart from nzmath or aladim? Thanks Phil "Philip Smith" <as006d4848@blueyonder.co.uk> writes: > Does anyone have/know of a python implementation of the el...
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Mplus Discussion >> Save fscores
Student 09 posted on Thursday, April 30, 2009 - 8:54 am
I wonder whether there is any possibility to save variables which are not part of the model with the savedata command.
I need to save both fscores and an Identifier-variable, which is not part of the model, and obviously cannot be saved together with the information on the fscores?
Linda K. Muthen posted on Thursday, April 30, 2009 - 11:40 am
See the FSCORES option of the SAVEDATA command and the IDVARIABLE option of the VARIABLE command. You can save other variables using the AUXILIARY option of the VARIABLE command.
Marissa Ericson posted on Thursday, June 23, 2011 - 9:27 am
would the SAVE FSCORES option save factor scores that are free from measurement error if you are specifying a latent variable with 5 predictors?
Marissa Ericson posted on Thursday, June 23, 2011 - 10:41 am
Also, I am confused about how I am receiving a factor score for those participants who have no data. Thanks!
Linda K. Muthen posted on Thursday, June 23, 2011 - 11:39 am
The default in Mplus is to estimate the model using all available information and missing data estimation. Therefore, all observations receive a factor score. For listwise deletion, add LISTWISE=ON;
to the DATA command.
Factor scores are not the same as the factors in a model. How close they are to each other can be seen in the factor determinacy score. See the FSDETERMINACY option of the OUTPUT command.
Marissa Ericson posted on Thursday, June 23, 2011 - 11:46 am
Thank you! One follow up question. If I have 5 measures and am creating a latent variable and then using that latent variable in subsequent analyses with other latent variables would the model use
the same factor scores as I would receive in the fscores output? Is that what you meant by the factor scores are not the same as the factors in the model? For example, if I created a mean composite
of the 5 measures would that differ the fscores output?
Marissa Ericson posted on Thursday, June 23, 2011 - 11:49 am
To clarify further: I am doing a 4 latent variable cross lagged analysis in mplus but first doing CFA analyses on each of the 4 variables. Would the fscores data be the same as what is being used in
this 4 latent variable cross lagged analysis? I would like to extract the same data without measurement error. Is that possible? Thank you again!
Linda K. Muthen posted on Friday, June 24, 2011 - 9:25 am
I do not understand your question. However, I would only use factor scores if it was unavoidable.
Marissa Ericson posted on Friday, June 24, 2011 - 9:58 am
Sorry if that was unclear. I am doing a basic 1-Factor CFA using 5 measures. When using the SAVE Fscores option, I am wondering whether the factor scores that are saved are free from measurement
error. Thank you!
Bengt O. Muthen posted on Friday, June 24, 2011 - 10:27 am
Factor scores can be seen as free of measurement error, but are not free of estimation error. Estimated factor scores do not behave as true factors (give different variances and relations to other
variables) unless you have many good indicators. See for instance
Skrondal, A. and Laake, P. (2001). Regression among factor scores. Psychometrika 66, 563-575.
Lisa M. Yarnell posted on Wednesday, February 15, 2012 - 8:43 am
When factor scores are saved, are they placed in the rightmost columns in the DAT file?
(This will affect the VARIABLE NAMES line in the program where I call in the saved data file.)
Jon Heron posted on Wednesday, February 15, 2012 - 8:47 am
Hi Lisa,
it should tell you at the end of your .out file, e.g. here it's the last 8 vars
Order and format of variables
TOT_T1 F10.3
TOT_T2 F10.3
TOT_T3 F10.3
TOT_T4 F10.3
TOT_T5 F10.3
AGE_T1 F10.3
AGE_T2 F10.3
AGE_T3 F10.3
AGE_T4 F10.3
AGE_T5 F10.3
INTCPT1 F10.3
INTCPT1_SE F10.3
SLOPE1 F10.3
SLOPE1_SE F10.3
INTCPT2 F10.3
INTCPT2_SE F10.3
SLOPE2 F10.3
SLOPE2_SE F10.3
Lisa M. Yarnell posted on Wednesday, February 15, 2012 - 9:06 am
Thanks, Jon--this is very helpful! I see in the Mplus manual how to merge files using an ID variable--but it is problematic that the saved factor score file does not have an ID in it.
How then do I merge these saved factor scores back into my main data set?
Thank you so much.
Jon Heron posted on Wednesday, February 15, 2012 - 9:18 am
You can push the ID through to the final file within your variable section using
idvariable = ID;
the data I use had more than one ID and this isn't permitted with this command. If you're like me then you'll need to push other variables through using "auxiliary"
so I do
idvariable = ID1;
auxiliary = ID2;
Don't put them in your usevariable list though - the ID variable rarely fits the model well!!
Lisa M. Yarnell posted on Wednesday, February 15, 2012 - 9:22 am
Exactly--I didn't want to put ID in the USEVARIABLE list because it would mess up the fit of my factor models!
Thanks so much, Jon, for your expertise on this.
Joanne Bradbury posted on Thursday, February 16, 2012 - 3:35 pm
I've just spent hours trying to save my factor scores through the MFILE command of the SAVEDATA function. I can save the factor scores no problem but I can't seem to get the factor scores to be
merged back into the original dataset. Even trying to take them into SPSS was a problem with the factor scores being saved over three columns/variables. With MFILE, should I be nominating the
original dataset that I want to merge the factor scores into?
Linda K. Muthen posted on Friday, February 17, 2012 - 2:01 pm
You should be using the SAVE option if you want the factor scores in the original data set:
SAVE = FSCORES;
Lisa M. Yarnell posted on Wednesday, February 22, 2012 - 11:36 pm
Hi Linda, can you tell me what is wrong in my Mplus script below, such that I get the error message at the bottom of this section of text? I was relying on p. 404 of the current Mplus manual as an
This script begins at the ANALYSIS line of my program, to save space.
TYPE = BASIC;
MNAMES ARE H4MH3R H4MH4R H4MH5R H4MH6R H4PE37R H4PE38R H4PE39R H4PE40R H4PE41R AID GENERAL CONCRETE;
MFORMAT IS 9F10.3 I9 2F10.3;
MMISSING = H4MH3R H4MH4R H4MH5R H4MH6R H4PE37R H4PE38R H4PE39R
H4PE40R H4PE41R AID GENERAL CONCRETE (. *);
FILE IS MERGED.sav;
MISSFLAG = .;
*** ERROR in SAVEDATA command
Unknown option:
Linda K. Muthen posted on Thursday, February 23, 2012 - 6:52 am
Please send the output and your license number to support@statmodel.com.
Lisa M. Yarnell posted on Thursday, February 23, 2012 - 10:10 am
Hi Linda, I took the advice of you and Jon Heron above, and saved the factor scores using the SAVE=FSCORES command, and selected other variables that I wanted in the saved data set using the
AUXILIARY option.
But when I work with the newly created data set, factorscores_022312.dat, I get the message. Why would this be?
*** ERROR
The number of observations is 0. Check your data and format statement.
Data file: D:\AH\locker\factorscores_022312.dat
*** ERROR
Non-missing blank found in data file at record #1, field #: 18
Linda K. Muthen posted on Thursday, February 23, 2012 - 11:16 am
Open the data file and look at record 1 filed 18.
Lisa M. Yarnell posted on Tuesday, February 28, 2012 - 9:50 am
Hi Linda, can I change the default missing data flag when saving FSCORES from * to another value such as -999 or 999?
I got an error message when I tried to change the default missing data flag in this script.
SAVEDATA: FILE IS FACTORSCORES_022812.DAT; SAVE = FSCORES;
MISSFLAG = -999;
Linda K. Muthen posted on Tuesday, February 28, 2012 - 10:28 am
Please send the output and your license number to support@statmodel.com.
Lisa M. Yarnell posted on Monday, March 05, 2012 - 12:32 pm
Hi Linda, thanks for all of your help. What does it mean that Mplus saved two versions of my latent variables in the DAT file with the fscores:
GENERAL F10.3
GENERAL_SE F10.3
CONCRETE F10.3
CONCRETE_SE F10.3
Does the "_SE" refer to a standard error?
Linda K. Muthen posted on Monday, March 05, 2012 - 12:40 pm
Lisa M. Yarnell posted on Tuesday, May 01, 2012 - 7:58 am
Hi Linda, I noticed that when I use SAVE FSCORES for a categorical factor analysis, SEs for the saved factor scores are missing from the list of variables in the DAT file that is created.
Is that true: There are no saved SEs for fscores from a categorical CFA?
Bengt O. Muthen posted on Thursday, May 03, 2012 - 9:56 am
Mplus 6.12 does not have that, but if you use ML you can get that information via the information curve that you get in the PLOT command. One divided by the square root of the information value is
the SE for the factor score at that factor value.
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Topic: How many rectangles are in an n by n square array of dots?
Replies: 4 Last Post: Sep 9, 2006 11:14 PM
Messages: [ Previous | Next ]
Re: How many rectangles are in an n by n square array of dots?
Posted: Sep 9, 2006 6:54 PM
On 9/9/06, Oliver Stemforn <olive79pop@yahoo.com> wrote:
> To:
> Lou Talman
> Department of Mathematical and Computer Sciences
> Metropolitan State College of Denver
> You are wrong about me having to be more careful when a rectangle counts. In the description for this problem, I never limited the orientation of the rectangles I was seeking to be counted within
an array. I stated to get a count for the total. If I had only wanted a total of vertical/horizontal rectangles, I would have had to mention that explicitly. Or if I had wanted only a total of
oblique rectangles, I would have had to mention that explicitly.
The original question was:
>Can you come up with a polynomial in terms of n for the total number
of rectangles for a general n by n array?
I think the implication is that if you wanted a polynomial, you should
have been more careful in your choice of problem.
for example.
--Joshua Zucker
Date Subject Author
9/8/06 How many rectangles are in an n by n square array of dots? Oliver Stemforn
9/9/06 Re: How many rectangles are in an n by n square array of dots? talmanl@mscd.edu
9/9/06 Re: How many rectangles are in an n by n square array of dots? Oliver Stemforn
9/9/06 Re: How many rectangles are in an n by n square array of dots? Joshua Zucker
9/9/06 Re: How many rectangles are in an n by n square array of dots? Oliver Stemforn
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2.3 Distance and Ruler Axioms Printout
Whereas at the outset geometry is reported to have concerned herself with measurement of muddy land, she now handles celestial as well as terrestrial problems: she has extended her domain to the
furthest bounds of space.
—W. B. Frankland, The Story of Euclid (1901)
SMSG Ruler Postulate defines a correspondence between the points on a line (markings on a meter stick)
Postulate 2. (Distance Postulate) To every pair of distinct points there corresponds a unique positive number. This number is called the distance between the two points.
Postulate 3. (Ruler Postulate) The points of a line can be placed in a correspondence with the real numbers such that:
i. To every point of the line there corresponds exactly one real number.
ii. To every real number there corresponds exactly one point of the line.
iii. The distance between two distinct points is the absolute value of the difference of the corresponding real numbers.
Postulate 4. (Ruler Placement Postulate) Given two points P and Q of a line, the coordinate system can be chosen in such a way that the coordinate of P is zero and the coordinate of Q is positive.
By Proposition 2.1 and the accompanying exercises, the Euclidean plane, Taxicab plane, Max-distance plane, Missing Strip plane, Poincaré Half-plane, Modified Riemann Sphere, and discrete
planes all satisfy the Distance Postulate. Tools for working with rulers in Geometer's Sketchpad are available in the Appendix B Prepared Geometer's Sketchpad and GeoGebra Sketches.
Definition. A ruler or coordinate system is a function mapping the points of a line into the real numbers, that satisfies SMSG Postulate 3.
Note the first and second conditions of the Ruler Postulate imply that f is a one-to-one and onto function. As a reminder, we write the definitions for one-to-one and onto functions.
Definition. A function f from A to B is onto B if for any b in B there is at least one a in A such that f(a) = b.
Definition. A function f from A to B is one-to-one if for any x and y in A with then (Note the contrapositive of this definition is often used in writing proofs.)
Proposition 2.4. The Euclidean Plane satisfies the Ruler Postulate.
Before we begin the proof, we do some scratch work to find the correct form for the rulers for the lines. We need a relationship between the distance and a ruler, so we begin with the distance
function. First, consider a vertical line l[a], which has all the first coordinates the same.
This motivates the definition for the standard ruler of a vertical line l[a] to be f(a, y) = y. Next, consider a nonvertical line l[m,b].
which motivates the definition for the standard ruler of a nonvertical line l[m,b] to be .
y = x. The points (0, 0), (1, 1), (2, 2), and (3, 3) are on the line. What is the distance from (0, 0) to (1, 1)? From (1, 1) to (2, 2)? From (1, 1) to (3, 3)? Note the standard ruler for this line
is . The coordinates for the four points determined by the standard ruler are 0, , respectively. By subtracting the appropriate coordinates of the ruler, do you obtain the distance between the
In the "real-world" sense, the standard ruler (coordinate system) is the placement of a meter stick such that the zero end is at the y-axis along any line through that point on the y-axis.
Proof. Let l be a line in the Euclidean Plane. Then l is either a vertical line or a nonvertical line.
Case 1. Assume l = l[a] a vertical line. Define by f(a, y) = y. We need to show the three conditions. First, show f is one-to-one. Let (x[1], y[1]) and (x[2], y[2]) be points on l[a]. We have x
[1] = x[2] = a. Suppose f(x[1], y[1]) = f(x[2], y[2]). Then y[1] = y[2] by the definition of f. Thus (x[1], y[1]) = (a, y[1]) = (a, y[2]) = (x[2], y[2]). Hence f is one-to-one. We next show f is
onto. Let r be any real number. Consider the point (a, r) on line l[a]. Note f(a, r) = r. Hence f maps the line onto the real numbers. Finally, we show the distance condition. Let P(x[1], y[1]) and
Q(x[2], y[2]) be points on line l[a]. We have x[1] = x[2] = a. Thus
Case 2. Assume l = l[m,b] is a nonvertical line. Define by . We first show f is one-to-one. Let (x[1], y[1]) and (x[2], y[2]) be points on l[m,b]. Suppose f(x[1], y[1]) = f(x[2], y[2]). Then .
Hence, x[1] = x[2]. We then have y[1] = mx[1] + b = mx[2] + b = y[2]. Thus (x[1], y[1]) = (x[2], y[2]). Hence f is one-to-one. We next show f is onto. Let r be any real number. Consider the point .
Hence f is onto. Finally, we show the distance condition. Let P(x[1], y[1]) and Q(x[2], y[2]) be points on line l[m,b].
Therefore, by Cases 1 and 2, an arbitrary line in the Euclidean plane has a ruler (coordinate system).//
As was discussed in Chapter 1, axioms need not be independent, which is the case with the Ruler Placement Postulate.
Theorem 2.5. The Ruler Placement Postulate is not independent of the other axioms.
Outline of the proof. We need to show that given two distinct points P and Q on a line l, there is a ruler that satisfies the conditions that the coordinate of point P is zero, and the coordinate of
point Q is positive.
Assume is a ruler. (Why do we know a line and a ruler exist?)
Let P and Q be two distinct points on l.
Set .
Define by g(A) = k[f(A) – f(P)] for all points A on l. (Why is g defined this way?)
Show g satisfies the conditions of the Ruler Postulate, i.e. show g is one-to-one, show g is onto, and show g satisfies the distance condition.
Show g(P) = 0 and g(Q) > 0.//
A point B is between points A and C, denoted A-B-C, if {A, B, C} is a collinear set of three distinct points and AB + BC = AC. (Here, AB represents the distance from A to B, i.e. d(A, B) = AB.)
A line segment is the union of two distinct points and all points between those two points, denoted either as segment AB or . The points A and B are called the endpoints of segment AB.
Two segments are congruent if they have the same measure, denoted .
A point M is the midpoint of segment AB if AM = MB and {A, M, B} is collinear.
A bisector of a segment is a line that contains the midpoint of the segment.
A ray AB is the union of the segment AB and the set of all points C such that B is between A and C, denoted either as ray AB or . The point A is called the endpoint of the ray AB. (Note ray AB
and ray BA are different rays.)
A triangle is the union of three segments determined by three noncollinear points, i,e., triangle ABC is the union of segment AB, segment AC, and segment BC. Each of the three noncollinear points
that determine a triangle is called a vertex of the triangle.
Exercise 2.17. Find the axioms from a high school geometry book that correspond to SMSG Postulates 2, 3, and 4.
Exercise 2.18. How do the SMSG Postulates 3 and 4 relate to "real-world" applications?
Exercise 2.19. For each model (Euclidean, Taxicab, Max-Distance, Missing Strip, and Poincaré Half-plane) find a ruler where f(P) = 0 and f(Q) > 0 for (a) P(3, 4) and Q(3, 7); and (b) P(–1, 3) and
Q(1, 2).
Exercise 2.20. Complete the proof that the Ruler Placement Postulate is not independent, Theorem 2.5.
Exercise 2.21. Show the stated model satisfies SMSG Postulate 3, the Ruler Postulate, for (a) Taxicab Plane; (b) Max-Distance Plane; (c) Missing Strip Plane; and (d) Poincaré Half-plane.
Exercise 2.22. Does the Modified Riemann Sphere satisfy SMSG Postulate 3, the Ruler Postulate? Explain.
Exercise 2.23. Explain why collinear is necessary in the definition of betweenness. (Hint. Look for an example in either the Taxicab or Max-distance plane where the distance condition is satisfied,
but the point would not be on the line.)
Exercise 2.24. Prove a segment has a unique midpoint.
Exercise 2.25. Find the midpoint of the segment AB for each model (Euclidean, Taxicab, Max-distance, Missing Strip, and Poincaré Half-Plane) where (a) A(1, 1) and B(1, 5); and (b) A(–1, 1) and B
(3, 2). (Show the work using the standard ruler for each model.)
Exercise 2.26. Find the ray AB for each model (Cartesian, Missing Strip, and Poincaré Half-plane) where (a) A(–3, 1) and B(–3, 7); and (b) A(–1, 5) and B(3, 1).
Exercise 2.27. An equivalence relation, ~, is a relation on a set that satisfies each of the following: (i) a ~ a (reflexive property) (ii) If a ~ b, then b ~ a. (symmetric property) (iii) If
a ~ b and b ~ c, then a ~ c. (transitive property).
Prove that is an equivalence relation for the set of all segments.
Don't measure yourself by what you have accomplished, but by what you should have accomplished with your ability.
—John Wooden (1910–2010)
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Charge Form Factor and Sum Rules of Electromagnetic and Neutral Current Response Functions in Carbon-12
October 28, 2013 3:30PM to 4:30PM
I will report on an ab initio calculation of the 12C elastic form factor and sum rules of longi-tudinal and transverse response functions measured in inclusive (e,e') scattering. Our calculations are
based on realistic nuclear potentials and electromagnetic currents. The longitudinal elastic form factor and sum rule are found to be in satisfactory agreement with available experimental data. As I
will show, a direct comparison between theory and experiment is difficult, particularly for the transverse sum rule. However, it is shown that the calculated one has large contributions from two-body
currents, indicating that these mechanisms lead to a significant enhancement of the quasi-elastic transverse response. This fact may have implications for the anomaly observed in recent neutrino
quasi-elastic charge-changing scattering data off 12C. Preliminary results on the neutral current sum rules, obtained within the same theoretical framework, will also be presented.
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Weierstrass half-periods: Introduction to the Weierstrass utility functions (subsection WeierstrassUtilities/02)
Definitions of Weierstrass utilities
The Weierstrass half‐periods and the Weierstrass invariants , the Weierstrass function values at half-periods , and the Weierstrass zeta function values at half-periods are defined by the following
where is the Klein invariant modular function, is Weierstrass elliptic function, and is theWeierstrass zeta function.
The previous four vector functions are sometimes called the Weierstrass utilities because they are the basic elements of the Weierstrass theory of elliptic functions.
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Developing a mathematical model of lipid titrations
The interactions of cyclodextrins binding with phospholipids have been expressed in the form of differential rate equations. Six rate equations were constructed to represent the interactions of
phospholipids with cyclodextrins. Using the computer program Mathematica, we solved and plotted these interactions given the time between injections, the number of injections, the equilibrium
constants, and the concentrations of the both the cyclodextrins and the phospholipids. The mathematical model generated similar results for the titration of phospholipids into cyclodextrin in
previous research. A binding constant of 8.1x10^-5 was the calculated equilibrium constant for the overall reaction. This model can determine plausible values for the equilibrium constants, the
enthalpy of the reaction, and can potentially lead to understanding of other areas of phospholipids research.
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Teaching First Grade Math: Money
Teachers can use the following resources for students that are in the first grade who are learning about money with a total value up to 100 cents or less (Virginia Standards of Learning for 1.7 a &
Text Annotations:
The Coin Counting Book written by Rozanne Lanczak Williams is a fun rhyming book for students to learn about counting money and it’s value. The book introduces pennies, nickels, dimes, quarters as
a way for students to do simple math with rhyme:
“Let’s count our five pennies just one more time. If we add five more pennies we’ll have…one dime.”
Actual size coins are spread out over the pages showing both front and back for student learning. If the book says to count five pennies as an example, there are five pennies laid out on the page
with a addition sign in between each coin to help with student visualization. The book ends by showing a hand placing coins in a piggy bank making the statement: “If we save some of it- the rest we
can spend!”
Pigs will be Pigs, Fun with Math and Money written by Amy Axelrod and illustrated by Sharon McGinley-Nally is about a family of hungry pigs looking for money in their house so they can go to their
favorite place to eat a snack. This is a great book for introducing students in the first grade to money. The pig family is hungry and realize they do not have enough money to go out to eat; so
Mrs. Pig decides that everyone will “Hunt for Money!”. The book describes where in the house and how much money everyone in the family finds while on the money hunt. In the end, the Pigs have
enough money to eat out and when they arrive home they find their house in a mess from their hunt. Pigs will always be pigs.
The book, 26 Letters and 99 Cents written by Tana Hoban provides photos of numbers from 1-30, counting by 5′s from 30-90 and 99. Beside of each number there is a photo of coins that shows the value
of the number when added. The book can be shown to the whole class while identifying each coin and the value. This would also be a great book for students to look at during the day as a center
activity, etc. The book shows students both the front and back of real American coins: pennies, nickels, dimes and quarters in their actual size. This helps students to visualize the size and
identification of each coin.
The book If You Made A Million written by David M. Schwartz and illustrated by Steven Kellogg is a book where students can really use their imagination. Readers are given different scenarios with
spending anything from one penny to purchase a peeble all the way up to one million dollars with the option of saving the money at the bank. This would be a great way to ask students for ideas about
what they would purchase with different amounts of money. Schwartz gives differnet forms of measurement for various amounts of money. For example, one hundred dollars in pennies stacked up would be
equal to fifty feet or a million dollars in quarters would equal a whale’s weight. This is a great book to help students realize that a paper bill is sometimes easier (and lighter) to carry around
instead of coins.
The last text would be a great resource for students who are in need of a more challenging way to think about money and its uses. Money Madness written by David A. Adler and illustrated by Edward
Miller explains how money first originated and how money is now used to purchase different items from around the world. The book starts off by asking:
“What’s all this money madness? People talk about money and work for it. They seem to always want more of it…”
The book gives examples of why people now use money to purchase a variety of items. If people did not have money then they would have to make their own bread. Adler explains at a child’s level how
people first started to trade by introducing the word, barter. An example that Adler used was when a person would trade an animal in exchange for berries. Even though the person receiving the
berries might not want them he knew that someone else would want to trade the berries for something that he wanted or needed in return. The book explains how rocks were used as an early form of
money and then replaced by metals (silver and gold). The silver and gold pieces were made into coins but were at times difficult to carry if someone had a lot. Paper money was then invented. Adler
explains how each country has it’s own form of money and that the value of the money can vary from place to place. ”You know with money you can buy things you want. With money you can buy things you
Web Annotations:
Students can play the game Change It for additional practice on adding up different coin values. Teachers can create each game to specifically fit each students instructional level.
GPB Kids has created a web-site for students to play a game where they are given nine different items that they need to buy. Players are instructed to buy one of the nine items by dragging the
correct coin(s) to the matching picture in the chart. If the player is right then they can move on to the next problem; if not, they have the chance to try again.
Teachers can create different tutorials for students by selecting any combination of pennies, nickels, dimes and quarters for practice. For each category chosen, students are provided a picture of
real money on the left side of the screen and need to select the correct value of the money from the right side of the screen. If the student selects the correct amount of money they can move on to
the next problem. If an incorrect answer is chosen, then the student can try again.
HMH School Publishers created a great money practice tool for students. For the activity, coins are lined up in decreasing value from largest to smallest. Students need to count the value of the
coins and type the correct amount of money in the blank provided. Students then need to click on “check” to see if they have typed in the correct amount. If so, the student will hear chimes, if an
incorrect amount is typed in then the student will see a screen flash up that explains the amount is either greater or less than the answer that was entered.
Kid 20/20 has an activity, Coin Sort that students can play on-line. Students are given 280 seconds to properly place different coins in the corresponding piggy bank. Each piggy bank is labeled
with either pennies, nickels or dimes on the side. Students must click on each coin and drag it to the proper piggy bank. If the coin is taken to the correct piggy bank then the coin will disappear
and the value of the coin will be added to the amount already in the piggy bank. Students can visually watch as the amount increases by either one, five or ten cents.
Additional Resources:
The United States Mint has a great web-site that teachers can use for various reasons. The site contains ideas for lesson plans, coin programs which give detailed information about each coin and
coin curricula. Teachers can also use the site for class activities: game centers, web gadgets (worksheets), learning centers (ways to bring in different areas of the curriculum and financial
Scholastic has a great web-site for teachers. Teachers can download different activities from worksheets, foldables, mini-books, and even lesson plans. Click on “Teachers Resources” and select
lesson plans, printables or mini-books. Narrow each search by selecting 1st grade, math and then money from each category on the left hand side of the screen. *This web-site requires a yearly paid
Teaching Money Skills by Grade Level: First Grade is an article that teachers can read prior to teaching first grade students about money. The article provides a review and instructional method for
teaching a lesson or unit on money. The article recommends that teachers use play/fake money with their students for a hands-on learning experience. After the unit lesson on money, students should
be prepared for second grade math: addition and subtraction of money.
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LOS ANGELES PIERCE COLLEGE ELECTRONICS PROGRAM
ELECTRONICS 72A AND 72B
DIGITAL CIRCUITS 1 LECTURE AND LAB
Room: VLGE 8109
Instructor: Ken Sharpe Office: VLGE 8110, Office Hours: TBA (will be posted by the second week of class)
Phone: (818) 719-6480 Email: sharpekj@piercecollege.edu
Text: "Digital Fundamentals" 10th ed., by Floyd
Electronics Department Web Page: http://info.piercecollege.edu/departments/electronics/
Sample Student Learning Outcomes for 72A:
1. Draw, recognize and analyze combinational logic circuits using system presentation tools including logic equations, truth tables, and logic diagrams.
2. Analyze sample sequential circuits including multivibrators, counters and registers.
3. Design synchronous sequential systems and validate the results using transition tables and timing diagrams.
Sample Student Learning Outcomes for 72B:
1. Build, and analyze combinational logic circuits using system presentation tools including logic equations, truth tables, and logic diagrams. Construct the circuit and measure logic levels.
2. Analyze a sample sequential circuit counter and/or register. Build and verify operation.
3. Design synchronous sequential systems and validate the results using transition tables and timing diagrams.
1. If you stop attending after the third week, it is your responsibility to drop the class to avoid a failing grade. No incomplete grades are given. You must take the final exam at the time and day
scheduled. Students missing more than 3 classes may be excluded by the instructor.
2. A quiz may be given once a week covering both lecture and lab material, so please prepare accordingly.
3. NO MAKEUP QUIZZES. Missed quizzes count as zero.
4. The lab grade is the same as the lecture grade. All labs must be completed and lab work signed by the instructor for credit. Each student will build his/her own circuits. All lab work must be done
in the scheduled lab during the published lab hours for Electronics 72B.
4. Grading: Weekly quizzes: 75% Final exam: 25% . Subtract 2% from final average for each missing homework assignment. Assignments are due at the beginning of the lecture period - late work is not
accepted. Homework must be 70% correct for credit. Incomplete grades are not given.
6. Keep all quizzes and handout materials. Please read the attached course outline for a list of topics. Many textbooks are available that cover the basics of Digital Electronics so be aware that
finding supplemental material in libraries, book stores, or the internet may be helpful to you.
7. Please turn off all phones and pagers during lecture. During lecture classes, please sit in the chairs and not at the lab tables. Do not enter the stockroom/office area unless invited by a staff
8. Please consult the Pierce College Electronics Department web site for catalog descriptions, course outlines, copies of syllabi and other important information.
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Einstein’s Theories About Time
Although Albert Einstein lacked the impressive educational background that most other famous philosophers and physicists had, he is still regarded as the “Father of Physics” and a renowned genius. He
was born in Württemberg, Germany, on March 14, 1879. Among his many theories are a few related to time. We at OnlineClock.net dedicate this blog post to Einstein’s theories of time and how they were
In 1905, Einstein proposed the theory of special relativity, outlined in a paper he wrote about the electrodynamics of moving bodies. One of the most well-known equations explaining the theory is
well-recognized: E = mc². The equation shows that small amounts of mass may be rendered into larger amounts of energy. This theory reflects the idea that every uniform motion is relative, with no
absolute resting state. Included in this concept are the laws of physics, mechanics and electrodynamics. In this theory, the speed of light is equal to all observers, regardless of the source’s
motion. In a Stanford University report published by SLAC Accelerator Laboratory for the U.S. Department of Energy Office of Science, the relation of this theory is connected to time also, causing a
form of “time dilation.”
In the time dilation aspect, as reported by Stanford University, the processes of particles have their own internal clock that is the determining factor of the decay half-life process. From the
observer’s eye, in a moving frame the clock’s ticking rate is slower than the ticking rate of a static clock. The result of this shows that from the perspective of the observer, the half-life of
moving particles seems to be increased by the gamma factor. An example demonstrated in this report uses a particle called tau, created at SLAC.
From the reference frame showing the tau particle resting, its lifetime is about 3.05 x 10-13 s. Using the distance = speed x time equation, the distance it travels before decaying can be calculated.
Since it is so close to the speed of light, the equation c = 3 x 108 meters per second is used for particle speed. The result shows that the distance traveled is 9.15 x 10-5 meters, shown as d = v t,
d = (3 x 108 m/sec)( 3.05 x 10-13 s) = 9.15 x 10-5 m. However, when the tau particle is measured correctly, it actually travels further than that. This contradicting result is explained by time
The tau’s decay time can be represented as a moving clock; in relativity theory, moving clocks tick slower than static clocks (!).
Used to multiply the time travel of the taus, this fact shows the moving frame by gamma, giving the measurable time. Gamma is dependent upon the tau’s energy, but SLAC reports that a regular tau
particle has a gamma of 20. The measurable time multiplied by the approximate speed of tau results in the distance a high-energy tau may travel. Resulting is 20 x (9.15 x 10-5 m) = 1.8 x 10-3 m,
which is approximately 1.8 millimeters, which is nearly 20 times further than expected using the perspective of classical physics. SLAC also notes that time dilation is a real effect; the time
dilation effect is the reason that cosmic ray muons reach the earth’s surface before decaying.
In 1916, Einstein developed another theory stemming from special relativity. This new theory, called general relativity, stated that matter causes space to bend or curve. When space curves, it was
proposed that time also could curve. This is supposed to affect the inertial motion of other matter also. General relativity is based on the Equivalence Principle, defining the equivalence between
gravitational mass and inertial mass. Explained in a report published by the University of California in San Diego, the inertial mass determines the difficulty required to alter an object’s motion,
also the mass in Newton’s Second Law. Gravitational mass is determined by the strength of gravitational attraction between two objects. This results in the uniformity of gravitational acceleration,
stating that all objects, independent of mass, fall at the same rate.
Although others thought of this result as coincidence, Einstein was sure enough about it to consider it a solid principle. With this principle, matter in space is thought to curve. Opposing the view
of Newton, who proposed that gravity is a force, this theory states that gravity is a curve or bend in space time’s fabric, resulting in an alteration of time also.
Along with the belief that time could bend, Einstein believed that time was essentially “timeless;” it could go on indefinitely.
Other theories were developed from Einstein’s relativity theories. One well-known example is the Big Bang Theory, developed by a Russian meteorologist and mathematician named Alexander Friedmann. In
a publication from PBS, it is noted that Friedmann used Einstein’s theory of general relativity to show that time and space would continue indefinitely.
Since Einstein believed that time was really an illusion and was relative only to the perspective of the observer, he also contemplated the notion of time travel, but he didn’t seem to think it was
absolutely possible – or impossible.
In a PBS NOVA article, the notion of time travel as contemplated by Einstein was written. The article explains the relationship between time, space and distance, the way time and distance can change
with speed remaining the same. The speed of something is also different in relation to time from a person’s perspective, depending on where they are in relation to the moving object. For example,
when a person views a passing train up close, it seems to travel faster than if they viewed it from a distance, even if it is actually traveling the same speed. Time and space obviously have a
cohesive relationship in Einstein’s theories, which is why many people used Einstein’s relativity theories to propose the possibility of time travel. When Einstein died in 1955, as reported by
Science Ray, he did not fully believe in the idea of time travel, but he was still considering his proposed idea of wormholes.
In the space time continuum, a wormhole is a proposed shortcut through it. In a picture diagram, it would be shown as a four-dimensional object with a tube-like structure in the middle, allowing a
“shortcut” from one point in space time to another. In an astronomy guide published by Cornell University, it is clarified that wormholes are purely theoretical, formulated as a solution to
Einstein’s general relativity field equation. Although the idea of time travel is still being tested and analyzed, nobody has discovered a verifiable wormhole in space or method of time travel,
however the Philadelphia Experiment controversy remains in question as to possible time travel. If the reports from that experiment are true, time travel could be very dangerous to health. Some
scientists believe that there may be a “black hole” in space that is a wormhole, but it has not been verified.
A person need not be a scientist to ponder the idea of time not existing. ( A scary thought, for all of us at OnlineClock.net – would the world need our alarm clock if time didn’t exist?!)
Consider, as Einstein did, that time is merely something humans use to measure the days and organize their activities or commitments.
If the days were simply being measured in increments, and time did not actually move forward, time would stand still and people would simply “decay” as Einstein noted in his special theory of
Science definitely proves the idea that matter deteriorates or decays. Humans, plants and animals age, die and decay. Time continues on after each generation dies and there are no certain
documentations or proof of the beginning of time – much less an end to time.
So is Einstein’s theory preposterous or genius?
His I.Q. certainly ranked very high on the genius scale. If time and space continue on, then surely the idea of their cohesion together to be considered “space time” is not preposterous either.
Until future proof is discovered, some of Einstein’s theories will continue to be simply that — theories; but they definitely do provide us with an entirely different perspective of time.
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The natural work-stealing algorithm is stable
The Library
The natural work-stealing algorithm is stable
UNSPECIFIED. (2003) The natural work-stealing algorithm is stable. SIAM JOURNAL ON COMPUTING, 32 (5). pp. 1260-1279. ISSN 0097-5397
Full text not available from this repository.
In this paper we analyze a very simple dynamic work-stealing algorithm. In the work-generation model, there are n (work) generators. A generator-allocation function is simply a function from the n
generators to the n processors. We consider a fixed, but arbitrary, distribution D over generator-allocation functions. During each time step of our process, a generator-allocation function h is
chosen from D, and the generators are allocated to the processors according to h. Each generator may then generate a unit-time task, which it inserts into the queue of its host processor. It
generates such a task independently with probability.. After the new tasks are generated, each processor removes one task from its queue and services it. For many choices of D, the work-generation
model allows the load to become arbitrarily imbalanced, even when. lambda < 1. For example, D could be the point distribution containing a single function h which allocates all of the generators to
just one processor. For this choice of D, the chosen processor receives around λn units of work at each step and services one. The natural work-stealing algorithm that we analyze is widely
used in practical applications and works as follows. During each time step, each empty processor ( with no work to do) sends a request to a randomly selected other processor. Any nonempty processor
having received at least one such request in turn decides ( again randomly) in favor of one of the requests. The number of tasks which are transferred from the nonempty processor to the empty one is
determined by the so-called work-stealing function f. In particular, if a processor that accepts a request has l tasks stored in its queue, then f(l) tasks are transferred to the currently empty one.
A popular work-stealing function is f(l) = [l/2], which transfers (roughly) half of the tasks. We analyze the long-term behavior of the system as a function of. and f. We show that the system is
stable for any constant generation rate. < 1 and for a wide class of functions f. Most intuitively sensible functions are included in this class ( for example, every monotonically nondecreasing
function f which satisfies 0 less than or equal to f(l) less than or equal to l/2 and f(l) = omega(1) as a function of l is included). Furthermore, we give upper bounds on the average system load (
as a function of f and n). Our proof techniques combine Lyapunov function arguments with domination arguments, which are needed to cope with dependency.
Data sourced from Thomson Reuters' Web of Knowledge
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Théorie des matrices en information quantique
David Kribs
(University of Guelph),
Rajesh Pereira
(University of Guelph) et
Sarah Plosker
(Brandon University)
Private information retrieval is a cryptographic scheme that allows a client to secretly query a database. We show that information-theoretic single-server quantum private information retrieval
requires a linear amount of communication to be secure against specious adversaries, which are the quantum analog of honest-but-curious adversaries. We also stress the importance of adequate
comparison between classical and quantum adversaries---unfair comparisons might lead to an unjustified advantage for the quantum case.
Quantum key distribution uses public discussion protocols to establish shared secret keys. In the exploration of ultimate limits to such protocols, the property of symmetric extendibility of
underlying bipartite states $\rho_{AB}$ plays an important role. A bipartite state $\rho_{AB}$ is symmetric extendible if there exits a tripartite state $\rho_{ABB'}$, such that the $AB$ marginal
state is identical to the $AB'$ marginal state, i.e. $\rho_{AB'}=\rho_{AB}$. For a symmetric extendible state $\rho_{AB}$, the first task of the public discussion protocol is to break this
symmetric extendibility. Therefore to characterize all bi-partite quantum states that possess symmetric extensions is of vital importance. We prove a simple analytical formula that a two-qubit
state $\rho_{AB}$ admits a symmetric extension if and only if $tr(\rho_B^2)\geq tr(\rho_{AB}^2)-4\sqrt{\det{\rho_{AB}}}$. Given the intimate relationship between the symmetric extension problem
and the quantum marginal problem, our result also provides the first analytical necessary and sufficient condition for the quantum marginal problem with overlapping marginals.
We are interested in the behavior of typical quantum channels with large input space and small output space. We show that the output set of these quantum channels is almost deterministic, and
that it can be described through free probability techniques. We compute the minimum output entropy of these typical quantum channels, and as an application, we obtain new bounds for the
violation of the minimum output entropy. This is a report on joint works with S. Belinschi and I. Nechita.
The recent representation theory surrounding locally compact groups has initiated several new connections between harmonic analysis and quantum information. In this talk, we will use this theory
to generate two ``dual'' classes of quantum channels for every locally compact group. Focusing mainly on finite groups we will explore their properties from the point of view of quantum
information such as noiseless subsystems, quantum capacities and entanglement preservation. Time permitting, we will present further manifestations of this duality, in particular its connection
to the complementarity of private and correctable subsystems. This is joint work with Matthias Neufang.
Surface codes, introduced by Kitaev, are quantum error-correcting codes defined from a tiling of surface. First, we recall how the parameters of the surface code are related with the properties
of the tiling of surface. Then, we observe the similarities between quantum erasures and percolation theory. Using these similarities, we derive an upper bound on the percolation threshold of a
family of hyperbolic lattices from results of quantum information theory. This talk is based on joint work in progress with Gilles Zémor.
A matrix system $\mathcal S$ is a $*$-vector space of complex $n\times n$ matrices that contains the identity matrix. By a theorem of Choi and Effros, the dual space $\mathcal S^d$ of a matrix
system $\mathcal S$ has the structure of an operator system. In this lecture I will report on joint work with A. Kavruk, V. Paulsen, and I.G. Todorov whereby Tsirelson's problem on quantum
correlations is cast in terms of states on certain tensor products of dual matrix operator systems.
The evaluation of many important quantities in quantum information theory involves finding the solution to a convex optimization problem, usually in the form of minimizing a convex function over
a convex subset of hermitian matrices. For example, determination of the relative entropy of entanglement (REE) for an arbitrary quantum state $\rho$ amounts to minimizing the relative entropy of
$\rho$ with respect to the convex set of separable states. While finding closed fomulae solutions to such convex optimization problems is usually impossible, solving the converse problem is often
instructive and enlightening in regard to the original problem. That is, given a family of convex functions and a state $\sigma$ on the boundary of a subset of hermitian matrices, we can find all
functions whose minimum value is achieved at $\sigma$. In particular, this allows us to determine explicit expressions for the REE and its variants, such as the Rains bound. This approach also
elucidates interesting facts about these quantities, such as, among others, that the Rains bound reduces to the REE when at least one subsystem is a qubit.
Multi-particle entanglement is an essential resource for a variety of quantum information processing tasks. Yet, despite an enormous amount of literature dedicated to its study, our current
understanding of it is still in its infancy. In this talk I will introduce a systematic classification of multiparticle entanglement in terms of equivalence classes of states under stochastic
local operations and classical communication (SLOCC). I will show that such an SLOCC equivalency class of states is characterized by ratios of homogenous polynomials that are invariant under
local action of the special linear group. I will then introduce a complete construction of the set of all such SL-invariant polynomials (SLIPs). The construction is based on Schur-Weyl duality
and applies to any number of qudits in all (finite) dimensions. In addition, I will introduce an elegant formula for the dimension of the homogenous SLIPs space of a fixed degree as a function of
the number of qudits. The expressions for the SLIPs involve in general many terms, but for the case of qubits can be written in a much simpler form.
The separability from spectrum problem asks for a characterization of the eigenvalues of the bipartite mixed states $\rho$ with the property that $U^*{\rho}U$ is separable for all unitary
matrices $U$. This problem has been solved when the local dimensions $m$ and $n$ satisfy $m = 2$ and $n \leq 3$. We solve all remaining qubit-qudit cases (i.e., when $m = 2$ and $n \geq 4$ is
arbitrary). In all of these cases we show that a state is separable from spectrum if and only if $U^*{\rho}U$ has positive partial transpose for all unitary matrices $U$. This equivalence is in
stark contrast with the usual separability problem, where a state having positive partial transpose is a strictly weaker property than it being separable.
A square matrix with non-negative entries, all of whose rows and columns sum to $1$ is called a doubly stochastic matrix. The set of such matrices of size $n \times n$ is denoted $\Omega_n$.
Doubly stochastic matrices are closely tied to majorization, a partial order on vectors in $\mathbb{R}^n$, a connection made explicit by the Hardy-Littlewood-Polya theorem. Majorization plays an
important role in quantum information, where it can be used to compare entanglement of two quantum states. In 1965, Perfect and Mirsky conjectured that the region of all possible eigenvalues of
all $n \times n$ doubly stochastic matrices (denoted $\omega_n$) would be the union of the regions $\Pi_k$ for $k \in \{ 1,2,\cdots ,n\}$, where $\Pi_k$ is the convex hull of the $k^{th}$ roots
of unity. They proved the conjecture for $n=1,2,3$. In 2007, Rivard and Mashreghi exhibited a counterexample for $n=5$. We prove the Perfect-Mirsky conjecture for $n=4$, and provide a new
conjecture for which Rivard and Mashreghi's example is not a counterexample. We also discuss some geometric interpretations of the problem of characterizing $\omega_n$.
In quantum information science, quantum gates acting on vector states are unitary transformations. It is desirable from the theoretical as well as practical point of view to decompose a general
unitary transformation into simple ones that are easy to control and implement. In this talk, we will describe some current research on this topic.
We consider a sequence of probes, sent to interact one by one with a fixed scatterer. Before interaction, the probes are independent, but they become entangled via the contact with the scatterer.
After a probe finishes interacting with the scatterer, a quantum measurement is performed on the probe. The measurement history, i.e., the collection of measurement outcomes, is a stochastic
process. We analyze the convergence and fluctuation properties of this process by linking its asymptotic evolution to spectral characteristics of the dynamics.
Majorization is a concept that emerges from the properties of stochastic matrices. The theory of majorization has been identified as an important tool in quantum information since its application
to the theory of entanglement by Nielsen. Later works have identified other areas of quantum information where it plays a role, including thermodynamics. In this talk, we describe the connection
between majorization theory and thermodynamics, with a summary of our recent results in this area. These include necessary and sufficient conditions for transitions between thermodynamic states
of quantum systems, under various conditions (with or without catalysts, costing or yielding work, small or large systems). Notably, our results imply the insufficiency of the traditional
formulation of the Second Law to decide the feasibility of state transitions.
Based on the work in http://arxiv.org/abs/1309.6586
In this work, we introduce the EPOSIC channels, a class of SU(2)-irreducibly covariant quantum channels. We show that if H and K are SU(2)-irreducible spaces then the EPOSIC channels from End(H)
into End(K) are the extreme points of the convex set of all SU(2)-irreducibly covariant channels from End(H) into End(K). We get a set of Kraus operators, the Choi matrix, a complementary
channel, and the dual map of EPOSIC channel. As an application of the EPOSIC channels, we get a new example of a positive map that is not completely positive. We obtain a bound for the minimal
output entropy of the tensor product of two SU(2)-irreducibly covariant channels. We also examine the E.B.T property of EPOSIC channels.
Majorization and trumping are two partial orders that have proved useful in entanglement theory. We show some relations between these two partial orders and generalized Dirichlet polynomials,
Mellin transforms, and completely monotone functions. These relations are used to prove a succinct generalization of Turgut's characterization of Trumping. This is joint work with R. Pereira.
One of the fundamental problems quantum information scientists concerned with, is whether one can design and construct a quantum device that transforms certain quantum states into other quantum
states. This task is physically possible if a specified quantum operation (transformation) of certain prescribed sets of input and output states can be found. The problem then becomes to
determine an existence condition of a trace preserving completely positive map sending $\rho_j$ to $\sigma_j$ for all $j$, for certain given sets of quantum states $\{\rho_1,\dots,\rho_k\}$ and $
\{\sigma_1,\dots,\sigma_k\}$. This is called the problem of state transformation. In this talk, recent results on this problem will be presented.
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Borlnad C++ eroor with if
First of all I do not know what is the "if function". Secondly before using some C++ statement please read a book on C++ how to write such a statement.
I think that the problem is that and is not defined as keyword in Borland C++. Change it to &&
Last edited on
Topic archived. No new replies allowed.
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Cardinals of transitive permutation groups acting on $\{1,\dots,n\}$
up vote 4 down vote favorite
Is there a nice description of all divisors of $n!$ which can be realized as cardinals of permutation groups acting transitively on $\{1,\dots,n\}$?
A necessary condition is of course that such a divisor is a multiple of $n$.
$n$ (cyclic) and $2n$ (dihedral) are always possible but, without mistake on my behalf, $3n$ [S:is not feasible unless $n\equiv 1\pmod 3$ (it can then be realized as a semi-direct product analogous
to the dihedral group).:S] can only be realized if $3$ divides the number of invertible integers modulo $n$.
gr.group-theory permutation-groups
9 This is almost certainly too difficult to answer in general. What you write about $3n$ is not true. There are transitive groups of order $3n$ for $n=9$ and $n=14$ for example. On the other hand,
there are none for $n=10$. The transitive groups have been enumerated for all $n \le 32$ by the way. Could you try asking a more specific question? – Derek Holt Apr 5 '12 at 12:23
Thank you for the correction: A semi-direct product needs of course divisibility by 3 of the number of invertible elements modulo $n$. I think you suggest that the answer is messy! – Roland Bacher
Apr 5 '12 at 13:25
add comment
1 Answer
active oldest votes
As Derek suggests in his comment, this question is too difficult to answer in general. However one could limit the question as follows: clearly if $K$ is a transitive permutation group
then $|K|$ divides $|M|$ where $M$ is a maximal transitive subgroup of ${\mathrm Sym}(n)$; thus we can ask about the cardinality of a maximal transitive subgroup $M$ of ${\mathrm Sym}(n)
The O'Nan-Scott theorem is the main tool here. Roughly speaking it asserts that such a subgroup $M$ is either imprimitive (and hence a wreath product, with order formula easy), or else it
is in a bunch of primitive families. Most of these families have a geometric description and, as such, it is easy to calculate their order.
up vote 3
down vote The `difficult' family in this regard is the family of primitive almost simple groups. In this case one basically needs an enumeration of the maximal subgroups of all almost simple
groups, which is a difficult problem but one which has received a great deal of attention.
Depending on what level of information you need, there are complete enumerations of maximal subgroups for many of the almost simples (although not all). However there are also some very
nice general statements about the possible sizes of maximal subgroups. One example is this paper by Martin Liebeck; there are many others like it (many by Liebeck and his collaborators).
add comment
Not the answer you're looking for? Browse other questions tagged gr.group-theory permutation-groups or ask your own question.
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Small Terms
Q: I have a very large polynomial that has several "small" variables (substantially less than unity) and I would like to expand it to lowest order in those small variables. For example:
The lowest-order expansion keeps only
Daniel Lichtblau (danl@wolfram.com) answers: You want to keep only those terms whose power-products are not divisible by those of other terms. These terms can be found using a Groebner basis of the
set of monomials.
After recovering the coefficients,
the lowest-order expansion is
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Re teaching genetics to undergrad non-scientists
moynihan at mbcl.rutgers.edu moynihan at mbcl.rutgers.edu
Sat Sep 4 09:27:36 EST 1993
A simple inexpensive tool I have found useful for demonstrating chromosome
reproduction and distribution in mitosis and meiosis is a couple packs
of ordinary playing cards.
In particular, if you use one suit to represent each haploid genome, with
one value representing each chromosome, you can get generate visual
representations that show that each parent contributes half, but that each
grandparent, while averaging half, contributes a variable number of chromosomes
to a particular individual, and so on. Because people are familiar with
playing cards they seem to have an easier time keeping track of the labels.
Still, I find it advisable to start with a small number of chromosome pairs,
and then increase the complexity. Many of my students appear to be paralyzed
by computations, but with simple examples the numbers of combinations can be
tabulated. Then you can point out the pattern of increase, and then calculate
the staggering number number of possible outcomes for heterozygous individuals
with n = 23.
Given enough playing cards, a class can work out simple examples in small
groups. If you want to show patterns of cards to larger groups of students,
try a tack-note type of glue stick (I don't have mine in front of me, and I
have forgotten the manufacturer, but it works like the the 3M post-it adhesive)
and stick the cards on the wall. If you want to take things farther you can
cut and past cards to make "mutants" or "recombinant" cards.
Have fun.
M. Moynihan
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Post a reply
dear everybody,
i'm donig MCA (Master in Computer Applications). there are so many math concepts which are required in computer programming. but i did maths only in 10th standard. its been many years. also, there
are many concepts which are in my current cirriculum too, like, computer based numerical methods, differential , integration etc. etc. i really want to be comfirtable in maths coz it will help me all
life in my field. please suggest me how should i go about it. thing is that, for my current curriculum, they assume i already know maths concepts of 12th standard. so, should i study maths of 12th
standard now, or is there any website also which can make me learn math concepts like a book.
i'll really appreciate if i'm given help.
many thanks,
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st: AW: Extracting the t-statistics from a regression output
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st: AW: Extracting the t-statistics from a regression output
From "Martin Weiss" <martin.weiss1@gmx.de>
To <statalist@hsphsun2.harvard.edu>
Subject st: AW: Extracting the t-statistics from a regression output
Date Thu, 8 Jul 2010 09:11:50 +0200
-ssc d parmest- is the ideal solution for this problem, as Eric points out.
If you want to take a look at the underlying arithmetic, try Maarten`s
-----Ursprüngliche Nachricht-----
Von: owner-statalist@hsphsun2.harvard.edu
[mailto:owner-statalist@hsphsun2.harvard.edu] Im Auftrag von Dani Tilley
Gesendet: Donnerstag, 8. Juli 2010 05:18
An: statalist@hsphsun2.harvard.edu
Betreff: st: Extracting the t-statistics from a regression output
After I -regress y x1 x2 x3, noconstant- I need to access the vector of
t-statistics for the null that beta_i=0. This corresponds to the third
column on
the regression output. I actually only need one of the t-stats but I don't
this simplifies the problem.
>From the help article, I know that e(b) extracts the coefficients but
find something similar for the t-stats.
Any help would be appreciated.
* For searches and help try:
* http://www.stata.com/help.cgi?search
* http://www.stata.com/support/statalist/faq
* http://www.ats.ucla.edu/stat/stata/
* For searches and help try:
* http://www.stata.com/help.cgi?search
* http://www.stata.com/support/statalist/faq
* http://www.ats.ucla.edu/stat/stata/
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Homogeneous coordinates
I am a newbie to the field of Digital Image Processing and Computer Graphics. I am reading about Geometry for 3D vision. I am finding very hard to understand the concepts of perspective projection
and homogeneous coordinates. I tried googling, various books, and I also read a similar post from one of the members of this forum, but could not get a satisfactory answer.
I have the following doubts and hope to have them cleared:
1. Why do we need homogeneous coordinates?
2. I also read that, points in the projective space are expressed in homogeneous coordinates (REF: Image Processing Analysis and Machine Vision. By, Milan Sonka, Vaclav Hlavac, Roger Boyle). Why
can’t we represent the points using the same Cartesian coordinates?
If there is/are any error(s) in the questions, I apologize. Please consider my case sympathetically.
Thank you.
1. Why do we need homogeneous coordinates?
You need them for affine transformations and projection. With an ordinary 3x3 matrix you can only apply linear transformations such as rotation, scaling and shearing. The point at (0, 0, 0) will
always remain at that position. A transformation matrix is essentially just a linear system of equations. A vector (x, y, z) transformed by a 3x3 matrix M is essentially just x*M.xAxis + y*M.yAxis +
z*M.zAxis. It’s ovious that you can’t have a translation if all the components of the vector are zero, so you need an extra term: x*M.xAxis + y*M.yAxis + z*M.zAxis + translation. By using homogeneous
coordinates, you include a w-component to the vector (which usually is an implied 1) in order to be able to include translation in the matrix: x*M.xAxis + y*M.yAxis + z*M.zAxis + w*M.translation. To
get to the actual 3d point in space, you should divide the resulting vector by it’s w-component. By making sure that w is always 1 this will save a divide. Therefore common calculations only involve
3d vectors with an implied w=1 and 3x4 or 4x3 matrices with an implied last row/column of (0, 0, 0, 1)
For perpective projections, you need to be able to divide. For a point in view space, the projected point on a 2d screen can be calculated by dividing x and y by z. This is an operation that is not
possible with matrices. However, using the properties of homogeneous coordinates, the division by z can be achieved simply by copying ‘z’ to ‘w’ using a 4x4 transformation matrix.
Note that while mathematically homogeneous coordinates are very well-defined, in computer graphics we merely use them as a convenience to be able specify the calculations that we need. Most computer
graphics applications will never use homogeneous coordinates to their full extent. Before perspectice transformation, ‘w’ will always equal 1, and the only place where an actual division by w takes
place is when projecting the points on a 2D surface before rendering. No one cares about the property that (1, 2, 3, 1) and (2, 4, 6, 2) are essentially the same point. In fact, having w=2 will break
most code as they simply assume that w=1, and it may not even be possible to explicitely store a w component.
Your coherent explanation has resolved all my doubts. Thank you very much, .oisyn.
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Items tagged with parameter
Hello experts..
The following is the IVP:
restart:Digits:=14:t0:=0.0:tN:=5000.0: N1:=5000;th:=evalf((tN-t0)/N1):
dsys1 :=diff(y(t),t)=y(t)*((1-y(t)/3-epsilon)-0.8*y(t)/(y(t)^2+0.5^2));
dsol1 :=dsolve({dsys1,ini1},var,numeric, output=listprocedure, abserr=1e-9, relerr=1e-8,range=0..1);
for i from 0 to N1 do t1[i]:=evalf(th*i):u1[i]:=evalf(dsolu(t1[i]));pt1[i]:=[t1[i],u1[i]]:
the above code is to plot y(t) against the time t for fixed epsilon
Now the question is how to plot epsilon against the time???
I do appriciated any comments
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A Chance to Write History: WWIII [Ongoing]
With 47 players and only 7 days remaining until the 30-day mark (at which point the tourney must be full), I've
reduced the number of players taking part from the original 98 to a more manageable 70.
The format remains virtually the same; there will still be a 16-player Medal of Honor Cup.
But in the main tournament play, the 7 fronts will not have 12 players, but 8 players (4 from each side).
Here are the official changes made in the Rules. The changes are clearly marked in
In this creative, community-driven tournament, 94 players will compete in 8 battles that make up our World War III.
Now reads,
In this creative, community-driven tournament, 70 players will compete in 8 battles that make up our World War III.
Each player on each side is randomly assigned a number.
Using the Sequence Generator at random.org, I will generate 7 "fronts" of 12 players each.
Each front has 6 players from the Alliance, and 6 players from the Confederation.
*Note that not all 98 players are taking part here: I will go into detail later about this.
Now reads,
Each player on each side is randomly assigned a number.
Using the Sequence Generator at random.org, I will generate 7 "fronts" of 8 players each.
Each front has 4 players from the Alliance, and 4 players from the Confederation.
*Note that not all 70 players are taking part here: I will go into detail later about this.
Each player will simultaneously play 6 1v1 games; one game against each opponent from the opposing faction.
Now reads,
Each player will simultaneously play 4 1v1 games; one game against each opponent from the opposing faction.
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College Algebra: Enhanced With Graphing Utilities
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DLESE Find a Resource
> Subject: Mathematics + Collections: DCC
The Fermilab Flora and Fauna Exhibit provides information on wildlife phenomena commonly found at Fermilab, in Illinois, such as: bird nests, beavers, buffalo, Canada geese, deer, fungi, lichens,
poison ivy, insects (including prairie insects in winter), red-tailed hawks, and woodchucks. A section on math patterns in nature explains what Fibonacci sequences, golden numbers and angles, and
fractals ...
Full description.
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Cumulative income/demand Percentage
Does anyone have any links where I can see graphs of either the cumulative demand or cumulative income. I would like for these graphs the percentage of population on the x axis? I understand it is
understand it is suppose to follow a
lorenz curve
and from it you can calculate the
Giini coefficient
which measures income inequality. Given all the talk about income inequality you would think such data would be easier to find.
|
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|
Cumulative income/demand Percentage
Does anyone have any links where I can see graphs of either the cumulative demand or cumulative income. I would like for these graphs the percentage of population on the x axis? I understand it is
understand it is suppose to follow a
lorenz curve
and from it you can calculate the
Giini coefficient
which measures income inequality. Given all the talk about income inequality you would think such data would be easier to find.
|
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Elements of Mathematics, Algebra, Volume 1
- Foundations of Secure Computation, NATO ASI Series, Series F: Computer & System Sciences , 2000
"... This article presents a theory of classes and inheritance built on top of constructive type theory. Classes are defined using dependent and very dependent function types that are found in the
Nuprl constructive type theory. Inheritance is defined in terms of a general subtyping relation over the und ..."
Cited by 15 (7 self)
Add to MetaCart
This article presents a theory of classes and inheritance built on top of constructive type theory. Classes are defined using dependent and very dependent function types that are found in the Nuprl
constructive type theory. Inheritance is defined in terms of a general subtyping relation over the underlying types. Among the basic types is the intersection type which plays a critical role in the
applications because it provides a method of composing program components. The class theory is applied to defining algebraic structures such as monoids, groups, rings, etc. and relating them. It is
also used to define communications protocols as infinite state automata. The article illustrates the role of these formal automata in defining the services of a distributed group communications
system. In both applications the inheritance mechanisms allow reuse of proofs and the statement of general properties of system composition. 1
- Proof and System-Reliability, Proceedings of International Summer School Marktoberdorf, July 24 to August 5, 2001, volume 62 of NATO Science Series III , 2002
"... The basic concepts of type theory are fundamental to computer science, logic and mathematics. Indeed, the language of type theory connects these regions of science. It plays a role in computing
and information science akin to that of set theory in pure mathematics. There are many excellent accounts ..."
Cited by 5 (1 self)
Add to MetaCart
The basic concepts of type theory are fundamental to computer science, logic and mathematics. Indeed, the language of type theory connects these regions of science. It plays a role in computing and
information science akin to that of set theory in pure mathematics. There are many excellent accounts of the basic ideas of type theory, especially at the interface of computer science and logic —
specifically, in the literature of programming languages, semantics, formal methods and automated reasoning. Most of these are very technical, dense with formulas, inference rules, and computation
rules. Here we follow the example of the mathematician Paul Halmos, who in 1960 wrote a 104-page book called Naïve Set Theory intended to make the subject accessible to practicing mathematicians. His
book served many generations well. This article follows the spirit of Halmos ’ book and introduces type theory without recourse to precise axioms and inference rules, and with a minimum of formalism.
I start by paraphrasing the preface to Halmos ’ book. The sections of this article follow his chapters closely. Every computer scientist agrees that every computer scientist must know some type
theory; the disagreement begins in trying to decide how much is some. This article contains my partial answer to that question. The purpose of the article is to tell the beginning student of advanced
computer science the basic type theoretic facts of life, and to do so with a minimum of philosophical discourse and logical formalism. The point throughout is that of a prospective computer scientist
eager to study programming languages, or database systems, or computational complexity theory, or distributed systems or information discovery. In type theory, “naïve ” and “formal ” are contrasting
words. The present treatment might best be described as informal type theory from a naïve point of view. The concepts are very general and very abstract; therefore they may
, 2002
"... This article follows the spirit of Halmos' book and introduces type theory without recourse to precise axioms and inference rules, and with a minimum of formalism. I start by paraphrasing the
preface to Halmos' book. The sections of this article follow his chapters closely. Every computer scientist ..."
Add to MetaCart
This article follows the spirit of Halmos' book and introduces type theory without recourse to precise axioms and inference rules, and with a minimum of formalism. I start by paraphrasing the preface
to Halmos' book. The sections of this article follow his chapters closely. Every computer scientist agrees that every computer scientist must know some type theory; the disagreement begins in trying
to decide how much is some. This article contains my partial answer to that question. The purpose of the article is to tell the beginning student of advanced computer science the basic type theoretic
facts of life, and to do so with a minimum of philosophical discourse and logical formalism. The point throughout is that of a prospective computer scientist eager to study programming languages, or
database systems, or computational complexity theory, or distributed systems or information discovery
"... Abstract This article presents a theory of classes and inheritance built on top of constructive typetheory. Classes are defined using dependent and very dependent function types that are found
in the Nuprl constructive type theory. Inheritance is defined in terms of a general subtypingrelation over ..."
Add to MetaCart
Abstract This article presents a theory of classes and inheritance built on top of constructive typetheory. Classes are defined using dependent and very dependent function types that are found in the
Nuprl constructive type theory. Inheritance is defined in terms of a general subtypingrelation over the underlying types. Among the basic types is the intersection type which plays a critical role in
the applications because it provides a method of composing program components.The class theory is applied to defining algebraic structures such as monoids, groups, rings, etc. and relating them. It
is also used to define communications protocols as infinite stateautomata. The article illustrates the role of these formal automata in defining the services of a distributed group communications
system. In both applications the inheritance mechanismsallow reuse of proofs and the statement of general properties of system composition. 1 Introduction The results presented here were created as
part of a broad effort to understand how to use computers to significantly automate the design and development of software systems. This is one of the main goals of the "PRL project " at Cornell1.
One of the basic tenants of our approach to this task is that we should seek the most naturally expressive formal language in which to specify the services, characteristics and constraints that a
software system must satisfy. If the formal expression of services is close to a natural one, then people can more readily use it. We also want to allow very compact notations for concepts used to
describe systems, and this effect is also a consequence of expressive richness. We have discovered that it is frequently the case that the system we have built to implement one formal language will
support an even richer one. So we have come to see our work as also progressively improving the reach of our tools.
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Computing a Pasture Rental Rate
Is there a simple and uniform method of figuring a rental rate for pasture and hay land? Probably not, but guidelines are available. Below are several methods you can use for computing a pasture
rental rate.
Several factors will influence the rental rate. Pasture rental rates vary according to the quality of stand, type of grass species, amount of timber, condition of the fences, availability of water,
and previous fertility practices on the pasture. Hay land rental rates depend mostly on the quality and age of the stand.
A rental rate can be based on:
• a return on investment in pastureland,
• forage value,
• rent per head per month,
• carrying capacity,
• rent per pound of gain.
Current Market Rates
Pasture rent per acre can be established by charging a rental rate similar to what others are charging. Average pasture rental rates by county and region of the state are shown in Information File
Cash Rental Rates. These rates are based on a survey that is conducted every spring.
Return on Investment
Another method is to compute a rental rate based on sale or market value of the pastureland. Pasture rent may range from 2.5 to 3.5 percent of market value. For example, pasture with a sale value of
$2,000 per acre will rent from $50 to $70 per acre ($2,000 x 2.5% to 3.5% = $50 to $70).
However, determining the market value of pastureland is difficult because pasture is seldom sold separately from the farm. Information Files Farmland Value Survey - Iowa State University and Farmland
Value Survey - Realtors Land Institute provide information on current pastureland values.
Forage Value
To compute a rental rate based on forage value, estimate the expected pasture or hay production per acre and multiply by either 25 percent of the price of grass hay during the grazing season for
pasture, or 35 percent of the price of hay for an established stand of hay. If the tenant supplied labor and machinery for establishing the hay crop and pays half of the seed and fertilizer costs,
then a rental rate equal to 50 percent of the value of the hay crop would be more appropriate. Use hay prices corresponding to the type and quality of the stand. Some typical pasture production
levels are shown in Table 1.
For example, assume a summer grass hay price of $100 per ton and an unimproved bluegrass pasture yield of from one to one and one-half tons per acre (Table 1). The rental rate per acre is from $25
($100 per ton x 25 percent x 1 ton per acre = $25.00) to $38 ($100 per ton x 25 percent x 1.5 tons per acre = $37.50).
An alfalfa/grass summer hay price of $120 per ton and an alfalfa/grass yield of from four to six tons per acre (Table 1) results in a rental rate per acre of from $168 ($120 per ton x 35 percent x 4
tons per acre = $168 per acre) to $252 ($120 per ton x 35 percent x 6 tons per acre = $252 per acre) for hay production.
Rent per Head per Month
With this method, the livestock owner pays rent according to the number of animals grazed and length of time the pasture is used. This is measured by computing the animal unit months (AUMs). An AUM
is the amount of forage required to support a 1,000 pound cow with a calf up to 4 months of age for one month. Table 2 can be used for figuring AUMs. For example, 10 cows (1,000 Ibs. each) and calves
pastured for three months equals 30 AUMs (10 x 1.0 x 3). Note that forage consumption normally parallels the weight of the animal.
Rent is figured by multiplying a rental rate per AUM by the number of AUMs. A rental rate per AUM can be figured by using the current hay price and the quality rating of the pasture. Four forage
quality ratings are shown in Table 3.
For example, let’s assume the pasture is brome (tallgrass) pasture. Also, assume the average grass hay price during the summer is $100 per ton. The rental rate per AUM is $20 ($100 x .20). If ten,
1,000 pound cows with calf by their side are pastured for three months, 30 AUMs of pasture are used during the summer. The rent is $600 (30 AUMs x $20 per AUM) for the summer.
Typical rates per AUM by county and region of the state are shown in Information File Cash Rental Rates.
Carrying Capacity
This method is based on the carrying capacity of the pasture. The rental rate per AUM is multiplied by the carrying capacity of the pasture in AUM’s per acre to estimate a pasture rental rate per
acre for the whole grazing season. The rental rate per AUM is computed by either multiplying the hay price during the grazing season by the pasture quality factor (Table 3), or by using a typical
rental rate from Information File Cash Rental Rates.
For example, a $100 grass hay price and a tallgrass pasture rating of .20 results in a rental rate per AUM of $20 ($100 x .20). A brome grass pasture may produce four AUMs per acre during the grazing
season (Table 1). Multiplying the rate per AUM by the AUMs per acre results in a rent of $80 per acre ($20 per AUM x 4 AUMs).
Rent per Pound of Gain
With this method, pasture rent is based on the added weight the livestock gain while they are on pasture. This approach is best suited for stocker and feeder cattle rather than beef cows. To
determine the rent payment, it is necessary for the cattle to be weighed or an average weight estimated before they are placed on pasture and after they are taken off pasture. This may not be
practical in some situations.
Gain from pasture forage can be valued at about two-thirds to three-fourths the feed costs of gain in a feedlot. In a normal year the value of gain of livestock on pasture is from 50 to 60 cents per
pound of gain. Rent is figured by multiplying the value of the gain by the total amount of gain.
For example, assume the average gain per animal is 1.25 pounds per day. The amount of gain for a month is 37.5 pounds (1.25 Ibs. x 30 days). If the rental rate is 50 cents per pound of gain, the
rental charge for a month is $18.75 per head (50 cents x 37.5 pounds).
, retired extension value added agriculture specialist, , extension economist, 515-294-6161,
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The Overlooked Phenomena in the Michelson-Morley Experiment
The Overlooked Phenomena in the Michelson-Morley Experiment
Paul Marmet
( Last checked 2012/06/03 - The estate of Paul Marmet )
We show that Michelson and Morley used an over simplified description and failed to notice that their calculation is not compatible with their own hypothesis that light is traveling at a constant
velocity in all frames. During the last century, the Michelson-Morley equations have been used without realizing that two essential fundamental phenomena are missing in the Michelson-Morley
demonstration. We show that the velocity of the mirror must be taken into account to calculate the angle of reflection of light. Using the Huygens principle, we see that the angle of reflection of
light on a moving mirror is a function of the velocity of the mirror. This has been ignored in the Michelson-Morley calculation. Also, due to the transverse direction of the moving frame, light does
not enter in the instrument at 90 degrees as assumed in the Michelson-Morley experiment. We acknowledge that, the basic idea suggested by Michelson-Morley to test the variance of space-time, using a
comparison between the times taken by light to travel in the parallel direction with respect to a transverse direction is very attractive. However, we show here that the usual predictions are not
valid, because of those two classical secondary phenomena, which have not been taken into account. When these overlooked phenomena are taken into account, we see that a null result, in the
Michelson-Morley experiment, is the natural consequence, resulting from the assumption of an absolute frame of reference and Galilean transformations. On the contrary, a shift of the interference
fringes would be required in order to support Einstein’s relativity. Therefore, for the last century, the relativity theory has been based on a misleading calculation.
1 - Assessment of the Problem.
The aim of the Michelson-Morley experiment ^(1-10) is to verify “experimentally” whether the time taken by light to travel a distance in a direction parallel to the velocity of a moving frame, is the
same as the time to travel the same distance in a perpendicular direction. The experiment is based on the assumption that the velocity of light is constant in an absolute frame considered at rest.
The Michelson-Morley apparatus ^(1) is illustrated on figure 1. After light is emitted by the light source, a central semi-transparent mirror M, splits the beam of light between two perpendicular
directions. The distance L traveled between point A (on mirror M) and point B on mirror M[2] is equal to the distance L between the point A on mirror M and point C on mirror M[1].
In our experiment, let us consider that light moves downward at velocity c, while the moving frame also moves down but at velocity v, as illustrated on figure 1. In order to verify the hypothesis
that the velocity of light is c with respect to an absolute frame of reference, (in opposition to a constant velocity equal to c in all moving frames), Michelson and Morley have calculated the time
interval taken by light to travel in the longitudinal direction (between A and B) compared with the time for light to travel in transverse direction between A and C. Therefore they suggested building
an interferometer to test their hypothesis as illustrated on figure 1. According to the Michelson-Morley predictions, who affirm that the optical distance traveled by light between each arms of the
interferometer must be different when in motion, consequently, there must be a drift of interference fringes on mirror M where the beams join together, when the apparatus is rotating. No such drift,
having the amplitude predicted by Michelson-Morley has ever been observed. Let us examine their calculation.
In the Michelson-Morley experiment, it is assumed that light travels at a constant velocity with respect to an absolute frame assumed at rest. In that experiment, Michelson and Morley calculate the
time t(A
Since the last term in brackets of equation (1) is larger than unity, light takes a longer time to complete that return trip, than when the frame velocity is zero. Therefore, light must travel an
extra distance between locations A
In equation (2), t[o] is the time taken by light to travel the distance 2L, when the frame velocity is zero. Also t[v], is the time when the frame velocity is v. We have t[o], is equal to:
When the light paths between locations A
Equation (4) gives the time interval t(A
According to Michelson and Morley, we have seen in equation (2) that, when light moves parallel to the frame velocity (between At[o](v^2/c^2), with respect to the system at rest. However, in equation
(5), when the direction of the frame is transverse to the velocity of light (between At[o]/2)(v^2/c^2). It is that difference of time interval between axes, which led Michelson and Morley to predict
that there should be a shift of interference fringes between the arms of the interferometer. From equations (2) and (5), we see that, between the parallel direction (A
There is a difference of distance DL, corresponding to the difference of time given in equation (6). That difference is equal to the velocity of light c, times the difference of time Dt (see equation
(6)). Therefore, the difference of distance traveled by light between the parallel and transverse arm of the interferometer, as given by equation (3) and (6) gives:
When rotating the interferometer through exchange lengths, giving a total path difference DL(rotation 90^o) between the two rotating perpendicular axes. Using equation (7), the difference of path
length DL(rotation 90^o) due to that rotation is:
According to Michelson-Morley, equation (8) gives the difference of distance traveled by light between the parallel and the transverse direction, when the apparatus is rotated by 90^o. Following
these calculations, the Michelson-Morley experiment was made and repeated by many researchers under various conditions and at different locations. Most importantly, it was observed experimentally
that the observed shift of interference fringes was, if any, quite negligible, and therefore much smaller than predicted by Michelson and Morley. Consequently, scientists decided to consider some
esoteric hypothesis to explain these experimental observations. We show here that this Michelson-Morley’s demonstration is seriously over simplified. In fact we show below that the unexpected result
is due to an erroneous prediction.
2 - Reflection of Light on a Moving Mirror.
We show here that there are at least two crucial physical phenomena, which have been ignored in the Michelson-Morley calculation. The importance of these phenomena changes radically the
Michelson-Morley prediction. One of these phenomenon takes place on the reflected light on the semi transparent mirror M of the interferometer.
In the Michelson-Morley experiment, it is considered that light is reflected at 90^o because the mirror is at 45^o. However, we show here that it cannot be so, because of the proper velocity of the
mirror. Whenever a mirror possesses a velocity with respect to the stationary frame in which light travels at velocity c, we see here that the usual laws of reflection on moving mirrors are not
compatible with a constant velocity of light in that frame.
On figure 2, let us consider first the motion of mirror M at 45^o, moving in the same downward direction as the incoming light. The position of the mirror at time (t[m] =1) is represented by the
narrow line between the pair of labels (t[m]=1). At time t[w]=0, (labeled with four 0’s in the wavefront) we see the incoming wave. We consider light arriving progressively on mirror M. At time t[w]=
1, (labeled with 1's), we see that the incoming wavefront, just reaches the left hand side of mirror M. Since it takes time, for that wavefront, to move downward and reach the right hand side of the
mirror, the mirror M moves a short distance downward, while the wavefront of light is moving down much faster. The continuous progression of the wavefront on the mirror (while the mirror is moving
down) is illustrated in four steps. During each step, the (moving down) mirror is shown between each pair of labels t[m]=1, t[m]=2, t[m]=3, and t[m]=4.
Let us now consider the motion of the wavefront. The labels on each wavefront (t[w]=0, 1, 2, 3 or 4) are repeated at each individual quarter of wavefront. After the initial time t[w]=0, that same
wavefront is shown at different later times at: t[w]=1, t[w]=2, t[w]=3 and t[w]=4, (inscribed on each segment) during light propagation. All wavefronts (labeled t[w]=1, t[w]=2, t[w]=3 and t[w]=4 )
drawn on figure 2 correspond to the same wavefront at different times.
Let us consider the progression of the wavefront. At time t[w]=1, (on figure 2) the left hand side of the wavefront of light just reaches the left hand side (bold segment) of mirror M, (segmented in
four parts). At that time, the first segment (bold line) of the mirror is at mirror location t[m]=1. Then, the wavefront keeps moving down. At time t[w]=2, the second quarter of the same wavefront
reaches the second quarter of the mirror, (see bold segment at mirror location t[m]=2), which has then moved downward due to the velocity of the mirror. Similarly, at time t[w]=3, the third segment
of the wavefront reaches the third section of the mirror, (see bold segment at mirror location t[m]=3), which moved still further down during that time. Finally, at time t[w]=4, the reflection of the
wavefront on the mirror is completed after the fourth quarter of the wavefront is reflected on the fourth section of the mirror (bold segment at mirror location t[m]=4), which has moved still further
down due to the mirror velocity. Consequently, even if the mirror makes exactly an angle of 45 degrees with respect to the incoming wave, that wave is reflected by a mirror making effectively a
larger angle, because the mirror has the time to move down, during the time of reflection on the whole surface of the mirror. The “effective moving mirror” is illustrated on figure 2, as the sum of
the four bold segments of the moving mirror, formed by the wide set of narrow lines (crossed by three parallel lines), covering the average location of the four bold sections of the mirror. That
effective mirror makes an affective angle a (with respect to 45 degrees) as illustrated on figure 2. The angle a between the instantaneous and the effective moving angle of the mirror is shown
separately on figure 2 (bottom left). It can be shown that this angle a, represents half of the increase of the angle q of reflection of light due to the mirror velocity. However, here, the angle of
reflection of light will be calculated using a more direct method.
Let us calculate the change of angle of the reflected wavefront, after reflection, due to the velocity of the moving mirror. The projected width of the wavefront on mirror M is equal to P (see fig.
2). The “instantaneous” position of the mirror with respect to the wavefront is exactly 45^o. Since light is moving downward at a velocity (c-v) with respect to the moving frame, let us calculate the
time interval T[1] needed to reach the opposite edge of the mirror. Since the mirror is at 45^o, the vertical distance (of projection) P is the same. The time T[1] taken by light to travel the
vertical distance P is:
We can see that the change of distance P in the vertical direction, due to the motion of the mirror leads to a correction implying a higher power of v/c, which is negligible. During the same time T
[1], while light travels downward toward the right hand side of the mirror, the previously reflected light on the left hand side of the same mirror travels horizontally toward the right hand side.
The horizontal velocity of light is equal to c. Let us calculate that horizontal distance D, traveled by light at velocity c, during the same time interval T[1]. Using equation (9), we find:
The distance D is illustrated on figure 2. In equation (10), we take into account that the relative velocity of the mirror (which is the velocity of the Earth around the sun) is very small (i.e. 1/
10000) compared with the velocity of light. Higher powers of v/c are neglected when appropriate (as seen in equation (10).
Let us use the Huygens method of light propagation. From figure 2, we see that, after reflection on the upper left corner of the mirror, the Huygens wave method show that light has traveled a larger
horizontal distance D, than the X-coordinates “P” on the right hand side of the mirror. Therefore this produces the angle q on the reflected wavefront. From equation (10), we find that the additional
distance (D-P) is:
From figure 2 and equations (10) and (11), the tangent of angle q is:
Therefore, light is reflected at (90^o+q), when the static angle of the mirror at 45 degrees is moving at velocity v. Figure 2 also illustrates the wavefront (of the wave drawing) (label t[w]=4)
after total reflection by the moving mirror with the additional angle q due to the mirror having an effective angle a.
It is important to realize that the angle q also appears when the moving frame travels in different directions. Instead of having the mirror moving downward in figure 2, using the same method, we can
show that the increase of angle q of the wavefront is the same when the mirror moves toward the left hand side. Using the same method as above, we can also show that when the mirror is moving in a
direction opposite to the velocity of light (upward) or toward the right hand side, the effective angle of deflection of light then decreases by the angle q degrees.
The demonstration which shows that the change of angle of deflection of light reflected on a mirror moving in the transverse direction is given in the appendix of this paper.
3– Shifted Direction of Light in a Transverse Direction.
There is a second phenomenon which also has been ignored in the Michelson-Morley experiment.
Let us consider figure 3. Just as hypothesized by Michelson and Morley, figure 3 illustrates light moving at velocity c with respect to a stationary frame. After emission, that light forms circular
wavefronts around the instantaneous location of the emitter. Then, the circular wavefronts get bigger with time. However in the problem here, the “light source” is not stationary, but moves sideways
on Earth at the same time as the interferometer. On figure 3, we illustrate that both, the light source and the interferometer move at velocity v toward the right-hand side.
Let us consider a wavefront of light emitted at time t(-2). Of course, at the instant light is emitted, the mirror M of the Michelson-Morley interferometer, is located just below the light source,
where the interferometer is shown (ghost image). Two units of time later, at t(0), that spherical wavefront of light [emitted at t(-2)] reaches mirror M of the Michelson-Morley interferometer (new
location of the interferometer drawn with dark lines). Simultaneously, the light emitting source also moves toward the right hand side. Therefore, both the source and the interferometer still have
similar relative positions (same vertical axis) as seen experimentally. This description corresponds to the Michelson-Morley experimental apparatus.
We see that light reaching location A on mirror M, originates from a location where the source of light was located two units of time previously. We see clearly that light makes an angle q with
respect to the Y-axis, in order to reach the mirror M at location A of the interferometer and beyond, (toward B’). Therefore due to the velocity of the frame, even if the source of light is
instantaneously always located exactly above the mirror M of the interferometer, it must be understood that light traveling toward mirror M[2], either can be considered to move at velocity c at the
angle q in the rest coordinates, or at velocity [c Cosq] along the Y axis of the moving coordinates. Of course, as seen on figure 3, these two calculations are indistinguishable. However, the
function Cosq has been ignored in the moving frame by Michelson and Morley. This will be taken into account below in figure 5.
Let us recall that the M-M calculation is a completely classical calculation (not relativistic). Recalling that the M-M calculation is totally classical, "with an absolute frame" whatever the
observations of the moving observer are, let us consider again how much time light takes to travel from point A on mirror M to mirror M[2] (figure 3), independently of the location on the surface of
mirror M[2]. We see that even if the observer in the moving frame perceives that light moves along his moving Y axis, the time interval taken by light to travel that distance is (L/Cosq)/c (because
the frame is moving). Since this is a classical calculation, there is no space-time distortion involved in that calculation, as expected for the M-M calculation.
It seems that we are so much involved with relativity theory, that we sometimes overlook "when" exactly we must apply relativity or classical physics. It is not because the moving observer cannot see
directly the angle q that the time interval between mirrors M and M[2] is changing! The transit time is longer because, in the M-M experiment, light traveling from mirrors M to M[2] must necessarily
travel at the angle q.
A similar phenomenon happens when two fast cars, emitting sounds in stationary air, are moving parallel. The observer in both cars will detect sounds, apparently coming from a direction perpendicular
to their velocity, but the time interval taken by sound before reaching the opposite car increases as (L/Cosq) with the velocity of the cars.
The fact that the light path reaching the interferometer makes an angle q with respect to the observed direction inside the moving frame is related to another well known phenomenon, discovered by
Bradley ^(11) in 1725. This phenomenon explains how an observer in the moving frame can see light coming from a direction which is parallel to the Y-axis, even if in fact, the light source is an
angle q. Consequently, it becomes obvious that light takes more time to travel between mirrors M and M[2] than when the frame is at rest. This is taken into account below.
4 - Application to the Michelson-Morley Apparatus.
Figure 4, represents the Michelson-Morley apparatus moving in a direction parallel to the velocity of light. On figure 4-A, the interferometer moves downward away from the light source.
The velocity of light is c in the background frame at rest. Therefore, as in the Michelson-Morley apparatus, both, the source of light and the interferometer move with respect to that background. On
figure 4-A, the emitted wavefront expands and form circles around the instantaneous position where the light source is located at the moment of emission. On figure 4, half the light is reflected from
location A on mirror M, toward mirror M[1], and the other half is transmitted through mirror M, toward mirror M[2]. Light paths are illustrated as bold dashes on a narrow line.
Figure 4 shows the moving interferometer and the wavefront as seen from the rest frame, at time t=0, at the instant light, emitted earlier, reaches the mirror M of the interferometer. That light was
previously emitted at time t=-1. On figure 4-A, the frame is moving downward. It is moving upward on figure 4-B.
Figure 4-A illustrates light emitted from the source at time t(-1). Later, at time t(0) we see that the frame has moved down. Then, at time t(0), the wavefront, which was emitted at t(-1), forms a
circle just reaching location A on mirror M of the Michelson-Morley interferometer. As illustrated on figure 4-A, when light moves through mirror M toward location B on mirror M[2], the velocity of
the frame is parallel to the velocity of light. Light passes directly from A on mirror M toward point B on mirror M[2]. We have seen above in equation (2) that when the moving frame moves parallel to
the velocity of light, the time taken by light between mirrors is equal to:
However, in the case of light reflected on mirror M toward mirror M[1], we have seen above on figure 2 that, due to the velocity of the mirror, light is not reflected at 90^o. As demonstrated in
section 2, (equation 12), that light is reflected at an additional angle q. Therefore light is not traveling from Aq instead of the horizonal path. The relationship between the distance A
Using a series expansion of Cos q, we get from equation (14)
Since the times t(A
which is equal to:
Substituting equation (5) in equation (17) gives:
which is equal to:
Equation (19) shows that the time for light to travel, in the transverse direction along A
Let us also consider the case when light and the observer’s frame are moving in the opposite direction, as illustrated on figure 4-B. Consequently, as explained on figure 2, due to the proper upward
velocity of mirror M, the angle q of the beam of light is in the opposite direction, with respect to the X-axis, compared with when the frame is moving downward. As a consequence of this shift in
light direction by the angle q, we get the same increase of distance in the direction of q and the same time interval as calculated in equation (19). Therefore, the total time interval t(A
5 – Moving Frame in the Transverse Direction to Incident Light.
We now use figure 5 to see the trajectory of light entering the interferometer moving in a transverse direction, at velocity V, with respect to the light source.
On figure 5-A, we see the light source at time t(-1), so that the spherical wavefront of light reaches the mirror M of the interferometer at time t(0). We see that, at the moment light reaches mirror
M, the source of light has now moved to location t(0). However, the new light just emitted at t=0 did not have the time to reach the interferometer yet. Light reaching the interferometer must always
be emitted previously so that light has enough time to travel to the interferometer.
Figure 5 shows the moving interferometer and the wavefront as seen from the rest frame, at time t=0, at the instant light, emitted previously at (t-1), reaches the mirror M of the interferometer. On
figure 5-A, the frame is moving toward the right hand side. That frame is moving toward the left hand side direction on figure 5-B. We notice that at the instant t=0, the light source (t[o]) and
mirror M, (at location A) are located exactly in a direction along the Y-axis, just as explained on figure 4. This is necessary to satisfy the Michelson-Morley description that requires that the
light source must be located at 90^o with respect to the frame velocity.
Consequently, in that case, contrary to figure 4, due to the frame velocity, light cannot then move parallel to that Y-axis, due to the transverse motion of the frame. On figure 5, we have now a new
angle q, with respect to the Y-axis, due to the velocity of the moving frame. Therefore, since light reaching the interferometer comes from a transverse direction, light necessarily arrives on the
interferometer at the angle q, with respect to the Y-axis, as illustrated also on figure 3.
On figure 5-A, after reflection on the moving mirror M, light travels in the direction of M[1]. Due to the velocity of the mirror explained on figure 2, and equation (12), light is reflected on
mirror M, with an angle of incidence (with respect to the mirror surface) which is no longer 45^o, but it is (45^o-q). Therefore, the angle of reflection is also (45^o-q). This is illustrated on
figure 5-A as the direction of the dashed line Aq. Consequently, the moving mirror reflects light in the direction of the X-axis along the path A
We must notice that due to the rotation of the apparatus, the notation (label) A
Let us now study on figure 5-A, light moving through mirror M, between mirrors M and M[2]. We have seen above that due to the transverse velocity of the moving frame with respect to light, light
reaching mirror M makes angle q with respect to the Y-axis.
That angle q is needed, to be compatible with the Michelson-Morley description which has located the light source exactly on the Y-axis. This is similar to figure 3 when the light source on the
Y-axis produces a light beam making an angle q with respect to the Y-axis. This is different of figure 4, in which case, a light source (at t[o]) on the Y-axis produces a light beam parallel to the
Y-axis. Therefore, light between mirrors M and M[2] travels between A
However, light does not travel exactly along direction Aq, it takes a longer time to complete that new path. Let us calculate the time taken by light, between Aq. Using figure 5, we find that the
relationship between the distance A
Using a series expansion of Cos q, equation (22) gives:
Since the time t(A
Equation (24) gives:
Considering the rotation, t(A
Equation (26) gives:
Equation (27) gives the time for light to travel between A
Using a similar demonstration as above, we can see on figure 5-B, that, when the direction of motion of the moving frame is reversed, the time for light to travel between A^o+q) due to the velocity
of the moving frame moving toward the left hand side direction. The light would be reflected along the direction Aq. Therefore, the reflected beam of light is reflected along the X-axis. This is
similar to the problem calculated on figure 5-A.
In the case of light moving downward on figure 5-B, the angle q with respect to the Y-axis is similar to the problem in figure 5-A. There is only a change of sign of the angle q. Consequently, the
time taken by light to travel the distance A
Consequently in all cases, the time taken by light to travel between mirrors is always the same.
6 - Analysis of the New Results.
We have shown here that, in the Michelson-Morley experiment, using classical physics, the time for light to travel between any pair of mirrors, in any direction, is always the same, independently of
the direction of the moving frame and also independently of having light moving either parallel or transverse to the frame velocity. In any direction, that time interval Dt between mirrors is always
equal to:
More specifically, in equations (20) and (27), when the velocity of the frame is perpendicular to the direction of light penetrating in the instrument, we have shown that the times for light to
travel between the horizontal and vertical mirrors are identical. We have shown that this is always true whether the frame is moving toward the right hand direction or the left hand direction.
Furthermore we have seen above in equations (13) and (19), that when light penetrates into the instrument in a direction parallel to the frame velocity, the times for light to travel between the
parallel or transverse arm of the interferometer are also always identical. We have also shown that this is always true, whether the frame is moving parallel or anti-parallel with respect to the
velocity of light. We must conclude that the times taken by light to travel between any pair of mirrors are always the same, independently of any rotation of the interferometer in space.
Therefore, according to classical physics, the rotation of the Michelson-Morley apparatus in space should never show any drift of interference lines. On the contrary, a positive shift of interference
fringes with the amplitude compatible with the Michelson-Morley predictions is required in order to be compatible with Einstein’s relativity. Such a shift of interference fringes due to a rotation
has never been observed. The absence of an observed drift of interference fringes invalidates Einstein's relativity.
We have seen above that the prediction presented by Michelson and Morley are based on a model which ignores two important fundamental phenomena. These disregarded phenomena are the law of reflection
of light on the moving mirror and also the deviation of the observed direction of light coming from a moving system.
Relativity theory, astrophysics, and most of modern physics in the 20^th century has been based on the belief that a null result in the Michelson-Morley experiment is an argument in favor of
relativity theory. We see now that the contrary is true. An enormous amount of human effort and an unbelievable amount of money for research has been based on that erroneous prediction published in
1887. It is inconceivable that the original demonstration has never been seriously reconsidered. This is the result of an extremely dogmatic attitude of the physics establishment against a few
scientists whose status were threatened and even ruined because they dared to reconsider some fundamental principles of physics.
It is also important to mention that the non-zero result observed in the Michelson-Morley experiment does not provide any proof of existence of ether. The presence of ether appears totally useless,
when an appropriate model is used. Without matter nor radiation, space is nothing. Other experiments^(12-17) have already shown that everything in physics can be explained using classical physics
without the ether hypothesis.
We acknowledge that, the basic idea suggested by Michelson-Morley to test the variance of space-time, using a comparison between the times taken by light to travel in the parallel direction with
respect to a transverse direction is very attractive. However, this test is not valid, because there are two classical secondary phenomena, which have not been taken into account. Just one year
before the commemoration of the 1905 Einstein’s paper, we must realize that the relativity theory relies on a ghost experiment.
The calculations above do not include all the possible physical mechanisms that can possibly perturb the light path in the Michelson-Morley apparatus. However, we strongly suspect that all other
mechanisms produce effects, which are enormously smaller than the phenomena overlooked by Michelson and Morley. Of course, we have seen in this paper that there exists a fourth order term (v^4/c^4),
that has been neglected here. This high order term is much too small to be observed. We can also mention the Fizeau effect, which is known to drag light traveling in a moving medium as a function of
the index of refraction. The empirical equation of the Fizeau effect is known in the case of a medium moving parallel to the direction of light. We have verified that these other phenomena also make
a negligible contribution to an assumed drift of fringes. However, that Fizeau drag phenomenon seems to be totally unknown when the medium moves perpendicular to light velocity. Finally, the
misalignment of the mirrors of the interferometer might also have some effect on the fringes observed^(18) in the Michelson-Morley experiment.
It is important to recall that the overlooked phenomena described here also have important implications in other fundamental experiments^(19) in relativity. For example, in the Lorentz transformation
^(19), which usually predicts length contraction along the velocity axis of moving matter with respect to the transverse axis, it has been shown that the predictions are also in error, due to a
secondary phenomenon explained in this present paper. We know also that the Brillet and Hall experiment ^(20) is also a test for the anisotropy of space. The Brillet and Hall experiment ^(20) has
also been carefully studied and similarly, it has been shown ^(21), that a corresponding phenomenon is changing the light path inside a Fabry-Pérot etalon. Consequently, in that case again, the null
change of frequency observed experimentally, corresponds to an absolute frame of reference, while an anisotropic relativist space would require an observed shift of frequency.
Reflection of Light on a Mirror Moving in a Transverse Direction.
On figure A, let us consider the motion of mirror, moving horizontally when light moves downward. The initial position of the mirror at time (t[m] =1) is represented on figure A by the narrow line
between the pair of labels (t[m]=1) at the moment the wavefront reaches the left hand side of the mirror. The continuous progression of the wavefront on the mirror, while the mirror is moving to the
right hand side, is illustrated in four steps. During each step, the sideways moving mirror is shown moving toward the right hand side direction, between each pair of labels t[m]=1, t[m]=2, t[m]=3,
and t[m]=4.
Let us now consider the motion of the wavefronts. The labels on each wavefronts (t[w]=0, 1, 2, 3, 4 and 5) are repeated at each individual quarter of wavefront. After the initial time t[w]=0, that
same wavefront is shown at different later times at: t[w]=1, t[w]=2, t[w]=3, t[w]=4 and t[w]=5 during light propagation. All wavefronts on figure A correspond to the same wavefront at different
Let us consider the progression of a wavefront. At time t[w]=1, (on figure A) the left hand side of the wavefront #1 of light just reaches the left hand side (bold segment) of mirror M. At that time,
the first segment (bold line) of the mirror is at mirror location t[m]=1. Then, the wavefront keeps moving down. At time t[w]=2, the mirror, is still moving to the right hand side, and the second
quarter of the same wavefront reaches the second quarter of the mirror.
However, due to the motion of the mirror toward the right hand side, the wavefront is reaching the mirror at an earlier time, as can be seen on figure A. Similarly, at time t[w]=3, the third segment
of the wavefront reaches the third section of the mirror, (see bold segment at mirror location t[m]=3), which moved still further to the right hand side. Finally, at time t[w]=4, the reflection of
the wavefront on the mirror is completed after the fourth quarter of the wavefront is reflected on the fourth section of the mirror (bold segment at mirror location t[m]=4), which has moved still
further to the right due to the mirror velocity.
Consequently, even if the mirror makes exactly an angle of 45^o with respect to the incoming wave, that wave is reflected by a mirror making effectively a different angle, because the mirror has the
time to move to the right hand side, during the total time of reflection on the whole surface of the mirror.
The “effective mirror” is illustrated on figure A as the sum of the four bold segments of the mirror, which makes an affective angle q. The effective angle of that moving mirror is illustrated on
figure A, by the wide set of narrow lines (crossed by three parallel lines), covering the average location of the four bold sections of the mirror.
The angle a between the real and the effective angle of the mirror is shown separately on figure A (bottom left). It can be shown that this angle a, represents half of the increase of the angle q of
reflection of light due to the mirror velocity. However here, the angle of reflection of light is calculated using the Huygens’ principle as seen in section 2 above.
Let us calculate the angle of the reflected wavefront, taking into account the velocity of the moving mirror. The projected width of the wavefront on mirror M is equal to P (see fig. A). The
“instantaneous” angle of the mirror with respect to the wavefront is exactly 45^o. However, since the mirror is moving to the right hand side, while the wavefront moves downward, light will not reach
the opposite side at the same time as when the mirror is stationary. In fact, since the mirror is moving, we see on figure A that, compared with a stationary mirror, light reaches the mirror at an
earlier time, at location H. Light travels only from G to H.
At rest, the vertical and horizontal components of the mirror at 45^o are equal to P.
We see that, since the angle of the moving mirror is 45^o, the horizontal distance (J-H) is equal to the vertical distance (H-K).
The time interval T[1] is equal to the time for light to travel the distance P. During that time interval, the mirror M travels the distance J-H. Therefore, using equation (1A), we have:
Equation (2A) gives:
Equation (3A) shows that the distance traveled by light is shorter, before hitting the mirror at location H, when the mirror is moving. Let us compare this time interval for light travel, with the
distance A-G, after a Huygens reflection on point A). Using the Huygens principle again, the wavefront re-emitted at location A cannot reach location G (after traveling distance P) during the same
time interval light reaches the mirror at H. Due to the mirror motion, (G-H) is now shorter than (A-G). After the Huygens reflection on point A, light travels only the shorter distance (A-F).
Therefore we have:
Therefore after reflection, at the moment light is finally reflected at location H (forming the wavefront #4), the reflected wavefront makes an angle q with respect to the vertical axis. This gives:
A moment later, the wavefront #5 escapes from the surface of the mirror at the angle q. One must conclude that light reflected on a moving mirror makes an extra angle of reflection q, as given in
equation (5A). Using the same demonstration, we see that changing the direction of the velocity v of the mirror also changes the sign of the angle q. This demonstration explains the behavior of light
on figure 5.
7 - References
1 - Albert A. Michelson, and Edward W. Morley, The American Journal of Science, “On the Relative Motion of the Earth and the Luminiferous Ether”. No: 203, Vol. 134, P. 333-345, Nov. 1887.
2 - W. M. Hicks, Phil. Mag. Vol. 3 , 9, (1902)
3 - D. C. Miller "The Ether-Drift Experiment and the Determination of the Absolute Motion of the Earth" Rev. Mod. Phys. Vol 5, 203, (1933)
4 - E. W. Morley and D. C. Miller, "Report on an Exp/riment to Detect the FitzGerald-Lorentz Effect." Phil. Mag. 9, 669, (1905)
5 - M. Consoli and E. Costando, ”The Motion of the Solar System and the Michelson-Morley Experiment”
in: http://www.arxiv.org/pdf/astro-ph/0311576, 26 Nov 2003
6 - Raymond A. Serway, Clement J. Moses, and Curt A. Moyer, Modern Physics, Saunders Golden Sunburst series, Saunders College Publishing, London (1989), ISBN 0-03-004844-3
7 - Miller, D. C. Ether-drift experiments at Mount Wilson, Proc. Nat. Acad. Sci. 11(1925) 306-314.
8 - Miller, D. C. The ether-drift experiment and the determination of the absolute motion of the earth, Rev. Mod. Phys. 5, (1933) 203-242.
9 - Piccard, A. and Stahel, E., Nouveaux résultats obtenus par l’expérience de Michelson, Compte rendus 184, (1927) 152.
10 - Kennedy, R. J., A Refinement of the Michelson-Morley experiment, Proc. Nat. Acad. Sci. 12, (1926) 621-629.
11 - James Bradley, “Aberration of light” at: http://brandt.kurowski.net/projects/lsa/wiki/view.cgi?doc=563 (This web page does not seem to be available anymore.)
12 - P. Marmet, Einstein's Theory of Relativity Versus Classical Mechanics, 200 p. Ed. Newton Physics Books, Ogilvie Road, Gloucester, Ontario, Canada, K1J 7N4
Also on the Web at: http://www.newtonphysics.on.ca/einstein/index.html
13 - P. Marmet, "Classical Description of the Advance of the Perihelion of Mercury" in Physics Essays Volume 12, No: 3, 1999, P. 468-487. Also on the Web at:
14 - P. Marmet, “Natural Length Contraction Mechanism Due to Kinetic Energy”, Journal of New Energy, ISSN 1086-8259, Vol. 6, No: 3, pp. 103-115, Winter 2002. Also on the Web at:
15 - P. Marmet, “Natural Physical Length Contraction Due to Gravity”.
16 - P. Marmet, “Fundamental Nature of Relativistic Mass and Magnetic Fields”.
Invited paper in: International IFNA-ANS Journal "Problems of Nonlinear Analysis in Engineering Systems" No. 3 (19), Vol. 9, 2003 Kazan University, Kazan city, Russia.
17 - P. Marmet, Einstein’s Mercury Problem Solved in Galileo’s Coordinates, Symposium “Galileo Back in Italy” Bologna, Italy, P. 335-351, May 26-29,1999,
18 - Héctor Múnera, Centro Internacional de Física, Michelson-Morley Experiment Revisited: Systematic Errors, Consistency among Different Experiments and Compatibility with Absolute Space. A. A.
251955, Bogotá D.C. Columbia, APEIRON, Vol 5, 37, (1998).
Also at: http://www.newtonphysics.on.ca/faq/michelson_morley.html
19 - P. Marmet, “The Collapse of the Lorentz Transformation” P. Marmet, To be published.
On the Web at: http://www.newtonphysics.on.ca/lorentz/index.html
20 – A. Brillet, and J. L. Hall, “Improved Laser Test on the Isotropy of Space”, Physical Review Letters, Vol. 42, No: 9, (February 26, 1979).
21 – P. Marmet “Design Error in the Brillet and Hall Experiment” to be published.
On the Web at: http://www.newtonphysics.on.ca/brillet_hall/index.html
To be published in Galilean Electrodynamics.
Ottawa, Original paper May 30, 2004
Updated Nov. 30, 2004.
Return To: List of Papers on the Web
Contact e-mail address.
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Journal of the Optical Society of America B
Performing reliable measurements in optical metrology, such as those needed in ellipsometry, requires a calibrated source and detector, or a well-characterized reference sample. We present a novel
interferometric technique to perform reliable ellipsometric measurements. This technique relies on the use of a nonclassical optical source, namely, polarization-entangled twin photons generated by
spontaneous parametric downconversion from a nonlinear crystal, in conjunction with a coincidence-detection scheme. Ellipsometric measurements acquired with this scheme are absolute, i.e., they
require neither source nor detector calibration, nor do they require a reference.
© 2002 Optical Society of America
OCIS Codes
(000.1600) General : Classical and quantum physics
(120.2130) Instrumentation, measurement, and metrology : Ellipsometry and polarimetry
(120.3940) Instrumentation, measurement, and metrology : Metrology
(190.0190) Nonlinear optics : Nonlinear optics
(270.0270) Quantum optics : Quantum optics
(350.4600) Other areas of optics : Optical engineering
Ayman F. Abouraddy, Kimani C. Toussaint, Jr., Alexander V. Sergienko, Bahaa E. A. Saleh, and Malvin C. Teich, "Entangled-photon ellipsometry," J. Opt. Soc. Am. B 19, 656-662 (2002)
Sort: Year | Journal | Reset
1. P. Drude, “Bestimmung optischer Konstanten der Metalle,” Ann. Physik Chemie 39, 481–554 (1890).
2. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).
3. H. G. Tompkins and W. A. McGahan, Spectroscopic Ellipsometry and Reflectometry (Wiley, New York, 1999).
4. A. Rothen, “The ellipsometer, an apparatus to measure thicknesses of thin surface films,” Rev. Sci. Instrum. 16, 26–30 (1945).
5. A. B. Winterbottom, “Optical methods of studying films on reflecting bases depending on polarisation and interference phenomena,” Trans. Faraday Soc. 42, 487–495 (1946).
6. M. Mansuripur, “Ellipsometry,” Opt. Photon. News 11(4), 52–56 (2000).
7. D. N. Klyshko, “Coherent decay of photons in a nonlinear medium,” Pis'ma Zh. Eksp. Teor. Fiz. 6, 490–492 (1967) [Sov. Phys. JETP Lett. 6, 23–25 (1967)].
8. S. E. Harris, M. K. Oshman, and R. L. Byer, “Observation of tunable optical parametric fluorescence,” Phys. Rev. Lett. 18, 732–735 (1967).
9. T. G. Giallorenzi and C. L. Tang, “Quantum theory of spontaneous parametric scattering of intense light,” Phys. Rev. 166, 225–233 (1968).
10. D. A. Kleinman, “Theory of optical parametric noise,” Phys. Rev. 174, 1027–1041 (1968).
11. D. N. Klyshko, Photons and Nonlinear Optics (Gordon and Breach, New York, 1988).
12. A. Zeilinger, “Experiment and the foundations of quantum physics,” Rev. Mod. Phys. 71, S288–S297 (1999).
13. E. S. Fry and T. Walther, “Fundamental tests of quantum mechanics,” in Advances in Atomic, Molecular, and Optical Physics, B. Bederson and H. Walther, eds. (Academic, Boston, 2000), Vol. 42, pp.
14. D. N. Klyshko, “Utilization of vacuum fluctuations as an optical brightness standard,” Kvant. Elektron. (Moscow) 4, 1056–1062 (1977) [Sov. J. Quantum Electron. 7, 591–595 (1977)].
15. A. Migdall, R. Datla, A. V. Sergienko, and Y. H. Shih, “Absolute detector quantum efficiency measurements using correlated photons,” Metrologia 32, 479–483 (1995).
16. D. Branning, A. L. Migdall, and A. V. Sergienko, “Simultaneous measurement of group and phase delay between two photons,” Phys. Rev. A 62, 063808 (2000).
17. A. F. Abouraddy, K. C. Toussaint, Jr., A. V. Sergienko, B. E. Saleh, and M. C. Teich, “Ellipsometric measurements by use of photon pairs generated by spontaneous parametric downconversion,” Opt.
Lett. 26, 1717–1719 (2001).
18. A. K. Ekert, J. G. Rarity, P. R. Tapster, and G. M. Palma, “Practical quantum cryptography based on two-photon interferometry,” Phys. Rev. Lett. 69, 1293–1295 (1992).
19. A. V. Sergienko, M. Atatüre, Z. Walton, G. Jaeger, B. E. A. Saleh, and M. C. Teich, “Quantum cryptography using femtosecond-pulsed parametric down-conversion,” Phys. Rev. A 60, R2622–R2625
20. T. Jennewein, C. Simon, G. Weihs, H. Weinfurter, and A. Zeilinger, “Quantum cryptography with entangled photons,” Phys. Rev. Lett. 84, 4729–4732 (2000).
21. D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Nature 390, 575–579 (1997).
22. D. Boschi, S. Branca, F. De Martini, L. Hardy, and S. Popescu, “Experimental realization of teleporting an unknown pure quantum state via dual classical and Einstein-Podolsky-Rosen channels,”
Phys. Rev. Lett. 80, 1121–1125 (1998).
23. B. E. A. Saleh, S. Popescu, and M. C. Teich, “Generalized entangled-photon imaging,” in Proceedings of the Ninth Annual Meeting of the IEEE Lasers and Electro-Optics Society, P. Zory, ed. (IEEE,
Piscataway, N.J., 1996), Vol. 1, pp. 362–363.
24. M. B. Nasr, A. F. Abouraddy, M. C. Booth, B. E. A. Saleh, A. V. Sergienko, M. C. Teich, M. Kempe, and R. Wollenschensky, “Biphoton focusing for two-photon excitation,” Phys. Rev. A 65, 023816
25. A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Entangled-photon Fourier optics,” J. Opt. Soc. Am. B (to be published).
26. M. C. Teich and B. E. A. Saleh, “Observation of sub-Poisson Franck–Hertz light at 253.7 nm,” J. Opt. Soc. Am. B 2, 275–282 (1985).
27. M. C. Teich and B. E. A. Saleh, “Photon bunching and antibunching,” Prog. Opt. 26, 1–104 (1988).
28. M. C. Teich and B. E. A. Saleh, “Squeezed and antibunched light,” Phys. Today 43(6), 26–34 (1990).
29. P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337–4341 (1995).
30. A. V. Sergienko, Y. H. Shih, and M. H. Rubin, “Experimental evaluation of a two-photon wave packet in type-II parametric downconversion,” J. Opt. Soc. Am. B 12, 859–862 (1995).
31. U. Fano, “Description of states in quantum mechanics by density matrix and operator techniques,” Rev. Mod. Phys. 29, 74–93 (1957).
32. A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Degree of entanglement for two qubits,” Phys. Rev. A 64, 050101(R) (2001).
33. R. A. Campos, B. E. A. Saleh, and M. C. Teich, “Quantum-mechanical lossless beam splitter: SU(2) symmetry and photon statistics,” Phys. Rev. A 40, 1371–1384 (1989).
34. R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev. 130, 2529–2539 (1963).
35. B. E. A. Saleh, A. F. Abouraddy, A. V. Sergienko, and M. C. Teich, “Duality between partial coherence and partial entanglement,” Phys. Rev. A 62, 043816 (2000).
36. R. J. Glauber, “Optical coherence and photon statistics,” in Quantum Optics and Electronics, Les Houches, C. DeWitt, A. Blandin, and C. Cohen-Tannoudji, eds. (Gordon and Breach, New York, 1965).
37. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).
38. D. N. Klyshko, “A simple method of preparing pure states of an optical field, of implementing the Einstein-Podolsky-Rosen experiment, and of demonstrating the complementarity principle,” Usp.
Fiz. Nauk 154, 133–152 (1988) [Sov. Phys. Usp. 31(1), 74–85 (1988)].
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FOM: recursive independent axiomatizations
Harvey Friedman friedman at math.ohio-state.edu
Thu Dec 18 05:54:58 EST 1997
This is a clarifiction to my e-mail of 6:02AM 12/17/97 which I reproduce
with a change in the proof of Theorem II. Sam Buss pointed out that the
proof is incomplete.
This concerns Tait, 9:08AM 12/13/97.
We prove that every r.e. extension of PA has an r.e. independent
Clarification: Any r.e. theory has a recursive axiomatization. Any theory
with an r.e. independent axiomatization has a recursive independent
THEOREM I. Let T be an r.e. theory in predicate calculus with equality, in
a recursively presented language. Then T has an infinite recursive
independent axiomatizaion if and only if there is a partial recursive
function which maps every consequence A of T to another consequence B of T
such that A does not logically imply B.
Proof: Let T be as given, and assume that T has an infinite recursive
independent axiomatization. Given any sentence A in the language of T, we
search for a proof using finitely many terms in the infinite recursive
independent axiomatization. If we find it then we output the next term
after the highest term among the finitely many terms found.
On the other hand, suppose f is a partial recursive function as described.
Let A1,... be any recursive enumeration of T. Define B1,B2,... as follows.
Take B1 = f(A1) & A1. Take Bi+1 = f(Bi) & Bi & Ai+1. Then B1,B2,... is a
recursive axiomatization of T where B1 is not logically valid, and for all
i, Bi+1 logically implies Bi and Bi does not logically imply Bi+1.
Now consider B1, B1 arrows B2, B2 arrows B3, etcetera. Clearly this is also
a recursive axiomatization of T. Now suppose B1 logically follows from
finitely many other terms. Note that every term other than B1 logically
follows from notB1. Hence B1 logically follows from notB1, and so B1 is
logically valid, which is a contradiction.
Now suppose Bi arrows Bi+1 follows logically from finitely many other
terms. The earlier terms all logically follow from Bi, and the later terms
all logically follow from notBi+1. So we have {Bi,notBi+1} logically
implies (Bi arrows Bi+1). Hence {Bi,notBi+1} logically implies Bi+1.
Therefore Bi logically implies Bi+1, which is a contradiction. QED.
THEOREM II. Every r.e. extension of PA has a recursive independent
axiomatization. Moreover, every r.e. extension of PA formulated in a
language expanding the language of PA with finitely many new symbols, with the
induction scheme for the expanded language, has a recursive independent
axiomatization. Furthermore, this holds even if we allow the addition of
finitely many new sorts, provided T also has the standard axioms for
encoding finite sequences of objects from the sorts.
Proof: PA is in the language of 0,S,+,x,= with basic axioms (successor
axioms and defining equations) and the induction scheme. For each n >= 1,
let PAn be the basic axioms together with the induction scheme with respect
to all formulas with at most n quantifiers and any number of bounded
quantifiers and any free variables. It is well known that PAn is finitely
axiomatizable, and the finite axiomatization is effectively given in n.
Let T be an r.e. extension of PA in the language of PA. Without loss of
generality, we may assume that T is consistent. Let A be a theorem of T.
Let n be the total number of quantifiers in A. Then it is well known that A
does not prove PAn+2 even in the presence of PAn, unless A + PAn is
inconsistent. However, A + PAn is consistent since T is consistent. (To see
this well known fact, note that PAn+2 + A proves the consistency of PAn +
A, [using formalized cut elimination and truth definitions], and apply
Godel's second incompleteness theorem. Now there's an application of some
REAL big gun f.o.m.!!). So we can take the value of a partial recursive map
at A to be simply PAn+2, and apply Theorem I.
Finally, we come to the second claim. Here we can define PAn analogously,
with analogous standard facts; so the argument goes through without
substantial change. The final claim is also analogous. QED,
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symplectic topology
symplectic topology
Every (paracompact Hausdorff) differentiable manifold can be equipped with a Riemannian structure. Therefore the existence of a metric does not impose constraints on a toplogy of a manifold. The
local Riemannian structure of a Riemannian manifold on the other hand can be very different with the same global topology: there are even locally many nonisometric Riemannian structures on the same
manifold/neighborhood. All even dimensional manifolds allow locally a (unique up to isomorphism) symplectic structure. Symplectic manifolds, due Darboux theorem, are all locally isomorphic, but
according to the spectacular findings of Gromov in 1985, there are still global invariants of symplectic structure as well as topological constraints on even-dimensional manifolds permitting
symplectic structure.
• Pseudo holomorphic curves in symplectic manifolds, Inventiones Mathematicae 82, n. 2 (1985) 307–347.
Revised on August 30, 2011 20:00:57 by
Zoran Škoda
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Optimization Software
Software Systems Optimization Laboratory
Stanford University
Dept of Management Science and Engineering (MS&E)
Huang Engineering Center
Stanford, CA 94305-4121 USA
Freely Available Optimization Software
The following software packages are provided by SOL under the terms of the OSI Common Public License (CPL). The software may alternatively be used under the terms of a BSD License (
Systems Optimization Laboratory
Stanford University
Dept of Management Science and Engineering (MS&E)
Huang Engineering Center
Stanford, CA 94305-4121 USA
The following software packages are provided by SOL under the terms of the OSI Common Public License (CPL). The software may alternatively be used under the terms of a BSD License (BSDlicense.txt).
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Bug 479 – Adjust the max iteration count wrt arithmetic precision
Currently, the maximal number of iterations allowed in the Schur decomposition
is fixed to 40. While this is good for double, this is clearly not enough for
floating point numbers with much higher precision. The same situation probably
arises in other iterative algorithms.
I see several options:
1 - find a way to automatically compute a max number of iterations based on
2 - let the user adjust the max number of iterations
3 - find another way to detect no convergence
Comment 1 Desire NUENTSA 2012-07-09 11:15:27 UTC
This parameter is really problem-dependent. The user should have the
possibility to adjust it. For the default value, we can follow the approach in
LAPACK and EISPACK where the maximum number of iterations is 30 *
For the point 3, iterative methods for linear systems stops when itmax is
reached or when the norm of the current residual is greater than the norm of
the right-hand side multiplied by some predefined constant dtol.
Comment 3 Jitse Niesen 2012-07-16 17:10:36 UTC
I thought that Eispack has a maximum of 30 iterations for reducing one
sub-diagonal element to zero, and that is also what Eigen used. However, I had
another look at the old Fortran code and I see now that it has a maximum of 30
* (number of rows) iterations for the whole reduction to triangular form.
Currently, after the change, Eigen allows for 30 * (number of rows) iterations
for reducing one sub-diagonal element. If that is indeed the intention - and I
don't see a problem with it - then the documentation for m_maxIterations should
probably be updated.
Comment 4 Desire NUENTSA 2012-07-16 18:21:44 UTC
Oooh No. I didn't see that iter is set to zero at the convergence of one
eigenvalue. I think 30 * (number of rows) is very large for one single
sub-diagonal. Can we keep 30 iterations to remove one sub-diagonal as it was
done before and 30 * (number of rows) iterations for the whole reduction ?
Comment 5 Gael Guennebaud 2012-07-17 12:55:54 UTC
I think we should do as Lapack, and not limit a single sub-diagonal removal to
30 iterations because 1- the complexity is clearly not constant for all
sub-diagonals, and 2- this strategy has been proved to fail on some examples.
Comment 6 Jitse Niesen 2012-07-23 15:32:01 UTC
Yesterday I changed the max number of iterations used to the same strategy as
Lapack/Eispack; see changeset 24a5443ecde9 for the dev branch and changeset
23f17dd6901e for the 3.1 branch.
I think we still need to give the user the opportunity to change this, for
instance when using very high precision numbers.
Comment 7 Jitse Niesen 2012-07-24 16:24:08 UTC
Changeset 58c321490416 (only in the dev branch) allows the user to specify the
maximum number of iterations using an extra argument of the compute() function:
eigensolver.compute(A, true, 100) computes the eigenvalues and eigenvectors of
A using at most 100 iterations.
In hindsight, I'm wondering whether I should have implemented a different API,
like: eigensolver.setMaxIterations(100).compute(A) . This makes the user code
perhaps less cryptic. A disadvantage would be that the EigenSolver object gets
an additional member variable, to store the maximum number of iterations,
making the object occupy slightly more memory.
Comment 8 Gael Guennebaud 2012-07-24 18:36:49 UTC
I prefer the setMaxIterations() API. It is consistent with the sparse solvers,
and permits to configure a solver and then pass it to a generic algorithm.
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Posts about star formation on In the Dark
I stumbled across this wonderful image (and associated description) yesterday and thought I’d share it. It’s a region of the Orion Nebula (which is located in the [DEL:Midlands:DEL] region of
Orion’s “sword”, i.e. the long thing hanging down below his belt). It’s a turbulent region of dust and gas in which stars are forming. This image was taken in the far-infrared part of the spectrum
by the Herschel Space Observatory, which is now defunct but much data remains to be analysed. Because the image was taken at wavelengths much longer than optical light, the colours are obviously
“false”. I don’t work on star formation so I tend to see images like this just as beautiful things to be enjoyed for themselves rather than as a subject for scientific research. In fact, I have no
difficulty at all in describing this picture as a work of art, slightly reminiscent of the cloudscapes and seascapes of J.M.W Turner in that it is, at the same time, both a representation of a
natural phenomenon and an abstract creation that transcends it. You can click on the image to make it larger…
UPDATE: I see that someone else has thought of the parallel with Turner!
Follow @telescoper
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Calculating Costs and Savings
Knowing how much it costs to operate a product is the first step to understanding what the potential for savings is. The annual cost of operating any electric item (whether it=s a light bulb,
refrigerator, etc.) can be calculated as follows:
((((Daily Hours x 365) x Watts) / 1000) x Rate per kWh) = Annual Electricity Cost
Divide by 1000 because electric rates are set per kilowatt hour, which is to say per thousand watt hours. The national average rate per kWh is $0.08 in the United States.
Annual Savings
While it is interesting to know how much energy it takes to run an item, energy efficiency is about savings. To calculate savings, the most obvious method is to subtract the annual electricity cost
of the new product from the annual electricity cost of the existing product. However, a quicker way is to insert the wattage difference between the two products in the “Watts” part of the formula
((((Daily Hours x 365) x Watts Saved) / 1000) x Rate per kWh) = Annual Electricity Savings
For example, A75" would be used if a 100 watt incandescent bulb is to be replaced by a 25 watt compact fluorescent. ((((6 x 365) x 75) / 1000) x $0.08) = $13.14 Annual Electricity Savings
Lifetime Savings
To calculate the savings over the expected lifetime of the new product (in the case of many compact fluorescent bulbs, 10,000 hours is often the rated lifetime), in place of ADaily Hours x 365"
insert the rated life (in hours) of the product (e.g., 10,000).
((((Rated Lifetime) x Watts Saved) / 1000) x Rate per kWh) = Lifetime Electricity Savings
Example: ((((10,000 x 75) / 1000) x $0.08) = $60.00 Lifetime Electricity Savings
As energy efficient products tend to cost more than conventional technologies, a common question is how quickly the savings will cover the product=s purchase price. By calculating the payback, we can
determine at what point this will occur through the realized electricity savings. The formula follows.
Cost of Energy Efficient Product / Annual Electricity Savings = Simple Payback
For example, is the energy efficient product costs $15.95, and the calculated Annual Electricity Savings is $13.14 (see above for formula), the simple payback is 1.2 years.
Simple payback does not take into account compounded savings, discount rates, or inflation rates, all of which play a role in calculating the precise payback over many years or decades.
The Federal Energy Management Program (FEMP) offers energy cost calculators on a number of items, including appliances and lighting technologies. http://www1.eere.energy.gov/femp/procurement/
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[FOM] The Irrelevance of definite descriptions in the Slingshot Argument?
[FOM] The Irrelevance of definite descriptions in the Slingshot Argument?
A.S.Virdi@lse.ac.uk A.S.Virdi at lse.ac.uk
Mon Oct 2 06:02:52 EDT 2006
Alonzo Church sees in Frege's thinking that the reference of a sentence
is that sentence's truth-value an exoneration of the idea that if
sentences designate propositions then all designate the very same
proposition. Church demonstrates this by providing a slingshot argument
in his 1943 review of Carnap's 'Introduction to Semantics' and he also
reiterates it in his 1956 'Introduction to Mathematical Logic'. Indeed,
Frege does not *use* any slingshot argument.
Around the same time that Church's review of Carnap appeared, Goedel
wrote "Russell's Mathematical Logic" (1944), footnote 5 of which has a
proof sketch for the Fregean conclusion that all true propositions have
the same signification. {Goedel uses a much weaker principle than
logical equivalence, one according to which "Fa" and "a = ix((x = a) &
Fx)" have 'the same meaning'). Neale's 1995 Mind paper (and the 2001
book 'Facing Facts' that grew out of that paper) is very good on all
The received wisdom in combating the slingshot argument is that we need
to undermine the referential status of definite descriptions and this is
aptly done by understanding Russell's reasons for thinking that definite
descriptions as syncategorematic terms. But, if we rearticulate the
slingshot argument in terms of set abstracts (taken as primitive, as
Heck suggests, so definite descriptions are definable in terms of these)
then the issue of the semantics of definite descriptions becomes an
irrelevant one. I suspect Goedel appreciated this very point. In
discussing the Russellian escape route (in the 1944 paper referred to
above) he says:
"I cannot help feeling that the problem raised by Frege's puzzling
conclusion [that all true sentences have the same signification]
only been evaded by Russell's theory of descriptions and that there
something behind it which is not yet completely understood. "
(page 215 in the 1964 edition of the Benacerraf and Putnam anthology
Philosophy of Mathematics)
Part of what was not "completely understood" then (I humbly suggest) is
the fact that set abstractions do the very job iota-expressions are
supposed to do without raising these semantic issues and, thus,
solidifying the slingshot argument.
Arhat Virdi
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Tensor fields and multiplication
Regarding the second question...
What Lee is saying is that a connection ∇: [itex]\Gamma(TM)\times \Gamma(TM)\rightarrow \Gamma(TM)[/itex] looks like a (2,1) tensor (compare with Lemma 2.4), but it is not one as it is not [itex]C^{\
infty}(M)[/itex]-linear in its second argument. Later, he defines the covariant derivative of a tensor, and remarks that if you take a tensor T of type (k,l), and take its covariant derivative ∇T,
you get a tensor of type (k+1,l). In particular, if you take a vector field Y (tensor of type (0,1)) and jam it up the second slot of the connection map like so: ∇Y, you get a tensor, because the
problem was in the second argument of ∇ and you've now eliminated that problem.
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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.
This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.
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Need help with formation of basis in R3
February 23rd 2011, 03:40 PM #1
Junior Member
Sep 2008
Need help with formation of basis in R3
The three vectors form a basis in R3 if and only if k does not equal ____?
Thank You,
What ideas have you had so far?
Well, i know that v(1) v(2) and v(3) have to be linearly independent they also have to span V....
and i tried putting them into reduced row echelon form but i dont know what to do with the k
What did you get for the rref? As for the k, just treat it like any other number.
February 23rd 2011, 03:41 PM #2
February 23rd 2011, 03:51 PM #3
Junior Member
Sep 2008
February 23rd 2011, 04:30 PM #4
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Fractional Sylvester-Gallai
Avi Wigderson was in town and gave a beautiful talk about an extension of Sylvester-Gallai theorem. Here is a link to the paper: Rank bounds for design matrices with applications to combinatorial
geometry and locally correctable codes by Boaz Barak, Zeev Dvir, Avi Wigderson, and Amir Yehudayoff.
The Sylvester-Gallai Theorem: Let X be a finite set of n points in an eulidean space such that for every two distinct points $x,y \in X$ the line through $x$ and $y$ contains a third point $z \
in X$. Then all points in $X$ are contained in a line.
I heard about this result when I took Benjy Weiss’s mathematics course for high-school students in 1970/1. a The Sylvester-Gallai theorem was the last question marked with (*) in the first
week’s homework. In one of the next meetings Benjy listened carefully to our ideas on how to prove it and then explained to us why our attempts of proving it are doomed to fail: What we tried to do
only relied on the very basic incidence axioms of Euclidean geometry but the Sylvester-Gallai theorem does not hold for finite projective planes. (Sylvester conjectured the result in 1893. The first
proof was given by Mechior in 1940 and Gallai proved it in 1945.)
My MO question
Befor describing the new results let me mention my third ever MathOverflow question that was about potential extensions of the G-S theorem. The question was roughly this:
Suppose that V is an r dimensional variety embedded into n space so that if the intersection of every j-dimensional subspace with V is full dimensional then this intersection is “complicated”.
Then $n$ cannot be too large.
I will not reproduce the full question here but only a memorable remark made by Greg Kuperberg:
If you claimed that Gil is short for Gilvester (which is a real first name although rare), then you could say that any of your results is the “Gilvester Kalai theorem”. – Greg Kuperberg Nov 24
2009 at 5:13
The result by Barak, Dvir, Wigderson and Yehudayoff
Theorem: Let X be a finite set of n points in an Euclidean space such that for every point $x \in X$ the number of $y, y\in X,y e x$ such that the line through $x$ and $y$ contains another point
of $X$ is at least $\delta (n-1)$. Then
$\dim (Aff(X))\le 13/\delta^2$
Some remarks:
1) The proof: The first ingredient of the proof is a translation of the theorem into a question about ranks of matrices with a certain combinatorial structure. The next thing is to observe is that
when the non zero entries of the matrix are 1′s the claim is simple. The second surprising ingredient of the proof is to use scaling in order to “tame” the entries of the matrix.
2) The context – locally correctable codes: A $q$-query locally correctable $(q,\delta)$-code over a field $F$ is a subspace of $F^n$ so that, given any element $\tilde y$ that disagrees with some
$y \in C$ in at most $\delta n$ positions and an index $i$, $1 \le i \le n$ we can recover $y_i$ with probability 3/4 by reading at most $q$ coordinates of $\tilde y$. The theorem stated above imply
that, for two queries, over the real numbers (and also over the complex numbers), such codes do not exist when $n$ is large. Another context where the result is of interest is the hot area of sum
product theorems and related questions in the geometry of incidences.
3) Some open problems: Is there a more detailed structure theorem for configurations of points satisfying the condition of the theorem? Can the result be improved to $\dim (Aff(X))=O(1/\delta )$? Can
a similar result be proved on locally correctable codes with more than two queries? This also translates into an interesting Sylvester-Gallai type question but it will require, Avi said, new ideas.
3 Responses to Fractional Sylvester-Gallai
1. From Avi: “Shubhangi Saraf recently solved the open problem – she proved that the dim of such fractional SG config O(1/\delta), so you can replace that open ‘?’ with a closed ‘!’ ”
As a substitute open problem what about this: If among any k points you can find two so that the line through then contain a third point (in the set) then the points lies on f(k) lines?
2. Melchior’s name is not spelled properly above. His proof of the Sylvester-Gallai Theorem is discussed in Branko Grunbaum’s monograph: Arrangements and Spreads, pg. 17.
3. thank :D
This entry was posted in Combinatorics, Computer Science and Optimization, Geometry and tagged Avi Wigderson, Codes, Greg Kuperberg, Sylvester-Gallai. Bookmark the permalink.
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50. Jahrestagung der Deutschen Gesellschaft für Medizinische Informatik, Biometrie und Epidemiologie (gmds)
Methods for the estimation of incremental cost effectiveness
Meeting Abstract
Search Medline for
Published: September 8, 2005
Introduction and Purpose
Meanwhile therapeutic and diagnostic procedures are not only evaluated from a clinical, but also from a health economic point of view to link their clinical efficacy to the underlying direct costs.
Discussions on ressource allocation and the founding of medical supplementation can be based on the results of such cost effectiveness evaluations and therefore provide an objective rationale for
The estimation of incremental cost effectiveness ratios (ICERs) has earnt increasing attention during this decade. ICERs relate the cost difference between therapeutical alternatives to the
corresponding difference in clinical efficacy or utility. Despite the intuitive interpretation of ICERs as additional costs per additional benefit unit, their statistical treatment imposes severe
problems because of the necessity to estimate the distribution of a ratio of at least two stochastically dependent distributions [Ref. 1]. Several attempts were made to modify the well-known Fieller
theorem for this purpose, other authors suggest the use of multivariate Bootstrap simulation or Bayesian approaches [Ref. 1], [Ref. 2] . Nevertheless, the problems in ratio estimation remain.
Therefore, the concept of ICER estimation is contrasted to the idea of net health benefit (NHB) estimation, which transforms the former ratio estimation problem into a linear estimation procedure
[Ref. 2].
Material and Methods
Model parametrization
The following will consider two therapeutic alternatives 1 and 2, where treatment 1 denotes an established standard procedure and treatment 2 is under discussion concerning possible recommendation
for founding by health care insurers. If then the random variables K[1] and K[2] denote the treatments’ costs and the corresponding random variables E[1] and E[2] the treatments’ respective
efficacies, the following will assume K[2] > K[1] and E[2] > E[1] (such a treatment alternative 2 is usually called “admissable” for ressource allocation). A cost effectiveness comparison of these
treatments may then provide a decision on when to found treatment 2 instead of treatment 1, or when to retain ressource allocation to the standard treatment 1.
The ratio K / E is refered to as the cost effectiveness ratio (CER) and describes a treatment’s marginal costs per gained clinical benefit unit. The incremental cost effectiveness ratio (ICER) of a
treatment 2 versus the standard treatment 1 is defined as ICER[21] = (K[2] – K[1]) / (E[2] – E[1]) and estimates the additional costs, which must be invested to achieve one additional clinical
benefit unit under treatment 2 instead of the standard. In this sense, the ICER concept allows for “health economical ranking” of different health care service offers, when being contrasted to the
same standard. Ressource allocation rules can now be formulated straigt-forward: If a health care insurer considers treatment 2 for founding, an allocation rule based on the ICER could use a
pre-specified benchmark µ, which characterizes the insurer’s maximum willingness to pay (WTP) additional treatment costs per gained benefit unit. Therefore treatment 2 would be founded as soon as µ >
ICER[21], whereas treatment 1 remains founded for CER[1] < µ < ICER[21].
A different concept for cost effectiveness evaluation is based on net health benefit estimation [3]: The net health benefit (NHB) of a treatment is defined as its clinical benefit after correction
for its incremental costs when being contrasted to a standard, NHB = E – K/µ, where µ denotes the above willingness to pay benchmark. Therefore the net health benefit approach directly involves the
WTP model parameter into cost effectiveness estimation. In this context the NHB measures a health service’s benefit after correction for the insurer’s willingness to pay philosophy. A new treatment
alternative’s incremental net health benefit versus a standard is then defined as INHB[21] = NHB[2] – NHB[1] and measures the additional clinical benefit of treatment 2 after correction for the two
treatment alternatives’ relative cost effectiveness. An NHB-based allocation rule suggests founding of treatment 2 as soon as INHB[21] > 0 (thereby NHB[2] > NHB[1]) and founding of treatment 1
otherwise. It is easy to show, that the ICER-based and the INHB-based allocation rules yield the same allocation decisions [Ref. 3].
The cost and benefit parameters K[i] and E[i] can be estimated by their population means and imputed into the above allocation rules as follows:
Determine the WTP benchmark µ > 0 and a significance level α > 0.
Estimate the sample cost and efficacy estimates K and E and compute the NHB-estimates in terms of K and E via NHB = E – K/µ for both therapies.
Compute the incremental net health benefit INHB = NHB[2] – NHB[1] of treatment 2 versus the standard treatment 1.
Obtain a one-sided (1-α/2) confidence interval for INHB based on the appropriate t-distribution, or perform a one-sided t-test to ensure INHB>0.
Cataract Surgery Data
The above will be illustrated by means of the cost effectiveness evaluation of cataract surgery with multifocal intraocular lenses [Ref. 3]. One drawback of monofocal intraocular lenses consists in
the frequent ongoing need for seeing aids after surgery, for example when reading or driving. Multifocal lenses often overcome this need; an increase in subjective quality of life can be expected.
German health care insurers reimburse the costs of monofocal lens supplementation. However, founding of multifocal cataract surgery is still under discussion. To assess the incremental cost
effectiveness of multifocal lens supplementation with respect to its putative quality of life benefit, data of a randomized trial in cataract patients have been re-analysed from a health economic
point of view. Quality of life was assessed by means of a questionnaire-based utility value ranging from 0.0 to 1.0. A total of 400 patients were equally assigned to monofocal or multifocal lens
supplementation, and a 6 months follow up on complications and post surgical quality of life was performed. The WTP benchmark was pre-specified at 800 € per gained QALY.
A remarkable difference in both costs and gained QALYs was found, whereas already the marginal cost effectiveness of supplementation with multifocal lenses (852 € per gained QALY) turned out worse
than the corresponding monofocal estimate of 786 € per gained QALY. If only applied to mean point estimates, the ICER-based allocation rule would still decide founding of monofocal intraocular
lenses, but no longer consider multifocal supplementation: ICER = 1060 € / QALY > 800 € / QALY = µ. The INHB-based allocation rule would yield the same decision: INHB = NHB(multifocal) – NHB
(monofocal) = -0.133 QALYs – 0.028 QALYs = -0.161 QALYs < 0, i.e. multifocal cataract surgery results in a (slight) loss in cost effectiveness when contrasted to the monofocal therapeutic standard.
If the strategy in Section 2.2 is applied at a 5% significance level, a p-value of 0.045 (one-sided two sample t-test) results. Since p>0.025, no statistically significant difference in net health
benefits was found between the multifocal and monofocal lens supplementation, where benefit was based on the number of QALYs achieved by the respective treatment. Therefore supplementation of
cataract patients with multifocal intraocular lenses cannot be ensured to show a significantly positive net health benefit when compared to the monofocal supplementation standard.
Whereas the ICER-based approach provides somewhat instructive information, its statistical feasibility has to be based on severe model assumptions. The NHB approach, however, transforms the problem
into standard linear estimation. In this context it is overly important, that the allocation rules based on the net health benefit approach yield the same allocation decisions as the ICER-based ones,
since interval estimation in the NHB context can be reduced to standard univariate significance testing and interval estimation. Therefore the rather difficult interpretation of (I)NHB estimates
becomes weakened by their advantages concerning statistical feasibility. On the other hand, communication of NHB estimates should be handled with care: Note, that the NHB point estimates in the
cataract surgery example do not even slightly mirror the order of the underlying willingness to pay parameter µ! This motivates the integration of methodologists into both the planning and evaluation
phase of cost effectiveness investigations [Ref. 4].
Wakker P, Klaassen MP. Confidence intervals for cost-effectiveness ratios. Health Economy 1995; 4: 373-81
Heitjan DF. Fieller's method and net health benefits. Health Economy 2000; 9: 327-35
Krummenauer F, Landwehr I. Incremental cost effectiveness evaluation in clinical research. European Journal of Medical Research 2005; 10: 18 - 23
Seither C. Vorschläge zur gesundheitsökonomischen Evaluation zahnärztlicher Präventionsprogramme im Kindesalter. Dissertation zur Erlangung des Grades "Dr. med. dent.", Fachbereich Medizin der
Universität Mainz; 2004
Laska EM, Meissner M, Siegel C, Wanderling J. Statistical cost effectiveness analysis of two treatments based on net health benefits. Stat Med 2001; 20: 1279-1302
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Uniform distribution
April 3rd 2009, 11:49 PM #1
Mar 2009
Uniform distribution
I'm not quite sure what part (a) is asking.
For part (b), I'm thinking that this has something to do with Chebyshev's inequality, but the examples for Chebyshev's inequality in my textbook and notes don't involve the use of 'max {...}'. So
I'm pretty much stuck. But intuitively, I think it will be 0 when n tends to infinity, but I don't really know how to prove it.
I think you'll find this thread interesting. It gives you a possible choice of what $k_n$ should be and there are hints for a proof; this should allow you to understand how $k_n$ must be chosen.
I think you'll find this thread interesting. It gives you a possible choice of what $k_n$ should be and there are hints for a proof; this should allow you to understand how $k_n$ must be chosen.
Here's what I've done, but I'm not sure how to continue.
$P(k_n(\theta-Y_n)\leq x)=P(Y_n\geq\theta-\frac{x}{k_n})=1-P(Y_n\leq\theta-\frac{x}{k_n})$
$=1-(1-\frac{x}{\theta k_n})^n$
How do I continue from here? What do they mean when they say $k_n$ is a sequence of constants?
Does it have anything to do with e^-x? If yes, then I guess I'm more or less get it.
Last edited by knighty; April 4th 2009 at 06:35 AM.
Here's what I've done, but I'm not sure how to continue.
$P(k_n(\theta-Y_n)\leq x)=P(Y_n\geq\theta-\frac{x}{k_n})=1-P(Y_n\leq\theta-\frac{x}{k_n})$
$=1-(1-\frac{x}{\theta k_n})^n$
How do I continue from here? What do they mean when they say $k_n$ is a sequence of constants?
Does it have anything to do with e^-x? If yes, then I guess I'm more or less get it.
What you did is fine. Now you are asked to choose $k_n$ to be any sequence such that $1-(1-\frac{x}{\theta k_n})^n$ has a limit when $n\to\infty$. There are many possible choices.
The most obvious would be $k_n=n$ or $k_n=\theta n$ (or $k_n=\alpha n$ with an arbitrary $\alpha$). You indeed get $1-e^{-\frac{x}{\theta}}$ or $1-e^{-x}$ (or $1-e^{-\alpha/\theta}$), which is
the distribution function of an exponential distribution with some parameter.
What happens for other choices? Suppose for instance $k_n=n^2$ or any sequence with $\frac{k_n}{n}\to\infty$. Then you can see that the limit would be 0, which is not a distribution function.
If to the contrary $\frac{k_n}{n}\to 0$ (still with $k_n\to\infty$), for instance $k_n=\sqrt{n}$, then the limit is 1, which is the distribution function of the Dirac probability measure at 0,
i.e. the distribution of a r.v. that is constant, equal to 0. This would be a correct answer, but it is called a "degenerate" limit, which is not very interesting. The best choice is when $k_n$
is on the order of $n$.
Of course, any sequence such that $k_n\sim_n \alpha n$ would give an exponential as the limit, not only $k_n=\alpha n$. But you're only asked for one, so let's choose simple.
Okay I get it fully now. Thanks very much.
Hi, is there anybody who can help to solve part b?
April 4th 2009, 02:22 AM #2
MHF Contributor
Aug 2008
Paris, France
April 4th 2009, 06:05 AM #3
Mar 2009
April 4th 2009, 11:19 AM #4
MHF Contributor
Aug 2008
Paris, France
April 4th 2009, 07:23 PM #5
Mar 2009
April 12th 2009, 10:50 AM #6
Apr 2009
April 12th 2009, 02:00 PM #7
MHF Contributor
Aug 2008
Paris, France
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A math or a simple fluid problem?
Hi all!
I am solving for the height of a tank as a function of time.
The tank has a constant inflow of a and is subjected to small fluctuation of bsin(wt).
So the inflow is simply a+bsin(wt).
The outflow should be proportional to the square root of the height, H.
So outflow = c*sqrt(H)
Therefore the below differential equation is obtained with k=the area of the tank:
but the problem is that I don't know how to solve this ODE...
Can anyone solve the problem?
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A characterization of functions that generate wavelet and related expansion
Results 1 - 10 of 12
, 2001
"... We discuss wavelet frames constructed via multiresolution analysis (MRA), with emphasis on tight wavelet frames. In particular, we establish general principles and specific algorithms for
constructing framelets and tight framelets, and we show how they can be used for systematic constructions of spl ..."
Cited by 129 (50 self)
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We discuss wavelet frames constructed via multiresolution analysis (MRA), with emphasis on tight wavelet frames. In particular, we establish general principles and specific algorithms for
constructing framelets and tight framelets, and we show how they can be used for systematic constructions of spline, pseudo-spline tight frames and symmetric biframes with short supports and high
approximation orders. Several explicit examples are discussed. The connection of these frames with multiresolution analysis guarantees the existence of fast implementation algorithms, which we
discuss briefly as well.
- J. GEOM. ANAL , 1999
"... Sets K in d-dimensional Euclidean space are constructed with the property that the inverse Fourier transform of the characteristic function 1K is a single dyadic orthonormal wavelet. The
construction is characterized by its generality in terms of a quantitative iterative procedure, by its computat ..."
Cited by 16 (2 self)
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Sets K in d-dimensional Euclidean space are constructed with the property that the inverse Fourier transform of the characteristic function 1K is a single dyadic orthonormal wavelet. The construction
is characterized by its generality in terms of a quantitative iterative procedure, by its computational implementation, and by its simplicity. The general case in which the inverse Fourier transforms
of the characteristic functions 1 K 1 ; : : : ; 1 K L are a family of orthonormal wavelets is treated in [Leo99].
- Technion Israel School of Technology. Downloaded on May 27,2010 at 17:56:21 UTC from IEEE Xplore. Restrictions apply. IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL , 2005
"... ABSTRACT. Let Tk denote translation by k ∈ Zd. Given countable collections of functions {φj}j∈J, { φj ˜}j∈J ⊂ L2 (Rd) and assuming that {Tkφj} j∈J,k∈Zd and {Tk ˜ φj} j∈J,k∈Zd are Bessel
sequences, we are interested in expansions f = ∑ ∑ j∈J k∈Zd 〈 f, Tk ˜φj Tkφj, ∀f ∈ span {Tkφj} k∈Zd,j∈J. Our mai ..."
Cited by 16 (15 self)
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ABSTRACT. Let Tk denote translation by k ∈ Zd. Given countable collections of functions {φj}j∈J, { φj ˜}j∈J ⊂ L2 (Rd) and assuming that {Tkφj} j∈J,k∈Zd and {Tk ˜ φj} j∈J,k∈Zd are Bessel sequences, we
are interested in expansions f = ∑ ∑ j∈J k∈Zd 〈 f, Tk ˜φj Tkφj, ∀f ∈ span {Tkφj} k∈Zd,j∈J. Our main result gives an equivalent condition for this to hold in a more general setting than described
here, where translation by k ∈ Z d is replaced by translation via the action of a matrix. As special cases of our result we find conditions for shift-invariant systems, Gabor systems, and wavelet
systems to generate a subspace frame with a corresponding dual having the same structure. 1.
, 1999
"... . This paper is devoted to the study of the dimension functions of (multi)wavelets, which was introduced and investigated by P. Auscher in [1]. Our main result provides a characterization of
functions which are dimension functions of a (multi)wavelet. As a corollary, we obtain that for every functio ..."
Cited by 10 (3 self)
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. This paper is devoted to the study of the dimension functions of (multi)wavelets, which was introduced and investigated by P. Auscher in [1]. Our main result provides a characterization of
functions which are dimension functions of a (multi)wavelet. As a corollary, we obtain that for every function D that is the dimension function of a (multi)wavelet, there is an MSF (multi)wavelet
whose dimension function is D. In addition, we show that if a dimension function of a wavelet not associated with an MRA attains the value K, then it attains all integer values from zero to K.
Moreover, we prove that every expansive matrix which preserves Z N admits an MRA structure with an analytic (multi)wavelet. 2 1. Introduction The dimension function of an orthonormal wavelet 2 L 2
(R) is dened as D () = 1 X j=1 X k2Z j ^ (2 j ( + k))j 2 : The importance of the dimension function was discovered by P. G. Lemarie, who used it to prove that certain wavelets are associated with ...
"... A countable collection X of functions in L 2 (IR ) is said to be a Bessel system if the associated analysis operator T # X : L 2 (IR (#f, x#) x#X is well-defined and bounded. A Bessel system is
a fundamental frame if T # X is injective and its range is closed. ..."
Cited by 8 (2 self)
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A countable collection X of functions in L 2 (IR ) is said to be a Bessel system if the associated analysis operator T # X : L 2 (IR (#f, x#) x#X is well-defined and bounded. A Bessel system is a
fundamental frame if T # X is injective and its range is closed.
- of R n . AMS Research Announcements , 1999
"... . In the context of a general lattice \Gamma in R n and a strictly expanding map M which preserves the lattice, we characterize all the wavelet families, all the MSF wavelets, all the
multiwavelets associated with a Multiresolution Analysis (MRA) of multiplicity d 1; and all the scaling functi ..."
Cited by 6 (1 self)
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. In the context of a general lattice \Gamma in R n and a strictly expanding map M which preserves the lattice, we characterize all the wavelet families, all the MSF wavelets, all the multiwavelets
associated with a Multiresolution Analysis (MRA) of multiplicity d 1; and all the scaling functions. Moreover, we give several examples: in particular, we construct a single, MRA and C 1 (R n )
wavelet, which is nonseparable and with compactly supported Fourier transform. 1. Introduction An orthonormal wavelet is a function / 2 L 2 (R) such that the set \Phi / j;k j 2 j=2 /(2 j x \Gamma k)
: j; k 2 Zg (1) is an orthonormal basis for L 2 (R): A complete characterization of these wavelets is given by the equations a) X j2Z j b /(2 j )j 2 = 1 a.e. 2 R; b) 1 X j=0 b /(2 j ) b /(2 j ( +
2k)) = 0 a.e. 2 R; k 2 2Z + 1; (I) together with the assumption k/k 2 1: These two equations have been known since the beginning of the theory of wavelets (see [L1] ...
- Math
"... The single underlying method of “averaging the wavelet functional over translates” yields first a new completeness criterion for orthonormal wavelet systems, and then a unified treatment of
known results on characterization of wavelets on the Fourier transform side, on preservation of frame bounds b ..."
Cited by 6 (2 self)
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The single underlying method of “averaging the wavelet functional over translates” yields first a new completeness criterion for orthonormal wavelet systems, and then a unified treatment of known
results on characterization of wavelets on the Fourier transform side, on preservation of frame bounds by oversampling, and on the equivalence of affine and quasiaffine frames. The method applies to
multiwavelet systems in all dimensions, to dilation matrices that are in some cases not expanding, and to dual frame pairs. The completeness criterion we establish is precisely the discrete Calderón
condition. In the single wavelet case this means we take invertible matrices a and b and a function ψ ∈ L 2 (R d), and assume either a is expanding or else a is amplifying for ψ. We prove that the
system { | det a | j/2 ψ(a j x − bk):j ∈ Z,k ∈ Z d} is an orthonormal basis for L 2 (R d) if and only if it is orthonormal and ∑ j∈Z | ˆ ψ(ξa j) | 2 = | det b | for almost every row vector ξ ∈ R d.
"... Abstract. In this paper, we study stationary and nonstationary nonhomogeneous dual wavelet frames with an arbitrary real dilation factor in the frequency domain by introducing and investigating
a pair of frequency-based nonhomogeneous dual wavelet frames in the distribution space. This notion of a p ..."
Cited by 3 (3 self)
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Abstract. In this paper, we study stationary and nonstationary nonhomogeneous dual wavelet frames with an arbitrary real dilation factor in the frequency domain by introducing and investigating a
pair of frequency-based nonhomogeneous dual wavelet frames in the distribution space. This notion of a pair of frequency-based nonhomogeneous dual wavelet frames in the distribution space enables us
to completely separate its perfect reconstruction property from its stability property in function spaces. The results in this paper lead to a natural explanation for the oblique extension principle
for constructing dual wavelet frames from refinable functions without any a priori condition on the generating wavelet functions and refinable functions. A pair of frequency-based nonhomogeneous dual
wavelet frames in the distribution space, that is not necessarily derived from refinable functions via a multiresolution analysis, has a natural multiresolution-like structure, which is closely
linked to the fast wavelet frame transform. Moreover, nonhomogeneous dual wavelet frames in the distribution space play a basic role in understanding dual wavelet frames in various function spaces
and have a close relation to nonstationary dual wavelet frames, which are of interest in applications. To illustrate the flexibility and generality of the results in this paper, we characterize a
pair of fully nonstationary dual wavelet frames in the distribution space. Our results naturally link a nonstationary dual wavelet frame filter bank having the perfect reconstruction property to a
pair of nonstationary dual wavelet frames in the distribution space.
"... In this paper we give a necessary and sufficient condition for a pair of wavelet families \Psi = f/ 1 ; : : : ; / L g; e \Psi = f ~ / 1 ; : : : ; ~ / L g; in L 2 (R n ), to arise from a pair of
biorthogonal MRA's. The condition is given in terms of simple equations involving the function ..."
Cited by 2 (1 self)
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In this paper we give a necessary and sufficient condition for a pair of wavelet families \Psi = f/ 1 ; : : : ; / L g; e \Psi = f ~ / 1 ; : : : ; ~ / L g; in L 2 (R n ), to arise from a pair of
biorthogonal MRA's. The condition is given in terms of simple equations involving the functions / ` and ~ / ` . To work in greater generality, we allow multiresolution analyses of arbitrary
multiplicity, based on lattice translations and matrix dilations. Our result extends the characterization theorem of G. Gripenberg and X. Wang for dyadic orthonormal wavelets in L 2 (R), and
includes, as particular cases, the sufficient conditions of P. Auscher and P.G. Lemari'e in the biorthogonal situation. 1
, 2005
"... Under certain assumptions we show that a wavelet frame {τ(Aj,bj,k)ψ}j,k∈Z: = {|det Aj | −1/2 ψ(A −1 j (x − bj,k))}j,k∈Z in L 2 (R d) remains a frame when the dilation matrices Aj and the
translation parameters bj,k are perturbed. As a special case of our result, we obtain that if {τ(A j,A j Bn)ψ} j ..."
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Under certain assumptions we show that a wavelet frame {τ(Aj,bj,k)ψ}j,k∈Z: = {|det Aj | −1/2 ψ(A −1 j (x − bj,k))}j,k∈Z in L 2 (R d) remains a frame when the dilation matrices Aj and the translation
parameters bj,k are perturbed. As a special case of our result, we obtain that if {τ(A j,A j Bn)ψ} j∈Z,n∈Z d is a frame for an expansive matrix A andaninvertiblematrixB, then{τ(A ′ j,Aj Bλn)ψ}
j∈Z,n∈Z d is a frame if �A −j A ′ j − I�2 ≤ ε and �λn − n� ∞ ≤ η for sufficiently small ε, η> 0.
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Mechanical Advantage ( Read ) | Physics
Students will learn about mechanical advantage and how to find the mechanical advantage of various tools.
Key Equations
Mechanical Advantage (MA)
$MA = \frac{d_1}{d_2} = \frac{F_{out}}{F_{in}}$
d1 is the distance of effort and d2 is the distance the object is moved
• Mechanical Advantage is the ability to lift or move objects with great force while utilizing only a little force. The trade-off is that you must operate the smaller input force for a large
distance. This is all seen through the work Equation. Work equals force times distance. Energy is conserved. Thus one can get a large force for a small distance equal to a small force for a large
• Mechanical advantage equals the distance of effort divided by the distance the object moves. It is also equal to the output force divided by the input force.
Example 1
You need to push a 500 kg grand piano onto a stage that is 3 m above the ground. If you can only apply a maximum force of 1000 N, what is the minimum distance from the stage that you should begin
building your ramp?
We should start this problem by determining the mechanical advantage required to move the piano based on the weight of the piano and the force you can apply. We'll define $F_{out}$$F_{in}$
$MA&=\frac{F_{out}}{F_{in}}\\MA&=\frac{500\;\text{kg} * 9.8\;\text{m/s}^2}{1000\;\text{N}}\\MA&=\frac{4900\;\text{N}}{1000\;\text{N}}\\MA&=4.9\\$
Now we can use this to find how long the ramp needs to be.
Now we can just use the Pythagorean theorem to determine how far away from the stage the ramp should start.
Watch this Explanation
Time for Practice
1. A crowbar is a very handy tool to get out tough nails, demolition of a wall or breaking into or out of jail. It can be used from either end. The man shown is prying out a board. He is applying 30
N of force, study the picture to see how he is using it and which dimensions should be used.
1. What is the Mechanical Advantage of this crowbar, shown above with its dimensions?
2. How much force is being applied to the wood plank?
2. A mover loads a 100-kg box into the back of a moving truck by pushing it up a ramp. The ramp is 4 m long and the back of the truck is 1.5 m high.
1. Calculate the potential energy gained by the box when it’s loaded into the truck.
2. Calculate the mechanical advantage of the ramp.
3. Calculate the force required to push the box up the ramp in the absence of friction.
1. a. M.A. = 12 b. Force out = 360 N
2. a. 1500 J b. 2.7 c. 375 N
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Solving scientific and engineering problems
Scientific computing, also known as computational science, uses computational methods to solve science and engineering problems. The modeling of natural systems using numerical simulation is an
important area of focus within scientific computing. These models are often computationally intensive and require high-performance computing resources.
Scientists and engineers often create models using applied mathematical methods for Fourier analysis, numerical linear algebra, and solving ordinary and partial differential equations. Models are
often implemented using programming languages or domain-specific modeling tools.
Most common among these is MATLAB^®, a high-level language and interactive development environment with prebuilt functions for scientific computing. For detail on solving specialized classes of
problems, see the toolboxes for statistics, optimization, and parallel computing.
Examples and How To
Software Reference
See also: random number generation, mathematical modeling, parallel computing, numerical analysis, Statistics Toolbox, Optimization Toolbox
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Create a realization of an AR(4) wide-sense stationary random process. Estimate the PSD using the Yule-Walker method. Compare the PSD estimate based on a single realization to the true PSD of the
random process.
Create an AR(4) system function. Obtain the frequency response and plot the PSD of the system.
A = [1 -2.7607 3.8106 -2.6535 0.9238];
[H,F] = freqz(1,A,[],1);
xlabel('Hz'); ylabel('dB/Hz');
title('True Power Spectral Density of AR(4) System Function')
Create a realization of the AR(4) random process. Set the random number generator to the default settings for reproducible results. The realization is 1000 samples in length. Assume a sampling
frequency of 1. Use pyulear to estimate the PSD for an 4-th order process. Compare the PSD estimate with the true PSD.
rng default;
x = randn(1000,1);
y = filter(1,A,x);
[Pxx,F] = pyulear(y,4,1024,1);
hold on;
plot(F,10*log10(Pxx),'r'); hold on;
legend('True Power Spectral Density','PSD Estimate')
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Functions, Graphs, and Limits Fun Fun Functions Quiz
Think You Got it Down?
Test your knowledge again:
Functions, Graphs, and Limits: Fun Fun Functions Quiz
Think you’ve got your head wrapped around Functions, Graphs, and Limits? Put your knowledge to the test. Good luck — the Stickman is counting on you!
f(x) increases.
f(x) decreases.
f(x) increases if x < 0 and decreases if x > 0.
f(x) decreases if x < 0 and increases if x > 0.
As x approaches 2, f(x) approaches
different numbers depending on whether x is approaching 2 from the right or from the left.
Q.The statement
f(a) = L
as x gets close to a, f(x) gets close to f(a)
as x gets close to a, f(x) gets close to L
as x gets close to a, f(a) gets close to L
The function
) is graphed below.
the limit does not exist
The function
) is graphed below.
the limit does not exist
The function
) is graphed below.
the limit does not exist
Q.Use the table to determine which is the best estimate for
The limit does not exist, since the left- and right-hand limits do not agree.
We don't get a number that exists, since we can't divide by zero.
We don't get a number that exists, since we can't divide by zero.
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We were asked to debate the value of [m]. Is the universe flat and marginally bound with [m] = 1 in accordance with the simplest cosmological model? Is [m] clearly smaller than unity as seems to be
indicated by several observations? Unfortunately, we cannot provide a clear answer at this point because there is conflicting evidence. Entertaining the audience with our biased views on the subject
might not be very constructive. Instead, it may be more interesting to lay out the various methods used to measure [m], mention new developments and current estimates, and focus on the promising
prospects versus the associated difficulties. In the critical discussion that follows we try to shed light on the nature of the uncertainties that may be responsible for the current span of estimates
for [m].
We divide the methods into the following four classes:
• Global measures. Based on properties of space-time that constrain combinations of [m] and the other cosmological parameters (H[0], t[0]).
• Virialized Systems. Methods based on nonlinear dynamics within galaxies and clusters on comoving scales 1 - 10 h^-1Mpc.
• Large-scale structure. Measurements based on mildly-nonlinear gravitational dynamics of fluctuations on scales 10 - 100 h^-1Mpc of superclusters and voids, in particular cosmic flows.
• Growth rate of fluctuations. Comparisons of present day structure with fluctuations at the last scattering of the cosmic microwave background (CMB) or with high redshift objects of the young
The methods and current estimates are discussed below and summarized in Figure 1 and Table 1. The estimates based on virialized objects typically yield low values of [m] ~ 0.2 - 0.3. The global
measures, large-scale structure and cosmic flows typically indicate higher values of [m] ~ 0.4 - 1.
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FW: st: Any way to annotate a Stata graph with LaTeX?
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FW: st: Any way to annotate a Stata graph with LaTeX?
From "Feiveson, Alan H. (JSC-SK311)" <alan.h.feiveson@nasa.gov>
To "statalist@hsphsun2.harvard.edu" <statalist@hsphsun2.harvard.edu>
Subject FW: st: Any way to annotate a Stata graph with LaTeX?
Date Fri, 7 Aug 2009 09:56:54 -0500
This just came to me from a colleague. It may be out of date - but here it is, if it's of help.
Al Feiveson
-----Original Message-----
From: Fiedler, James (JSC-SK)[USRA]
Sent: Thursday, August 06, 2009 5:44 PM
To: Feiveson, Alan H. (JSC-SK311)
Subject: RE: st: Any way to annotate a Stata graph with LaTeX?
[A year later...] You know, I think I found exactly the right LaTeX package for this a while ago. It's called overpic:
http://www.nada.kth.se/~carsten/latex/overpic.html .
What's suggested below seems like more work.
From: Feiveson, Alan H. (JSC-SK311)
Sent: Monday, August 25, 2008 2:06 PM
To: Fiedler, James (JSC-SK)[STU]
Subject: FW: st: Any way to annotate a Stata graph with LaTeX?
Hi James - FYI -
-----Original Message-----
From: owner-statalist@hsphsun2.harvard.edu [mailto:owner-statalist@hsphsun2.harvard.edu] On Behalf Of Daniel Becker
Sent: Monday, August 25, 2008 1:59 PM
To: statalist@hsphsun2.harvard.edu
Subject: Re: st: Any way to annotate a Stata graph with LaTeX?
Am 25.08.2008 um 10:52 schrieb Richard Sperling:
> Has anyone developed a way or method to incorporate LaTeX formatted
> text into a Stata graph? It would be great if the Stata graph editor
> had the capability to generate LaTeX formatted annotations. For Mac
> users, I have in mind something along the lines of the program LaTeXiT
> that allows one to add snippets of LaTeX text to Powerpoint or Keynote
> presentations.
My way is: Exporting the Stata graph as an eps, then using a .tex file like the one posted below. This file is then typeset by the following sequence of commands aslo posted below (in TeXShop, a Mac-Frontend for LaTeX, I do this via an "engine file" epsoverlay)
If anyone is interested, I could provide more details how to prepare your system to do all this. It assumes that the command epspdf exists, for instance. Of course this is not very convenient, at first sight, but works for me. It is even possible to use pstricks ...
++++++++++++++++++++++ start sequence of commands
set path= ($path /usr/texbin /usr/local/bin) set filename = "$1"
set dviname = "${filename:r}.dvi"
set psname = "${filename:r}.ps"
set pdfname = "${filename:r}.pdf"
set pdfnamebb = "${filename:r}-bb.pdf"
set epsnamebb = "${filename:r}.eps"
set auxname = "${filename:r}.aux"
set logname = "${filename:r}.log"
# run latex via ghostscript on the tex-file latex --shell-escape "$1"
dvips -Ppdf -o "$psname" "$dviname"
ps2pdf13 -dAutoRotatePages=/None "$psname" "$pdfname"
# calculate bounding box
epspdf -bb "$pdfname" "$epsnamebb"
epspdf -bb "$pdfname" "$pdfnamebb"
# clean up
/bin/rm "$pdfname"
/bin/rm "$dviname"
/bin/rm "$auxname"
/bin/rm "$psname"
/bin/rm "$logname"
#rename pictures
mv "$pdfnamebb" "$pdfname"
+++++++++++++ start LaTeX file
%!TEX TS-program = epsoverlay
%%%% HERE: Load the eps-file
%%%% HERE: add LaTeX/pstricks
% grid to see coordinates, comment out when done \psgrid[subgriddiv=0,griddots=1,gridlabels=7pt]%
% Add labels etc here:
\rput{0}(5.9,3.9){$b = 1$}
\rput{0}(3.4,2.2){$b = 5 $ nicht verwendet} \rput{0}(2.9,1.2){$b = 10 $}
* For searches and help try:
* http://www.stata.com/help.cgi?search
* http://www.stata.com/support/statalist/faq
* http://www.ats.ucla.edu/stat/stata/
* For searches and help try:
* http://www.stata.com/help.cgi?search
* http://www.stata.com/support/statalist/faq
* http://www.ats.ucla.edu/stat/stata/
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C99 integer types
Eric Sosman <(E-Mail Removed)> wrote:
> On 7/30/2012 2:26 PM, Ronald Landheer-Cieslak wrote:
>> Eric Sosman <(E-Mail Removed)> wrote:
>>> On 7/30/2012 9:53 AM, Ronald Landheer-Cieslak wrote:
>>>> Barry Schwarz <(E-Mail Removed)> wrote:
>>>>> [...]
>>>>> While the contributors to this group may not be able to infer why the
>>>>> designers chose what they did does not imply the absence of a rationale.
>>>>> For example, while they are the same size, unsigned int and unsigned long
>>>>> have different conversion ranks. This may make a difference in the
>>>>> generated code and may have driven the compiler writers to choose one over the other.
>>>> Excuse my ignorance, but what's a "conversion rank"?
>>> Shorthand for "integer conversion rank," of course.
>>> C sometimes needs to convert values from one type to another
>>> before working with them. For example, you cannot compare an
>>> `unsigned short' and a `signed int' as they stand; you must first
>>> convert them to a common type and then compare the converted values.
>>> But what type should be chosen? Plausible arguments could be made
>>> for any of `unsigned short' or `signed int' or `unsigned int' or
>>> even `unsigned long', depending on the relative "sizes" of these
>>> types on the machine at hand.
>>> "Integer conversion rank" formalizes this notion of "size."
>>> In the old days there were only a few integer types and it was
>>> easy to enumerate the possible combinations. Things got more
>>> complicated when C99 not only introduced new integer types, but
>>> made the set open-ended: An implementation might support types
>>> like `int24_t' or `uint_least36_t', and we need to know where
>>> these fit with respect to each other and to generic types like
>>> `long'. For example, when you divide a `uint_least36_t' by a
>>> `long', what conversions occur? Inquiring masochists want to know.
>>> To this end, each integer type has an "integer conversion rank"
>>> that establishes a pecking order. Roughly speaking, "narrow" types
>>> have low ranks and "wide" types have higher ranks. It's all in
>>> section 6.3.1.1 of the Standard, which takes quite a bit of prose
>>> to express this "narrow versus wide" idea precisely -- but that's
>>> really all it's doing: narrow versus wide, and how to handle ties.
>>> Eventually, when C needs to perform "integer promotions" or
>>> "usual arithmetic conversions," its choice of target type for
>>> integers is driven by the integer conversion rank(s) of the original
>>> type(s) involved.
>> OK, so it basically formalizes the conversions that the integer types goes
>> through to end up with either something useful or something
>> implementation-defined, or both.
>> Reading the draft Barry pointed to, it doesn't seem to actually change any
>> of the rules as they were before - just formalize them, is that right?
> Pretty much, yes: It's formalized to make it work with
> implementation-defined integer types the Standard doesn't know
> about, or doesn't require, or doesn't fully specify.
>> (and
>> comparing a negative signed short to an unsigned long still yields
>> implementation-defined results).
> Within limits, yes. Let's work through it:
> - First, we consult 6.5.8 for the relational operators, and
> learn that the "usual arithmetic conversions" apply to
> both operands.
> - Over to 6.3.1.8 for the UAC's, where we learn that the
> "integer promotions" are performed on each operand,
> independently.
> - 6.3.1.1 describes the IP's. We find that `unsigned long'
> is unaffected. It takes a little more research, but we
> eventually find that `signed short' converts to `int'.
> - 6.3.1.3 tells us that this conversion preserves the
> original value, so we now have the `int' whose value
> is the same as that of the original `signed short'.
> - Back to 6.3.1.8 again to continue with the UAC's, now with
> an `unsigned long' and an `int' and working through the
> second level of "otherwise." There we find that we've got
> one signed and one unsigned operand, *and* the unsigned
> operand has the higher rank (consult 6.3.1.1 again). This
> tells us we must convert the `int' again, this time to
> `unsigned long'.
> - Over to 6.3.1.3 again for the details of the conversion,
> and if the `int' is negative we must use "the maximum
> value that can be represented in the new type" to finish
> converting. ULONG_MAX is an implementation-defined value,
> so this is where implementation-definedness creeps in.
> - ... and we're back to 6.5.8, with two `unsigned long' values,
> which we know how to compare.
A very thorough walk-through of the conversions indeed, thanks.
> Seems like quite a lot of running around for a "simple" matter,
> but consider: Before the first ANSI Standard nailed things down,
> different C implementations disagreed on how some comparisons should
> be done! Both the "unsigned preserving" and "value preserving" camps
> (see the Rationale) would have agreed on the particular example we've
> just worked through, but would have produced different results for
> some other comparisons. The Standard's complicated formalisms --
> including "integer conversion rank" -- are part of an attempt to
> eliminate or at least minimize such disagreements.
I didn't want to imply that I had any problem with the added complexity
(and don't think I had): I understand very well that there's a real need to
specify in detail how these sorts of conversions need to work.
However, I think it only works as expected if the signed integer type is a
two's complement type. 6.2.6.2s2 allows for three representations, two of
which won't work as expected when ULONG_MAX + 1 is "repeatedly added" as in
I've never worked with hardware that had anything other than two's
complement integers, but that is what I meant with the
"implementation-defined" bit.
Software analyst & developer --
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CS276 Lecture 27: Computational Zero Knowledge
Scribed by Madhur Tulsiani
In this lecture we begin the construction and analysis of a zero-knowledge protocol for the 3-coloring problem. Via reductions, this extends to a protocol for any problem in NP. We will only be able
to establish a weak form of zero knowledge, called “computational zero knowledge” in which the output of the simulator and the interaction in the protocol are computationally indistinguishable
(instead of identical). It is considered unlikely that NP-complete problem can have zero-knowledge protocols of the strong type we defined in the previous lectures.
As a first step, we will introduce the notion of a commitment scheme and provide a construction based on any one-way permutation.
1. Commitment Scheme
A commitment scheme is a two-phase protocol between a Sender and a Receiver. The Sender holds a message ${m}$ and, in the first phase, it picks a random key ${K}$ and then “encodes” the message using
the key and sends the encoding (a commitment to ${m}$) to the Receiver. In the second phase, the Sender sends the key ${K}$ to the Receiver can open the commitment and find out the content of the
message ${m}$.
A commitment scheme should satisfy two security properties:
• Hiding. Receiving a commitment to a message ${m}$ should give no information to the Receiver about ${m}$;
• Binding. The Sender cannot “cheat” in the second phase and send a different key ${K'}$ that causes the commitment to open to a different message ${m'}$.
It is impossible to satisfy both properties against computationally unbounded adversaries. It is possible, however, to have schemes in which the Hiding property holds against computationally
unbounded Receivers and the Binding property holds (under appropriate assumptions on the primitive used in the construction) for bounded-complexity Senders; and it is possible to have schemes in
which the Hiding property holds (under assumptions) for bounded-complexity Receivers while the Binding property holds against any Sender. We shall describe a protocol of the second type, based on
one-way permutations. The following definition applies to one-round implementations of each phase, although a more general definition could be given in which each phase is allowed to involve multiple
Definition 1 (Computationally Hiding, Perfectly Binding, Commitment Scheme) A Perfectly Binding and ${(t,\epsilon)}$-Hiding Commitment Scheme for messages of length ${\ell}$ is a pair of
algorithms ${(C,O)}$ such that
□ Correctness. For every message ${m}$ and key ${K}$,
$\displaystyle O(K,C(K,m)) = m$
□ ${(t,\epsilon)}$-Hiding. For every two messages ${m,m' \in \{ 0,1 \}^\ell}$, the distributions ${C(K,m)}$ and ${C(K,m')}$ are ${(t,\epsilon)}$-indistinguishable, where ${K}$ is a random key,
that is, for every algorithm ${A}$ of complexity ${\leq t}$,
$\displaystyle | \mathop{\mathbb P} [ A(C(K,m))=1] - \mathop{\mathbb P} [ A(C(K,m'))=1] | \leq \epsilon$
□ Perfectly Binding. For every message ${m}$ and every two keys ${K,K'}$,
$\displaystyle O(K',C(K,m)) \in \{ m, FAIL \}$
In the following we shall refer to such a scheme ${(C,O)}$ as simply a ${(t,\epsilon)}$-secure commitment scheme.
Given a one-way permutation ${f: \{ 0,1 \}^n \rightarrow \{ 0,1 \}^n}$ and a hard-core predicate ${P}$, we consider the following construction of a one-bit commitment scheme:
• ${C(K,m):= f(K) , m \oplus P(K)}$
• ${O(K,(c_1,c_2))}$ equals ${FAIL}$ if ${f(K) eq c_1}$, and ${P(K) \oplus c_2}$ otherwise.
Theorem 2 If ${P}$ is a ${(t,\epsilon)}$-secure hard core predicate for ${f}$, then the above construction is a ${(t-O(1),2\epsilon)}$-secure commitment scheme.
Proof: The binding property of the commitment scheme is easy to argue as the commitment is a permutation of the key and the message. In particular, given ${C(K,m) = (x,y)}$, we can find the unique $
{K}$ and ${m}$ that generate it as
$\displaystyle K = f^{-1}(x) ~~~~\mbox{and}~~~~ m = y \oplus P(K) = y\oplus P(f^{-1}(x))$
To prove the hiding property in the contrapositive, we want to take an algorithm which distinguishes the commitments of two messages and convert it to an algorithm which computes the predicate ${P}$
with probability better than ${1/2 + \epsilon}$. Let ${A}$ be such an algorithm which distinguishes two different messages (one of which must be 0 and the other must be 1). Then, we have that for $
$\displaystyle \begin{array}{rcl} &&\left\lvert \mathop{\mathbb P}[A(C(K,m)) = 1] - \mathop{\mathbb P}[A(C(K,m) = 1]\right\rvert > 2\epsilon \\ \implies && \left\lvert \mathop{\mathbb P}[A(f(K), P(K)
\oplus 0) = 1] - \mathop{\mathbb P}[A(f(K), P(K)\oplus 1) = 1]\right\rvert > 2\epsilon \end{array}$
Assume without loss of generality that the quantity in the absolute value is positive i.e.
$\displaystyle \mathop{\mathbb P}[A(f(K), P(K)) = 1] - \mathop{\mathbb P}[A(f(K), P(K)\oplus 1) = 1] > 2\epsilon$
Hence, ${A}$ outputs 1 significantly more often when given the correct value of ${P(K)}$. As seen in previous lectures, we can convert this into an algorithm ${A'}$ that predicts the value of ${P(K)}
$. Algorithm ${A'}$ takes ${f(K)}$ as input and generates a random bit ${b}$ as a guess for ${P(K)}$. It then runs ${A(f(K),b)}$. Since ${A}$ is correct more often on the correct value of ${P(K)}$, $
{A'}$ outputs ${b}$ if ${A(f(K),b) = 1}$ and outputs ${b \oplus 1}$ otherwise. We can analyze its success probability as below
$\displaystyle \begin{array}{rcl} && \mathop{\mathbb P}[A'(f(K)) = P(K)] = \mathop{\mathbb P}[b = P(K)] \cdot \mathop{\mathbb P}[A(f(K),P(K)) = 1] \\ &&+ \mathop{\mathbb P}[b eq P(K)] \cdot \mathop{\
mathbb P}[A(f(K),P(K)\oplus1) = 0] \\ &=& \frac12 \cdot \mathop{\mathbb P}[A(f(K),P(K)) = 1] \\ &&+ \frac12 \cdot \left( 1- \mathop{\mathbb P}[A(f(K),P(K)\oplus1) = 1]\right) \\ &=& \frac12 + \frac12
\cdot \left(\mathop{\mathbb P}[A(f(K),P(K)) = 1] - \mathop{\mathbb P}[A(f(K),P(K)\oplus1) = 1]\right)\\ &\geq& \frac12 + \epsilon \end{array}$
Thus, ${A'}$ predicts ${P}$ with probability ${1/2 + \epsilon}$ and has complexity only ${O(1)}$ more than ${A}$ (for generating the random bit) which contradicts the fact that ${P}$ is ${(t, \
epsilon)}$-secure. $\Box$
There is a generic way to turn a one-bit commitment scheme into a commitment scheme for messages of length ${\ell}$ (just concatenate the commitments of each bit of the message, using independent
Theorem 3 Let ${(O,C)}$ be a ${(t,\epsilon)}$-secure commitment scheme for messages of length ${k}$ such that ${O(\cdot,\cdot)}$ is computable in time ${r}$. Then the following scheme ${(\
overline C, \overline O)}$ is a ${t-O(r\cdot \ell), \epsilon \cdot l)}$-secure commitment scheme for message of length ${k\cdot \ell}$:
□ ${\overline C (K_1,\ldots,K_\ell,m):= C(K_1,m_1),\ldots, C(K_\ell,m_\ell)}$
□ ${\overline O (K_1,\ldots,K_\ell,c_1,\ldots,c_\ell)}$ equals ${FAIL}$ if at least one of ${O(K_i,c_i)}$ outputs ${FAIL}$; otherwise it equals ${O(K_1,c_1),\ldots,O(K_\ell,c_\ell)}$.
Proof: The commitment to ${m}$ is easily seen to be binding since the commitments to each bit of ${m}$ are binding. The soundness can be proven by a hybrid argument.
Suppose there is an ${A}$ algorithm distinguishing ${\overline C(K_1, \ldots, K_\ell,m)}$ and ${C(K_1, \ldots, K_\ell, m)}$ with probability more than ${\epsilon \cdot \ell}$. We then consider
“hybrid messages” ${m^{(0)}, \ldots, m^{(\ell)}}$, where ${m^{(i)} = m_1' \ldots m_i' m_{i+1}, \ldots, m_{\ell}}$. By a hybrid argument, there is some ${i}$ such that
$\displaystyle \left\lvert\mathop{\mathbb P}[A(K_1, \ldots, K_\ell, m^{(i)}) = 1] - \mathop{\mathbb P}[A(K_1, \ldots, K_\ell, m^{(i+1)}) = 1] \right\rvert > \epsilon$
But since ${m^{(i)}}$ and ${m^{(i+1)}}$ differ in only one bit, we can get an algorithm ${A'}$ that breaks the hiding property of the one bit commitment scheme ${C(\cdot,\cdot)}$. ${A'}$, given a
commitment ${c}$, outputs
$\displaystyle A'(c) ~=~ A(C(K_1,m_1), \ldots, C(K_i,m_i), c, C(K_{i+2}, m_{i+2}'), \ldots, C(K_\ell, m_\ell'))$
Hence, ${A'}$ has complexity at most ${t + O(r\cdot l)}$ and distinguishes ${C(K_{i+1}, m_{i+1})}$ from ${C(K_{i+1}, m_{i+1}')}$. $\Box$
There is also a construction based on one-way permutations that is better in terms of key length.
2. A Protocol for 3-Coloring
We assume we have a ${(t,\epsilon)}$-secure commitment scheme ${(C,O)}$ for messages in the set ${\{1,2,3\}}$.
The prover ${P}$ takes in input a 3-coloring graph ${G=([n],E)}$ (we assume that the set of vertices is the set ${\{1,\ldots,n\}}$ and use the notation ${[n] := \{1,\ldots,n\}}$) and a proper
3-coloring ${\alpha : [n] \rightarrow \{ 1,2,3\}}$ of ${G}$ (that is, ${\alpha}$ is such that for every edge ${(u,v)\in E}$ we have ${\alpha (u) eq \alpha(v)}$). The verifier ${V}$ takes in input $
{G}$. The protocol, in which the prover attempts to convince the verifier that the graph is 3-colorable, proceeds as follows:
• The prover picks a random permutation ${\pi: \{1,2,3\} \rightarrow \{1,2,3\}}$ of the set of colors, and defines the 3-coloring ${\beta(v) := \pi(\alpha(v))}$. The prover picks ${n}$ keys ${K_1,\
ldots,K_n}$ for ${(C,O)}$, constructs the commitments ${c_v := C(K_v,\beta(v))}$ and sends ${(c_1,\ldots,c_n)}$ to the verifier;
• The verifier picks an edge ${(u,v) \in E}$ uniformly at random, and sends ${(u,v)}$ to the prover;
• The prover sends back the keys ${K_u,K_v}$;
• If ${O(K_u,c_u)}$ and ${O(K_v,c_v)}$ are the same color, or if at least one of them is equal to ${FAIL}$, then the verifier rejects, otherwise it accepts
Theorem 4 The protocol is complete and it has soundness error at most ${(1-1/|E|)}$.
Proof: The protocol is easily seen to be complete, since if the prover sends a valid 3-coloring, the colors on endpoints of every edge will be different.
To prove the soundness, we first note that if any commitment sent by the prover opens to an invalid color, then the protocol will fail with probability at least ${1/|E|}$ when querying an edge
adjacent to the corresponding vertex (assuming the graph has no isolated vertices – which can be rivially removed). If all commitments open to valid colos, then the commitments define a 3-coloring of
the graph. If the graph is not 3-colorable, then there must be at least one edge ${e}$ both of whose end points receive the same color. Then the probability of the verifier rejecting is at least the
probability of choosing ${e}$, which is ${1/|E|}$. $\Box$
Repeating the protocol ${k}$ times sequentially reduces the soundness error to ${(1-1/|E|)^k}$; after about ${27\cdot |E|}$ repetitions the error is at most about ${2^{-40}}$.
3. Simulability
We now describe, for every verifier algorithm ${V^*}$, a simulator ${S^*}$ of the interaction between ${V^*}$ and the prover algorithm.
The basic simulator is as follows:
Algorithm ${S_{1round}^*}$
• Input: graph ${G=([n],E)}$
• Pick random coloring ${\gamma : [n] \rightarrow \{1,2,3\}}$.
• Pick ${n}$ random keys ${K_1,\ldots,K_n}$
• Define the commitments ${c_i := C(K_i, \gamma(i))}$
• Let ${(u,v)}$ be the 2nd-round output of ${V^*}$ given ${G}$ as input and ${c_1,\ldots,c_n}$ as first-round message
• If ${\gamma(u) = \gamma(v)}$, then output FAIL
• Else output ${((c_1,\ldots,c_n),(u,v),(K_u,K_v))}$
And the procedure ${S^*(G)}$ simply repeats ${S^*_{1round} (G)}$ until it provides an output different from ${FAIL}$.
It is easy to see that the output distribution of ${S^*(G)}$ is always different from the actual distribution of interactions between ${P}$ and ${V^*}$: in the former, the first round is almost
always a commitment to an invalid 3-coloring, in the latter, the first round is always a valid 3-coloring.
We shall prove, however, that the output of ${S^*(G)}$ and the actual interaction of ${P}$ and ${V^*}$ have computationally indistinguishable distributions provided that the running time of ${V^*}$
is bounded and that the security of ${(C,O)}$ is strong enough.
For now, we prove that ${S^*(G)}$ has efficiency comparable to ${V^*}$ provided that security of ${(C,O)}$ is strong enough.
Theorem 5 Suppose that ${(C,O)}$ is ${(t+O(nr),\epsilon/(n\cdot |E|))}$-secure and ${C}$ is computable in time ${\leq r}$ and that ${V^*}$ is a verifier algorithm of complexity ${\leq t}$.
Then the algorithm ${S^*_{1round}}$ as defined above has probability at most ${\frac 13 + \epsilon}$ of outputting ${FAIL}$.
The proof of Theorem 5 relies on the following result.
Lemma 6 Fix a graph ${G}$ and a verifier algorithm ${V^*}$ of complexity ${\leq t}$.
Define ${p(u,v,\alpha)}$ to be the probability that ${V^*}$ asks the edge ${(u,v)}$ at the second round in an interaction in which the input graph is ${G}$ and the first round is a commitment to
the coloring ${\alpha}$.
Suppose that ${(C,O)}$ is ${(t + O(nr) ,\epsilon/n)}$-secure, and ${C}$ is computable in time ${\leq r}$.
Then for every two colorings ${\alpha,\beta}$ and every edge ${(u,v)}$ we have
$\displaystyle | p(u,v,\alpha) - p(u,v,\beta) | \leq \epsilon$
Proof: If ${p(u,v,\alpha)}$ and ${p(u,v,\beta)}$ differ by more than ${\epsilon}$ for any edge ${(u,v)}$, then we can define an algorithm which distinguishes the ${n}$ commitments corresponding to $
{\alpha}$ from the ${n}$ commitments corresponding to ${\beta}$. ${A}$ simply runs the verifier given commitments for ${n}$ colors and outputs 1 if the verifier selects the edge ${(u,v)}$ in the
second round.
Then, by assumption, ${A}$${\epsilon}$-distinguishes the ${n}$ commitments corresponding to ${\alpha}$ from the ${n}$ commitments corresponding to ${\beta}$ in time ${t + O(nr)}$. However, by Theorem
3, this means that ${(C,O)}$ is not ${(t + O(nr), \epsilon/n)}$-secure which is a contradiction. $\Box$
Given the lemma, we can now easily prove the theorem.
Proof: (of Theorem 5) The probability that ${S^*_{1round}}$ outputs ${FAIL}$ is given by
$\displaystyle \mathop{\mathbb P}\left[ S^*_{1round} ~=~ FAIL\right] ~=~ \frac{1}{3^n} \cdot \sum_{c \in \{1,2,3\}^n} \sum_{(u,v) \in E \atop c(u) eq c(v)} p(u,v,c)$
Let ${{\bf 1}}$ denote the coloring which assigns the color 1 to every vertex. Then using Lemma 6 we bound the above as
$\displaystyle \begin{array}{rcl} \mathop{\mathbb P}\left[ S^*_{1round} = FAIL\right] &\leq& \frac{1}{3^n} \cdot \sum_{c \in \{1,2,3\}^n} \sum_{(u,v) \in E \atop c(u) eq c(v)} (p(u,v,{\bf 1}) + \
epsilon)\\ &=& \sum_{(u,v) \in E} p(u,v,{\bf 1}) \left(\sum_{c:c(u) eq c(v)} \frac{1}{3^n}\right) ~+~ \epsilon\\ &=& \frac{1}{3} \sum_{(u,v) \in E} p(u,v,{\bf 1}) + ~\epsilon~\\ &=& \frac{1}{3} + \
epsilon \end{array}$
where in the second step we used the fact that ${c(u) eq c(v)}$ for a ${1/3}$ fraction of all the colorings and the last step used that the probability of ${V^*}$ selecting some edge given the
coloring ${{\bf 1}}$ is 1. $\Box$
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Series (2 related exercises)
August 1st 2010, 02:43 PM #1
Series (2 related exercises)
Given the series $\sum _{n=-\infty}^{+ \infty} \frac{z^n}{10^{|n|}}$, calculate its region of convergence and the function it represents.
Attempt: $\sum _{n=-\infty}^{\infty} \frac{z^n}{10^{|n|}}=2 \sum _{n=0}^{+ \infty} \frac{z^n}{10^n}-1=2\left (\frac{1}{1-\frac{z}{10}} \right ) -1$ for $|z|<10$. Which is equal to $\frac{10+z}
{10-z}=f(z)$. The radius of convergence is 10, so the region is a disk (without the border) centered at 0 with radius 10. Is that good?
2)Is $\sum _{n=-\infty}^{\infty} z^n$ a Laurent series of a function? Why?
Attempt: No, it isn't. Because $\sum _{n=-\infty}^{\infty} z^n=\sum _{n=0}^{+\infty} z^n+ \left ( \sum _{n=-\infty}^{0} z^n \right ) -1$ which would be equal to $\frac{1}{1-z}+\frac{z}{z-1}-1$.
The problem is that $\sum _{n=0}^{+\infty} z^n=\frac{1}{1-z}$ for $|z|<1$ and $\sum _{n=-\infty}^{0} z^n = \frac{z}{z-1}$ for $|z|>1$. Since the modulus cannot be greater and lesser than 1 at a
time, the series doesn't converge to any function. (Yeah I know, I didn't prove it well. How would you solve the exercise?)
You have a mistake in your first series since when you reduced it from a doubly infinite to an infinite series, you forgot that the sign flips on the $z^n$.
The calculation is actually:
[LaTeX ERROR: Convert failed]
[LaTeX ERROR: Convert failed]
Your analysis on the radius of convergence is correct for the calculation you made. Now for this one that I've shown, what is the radius of convergence? I like your analysis in the attempt for
the second part and as far as I can tell it has the right idea. Here is what I think:
[LaTeX ERROR: Convert failed]
Now, the left sum only converges for |z|<1 and the right sum only converges if |z|>1, so the overal expression diverges for all z in the complex plane. The series doesn't converge at all, let
alone to some function.
P.S. I notice you are a bit iffy with the indices of summation, so be careful.
Last edited by Vlasev; August 1st 2010 at 10:13 PM.
Ok thanks, I see.
Your analysis on the radius of convergence is correct for the calculation you made. Now for this one that I've shown, what is the radius of convergence?
Hmm which series? For $\sum _{n=0}^{\infty} \frac{1}{(10z)^{n}}$, I get that it's worth 1/10. While for $\sum _{n=0}^{\infty} \frac{z^n}{10^{n}}$, I get 10... I'm obviously wrong. Hmm it's
impossible, what do you get?
I like your analysis in the attempt for the second part and as far as I can tell it has the right idea. Here is what I think:
[LaTeX ERROR: Convert failed]
Now, the left sum only converges for |z|<1 and the right sum only converges if |z|>1, so the overal expression diverges for all z in the complex plane. The series doesn't converge at all, let
alone to some function.
P.S. I notice you are a bit iffy with the indices of summation, so be careful.
You are partially right. To get the answer though, consider the region of convergence for each series. The second series converges for $|z|<1$0 as you said, because if $0<|z|<10$, then $0<|z/10|
<1$ and then the geometric sum converges. Now for the other one you have to be careful. Here's how I would do it:
You want $0<|1/(10z)|<1$. Then take the exponent -1 of it to get:
$\infty >|10 z| > 1$
Then divide out by 10 to finally get
$\infty > |z |> 1/10$
So in this case you want $|z| > 1/10.$
Finally, you just have the region of convergence being $1/10 < |z| < 10$, since this is where both series converge.
Wow, thanks a lot, that was very helpful and useful to know.
August 1st 2010, 09:46 PM #2
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Jul 2010
August 2nd 2010, 01:24 PM #3
August 2nd 2010, 01:36 PM #4
Senior Member
Jul 2010
August 2nd 2010, 01:43 PM #5
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Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, Vol. 42, No. 2, pp. 557-573 (2001)
Finite and Infinite Collections of Multiplication Modules
Majid M. Ali, David J. Smith
Department of Mathematics, University of Auckland, Auckland, New Zealand, e-mail: majid@math.auckland.ac.nz; smith@math.auckland.ac.nz
Abstract: All rings are commutative with identity and all modules are unitary. In this note we give some properties of a finite collection of submodules such that the sum of any two distinct members
is multiplication, generalizing those which characterize arithmetical rings. Using these properties we are able to give a concise proof of Patrick Smith's theorem stating conditions ensuring that the
sum and intersection of a finite collection of multiplication submodules is a multiplication module. We give necessary and sufficient conditions for the intersection of a collection (not necessarily
finite) of multiplication modules to be a multiplication module, generalizing Smith's result. We also give sufficient conditions on the sum and intersection of a collection (not necessarily finite)
for them to be multiplication. We apply D. D. Anderson's new characterization of multiplication modules to investigate the residual of multiplication modules.
Keywords: multiplication module, multiplication ideal, Pr\"{u}fer domain, arithmetical ring, radical, direct sum, prime submodule, residual, torsion module
Classification (MSC2000): 13C05; 13A15
Full text of the article:
[Previous Article] [Next Article] [Contents of this Number] © 2001 ELibM for the EMIS Electronic Edition
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Group items tagged
A great new resource from the creator of 'A Maths Dictionary for Kids'. Download and print beautifully designed and wonderfully useful maths posters on a good range of topics. Your classroom
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Martin Burrett on 11 Jan 12
A good maths game where you need to find the unknown numbers. Good as an introduction to algebra.
Martin Burrett on 02 Aug 11
A superb site with lots and fun maths games on a wide range of topics across the maths curriculum.
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Application of Perturbation Theory in Classical Mechanics
Application of Perturbation Theory
Classical Mechanics
- Shashidhar Guttula
• Classical Mechanics
• Perturbation Theory
• Applications of the theory
• Simulation of Mechanical systems
• Conclusions
• References
Classical Mechanics
• Minimum Principles
• Central Force Theorem
• Rigid Body Motion
• Oscillations
• Theory of Relativity
• Chaos
Perturbation Theory
• Mathematical Method used to find an
approximate solution to a problem which
cannot be solved exactly
• An expression for the desired solution in terms
of a *power series
Method of Perturbation theory
• Technique for obtaining approx solution based
on smallness of perturbation Hamiltonian and
on the assumed smallness of the changes in the
– If the change in the Hamiltonian is small,
the overall effect of the perturbation on the
motion can be large
• Perturbation solution should be carefully
analyzed so it is physically correct
Classical Perturbation theory
• Time Dependent Perturbation theory
• Time Independent Perturbation theory
– Classical Perturbation Theory is more
complicated than Quantum Perturbation
– Many similarities between classical
perturbation theory and quantum perturbation
Solve :Perturbation theory problems
• A regular perturbation is an equation of the form : D (x; φ)=0
– Write the solution as a power series :
• xsol=x0+x1+x2+x3+…..
– Insert the power series into the equation and rearrange to a
new power series in
• D(xsol;”)=D(x0+x1+x2+x3+…..);
– Set each coefficient in the power series equal to zero and
solve the resulting systems
• P0(x0;0)=D(x0;0)=0
• P1(x0;x1)=0
• P2(x0;x1;x2)=0
Idea applies in many contexts
• To Obtain
– Approximate solutions to algebraic and
transcendental equations
– Approximate expressions to definite
– Ordinary and partial differential equations
Perturbation Theory Vs Numerical Techniques
• Produce analytical approximations that reveal
the essential dependence of the exact solution
on the parameters in a more satisfactory way
• Problems which cannot be easily solved
numerically may yield to perturbation method
• Perturbation analysis is often Complementary
to Numerical methods
Applications in Classical Mechanics
• Projectile Motion
• Damped Harmonic Oscillator
• Three Body Problem
• Spring-mass system
Projectile Motion
• In 2-D,without air resistance parameters
– Initial velocity:V0 ; Angle of elevation :θ
• Add the effect of air resistance to the motion
of the projectile
– Equations of motion change
– The range under this assumption decreases.
– *Force caused by air resistance is directly
proportional to the projectile velocity
Force Drag k << g/V
Effect of air resistance : projectile motion
U kT
R (1 e )
kV g kT
T (1 e )
4k V
R Ro (1 )
Range Vs Retarding Force Constant ‘k’ from P.T
Damped Harmonic Oscillator
• Taking
• Putting
Harmonic Oscillator (contd.)
• First Order Term
• Second Order Term
• General Solution through perturbation
• Exact Solution
Three Body Problem
• The varying perturbation of the Sun’s gravity on the
Earth-Moon orbit as Earth revolves around the Sun
– Secular Perturbation theory
• Long-period oscillations in planetary orbits
• It has the potential to explain many of the
orbital properties of these systems
• Application for planetary systems with three or
four planets
• It determines orbital spacing, eccentricities and
inclinations in planetary systems
Spring-mass system with no damping
d 2x
m 2 kx F
Input :Impulse Signal
Displacement Vs Time
Spring-mass system with damping factor
d 2x
m 2 b bo k x F
dt dt
Input Impulse Signal
Displacement Vs Time
• Use of Perturbation theory in mechanical systems
• Math involved in it is complicated
• Theory which is vast has its application
– Quantum Mechanics
– High Energy Particle Physics
– Semiconductor Physics
• Its like an art must be learned by doing
• Classical Dynamics of particles and systems ,Marion
&Thornton 4th Edition
• Classical Mechanics, Goldstein, Poole & Safko,
Third Edition
• A First look at Perturbation theory ,James
G.Simmonds & James E.Mann,Jr
• Perturbation theory in Classical Mechanics, F M
Fernandez,Eur.J.Phys.18 (1997)
• Introduction to Perturbation Techniques ,Nayfeh. A.H
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• You can poke a $a+b = c$ anywhere in your text; typing $$a+b = c$$ will make the formula behave like a block between paragraphs % is used to include comments.
Netvouz - new bookmarks
• April 4th, 2009 at 6: 31 pm a+b = what my mom feels ..
Well then, never you mind.
• Take two vectors a and b in a Eucledian space and add them to form vector c = a+b.
Are Changes Brewing and How Does the Mind Fit In?
• If two variables a and b are highly correlated, orthogonalization would just replace them by something like a+b and a-b.
Multicollinearity and Micronumerosity, Bryan Caplan | EconLog | Library of Economics and Liberty
• When ai tole himz ai seed it he shook himz hed an said da upper mafs nawt fore mi cause dere was no wae to “see da a+b=x to da 9th power”.
AHHHH… - Lolcats 'n' Funny Pictures of Cats - I Can Has Cheezburger?
• The ring of numbers of the form a+b 2 where a and b are rational, because a+b 2 is zero only when
• The group of numbers of the form a+b 2 where a and b are integer, because
• Let the ages at first be x, y, (x + y) Now, if a+b = 2c, then (a-n) + (b-n) = 2 (c-n), whatever be the value of n.
A Tangled Tale
• The song of a yellow warbler in the school maples, the whirl of scarlet leaves across the window pane, or the gleam of snow on the far-off hilltops, would drive away every item of knowledge
concerning the value of (a+b) 2 or the characteristics of a parallelogram.
'Lizbeth of the Dale
• Apres cela vient un mathematicien qui vous bourre avec des a+b et vous rapporte enfin un x+y, dont vous n'avex pas besoin et qui ne change nullement vos relations avec la vie.
Autocrat of the Breakfast Table
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Michel Baudin's Blog
See on Scoop.it - lean manufacturing
Blog post at Lean Blog : The price paid for most management consulting work is based on either a daily rate or some variation of a flat-rate fee based on what is being delivered. Enterprise software
pricing is also often fixed. In both cases, the client pays this with some expectation of benefits and even an “ROI” for the customer.[..]
Michel Baudin's comments:
I agree with Mark, and I am happy when clients report that they get ten times in benefits what our services cost. A daily fee for work done on site and a fixed fee for deliverables for offsite work
are simple arrangements; paying a percentage of benefits, whether cost savings or revenue increases, is a complicated arrangement, conducive to misunderstandings and disagreements.
See on www.leanblog.org
Renault: An international school of Lean Manufacturing opens at Flins | Automotive World
See on Scoop.it - lean manufacturing
"Jose-Vicente de Los Mozos, Executive Vice President, Manufacturing and Supply Chain, of the Renault Group, inaugurated the International School of Lean Manufacturing yesterday."
Michel Baudin's insight:
When I visited this plant in 1994, I never imagined that it would be the site of an international school of Lean 20 years later.
We were working at the time on Lean implementation with CIADEA, the Renault licensee in Argentina. It had originally been a subsidiary, was sold to local entrepreneur Manuel Antelo in 1992, and was
repurchased by Renault in 1997.
At the time, my hosts in Flins thought that Lean was just a way to cut heads and that implementing 5S would cause production to drop.
Times change.
See on www.automotiveworld.com
Is it Lean’s Fault or the Old Management System’s? | Mark Graban
See on Scoop.it - lean manufacturing
Blog post at Lean Blog :
"[...]The problem is the culture doesn’t change overnight. Leaders have years or decades of old habits (bad habits) that run counter to Lean thinking. They might be (might!) be trying to change,
but people will still fall back into old habits, especially when under pressure.
I hear complaints (in recent cases) coming from different provinces in Canada that say things like:
Lean is causing hospitals to be “de-skilled” by replacing nurses with aides. Lean drives a focus on cost and cost cutting, including layoffs or being understaffedLean is stressing out managers by
asking them to do more and taking nothing off their plateNurses hate Lean because they aren’t being involved in changes[...]"
Michel Baudin's comments:
In this post, Mark Graban explains how the leadership in Canadian hospitals is slapping the "Lean" label on ancient and counterproductive "cost-cutting" methods, and how the victims of these
practices unfairly blame Lean.
This is definitely L.A.M.E., Mark's apt term for "Lean As Misguidedly Executed," and is found in Manufacturing as well as Health Care. Much of the article -- and of the discussion that follows -- is
about what I call yoyo staffing: you hire more than you should in boom times, and lay off in recessions.
Of course, it isn't what Toyota did, and churning your work force in this fashion not only disrupts people's lives but is bad business. Hiring, training and firing repeatedly prevents your
organization from accumulating the knowledge and skills it needs.
Mark makes the case that Lean should not be blamed for mistakes that have nothing to do with it. Other than raising consciousness, however, the post does not propose solutions to keep this from
While there have been studies published on Toyota's approach to Human Resources (HR), I don't recall seeing much in the American Lean literature on topics like career planning for production
In his comments, Bob Emiliani paints the current generation of leaders as "a lost cause," and places his hopes on the next. He seems to suggest that the solution is to wait out or fire the current,
baby-boomer leadership and replace it with millenials. I don't buy it and, deep down, neither does Bob, because he ends by saying "While one always hopes the “next generation will do better”, it
could turn out to be a false hope."
Like everything in HR, generational change has to be planned carefully. The people who rose to leadership positions presumably did so not just because of bad habits but because they also had
something of value to offer. And the way the baton is passed is also a message to the incoming leaders: it tells them what to expect when their turn comes.
See on www.leanblog.org
A Definition of Lean | Mike Rother
See on Scoop.it - lean manufacturing
Maybe it's time for a better definition of "Lean." Here's one for you to consider and build on.
Michel Baudin's comments:
The proposal is "Lean is the permanent struggle to flow value to one customer."
Permanent struggle is fine, but I prefer pursuit. It means the same thing but it is shorter and "pursuit of happiness" sounds better than "permanent struggle for happiness."
On the other hand, I have a problem with "flow value," which I see as the sort of vague abstraction that would prompt Mike Harrison to ask whether it come in bottles. It is exactly what Dan Heath is
warning against in the video included in the slideshare.
I also have a problem with the exclusive focus on customers, which I see as Business 101 rather than Lean. Lean includes many features like heijunka, that are intended to make life easier for
suppliers and are transparent to customers. Going Lean means looking after all the stakeholders of the business, not just its customers.
This is why I define it instead as the pursuit of concurrent improvement in all dimensions of manufacturing performance through projects that affect both the production shop floor and support
Yes, I know, it is specific to manufacturing, but that is not my problem.
See on www.slideshare.net
Averages in Manufacturing Data
The first question we usually ask about lead times, inventory levels, critical dimensions, defective rates, or any other quantity that varies, is what it is "on the average." The second question is
how much it varies, but we only ask it if we get a satisfactory answer to the first one, and we rarely do.
When asked for a lead time, people usually give answers that are either evasive like "It depends," or weasel-worded like "Typically, three weeks." The beauty of a "typical value" is that no such
technical term exists in data mining, statistics, or probability, and therefore the assertion that it is "three weeks" is immune to any confrontation with data. If the assertion had been that it was
a mean or a median, you could have tested it, but, with "typical value," you can't.
For example, if the person had said "The median is three weeks," it would have had the precise meaning that 50% of the orders are delivered in less than 3 weeks, and that 50% take longer. If the
3-week figure is true, then the probability of the next 20 orders all taking longer, is $0.5^{20}= 9.6\,ppm$. This means that, if you do observe a run of 20 orders with lead times above 3 weeks, you
know the answer was wrong.
In Out of the Crisis, Deming was chiding journalists for their statistical illiteracy when, for example, they bemoaned the fact that "50% of the teachers performed beneath the median." In the US,
today, the meaning of averages and medians is taught in Middle School, but the proper use of these tools does not seem to have been assimilated by adults.
One great feature of averages is that they add up: the average of the sum of two variables is the sum of their averages. If you take two operations performed in sequence in the route of a product,
and consider the average time required to go through these operations by different units of product, then the average time to go through operations 1 and 2 is the sum of the average time through
operation 1 and the average time through operation 2, as is obvious from the way an average is calculated. If you have n values $X_{1},...,X_{n}$
the average is just
$\bar{X}= \frac{X_{1}+...+X_{n}}{n}$
What is often forgotten is that most other statistics are not additive.
To obtain the median, first you need to sort the data so that $X_{\left(1\right)}\leq ... \leq X_{\left(n\right)}$. For each point, the sequence number then tells you how many other points are under
it, which you can express as a percentage and plot as in the following example:
Graphically, you see the median as the point on the x-axis where the curve crosses 50% on the y-axis. To calculate it, if n is odd, you take the middle value
$\tilde{X}= X_{_{\left (\frac{n}{2}+1\right )}}$
and, if n is even, you take the average of the two middle values, or
$\tilde{X}= \frac{\left[ X_{_{\left (\frac{n}{2}\right )}}+X_{_{\left (\frac{n}{2}+1\right )}}\right]}{2}$
and it is not generally additive, and neither are all the other statistics based on rank, like the minimum, the maximum, quartiles, percentiles, or stanines.
An ERP system, for example, will add operation times along a route to plan production, but the individual operation times input to the system are not averages but worst-case values, chosen so that
they can reliably be achieved. The system therefore calculates the lead time for the route as the sum of extreme values at each operation, and this math is wrong because extreme values are not
additive. The worst-case value for the whole route is not the sum of the worst-case values of each operation, and the result is an absurdly long lead time.
In project management, this is also the key difference between the traditional Critical Path Method (CPM) and Eli Goldratt's Critical Chain. In CPM, task durations set by the individuals in charge of
each task are set so that they can be confident of completing them. They represent a perceived worst-case value for each task, which means that the duration for the whole critical path is the sum of
the worst-case values for the tasks on it. In Critical Chain, each task duration is what it is actually expected to require, with a time buffer added at the end to absorb delays and take advantage of
early completions.
That medians and extreme values are not additive is experienced, if not proven, by a simple simulation in Excel. Using the formula "LOGNORM.INV(RAND(),0,1)" will give you in about a second, 5,000
instances of two highly skewed variables, X and Y, as well as their sum X+Y. On a logarithmic scale, their histograms look as follows:
And the summary statistics show the Median, Minimum and Maximum for the sum are not the sums of the values for each term:
Averages are not only additive but have many more desirable properties, so why do we ever consider medians? There are real problems with averages, when taken carelessly:
1. Averages are affected by extreme values. It is illustrated by the Bill Gates Walks Into a Bar story. Here we inserted him into a promotional picture of San Fancisco's Terroir Bar:
2. Averages are meaningless over heterogeneous populations. The statement that best explains this is "The average American has exactly one breast and one testicle." It says nothing useful about
the American population. In manufacturing, when you consider, say, a number of units produced, you need to make sure you are not commingling 32-oz bottles with minuscule free samples.
3. Averages are meaningless for multiplicative quantities. If you data is the sequence $Y_{1}, ...,Y_{n}$ of yields of the n operations in a route, then the overall yield is $Y= Y_{1}\times ...\
times Y_{n}$, and the plain average of the yields is irrelevant. Instead, you want the geometric mean $\bar{Y}=\sqrt[n]{Y_{1}\times ...\times Y_{n}}$.
The same logic applies to the compounding of interest rates, and the plain average of rates over several years is irrelevant.
4. Sometimes, averages do not converge when the sample size grows. It can happen even with a homogeneous population, it is not difficult to observe, and it is mind boggling. Let us say your
product is a rectangular plate. On each one you make, you measure the differences between their actual lengths and widths and the specs, as in the following picture:
Assume then that, rather than the discrepancies in length and width, you are interested in the slope ΔW/ΔL and calculate its average over an increasing number of plates. You are then
surprised to find that, no matter how many data points you add, the ratio keeps bouncing around instead of converging as the law of large numbers has led you to expect. So far, we have looked
at the averages as just a formula applied to data. To go further, we must instead consider that they are estimators of the mean of an "underlying distribution" that we use as a model of the
phenomenon at hand. Here, we assume that the lengths and widths of the plates are normally distributed around the specs. The slope ΔW/ΔL is then the ratio of two normal variables with 0 mean,
and therefore follows the Cauchy distribution. This distribution has the nasty property of not having a mean, as a consequence of which the law of large numbers does not apply. But it has a
median, which is 0.
The bottom line is that you should use averages whenever you can, because you can do more with them than with the alternatives, but you shouldn't use them blindly. Instead, you should do the
1. Clean your data.
2. Identify and filter outliers.
3. Make sure that the data represents a sufficiently homogeneous population.
4. Use geometric means for multiplicative data.
5. Make sure that averaging makes sense from a probability standpoint.
As Kaiser Fung would say, use your number sense.
Forthcoming book: The Deming Legacy
About two years ago, I started posting essays on this blog about Deming's 14 points and their current relevance. Now I am writing on Points 11.a and 12 through 14, which I have not covered yet,
organizing the material, and editing it into an eBook entitled The Deming Legacy, that will be available shortly in PDF, iBook and Kindle formats. If you are interested, please visit the site and let
me know. Comments here are also welcome.
The posts on the topic to date are as follows:
The title is a ploy to convince Matt Damon to play Deming in the movie version.
Lean Handbags and Micro Failures | Mark Graban
See on Scoop.it - lean manufacturing
Blog post from Mark Graban at Lean Blog :
"I enjoy reading the magazine Inc. for my interests in startups and entrepreneurship. There are often examples and case studies that directly reference Lean thinking or just sound like Lean and
Kaizen with another label..."
Michel Baudin's comments:
Well run businesses are always good reading, even if their stories are usually embellished. Starting the design of fashion accessories from a market price or organizing to allow chefs in a restaurant
chain to experiment with new dishes, however, just sounds like good management, not examples of "Lean Thinking."
I have never found much depth in the contrasting of "Margin = Price – Cost" with "Price = Cost + Margin," maybe because I have never worked in a cost-plus business. Commercial manufacturers usually
do not have the power to set prices this way. Perhaps, the Big Three US automakers did have that power in the 1950s, and Toyota didn't.
In Tracy Kidder's 1985 documentary book House, a Boston lawyer hired a local contractor named to build a house in the suburbs. The contractor rigorously calculated the costs of the materials and
labor, tacked on a 10% profit, and presented a bid with no wiggle room. It was not intended for negotiation, but the lawyer just had to wrangle some concession out of the contractor. The culture
clash between the two makes great reading, but also throws light on how "cost-plus" works in practice.
The equation "Margin = Price - Cost" is based on the assumption that Price and Cost are characteristics of the same nature, both attached to each unit of product. It is true of Price: whenever a unit
is sold -- in whatever form and however it is financed -- it has a unit price, and it is not ambiguous.
Unit cost, on the other hand, is the result of allocations among products and over time done in a myriad different ways, with different results. By shifting overhead around, managers make the
products they like appear cheap, and the ones they want to kill appear expensive. Once the "expensive" products are terminated, the same overhead is spread among fewer survivors, thus making new ones
unprofitable, and the death spiral ends only with closure of the factory.
Instead of the simplistic "Margin = Price - Cost" for each unit, a sound economic analysis of manufacturing considers the flows of revenues and expenses associated with making a product in given
volumes over its life cycle, and sometimes a product family rather than an individual products with, for example, some products given away as free samples to promote the sale of other products.
See on www.leanblog.org
Is OEE a Useful Key Performance Indicator? | Jeffrey Liker
See on Scoop.it - lean manufacturing
"For manufacturing that is equipment-intensive, how the equipment works is often the main factor in productivity. Total Productive Maintenance (TPM) has become a buzzword in lean and a generally
accepted metric is Overall Equipment Effectiveness (OEE). This is measured as the product of three factors:
• OEE = Availability x Performance x Quality
• Availability = run time/total time
• Performance = Total count of parts/target count (based on a standard)Quality = Good count/Total count
Ignacio S. Gatell questions whether companies using OEE really understand it, can explain it clearly to their customers, and understand what it means to compare OEE as a KPI across plants. He
questions whether even plant managers understand how it is calculated and what it means.
The only good argument for OEE is that at a macro-level in a plant it provides a high level picture of how your equipment is functioning."
Michel Baudin's insight:
About 15 years ago, a summer intern came to work at a client plant in aerospace machining. I thought a great project for him would have been to identify a common tooling package for machining centers
that were grouped in a "Flexible Manufacturing System" (FMS). It was challenging, but it would have actually given the FMS the flexibility it was supposed to have. It was a real engineering project
that would have improved performance.
Management, however, decided that a better use of his time was to collect data and calculate OEEs for another set of machines. It did keep the student busy all summer, but resulted in no change, and
no improvement bragging rights for the student.
I have had a problem with OEE ever since. It is an overly aggregated and commonly gamed metric that you can only use by breaking it down into its constituent factors; you might as well bypass this
step and go straight to the factors.
Among these factors, I find Availability to be most often confused with Uptime. The availability of a device is the probability that it works when you need it, and the total time in the denominator
has to be the time you need it for. For example, if you work two shifts a day, the availability of a machine is not affected by your taking it down for maintenance on third shift. There have been
cases of managers overproducing to increase run time and thereby boost the OEE of their machines...
See on www.industryweek.com
Shortage of skills, not yet - but very soon - a wake up call (part 2) | Wiegand's Watch
This is a translation of the bulk of Bodo Wiegand's latest newsletter, about Lean in Germany, followed by my comments:
"In Part 1, we discussed the possibility of becoming more effective in your own work environment by stemming the flood of email and reducing the extent of meetings. In Part 2 , we want to focus
on how you can optimize cooperation between employees and departments.
In production, there are precise procedures and instructions , on how a product is to be made. There, the processes are stable , documented and visual. We have not considered this to be necessary
in support departments. Everyone works as he sees fit , then delivers when he is ready and at the quality he is capable of.
Sorry - in administration, we produce nothing .
We don't! Or do we?
In any case, work is not done according to a plan or delivered just in time at a precisely defined quality. Don't we need to? We do! We need to gradually start to handle administrative processes
like production processes - because we need more effectiveness and efficiency on our office floors to reduce skill shortages and remain competitive .
It is not about takt in administration but about flow and on-time delivery. Run time, interfaces, and flexibility are the principles. I can already hear the staff complain in Development or in
Construction: "For us no project is like any other - so you can't define processes , let alone standardize. And yet 7o% to 80 % of the activities are routine and repetitive, consisting of
foolishly long meetings and secretarial or travel agency work that is unrelated to project content.
Defining and standardizing the processes of development and construction saves employees valuable time , while proceeding with fixed rules and checkpoints prevents errors or detects them faster,
improves the quality and timeliness of the work, and avoids interface problems, for example in making prototypes or starting up manufacturing.
I can already hear the complaints of managers in Human Resources , Information Technology, or Accounting : "We produce nothing - we can't optimize anything." The most beautiful expression I
frequently hear from this faction is "Mr. Wiegand, without us, nothing runs here ." And then when I ask , what products do you make or what services do you render ? Then I see usually only blank
Hello! Is hiring, challenging, and coaching employees not a service? Are indicators that show facts, or figures that support decisions not defined products? Or implementing software , delivering
training, and other support functions? Of course, these are products and services. Can we describe these products, deliver them more efficiently, standardize them, define quality requirements,
and visualize their processes?
Yes, we can !
So what is the difference between the production of goods and the products in the so-called indirect areas?
None - except for the fact that the first are visible, tangible, and palpable, while the product of Administration is information - to interpret, invisible and intangible. If it is possible,
therefore, to make the information visible and to define it , then you can treat it like a product and make the processes more effective and efficient. And why do we not do it?
We had the same problem in production 20 to 30 years ago. Processes were previously under the responsibility of master craftsmen who delivered as they saw fit. We had to define the processes,
specify interfaces, and establish quality, formulate work orders and convert from the functional organization of workshops and production areas to an organization along manufacturing processes.
I remember vividly how the Craftsmen, Workshop Supervisors , and Production Area Managers fought and defended their kingdoms. It was a long, hard struggle. Today, however, less than 10% of
companies are still aligned functionally in production. They all fought to the end, against better judgment, against the greater economic performance, and for their kingdoms.
This is what we face today every day on our office floors. The same arguments are repeated. As an acccountant said, "If we move to a process-oriented organization, the specific know-how goes down
the drain." By the way - the last major innovation in accounting -- breakeven analysis -- is more than half a century old. So what kind of know-how must be centrally held, promoted, and
protected ?
Do not get me wrong -- we need accounting to measure our success , but not in an ivory tower, but on the spot, so you know what you need to measure and therefore can support the decision makers ,
thus giving guidelines to your trade (see also my article in the Book: The accountant as in-house consultant).
So we anchor the controls in the process , where needed , and not in a functional department. If we want to raise the potential in the indirect areas , we must not look at the individual
functions , but at processes across functions and optimize the functions themselves. Now you know now why it is so hard to find support for Lean Administration. But, as 12 years of Lean
Administration consulting have shown , it pays. Here are a few examples :
□ Today, 900 employees in Development and Administration are doing work that used to require 1,300.
□ Capital goods are shipped six months earlier.
□ A service center saves €17M.
□ A pharmaceutical company handle 20% in sales without adding employees.
□ A government office reduced processing time from three weeks to two days .
Now how is this done? It starts with process mapping, defining products , analyzing the task structure and the job structure, and then optimizing the value streams . Quite simple - or not?
Unfortunately, not quite that simple. You can make many mistakes. I have seen many process maps. Some were created from an IT perspective, others from the organization's point of view -- but why
not from a customer perspective?
Others avoid analyzing the structure of the activity usually with the argument "Not acceptable to the Works Council."
What a joke!
We have been implementing Lean administration in companies for 12 years and have never had problems with the Works Councils due to an activity structure analysis. Mostly we were rather supported
with the motto: "Finally in this area something is happening."
Often the products are not defined from a customer perspective. The optimized value streams are contradictory and watered down by compromise at the interfaces and turned into overcomplex
Why ?
Out of consideration to individuals and functions. Lean Administration projects rarely succeed from the inside out , but require external coaches to bring to light self-interests and put the
process in the foreground.
You should however not be deterred by these difficulties . Especially with projects in Administration, the five success factors I so often stress are:
□ Planning
□ Leadership commitment
□ Holistic approach
□ Resolute implementation /change in mindset
□ Measurement
The potential is large and success easy to achieve. You and your colleagues just have to really want it and, of course, start properly. "
Michel Baudin's comments:
As many discussions of the "Lean office" do, Wiegand's lumps together all activities other than production. Much of his letter is devoted to the standardization of office work, which he presents as
essential to avoiding a skill shortage by increasing productivity. While a case can be made for the value of following documented procedures in transaction processing like rental car issue and
return, it is far-fetched for creative knowledge work like R&D.
In product development, it helps to have some discipline in managing the flow of projects through phases, with appropriate validation at various checkpoints, but there is little evidence that it is
essential. The history of product development is replete with cases where all the procedures were in place but the products failed, and, on the contrary, of cases of product developers who broke the
rules and succeeded.
Wiegand describes the transition from craft control to controlled, documented processes in production as a battle fought won in the past 20 to 30 years. I view it instead as a struggle that started
with the industrial revolution about 1750 and is still going on, with the Lean approach to it being only the last of a long list. And it does not involve standardizing everything. If you have
machines with controls that are visually obvious and mistake-proof, you don't need instructions.
Another theme of Wiegand's letter is the change from organization by function, where employees are in departments focused on one operation, to organization by process, where they are in teams in
charge of all the operations needed to generate a finished output. It is like the change from a machining job-shop with departments for turning, milling, heat treatment, grinding, etc. to a flow shop
with lines or cells that machine blanks from start to finish.
Wiegand asserts that only 10% of companies still have functional organizations in production. It is a number I have a hard time believing. I don't believe it's true even in Japan. In fact, the
functional, or job-shop, organization is not wrong for everything. Once you have done your Runner/Repeater/Stranger analysis, it is actually what you need for Strangers. And it is not always wrong in
office work either. Product development at Toyota, for example, is done by functional departments.
I am also puzzled by his description of "break-even analysis" as the last great innovation in accounting. It does not strike me as particularly advanced. What about discounted cash flows, internal
rates of return, activity-based costing, and other concepts that shine a light on different aspects of operations than just break-even points?
One last comment is that Wiegand mentions "optimization" six times and "improvement" never. One of my pet peeves is that, in Lean, you always improve but never optimize, because it is, by definition,
the end of improvement. I have been assured both in Germany and France, that they mean "improvement" when they say "optimization," which begs the question of what they use when they actually mean
When One-Piece Flow Restricts Capacity
Philip Marris told me of the case of a machining cell in an auto parts plant where management was ready to buy more machines because it was "lacking capacity," but he was able to find a cheaper way
to increase capacity by 17% in 15 minutes.
Unlike manual assembly cells, in which work can be balanced among stations, cells that involve machines always have one that is slower than all others, and, reallocating work among machines with
different capabilities is not an option. In particular, almost all machining cells have a bottleneck, and the situation Philip described involved this bottleneck and the machine feeding it. The cell
practiced one-piece flow. Therefore, if the feeder machine had worked perfectly, the timelines of the Feeder and the Bottleneck would have been as follows:
The Feeder would have started one piece at the beginning of each takt interval, and, since it is faster than the Bottleneck, it would have finished the piece before the end of the interval. The
Feeder then would have waited for the bottleneck to pick up the piece before starting the next one. The Bottleneck would have been working 100% of the time; the Feeder would not.
But what Philip discovered by observing operations was that the Feeder had microstoppages. When the Feeder was hit by a microstoppage, the delay it caused passed to the bottleneck, which was
prevented from working 100% of the time, as shown below:
This reduced the capacity of the entire cell. In the actual case, even with its microstoppages, the Feeder had enough capacity to feed the Bottleneck, on the average, just not on a takt basis. The
microstoppages caused the output of the Feeder to fluctuate and disrupt the operation of the Bottleneck.
To anyone trained in Lean, the only appropriate solution was to eliminate the microstoppages... But it was easier said than done. Sometimes, all it takes is slowing down the machine, or changing a
maintenance policy from "clean for one minute" to "clean until it is clean." But it is not always that simple.
Microstoppages are often unreported because they are fixed on the fly by production operators. To understand microstoppages, you need to monitor the machine to observe when they occur and trace their
causes. Eliminating them may require you to modify chutes, fixtures, jigs or dies, or even the basic process, and it can take time, but you need to do it if you want one-piece flow to work.
In the meantime, what do you do? Buying more equipment is an expensive solution, especially when you don't expect to need it once you are rid of the microstoppages. A cheaper countermeasure is
to protect the supply of parts to the bottleneck against fluctuations by decoupling the two machines with a buffer of WIP. You can set the size of this buffer by trial and error, knowing that it is
not a long-term solution.
Of course, manufacturing engineers understand that you cannot have one-piece flow with microstoppages. So why did they ignore their own wisdom? The most likely explanation is a demand from a
corporate "Lean group" to implement one-piece flow everywhere and "damn the torpedoes!" These engineers had complied not because they thought it was a good idea, but because it was required to keep
their jobs.
Technically, Philip sees this story as a case study in the addition of Theory of Constraints (TOC) thinking to Lean; I just see it as due consideration of equipment issues in cell design, as I was
taught it more than 25 years ago. From a management standpoint, I see it as an example of the local consequences of half-baked corporate mandates.
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Re: Differentiation
Okay, it will be tedious at first but easier later. Post if you get stuck.
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.
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History of
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Obligatory Note: This matter is created/compiled by Sarvesh Srivastava from various authentic resources for the site titled "Facets of India : Ancient and Modern". Please feel free to link the page
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History of Ganit (Mathematics)
Ganit (Mathematics) has been considered a very important subject since ancient times. We find very elaborate proof of this in Vedah (which were compiled around 6000 BC). The concept of division,
addition et-cetera was used even that time. Concepts of zero and infinite were there. We also find roots of algebra in Vedah. When Indian Beez Ganit reached Arab, they called it Algebra. Algebra was
name of the Arabic book that described Indian concepts. This knowledge reached to Europe from there. And thus ancient Indian Beez Ganit is currently referred to as Algebra.
The book Vedang jyotish (written 1000 BC) has mentioned the importance of Ganit as follows-
Just as branches of a peacock and jewel-stone of a snake are placed at the highest place of body (forehead), similarly position of Ganit is highest in all the branches of Vedah and Shastras
has said the following-
What is the use of much speaking. Whatever object exists in this moving and nonmoving world, can not be understood without the base of Ganit(Mathematics).
This fact was well known to intellectuals of India that is why they gave special importance to the development of Mathematics, right from the beginning. When this knowledge was negligible in Arab and
Europe, India had acquired great achievements.
People from Arab and other countries used to travel to India for commerce. While doing commerce, side by side, they also learnt easy to use calculation methods of India. Through them this knowledge
reached to Europe. From time to time many inquisitive foreigners visited India and they delivered this matchless knowledge to their countries. This will not be exaggeration to say that till 12th
century India was the World Guru in the area of Mathematics.
The auspicious beginning on Indian Mathematics is in Aadi Granth (ancient/eternal book) Rigved. The history of Indian Mathematics can be divided into 5 parts, as following.
1) Ancient Time (Before 500 BC)
a)Vedic Time (1000 BC-At least 6000 BC)
a)Later Vedic Time (1000 BC-500BC)
2) Pre Middle Time (500 BC- 400 AD)
3) Middle Time or Golden Age (400 AD - 1200 AD)
4) Later Middle Time (1200 AD - 1800 AD)
5) Current Time (After 1800 AD)
1) Ancient Time (Before 500 BC)
Ancient time is very important in the history of Indian Mathematics. In this time different branches of Mathematics, such as Numerical Mathematics; Algebra; Geometrical Mathematics, were properly and
strongly established.
There are two main divisions in Ancient Time. Numerical Mathematics developed in Vedic Time and Geometrical Mathematics developed in Later Vedic Time.
1a) Vedic Time (1000 BC-At least 6000 BC)
Numerals and decimals are cleanly mentioned in Vedah (Compiled at lease 6000 BC). There is a Richa in Veda, which says the following-
In the above mentioned Richa , Dwadash (12), Treeni (2), Trishat (300) numerals have been used. This indicates the use of writing numerals based on 10.
In this age the discovery of ZERO and "10th place value method"(writing number based on 10) is great contribution to world by India in the arena of Mathematics.
If "zero" and "10 based numbers" were not discovered, it would not have been possible today to write big numbers.
The great scholar of America Dr. G. B. Halsteed has also praised this. Shlegal has also accepted that this is the second greatest achievement of human race after the discovery of Alphabets.
This is not known for certain that who invented "zero" and when. But it has been in use right from the "vedic" time. The importance of "zero" and "10th place value method" is manifested by their wide
spread use in today's world. This discovery is the one that has helped science to reach its current status.
In the second section of earlier portion of Narad Vishnu Puran (written by Ved Vyas) describes "mathematics" in the context of Triskandh Jyotish. In that numbers have been described which are ten
times of each other, in a sequence (10 to the power n). Not only that in this book, different methods of "mathematics" like Addition, Subtraction, Multiplication, Addition, Fraction, Square, Square
root, Cube root et-cetera have been elaborately discussed. Problems based on these have also been solved.
This proves at that time various mathematical methods were not in concept stage, rather those were getting used in a methodical and expanded manner.
"10th place value method" dispersed from India to Arab. From there it got transferred to Western countries. This is the reason that digits from 1-9 are called "hindsa" by the people of Arab. In
western countries 0,1,2,3,4,5,6,7,8,9 are called Hindu-Arabic Numerals.
1b) Later Vedic Time (1000 BC - 500 BC)
1b.1) Shulv and Vedang Jyotish Time
Vedi was very important while performing rituals. On the top of "Vedi" different type of geomit(geometry: as you notice this word is derived from a Sanskrit word)) were made. To measure those
geometry properly, "geometrical mathematics" was developed. That knowledge was available in form of Shulv Sutras (Shulv Formulae). Shulv means rope. This rope was used in measuring geometry while
making vedis.
In that time we had three great formulators-Baudhayan, Aapstamb and Pratyayan. Apart from them Manav, Matrayan, Varah and Bandhul are also famous mathematician of that time.
The following excerpt from "Baudhayan Sulv Sutra (1000 BC)" is today known as Paithogorus Theorem (amazing, isn't it ?)
In the above formula , the following has been said. In a Deerghchatursh (Rectangle) the Chetra (Square) of Rajju (hypotenuse) is equal to sum of squares of Parshvamani (base) and Triyangmani
In the same book Baudhayan has discussed the method of making a square equal to difference of two squares. He has also described method of making a square shape equal to addition of two squares. He
has also mentioned the formula to find the value (upto five decimal places) of a root (square root, cube root ...) a number, according to that the square root of 2 can be found as below-
While Geometric Mathematics was developed for making Vedi in Yagya , in parallel there was a need to find appropriate timing for Yagya. This need led to development of Geotish Shastra (Astrology) In
Geotish Shastra (Astrology) they calculated time, position and motion of stars. By reading the book Vedanga Jyotish (At least 1000 BC) we find that astrologers knew about addition, multiplication,
subtraction et-cetera. For example please read below-
Multiply the date by 11, then add to it the "Bhansh" of "Parv" and then divide it by "Nakshatra" number. In this way the "Nakshtra" of date should be told.
1b.2) Surya Pragyapti Time
We find elaborated description of Mathematics in the Jain literature. In fact the clarity and elaboration by which Mathematics is described in Jain literature, indicates the tendency of Jain
philosophy to convey the knowledge to the language and level of common people (This is in deviation to the style of Veda which told the facts indirectly).
Surya Pragyapti and Chandra Pragyapti (At least 500 BC) are two famous scriptures of Jain branch of Ancient India. These describe the use of Mathematics.
Deergha Vritt (ellipse) is clearly described in the book titled Surya Pragyapti. "Deergha Vritt" means the outer circle (Vritta) on a rectangle(Deergha), that was also known as Parimandal.
This is clear that Indians had discovered this at least 150 years before Minmax (150 BC). As this history was not known to the West so they consider Minmax as the first time founder of ellipse.
This is worth mentioning that in the book Bhagvati Sutra (Before 300 BC) the word Parimandal has been used for Deergha Vritt (ellipse). It has been described to have two types 1) Pratarparimandal and
Jain Aacharyas contributed a lot in the development of Mathematics. These gurus have described different branches of mathematics in a very through and interesting manner. They are examples too.
They have described fractions, algebraic equations, series, set theory, logarithm, and exponents .... Under the set theory they have described with examples- finite, infinite, single sets. For
logarithm they have used terms like Ardh Aached , Trik Aached, Chatur Aached. These terms mean log base 2, log base 3 and log base 4 respectively. Well before Joan Napier (1550-1617 AD), logarithm
had been invented and used in India which is a universal truth.
Buddha literature has also given due importance to Mathematics. They have divided Mathematics under two categories- 1) Garna (Simple Mathematics) and 2)Sankhyan (Higher Mathematics). They have
described numbers under three categories-1)Sankheya(countable),2)Asankheya(uncountable) and 3)Anant(infinite). Which clearly indicates that Indian Intellectuals knew "infinite number" very well.
2) Pre Middle Time (500 BC- 400 AD)
This is unfortunate that except for the few pages of the books Vaychali Ganit, Surya Siddhanta and Ganita Anoyog of this time, rest of the writings of this time are lost. From the remainder pages of
this time and the literature of Aryabhatt, Brahamgupt et-cetera of Middle Time, we can conclude that in this time too Mathematics underwent sufficient development.
Sathanang Sutra, Bhagvati Sutra and Anoyogdwar Sutra are famous books of this time. Apart from these the book titled Tatvarthaadigyam Sutra Bhashya of Jain philosopher Omaswati (135 BC) and the book
titled Tiloyapannati of Aacharya (Guru) Yativrisham (176 BC) are famous writings of this time.
The book titled Vaychali Ganit discusses in detail the following -the basic calculations of mathematics, the numbers based on 10, fraction, square, cube, rule of false position, interest methods,
questions on purchase and sale... The book has given the answers of the problems and also described testing methods. Vachali Ganit is a proof of the fact that even at that time (300 BC) India was
using various methods of the current Numerical Mathematics. This is noticeable that this book is the only written Hindu Ganit book of this time that was found as a few survived pages in village
Vaychat Gram (Peshawar) in 1000 AD.
Sathanang Sutra has mentioned five types of infinite and Anoyogdwar Sutra has mentioned four types of Pramaan (Measure). This Granth(book) has also described permutations and combinations which are
termed as Bhang and Vikalp .
This is worth mentioning that in the book Bhagvati Sutra describes the following. From n types taking 1-1,2-2 types together the combinations such made are termed as Akak, Dwik Sanyog and the value
of such combinations is mentioned as n(n-1)/2 which is used even today.
Roots of the Modern Trignometry lie in the book titled Surya Siddhanta . It mentions Zya(Sine), Otkram Zya(Versesine), and Kotizya(Cosine). Please remember that the same word (Zia) changed to "Jaib"
in Arab. The translation of Jaib in Latin was done as "Sinus". And this "Sinus" became "Sine" later on.
This is worth mentioning that Trikonmiti word is pure Indian and with the time it changed to Trignometry. Indians used Trignometry in deciding the position , motion et-cetera of the spatial planets.
In this time the expansion of Beezganit (When this knowledge reached Arab from India it became Algebra)was revolutionary. The roots of Modern Algebra lie in the book Vaychali Ganit. In this book
while describing Isht Karma Isht Karm "Rule of False" as the origin of expansion of Algebra. Thus Algebra is also gifted to world by Indians
Although almost all ancient countries used quantities of unknown values and using them found the result of Numerical Mathematics. However the the expansion of Beez Ganit (Now known as Alzebra) became
possible when right denotion method was developed. The glory for this goes to Indians who for the first time used Sanskrit Alphabet to denote unknown quantities. Infact expansion of Beez Ganit (Now
known as Alzebra) became possible when Indians realized that all the calculations of Numerical Mathematics could be done by notations. And that +, - these signs can be used with those notations.
Indians developed rules of addition, subtraction, multiplication with these signs (+,-,x). In this context we can not forget the contribution of great mathematician Brahmgupt (628 AD). He said-
The multiplication of a positive number with a negative number comes out to be a negative number and multiplication of a positive number with a positive number comes out to be a positive number.
He further told:
When a positive number is divided by a positive number the result is a positive number and when a positive number is divided by a negative number or a negative number is divided by a positive number
the result is a negative number.
Indians used notations for squares, cube and other exponents of numbers. Those notations are used even today in the mathematics. They gave shape to Beezganit Samikaran(Algebraic Equations). They made
rules for transferring the quantities from left to right or right to left in an equation. Right from the 5th century AD, Indians majorly used aforementioned rules.
In the book titled Anoyogdwar Sutra has described some rules of exponents in Beez Ganit (Later the name Algebra became more popular).
Thus it proves that Beez Ganit (Later the name Algebra became more popular) was well expanded by the mathematicians of Pre-middle Time. This was more expanded in the Middle Time.
It is without doubt that like Aank Ganit (Numerical Mathematics) Beez Ganit (Later the name Algebra became more popular) reached Arab from India. Arab mathematician Al-Khowarizmi (780-850 AD) has
described topics based on Indian Beez Ganit in his book titled "Algebr". And when it reached Europe it was called Algebra.
As for as other countries are concerned we find that in the golden time of Greece Mathematics there was no sign of Algebra with respect to modern concept of Algebra. In classical period Greece people
had ability to solve tough questions of Beez Ganit (Later the name Algebra became more popular) but there all solutions were based on Geometrical Mathematics. For the first time in Greece world, the
concept of Beez Ganit (Later the name Algebra became more popular) is described in a books of Diofantus (275 AD). By that time Indians were far ahead. This is worth noting that the shape and form of
current Beez Ganit (Later the name Algebra became more popular) is originally Indian.
3) Middle Time or Golden Age 400 AD- 1200 AD)
This period is called golden age of Indian Mathematics. In this time great mathematicians like Aryabhatt, Brahmgupt, Mahaveeracharya, Bhaskaracharya who gave a broad and clear shape to almost all the
branches of mathematics which we are using today. The principles and methods which are in form of Sutra(formulae) in Vedas were brought forward with their full potential, in front of the common
masses. To respect this time India gave the name "Aryabhatt" to its first space satellite.
The following is the description about great mathematicians and their creations.
Aryabhatt (First) (490 AD)
He was a resident of Patna in India. He has described, in a very crisp and concise manner, the important fundamental principles of Mathematics only in 332 Shlokas. His book is titled Aryabhattiya. In
the first two sections of Aryabhattiya, Mathematics is described. In the last two sections of Aryabhattiya, Jyotish (Astrology) is described. In the first section of the book, he has described the
method of denoting big decimal numbers by the alphabets.
In the second section of the book Aryabhattiya we find difficult questions from topics such as Numerical Mathematics, Geometrical Mathematics, Trignometry and Beezganit (Algebra). He also worked on
indeterminate equations of Beezganit (Later in West it was called Algebra). He was the first to use Vyutkram Zia (Which was later known as Versesine in the West) in Trignometry. He calculated the
value of pi correct upto four decimal places.
He was first to find that the sun is stationary and the earth revolves around it. 1100 years later, this fact was accepted by Coppernix of West in 16th century. Galileo was hanged for accepting this.
Bhaskar (First) (600 AD)
He did matchless work on Indeterminate equations. He expanded the work of Aryabhatt in his books titled Mahabhaskariya, Aryabhattiya Bhashya and Laghu Bhaskariya .
Brahmgupt (628 AD)
His famous work is his book titled Brahm-sfut. This book has 25 chapters. In two chapters of the book, he has elaborately described the mathematical principles and methods. He threw light on around
20 processes and behavior of Mathematics. He described the rules of the solving equations of Beezganit (Algebra). He also told the solution of indeterminate equations with two exponent. Later Ailer
in 1764 AD and Langrez in 1768 described the same.
Brahmgupt told the method of calculating the volume of Prism and Cone. He also described how to sum a GP Series. He was the first to tell that when we divide any positive or negative number by zero
it becomes infinite.
Mahaveeracharya (850 AD)
He wrote the book titled "Ganit Saar Sangraha". This book is on Numerical Mathematics. He has described the currently used method of calculating Least Common Multiple (LCM) of given numbers. The same
method was used in Europe later in 1500 AD. He derived formulae to calculate the area of ellipse and quadrilateral inside a circle.
Shridharacharya (850 AD)
He wrote books titled "Nav Shatika", "Tri Shatika", "Pati Ganit". These books are on Numerical Mathematics. His books on Beez Ganit (Algebra) are lost now, but his method of solving quadratic
equations is still used. This is method is also called "Shridharacharya Niyam". The great thing is that currently we use the same formula as told by him. His book titled "Pati Ganit" has been
translated into Arabic by the name "Hisabul Tarapt".
Aryabhatta Second (950 AD)
He wrote a book titled Maha Siddhanta. This book discusses Numerical Mathematics (Ank Ganit) and Algebra. It describes the method of solving algebraic indeterminate equations of first order. He was
the first to calculate the surface area of a sphere. He used the value of pi as 22/7.
Shripati Mishra (1039 AD)
He wrote the books titled Siddhanta Shekhar and Ganit Tilak. He worked mainly on permutations and combinations. Only first section of his book Ganit Tilak is available.
Nemichandra Siddhanta Chakravati (1100 AD)
His famous book is titled Gome-mat Saar. It has two sections. The first section is Karma Kaand and the second section is titled Jeev Kaand. He worked on Set Theory. He described universal sets, all
types of mapping, Well Ordering Theorems et-cetera.One to One Mapping was used by Gailileo and George Kanter(1845-1918) after many centuries.
Bhaskaracharya Second (1114 AD)
He has written excellent books namely Siddhanta Shiromani,Leelavati Beezganitam,Gola Addhaya,Griha Ganitam and Karan Kautoohal. He gave final touch to Numerical Mathematics, Beez Ganit (Algebra), and
Trikonmiti (Trignometry).
The concepts which were in the form of formulae in Vedah. He has also described 20 methods and 8 behaviors of Brahamgupt.
Great Hankal has praised a lot Bhaskaracharya's Chakrawaat Method of solving indeterminate equations of Beezganit (Algebra). This Bhaskaracharya's Chakrawaat Method was used by Ferment in 1667 to
solve indeterminate equations.
In his book Siddhanta Shiromani, he has described in length the concepts of Trignometry. He has described Sine, Cosine, Versesine,... Infinitesimal Calculus and Integration. He wrote that earth has
gravitational force.
3) Later Middle Period (1200 AD- 1800 AD)
Not much original work was done after Bhaskaracharya Second. Comments on ancient texts are the main contribution of this period.
In his book (1500 AD), the mathematician Neel Kantha of Kerla has given the formula to calculate Sine r -
The same formula is given in the Malyalam book Mookti Bhaas. These days this series is called Greygeries Series. The following is a descriptions of the famous mathematicians of this period.
Narayan Pundit (1356 AD)
He wrote the book titled Ganit Kaumidi. This book deals with Permutations and Combinations, Partition of Numbers, Magic Squares.
Neel Kanta (1587 AD)
He wrote the book titled Tagikani Kanti. This book deals with Zeotish Ganit(Astrological Mathematics).
Kamalakar (1608 AD)
He wrote a book titled Siddhanta Tatwa Viveka.
Samraat Jagannath (1731 AD)
He wrote two books titled Samraat Siddhanta and Rekha Ganit (Line Mathematics)
Apart from the above-mentioned mathematicians we have a few more worth mentioning mathematicians. From Kerla we have Madhav (1350-1410 AD). Jyeshta Deva (1500-1610 AD) wrote a book titled Ukti Bhasha
. Shankar Paarshav (1500-1560 AD) wrote a book titled Kriya Kramkari.
3) Current Period (1800 AD- Current)
Please find below a list of famous mathematicians and their writings.
Nrisingh Bapudev Shastri (1831 AD)
He wrote books on Geometrical Mathematics, Numerical Mathematics and Trignometry.
Sudhakar Dwivedi (1831 AD)
He wrote books titled Deergha Vritta Lakshan(which means characteristics of ellipse), Goleeya Rekha Ganit(which means sphere line mathematics),Samikaran Meemansa(which means analysis of equations)
and Chalan Kalan.
Ramanujam (1889 AD)
Ramanujam is a modern mathematics scholar. He followed the vedic style of writing mathematical concepts in terms of formulae and then proving it. His intellectuality is proved by the fact it took all
mettle of current mathematicians to prove a few out of his total 50 theorems.
Swami Bharti Krishnateerthaji Maharaj (1884-1960 AD)
He wrote the book titled Vedic Ganit.
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Transient Heat Transfer Analysis
I have put together an Excel spreadsheet that calculates the temperature profile through a wall that is heated by the sun.
It use the matrix [K]{T} + [C]{dT/dt} = {S}, where [K] is the heat transfer, {T} is the temperature at the various nodes, [C] is the thermal capacitance of the node, and {S} is the thermal heat gain
of the outer node. A copy of the spreadsheet and matrix math is attached.
The math equations have not been verified for accuracy or errors, but it is kind of fun to play around with the variables to see how the temperature profile in the wall changes throughout the day.
(Note that an earlier version of this post had the attached Excel file as a *.zip file. It has been replaced with an Excel version.)
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Asymptotic bounds for a confluent hypergeometric function $F_{1}[;1;x]$
up vote 4 down vote favorite
I know that for infinite series and $|z|<1$ there exists a confluent hypergeometric expression
$ \sum_{k=0}^{\infty} \frac{z^k}{k!k!} = F_{1}[;1;z] $
This is not very helpful though, and I 'd like to know if it is possible to get some asymptotic expansion for this function and if there exists some general approach to bounding hypergeometric
functions asymptotically.
hypergeometric-functions asymptotics special-functions
2 In comments, you keep saying $|z|<1$, but then the series you give is a perfectly good approximation. But you ask for asymptotic expansions, which usually means positive real infinity. But you
don't seem to mean that. What do you want? – Jacques Carette Nov 6 '11 at 13:20
@Jacques: yes, now I understand what you mean, thanks. – sigma_z_1980 Nov 6 '11 at 20:17
add comment
3 Answers
active oldest votes
This particular function can be expressed in terms of Bessel's I function as $I(0,2\sqrt{z})$, and from there an asymptotic expression (at $\infty$) is easily derived. It starts $$\frac
{e^{\frac{2}{\sqrt{\frac{1}{z}}}}\left(\frac{1}{z}\right)^{\left(\frac{1}{4}\right)}}{2\sqrt{\pi}} + O\left(e^{\frac{2}{\sqrt{\frac{1}{z}}}}\left(\frac{1}{z}\right)^{\left(\frac{3}{4}\
right)}\right)$$ where the roots are chosen to have the correct branching behaviour.
up vote 6 The easiest way to obtain such results is actually from the ODE satisfied by your function, in this case $zy'' + y' - y$ with $y(0)=1$. It is easy to get an ODE at $\infty$ from there,
down vote and from there one gets the asymptotic expansion. The hardest part is getting the singular behaviour 'just right', as well as the branches. Those who have voted to close this likely
accepted have never had to compute the asymptotic expansion along a branch cut with the expansion point being an irregular singularity. While it is known how to do this, it is practiced by very
very few.
Thanks, though with $|z|<1 I(0,2 \sqrt(z)$ seems to behave like $1+z+\frac{z^2}{4} +O(z^3)$. Can you suggest some literature that treats asymptotic expansion of hypergeometric
functions? – sigma_z_1980 Nov 6 '11 at 5:19
Did you want the series at 0 ? – Jacques Carette Nov 6 '11 at 13:16
@sigma_z_1980: and that is correct, since that is exactly your series. – Jacques Carette Nov 6 '11 at 13:18
@Jacques: yes, now I see what you mean by 'exactly my series'. Unfortunately I'm not so familiar with asymptotic expansion of hyper geometric functions, so I didn't see it at first
glance. – sigma_z_1980 Nov 7 '11 at 1:52
add comment
Although not directly for $z < 1$, here is a general paper that might help with some of the techniques that one can use:
The confluent hypergeometric functions $M(a; b, z)$ and $U(a; b, z)$ for large $b$ and $z$ by J. L. López and P. J. Pagola.
up vote 2 down vote
Also, if not this paper, please take a look at other papers by López; from what I heard, he is an expert on asymptotics of hypergeometric functions.
great, thanks, I regret I can't accept two answers. – sigma_z_1980 Nov 6 '11 at 20:18
no worries; besides Jacques' answer is muccccch nicer :-) – Suvrit Nov 6 '11 at 21:35
add comment
The OP is unclear about the domain of $z$ required. If only real $z\to\infty$ is of interest, just realise that the terms near $k=\sqrt z$ dominate the rest and their shape is close to
up vote 0 a normal density with mean and variance both asymptotically $\sqrt z$. Apply Stirling's formula and that's it.
down vote
no it doesn't, $|z|<1$, – sigma_z_1980 Nov 6 '11 at 5:08
1 You asked for an asymptotic expansion. You gave a Taylor series showing it to be an entire function and you maintain $|z|>1$. Maybe the problem is that you don't know what
"asymptotic expansion" means. – Brendan McKay Nov 6 '11 at 14:04
add comment
Not the answer you're looking for? Browse other questions tagged hypergeometric-functions asymptotics special-functions or ask your own question.
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(Anglo-Saxon, lige, a falsehood.)
Father of lies. Satan (John viii. 44).
The greatest lie. The four P's (a Palmer, a Pardoner, a Poticary, and a Pedlar) disputed as to which could tell the greatest lie. The Palmer said he had never seen a woman out of patience; whereupon
the other three P's threw up the sponge, saying such a falsehood could not possibly be outdone. (Heywood: The Four P's.)
White lies. (See White.)
Lie Circumstantial
(The) or The lie with circumstance. Sir, if you said so, it was a lie. As Touchstone says, this insult is voidable by this means- “If you said so, I said it was a lie,” but the word “if” makes
the insult hypothetical. This is the lie direct in the second degree or once removed. (See Countercheck.)
Lie Direct
(The). Sir, that's a lie. You are a liar. This is an offence no gentleman can take.
One day as I was walking, with my customary swagger,
Says a fellow to me, `Pistol, you're a coward, though a bragger.'
Now, this was an indignity no gentleman could take, sir;
So I told him flat and plump. `You lie- (under a mistake sir).'
Lie Quarrelsome
(The). To tell one flat and plump “You lie.” Touchstone calls this “the countercheck quarrelsome.”
“If again [the fifth time] it was not well cut, he would say I lied: this is called the countercheck quarrelsome.” —Shakespeare: As You Like It, v 4
Lie hath no Feet
(A). Because it cannot stand alone. In fact, a lie wants twenty others to support it, and even then is in constant danger of tripping
(Anglo-Saxon, licgan, to `bide or rest; but lie, to deceive, is the Anglo-Saxon verb leog-an.)
Lie heavy on him, earth, for he
Laid many a heavy load on thee
This is part of Dr. Evan's epitaph on Sir John Vanbrugh, the comic poet, herald, and architect. “The heavy loads” referred to were Blenheim, Greenwich Hospital (which he finished), Castle Howard
in Yorkshire, and other massive buildings. (1666-1726.)
Source: Dictionary of Phrase and Fable, E. Cobham Brewer, 1894
Lie Low
More on Lie from Fact Monster:
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Mathematical Software Group
Ronald Boisvert mathematical software, repositories, testing and evaluation, WWW
Robert Lipman scientific visualization, WWW
Daniel Lozier special functions, computer arithmetic
Marjorie McClain programming support
Bruce Miller nonlinear dynamics, symbolic computing, Lisp, Java
William Mitchell adaptive multigrid methods for PDEs, parallel computing
Roldan Pozo object oriented software design, numerical linear algebra, parallel computing
Karin Remington numerical linear algebra, numerical solution of PDEs, parallel computing
Bert Rust inverse and ill-posed problems, deconvolution, statistical computing
Pete Stewart (UMd) numerical linear algebra, perturbation theory
Frank Olver (UMd) asymptotics, special functions
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space-time (physics)
Article Free Pass
space-time, in physical science, single concept that recognizes the union of space and time, posited by Albert Einstein in the theories of relativity (1905, 1916).
Common intuition previously supposed no connection between space and time. Physical space was held to be a flat, three-dimensional continuum—i.e., an arrangement of all possible point locations—to
which Euclidean postulates would apply. To such a spatial manifold, Cartesian coordinates seemed most naturally adapted, and straight lines could be conveniently accommodated. Time was viewed
independent of space—as a separate, one-dimensional continuum, completely homogeneous along its infinite extent. Any “now” in time could be regarded as an origin from which to take duration past or
future to any other time instant. Within a separately conceived space and time, from the possible states of motion one could not find an absolute state of rest. Uniformly moving spatial coordinate
systems attached to uniform time continua represented all unaccelerated motions, the special class of so-called inertial reference frames. The universe according to this convention was called
By use of a four-dimensional space-time continuum, another well-defined flat geometry, the Minkowski universe (after Hermann Minkowski), can be constructed. In that universe, the time coordinate of
one coordinate system depends on both the time and space coordinates of another relatively moving system, forming the essential alteration required for Einstein’s special theory of relativity. The
Minkowski universe, like its predecessor, contains a distinct class of inertial reference frames and is likewise not affected by the presence of matter (masses) within it. Every set of coordinates,
or particular space-time event, in such a universe is described as a “here-now” or a world point. Apparent space and time intervals between events depend upon the velocity of the observer, which
cannot, in any case, exceed the velocity of light. In every inertial reference frame, all physical laws remain unchanged.
A further alteration of this geometry, locally resembling the Minkowski universe, derives from the use of a four-dimensional continuum containing mass points. This continuum is also non-Euclidean,
but it allows for the elimination of gravitation as a dynamical force and is used in Einstein’s general theory of relativity (1916). In this general theory, the continuum still consists of world
points that may be identified, though non-uniquely, by coordinates. Corresponding to each world point is a coordinate system such that, within the small, local region containing it, the time of
special relativity will be approximated. Any succession of these world points, denoting a particle trajectory or light ray path, is known as a world line, or geodesic. Maximum velocities relative to
an observer are still defined as the world lines of light flashes, at the constant velocity c.
Whereas the geodesics of a Minkowski continuum (without mass-point accelerations) are straight lines, those of a general relativistic, or Riemannian, universe containing local concentrations of mass
are curved; and gravitational fields can be interpreted as manifestations of the space-time curvature. However, one can always find coordinate systems in which, locally, the gravitational field
strength is nonexistent. Such a reference frame, affixed to a selected world point, would naturally be in free-fall acceleration near a concentrated mass. Only in this region is the concept well
defined—i.e., in the neighbourhood of the world point, in a limited region of space, for a limited duration. Its free-fall toward the mass is due either to an externally produced gravitational field
or to the equivalent, an intrinsic property of inertial reference frames. Mathematically, gravitational potentials in the Riemannian space can be evaluated by the procedures of tensor analysis to
yield a solution of the Einstein gravitational field equations outside the mass points themselves, for any particular distribution of matter.
The first rigorous solution, for a single spherical mass, was carried out by a German astronomer, Karl Schwarzschild (1916). For so-called small masses, the solution does not differ appreciably from
that afforded by Newton’s gravitational law; but for “large” masses the radius of space-time curvature may approach or exceed that of the physical object, and the Schwarzschild solution predicts
unusual properties. Astronomical observations of dwarf stars eventually led the American physicists J. Robert Oppenheimer and H. Snyder (1939) to postulate super-dense states of matter. These, and
other hypothetical conditions of gravitational collapse, were borne out in later discoveries of pulsars and neutron stars. They also have a bearing on black holes thought to exist in interstellar
space. Other implications of space-time are important cosmologically and to unified field theory.
Do you know anything more about this topic that you’d like to share?
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Diameter of circle--equilateral triangle
December 16th 2012, 08:36 PM #1
Dec 2012
Top of Utah
Diameter of circle--equilateral triangle
If I have an equilateral triangle with sides of 4", how do I figure out what the diameter of a circle would be that touches the three points?
Re: Diameter of circle--equilateral triangle
Draw it out; make sure the diagram isn't too small. Write EVERYTHING you know about the circle and the triangles (notice I said triangles--there's a hint).
Re: Diameter of circle--equilateral triangle
I should have said that this is not homework. I am 71 years old. If I had a compass I would try to figure it out geometrically as follows:
Bisect a side, set the compass to one half the side, draw an arc. Do the same thing with an adjacent--oops, that's not going to work. I would have to have the circle which is what I am trying to
Well, I've read some similar discussions (below) and apparently this can't be solved just using geometry--you have to use Pythagorean theorem to solve for r. Thanks anyway.
Re: Diameter of circle--equilateral triangle
If we deconstruct the equilateral triangle into 3 congruent isosceles 30°-30°-120° triangles, we find the two equal sides of these isosceles triangles is the radius of the circle.
Then, we may find the radius r in inches is:
And so the diameter d is:
$d=2r=\frac{8}{\sqrt{3}}\approx4.62\text{ in}$.
Re: Diameter of circle--equilateral triangle
I should have said that this is not homework. I am 71 years old. If I had a compass I would try to figure it out geometrically as follows:
Bisect a side, set the compass to one half the side, draw an arc. Do the same thing with an adjacent--oops, that's not going to work. I would have to have the circle which is what I am trying to
Well, I've read some similar discussions (below) and apparently this can't be solved just using geometry--you have to use Pythagorean theorem to solve for r. Thanks anyway.
This page answers your question.
You want the radius of the inscribed circle. It is right there.
Re: Diameter of circle--equilateral triangle
This page answers your question.
You want the radius of the inscribed circle. It is right there.
Your link seems to provide the simplest solution.
(I wanted the circumscribed circle, not inscribed.)
The square root of 3, divided by 3, times 4"; times 2 for diameter.
My answer: 4.6188".
Thanks everyone.
Re: Diameter of circle--equilateral triangle
When most mathematicians see "circle would be that touches", we think tangent. That would be an incircle.
December 16th 2012, 09:21 PM #2
Jun 2012
December 17th 2012, 08:42 AM #3
Dec 2012
Top of Utah
December 17th 2012, 09:00 AM #4
December 17th 2012, 09:02 AM #5
December 17th 2012, 09:53 AM #6
Dec 2012
Top of Utah
December 17th 2012, 12:19 PM #7
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Applying Network Theory to Epidemics: Control Measures for
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More information
Emerg Infect Dis. Feb 2003; 9(2): 204–210.
Applying Network Theory to Epidemics: Control Measures for Mycoplasma pneumoniae Outbreaks
We introduce a novel mathematical approach to investigating the spread and control of communicable infections in closed communities. Mycoplasma pneumoniae is a major cause of bacterial pneumonia in
the United States. Outbreaks of illness attributable to mycoplasma commonly occur in closed or semi-closed communities. These outbreaks are difficult to contain because of delays in outbreak
detection, the long incubation period of the bacterium, and an incomplete understanding of the effectiveness of infection control strategies. Our model explicitly captures the patterns of
interactions among patients and caregivers in an institution with multiple wards. Analysis of this contact network predicts that, despite the relatively low prevalence of mycoplasma pneumonia found
among caregivers, the patterns of caregiver activity and the extent to which they are protected against infection may be fundamental to the control and prevention of mycoplasma outbreaks. In
particular, the most effective interventions are those that reduce the diversity of interactions between caregivers and patients.
Keywords: epidemiology, models, theoretical, network, respiratory tract infections, Mycoplasma pneumonia
Mathematical modeling has a rich and growing tradition in epidemiology (1-3). Because experimental approaches to epidemic interventions are often impractical, and in some cases unethical,
mathematical models can provide otherwise unobtainable insights on the spread and control of disease. Recently, considerable interest has been shown in the effect of contact networks on the spread of
disease, and particularly in using the so-called percolation theory to model epidemics (4-10). Agent-based simulation is also being used increasingly to help epidemiologic investigations (11). In
this paper, we use both of these tools to assess the effects of epidemic interventions in closed health-care facilities.
Mycoplasma pneumoniae is a major cause of bacterial pneumonia in the United States (12). This bacterium, the smallest self-replicating organism capable of cell-free existence, is spread both by
direct contact between an infected person and a susceptible person, and by airborne droplets expelled when an infected person sneezes, coughs, or talks. Large, sustained outbreaks of M. pneumoniae
have occurred in closed and semi-closed populations such as hospitals, psychiatric institutions, military and religious communities, and prisons (13-15). Public health officials and health-care
providers struggle, often with little success, to control mycoplasma outbreaks because of the long incubation period of the organism, late detection of outbreaks, and an incomplete understanding of
the effectiveness of various infection control strategies.
Effective measures to control mycoplasma outbreaks are needed to limit the associated illness and substantial costs. Previous work has addressed candidate strategies, including infection control
practices to prevent the exchange of respiratory droplets between patients and caregivers, cohorting members of the community who display symptoms of a respiratory infection, and antibiotic
prophylaxis of asymptomatic members of the community (14-16). The costs of these strategies include curtailed social interactions because of cohorting, undesirable side effects or allergic reactions
to prophylactic antibiotics, and a potential increase in the risk for infections caused by antibiotic-resistant bacteria. Studies of these control measures have been limited by incomplete information
and participation.
Using a network model approach, we show how data on interactions in real-world communities can be translated into graphs—mathematical representations of networks—and how to predict the course of an
epidemic from the structure of a graph. We found that the assignment of caregivers to patient groups is more critical to the course of an epidemic than the cohorting of patients. Within our models,
the most effective interventions are those that reduce the diversity of interactions that caregivers have with patients. For example, an institution with many wards can avoid a large outbreak by
confining caregivers to work in only one or very few wards.
The Model
Here we model an institution with spatially disjointed wards. Patients are confined to a single ward, and caregivers work in one or more wards. Each person or ward is represented by a “vertex” in the
graph. “Edges” connect people to the wards in which they reside or work. Figure 1 shows the graph for an institution with four wards, each with three or four patients and two to four caregivers.
Health-care institution network. Each vertex represents a patient, caregiver, or ward, and edges between person and place vertices indicate that a patient resides in a ward or a caregiver works in a
A key property of graphs is their degree distribution. The degree of a vertex is the number of other vertices to which it is connected. In Figure 1, for example, the degree of all patients is one;
the degree of each caregiver ranges from one to four; and the degree of the wards ranges from six to seven, indicating the number of inhabitants and caregivers working there. Direct transmission of
M. pneumoniae can only occur between two vertices if an edge connects them.
Throughout this model, we allow transmission to occur between people and places. We do not mean that bacteria actually infect a space by residing on inanimate objects or in the air. Rather, we mean
that the person has transmitted the bacteria to another person who resides or works in that place. Conversely, when a place transmits to a person, we mean that the bacterium is transmitted to an
uninfected person living or working in that place.
We begin by considering only the caregivers and wards. Later we add the patients to the model. (All notations are defined in the Table.) A probability generating function (pgf) is a mathematical
quantity that describes a probability distribution, and thereby summarizes a large amount of useful information about the network architecture. We can define pgfs that capture the distribution of the
number of wards assigned to each caregiver and the distribution of the number of caregivers working in each ward.
Notation for epidemiologic interaction network model
Pgfs can be mathematically manipulated to give many useful results. For example, the derivative gives the average of the distribution, e.g., the mean number of wards assigned to a caregiver, or the
mean number of caregivers working in a ward. We can also answer the following question using pgfs: If an infected caregiver exposes a ward, how many other caregivers, on average, will be vulnerable
to infection because they also work in that ward? Appendix A defines our pgfs and describes the derivations that answer this question.
Transmission through the Graph
Transmission of M. pneumoniae occurs when people occupy the same physical space for some period of time. Therefore, in our model, transmission can occur between persons if the vertices representing
them are connected to the same ward.
We derive two complementary estimates for the size of an outbreak. The first is appropriate for conditions not conducive to large outbreaks, such as a pathogen with low transmissibility, or an
institution with few interpersonal interactions. The second applies to conditions that favor large outbreaks.
We begin with two questions. If a healthy caregiver works in an infected ward, how many other wards will eventually become infected as a result of that caregiver’s interaction with that ward?
Similarly, if an infected caregiver works in a yet uninfected ward, how many other caregivers will eventually become infected as a result of that caregiver’s activity in that ward? Answers to these
questions vary from ward to ward and from caregiver to caregiver. Therefore, we calculate probability distributions for the spread, which we represent by using pgfs.
First, consider an edge linking an infected ward to a caregiver. Figure 2 breaks down the possible scenarios. First, the caregiver may not become infected. Second, the caregiver might become infected
but not transmit to any other wards. Third, the caregiver might transmit infection to one or more other wards in which he or she works. In Appendix B, we construct a pgf by summing up the
probabilities of these different outcomes.
Future transmission diagram I, summing all possible future transmissions stemming from a caregiver who works in an infected ward.
Next, we start with an edge from an infected caregiver to a ward. As shown in Figure 3, there may be no transmission along the edge in question to the ward, no further transmission from the ward to
other people, or transmission to one or more other people who spend time in the ward.
Future transmission diagram II, summing all possible future transmissions stemming from a ward in which an infected caregiver works.
With these two pgfs, we derive the average size of a small outbreak, starting from a single infection:
Where ƒ′ denotes the first derivative of ƒ with respect to its argument. Thus, the average size of the outbreak is 1 (the original patient) plus a function of the two transmission rates (from
caregivers to wards, τ[c] , and from wards to caregivers, τ[w]), and the average number of wards assigned to a caregiver ( ƒ′[0](1) ). The term ƒ′[1](1) assumes that we choose any ward at random from
the entire network, then choose one of the edges connected to that ward at random, then follow that edge to a caregiver, and finally calculate the number of other wards assigned to the caregiver. On
average, that will be ƒ′[1](1). Likewise g′[1](1) is the average number of other caregivers working in a ward that we reach by first choosing a caregiver at random and then randomly choosing one of
the wards in which the caregiver works. These terms contain information not only about the average degrees of caregivers and wards but also about the probability that a given caregiver or node will
become infected in the first place.
The expression for [] diverges when
This expression represents the transition between a regime in which only small isolated outbreaks of disease can occur and one in which a full-blown community-wide epidemic can occur. A community
will cross that transition point if transmission rates are sufficiently high (τ[w] and τ[c]) or the interactions among wards and caregivers are sufficiently dense ([ƒ]′[1](1)and g′[1](1) ). Equation
no. 1 provides an estimate of the epidemic size below the threshold only. It is based on the assumption that interactions are rare enough that a person or a place only encounters the infection once.
When interactions are more common and the community lies above the epidemic transition, we must use a different estimate for the size of the outbreak.
The “giant component” of the graph is the largest connected set of vertices that have all been infected. The size of the outbreak above the epidemic transition is exactly equal to the number of
vertices in this giant component. We calculate the size of the giant component Ѕ[c] (the number of caregivers affected) by calculating the fraction of vertices not contained in it:
where Ф[0](1) is the probability that an infected caregiver will produce no further infections (Appendix B). A similar expression describes the number of wards affected in an epidemic:
These expressions reflect both the fraction of the population infected and the probability that an outbreak will reach epidemic proportions in the first place. Since Ѕ[c] and Ѕ[w] are often much less
than 1, not all outbreaks turn into epidemics, even above the epidemic transition.
Degree Distributions
Equation nos. 3 and 4 allow us to estimate the size of an epidemic on the basis of transmission probabilities and the degree distribution of caregivers to wards. To make specific numerical
predictions, we must first calculate pgfs for the degree distributions. Here we make the simple assumption that the degree distributions follow a Poisson distribution for both the number of wards
associated with a given caregiver and the number of caregivers associated with a given ward. This assumption is equivalent to requiring that all caregivers have an equal likelihood of working in any
ward and that a caregiver is assigned to any given ward independent of his or her other ward assignments. In the absence of more specific information about assignment to wards, this assumption seems
a reasonable first step. This distribution assumes an infinite population and is generally applied to very large populations. Although perhaps not the ideal model for small institutions, this
distribution is used here because it yields pgfs with convenient mathematical properties (see Appendix C).
Case Study
Data gathered by the Centers for Disease Control and Prevention (CDC) during a recent mycoplasma outbreak allowed us to extract values for the parameters in our theory. In 1999, an outbreak of
mycoplasma pneumonia occurred in a psychiatric institution (14). All 15 wards at the institution were affected, with 60 of 257 residents and 82 of 440 employees diagnosed with mycoplasma-like
illness. In the following sections, we predict the epidemic threshold for this institution. The threshold is a function of the degree distribution of caregivers and transmission rates, the size of
the epidemic above the threshold, and a range of realistic transmission rates for M. pneumoniae in this outbreak.
We assumed that each patient was confined to a single ward. While this was not true for all patients at the institution, it simplified the mathematics and allowed us to make a reasonable
approximation of the epidemiology. Interactions between patients in separate wards will increase the threat of a full-blown epidemic and make early intervention all the more critical. Including such
interactions in the model is possible by adding edges to the graph that connect patients to multiple wards. This scenario can be solved exactly by using techniques similar to those presented here.
Epidemic Threshold
If we assume that the degree distributions for wards and caregivers are Poissonian, the epidemic threshold (equation no. 2) is equivalent to τ[w] τ[c] μ[w]μ[c]=1.
In other words, when the product of the transmission rates, the average number of caregivers per ward, and the average number of wards per caregiver exceeds 1, epidemics become possible. In the
psychiatric institution, W = 15 and C = 440, hence [] and the threshold becomes [].
Figure 4 illustrates the epidemic threshold for five different demographic scenarios ( μ[c] = 1,2,3,4,5 ). For the most densely connected case, when each caregiver works in five wards on average, the
epidemic threshold is crossed at very low rates of transmission. When the community is less densely connected, it can withstand much higher infectivity without giving rise to epidemics.
Epidemic thresholds. Each line assumes a different value for μ[c](the average number of wards per caregiver), and graphs the combination of τ[c] and τ[w](transmission parameters) above which the
population crosses the epidemic ...
Calculating the Size of the Epidemic
Combining equation no. 2 with equations 5, 6, 7, and 8 from Appendix C, we derived the following:
Ф[0](1) = exp[μ[w] (1- τ[c +] τ[c] exp[ μ[c]( 1-τ[w +] τ[w]Ф[0](1) - 1 ] -1) ].
Given values for demographic parameters μ[c] and μ[w] , we search for the value of Ф[0](1) that satisfies equation no. 9 numerically. Then, the predicted number of caregivers infected during an
epidemic is Ѕ[c] = 1- Ф[0](1). (The number of affected wards is similarly derived.) Since we know neither the exact distribution of caregivers in wards nor the transmission rates between caregivers
and wards, we solve for the size of the epidemic outbreak in a range of values of the three independent parameters μ[c], τ[c] and τ[w] .
Figure 5 shows both the fraction of wards and caregivers infected in our model as a function of the number of wards per caregiver ( μ[c]), and the fraction of wards and caregivers infected in the
actual outbreak. We assume transmission rates of τ[c] = 0.6 and τ[w] =0.06 (discussed below). The top dashed line indicates that 100% of the wards were affected during the actual epidemic. The lower
horizontal lines depict the upper and lower bound empirical estimates for the number of caregivers affected (TB Hyde, unpub. data). As μ[c] increases, so does the possibility of transmission from one
ward to another through caregivers that work in both. The number of wards affected climbs sharply to 100% (as actually occurred in this outbreak), whereas the number of caregivers climbs more
gradually, passing through the realistic range at relatively low values of μ[c].
Size of epidemic. Predicted and actual number of caregivers and wards affected in an outbreak. These predictions assume that the transmission rate from caregivers to wards is τ[c] = 0.6 and from
wards to caregivers is τ[w]= 0.06.
This analysis suggests that the likelihood of an epidemic and the eventual size of an epidemic, should one occur, are highly sensitive to the degree distribution for caregivers. Transmission of M.
pneumoniae is limited, and the extent and duration of the outbreak are reduced if each caregiver’s activities are confined to just a few wards.
The derivations given here are exact in the limit of large network size. To assess their accuracy on networks like these with a few hundred vertices, we have constructed specific graphs that realize
these distributions and performed computer simulations of the spread of epidemics on them. Each simulation constructs a network with 15 wards and 440 caregivers, where the degree distribution of each
caregiver is binomial with n = 15 , and p such that np = μ[c]. We assume constant infection periods of δ[c]= 14 days (for caregivers) and δ[w] = 21 days (for wards) and that contact between a
caregiver and a ward occurs independently of any other such contact. Initially a single, randomly chosen caregiver is infected. Every day, transmission occurs from an infected caregiver to a
connected ward with probability τ[c]. Thus, the probability that the caregiver will transmit the infection to the connected ward at all is []. Likewise, the daily transmission rate from an affected
ward to a healthy caregiver that works there is [].
Figure 6 shows a frequency distribution of the sizes of epidemics for 1,000 runs of the simulation. Figure 7 compares these results with the predictions of our analytic theory. As the figure clearly
shows, the agreement between simulation and theory is excellent.
Simulated outbreak sizes. Frequency distributions of the numbers of wards and caregivers affected in 1,000 epidemic simulations are shown for μ[c]= 1,2,3.
Comparing derivations to simulation. This graph compares the analytical predictions to the size of a simulated outbreak averaged over 1,000 simulations for each value of μ[c].
Inferring Transmission Rates
Our numeric method also allows us to pinpoint transmission rates that are consistent with the empirical observations. Assuming the average caregiver works in one to four wards, we identify
transmission rates that predict the observed numbers of affected caregivers and wards. We find that τ[c] τ[w]
The Patients
Based on the outbreak data, the probability that a particular patient will become infected if at least one other patient in the ward is infected is 0.15 (0.02) for confirmed cases or 0.23 (0.02) when
probable cases are included.^1 Figure 8 shows the within-ward transmission rates and ward size for the 15 wards. Although not shown, ward size and the transmission rate are not correlated.
Distribution of transmission rates and ward sizes in the psychiatric institution.
We simulate the spread of M. pneumoniae among patients, assuming the ward size distribution shown in Figure 8, and assuming that the number of patients infected per ward follows a binomial
distribution with probability parameter p. (The Poisson approximation is inappropriate as it only applies to very large wards with small transmission rates.) That is, all 15 wards are assumed to be
affected, and each patient in a ward becomes infected with probability p. Figure 9 shows frequency distributions for the fraction of patients infected in 100,000 simulations at three values of p (p =
0.2,0.25,0.3). These distributions resemble the actual frequency distribution shown in Figure 8, and thereby support the binomial approximation.
Simulated spread of Mycobacterium pneumoniae among patients within a ward.
Network theory enables epidemiologists to model explicitly and analyze patterns of human interactions that are potential routes for transmission of an infectious disease. The statistical properties
of an epidemic graph determine the extent to which an infectious agent can spread. By manipulating the structure of a graph, we can identify interventions that may dramatically alter the course of an
epidemic, or even prevent one altogether, and translate them into measures that make sense in a real community. In this paper, we have used network methods to model the spread of a respiratory tract
infection in a health-care facility.
How might this be applied to a real outbreak? We have considered data from a recent investigation of an outbreak of M. pneumoniae in a residential psychiatric institution (14). In that investigation,
standard infection control practices, including strict respiratory droplet precautions, cohorting of ill patients, and employee education about mycoplasma illness and symptoms were instituted at the
facility. Unfortunately, M. pneumoniae has a long incubation period (1–4 weeks), during which time an asymptomatic, infected person can transmit the bacterium to an uninfected person. This long
incubation period limits the beneficial effect of cohorting, since infected persons are only identified and taken out of the community after they have passed through the incubation period.
In both the outbreak and our model (assuming parameters based on this particular institution), caregivers are less likely to become infected than are patients. This observation may mislead
investigators and lead to inappropriate recommendations. Although caregivers are less likely to become ill, they are the primary vectors of infection in the facility. Our model suggests that
transmission rates from patients to caregivers are lower than transmission rates from caregivers to patients. Therefore, once a caregiver is infected with M. pneumoniae, the likelihood is high that
they will transmit the infection to their patients. These data support infection control strategies that limit transmission of M. pneumoniae to caregivers.
We suggest two complementary strategies: limit the number of wards with which caregivers interact, and reduce the probability that caregivers become infected through, for example, respiratory droplet
precautions. This strategy limits the time and cost of laboratory testing as well as the risks for antibiotic use in uninfected persons. The activity of some ancillary staff (e.g., physical
therapists and nutritionists) cannot be limited to a select number of wards. In these cases, alternative precautions against transmission of M. pneumoniae are required.
We conclude with three caveats. First, the epidemic model includes all infections, even those that do not result in symptoms. Most persons with M. pneumoniae infections have relatively mild disease,
only a cough or sore throat or no symptoms at all (17). When applying the model to the outbreak investigation, we considered only symptomatic carriers. While including asymptomatic carriers would
change the estimates for the rates of transmission, our qualitative recommendations for intervention would remain the same.
Second, for mathematical tractability, our model assumes random (Poissonian) assignment of caregivers to wards. The quantitative (but probably not qualitative) results would differ under different
degree distributions. In the future, we hope to analyze distributions taken from actual health-care institutions, when available.
Third, because of the long incubation period of M. pneumoniae infection, interventions are often initiated well into the outbreak. Since epidemics can last months, and in the psychiatric institution
at least half of the wards were not affected until 6 weeks after the first case-patient was diagnosed, we are optimistic that intervention of the type proposed will have a positive impact.
The theoretical tools are in place for building community-specific networks and analyzing the transmission of infectious diseases on these networks. Our approach enables mathematical experiments, in
which the inputs are interventions—structural reorganization, cohorting, treatment, and the like—and the output is predictions about the spread of a disease (or lack thereof) on the network. This
approach can both aid the development of general measures and lend insight into specific scenarios in which intervention is still possible.
Appendix A
Probability Generating Functions
Let Ρ[χ] be the normalized probability that a randomly chosen caregiver is working in k wards and q[χ] the probability that a randomly chosen ward has k caregivers working in it. We define
probability generating functions (pgfs) for these degree distributions thus:
Caregivers: ƒ[0](x) = ∑Ρ[Χ]Χ^Χ
Since Ρ[χ] and q[χ] are each properly normalized probability distributions, [] and g[0](1)=1. The generating functions contain all the same information as the probability distributions but in a form
that will be more convenient for our purposes. We can always recover the probability distributions again by differentiation [].
If we assume that each of W wards has on average μ[w] caregivers working in it, and each of C caregivers interact with μ[c] wards on average, then, ƒ′[0](1) = μ[c] and g′[0](1) = μ[w]. (In general,
the moments of the probability distributions are given by derivatives of the generating functions evaluated at one.)
Suppose we now choose a caregiver at random and follow an edge to a ward in which the caregiver works. The pgf for the number of caregivers working this ward is []. Hence the distribution of
caregivers working in this ward other than the originally selected caregiver is described by [].
Likewise, if we start from a specific ward and choose a random caregiver working in that ward, then the number of other wards in which the caregiver works is given by [].
Appendix B
Deriving [] and Ѕ[c]
We denote the probability of transmission from a caregiver to a ward as τ[c] and the probability of transmission from a ward to a caregiver as τ[w]. By summing the probabilities for the different
outcomes depicted in Figure 2, we arrive at a generating function for the number of wards that will ultimately be affected:
Ф[1](x) = (1-τ[w]) + τ[w]x[1] + τ[w]x[2]Γ[1](x) + … = (1-τ[w]) + τ[w]xƒ[1] (Γ[1] (x)),
where [i] is the probability that the caregiver transmits the infection to i new wards. Each term in this expression corresponds to a pictorial term in Figure 2. Recall that ƒ[1](x) is a generating
function for the number of wards with which a caregiver interacts (other than the ward from which transmission occurred). Γ[1](x) is the generating function (discussed below) for the number of future
infections starting with an edge going from a caregiver to a chosen ward. The generating function for the number of infections starting with a randomly chosen infected caregiver is Ф[0](x) = xƒ[0(]Γ
Next, the generating function for the cluster of infections arising from a randomly chosen edge from a person to a ward is thus Γ[1](x) = (1-τ[c]) + τ[c](g[1] (Ф[1](x))) and Γ[0](x) = g[0] (Ф[1](x)).
Substituting into the formulas for Ф[0](x) and Ф[1](x), we find Ф[0](x) = Χƒ[0][1-τ[c] + τ[c]g[1](Ф[1](x))] and Ф[1](x) = 1 -τ[w] + τ[w]xƒ[1] [1- τ[c]+ τ[c]g[1](Ф[1](x))]. To calculate average
outbreak size [], we differentiate Ф[0](1):
[] Ф′[0]= ƒ[0](1-τ[c] + τ[c]g[1](1)) +ƒ′[0] (1- τ[c] + τ[c]g[1]+ (1))τ[c]g′[1](1)Ф′[1](1) = 1 + τ[c]ƒ′[0](1) g′[1](1)Ф′[1](1)
Now, solving for Ф′[1](x), we find Ф′[1](x) = τ[w]ƒ[1][ 1- τ[c+] τ[c]g[1](Ф[1](x))] + τ[w]xƒ′[1][ 1 - τ[c]+ τ[c]g[1](Ф[1](x))]τ[c]g′[1] (Ф[1](x))[1](x). Hence, []. We thereby arrive at the following
expression for average outbreak size:
Turning next to the size of the giant component, we know that 1 - Ѕ[c =] Ф[0](1) = ƒ[0](1 -τ[c] + τ[c]g[1](Ф[1](1). Hence Ѕ[c] = 1 – ƒ[0](1 -τ[c] + τ[c]g[1](Ф[1](1). Likewise 1 - Ѕ[w] = Γ[0](1) = g
[0](1-τ[w] +τ[w]ƒ[1] ( Γ[1](1)) implies Ѕ[w] = 1 – g[0](1 - τ[w]+τ[w]ƒ[1] (Γ[1] (1)).
Appendix C
The Poisson Generating Function
If the probability that a given caregiver works in some ward is r, then the generating function for the number of wards per caregiver would be
Substituting for r , we find []. In the limit of a large number of wards, the binomial distribution approaches a Poisson distribution, and the generating function for the Poisson distribution is
Likewise, in the limit of many caregivers, g[0] (x) =e^μw (x-1). [6]
Performing a bit more mathematical legwork, we find that
and similarly g[1](x) = g[0](x). Note also that if we know the values of W,C , and μ[c], we can derive the average number of caregivers per ward:
We thank Joel Ackelsberg, Rich Besser, Terri Hyde, Catherine Macken, Mary Reynolds, and Deborah Talkington for their valuable insights and their help interpreting data from previous mycoplasma
This work was supported in part by a National Science Foundation Postdoctoral Fellowship in Biological Informatics to L.A.M. and National Science Foundation Grant DMS-0109086 to M.E.J.N.
Dr. Meyers is an assistant professor in the Section of Integrative Biology at the University of Texas at Austin. She uses a combination of theoretical, computational, and experimental approaches to
research the evolution and spread of microbial communities.
Suggested citation for this article: Ancel Meyers L, Newman MEJ, Martin M, Schrag S. Applying network theory to epidemics: control measures for Mycoplasma pneumoniae outbreaks. Emerg Infect Dis
[serial online] 2003 Feb [date cited]. http://dx.doi.org/10.3201/eid0902.020188
^1We calculate these rates by averaging the fraction of infected patients per ward across the 15 wards and compute the error by taking the standard deviation of these fractions, divided by the square
root of the sample size.
1. Bailey NTJ The mathematical theory of infectious diseases. New York: Hafner Press; 1975.
2. May R, Anderson R Infectious diseases of humans: dynamics and control. Oxford: Oxford University Press; 1992.
Hethcote HW Mathematics of infectious diseases. SIAM Rev. 2000;42:599–653.10.1137/S0036144500371907 [Cross Ref]
Sattenspiel L, Simon CP The spread and persistence of infectious diseases in structured populations. Math Biosci. 1988;90:341–66.10.1016/0025-5564(88)90074-0 [Cross Ref]
Longini IM A mathematical model for predicting the geographic spread of new infectious agents. Math Biosci. 1988;90:367–83.10.1016/0025-5564(88)90075-2 [Cross Ref]
Kretzschmar M, Morris M Measures of concurrency in networks and the spread of infectious disease. Math Biosci. 1996;133:165–95.10.1016/0025-5564(95)00093-3 [PubMed] [Cross Ref]
Ball F, Mollison D, Scalia-Tomba G Epidemics with two levels of mixing. Ann Appl Probab. 1997;7:46–89.10.1214/aoap/1034625252 [Cross Ref]
Newman MEJ Spread of epidemic disease on networks. Phys Rev E Stat Nonlin Soft Matter Phys. 2002;66:016128.10.1103/PhysRevE.66.016128 [PubMed] [Cross Ref]
Newman MEJ, Strogatz SH, Watts DJ Random graphs with arbitrary degree distributions and their applications. Phys Rev E Stat Nonlin Soft Matter Phys. 2001;64:026118.10.1103/PhysRevE.64.026118 [PubMed]
[Cross Ref]
10. Andersson H Epidemic models and social networks. The Mathematical Scientist. 1999;24:128–47.
Smith DJ, Forrest S, Ackley DH, Perelson AS Variable efficacy of repeated annual influenza vaccination. Proc Natl Acad Sci U S A. 1999;96:14001–6.10.1073/pnas.96.24.14001 [PMC free article] [PubMed]
[Cross Ref]
12. Talkington DF, Waites KB, Schwartz SB, Besser RE Emerging from obscurity: understanding pulmonary and extrapulmonary syndromes, pathogenesis, and epidemiology of human Mycoplasma pneumoniae
infections. In: WM Scheld, WA Craig, JM Hughes, editors. Emerging infections. Washington (DC): ASM Press; 2001.
Feikin DR, Moroney JF, Talkington DF, Thacker WL, Code JE, Schwartz LA, et al. An outbreak of acute respiratory disease caused by Mycoplasma pneumoniae and Adenovirus at a federal service training
academy: new implications from an old scenario. Clin Infect Dis. 1999;29:1545–50.10.1086/313500 [PubMed] [Cross Ref]
Hyde TB, Gilbert M, Schwartz SB, Zell ER, Watt JP, Thacker WL, et al. Azithromycin prophylaxis during a hospital outbreak of Mycoplasma pneumoniae pneumonia. J Infect Dis. 2001;183:907–12.10.1086/
319258 [PubMed] [Cross Ref]
Gray GC, McPhate DC, Leinonen M, Cassell GH, Deperalta EP, Putnam SD, et al. Weekly oral azithromycin as prophylaxis for agents causing acute respiratory disease. Clin Infect Dis. 1998;26
:103–10.10.1086/516275 [PubMed] [Cross Ref]
Klausner JD, Passaro D, Rosenberg J, Thacker WL, Talkington DF, Werner SB, et al. Enhanced control of an outbreak of Mycoplasma pneumoniae pneunomia with azithromycin prophylaxis. J Infect Dis. 1998;
177:161–6.10.1086/513818 [PubMed] [Cross Ref]
Foy HM, Grayston JT, Kenny GE Epidemiology of Mycoplasma pneumoniae infection in families. JAMA. 1966;197:859–66.10.1001/jama.1966.03110110083019 [PubMed] [Cross Ref]
Articles from Emerging Infectious Diseases are provided here courtesy of Centers for Disease Control and Prevention
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[Numpy-discussion] Python scalar float indices into array work - but not for array indices - why ?
Robert Kern robert.kern@gmail....
Wed Nov 17 13:15:44 CST 2010
On Wed, Nov 17, 2010 at 13:11, Sebastian Haase <seb.haase@gmail.com> wrote:
> On Wed, Nov 17, 2010 at 7:48 PM, Nathaniel Smith <njs@pobox.com> wrote:
>> On Wed, Nov 17, 2010 at 10:32 AM, Sebastian Haase <seb.haase@gmail.com> wrote:
>>> On Wed, Nov 17, 2010 at 7:26 PM, Robert Kern <robert.kern@gmail.com> wrote:
>>>> On Wed, Nov 17, 2010 at 12:20, Sebastian Haase <seb.haase@gmail.com> wrote:
>>>>> Why does numpy not accept float arrays as indices ?
>>>>> I was very happy and quite surprised once I found out that it worked
>>>>> at all for Python float scalars,
>>>>> but would it not just be consequent to also allow float ndarrays then ?
>>>> It only works for float scalars by accident. Do not rely on it.
>>> Could you be more specific ? As a feature, it for sure can be useful.
>> I think Robert Kern has the same intuition as me: that supporting
>> float indices is pointless. So, can you give any *specific examples*
>> of things you can do with float indices that would be difficult or
>> more expensive using integer indices? That's probably the best way to
>> convince people.
>> -- Nathaniel
> Well,
> suppose you have 2 vectors of floating point coordinates `x` and `y`
> and you want to do operations utilizing fancy indexing like
> image[ [x,y] ] += 1
> As I just realized, this specific case seems to be addressed by histogram2d,
> however, if float indices would work this would of course be much more
> general: higher dimensionality and not just '+=' operations.
Actually, it wouldn't work even if x and y were integers.
Robert Kern
"I have come to believe that the whole world is an enigma, a harmless
enigma that is made terrible by our own mad attempt to interpret it as
though it had an underlying truth."
-- Umberto Eco
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Ascent of Sap
What is the mechanism of it?
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'Who are you to judge everything?' -Alokananda
Re: Ascent of Sap
Could be capillary action. If you have ever seen a micropipette...
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: Ascent of Sap
And how does capillary action work?
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'Who are you to judge everything?' -Alokananda
Re: Ascent of Sap
They say it is caused by adhesion and cohesion of the liquid. They describe it as a fluid moving against gravity as long as the tube is very narrow.
Trouble is when you are using a micropipette which is a very, very thin tube. If you keep your thumb over the top no capillary action takes place. When you remove your thumb, presto liquid is drawn
into the tube.
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: Ascent of Sap
Doesn't it seem very contradictory to common sense?
When the liquid goes up, some work is done, where does the energy come from?
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'Who are you to judge everything?' -Alokananda
Re: Ascent of Sap
Doesn't it seem very contradictory to common sense?
To a theoretician maybe. But every lab student in chemistry and microbiology used to be able to make a pipette. I have seen the effect occur many times.
In a tree, the channels are very thin. this would allow for capillary action. The action would be assisted by the fact that evaporation occurs at the the leaves creating a pressure difference between
the top and bottoms of the long canals. SAp would thenm be forced up through them.
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: Ascent of Sap
The action would be assisted by the fact that evaporation occurs at the the leaves creating a pressure difference between the top and bottoms of the long canals. SAp would thenm be forced up
through them.
That is what exactly in my book. I just can't agree with that. There must be some pumping somewhere in the xylem
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'Who are you to judge everything?' -Alokananda
Re: Ascent of Sap
There might be, also in living things unknown processes might be taking place. Even a plant has consciousness and will.
But a micropipette is nothing but a very thin glass tube. There is no pumping just what they describe as capillary action. We assume the same effect is happening in the plant.
In science, we seek the simplest solutions.
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: Ascent of Sap
Even a plant has consciousness and will.
Do they have brains?
I still do not understand the micropipette action. I've seen such things though. I still do not understand why it should work.
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'Who are you to judge everything?' -Alokananda
Re: Ascent of Sap
There was a researcher named Cleve Backster whose work I read about when I was very young claimed that they did.
I still do not understand the micropipette action. I've seen such things though. I still do not understand why it should work.
Have you ever seen a meniscus?
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: Ascent of Sap
No. Whazzat?
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'Who are you to judge everything?' -Alokananda
Re: Ascent of Sap
When a liquid is poured into a test tube the top behaves like one of those pictures. That is called a meniscus. In chemistry you measure the volume of the liquid using the meniscus.
Wikipedia wrote:
Capillary action acts on concave menisci to pull the liquid up, increasing favorable contact area between liquid and container, and on convex menisci to pull the liquid down, reducing the amount
of contact area. This phenomenon is important in transpirational pull in plants. Honey, water, milk etc. have a lower meniscus. When a tube of a narrow bore, often called a capillary tube, is
dipped into a liquid and the liquid wets the tube (with zero contact angle), the liquid surface inside the tube forms a concave meniscus, which is a virtually spherical surface having the same
radius, r, as the inside of the tube. The tube experiences a downward force of magnitude 2πrdσ
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: Ascent of Sap
The force comes from where?
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'Who are you to judge everything?' -Alokananda
Re: Ascent of Sap
Beats me! Physicists love to talk jargon.
From the point of an experimentalist, capillary action works. With it we used to catch a single Paramecium while he was happily swimming in a drop of water.
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: Ascent of Sap
I would say, its a miracle!
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'Who are you to judge everything?' -Alokananda
Re: Ascent of Sap
Yes, it is amazing. As I often say, you do not have to understand something to admire and use it.
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: Ascent of Sap
Isn't science more about why than how?
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'Who are you to judge everything?' -Alokananda
Re: Ascent of Sap
It is not supposed to be. Science is observation. It is the how. The why can never be known.
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: Ascent of Sap
Do we call the why mathematics?
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'Who are you to judge everything?' -Alokananda
Re: Ascent of Sap
Mathematics could be described as a game. That is called Formalism. Some mathematicians know that it has no real basis in reality.
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: Ascent of Sap
I would rather add Ascent of sap to one of the most miraculous ideas list
Last edited by Agnishom (2013-07-06 12:55:29)
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'Who are you to judge everything?' -Alokananda
Re: Ascent of Sap
It sounds kind of sappy to me.
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: Ascent of Sap
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'Who are you to judge everything?' -Alokananda
Re: Ascent of Sap
Ascent of Sap
Sappy: Containing a lot of sap, foolish.
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: Ascent of Sap
Well, when the water is pulled up during capillary action some work is done. Who gives the energy?
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'Who are you to judge everything?' -Alokananda
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Recursive equations in finite Markov chain imbedding
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June 2013
Volume 65
Issue 3
pp 513-527
Recursive equations in finite Markov chain imbedding
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In this paper, recursive equations for waiting time distributions of r-th occurrence of a compound pattern are studied via the finite Markov chain imbedding technique under overlapping and
non-overlapping counting schemes in sequences of independent and identically distributed (i.i.d.) or Markov dependent multi-state trials. Using the relationship between number of patterns and r-th
waiting time, distributions of number of patterns can also be obtained. The probability generating functions are also obtained. Examples and numerical results are given to illustrate our theoretical
Within this Article
1. Introduction
2. Notations and preliminary results
3. Recursive equations for distributions of waiting time ${\varvec{W(r}}\mathbf {,\Lambda )}$
4. Numerical examples
5. Summary and discussion
6. References
7. References
Other actions
1. Aki, S. (1992). Waiting time problems for a sequence of discrete random variables. Annals of the Institute of Statistical Mathematics, 44, 363–378. CrossRef
2. Chang, Y. M. (2005). Distribution of waiting time until the $r$ th occurrence of acompound pattern. Statistics and Probability Letters, 75, 29–38. CrossRef
3. Chang, Y. M., Wu, T. L. (2011). On average run lengths of control charts for autocorrelated processes. Methodology and Computing in Applied Probability, 13, 419–431.
4. Chang, Y. M., Fu, J. C., Lin, H. Y. (2012). Distribution and double generating function of number of patterns in a sequence of Markov dependent multistate trials. Annals of the Institute of
Statistical Mathematics, 64, 55–68.
5. Cui, L., Xu, Y., Zhao, X. (2010). Developments and applications of the finite Markov chain imbedding approach in reliability. IEEE Transactions on Reliability, 59, 685–690.
6. Fu, J. C. (1996). Distribution theory of runs and patterns associated with a sequence of multi-state trials. Statistica Sinica, 6, 957–974.
7. Fu, J. C., Koutras, M. V. (1994). Distribution theory of runs: a Markov chain approach. Journal of the American Statistical Association, 89, 1050–1058.
8. Fu, J. C., Lou, W. Y. W. (2003). Distribution theory of runs and patterns and its applications. River Edge, NJ: World Scientific Publishing Co. Inc.
9. Fu, J. C., Lou, W. Y. W., Wang, Y. J. (1999). On the exact distributions of Eulerian and Simon Newcomb numbers associated with random permutations. Statistics and Probability Letters, 42,
10. Fu, J. C., Shmueli, G., Chang, Y. M. (2003). A unified Markov chain approach for computing the run length distribution in control charts with simple or compound rules. Statistics and Probability
Letters, 65, 457–466.
11. Han, Q., Hirano, K. (2003). Sooner and later waiting time problems for patterns in Markov dependent trials. Journal of Applied Probability, 40, 73–86.
12. Hirano, K., Aki, S. (1993). On number of occurrences of success runs of specified length in a two-state Markov chain. Statistica Sinica, 3, 313–320.
13. Inoue, K., Aki, S. (2005). A generalized Pólya urn model and related multivariate distributions. Annals of the Institute of Statistical Mathematics, 57, 49–59.
14. Inoue, K., Aki, S. (2007). On generating functions of waiting times and numbers of occurrences of compound patterns in a sequence of multistate trials. Journal of Applied Probability, 44, 71–81.
15. Inoue, K., Aki, S. (2009). On waiting time distributions associated with compound patterns in a sequence of multi-state trials. Annals of the Institute of Statistical Mathematics, 61, 499–516.
16. Koutras, M. V. (1997). Waiting times and number of appearances of events in a sequence of discrete random variables (pp. 363–384). In Advances in combinatorial methods and applications to
probability and statistics. Statistics for Industry and Technology. Boston, MA: Birkhäuser Boston
17. Koutras, M. V., Milienos, F. S. (2012). Exact and asymptotic results for pattern waiting times. Journal of Statistical Planning and Inference, 142, 1464–1479.
18. Lou, W. Y. W. (1996). On runs and longest run tests: a method of finite Markov chain imbedding. Journal of the American Statistical Association, 91, 1595–1601. CrossRef
19. Nuel, G. (2008). Pattern Markov chains: Optimal Markov chain embedding through deterministic finite automata. Journal of Applied Probability, 45, 226–243. CrossRef
20. Zhao, X., Cui, L. (2009). On the accelerated scan finite Markov chain imbedding approach. IEEE Transactions on Reliability, 58, 383–388.
Recursive equations in finite Markov chain imbedding
Cover Date
Print ISSN
Online ISSN
Springer Japan
Additional Links
□ Recursive equation
□ Simple and compound patterns
□ Waiting time
□ Finite Markov chain imbedding
□ Probability generating function
Industry Sectors
Author Affiliations
□ 1. Department of Statistics, University of Manitoba, Winnipeg, MB, R3T 2N2, Canada
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David L.
My teaching schedule is very congested right now. I will try to respond to inquiries late at night and sometimes early in the morning.
Students in math and science sometimes struggle on exams even after carefully completing all homework. As students advance through coursework, they are asked not only to mimic homework problems, but
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I coach math, science, writing, scientific illustration, and presentation. I want to help you to "learn to learn," to digest concepts, and to practice skills. My primary focus is high school- and
college-level physics and math. I also teach graduate students, as well as research scientists from traditionally less quantitative disciplines interested in strengthening their skills in
mathematical modeling.
I have unusual experience communicating mathematical and physical sciences concepts to diverse audiences.
I am part of the National Cancer Institute's Physical Sciences-Oncology Centers (PSOC) Network, where I contribute to knowledge transfer between physical scientists, biologists, clinicians, and
patient advocates. My participation in the PSOCs began while I earned my PhD in physics (2010) at Princeton University (NSF and NDSEG Graduate Research Fellow), continued during my postdocship in
cancer biology with the University of California, San Francisco (2010-2012), and continues through my affiliation as an Analyst at UCSF. My interests have included studying the consequences of
dynamic heterogeneity for optimizing therapy and developing a video tutorial course to help interdisciplinary scientists model biological systems mathematically. I received my BS in Physics from
Harvey Mudd College, where I earned a 990 on the Physics GRE and was employed as a tutor in quantum mechanics.
My illustrations have been published in journals including Science, Phys. Rev. Lett., Proc. Natl Acad. Sci. USA, and Phys. Biol.
Only serious inquiries please. (Students outside my posted commuting range may be asked to agree to 2-hr minimum sessions.)
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Integral kernel for the resolvent of the laplace operator
up vote 3 down vote favorite
Consider the Laplace operator defined in the biggest possible subset of $L^2(\mathbb{R}^2)$ and let $z \in \mathbb{C}\backslash\mathbb{R}$. Therefore $z \notin \sigma (\Delta)$ the spectrum of $\
Delta$, and the resolvent $R=(-\Delta - zI)^{-1}$ is well defined and bounded in all of $L^2(\mathbb{R}^2)$.
I'm trying to find the integral kernel for this operator, that is a function $K(x,y)$ such that almost everywhere in $\mathbb{R}^2$:
$$(Ru)(x)=\int_{\mathbb{R}^2} u(y)K(x,y) dy$$
Let $f := ( - \Delta - z I)^{- 1} u$ and let $\mathcal{F}$ be the Fourier transform. Now
$$ ( - \Delta - z I) f = u \Rightarrow \mathcal{F} ( ( - \Delta - z I) f) =\mathcal{F} ( u) \Rightarrow ( \xi^2 - z) \hat{f} ( \xi) = \hat{u} ( \xi) . $$
Solving for $\hat{f}$ and applying $\mathcal{F}^{- 1}$ we arrive at (modulo some constant depending on your favourite definition of $\mathcal{F}$)
$$ f ( x) =\mathcal{F}^{- 1} \left( \frac{1}{\xi^2 - z} \hat{u} ( \xi) \right) =\mathcal{F}^{- 1} \left( \frac{1}{\xi^2 - z} \right) \ast u ( x) $$
So I need to calculate
$$ \int_{\mathbb{R}^2} \frac{e^{ix \cdot \xi}}{\xi^2 - z} d \xi $$
and I'm completely stuck. Does anybody have any ideas or references? Thanks.
fa.functional-analysis sp.spectral-theory fourier-analysis laplacian
1 If n equal to 1 or 3,then the inverse fourier transform of $(\xi^{2}-z)^{-1}$ is $c_{n}\frac{e^{-\sqrt{z}|x|}}{|x|}$,for other values of dimension,it can be an expression in terms of bessel
functions. For instance,it can be found in stein's book "Singular Integrals and Differentiability Properties of Functions" – user23078 Nov 23 '12 at 3:40
add comment
3 Answers
active oldest votes
At least when $\Re z < 0$, and assuming that you're taking $\Delta$ to be a negative operator (i.e., $-\Delta \geq 0$), you can write $(s-z)^{-1} = \int_0^\infty e^{zt} e^{-st} dt$, so
that by the functional calculus, you should be able to write your resolvent as $Ru = \int_0^\infty e^{zt} e^{\Delta t}u dt$, and hence your integral kernel as $$ K(x,y) = \int_0^\infty e
^{zt} K_H(x,y;t) dt, $$ where $$K_H(x,y;t) = \frac{1}{4\pi t}e^{-|x-y|^2/4t}$$ is the heat kernel for $\mathbb{R}^2$. Even if this sketch is complete nonsense, I still suspect that heat
up vote 1 kernel methods might nonetheless be helpful, at least in the regime $\Re z < 0$.
down vote
accepted EDIT: Helpful perhaps with regards to any estimates you might want--not so helpful with regards to evaluating your perfectly concrete integral...
Why complete nonsense? Couldn't this be made to work? However, Evans' technique (see my answer) doesn't need the functional calculus and is simpler to justify. – Mike Nov 23 '12 at
Not complete nonsense, but certainly requires more care than what you outlined. On the other hand, if you should ever need to carry out these calculations in the curved case... –
Branimir Ćaćić Nov 24 '12 at 3:35
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This is a classical computation' going back to the 19th century. The formula for the integral kernel of the resolvent in any dimension is a function of $r = \mbox{dist}(x,y)$. In three
dimensions it is particularly simple: $r^{-1} e^{i\lambda r}$ (up to some constant factors) where $\lambda^2 = z$. The formula in any odd dimension is only a bit harder and less explicit, and
up vote in even dimensions (including dimension 2) has a slightly different behaviour so that it must be expressed in terms of Bessel functions. This all appears in almost any good book on classical
5 down mathematical physics. One good reference with many explicit formulas, is the book by J.C. Nedelec ``Acoustic and Electromagnetic Equations" (Springer). Anyway, you need to write the integral
vote you have in polar coordinates and do some contour-shifting in the $r$-integral.
add comment
I finally found the solution in Evans' book on Partial Differential equations, Chapt. 4.3. Using that
$$ \frac{1}{\xi^2 - z} = \int_0^{\infty} e^{- t ( \xi^2 - z)} d t, $$
and substituting into
up vote 1 down vote $$ \int_{\mathbb{R}^2} \frac{e^{ix \cdot \xi}}{\xi^2 - z} d \xi $$
then using Fubini, completing the square in the exponential function and evaluating a complex integral along a line $ \{ Im = const. \} $, one arrives at the solution.
Thanks everybody for the fine answers.
add comment
Not the answer you're looking for? Browse other questions tagged fa.functional-analysis sp.spectral-theory fourier-analysis laplacian or ask your own question.
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A noncommutative closed Friedman world model
Seminar Room 1, Newton Institute
In J. Math. Phys. 46, 2005, 122501, we have proposed a model unifying general relativity and quantum mechanics based on a noncommutative algebra A defined on a groupoid having the framer bundle over
space-time as its base space. The generalized Einstein equation is defined in terms of the algebra A and its derivations; matter sources are assumed to vanish. The closed Friedman world model, when
computed in this formalism, exhibits two interesting properties. First, additional components of the generalized Einstein equation turn out to be identical with the components of the energy-momentum
tensor for the usual Friedman model and the corresponding equation of state. One could say that, in this case, matter is produced out of pure (noncommutative) geometry. Second, owing to probabilistic
properties of the model, in the noncommutative regime (on the Planck level) singularities are irrelevant. They emerge in the process of transition to the usual space-time geometry. These results will
be briefly discussed.
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Fountain Valley Prealgebra Tutor
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24 Subjects: including prealgebra, chemistry, ACT Math, algebra 1
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2 Subjects: including prealgebra, algebra 1
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my life to teaching and tutoring. All to see students succeed.
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Quilt St. George Blog Hop and Giveaway
I am so excited to be teaching at
Quilt St. George
in just a little under 2 weeks!!!! What a great way to start the new year. This is the first year of this retreat and for a little first year, new year fun, the creators of Quilt St. George have put
together a great little
Blog Hop
hosted by the teachers of Quilt St. George. And today's my turn!!!
I will be teaching 3 classes at
Quilt St. George
. First off is my Braided Irish Chain pattern on Tuesday morning, followed by my Summer Breeze pattern on Tuesday afternoon. That evening there will be a wonderful Meet the Teachers event as well as
a seminar hosted by
Dr. Bob of Superior Threads
. (So very excited!!!) And lastly, on Wednesday morning I will be teaching my The Way The Crow Flies pattern.
Each of these quilts are so fun to put together and I am so excited to be teaching classes on them. As you can see, they all have their own feel and provide for lots and lots of color options. I
can't wait to see what everyone chooses. Seeing all of the different fabrics is one of my favorite parts of teaching ;) My Braided Irish Chain Class is full but there is still room in the other 2
classes. It's not to late to sign up, I would love to see you there. We are going to have such a great time!!!!
And for more great times and classes, be sure to check out all of the stops along the blog hop.
Superior threads January 2nd
The Superior Blog
Melissa Corry January 3rd
Happy Quilting Blog
Sandra Starley January 4th
Antique and Vintage Quilts Blog from Textile Time Travels
Yvette Jones January 5th
Annie Unrein January 6th
Blogs By Annie
Lynette Bingham January 7th
Main Street Quilt Cottage Blog
Sandra Starley January 8th
Art Quilt and Reproduction Quilt Blog from Quilt Art Star
And let's finish out with another little great time opportunity . . . lets do a Giveaway. Oh yes, there will be some super fun giveaway along this blog hop :)
For 3 lucky readers, I will be giving away 3 patterns each, you can choose your favorites :)
For a chance to win . . . Just leave me a comment telling me which three patterns you would choose :)
And that's it. 1 entry per person max and everyone is welcome to enter. I will choose the winners Thursday the 9th, at the close of the blog hop :) Good Luck and Happy Quilting!!!
196 comments:
1. The Way The Crow Flies, In Your Neighborhood and Wonky Irish Chain are my favorites. Thanks for the giveaway!
2. i would choose turnovers, simply by design and the way the crow flies
3. my favorites are ♥ Braided Irish Chain ♥ Summer Breeze ♥ and ♥ Turnovers ♥
thank you so much for this giveaway and dearest greetings
4. I like Summer Breeze, Braided Irish Chain and In Your Neighborhood the best of all the patterns. Lovely quilts! --Tami
5. My very favorite is Braided Irish Chain, then I like Summer Breese and Turn Overs
6. I would choose the Braided Irish Chain, In Your Neighborhood and Summer Breeze. Great patterns.
7. So hard to choose but I love summer breeze such a pretty pattern,I do love all your patterns
8. Hurray!! My favorites are Braided Irish Chain, Summer Breeze and Salutations!! :)
Thanks for hosting.
I'm eagerly awaiting the release of the Quilt Along starting later this month. Already have my eye on the perfect fabrics for it. :)
9. I would choose Braided Irish Chain, Turnovers, and Summer Breeze! Thanks for the giveaway!
10. Your designs are so wonderful!! I would choose Summer Breeze, Turnovers and Boxed In and Out. Thanks for the chance to win!
11. What a wonderful giveaway! I'd pick braided irish chain, the way the crow flies and boxed in & out. Thank you!
12. Summer Breeze, Turnovers and Simply by Design would be my choices. The retreat sounds like fun!
13. I already have Braided Irish Chain and Summer Breeze so I would choose Turnovers, Salutations and Dreaming of Paris. Thanks for the chance to win!
14. Great giveaway. I'd choose Braided Irish Chain, Wonky Irish Chain, and In Your Neighborhood. Thanks.
15. I think my favourites would be Wonky Irish Chain, In your neighbourhood and Simply by design. Fab giveaway, thanks.
16. Wonderful chance! Braided Irish Chain, The way the crow flies, In your neighbourhood. yay!
17. Simply by Design
Braided Irish Chain
18. The Way the Crow Flies, Summer Breeze, In Your Neighborhood.
19. Braided Irish Chain, Summer Breeze and Dreaming of Paris would be my picks! Wish I was closer and could take a class, too!
Sandy A
20. I would choose In Your Neighborhood, Braided Irish Chain and Summer Breeze. Thanks for the chance!
21. Summer Breeze, Salutations and Simply by Design! Thanks for the chance to win
22. Summer Breeze, the Wat the Crow Flies and In the Neighborhood would be my choices
23. In Your Neighborhood, Boxed in and out, and Dreaming of Paris. TIA Nancy
24. Simply a boxed paris. That involves all 3 I want, ha!
25. I like the ones you are teaching-- the way the crow flis, summer breeze, and braided Irish chain.
26. I like the way the crow flies, in your neighbourhood and summer breeze!
27. I love the braided Irish Chain, The Way the Crow Flies and Wonky Irish Chain.
28. your retreat sounds fun-thanks for the giveaway chance
My favorites are The way the crow flies, wonky irish chain, and braided irish chain
29. The 3 I like "The Way the Crow Flies,Wonky Irish Chain and The Braided Irish Chain".
Thank you for the chance :)
30. Well thank goodness you are letting us choose 3, picking one would be too hard! My picks would be: summer breeze, boxed in and out, and wonky Irish chain.
Thanks for the giveaway, and sharing your talent via teaching. Wish I could be there.
31. love all your patterns and quilts. i would choose "the way the crow flies", "boxed in and out" and "turnovers". thanks for the chance to win.
32. Braided Irish chain, summer breeze, and simply by design. Thank you for this chance
33. I like Summer breeze, salutations and the way the crow flies. Thanks
34. I'd choose the braided Irish chain, summer breeze, and in your neighborhood.
35. I would choose Wonky Irish Chain, Summer Breeze, and As the Crow Flies! Thanks for the chance!
36. Oh my goodness, so many to choose from, I love them all! I would probably choose the braided Irish chain (being as I am a bit irish, as my last name is Connors), Summer Breeze as I love stars and
In your Neighborhood, I know exactly where to put that one. Thanks for such a wonderful opportunity!
1. mtnfamily2013@yahoo.com
37. Thanks for the chance!!!
I'd choose
Simply by design
In your neighborhood
Boxed in and out
38. I would choose turnovers, summer breeze and in your neighborhood
39. Sounds like so much fun. Looks like some fun classes. Wish I could make it.
40. Wonderful designs. My picks are: The Way the Crow Flies,In Your Neighborhood and Summer Breeze.
41. My picks are: In your Neighborhood, The Way The Crow Flies, and Wonky Irish Chains. Thank you for doing this giveaway.
42. These are great! I love summer breeze, the way the crow flies & in your neighborhood
43. Wonky Irish Chain, Braided Irish Chain and Summer Breeze. Thanks for the chance!
44. I would choose Summer Breeze, Wonky Irish Chain and Simply by Design
45. So hard to pick just three! Well, here goes... Simply By Design, Wonky Irish Chain and Summer Breeze.
Or maybe...
46. Thank you for the opportunity to win! I love the Braided Irish Chain, Simply by Design, and Dreaming of Paris patterns!
47. Hi Melissa,
My choices would be:
1. In Your Neighborhood
2. Summer Breeze
3. The Way The Crow Flies.
Thanks for the chance to win!
48. Simply by Design, Braided Irish Chain and In Your Neighborhood! Thanks for the chance!
49. I would choose Braided Irish Chain, In Your Neighborhood and Boxed In and Out.
50. my favorites are in your neighborhood, wonky irish chain and braided irish chain - thanks for the great post today
51. Turnovers, Dream of Paris, and Summe Breeze would be my choices
52. Thanks for the chance to win Melissa. My choices would be
1. Summer Breeze
2. In Your Neighborhood
3. Boxed In and Out
53. Salutations, braided Irish chain, and dreaming of Paris!
54. The Way The Crow Flies, In Your Neighborhood, Boxed In And Out
Thanks for a chance
55. I especially like your Summer Breeze pattern and since I can't come to Quilt St. George I'll round out my pattern choices with Braided Irish Chain and The Way the Crow Flies. If I win it will
almost be like I was there! Thanks for sharing
56. The way the crow flies, turnovers, and braided Irish chain :)
57. I hope you have a blast teaching! Thanks for including us in the fun. If I win, I'll choose Summer Breeze, Braided Irish Chain and Wonky Irish Chain. A few of my favorites. You do beautiful work.
58. Turnovers, Wonky Irish Chain and Braided Irish Chain. Can't wait to take your class at the retreat!
59. I would choose Dreaming of Paris, Braided Irish Chain and Boxed In and Out. Thanks for the give away!
60. Braided Irish Chain, In Your Neighborhood, and Boxed In and Out are 3 of my favorites. Thanks
61. I love all of your patterns, but I would choose braided Irish Chain, Turnovers and Summer Breeze (which I think I already own!)
62. Good luck with your class. I bet it would be so fun to go to a class you are teaching! My favorites are Summer Breeze, Salutations, and Turnover.
63. Have a great time at the retreat! I am sure it will be a fun time. My favorites are Simply by Design, Boxd In and Out & The Way the Crow Flies. dragonfly9716 at yahoo dot com
64. I love all of your patterns. I would choose Summer Breeze, Braided Irish Chain and Dreaming of Paris. Thank you for the chance to win.
65. Wish I could be there. Have fun. Bet it will be a blast. In Your Neighborhood, Simply by Design, and Summer Breeze are my favorite patterns.
66. looks like you will be having sew much fun. hop looks fun and I'll be watching. thanks for the chance to win...sew many great patterns to choose from!
67. I already have In your Neighborhood.... So now I would choose Simply by Design, Summer Breeze and Turnovers.
Thanks for the giveaway
68. What a great opportunity for you to teach! Hope you have a good time! The patterns I would choose are Turnover, I Spy Diamonds, and Braided Irish Chain.
69. I would choose the Wonky Irish Charm, Simply by Design, and Turnovers.
70. Simply by Design, In your neighborhood and Dreaming of Paris
71. Love all of your patterns, but if limited to 3 they would be, Braided Irish Chain, In Your Neighborhood, and Boxed In and Out. Thanks for the giveaway!
72. Simply by Design, The Way the Crow Flies, and In Your Neighborhood are my top three favorites! Thanks for the giveaway.
73. I like all patters, but Love The Way Crow flies, Braided Irish Chain, and wonky Irish chain
74. How can I choose? Braided Irish Chain, Simply by Design and In Your Neighborhood would be my first picks!
75. Boxed in and Out, In your Neighborhood and Summer Breeze would be my three - and wow - well done you for all that teaching coming up!!!
76. Such beautiful patterns, it's so hard to choose! If I had to narrow it down, I would pick Boxed In and Out, In Your Neighborhood, and Braided Irish Chain. Thank you so much for the opportunity !
77. So hard to choose ... I think Summer Breeze, Simply by Design, and The Way the Crow Flies. Have fun in St. George!
78. I've had my eye on "In Your Neighborhood" for awhile now, but I also love "Salutations" and "Simply By Design." Good luck teaching!
79. I think turnover or simply by design. But they are all so cute it is hard to pick
80. In The Neighborhood, Salutations, The Way The Crow Flies.......but honestly, it is hard to narrow it down to three. Your designs are always appealing to me.
81. My favorite are Braided Irish Chain, In Your Neighborhood, and Simply by Design! I especially like the Braided Irish Chain quilt you made with a grey background - beautiful!
82. If I was ever so lucky to win then I would choose Dreaming Of Paris; Summer Breeze and Boxed In and Out. Thank you for the chance to win.!
83. I'd like to try Boxed in and Out, Dreaming of Paris and Simply by Design, Thank You for the chance to win.
84. Summer Breeze, Dreaming of Paris and In Your Neighborhood are my favorites!
Thanks or the giveaway!
85. I would choose Summer Dreams, Braided Irish Chain and Wonky Irish Chain.
Thanks for the chance!
86. Awesome patterns and I would choose Simply By Design, Summer Breeze, and Braided Irish Chain. Thanks!
87. I would love braided Irish Chain, wonky Irish Chain and In your neighborhood
88. If I was lucky enough to win, I would choose Summer Breeze, Simply by Design and Turnovers. Thanks for the lovely giveaway.
89. I would choose In Your Neighborhood, summer Breeze, and the Way The crow Flies. Your patterns are really great, and you are generous to have a giveaway. Happy 2014!
90. Sorry read it wrong so will try again apart from summer breeze I would choose boxed in and out and the way the crow flies all so lovely
91. Hard choice. I would choose Summer Breeze, Turnover, and Braided Irish Chain
92. Melissa, you have some great patterns (I already have Summer Breeze), and I'm signing up for Quilt St. George today!!! See you there!! Hugs, H in Healdsburg
93. 1) Irish Chain!
2) summer breeze!
3) the way the crow flies!
But I could change my mind... because
1)I'm a woman and its my prerogative
2) I like WAY to many of them!
94. Such beautiful patterns! I like Summer Breeze, Boxed In and Out ,and Braided Irish Chain.
95. Braied Irish Chain, Summer Breeze and In Your Neighborhood…but they're all gorgeous!
96. The Way the Crow Flies, In Your Neighborhood, and Simply By Design. But I like so many of your other ones, too!
97. Summer Breeze, The Way the Crow Flies, and Braided Irish Chain.
Thanks for the giveaway and I hope you have a great time teaching!
98. I like simply by design, in your neighborhood and summer breeze!
99. Oh WOW - now this is a hard decision!
What really calls out to me though is In your Neighbourhood, The Way the Crow Flies and Turnovers!
100. Thanks for the giveaway... Don't know if I could decide on only 3. Summer Breeze, In Your Neighborhood, Boxed In & Out, Wonky Irish Chain, & ... see more than 3 :)
101. Great giveaway! Hard to choose, but Boxed In and Out; Braided Irish Chain; and Turnovers. All are great! Wish these classes were closer. They sound like a good group of teachers.
102. The way the crow flies, salutations and simply by design are my top three, but they are all fabulous!
103. Thanks for the give away - tough choices, they are all beautiful: Turnovers;the way the crow flies; braided Irish chain.
104. So many great patterns! I would choose Braided Irish Chain, Summer Breeze and Turnovers. Thanks for the great giveaway!
105. Love, love, love your Summer Breeze pattern. The other 2 I'd pick are In Your Neighborhood, and Boxed In & Out.
106. We wanted to take the IN YOUR NEIGHBORHOOD class it was our favorite. I also like Dreaming of Paris, and Simply by Design
I am excited to come to the classes.
107. I would choose In Your Neighborhood, The Way the Crow Flies, and Boxed In and Out -- I have all the others already!! :-) (Summer Breeze is my favorite -- I have both the paper and PDF versions).
108. I would choose The Way the Crow Flies, Salutations and Summer Breeze! Actually any of them would be great!!
109. Thanks for the chance to win. I would choose "dreaming of paris", "summer breeze" and "wonky irish chain"
110. My favorites are Summer Breeze, In Your Neighborhood, and Boxed in and Out. Thank you for the giveaway.
111. Dreaming of Paris, Summer Breeze and in your neighborhood would be my choices!
Jessica x
112. See you soon in St. George. I'd choose The Way the Crow Flies, Braided Irish Chain and Summer Breeze
113. Thanks Melissa, I ameba looking for info on a new QAL I heard you were hosting and found a giveaway, yay!! I would choose braided Irish chain, summer breeze, and in your neighborhood!!! Off to
find what I came for!!!
114. I would choose braided Irish chain, in your neighborhood, and simply by design. Thanks for the chance and have fun at St George.
115. THe way the crow flies, turnovers, in your neighborhood
116. Wish I could be taking classes with you ~ sounds like fun!
I'd choose Summer Breeze, Salutations and Dreaming of Paris ~ love all your quilts tho ~
Thanks for the lovely giveaway!
117. I would pick summer breeze, simply by design, and turnovers!
118. Let's see, so many wonderful ones, but loves the boxed in and out, braided Irish chain, and probably turnovers although dreaming of Paris is calling to me also.
119. I love them all, but I would choose Dreaming of Paris, Braided Irish Chain, and the Way the Crow Flies! Thanks, Mary
120. Summer Breeze, Salutations and In your neighborhood. Thanks, these are awesome!
121. i would love to have dreaming of paris, simple by design and salutation babscorbitt at gmail dot com
122. Its hard to choos, but I have settled on these 3: The way the Crow Flies, In Your Neighborhood and Summer Breeze!
123. Love all the patterns -lets choose Braiderd Irish Chain- In Your Neighborhood- Summer Breeze- Thank you or a chance to see your beautiful Quilts
124. I like Summer Breeze, Simply By Design & Turnovers. mrscwatson81@hotmail.com
125. My favorites: Summer Breeze, Braided Irish Chain, and either Boxed In or Turnovers!
Can't wait for the new QA!
126. I would choose Baided Irish Chain, Summer Breeze and Simply by Design.
127. They are all beautiful but my favorites are "Simply by Design, Wonky Irish Chain, and Turnovers."
128. The Way The Crow Flies, In Your Neighborhood and Summer Breeze.
129. any 3 girl......... I love'em all, but you know me.. the less HST the better.. LOL
130. Summer Breeze, as the crow flies and simply by a Design. Fabulous!
131. I love the braided Irish pattern.
132. Either Wonky Irish Chain or Turnovers. I'm so indecisive.
133. Turnovers, Braided Irish Chain, and Summer Breeze. Not necessarily in that order. hehe.
134. I love the Braided Irish Chain, Wonky Irish Chain, and Summer Breeze patterns.
135. I would pick Summer breeze, Turnovers, and Braided Irish Chain.
136. I Have To Choose?! :) If I Had To Narrow It Down It Would be, Braided Irish Chain, Summer Breeze And Simply By Design.
137. I would choose Simple by design, the way the crow flies and In your neighborhood.
138. I would choose Braided Irish Chain, Summer Breeze, and Wonky Irish Chain. Thanks for the chance to win. Have a wonderful time teaching.
139. Dreaming of Paris, The way the Crow Flys and Braided or Wonky Irish Chain! I love your patterns!
140. Turnovers, In your neighborhood, The way the crow flies are my favorites. Thanks for sharing.
141. Summer Breeze, Turnovers and Braided Irish Chain are my favorites. All fun!
142. In your neighborhood, braided Irish chain and Summer Breeze are 3 I would love to make.
143. In your neighborhood, braided Irish chain and Summer Breeze are fantastic!
144. Thank you for this great giveaway. Love these patterns! I would like Summer Breeze, Salutations, and Simple by Design. Thank you again.
Have a super great sewing and stitching day!
145. I wish I could be there that is close to my hope town. I even went to Dixie College. I would love to make "In your neighborhood, braided Irish chain and Summer Breeze." Thanks for the chance for
the giveaway.
146. Love your patterns...I would choose: WONKY IRISH CHAIN, SIMPLY BY DESIGN, BRAIDED IRISH CHAIN
147. Oh, please pick me! I would choose Summer Breeze, Turnovers and Braided Irish Chain. Thanks for the chance!
148. I would pick Simply by Design, Boxed In and Out and Dreaming of Paris. Thanks
149. Oh what a lovely giveaway. I would choose Braided Irish Chain, The Way The Crow Flies and Turnovers. Thank you for a chance to win these great patterns.
150. Love your designs and blog. I'd choose Wonky Irish Chain, Simply by Design, and In Your Neighborhood. Thanks for the chance to win.
151. I like Blocked In, Braided Irish Chain (would make it for a friend from Ireland) and Summer Breeze. Thank you for this chance to win!!
152. I like Summer Breeze, Braided Irish Chain, and Turnovers. Thank you for a chance to win them!
153. I like Boxed In and Out, In your Neighborhood and Braided Irish Chain. Thanks for a chance to win! :)
154. I'd choose In Your Neighborhood, Braided Irish Chain, and my favorite, Simply by Design. Thanks for your generosity.
155. I would choose Braided Irish Chain, The Way the Crow Flies and Simply by Design.
156. My choices are In Your Neighbourhood, Summer Breeze and Simple by Design.
157. Oh how I wish I could attend a class! Sounds like some amaZing fun. If I were the lucky winner, I'd choose the boxed in pattern, Summer Breeze, or the Braided Irish Chain. Have fun, can't wait
to hear about it all. :D
158. I love your wonky irish chain, your braided irish chain, and the way the crow flies. You're a talented lady! Thanks for the giveaway!!!
159. I love Summer Breeze, the Way the Crow Flies, and In Your Neighborhood. Thanks!
160. I'd love to win Summer Breeze or even Braided Irish Chain oh and In Your Neighborhood looks fun too!
161. I love the Dreaming of Paris, Simply by Design, and In your Neighborhood. Thank you for giving me a chance to win a new quilt pattern.
162. I would pick Braided Irish Chain, Summer Breeze and In Your Neighborhood.
163. Summer Breeze, Neighborhood, Dreaming of Paris! Thanks for the opportunity to win these. :)
164. Summer Breeze, Salutations and Dreaming of Paris. You sure know how to make things hard on a person! Pick 3? Only 3?
165. Braided Irish Chain, Turnovers, and Summer Breeze on a cold winter night in KS.
166. Summer Breeze (sure wish we had one now!) As the Crow Flies and In Your Neighborhood!
167. Turnovers, simply by design and braided Irish chain :)
168. braided Irish chain, summer breeze and turnovers are my favorites!! thank you for a lovely giveaway!! :)
169. Which to choose? Such a fun dilemma. I'd pick Braided Irish Chain, Wonky Irish Chain, and Summer Breeze.
170. Braided Irish Chain, Turnovers and Dreaming of Paris are my choices. 24Tangent@gmail.com
171. Braided Irish Chain
Simple by Design
Dreaming of Paris
172. I'd pick In Your Neighborhood, The Way the Crow Flies, and your pick for the 3rd one. Thanks Melissa!
173. As the Crow Flies
Summer breeze
Those are the 3 I would choose
174. The Way the Crow Flies ♥♥♥♥
175. Braided Irish Chain, Summer Breeze, and In Your Neighborhood are my choices. Happy New Year from Oklahoma, USA. crystalbluern at onlineok dot com
176. Summer Breeze, and the two Irish Chain quilts are my favorites.
177. In your neighborhood, turnovers, and boxed in and out. Awesome giveaway!
178. I would choose Summer Breeze, Salutations, and In Your Neighborhood. midwayfarms@hughes.net
179. Wonderful patterns! I like The Way the Crow Flies, Braided Irish Chain, Salutations. Thanks!
180. My choices would be Summer Breeze, In Your Neighborhood and Simply By Design. Stay warm!
181. I would choose the following: Wonky Irish Chain, I Spy Diamonds and The Way the Crow Flies
182. I love Salutations, Summer Breeze and Turnovers.
183. Dreaming of Paris, Simply by Design and Braided Irish Chain. Thanks for the chance to win!
184. Summer breeze, Braided Irish Chain and Turnovers. Such lovely patterns!
185. Oooooohhhh...I love them all! But my three favorites are: "Turnovers", "Simply by Design" and "The Way the Crow Flies". Thanks for the chance to win!
186. I like braided Irish chain, in your neighborhood and simply by design.
187. My choices are Braided Irish Chain, Simply by Design, and Wonky Irish Chain. -- soparkaveataoldotcom
188. During these dark and cold days, I would choose Summer Breeze, In Your Neighborhood, and Braided Irish Chain. They all look sunny and cheery!
189. wonderful giveaway, thank you! I'd choose Turnovers, Simply by Design and The Way the Crow Flies. have fun at retreat!
190. The Braided Irish Chain is my "definitely gotta make that" favorite. I'd also choose Simply By Design and The Way The Crow Flies. Thanks for the chance!
191. Braided Irish Chain would be my first choice and then I'm wavering but The Way the Crow Flies and Simply by Design are front runners! pbstrand@msn.com
192. Your patterns are beautiful. I would choose:
The Way the Crow Flies
Boxed In and Out
In Your Neighborhood
193. Top 3 on my wish list are Summer Breeze, Turnovers, and Simply by Design. Thanks for the giveaway and best wishes for a fabulous time at Quilt St. George!
It's only 6 hours away from here! I so wish I could make this one. :(
194. I would choose Braided Irish Chain;The Way the Crow Flies and all of the other beautiful patterns.
195. I would choose braided and wonky Irish train and either summer breeze or in the neighborhood. Thanks for the chance to win!
Thank you so much for your sweet comments. I just love hearing from each and every one of you!!
Have a Happy Quilting Day :)
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functions - please help!!!!
08-21-2002 #1
I am in the second semester of computer programming and am not so familiar with functions. Arrays I know but don't know how to use them in functions either.
I am looking for someone who can help me with the following:
The problem is as follows:
Read two numbers, representing a rate of pay ($ per hour) and a number of hours. Print out the total pay, with hours up to 40 being paid at a basic rate, from 40 to 60 at a rate-and-a-half, above
60 at double-rate. Print the pay as $ to 2 decimal places.
Terminate the loop when a zero rate is encountered. At the end of the loop, print out total pay.
The code for computing the pay from the rate and hours is to be written as a function.
A loop , arrays and at least two functions should be used.
One function for calculation and another for printing the total pay.
The output format should look something like this:
Pay at $2.10 / hr for 12 hours is $25.20
Pay at $2.20 /hr for 48 hours is $114.40
Pay at $2.40 / hr for 68 hours is $206.40
Pay at $2.60 / hr for 48 hours is $135.20
Pay at $2.80 / hr for 68 hours is $240.80
Pay at $3.00 / hr for 48 hours is $156.00
Total Pay is 928.80
I've tried to write the program without arrays, and it kind of works , but don't know how to use arrays and functions at the same time
This is how I started:
Double pay_calc(double[10], double rate[10]);
Double total_pay(double);
Main ()
double num1[10], num2[10];
double j=0;
for (j=0;j<10;j++){
printf("please enter hours for employee:",j+1);
for (j=0;j<10;j++){
printf("\nplease enter rate for employee:"j+1);
if (num1[j]<0 || num2[j]<0)
printf("\nplease enter a positive value!!!");
return 0;
Double pay_calc(double hour[10], double[10])
double j=0, pay_calc[10];
for (j=0;j<10;j++){
if hour[j]>60
else if
else if
printf("\n\n Pay at %.2lf for %.2lf hours is %.2lf", rate[j],hour[j], pay_calc[j]);
return pay_calc[10];
Now I need another function that prints the total pay, how do I terminate the loop when a zero is entered?
PLEASE HELP ! ! ! Your urgent assistancewill be highly appreciated.
Ok, first thing, when using one dimensional arrays with functions you don't pass the bounds of the array, so your original function should become:
Double pay_calc(double hour[10], double[10]){...}
// should be:
double pay_calc(double hour[], double rate[], int hourArraySize, int rateArraySize){...}
// and the declaration would be:
double pay_calc(double[], double[], int, int);
Then, within your function the 'for' loops would look like:
for(i=0; i<hourArraySize, i++)
do stuff to hour[i];
Another thing I can notice is your return value in the pay_calc function is incorrect:
return pay_calc[10];
This would return the value held at pay_calc[10], which is a piece of memory you don't own!
If you want to return an array you will have to return a pointer to it. So the function becomes:
double *pay_calc(......)
and you return:
return pay_calc;
The most important thing about learning to use arrays with functions is realising the connection between arrays and pointers, look this up and that should point you in the right direction (no pun
As for your problem of not being able to check for an input of zero, your error checking code:
if (num1[j]<0 || num2[j]<0)
printf("\nplease enter a positive value!!!");
should be in your loops because, as it is, you are only checking a single element of each array (num1[j]). You should check that value just after it has been entered (still inside the for loop).
So you get:
for (j=0; j<10; j++)
printf("\nplease enter rate for employee:"j+1); // j+1??? what is this intended to do?
if(num2[j] < 0)
printf("\nplease enter a positive value!!!");
}while(num2[j] < 0);
hope that helps
"The most important thing about acting is honesty. If you can fake that you've got it made" - George Burns
08-21-2002 #2
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For an electron moving in a direction of the electric field its potential energy _______ and its electric potential _________. Wouldn't the blanks be "increases" and "decreases"? because PE=mgh while
The equipotential lines inside a parallel plate capacitor a) circles clockwise b)circles counterclockwise c)radiates inwards d)radiates outwards e)runs parallel to capacitor plates f)runs
perpendicular to capacitor plates I have narrowed it down to either e) or f) but I think ...
When discharging a capacitor it will a) loose half of its charge in equal time intervals b) loose a third of its charge in equal time intervals c) loose a quarter of its charge in equal time
intervals d) gain half of its charge in equal time intervals I know that d) isn't ...
1)Several resistors are connected in series. If a battery provides constant voltage in this circuit the addition of another resistor in series a)increases current through each resistor b)decreases
current through each resistor c)doesn't affect the current throught any resi...
A circuit with a capacitor took .138s to discharge from its max charge Qmax to half of that max charge once the battery was removed. In this series circuit there also is one resistor with 100 G(ohm
symbol). The voltage drop is 12V across the battery. a)How much time would it t...
A circuit with a capacitor took .128s to discharge from its max charge Qmax to half of that max charge once the battery was removed. In this series circuit there also is one resistor with 100 G(ohm
symbol). The voltage drop is 12V across the battery. What is the time constant ...
m-450=1/4m -450=1/4m-1m -450=1/4m-4/4m -450=-3/4m -450*(-4/3)=m 600=m
If the TV draws a 150W of power, the dishwasher 1500 W of power and the computer 100 W of power, will a typical 15 A circuit breaker trip if they operate simultaneously on one line? Explain why.
Isn't P= IV and P=I^2R and P- V^2/R So wouldn't that mean that as these th...
When Gorix escaped from Dr. Sloth's ship, his ship started at rest. Gorix immediately set his thrusters to full power, and his ship's acceleration increased by 2 m/s2 every second for 15 seconds. At
that point, the ship remained at constant acceleration for another 30 ...
A sexond smaller metal sphere was positioned near the van de Graff machine. THe sphere was grounded. Did this snaller sphere become charged? How and why? was the interaction between the two spheres
attractive or repulsive? Explain. Yes the smaller sphere was charged because th...
If the TV draws a 150W of power, the dishwasher 1500 W of power and the computer 100 W of power, will a typical 15 A circuit breaker trip if they operate simultaneously on one line? Explain why.
The following appliances are being used simultaneously: toster oven, TV, computer, three 100 W light bulbs and an iron. Yet the circuit breaker (15 A) has not tripped. Does that necessarily mean that
not all of these appliacnces are connected to a single circuit? I am having t...
A battery with voltage drop of 120V is connected in a ciruit containing both series and parallel. Moving in a clockwise direction. R1= 3 ohms. Suddenly there is a connection of point A to point B. In
between these two points is a parallel circuit with R2= 10 ohms and R3=2ohms ...
Anatomy and Physiology
A. Right to left B. proximal to distal C. anterior to ventral D. coronal to transverse The answer is B.
A battery with voltage drop of 120V is connected in a ciruit containing both series and parallel. Moving in a clockwise direction. R1= 3 ohms there is a connection of point A to point B. In between
there is a parallel circuit with R2= 10 ohms to R3=2ohms on the top. On the bot...
I need help doing my sentences By Andrea
You came across an egg that was a perfect sphere. You placed it in a tub of water and found that it displaced exactly 280,000 cubic centimetres. However, it was floating on the water, and only 72% of
the surface area of the egg was below the water line. What was the total volu...
We're not supposed to find it using arctan. We have to get one side to be 2 factors that equal zero when multiplied.
Solve sinx+cosx=0 Thanks!
You came across an egg that was a perfect sphere. You placed it in a tub of water and found that it displaced exactly 280,000 cubic centimetres. However, it was floating on the water, and only 72% of
the surface area of the egg was below the water line. What was the total volu...
An electric charfe of -3 C is moved from a position where the electric potential is 1 V to a position where the electric potential is -6V. What is the chsange in potential energy of the charge with
this change in position? I know that change in PE = k(q1q)/r_2 - k(q1q)/r_1 I a...
A block oscillates on a spring and passes its equilibrium position with a speed of .157m/s. It's kinetic energy is zero when the block is at the distance of .1m from equilibrium. Assume no friction
between the block and the table. v=.157m/s KE =0J when x=.1m a)First I need...
A violin string is .32m and has a mass of .0005kg. The freqency of the note played is 440Hz. What is the speed of the wave in the string? .5*wavelenght = L wavelength= 2*.32m =.64m v=f(wavelength) =
440hz*.64m =281.6m/s b) where must one put one's finger to play a note with...
A guitar string is .75m and has a mass of .005kg. A standing wave is produced when the string is plucked. a) what is the fifth harmonic if the tension in the string is 90N? v=sqrt(F_t/mass density)=
116m/s f1=v/2L = 116m/s/(2*.75m) =77.3Hz fn=n*f1 f5=5*77.3Hz =386.5Hz and to f...
environmental science
Why does the FEMA (Federal Emergency Management Area) only allow property owners to collect insurance if they rebuild in the same place and same way as before? Thanks!
A block oscillates on a spring and passes its equilibrium position with a speed of .157m/s. It's kinetic energy is zero when the block is at the distance of .1m from equilibrium. Assume no friction
between the block and the table. v=.157m/s KE =0J when x=.1m a)First I need...
A block oscillates on a spring and passes its equilibrium position with a speed of .157m/s. It's kinetic energy is zero when the block is at the distance of .1m from equilibrium. Assume no friction
between the block and the table. Force is 8.67N. Also mass displacement is ...
Why do your vowels examples ("aaah", "eee", "uhh") sound different then when I make them? Explain. Tension, mass density and the length of our vocal cords differ so would that be the reason why they
sound different? For instance: a child's voi...
A musician plays the middle C note (262 Hz) on a guitar, a piano, and a violin at the same loudness. Will all three instruments produce exactly the same sound? Why or why not explain. Frequency
(262Hz) is the same so that means that the pitch of the sound wave is the same. Als...
Why do ocean waves break as they approach the shore? Is it because of the principle of superposition. Would it be described as constructive or destructive interference? I believe it's constructive
interference as the two waves combine the amplitude is large which causes th...
A block oscillates on a spring and passes its equilibrium position with a speed of .157m/s. It's kinetic energy is zero when the block is at the distance of .1m from equilibrium. Assume no friction
between the block and the table. What is the period of its oscillation? v=....
The frequency of a stationary siren is 1500 Hz when measured by a stationary observer. If an observer moves away from the siren at mach number .85 what is the frequency he hears? (assume room
temperature so that velocity of the air is 340m/s) I know that I would use f'=f (...
Verify that each of the following is an identity. tan^2x-sin^2x=tan^2xsin^2x I can get it down to cos^2 on the right, but cannot get it to work out on the left. secx/cosx - tanx/cotx=1 On the left I
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what is the slaves experiences in mid19th century america
american history
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Hi. Can someone help me with this? Simplify: (sin^2x)(cos^2x)+(sin^4x) The answer is sin^2x, but I have no idea how to get there! Thanks!
Suppose you have monohybrid pea plants in your garden and find that they produce round seed to wrinkled seeds in the ratio of 3:1. If the allele are designated (R & r) respectively, what is the
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Find the value of tan(1/2sin^-1 15/17) and sin[pi/2-cos(1/2)] Thanks.
how would you factor 4x-x^2-3/ x-1...so that x-1 would cancel out? could you explain step by step..plz
A string of mass m=.1kg is stretched to where its length is L=20m. What is the tension force applied to the string if it takes a pulse 2 seconds to get from one end to the other? m=kg L=20m T=4s mass
density =m/L = .1/20 = kg/m Tension =? I know v=sqrt(F_t/mass density) But I&...
Find the instantaneous velocity of a pendulum at the instant when its bob is at the height equal to half of its maximum height h=(h_max/2) above teh equilibrium point. Assume the max velocity of the
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A swing is 2m long. how often do you need to push it to increase its swing's amplitude the most? L=2m When amplitude is the most KE=0 and PE is at its maximum. But i'm not completely sure how to
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A mass m=.5kg on a spring with a spring constant. It is displaced by the distance x_max =20cm (.2m) from equilibrium on a frictionless horizontal surface. At the time t=0s the mass is released and it
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Find the instantaneous velocity of a mass on a spring oscillating on a horizontal frictionless surface at the instant when its displacement is half of its maximum displacement x=(x_max/2). Assume the
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how would you solve for x? can you explain step by step? 40 = [(x+2.8)/2]*8.5
Math : Algebra
how would you solve for x? can you explain step by step? 40 = [(x+2.8)/2]*8.5
Math : Algebra
how would you solve for x in this equation? 40.0 = 0+ (x + 2.8)/2multiplied by(8.50)
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How does my elegantly cradfted proof look?
November 7th 2009, 09:44 AM #1
How does my elegantly cradfted proof look?
Haha, how does my proof look?
Q- If K is normal to H and H is normal to G show that $aKa^{-1}\lhd H$ for all a in G.
Let $h\in H$ and $aka^{-1}\in aKa^{-1}$. Now, if $aka^{-1}\lhd H$ then $haka^{-1}h^{-1}\in aKa^{-1}$. Now, pull out the inverse $hak(ah)^{-1}$. This gives $ah_1k(h_1a)^{-1}$. Since $h_1kh_1^{-1}=
k_1$ we have $ak_1a^{-1}\in aKa^{-1}$. and thus $aKa^{-1} \lhd H$ as desired.
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The Capture-Recapture Method to Estimate Class Size
Written by Daniel T. Bobrowski, University of North Carolina at Chapel Hill
Learning Outcomes:
- To solidify student understanding of the capture-recapture method by kinesthetic reinforcement
- To reinforce the idea that the capture-recapture method provides only an estimation of the total population size
Activity Description: Student volunteers will apply the capture-recapture method to a classroom filled with their peers. (This is an activity that works well in large classes.) The data collected in
this procedure will then be applied to the Lincoln-Petersen equation to determine an approximate total population size for the class.
Time Needed: 15 minutes
Materials Needed: Student volunteers and squares of paper as “tags”
Activity Instructions:
1. Give a preliminary description of the capture-recapture method. Students should have read the pertinent textbook sections prior to this class meeting.
2. Ask for two student volunteers to act as scientists attempting to determine the population size of college students in the lecture hall.
3. Explain to the class that one student volunteer will be conducting the first sampling of the population by capturing individuals of the population, tagging them, and then releasing them back into
the wild.
4. Ask the class what some qualities of an ideal tag would be. (An ideal tag would not impede the normal life of the animal but would be easy for scientists to spot and identify.)
5. Have student 1 walk around the classroom selecting classmates. A “tag,” or square of paper, is handed to each student that is considered captured. While student 1 is conducting the sampling,
student volunteer 2 should be outside the classroom to avoid sampling bias during the recapture procedure. An alternative procedure to having a student volunteer conduct this first sampling would
be to have teaching assistants tape pieces of paper that read “trap” on the bottom of seats throughout the lecture hall. The first sampling would then consist of notifying students that if they
had sat in a “trap,” they were considered captured and tagged.
6. Ask the class why some time should be allowed to pass between the first and the second samplings of the population. (Sufficient time must be allowed for the tagged individuals to redistribute
themselves among the population. This allows for the assumption that all individuals have the same probability of being captured in the second sampling.)
7. Student 2 will then conduct the second sampling of the population by walking around the room and selecting students for capture. Students captured in this second sampling that are already tagged
are considered recaptured students. Students that are captured for the first time are just factored into the total number of the second catch. The second sampling should be proportionally larger
than the first one.
8. Use the numbers from the demonstration to demonstrate on the whiteboard how to solve for the total population size in the Lincoln-Petersen equation: Where: M = The number of individuals caught
and marked in the first sampling
N = The estimate of the total population size
R = The number of tagged animals that were recaptured
C = The total number of individuals captured in the second sampling
9. While the capture-recapture demonstration is taking place, have the teaching assistants take a precise head count of students in attendance (or take attendance via a clicker system). Compare this
number to the experimentally determined figure. Use this discrepancy to illustrate the fact that this method provides an estimation of the total population size.
10. Ask the students why the Lincoln-Petersen method is only an approximation of the total population size. (Realistically, not all individuals have the same chance of being caught in the second
sampling because of the possibility of a learning curve, death, etc. Also, stochasticity or chance could factor into and distort the final result.)
Possible discussion/test questions:
1. Suppose 40 fish in a pond are captured and tagged. A week later, 100 are caught, and of these, 5 have tags. What is the estimation of the population? N = 4000 / 5 = 800
2. What would happen to our estimation if organisms with tags made the recapture easier? Would the estimate be too high or too low? The estimation would be too low. For the example above, the
recaptured would be more than 5, making N = a lower number.
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Euclid-Euler-Jiang Prime Theorem
Authors: Chun-Xuan Jiang
Santilli's prime chains: (see paper for equations) There exist infinitely many primes such that are primes for arbitrary length . It is the Book proof. This is a generalization of Euclid-Euler proof
for the existence of infinitely many primes. Therefore Euclid-Euler-Jiang theorem in the distribution of primes is advanced. It is the Book theorem.
Comments: 13 pages
Download: PDF
Submission history
[v1] 16 Mar 2010
Unique-IP document downloads: 191 times
Add your own feedback and questions here:
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Quantum Gravity and String Theory
0903 Submissions
[1] viXra:0903.0006 [pdf] submitted on 28 Mar 2009
Conformal Gravity, Maxwell and Yang-Mills Unification in 4D from a Clifford Gauge Field Theory
Authors: Carlos Castro
Comments: recovered from sciprint.org
A model of Emergent Gravity with the observed Cosmological Constant from a BF-Chern-Simons-Higgs Model is revisited which allows to show how a Conformal Gravity, Maxwell and SU(2) x SU(2) x U(1) x U
(1) Yang-Mills Unification model in four dimensions can be attained from a Clifford Gauge Field Theory in a very natural and geometric fashion.
Category: Quantum Gravity and String Theory
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