content stringlengths 86 994k | meta stringlengths 288 619 |
|---|---|
Math Help
April 19th 2008, 10:05 AM #1
Apr 2008
I don't know how to work out this problem...
There are 12 football matches each with 3 outcomes. Each match either ends up with the A team winning, the B team winning or they both draw.
How many total possibilities are there?
If anyone knows a solution please help!!
Thank you!!
You have 3 possibilities for each of 12 trials. This gives you $\underbrace{3\times3\times3\times\ldots\times3}_{1 2 \; times}=3^{12}=531441$
April 19th 2008, 12:50 PM #2
Dec 2007 | {"url":"http://mathhelpforum.com/statistics/35112-possibilities.html","timestamp":"2014-04-16T20:22:38Z","content_type":null,"content_length":"30638","record_id":"<urn:uuid:e4ef2404-c848-4a22-bbac-250856498ec5>","cc-path":"CC-MAIN-2014-15/segments/1398223202548.14/warc/CC-MAIN-20140423032002-00004-ip-10-147-4-33.ec2.internal.warc.gz"} |
A property of some sequences and series. A sequence u[i] is said to be convergent if there exists a value u with the property that by choosing a large enough value of i, we can make u[i] as close as
we wish to u. In other words, convergence is the tendency of a sequence toward a limit. In the case of a series, it is the tending toward of the consecutive partial sums of the series toward a limit.
A sequence or series which does not converge is said to diverge.
For example, for the series
1 + (½) + (½)^2 + ((½)^3 + ...
the sum of the first two terms is 1.5, the first three 1.75, and the first four 1.875; as more and more terms are evaluated, the sum approaches the limiting value of (i.e., converges on) 2.
Related categories | {"url":"http://www.daviddarling.info/encyclopedia/C/convergence.html","timestamp":"2014-04-19T20:13:57Z","content_type":null,"content_length":"6201","record_id":"<urn:uuid:46af2d8d-f8db-44c1-947e-f8dedbb3a1f4>","cc-path":"CC-MAIN-2014-15/segments/1398223207046.13/warc/CC-MAIN-20140423032007-00481-ip-10-147-4-33.ec2.internal.warc.gz"} |
Exponential form - Math Central
Hi Motaz. Let me demonstrate with a similar question:
Let z = log[5] (1/625).
Start by realizing that if the log is in base 5, we want to exponentiate with base 5 on both sides:
5^z = 5^ log[5] (1/625)
On the right hand side, the exponentiation and the log cancel each other, because they have the same base of 5. So this simplifies to
5^z = 1/625
From here, it is just normal work with exponents:
5^z = 625^-1
5^z = (25^2)^-1
5^z = [(5^2)^2]^-1
5^z = 5^-4
z = -4.
Now you can do your question exactly the same way.
Stephen La Rocque. | {"url":"http://mathcentral.uregina.ca/QQ/database/QQ.09.08/h/motaz1.html","timestamp":"2014-04-16T18:56:55Z","content_type":null,"content_length":"7264","record_id":"<urn:uuid:c8e7a483-ff5a-4ae7-b3e8-e27f85f2052a>","cc-path":"CC-MAIN-2014-15/segments/1397609524644.38/warc/CC-MAIN-20140416005204-00207-ip-10-147-4-33.ec2.internal.warc.gz"} |
Wolfram Demonstrations Project
Attractive snowflake-like designs can be created by iteratively arranging -gons around an initial -gon. A famous example of such a construction is the pentaflake, which was first noticed by Albrecht
Dürer. This Demonstration allows you to experiment with several different types of -flakes. The "scale factor" slider only affects -flakes of the type "variation 2" and controls the relative size of
the secondary -gons.
Snapshot 1: the first iteration of the standard pentaflake construction
Snapshot 2: the fourth iteration of the standard pentaflake construction
Snapshot 3: the first iteration of the hexaflake construction
Snapshot 4: the fifth iteration of a variation on the standard pentaflake construction. Whereas the standard pentaflake construction uses identical pentagons throughout, this variation uses scaled
pentagons for each iteration.
Snapshot 5: a similar construction to the pentaflake is this, the Vicsek fractal, which is constructed from squares rather than pentagons. This is the fourth iteration of the Vicsek fractal.
Snapshot 6: a snowflake constructed using scaled hexagons | {"url":"http://demonstrations.wolfram.com/NFlakes/","timestamp":"2014-04-16T21:56:19Z","content_type":null,"content_length":"43987","record_id":"<urn:uuid:37227a7f-ab35-4707-9fa9-ac2fc1a02b18>","cc-path":"CC-MAIN-2014-15/segments/1398223211700.16/warc/CC-MAIN-20140423032011-00140-ip-10-147-4-33.ec2.internal.warc.gz"} |
5 Tips for Faster Mental Division (Part 1)
Have you ever wished you could divide numbers quickly and easily in your head? Believe it or not, you can! Over the next two weeks, we'll be learning my top 5 tips to help you become a mental
division maestro.
July 27, 2013
...check if you can simplify the problem before you start doing any division. In particular, if the number you're dividing into (aka, the dividend) and the number you're dividing by (aka, the
divisor) are both even, you should start by dividing both by 2. For example, in the problem 164 / 26, start by dividing both the top and bottom by 2 to obtain the simplified (but otherwise identical)
problem 82 / 13. Why is that helpful? Simply because it's almost always easier to work with smaller numbers.
But that's not the only way a division problem can be simplified. In a problem like 93 / 27, the dividend and divisor are not both even numbers—so we can't simplify the problem by dividing both by 2.
But both are divisible by 3 (in other words, 3 is a factor of both 93 and 27). Which means that in this case we can divide both the dividend and divisor by 3 to obtain the simplified problem 31 / 9.
The quick and dirty tip here is to always look for factors common to both the dividend and the divisor, and then to divide both by each common factor before you do anything else.
Tip #3: Multiply Before Dividing
There are times in life when doing the opposite of what you should be doing is bad. But there are other times when it's actually very, very good. This is one of those times because when it comes to
mental division, it turns out that it's often helpful to multiply instead.
Like many mental math tricks before it, this one relies upon the fact that dividing by 10 (or 100, or any other power of 10) is easy—just move the decimal point one position to the left for each
power of 10. And since it's so easy to divide by powers of 10, we should strive to do it as often as possible—even when we're supposed to be doing something like dividing a number by 5. But how can
we possibly do that? We use the fact that 5 = 10 / 2.
For example, let's imagine you're back working in your coffee shop and you've decided to calculate the average amount of money you make each day of a 5-day work week. In other words, you need to
divide the total amount of money you bring in over those 5-days—let's say it's $1,677—and divide it by 5. But instead of calculating $1,677 / 5, let's use the fact that 5 = 10 / 2 to turn this
problem into $1,677 / (10 / 2). Using what we've learned about how to divide fractions (remember invert and multiply?), we see that this is the same as $1,677 × 2 / 10. In other words, we've managed
to turn a hard problem of mentally dividing by 5 into two simple problems. First, multiply $1,677 by 2 to get $3,354. And second, divide the result by 10 to find that the average income for each day
of the week is $3,354 / 10 = $335.40.
Does this trick only work for division by 5? No! You can use the same idea to turn division by 20 into multiplication by 5 followed by division by 100, or division by 25 into multiplication by 4
followed by division by 100, and lots and lots of other things, too. And when combined with today's first tip about making approximations, this trick becomes even more powerful. Just remember to stop
and think before you start working on a problem, and you'll often see that there's a much easier way to the solution.
Wrap Up
Okay, that's all the math we have time for today. Of course, those are only 3 of my top 5 tips for faster mental division. So be sure to check back next time to learn the last—and most powerful—pair
of tips!
Be sure to check out my mental math audiobook called The Math Dude’s 5 Tips to Mastering Mental Math. And for even more math goodness, check out my book The Math Dude’s Quick and Dirty Guide to
Remember to become a fan of the Math Dude on Facebook where you’ll find lots of great math posted throughout the week. If you’re on Twitter, please follow me there, too. Finally, please send your
math questions my way via Facebook, Twitter, or email at mathdude@quickanddirtytips.com.
Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!
Mental division image from Shutterstock.
You May Also Like... | {"url":"http://www.quickanddirtytips.com/education/math/5-tips-for-faster-mental-division-part-1?page=1","timestamp":"2014-04-17T15:45:06Z","content_type":null,"content_length":"94317","record_id":"<urn:uuid:6e60d366-4e38-4e16-91ec-11ca164836c0>","cc-path":"CC-MAIN-2014-15/segments/1397609530136.5/warc/CC-MAIN-20140416005210-00128-ip-10-147-4-33.ec2.internal.warc.gz"} |
A New Tractable Class of Constraint Satisfaction Problems
Victor Dalmau
Annals of Mathematics and Artificial Intelligence Volume 44, Number 1-2, , 2005.
In this paper we consider constraint satisfaction problems where the set of constraint relations is fixed. Feder and Vardi (1998) identified three families of constraint satisfaction problems
containing all known polynomially solvable problems. We introduce a new class of problems called {\em para-primal} problems, incomparable with the families identified by Feder and Vardi (1998) and we
prove that any constraint problem in this class is decidable in polynomial time. As an application of this result we prove a complete classification for the complexity of constraint satisfaction
problems under the assumption that the basis contains all the permutation relations. In the proofs, we make an intensive use of algebraic results from clone theory about the structure of para-primal
and homogeneous algebras.
Postscript - Requires a viewer, such as GhostView | {"url":"http://eprints.pascal-network.org/archive/00001833/","timestamp":"2014-04-20T15:55:39Z","content_type":null,"content_length":"6967","record_id":"<urn:uuid:d575c972-fcc5-40c8-9ebd-df632f60893a>","cc-path":"CC-MAIN-2014-15/segments/1397609538824.34/warc/CC-MAIN-20140416005218-00659-ip-10-147-4-33.ec2.internal.warc.gz"} |
Hemet Math Tutor
Find a Hemet Math Tutor
...Another year I worked for the private company Friendly Community Outreach Center (FCOC). My main goals as a tutor are to help students increase their grade, to establish study techniques, and
to understand basic concepts that will lead them to understand more abstract ones. I am enrolled in the ...
7 Subjects: including algebra 1, algebra 2, geometry, prealgebra
I am a retired math teacher with 25 years experience teaching basic math, algebra, geometry, trigonometry, pre-calculus and first semester calculus. I have also taught SAT prep and study skills.
My experience has been mainly in public schools in San Diego, Boston, and Chicago, but I have also taug...
9 Subjects: including algebra 1, algebra 2, geometry, prealgebra
...This is useful when analyzing court cases etc as one does in Government courses. I have a B.A. from Pepperdine University. At Pepperdine, I took a number of literature courses, including a
series of four Great Books Colloquia and majored in philosophy.
38 Subjects: including algebra 2, physics, precalculus, study skills
...I can teach writing, reading, some statistics, math, biology, botany, and a few other subjects. The majority of the classes that I take are during the day so in the afternoons I tend to be free
and likewise on weekends. I am a very patient person who loves to share my passion with others.
18 Subjects: including algebra 2, chemistry, algebra 1, biology
...My primary focus as a tutor is to first help struggling learners overcome the fear of math. Then, I provide the missing tools necessary to help them reach their top potential and achieve their
highest academic goals. I employ both traditional and non-traditional teaching methods that reinforce ...
6 Subjects: including calculus, algebra 1, algebra 2, geometry | {"url":"http://www.purplemath.com/Hemet_Math_tutors.php","timestamp":"2014-04-20T10:56:10Z","content_type":null,"content_length":"23519","record_id":"<urn:uuid:2f7bb465-3fab-4cca-8d83-a933d88f1862>","cc-path":"CC-MAIN-2014-15/segments/1397609538423.10/warc/CC-MAIN-20140416005218-00134-ip-10-147-4-33.ec2.internal.warc.gz"} |
embedded extended visual cryptography schemes uml diagrams
Some Information About
embedded extended visual cryptography schemes uml diagrams
is hidden..!! Click Here to show embedded extended visual cryptography schemes uml diagrams's more details..
Do You Want To See More Details About
"embedded extended visual cryptography schemes uml diagrams"
? Then
.Ask Here..!
with your need/request , We will collect and show specific information of embedded extended visual cryptography schemes uml diagrams's within short time.......So hurry to Ask now (No Registration ,
No fees ...its a free service from our side).....Our experts are ready to help you...
.Ask Here..!
In this page you may see embedded extended visual cryptography schemes uml diagrams related pages link And You're currently viewing a stripped down version of content. open "Show Contents" to see
content in proper format with attachments
Page / Author tags
Posted by:
Created at: Thursday uml diagrams for visual cryptography , visual cryptography schemes, uml diagram for cryptography , code visual cryptography scheme for color images, cryptography uml
27th of September 2012 diagrams , uml diagrams for cryptography, uml for visual cryptography matlab based project , visual cryptography schemes uml diagrams, uml diagrams for image cryptography ,
07:45:48 AM uml diagrams of visual cryptography, visual cryptography , embedded extended visual cryptography uml diagrams, embedded extended visual cryptography schemes uml diagrams ,
Last Edited Or Replied uml diagrams for visual cryptography pdf, uml diagrams of visual cryptography as a java project , visual cryptography uml diagrams,
at :Thursday 27th of
September 2012 07:45:48
i ..................[:=> Show Contents <=:]
Posted by:
Created at: Saturday embedded extended visual cryptography schemes implementation of java code , embedded extended visual cryptography schemes, visual cryptography java code , embedded extended
08th of December 2012 visual cryptography schemes source code, download of the code for embedded extended visual crytptography schemes , download of the code for embedded extended visual
05:18:02 AM cryptography schemes, embedded extended visual cryptography schemes project source code , visual cryptography code in java, embedded extended visual cryptography schemes
Last Edited Or Replied source code in java , java pdf for visual cryptography, embedded extended visual cryptography schemes project code in java , jiro java based technology pdf,
at :Wednesday 17th of
April 2013 01:12:40 AM
plz provide the java code for embedded extended visual cryptography schemes pdf
source code on embedded extended visual cryptography schemas
sour..................[:=> Show Contents <=:]
Posted by:
Created at: Friday 26th
of October 2012 data flow diagram foe video streamlining mobile phones, dfd of online video streaming , data flow diagram for project game theoretic pricing for video streaming in mobile
12:45:01 PM networks, video streaming project dfd , dfd for embedded extended visual cryptography, online video streaming project dfd , data flow diagram for video streaming,
Last Edited Or Replied
at :Friday 26th of
October 2012 12:45:01
[size=..................[:=> Show Contents <=:]
Posted by:
Created at: Thursday uml diagrams for visual cryptography, visual cryptography schemes , uml diagram for cryptography, code visual cryptography scheme for color images , cryptography uml
27th of September 2012 diagrams, uml diagrams for cryptography , uml for visual cryptography matlab based project, visual cryptography schemes uml diagrams , uml diagrams for image cryptography,
07:45:48 AM uml diagrams of visual cryptography , visual cryptography, embedded extended visual cryptography uml diagrams , embedded extended visual cryptography schemes uml diagrams,
Last Edited Or Replied uml diagrams for visual cryptography pdf , uml diagrams of visual cryptography as a java project, visual cryptography uml diagrams ,
at :Thursday 27th of
September 2012 07:45:48
i need uml diagrams for the project embedded extented visual ..................[:=> Show Contents <=:]
Posted by: ksumanth225
Created at: Tuesday embedded extended cryptograhy scheme , ppts on embedded extended visual cryptography, embedded extended visual cryptography schemes ppt , embedded extended visual
21st of February 2012 cryptography schemes pdf, ppts on embedded extended , embedded extended visual cryptograpy, ppt for embedded extended , ppt for embedded extended visual cryptograpy,
10:02:09 AM embedded extended visual cryptography , embedded extended visual cryptography schemes, extended visual cryptography schemes for be project , embedded extended visual
Last Edited Or Replied cryptography schemes source code, embedded extended visual cryptography schemes project , source code embedded visual cryptography schemes, visual cryptography schemes ppt ,
at :Thursday 20th of
December 2012 11:46:26
nth studying final b.tech at sri vidyanikethan college.
i want embedded extended visual cryptography schem ppt and source code.please send to me sir.
thanking you sir,
this is my mail id: theultimate.sumanth@gmail.com
..................[:=> Show Contents <=:]
Posted by:
Created at: Tuesday embedded extended cryptograhy scheme, algarithms used in extended embded visual cryptography , embedded extended visual cryptography scheme algorithms, embedded extended
21st of February 2012 visual cryptography schemes , difference between visual cryptography and embedded visual cryptography, what algorithm is used in embedded extended visual cryptography
05:22:06 AM schemes , algorithm used in embeeded extended, extended embedded algorithm in visual cryptography , extended embedded algorithm, difference between visual cryptography
Last Edited Or Replied schemes and embedded extended visual cryptography schemes , visual cryptography algorithm, extended embeded algorithm , extended embedded algorithms,
at :Tuesday 21st of
February 2012 05:22:06
what is e..................[:=> Show Contents <=:]
Posted by: jp16586 Embedded Extended Visual Cryptography Schemes, Embedded , Extended, Visual , applications of visual cryptography, shamir threshold scheme , visual cryptography applications,
Created at: Tuesday visual cryptography for color images , application of visual cryptography, visual cryptography , Cryptography, Schemes , embedded extended visual cryptography schemes
27th of December 2011 usecases, embedded extended visual cryptography schemes ppt , embedded extended visual cryptography schemes, applications of extended embedded cryptography , document of
07:23:42 AM embedded extended visual crptography schemes, embedded extended visual cryptography , use case view for embedded visual cryptography scheme, ppt on embedded extended visual
Last Edited Or Replied cryptography schemes , embedded extended visual cryptography schemes project powerpoint slides, full report of embedded extended visual cryptography schemes , ppt for
at :Thursday 12th of embedded extended vc scheme, embedded external visual cryptography scheme free download abstract ,
September 2013 11:13:57
Check ..................[:=> Show Contents <=:]
Posted by: jp16586 Embedded Extended Visual Cryptography Schemes , Embedded, Extended , applications of encryption, cryptography digital signatures , private key vs public key, visual
Created at: Monday 24th cryptography applications , visual encryption, key cryptographic , private key and public key, keys encryption , secret key encryption, encryption techniques , asymmetric
of October 2011 key encryption, private key encryption , Visual, Cryptography , Schemes, embedded extended visual cryptography schemes ppt , embedded extended visual cryptography schemes
11:49:13 AM code, embedded extended visual cryptography schemes pdf , embedded extented visual cryptography schemes, embedded extended visual cryptography schemes , what is the meaning
Last Edited Or Replied of embedded extended visual cryptography schemes, visual cryptography schemes for secret images ppt , embedded extend visual cryptography scchemes, gray level extended
at :Wednesday 20th of visual cryptography , extended visual cryptography, source code for embedded extended visual cryptography scheme , interface design using applet framework visual
February 2013 07:42:02 cryptography schemes, ppt for embedded extended vc scheme ,
e image in such a way that no-one apart from the sender and intended recipient even realizes the original image, a form of security through obscurity. By contrast, cryptography obscures the
original image, but it does not conceal the fact that it is not the actual image.
Limitations of the existing System:-
The existing system does not provide a friendly environment to encrypt or decrypt the data (images).
The existing system supports with only one type of image format only. For example, if it is .jpg, then it supports only that same kind of image format only.
Proposed Sys..................[:=> Show Contents <=:]
Cloud Plugin by Remshad Medappil | {"url":"http://seminarprojects.net/c/embedded-extended-visual-cryptography-schemes-uml-diagrams","timestamp":"2014-04-19T19:35:57Z","content_type":null,"content_length":"36574","record_id":"<urn:uuid:e86e731c-95a6-4da0-9ed5-345b6d717ef5>","cc-path":"CC-MAIN-2014-15/segments/1397609537376.43/warc/CC-MAIN-20140416005217-00063-ip-10-147-4-33.ec2.internal.warc.gz"} |
Re-Calculating the Standard Deviation
Date: 03/20/2002 at 11:38:49
From: Gareth Williams
Subject: Re-Calculating the Standard Deviation
Hi Dr. Math,
How can I re-calculate the standard deviation of a set of values when
I insert a new value but I don't know what the existing values are?
For instance, I have a set of 5 values. They have an average of 20
with a standard deviation of 12. If I add the value 26 to the set I
can re-calculate the new average of 21, but how do I re-calculate the
new standard deviation?
Thanks in advance,
Date: 03/21/2002 at 03:24:51
From: Doctor Twe
Subject: Re: Re-Calculating the Standard Deviation
Hi Gareth - thanks for writing to Dr. Math.
Yes, this can be done.
If you are working with population mean and standard deviation, an
alternate formula for the population standard deviation (the one
calculators and computers actually use) is
n*Sum[x^2] - (Sum[x])^2
s = Sqrt[ ----------------------- ]
n = the population size (number of data points)
Sum[x^2] = the sum of the squares of all data points
Sum[x] = the sum of all data points
So if you could recover the values of Sum[x^2] and Sum[x], then
the standard deviation after adding a new value, v, would be
(n+1)*(v^2 + Sum[x^2]) - (v + Sum[x])^2
s' = Sqrt[ --------------------------------------- ]
Recovering Sum[x] is easy: it's just n times the mean, mu:
Sum[x] = n*mu
Recovering Sum[x^2] is a little trickier. Starting from the formula,
n*Sum[x^2] - (Sum[x])^2
s = Sqrt[ ----------------------- ]
n*Sum[x^2] - (Sum[x])^2
s^2 = -----------------------
n^2 s^2 = n*Sum[x^2] - (Sum[x])^2
n^2 s^2 + (Sum[x])^2 = n*Sum[x^2]
n^2 s^2 + (n*mu)^2 = n*Sum[x^2]
n^2 s^2 + n^2 mu^2 = n*Sum[x^2]
n^2(s^2 + mu^2) = n*Sum[x^2]
n(s^2 + mu^2) = Sum[x^2]
So from n, mu, and s, we can recover Sum[x] and Sum[x^2], which means
we can compute s'.
A similar computation can be made if you have a sample mean and
standard deviation. The alternate formula for the sample standard
deviation is:
s = Sqrt[ --------------------- ]
So you start from there, but the rest is the same.
I hope this helps. If you have any more questions, write back.
- Doctor TWE, The Math Forum
Date: 03/22/2002 at 06:26:19
From: Gareth Williams
Subject: Re-Calculating the Standard Deviation
No need for a follow up; your well-explained answer told me all I
needed to know. Thanks, Dr. Math. | {"url":"http://mathforum.org/library/drmath/view/52820.html","timestamp":"2014-04-16T19:53:47Z","content_type":null,"content_length":"7994","record_id":"<urn:uuid:6ce4add6-14ec-4082-af55-70c207d2aef1>","cc-path":"CC-MAIN-2014-15/segments/1397609524644.38/warc/CC-MAIN-20140416005204-00390-ip-10-147-4-33.ec2.internal.warc.gz"} |
Fast Toeplitz Solvers (Part I of II)
Thursday, July 17
10:30 AM-12:30 PM
Meyer Library, Forum Room
Fast Toeplitz Solvers (Part I of II)
This minisymposium is centered on the development of fast Toeplitz solvers. There are many applications of the Toeplitz systems such as (1) numerical PDE, (2) numerical integral equation, (3) time
series, (4) signal processing: filter design, (5) image restoration, and (6) control theory.
In recent years, some fast iterative methods for Toeplitz systems have been developed. The main advantage of these methods is that the convergence rate is superlinear and hence is faster that any
existing ones by at least an O(log n) factor, where n is the size of the systems. The purpose of this minisymposium is to invite some leading experts in this area to give the talks on their recent
research results.
Outstanding scholars and experts in the field of numerical linear algebra from leading universities and research institutes around the world will be invited to give some talks on their research of
Toeplitz solvers. The people who are interested in the fast Toeplitz solvers will greatly benefit from their studies.
Organizer: Xiao-Qing Jin
University of Macau, Macau
10:30 A Korovkin-Weierstrass Matrix Theory for the Approximation of Toeplitz Matrices via Banach Matrix Algebra
Stefano Serra, University of Pisa, Italy
11:00 Numerical Solution of Eigenvalue Problem for Hermitian Toeplitz-like Matrices
Michael K. Ng and William F. Trench, The Australian National University, Australia
11:30 A Unifying Framework for Preconditioners Based on Fast Transforms
Kurt Otto, Uppsala University, Sweden
12:00 Effective Methods for Solving Banded Toeplitz Systems
Dario Andrea Bini and Beatrice Meini, University of Pisa, Italy
AN97 Homepage | Program Updates|
Registration | Hotel and Dormitory Information | Transportation | Program-at-a-Glance | Program Overview
MMD, 4/21/97
tjf, 5/29/97
MMD, 5/30/97 | {"url":"http://www.siam.org/meetings/archives/an97/ms55.htm","timestamp":"2014-04-19T14:32:56Z","content_type":null,"content_length":"2707","record_id":"<urn:uuid:ff98c3e2-bb89-4a08-b5cd-e19c7e9db176>","cc-path":"CC-MAIN-2014-15/segments/1398223203422.8/warc/CC-MAIN-20140423032003-00663-ip-10-147-4-33.ec2.internal.warc.gz"} |
Checker Tricks - Magic Tricks
( Article orginally published July 1927 )
[Coin Tricks] [Card Tricks] [Checker Tricks] [Conjuring With Cigarettes] [Cork Tricks] [Hand Tricks] [Handkerchief Tricks] [Match Tricks] [Miscellaneous Tricks] [Number Tricks] [Optical Tricks]
[Paper Tricks] [Spirit Tricks] [Table Tricks] [Thimble Tricks] [Tumbler Tricks] [More Magic Tricks]
Checkers are common objects that are suited to many impromptu tricks; yet somehow they have been neglected in the past. There are, however, quite a few good checker tricks in existence, and some of
them are explained in this chapter.
Every household has its checker-board and set of checkers. They are inexpensive articles that can furnish many minutes of diversion and entertainment.
1. The Magic Knock-out.
Ten checkers are stacked up, and all are red except the fourth from the bottom, which is black.
The magician stands another checker on edge, and by pressing down with his finger, snaps it so that it shoots rapidly on edge against the stack of checkers.
Instead of the stack falling, or the bottom checker going out, the one black checker, fourth from the bottom, flies from the stack, while the other checkers do not fall.
This is a very surprising experiment, and it is hard to believe, even after one has seen it performed. It is because the black checker is just high enough to receive the blow from the edge of the
checker. Note: If unusually thick checkers are used, it is possible that the third checker from the bottom may be the one ejected. This can be determined by experiment, and the black checker should
be placed at the proper position.
2. A Checker Trick.
Lay three checkers in a row, a black between two reds. Then ask someone to move one red one so that it comes between the other red and the black; yet the second red checker must not be touched, and
the black checker must not be moved!
These conditions make the trick sound impossible; but the procedure is very simple.
Place a forefinger on the black checker, and with the other hand slide the first red checker forcibly against the black. The blow will cause the second red checker to slide away, although the black
checker is not moved.
Then the first red checker may be placed between the second red and the black.
3. Making Kings.
The magician lays ten checkers in a row, and starts to make kings in a peculiar fashion. He lifts one checker, passes it over two, and sets it on the next checker. He lifts another, passes it over
two and sets it on the next checker, and continues thus until he has made five kings with ten checkers.
He must always pass the lifted checker over two, whether those two are separate or have been made into a king.
People who try to duplicate the quick and certain moves of the magician will generally make a mistake before they accomplish the trick.
Here are the correct moves: Pick up 4 and pass it over 3 and a setting it on 1; 6 over 7 and 8, placed on 9; 8 over 7 and 5, placed on 3 ; 2 over the 3 and 8, placed on 5 ; 10 over the 9 and 6,
placed on 7. The numbers refer to the positions of the checkers in the row.
4. The Mystic Nine.
With a number of checkers make a figure 9. At least a dozen checkers should be used. Tell a person to think of a number which must exceed the number of checkers in the bottom of the figure 9. Then,
commencing at the end checker, he is to count mentally up the base of the q, and around the circle until he reaches the number thought of. Then, starting with that checker, he must reverse his count,
this time avoiding the bottom of the 9, and continuing around the circle, until he stops at the number chosen mentally.
This is done while the magician's back is turned; but the magician can immediately point out the checker upon which the count is ended.
Here is the system: If the person counted up to his number, and then back again the way he came, he would end where he began. But instead he takes another course. So the magician simply notes the
number of checkers in the bottom part of the q, and counts that many around the circle to the left of its junction with the base. The person's count will always end on that checker, no matter what
number he selects.
The trick may be repeated by changing the number of checkers in the base of the 9.
5. The Magnetic Checker.
The magician takes a checker and sets it against the door. The checker remains there, and does not fall. Apparently it is magnetized to the door, thus proving that wood is magnetic!
Method: It is best to use a checker with a smooth side for this trick. In setting it against the door, or the wood-work, press it upward, sliding it a short distance. The friction will cause the
checker to adhere to the wood.
Do not work this trick on any highly polished wood-work, as it may scratch the surface.
6. The Color Changing Checker.
A stack of about seven checkers is set up with a black checker in the midst of red ones. The stack is covered with a paper tube; when the tube is lifted, the black checker is gone, and only red ones
Method: Cut a loose ring of black paper that will fit around a checker. ALL of the checkers in the stack are red ones, but the center one has the ring around it, and appears to be black. The stack
should be slightly uneven.
The paper tube is used to straighten the stack, and the tube when lifted carries away the black ring inside, leaving all red checkers.
7. Moving the Checkers.
The magician places eight checkers in a row, alternating red and black. He makes four quick moves, moving two checkers at a time, and at the finish, all the reds are together and so are all the
To learn this quick little trick, number the checkers from 1 to 8, and assume that there are two other spaces, 9 and 10, which are not filled with checkers.
Then move the checkers thus: 2 and 3 to 9 and ten; 5 and 6 to 2 and 3 ; 8 and 9 to 5 and 6; 1 and 2 to 8 and 9.
All the reds will then be together; and so will all the blacks.
8. Eleven or Twelve?
Lay three checkers on the table.Pick them up, counting "one, two, three", and lay them down, one at a time, counting "four, five, six". Pick them up, counting "seven, eight, nine", and lay them down
counting "ten, eleven, twelve".
This appears quite fair; but when the count is repeated, it ends at eleven instead of twelve, and no one can tell why!
Method: Picking up the checkers, count "one, two, three", and laying them down count "four, five, six". Then pick them up, counting "seven, eight-" but as you pick up the last checker, immediately
lay it down as you say "nine". Then follow with the two checkers in your hand, counting "ten, eleven".
This is very deceptive, and it will completely baffle people. When they want to try it, give them the checkers, and they will start the count by laying the checkers one at a time on the table. This
means failure, as the checkers must be on the table at the start.
Of course everyone will want to see this trick repeated. To repeat it might give away the secret; so instead, the following trick should be performed:
9. Nine or Ten?
Three checkers are laid on the table. A person is told to pick them up counting "one, two, three", and to lay them down counting "four, five, six", and then to pick them up counting "seven, eight,
But when the magician counts the checkers, his total is ten!
Method: The magician starts with the checkers in his hand, and lays them down counting "one, two three". He points at an end checker and says "four", then picks up the other two counting "five-six".
He imme, diately picks up the checker still on the table, saying "seven", and lays down the checkers from his hand, one at a time, counting "eight, nine, ten".
10. Right and Left.
Take a piece of paper and on it mark seven squares in a row. Place three black checkers in the three squares at the left, and three red checkers in the squares at the right.
The trick is to transpose the checkers, putting red on the left and black on the right, in accordance with the following rules: black can move only to the right, red only to the left; each checker
can be moved only one square at a time; single jumps are allowed.
This is a very perplexing problem, which cannot be performed in less than fifteen moves. There is a system to it, and the magician can execute it quickly and perfectly by following two simple rules:
First: Start with any color checker and move it, but after every single move by one color, make a jump with the other color.
After a jump, advance with the same color that made the jump. The positions will indicate whether you must make another jump or just a single move.
After the ninth move, the rules do not apply, but from that point on the moves are easy and obvious.
11. The Vanishing Checker.
The magician takes twelve checkers and counts them. He counts them a second time, and asks a person to hold them.
The magician then produces one of the checkers from his pocket, and when the spectator counts the checkers, he finds that he has only eleven !
Method: This is done by a clever method of counting. First count the twelve checkers on the table. In stacking them up, secretly hold one in the right hand, which is closed, only the right thumb and
forefinger being extended, to count the stack of checkers.
As the checkers have been counted up to twelve, the counting is now reversed. As each checker is laid in a new stack, it is counted thus: "twelve-eleven - ten-nine-eightseven-". Then the left hand
picks up the remaining checkers, shows them and says"and five makes twelve". Those checkers are then added to the stack which is held by the spectator.
By this count, eleven checkers have been made to appear as twelve. The magician puts his right hand in his pocket and brings out the odd checker. When the holder counts his checkers he will be
surprised to find only eleven.
12. A Trick with a Checker-board.
Eight checkers and the checker-board are used in this trick. The object is to lay eight checkers on the board in such a way that no two will be in the same line, vertical, horizontal or diagonal.
People will try this puzzler for a long time with no success; but one who knows the secret can do it in an instant.
Simply remember the following numbers: 5, 2, 4, 6, 8, 3, I, 7. Note how the even numbers run in rotation.
Place the checker-board in front of you and lay the first checker on the fifth square of the top row; the next on the second square of the second row, and so on, according to your formula. Then the
conditions of the trick will be fulfilled.
13. Picking Out the Black.
All the red checkers are thrown into a hat, along with a black checker-the spectators selecting any black checker that they wish. The magician shakes the hat, and holds it behind his back. Then
reaching in, he immediately draws out the black checker from among the red ones.
Method: The magician secretly obtains possession of a black checker. This he holds beneath his fingers which are holding the inside of the hat brim-or the checker can be put beneath the inside band
of the hat. When the hat is behind his back, the magician shifts hands and brings out the duplicate black checker.
He immediately turns over the hat on the table, and lets the red checkers fall out upon the remaining black checkers which are lying there. Thus the original black checker joins its companions, and
no one suspects that it was not taken from the hat at all!
14. Changing Checkers.
Two stacks of checkers are used in this trick - one stack red, the other black.
Each stack is wrapped up in a piece of paper, the paper being made into a cylinder which surrounds the stack, and the top being twisted over to hide the checkers from above.
The red stack is placed several feet away from the black, and the magician commands them to change places. When he lifts the paper cylinders, the checkers have obeyed the, order, the black being
where the red were supposed to be, and vice versa.
Two special checkers are required for this trick. One is red, but with black on the bottom; the other is black, with the bottom col ored red. It is best to have the bottoms painted the opposite
color, but a circular piece of paper may be glued beneath each checker, instead.
The prepared checkers are the bottom ones of the stacks. After each stack is surrounded with a cylinder, the magician closes the tops of the cylinders and mixes them around.
Then, to learn which color is in a cylinder, he tilts up the cylinder and lets people glimpse the bottom of the lowermost checker. In this manner, the black stack is identified as red, and the red
stack is supposed to be the black.
The magician merely commands, lifts the paper cylinders and shows the marvelous transposition. | {"url":"http://www.oldandsold.com/articles02/magictricks1.shtml","timestamp":"2014-04-19T02:52:44Z","content_type":null,"content_length":"19546","record_id":"<urn:uuid:cf7a995f-a316-489c-b596-fb3a4371092c>","cc-path":"CC-MAIN-2014-15/segments/1397609535745.0/warc/CC-MAIN-20140416005215-00460-ip-10-147-4-33.ec2.internal.warc.gz"} |
Question About Parabola's
December 10th 2005, 09:24 AM #1
Dec 2005
plz help me out in this
The support for a bridge over a river is in the shape of an upside down (inverted) parabolic arch. At water level the arch is 30m wide. Max height of 10m above water. Equation?
y=a(x-h)2+k **the little 2 represents squared*
what is the equation plzz need the answer quick
plz help me out in this
The support for a bridge over a river is in the shape of an upside down (inverted) parabolic arch. At water level the arch is 30m wide. Max height of 10m above water. Equation?
y=a(x-h)2+k **the little 2 represents squared*
what is the equation plzz need the answer quick
(h,k) is the vertex of the parabola.
Here, the parabola is vertical and opens downward, so the vertex is the maximum height.
(h,k) = (0,10) -------if the origin (0,0) of the x,y axes is at water level and vertically below the max height.
y = a(x-h)^2 +k ---------(i)
y = a(x-0)^2 +10
y = a(x^2) +10 --------(ii)
At righthand end of the parabola,
x = 30/2 = 15
y = 0
Substitute those into (ii),
0 = a(15^2) +10
0 = 225a +10
-225a = 10
a = -10/225 = -2/45 = -1/22.5
Therefore, if the origin (0,0) of the x,y axes is at water level and vertically below the max height, the parabola is
y = (-2/45)x^2 +10. -------answer.
December 10th 2005, 11:06 AM #2
MHF Contributor
Apr 2005 | {"url":"http://mathhelpforum.com/algebra/1430-question-about-parabola-s.html","timestamp":"2014-04-20T15:08:48Z","content_type":null,"content_length":"32542","record_id":"<urn:uuid:933892e6-1ff6-431f-8538-fe3e16831a6f>","cc-path":"CC-MAIN-2014-15/segments/1397609538787.31/warc/CC-MAIN-20140416005218-00619-ip-10-147-4-33.ec2.internal.warc.gz"} |
Frequency Domain Analysis Explained
Frequency Domain Analysis Explained
Predicting the future behavior of a process is key to the analysis of feedback control systems. Knowing how the controlled process will react to the controller's efforts allows the controller to
choose the course of action required to drive the process variable towards the setpoint. Linear processes are particularly predictable since a combination of two control efforts applied
Other useful reading
• Linear process behavior
• Fourier's Theorem and Bode plots
• Predicting future process performance
• Simple examples
Predicting the future behavior of a process is key to the analysis of feedback control systems. Knowing how the controlled process will react to the controller's efforts allows the controller to
choose the course of action required to drive the process variable towards the setpoint.
Linear processes are particularly predictable since a combination of two control efforts applied simultaneously to the process will produce a process variable equal to the sum of the outputs that
would have resulted had the two control efforts been applied separately. A linear process also demonstrates a constant, steady-state gain K. That is, if B is the value of the process variable when
the control effort is zero, then the process variable eventually will settle out at a value of Y=KX+B when the control effort is fixed at a value of X. This relationship yields a straight line when Y
is plotted against X (hence the expression 'linear process').
Linear processes also respond to non-constant inputs in predictable ways. Most importantly, sinusoidal inputs always yield sinusoidal outputs. In fact, if the input from the controller happens to be
a sine wave with frequency v, then the process variable output by the process also will be a sine wave with the same frequency. Although real-life controllers rarely generate purely sinusoidal
control efforts, this phenomenon is the basis for frequency domain analysis of a process' behavior.
Linear process example
A simple example of frequency domain analysis can be demonstrated by means of the child's toy shown in 'A simple linear process' graphic. This linear process consists of a weight hanging from a
handle-mounted spring. A child controls the position of the weight by moving the handle up and down.
Anyone who has ever played with such a toy knows that if the handle is moved in a more-or-less sinusoidal manner, the weight will start oscillating at the same rate, though out of synch with the
handle. Only at relatively low frequencies where the spring doesn't stretch will the handle and weight move in lock step.
At higher and higher frequencies, the weight will begin to oscillate more than the handle yet lag further and further behind it. At the natural frequency of the process, the weight's oscillations
will reach their maximum height. The natural frequency is determined by the mass of the weight and the stiffness of the spring.
A toy comprised of a weight attached to a handle-mounted spring can illustrate frequency domain analysis. If the handle is moved in a more-or-less sinusoidal manner, the weight will oscillate at the
same rate, though out of synch with the handle.
Above the natural frequency, the amplitude of the weight's oscillations will decrease and its phase will grow more negative (that is, oscillations will grow smaller and smaller and lag further
behind). At very high frequencies, the weight will move only slightly, in exactly the opposite direction of the handle.
Bode plots
All linear processes demonstrate similar behavior. They transform a sinusoidal input into a sinusoidal output with the same frequency but a different amplitude and phase. Just how much the amplitude
and phase change depends on the gain and phase lag of the process. Gain is the factor by which the process amplifies the sine wave en route from input to output, and the phase lag is the degree by
which the sine wave is delayed.
A Bode plot shows how a sine wave of frequency v radians per second passing through a linear process will change its amplitude by a factor of K(v) and lose phase by F (v) degrees. K(v) and v are
generally graphed on logarithmic scales. Bode plots vary in shape for different processes, but K(v) always approaches the steady-state gain as v tends to zero. For very high frequencies, K(v)
generally tends towards zero. A Bode plot can be derived empirically by exercising the process with a sinusoidal control effort at various frequencies or by analyzing the physical characteristics of
the process, such as the stiffness of the spring and the mass of the weight in the example process.
Unlike the steady-state gain K, the gain and phase lag of the process vary depending on the frequency of the incoming sine wave. The weight-and-spring process does not change the amplitude of a low
frequency sine wave much. It is said to have a low frequency gain of one. Near the natural frequency, the gain is greater than one since the amplitude of the output is greater than the amplitude of
the input. The high frequency gain of the process is almost zero since the weight barely oscillates at all when the toy is shaken rapidly.
The process's phase lag is an additive factor. In this example, it starts near zero for low frequency inputs since the weight and the handle oscillate in synch when the handle is moved very slowly.
The phase lag drops to -180 degrees at high frequencies where the handle and weight move in opposite directions (hence the expression '180 degrees out of phase' used to describe any situation
involving complete opposites).
'Bode plot' graphic shows the complete spectrum of gains and phase lags for the weight-and-spring process at all frequencies between 0.01 and 100 radians per second. This is an example of a Bode plot
, a graphical analysis tool developed by Hendrick Bode at Bell Labs in the 1940s. It can be used to determine the amplitude and phase of the output that results when the process is driven by a
sinusoidal input with a particular frequency. To get the output amplitude, simply multiply the input amplitude by the gain shown at that frequency. To get the output phase, add the phase lag to the
input phase.
Fourier's Theorem
Gains and phase lags shown in a process's Bode plot are its defining characteristics. They tell an experienced control engineer everything he needs to know about the behavior of the process and how
it will respond in the future not only to sinusoidal control efforts but to any control effort.
Such an analysis is made possible by Fourier's Theorem , which states that any continuous sequence of measurements or signal can be expressed as an infinite sum of sine waves. Mathematician Joseph
Fourier proved his famous theorem in 1822 and produced an algorithm known as the Fourier Transform for computing the frequency, amplitude, and phase of each sinusoid in that sum from measurements of
the original signal.
A ship oscillating at a frequency of v and an amplitude of A, a teeter-totter oscillating at a frequency of 3v and an amplitude of A/3, and a child’s toy oscillating at a frequency of 5v and an
amplitude of A/5 would each generate a sine wave if their motions were plotted on separate trend charts.
Theoretically, Fourier Transforms and Bode plots can be used together to predict how a linear process would react to a proposed sequence of control efforts. Here's how:
1) Use the Fourier Transform to mathematically decompose the proposed control effort into its theoretical sine wave components or frequency spectrum .
2) Use the Bode plot to determine how each of those sine waves would have been modified had it been passed through the process by itself. That is, apply the appropriate amplitude and phase changes
dictated by each sine wave's frequency.
3) Use an inverse Fourier Transform to recombine the modified sine waves into one signal.
The combined motion of the toy, the teeter-totter, and the ship yields a square wave with a period (inverse frequency) of 1/v and amplitude just shy of A.
Since the inverse Fourier Transform is essentially an addition operation, the linearity of the process will guarantee that the combined effect of the theoretical sine waves computed in step 1 will be
the same as if they had remained summed together. Thus, the combined signal computed in step 3 will represent the process variable that would have resulted had the proposed control efforts been input
to the process.
Note that at no point in this procedure are any individual sine waves generated by the controller nor plotted on paper. All such frequency domain analysis techniques are conceptual. It is a matter of
mathematical convenience to translate signals from the time domain into the frequency domain with the Fourier Transform (or the closely related Laplace Transform ), solve the problem at hand using
Bode plots and other frequency domain analysis tools, then transform results back into the time domain.
Most control-design problems that can be solved in this manner can also be solved by direct manipulations in the time domain, but the calculations are generally easier in the frequency domain. In the
above example, it was a matter of multiplication and subtraction to compute the frequency spectrum of the process variable given the Fourier Transform of the proposed control efforts and the Bode
plot of the process.
It is not altogether obvious that adding up the right combination of sine waves will yield a signal with a desired shape as Fourier posited. An example may help.
Consider again the child's spring/weight toy, a playground teeter-totter, and a ship on the open ocean. Suppose that the ship happens to be rising and falling on the waves in a sinusoidal manner at a
frequency ofù and amplitude of A. Also suppose that the teeter-totter is oscillating at three times that frequency and one-third that amplitude while a child bounces the toy at five times that
frequency and one-fifth that amplitude. 'Three individual sine waves' graphic shows what those three sinusoidal motions would look like if observed separately.
Now suppose that the child is sitting on the end of the teeter-totter, which in turn is bolted to the deck of the ship. If the three individual sine waves happen to line up just right, the toy's
total movement would approximate a square wave as shown in the 'Three combined sine waves' graphic.
This isn't exactly a practical example, but it does demonstrate that the sum of a base sine wave plus one-third of its third harmonic plus one-fifth of its fifth harmonic approximates a square wave
with a frequency of v and an amplitude just shy of A. The approximation gets even better when one-seventh of the seventh harmonic is added plus one-ninth of the ninth harmonic. In fact, Fourier's
Theorem shows that if such a summation were to be continued ad infinitum , the total would match a square wave of amplitude A exactly . Fourier's Theorem can also be used to decompose a non-periodic
signal into an infinite sum of sine waves.
The Engineers' Choice Awards highlight some of the best new control, instrumentation and automation products as chosen by...
Each year, a panel of Control Engineering editors and industry expert judges select the System Integrator of the Year Award winners.
Control Engineering Leaders Under 40 identifies and gives recognition to young engineers who...
Big plans for small nuclear reactors: Simpler, safer control designs; Smarter manufacturing; Industrial cloud; Mobile HMI; Controls convergence
Virtualization advice: 4 ways splitting servers can help manufacturing; Efficient motion controls; Fill the brain drain; Learn from the HART Plant of the Year
Two sides to process safety: Combining human and technical factors in your program; Preparing HMI graphics for migrations; Mechatronics and safety; Engineers' Choice Awards
The Ask Control Engineering blog covers all aspects of automation, including motors, drives, sensors, motion control, machine control, and embedded systems.
Join this ongoing discussion of machine guarding topics, including solutions assessments, regulatory compliance, gap analysis...
News and comments from Control Engineering process industries editor, Peter Welander.
IMS Research, recently acquired by IHS Inc., is a leading independent supplier of market research and consultancy to the global electronics industry.
This is a blog from the trenches – written by engineers who are implementing and upgrading control systems every day across every industry.
Anthony Baker is a fictitious aggregation of experts from Callisto Integration, providing manufacturing consulting and systems integration.
Integrator Guide
Search the online Automation Integrator Guide
Visit the System Integrators page to view past winners of Control Engineering's System Integrator of the Year Award and learn how to enter the competition. You will also find more information on
system integrators and Control System Integrators Association.
Case Study Database
Get more exposure for your case study by uploading it to the Control Engineering case study database, where end-users can identify relevant solutions and explore what the experts are doing to
effectively implement a variety of technology and productivity related projects.
These case studies provide examples of how knowledgeable solution providers have used technology, processes and people to create effective and successful implementations in real-world situations.
Case studies can be completed by filling out a simple online form where you can outline the project title, abstract, and full story in 1500 words or less; upload photos, videos and a logo.
Click here to visit the Case Study Database and upload your case study. | {"url":"http://www.controleng.com/single-article/frequency-domain-analysis-explained/3d9622cbe730a1dbb8c5d52377ddc90e.html","timestamp":"2014-04-25T09:35:49Z","content_type":null,"content_length":"92282","record_id":"<urn:uuid:99483aa2-4700-4fe6-b15f-4042935ac8f1>","cc-path":"CC-MAIN-2014-15/segments/1398223211700.16/warc/CC-MAIN-20140423032011-00149-ip-10-147-4-33.ec2.internal.warc.gz"} |
Got Homework?
Connect with other students for help. It's a free community.
• across
MIT Grad Student
Online now
• laura*
Helped 1,000 students
Online now
• Hero
College Math Guru
Online now
Here's the question you clicked on:
Draw a parallelogram whose adjacent sides are determined by the vectors [4,6] and [3,5]. Find the area of the parallelogram.
• one year ago
• one year ago
Best Response
You've already chosen the best response.
Isn't it dot product which finds the area?
Best Response
You've already chosen the best response.
You can use the origin (0,0) as the common start point for those two vectors; that would make it easiest to draw.
Best Response
You've already chosen the best response.
No, I don't think it's dot product, as dot product is a measure of how much the vectors are in each other's direction, so it would have high dottiness if they were like this|dw:1350760137445:dw|
but very low area (dottiness=cos(angle between vectors))
Best Response
You've already chosen the best response.
Cross product most likely, but forget about the result being a vector.
Best Response
You've already chosen the best response.
Oh, that's right, cross product because it is a*b*sin(Θ).
Best Response
You've already chosen the best response.
b*sin(Θ) is the height of the parallelogram. Duh, should have remembered the basic trig on that one.
Best Response
You've already chosen the best response.
Is there any way to solve for the area without trig?
Best Response
You've already chosen the best response.
Not really...
Best Response
You've already chosen the best response.
Well, doing the cross-product has the trig built-in, so you wouldn't actually be using a trig function explicitly.
Best Response
You've already chosen the best response.
I suppose you could try to use pythagoras
Best Response
You've already chosen the best response.
^ what do you mean? By stacking up right triangles and using subtraction?
Best Response
You've already chosen the best response.
|dw:1350764732234:dw| find B perpendicular with A by finding the 'gradient' of A... you can do it, but it's really fiddly
Best Response
You've already chosen the best response.
No- there must be an elegant way of finding B perp A
Best Response
You've already chosen the best response.
as a function of Ax, Bx, Ay and By
Best Response
You've already chosen the best response.
I'd rather just do the cross-product.
Best Response
You've already chosen the best response.
\[\left[\begin{matrix}i & j & k \\ 4 & 6 & 0 \\ 3 & 5 & 0\end{matrix}\right] = 4\times5 - 6\times3=2\]
Your question is ready. Sign up for free to start getting answers.
is replying to Can someone tell me what button the professor is hitting...
• Teamwork 19 Teammate
• Problem Solving 19 Hero
• Engagement 19 Mad Hatter
• You have blocked this person.
• ✔ You're a fan Checking fan status...
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. | {"url":"http://openstudy.com/updates/5082f432e4b0dab2a5ebfb74","timestamp":"2014-04-21T12:54:11Z","content_type":null,"content_length":"86720","record_id":"<urn:uuid:eabed695-f581-47cc-818b-9974396d7f48>","cc-path":"CC-MAIN-2014-15/segments/1397609539776.45/warc/CC-MAIN-20140416005219-00233-ip-10-147-4-33.ec2.internal.warc.gz"} |
Posts by
Total # Posts: 62
Romeo is trying to climb a ladder placed 3 meters away from a 4 meter high tower where Juliet is stuck. He weighs 110 kg, the ladder weighs 25 kg. Can Romeo climb to the top of the ladder? If so, can
he bring Juliet back down with him?
1)??? 2)he retired in 1987
Im still not sure
Why did George Wallace want to become president? and What did he do after 1987?? Please help. Thank you.
College Physics
13. What is the magnitude of the gravitational force that acts on two particles? Assume that particle 1 (m1) is 12 kg and particle 2 (m2) is 25 kg, they are both separated by a distance of 1.2m
Cry, the Beloved Country refers at various points to the fear that exists in South African society. According to the novel, what are the causes of this fear and what are the consequences of this
fear? Include evidence from the novel to support your response. Write at least fiv...
Calculate the acceleration of a turtle whose speed changes from 3 m/s to 72 m/s over a period of 17 seconds.
Out of Every 19 People, Only 9voted. So That Leaves 1 Person From Every 10 People. So, 72 People Voted.(:
after passing therough a material t centimeters thick, the intensity I(t) of a light beam is given by I(t) =(4^-ct)Io, where Io is the initial intensity and c is a constant called the absorption
factor. Ocean water absorbs light with an absorption factor of c= .0101. at what d...
Find the number of decibels for the power of the sound. Round to the nearest decibel. A rock concert, 5.4 10-6 watts/cm2
An artifact was found and tested for its carbon-14 content. If 84% of the original carbon-14 was still present, what is its probable age (to the nearest 100 years)? Use that carbon-14 has a half-life
of 5,730 years.
A seismograph 300 km from the epicenter of an earthquake recorded a maximum amplitude of 5.2 102 µm. Find this earthquake's magnitude on the Richter scale.
How many inches are there in 355 millimeters?
how to find all real zeros of "4x^5+8x^4-15x^3-23x^2+11x+15
math 208
5 times x over y...when x=8 and y=2
What is the relationships or patterns among regular polyhedrons - tetrahedron, cube, octahedron, dedcahedron and icosahedron regarding their number of faces, vertices, and edges?
geometry math
If you put two dodecagons together with another polygon that will exactly surround a point what type of polygon could be used?
What is the connection between the square numbers and triangular numbers in arrays as shown below with a line drawn to show the array as 3/6? ... ... ... and .... .... .... .... array shows 6/10?
the perimeters of 2 similar polygons are 20 and 28. One side of the smaller polygons is 4. What is the congruent side of the larger polygon?
how do you write seven names for 8.Using only numbers less then 10 and using at least three different operations in each name.Using also parentheses.?
Literary Analysis and Composition
I got 10/17 by using the above answers. it was a 58.8 and 40 out of 68 points. Its okay though because I lost my book and never read the stories so I probably would have gotten worse if it wasn't for
your answers. Thanks :)
Thanks, I was on the right track
a salesperson who must decide between two monthly income options: Option A: Salary of $1642 per month, plus 8% of monthly sales or Option B: Salary of $1900 per month, plus 6.5% of monthly sales For
what amount of monthly sales is Option A the better choice for Mike than Optio...
Given two functions as: f(x) = x2-x-1 and g(x) =3/x Find fog(x) also find the domain of f, g and fog
1 hour and 57 minutes
chemistry - helium
what colour does helium burn? what colour is helium in the spectra? thanks :)
what colour does helium burn? what colour is helium in the spectra? thanks :)
Financial Management
Managerial Economics
Tots-R-Us operates the only day-care center in an exclusive neighborhood just outside of Washington, D.C. Tots-R-Us is making substantial economic profit, but the owners know the new day-care centers
will soon learn of this highly profitable market and attempt to enter the mar...
Human Services
decribe current regulations or standards? what kind of impact does these regulations have on the issue
how many atoms of C are in 2 moles of sucrose C12H22O11? please help
critical thinking
what are the fallacies in 5. Letter to the editor: I strongly object to the proposed sale of alcoholic beverages at County Golf Course. The idea of allowing people to drink wherever and whenever
they please is positively disgraceful and can only lead to more alcoholism a...
Fiance applications
A stock s 2009 Earnings-per-share is $4.50. Its Payout Ratio is 30%. Next year in 2010 it expects its Earnings-per-share to be $4.75. Answer the following two questions please, and select the correct
choice: 1. What is the current 2009 dividend? 2. What will next year...
A race car traveling at +44m/s is accelerated to a velocity of +22 m/s over an 11 s interval. What is the displacement during this time?
A race car traveling at +44m/s is accelerated to a velocity of +22 m/s over an 11 s interval. What is the displacement during this time?
Diagnostic mammogram, left breast: GG
chemistry(total pressure)
My question is how do I find the total pressue, I found the number of moles for each situation do i add them and plug it into the Total pressure =(n1+n2..)(RT/V) This is what I have is this correct,
if so then I do not know what to do after ths. (0.44496)(27)(0.0821)/V(I do no...
chemistry(total pressure)
Two cylinders at 27C are connected by a closed stopcock ystem. One cyclinder contains 2.4 L of hydrogen gas at o.600 atm; the other cylinder contains 6.8L of helium at 1.40 atm. Assume valve take up
no room. what is the total pressure when the valve is open ? I dont know how t...
chem -help
Two cylinders at 27C are connected by a closed stopcock ystem. One cyclinder contains 2.4 L of hydrogen gas at o.600 atm; the other cylinder contains 6.8L of helium at 1.40 atm. Assume valve take up
no room. what is the total pressure when the valve is open ? (0.600 atm)(L)/(0...
How many electrons are transferred when C3H7OH burns in oxygen to form carbon dioxide and water? I have no idea where to start. Do I need to do half reactions? Can I even do half reactions for this?
how does granite turn to mud
what is the oldest to the youngest of these 4 rocks? granite, limestone, mudstone and conglomerate.
In 2003, the probability that a randomly chosen bag of cement from the world supply was consumed in the U.S. was .05, while the probability that it was consumed in China was .45. What is the
probability that a randomly chosen bag of cement was consumed in neither country?
A bag contains 2 red marbles, 4 green ones, 1 lavender one, 2 yellows, and 6 orange marbles. How many sets of four marbles include all the red ones?
What is 2.816*10^8 to 2 significant figures
A bag contains 2 red marbles, 4 green ones, 1 lavender one, 2 yellows, and 6 orange marbles. How many sets of four marbles include all the red ones?
How many six-letter sequences are possible that use the letters c, i, c, c, i, y?
How many ordered lists are there of 10 items chosen from 12?
Evaluate C(13, 6).
I will promote effective teamwork by creating open communication with team members. I will encourage team members to come forward with ideas and also praise them for hard work. My main issue here is:
who do I choose to be on my team?
employees from the billing department are not receiving the accurate codes and information needed for data entry, slowing production and payment for the doctor. As head of the billing department, you
have been delegated to lead a problem-solving team to resolve this issue. How...
How to find the decimal equivalent of the mixed number and the mixed number is 4 2/5
I am in Calculus and am currently learning how to find the Area of a Surface of Revolution. I cannot understand what the surface of revolution (whether it's the x-axis, y-axis, or y=6) is. For
example, I had a problem saying to use the washer method to find the volume of t...
the replication fo DNA occurs?
NOOOOO thats all wrongg
i dont understand the prime interlude *** how can 18 be a prime number ??!!
Which type of muscle is resposible for moving fluids through the urinary system Since this is not my area of expertise, I searched Google under the key words "urinary system muscle" to get these
possible sources: http://ect.downstate.edu/courseware/histomanual/urinar...
what are the form and content in Michaelangelos Creation of man? How do you pick out line, balance, shape, color, contrast, texture, space, and proportion to size in this painting. Posted for Guru
Blue These sites will give you some information on that painting: http://en.wiki...
what are the form and conent in Michaelangelos Creation of man? http://www.google.com/search?hl=en&client=firefox-a&rls=org.mozilla:en-US:official&hs=Xd9&sa=X&oi=spell&resnum=0&ct=result&cd=1&q=
form+content+michelangelo%27s+creation+man&spell=1 There are many sites among these...
Find the solution to the differential equation dy/dx = (x^3)/(y^2), where y(2) = 3 I have: The integral of: dy(y^2) = the integral of: dx(x^3) (y^3)/3 = (x^4)/4 The answer is supposed to come out to
be y= the cube root of: ( 3/4x + 15 ) Thanks The general solution is (y^3)/3 =... | {"url":"http://www.jiskha.com/members/profile/posts.cgi?name=suzie","timestamp":"2014-04-19T00:31:29Z","content_type":null,"content_length":"18953","record_id":"<urn:uuid:b836760c-9ef9-47db-a490-ae49d5f74500>","cc-path":"CC-MAIN-2014-15/segments/1397609539066.13/warc/CC-MAIN-20140416005219-00007-ip-10-147-4-33.ec2.internal.warc.gz"} |
Department of Mathematics
Math Awareness Month 2014
Past Math Awareness Month Pages
Math Awareness Month 2013
Math Awareness Month 2012
Math Awareness Month 2011
Math Awareness Month 2010
Math Awareness Month 2009
Math Awareness Month 2008
Math Awareness Month 2007
Math Awareness Month 2006
Math Awareness Month 2005
Math Awareness Month 2004
Math Awareness Month 2003
Math Awareness Month 2002
Math Awareness Month 2001
Math Awareness Month 2000
3-dimensional view of a 4-dimensional fractal
Math Awareness Month 1999
Fourier analysis of biological tissues
Math Awareness Week 1998 Imaging and resolution
Math Awareness Week 1997
Math, internet, and the weather
Past Activities for Elementary Schools
Sample Problems
Sample problems and previous math competition exams are available on the Sample Problems for the Math Competition page. | {"url":"http://www.math.ku.edu/info/mathawareness/","timestamp":"2014-04-16T21:59:40Z","content_type":null,"content_length":"36909","record_id":"<urn:uuid:ef6c04d0-389c-4b43-9059-74b33380d59d>","cc-path":"CC-MAIN-2014-15/segments/1398223202774.3/warc/CC-MAIN-20140423032002-00612-ip-10-147-4-33.ec2.internal.warc.gz"} |
Math 310 Introduction to Abstract Algebra
Mark Reeder
Spring 2010
Text: Abstract Algebra: Theory and Applications, by Thomas W. Judson (free online in .pdf)
Same text, reformatted with 100 fewer pages
Homework 1 solutions
Homework 2 solutions
Homework 3 solutions
Exam 1 study guide
Homework 4 solutions
Homework 5 solutions
Homework 6 solutions
Homework 7 solutions
Exam 2 study guide
Exam 2 solutions
Homework 8 solutions
Homework 9 solutions
Homework 10 solutions
Homework 11 solutions
LaTeX resources:
Mac OSX: Download MacTeX. For an introduction, see "Trying out TeX" and its links.
Windows: Download proTeXt and follow the installation instructions.
LaTeX template file for homework 1
LaTeX manual (math symbols on page 60)
LaTeX Wiki Book
You can also learn a lot of TeX-niques by copying things from the source file of our text at the link above. | {"url":"https://www2.bc.edu/~reederma/310Sp2010.html","timestamp":"2014-04-21T12:47:12Z","content_type":null,"content_length":"2665","record_id":"<urn:uuid:1d911932-6e12-4776-90c0-8ef04cf7c99f>","cc-path":"CC-MAIN-2014-15/segments/1398223207046.13/warc/CC-MAIN-20140423032007-00150-ip-10-147-4-33.ec2.internal.warc.gz"} |
Physics Forums - View Single Post - Learn something new everyday
a cannon fires a projectile (no air or wind) at 100 m/s at a 20 degrees angle north of west. it is fired down a "hill" (which is impossibly straight) the hill has a 20 degree angle north of east.
find d (see picture.
You do understand, don't you, that we CAN'T "see picture" because you didn't post it! I assume that "d" is the distance from the point at which the projectile is fired to where it hits the ground.
The real point of the problem is to find where it hit.
You say "at 20 degrees angle north of west". That sounds like you mean it is aimed 20 degrees to the north of due west but if that is true, we don't know the upward angle at which the projectile is
I'm going to take it that the cannon is aimed "north-west" (that is "45 degrees north of west", and that it is pointed 20 degrees above the horizontal. Also that the hill slopes downward in the same
direction as the cannon is aimed (which makes the problem MUCH easier) with a 20 degree downward slope.
Here's how I would do the problem: We can ignore the direction. Since the cannon is fired in the same direction as the hill slopes, just do it as a standard two dimensional problem- take (0,0) as the
position of the cannon and the x axis in the direction the cannon is fired. First ignore the hill. Use the standard (parabolic) formula for a projectile launched at angle [theta] with initial speed
x= v0 cos[theta] t, y= -(g/2) t2+ v0 sin[theta] t.
You can easily solve solve for t in terms of x and replace it in the formula for y to get y as a function of x. In a simple, "standard" problem, the projectile would hit where y= 0 so you would solve
y= 0 for x.
Here, the ground slopes. Knowing that the ground slopes downward in the x direction, find the equation of the straight line:
it is y= (tan[theta])x where [theta] is the angle the line makes with the x axis (in this problem it is -20 degrees).
The projectile "hits" the ground when it crosses that line.
Solve the two equations y= f(x) for the projectiles motion and
y= mx for the ground simultaneously to find the point at which the projectile hits the ground. Once you know the coordinates of that point, you can find d, the distance between (0,0) and that point. | {"url":"http://www.physicsforums.com/showpost.php?p=1339274&postcount=16","timestamp":"2014-04-18T00:25:29Z","content_type":null,"content_length":"10875","record_id":"<urn:uuid:bb7218e3-3ade-42cf-836a-3c739a718068>","cc-path":"CC-MAIN-2014-15/segments/1397609532374.24/warc/CC-MAIN-20140416005212-00612-ip-10-147-4-33.ec2.internal.warc.gz"} |
Physics Forums - View Single Post - Emitted radiation and absorbed solar radiation
Jupiter has a black body temperature of 125K, but this is 20 K higher than the temperature that would be calculated from absorbed solar radiation alone. From this information, calculate the ratio of
emitted radiation to absorbed solar radiation on Jupiter.
I started using
W = sigma * area * temp^4
Is that of any help?
Or are they any hints that I might try to start with?
Stefan–Boltzmann law is correct.
No need to include area; just use the ratio of the two in terms of their irradiance, comparing the 4th power of the temperature of the two. | {"url":"http://www.physicsforums.com/showpost.php?p=2142364&postcount=2","timestamp":"2014-04-19T09:47:18Z","content_type":null,"content_length":"8032","record_id":"<urn:uuid:92e00f6d-adc7-4e23-a488-94768d9fcbd4>","cc-path":"CC-MAIN-2014-15/segments/1397609537097.26/warc/CC-MAIN-20140416005217-00240-ip-10-147-4-33.ec2.internal.warc.gz"} |
Formalizing "no junk, no confusion"
up vote 5 down vote favorite
Goguen has popularized the initial algebra view of semantics via his "no junk, no confusion" slogan. By "no junk", he means that models of a theory presentation should not have unnecessary elements,
and "no confusion" that terms should not be mapped to equal values unless they are provably equal. Sometimes, "no junk" is also interpreted as every element in the model is a denotation of a term,
while "no confusion" as two different terms denote different elements in the model. [These are classically equivalent statements, but they are not intuinistically equivalent, so I mention both].
My questions are:
1. What is a 'good' formalization of this slogan? By this I mean an explicit statement of "no junk, no confusion" in the meta-logic (since we're talking about models), where the logical strength of
the corresponding statement is well understood.
2. Are there logics in which these requirements can be internalized?
3. What would be the corresponding slogan to "no junk, no confusion" for final coalgebras?
lo.logic model-theory ct.category-theory
I don't understand this question. What are the "explicit" and "corresponding" statements? Your "sometimes" interpretation admits a familiar formalisation, so I assume that is not what you are
after. The classical and intuitionistic renderings are both pi-0-2 properties, so it is quite likely that one is provable for a model iff the other is. – Charles Stewart Feb 24 '10 at 8:35
I tried to clarify by editing the question. I mean an explicit statement of "no junk, no confusion" - for example, your classical and intuinistics renderings as explicit pi-0-2 properties. Yes, a
statement like 'every element in the model is a denotation of a term' is easy to formalize in the meta-theory -- but is it the 'right one? And is there some way to internalize that statement? –
Jacques Carette Feb 24 '10 at 12:53
add comment
3 Answers
active oldest votes
The way I understand each of the slogans is as follows:
1. "No junk" I just take to mean that an appropriate induction principle is valid -- that is, we should look for initial models in the appropriate category of algebras for the theory.
This also implies that every element of the model is in the image of the interpretation of the the algebraic theory.
Presumably the dualization to coalgebras would just be the validity of using bisimilarity to prove equality.
2. "No confusion" is traditionally interpreted to mean that we should look for models in which two elements of the model are semantically equal if and only if the corresponding syntaxes
are provably equal. This is the really bizarre requirement, since it amounts to requiring that the model be isomorphic to the term model! And yet Goguen and the algebraic
up vote 5 specification community were emphatically not happy with decreeing the term model to be the intended model -- they work very hard to get the "right" model.
down vote
accepted I personally (ie, I don't know that anyone else believes this) take the way this requirement is phrased to be an artifact of the history of algebraic specification. IIRC, they
started out with purely algebraic theories -- that is, theories in which the equational axioms are all pure equalities. (E.g., the axioms for groups.) Now, of course every such
algebraic theory has a degenerate model, since the one-element model validates all equalities. So the no-confusion principle is intended to rule out such degenerate models.
These days, of course, the algebraic specification crowd has no problem with theories with inequalities (e.g., the field axioms), and I think this additional freedom lets us state
the no-confusion principle in a better way. Namely, we should design algebraic theories whose models are categorical. That is, we want theories for which all models are isomorphic.
This implies the traditional no-confusion criterion, and also explains why people try to adjust the signature when they can't prove it. (Of course, this is a non-first-order property
in general, as you need higher-order logic or set theory to quantify over models.)
A small point: when an induction principle is not available it may often be replaced by initiality or universality. What I mean is for example the universal property of classifying
models in categorical logic. – Andrej Bauer Feb 24 '10 at 22:05
Thank you, this definitely answers some of my questions. My impression of the history of algebraic specifications is akin to yours: there was too much early emphasis on equational
theories and equational logic. Luckily moving to institutions seemed to help in that respect. I am still looking for ideas on how to internalize (a proper interpretation of) as much of
"no junk, no confusion" into a logic as possible [like Bill Farmer's Chiron, which allows one to talk about both syntax and semantics]. – Jacques Carette Feb 25 '10 at 0:17
"No junk" I just take to mean that an appropriate induction principle is valid - This isn't right for the most obvious interpretation of "appropriate": nonstandard models of PA satisfy
the (first-order) induction principle, but they are the very picture of "has junk". You need something like "freely generated", I think. – Charles Stewart Feb 25 '10 at 8:42
"No confusion" - I like what you say very much. It's obviously analogous to the question of what role "abstractness" plays in the term "fully abstract model". – Charles Stewart Feb 25
'10 at 8:44
add comment
Belatedly, an answer in set-based situations to
1. What would be the corresponding slogan to "no junk, no confusion" for final coalgebras?
Given an initial algebra, any algebra will have a special subobject which is the image of the initial structure. The subobject may have confused elements (terms) of the initial algebra, and
the object may have extra junk. A map between objects will map the first special subobject to the second, possibly confusing more. The initial algebra has no confusion and no junk.
up vote 3
down vote Given a final/terminal coalgebra, the elements of any coalgebra will have a special colouring in terms of images in the elements of the terminal structure. The object may have more than one
element with the same colour, and the object may not use all the colours. A map between objects will preserve the colouring, the domain possibly using fewer colours. The terminal coalgebra
colours without ambiguity and without redundancy. If two things behave the same way, they are the same; all behaviours are covered.
No junk, no confusion; No redundancy, no ambiguity.
Late, but a nice contribution. I like your co-slogan. – Jacques Carette Mar 13 '12 at 12:04
add comment
The slogan is a meaningful english phrase. After removing negation, we may obtain this phrase: "stay clean, stay clear". So then, here is an actual theorem of quantified boolean formulas
that describes both qualities formally:
Linear Corollary: Quantified monotone boolean formulas are linearly decidable.
up vote
-4 down That is, no matter how many alternating quantifiers are in the prefix, when the body of the formula has "zero negations," then the monotonicity of the formula makes any quantifier prefix
vote linearly decidable; plug T for existentially quantified variables, NIL for universally quantified variables, then evaluate the boolean form, entirely linear in the size of the QBF.
Goguen may also enjoy the car/cdr Structure of Common Sense: Good ideas Usually have Two words.
1 I do not think this is helpful/topical (although the linear corollary is interesting as a result on its own). And I do not believe that your negation-removal was meaning-preserving! –
Jacques Carette Feb 23 '10 at 22:51
The Plus Transfrom gave a nearly equivalent english phrase; but the minus votes indicate an error somewhere...hmmm. "monotone" does also include formulas where every proposition is
negative; however, I considered that presentation to be confusing junk, in a simple propositional formalism... I use monotone formulas for loop management in complex programs that solve
QBFs; "correctness and clarity" are both of the utmost importance in such programs. Correctness is related to "no junk" and "clarity" allows code maintenance and improvements. Removing
negation is standard treatment for me. – daniel pehoushek Feb 26 '10 at 18:58
add comment
Not the answer you're looking for? Browse other questions tagged lo.logic model-theory ct.category-theory or ask your own question. | {"url":"http://mathoverflow.net/questions/16180/formalizing-no-junk-no-confusion/16263","timestamp":"2014-04-18T00:53:20Z","content_type":null,"content_length":"72696","record_id":"<urn:uuid:6909541d-ba56-4c51-9480-2619d8274e19>","cc-path":"CC-MAIN-2014-15/segments/1398223206120.9/warc/CC-MAIN-20140423032006-00211-ip-10-147-4-33.ec2.internal.warc.gz"} |
Rotation - of a polygon
From Latin: rotare - "revolve, roll"
A transformation where a figure is turned about a given point.
Try this Drag any orange dot to move P, the center of rotation. Drag the polygon to rotate it about P.
The "rotation" transformation is where you turn a figure about a given point (P in the diagram above). The point about which the object is rotated can be inside the figure or anywhere outside it. The
amount of rotation is called the angle of rotation and is measured in degrees. By convention a rotation counter-clockwise is a positive angle, and clockwise is considered a negative angle.
Rays from the point of rotation to any vertex all turn through the same angle as the image is rotated. Click on "show rays" and rotate the image to see this.
Things to try
In the diagram above - click 'reset'
1. Rotate the polygon by dragging anywhere inside it. Note how it rotates about the point P.
2. Drag the the point P inside the polygon. Now when you rotate it, note how it turns around the point P as if P was a pin holding it to the screen.
3. Rotate the polygon a full 360° and note how it is now back to its original position.
4. Click "hide details". Then rotate the polygon to some new position and estimate the angle of rotation. Click "show details" to see how close you got.
Other transformation topics
(C) 2009 Copyright Math Open Reference. All rights reserved | {"url":"http://www.mathopenref.com/rotate.html","timestamp":"2014-04-18T18:10:37Z","content_type":null,"content_length":"8724","record_id":"<urn:uuid:4e172dd4-e52a-43c8-b82d-e40ace717784>","cc-path":"CC-MAIN-2014-15/segments/1397609535095.7/warc/CC-MAIN-20140416005215-00323-ip-10-147-4-33.ec2.internal.warc.gz"} |
A Forgotten Model Of The Universe
A paper published in EPJ H provides the first English translation and an analysis of one of Albert Einstein's little-known papers, "On the cosmological problem of the general theory of relativity."
Published in 1931, it features a forgotten model of the universe, while refuting Einstein's own earlier static model of 1917. In this paper, Einstein introduces a cosmic model in which the universe
undergoes an expansion followed by a contraction. This interpretation contrasts with the monotonically expanding universe of the widely known Einstein-de Sitter model of 1932.
The authors, Cormac O'Raifeartaigh and Brendan McCann from the Waterford Institute of Technology, Ireland, provide insights into Einstein's view of cosmology. At that time, the first pieces of
evidence for an expanding universe emerged, among others, stemming from Hubble's observations of the expanding universe.
Einstein was keen to investigate whether a relativistic model could account for the new observations, by removing the so-called cosmological constant introduced in his 1917 cosmological model.
Einstein sets the constant to zero. He then arrives at a model of a universe that first expands and then contracts. This model is also characterised by singularity-like behaviour at either end.
In this paper, the authors also discuss Einstein's view of issues such as the curvature of space and the timespan of the expansion, while also uncovering some anomalies in Einstein's calculations.
For example, they highlight a numerical error in the calculation of the present radius and matter density of the universe. They also believe that Einstein's estimate of the age of the universe is
based on a questionable calculation of Friedmann's analysis of a relativistic universe of spherical curvature and time-varying radius. Finally, they argue that Einstein's model is not periodic,
contrary to what is often claimed. | {"url":"http://www.science20.com/news_articles/forgotten_model_universe-130045","timestamp":"2014-04-19T14:42:09Z","content_type":null,"content_length":"43440","record_id":"<urn:uuid:cbb0c4e3-606c-451b-94d8-65f9f763f7fb>","cc-path":"CC-MAIN-2014-15/segments/1398223206147.1/warc/CC-MAIN-20140423032006-00168-ip-10-147-4-33.ec2.internal.warc.gz"} |
The Piezo Linear Motor block represents the force-speed characteristics of a linear piezoelectric traveling wave motor. The block represents the force-speed relationship of the motor at a level that
is suitable for system-level modeling. To simulate the motor, the block uses the following models:
Mass and Friction Model for Unpowered Motor
The motor is unpowered when the physical signal input v is zero. This corresponds to applying zero RMS volts to the motor. In this scenario, the block models the motor using the following elements:
● An mass whose value is the Plunger mass parameter value.
● A friction whose characteristics you specify using the parameter values in the Motor-Off Friction tab.
The block uses a Simscape™ Translational Friction block to model the friction component. For detailed information about the friction model, see the Translational Friction block reference page.
Resonant Circuit Model for Powered Motor
When the motor is active, Piezo Linear Motor block represents the motor characteristics using the following equivalent circuit model.
In the preceding figure:
● The AC voltage source represents the block's physical signal input of frequency f and magnitude v.
● The resistor R provides the main electrical and mechanical damping term.
● The inductor L represents the rotor vibration inertia.
● The capacitor C represents the piezo crystal stiffness.
● The capacitor C[p] represents the phase capacitance. This is the electrical capacitance associated with each of the two motor phases.
● The force constant k[f] relates the RMS current i to the resulting mechanical force.
● The quadratic mechanical damping term, , shapes the force-speed curve predominantly at speeds close to maximum RPM. is the linear speed.
● The term represents the plunger inertia.
At model initialization, the block calculates the model parameters R, L, C, k[t] and λ to ensure that the steady-state force-speed curve matches the values for the following user-specified
● Rated force
● Rated speed
● No-load maximum speed
● Maximum (stall) force
These parameter values are defined for the Rated RMS voltage and Motor natural frequency (or rated frequency) parameter values.
The quadratic mechanical damping term produces a quadratic force-speed curve. Piezoelectric motors force-speed curves can typically be approximated more accurately using a quadratic function than a
linear one because the force-speed gradient becomes steeper as the motor approaches the maximum speed.
If the plunger mass M is not specified on the datasheet, you can select a value that provides a good match to the quoted response time. The response time is often defined as the time for the rotor to
reach maximum speed when starting from rest, under no-load conditions.
The quality factor that you specify using the Resonance quality factor parameter relates to the equivalent circuit model parameters as follows:
This term is not usually provided on a datasheet. You can calculate its value by matching the sensitivity of force to driving frequency.
To reverse the motor direction of operation, make the physical signal input v negative. | {"url":"http://www.mathworks.se/help/physmod/elec/ref/piezolinearmotor.html?s_tid=gn_loc_drop&nocookie=true","timestamp":"2014-04-24T06:25:17Z","content_type":null,"content_length":"42881","record_id":"<urn:uuid:ceb6d057-8a1e-417a-a367-e2ed7d245310>","cc-path":"CC-MAIN-2014-15/segments/1398223205375.6/warc/CC-MAIN-20140423032005-00573-ip-10-147-4-33.ec2.internal.warc.gz"} |
Math Forum Discussions
Math Forum
Ask Dr. Math
Internet Newsletter
Teacher Exchange
Search All of the Math Forum:
Views expressed in these public forums are not endorsed by Drexel University or The Math Forum.
Topic: Let's Have Some Fun With Math!!
Replies: 6 Last Post: Nov 22, 1996 9:59 PM
Messages: [ Previous | Next ]
Gary Let's Have Some Fun With Math!!
Posted: Nov 13, 1996 6:09 AM
Posts: 26
Registered: 12/6/04 Hello,
Sometimes those of us who love math need a comic diversion. Why
don't you contribute with a math joke or riddle? You may have to
provide an explanation. I'll start off.
Q: What did the acorn say when it finally grew up?
A: Geometry!
E: (Gee, I'm a tree!)
Date Subject Author
11/13/96 Let's Have Some Fun With Math!! Gary
11/19/96 Re: Let's Have Some Fun With Math!! Gary
11/21/96 Re: Let's Have Some Fun With Math!! The Traveller
11/21/96 Re: Let's Have Some Fun With Math!! Vikki Kowalski
11/22/96 Re: Let's Have Some Fun With Math!! Le Compte de Beaudrap
11/22/96 Re: Let's Have Some Fun With Math!! Brian Daniel Sammon
11/21/96 Re: Let's Have Some Fun With Math!! Le Compte de Beaudrap | {"url":"http://mathforum.org/kb/thread.jspa?threadID=179393","timestamp":"2014-04-20T03:42:06Z","content_type":null,"content_length":"23211","record_id":"<urn:uuid:2ef8c09f-dbe2-4e27-b744-1a5ed6ee1040>","cc-path":"CC-MAIN-2014-15/segments/1398223206118.10/warc/CC-MAIN-20140423032006-00498-ip-10-147-4-33.ec2.internal.warc.gz"} |
Square root
bret80 First of all, to do LaTeX, you need to put your expression between [ math ] and [ /math ] . (Without the spaces!) $0 = \sqrt{4-x^2}$. Square both sides: $0 = 4 - x^2$ $x^2 = 4$ $x = \pm 2$
Whenever you square both sides of an equation ALWAYS check to make sure that your solution set satisfies your original equation as squaring occasionally adds extra solutions. In this case, both
solutions work. -Dan
topsquark Not neccesarily. If all of your steps are equivalent to each other (if and only if; you might have seen me do that a number of times) then no check is required.
ThePerfectHacker Well, it's a good idea for us "lesser mortals" anyway. We can't ALL be "perfect." -Dan | {"url":"http://mathhelpforum.com/algebra/4659-square-root.html","timestamp":"2014-04-21T14:07:12Z","content_type":null,"content_length":"41520","record_id":"<urn:uuid:364215d1-2411-4bb5-892d-189f695af088>","cc-path":"CC-MAIN-2014-15/segments/1397609539776.45/warc/CC-MAIN-20140416005219-00218-ip-10-147-4-33.ec2.internal.warc.gz"} |
Problem Set In
Math & Art Problem Set 7-Individual Due Friday 3/25/11
Part I
In this portion of the assignment, you are going to draw the same cube in
different positions, using the Perspective Theorem. You will then use these
pictures to make observations that should reinforce the conclusions about the
perspective images of various types of lines that are being discussed in class.
You will need graph paper! (I would suggest having each unit be several
squares long, so your pictures are big enough to really appreciate.)
In both cases, our cube will have side of length 4, and the viewing distance
d (how far the viewer’s eye is from the picture plane) will be 8.
We will call our cube ABCDEF GH with the base being square ABCD and
the top being square EF GH. (Note that in the base square, A is connected
to B and D, B is connected to A and C, etc; and that E is directly above
A, F is directly above B, etc).
1. What are the coordinates for the viewer’s eye?
2. We’ll begin with a cube whose top and bottom are horizontal and whose
front and back are parallel to the picture plane. The bottom will be
above the viewer’s eye.
Use the following coordinates for the corners of the cube:
Base=ABCD Top=EFGH
A (8, 3, 4) E (8, 7, 4)
B (12, 3, 4) F (12, 7, 4)
C (12, 3, 8) G (12, 7, 8)
D (8, 3, 8) H (8, 7, 8)
(a) Using the Perspective Theorem, find the coordinates for each of
the 8 corners (shown again below) of the image in the picture
plane (that is, find (x , y )). You may do these calculations by
hand or, if you’re comfortable with it, you may use a spreadsheet
like Excel. If you do it by hand, include your work on a separate
sheet; if you use a spreadsheet, please include it with your work.
Sklensky Spring, 2011
Math & Art Problem Set 7-Individual Due Friday 3/25/11
(b) Carefully plot the points you found in part 2a in the xy plane
on graph paper. (Remember you are not using 3D space axes for
this!) Then (paying attention to the right order), neatly connect
them with straight lines (use a straight edge, and use dashed lines
to indicate the hidden edges) to obtain the perspective image.
(c) Get a good idea of what the viewing distance is in the scale you
used (that is, how long is 8 units?), and then put one eye at that
distance from the page, directly opposite the origin. Look at your
perspective image with that one eye. Do you see a cube, with the
illusion of depth?
(d) Your cube has one set of four parallel lines which are not parallel
to the picture plane. Do those lines look parallel in the perspective
image you’ve created? Using however many straight edges (pieces
of paper, for instance) you need, see where they intersect (this
may not be on your piece of graph paper). What can you say
about where these four lines intersect?
3. We’ll continue with the same cube, but we’ll turn it so that while the
top and bottom are still horizontal, now one edge is facing us, rather
than the front and back being parallel to the picture plane. We’ll also
move it so that the top is below the viewer’s eye.
Use the following coordinates for the corners of the cube:
Base=ABCD Top=EFGH
A (0, −6, 4) E (0, −2, 4)
B (2.8, −6, 6.8) F (2.8, −2, 6.8)
C (0, −6, 9.7) G (0, −2, 9.7)
D (−2.8, −6, 6.8) H (−2.8, −2, 6.8)
(a) Find the coordinates for each of the corners of the image in the
picture plane (include your work or spreadsheet). Carefully plot
Sklensky Spring, 2011
Math & Art Problem Set 7-Individual Due Friday 3/25/11
them in the xy plane on graph paper, then neatly connect them
with straight lines to obtain the perspective image.
(b) As with the previous exercise, put one eye at the viewing distance
opposite the origin. Look at your perspective image with that one
eye. Does it leap off the page at you?
(c) Again, can you get a sense of where the parallel lines intersect?
Part II:
For the next 3 problems, you will be investigating the use of perspective in 3
paintings by Renaissance masters. You will need print-outs of each of these
paintings; because you may need several of each, I have put links to the
paintings with the problem set rather than include the pictures in this file.
There’s no need for the print-outs to be in color, although you may enjoy the
process more. Hand in the print-outs (along with any additional pieces of
paper that you needed), as this is where most of your work for these problems
will be. You may find you need a couple print-outs of each painting.
4. Consider Leonardo’s The Last Supper (1495-1498), 460 cm × 880 cm.
(a) Finding the primary vanishing point
i. Locate and highlight at least two lines parallel to the picture
plane and parallel to each other.
ii. Locate and highlight in a different color (or on a different
print-out) at least three lines orthogonal to the picture plane.
(Remember, in real-life, these would be perpendicular to lines
you found in part (a) and be receding).
Sklensky Spring, 2011
Math & Art Problem Set 7-Individual Due Friday 3/25/11
iii. Extend the lines you found in part (a) to find the primary
vanishing point. In my experience, this may take several tries,
as it can be difficult to line your straight-edge up exactly with
an orthogonal, and some may be off initially.
iv. Is the primary vanishing point in the picture or off the picture?
Is it used to draw the eye anywhere important, or is it just
used to give an illusion of depth?
(b) Finding the ideal viewing position of your print-out
i. Draw the horizon line through the primary vanishing point.
ii. Locate a secondary vanishing point by finding a square lying
parallel to the floor, drawing its diagonal, and extending that
diagonal until it intersects the horizon line.
iii. Determine the intended viewing distance for your print-out by
measuring the distance between the primary and secondary
vanishing point.
iv. Determine the ideal viewing position for your copy, and de-
scribe it.
v. Try viewing it from the correct viewing position. Does it
improve the illusion of depth in the picture? (Of course, with
a print-out it’s not the same experience as looking at it in
(c) Estimating the ideal viewing position for the painting it-
i. Measure the height and width of your print-out.
ii. Use the measurements, along with the dimensions of the orig-
inal painting, the viewing distance you found in (b), and your
knowledge of proportion to get a pretty good estimate of the
ideal viewing position of the actual painting.
5. Next, consider Rafael’s School of Athens (1509-1511), 500 cm × 770
Sklensky Spring, 2011
Math & Art Problem Set 7-Individual Due Friday 3/25/11
(a) Find the primary vanishing point, by first locating at least two
lines that are parallel to the picture plane and parallel to each
other and then using those lines and your experience to deduce at
least three lines that must be orthogonal to the picture plane. Is
the primary vanishing point in the picture or off the picture? Is
it used to draw the eye anywhere important, or is it just used to
give an illusion of depth?
(b) Finding the ideal viewing position of your print-out. Again, try
viewing your print-out from that position – does it improve the
illusion of depth?
(c) Use the dimensions of your print-out, along with the dimensions
of the original painting, the viewing distance you found in (b),
and your knowledge of proportion to get a pretty good estimate
of the ideal viewing position of the actual painting.
6. Finally, consider Masaccio’s Trinity (1427-1428), 667 cm times 317 cm,
the painting that motivated our looking into finding the correct viewing
(a) Find the primary vanishing point. Is it used to draw the eye
anywhere important?
(b) Find the ideal viewing position of your print-out; does viewing
your print-out from that position improve the illusion of depth?
Note: Finding a square to work with takes a bit more work in this
painting. Try looking at the top of the columns. You will have to
finish off the squares for yourself – be sure to use the vanishing
point to draw in the missing orthogonal.
(c) Use your results and the dimensions of the painting to estimate
the ideal viewing position of the actual painting.
Part III:
These final two exercises are from Lessons in Mathematics and Art; they
extend the ideas we used to come up with the ideal viewing position to
different situations.
Sklensky Spring, 2011
Math & Art Problem Set 7-Individual Due Friday 3/25/11
7. Drawing your own cube:
In the figure below, a start has been made on the drawing of a cube
in one-point perspective. The front face is a square, V is the vanishing
point, and the dashed lines are guidelines for drawing receding edges of
the cube. Suppose you want to choose the viewing distance first, and
you choose it to be 6 inches. Finish drawing the cube.
Hint: For help in thinking about it, look at Figure 7 from Lesson 3.
The idea is to, in a sense, work backwards.
Sklensky Spring, 2011
Math & Art Problem Set 7-Individual Due Friday 3/25/11
8. If the box below represents a cube, then we can use our usual techniques
to find the correct viewing distance, and it would end up being the
distance between the two trees, as illustrated.
But suppose the box below is not a cube – suppose its front is a square,
but its top face is in reality twice as long as it is wide from left to right.
In this case, the viewing distance is not equal to the distance between
the two trees. What is the viewing distance in that cae? (You may
give your answer in terms of the distance between the trees, if that’s
Hint: Go through the same process we went through in class to find
the viewing distance with a cube, but make appropriate adjustments.
Sklensky Spring, 2011 | {"url":"http://www.docstoc.com/docs/75312116/Problem-Set-In","timestamp":"2014-04-16T21:13:08Z","content_type":null,"content_length":"64548","record_id":"<urn:uuid:818a2dec-5fa0-4678-8084-b0dc05abaae6>","cc-path":"CC-MAIN-2014-15/segments/1397609524644.38/warc/CC-MAIN-20140416005204-00208-ip-10-147-4-33.ec2.internal.warc.gz"} |
dafis / floatshow - be67271
-The Show instances for RealFloat types provided in base are very elegant, as they produce the shortest string which 'read' converts back to the original number.
-That, however, involves a check after each digit has been determined and arithmetic of usually fairly large Integers, which makes these Show instances rather slow.
-For cases where having the conversion fast is more important than having it produce elegant output, this package provides alternative conversions, which avoid the checks and reduce the occurrences of large Integers by just producing a sufficiently long output string.
-The speed gain can be substantial if the numbers have exponents of large absolute modulus, but for the more common case of numbers whose exponents have small absolute modulus, the difference is (although still significant for Double) too small in my opinion to seriously consider replacing the Show instances.
-Some benchmarks produced with criterion (http://hackage.haskell.org/package/criterion) show a considerable speedup, or example, converting 10,000 numbers to String and calculating the length produced:
-The difference is far smaller however if we consider only numbers with exponents of small absolute modulus (numbers between 1e-8 and 1e8 here):
-Another benchmark, calculating d*sin(d^2) for d = 1, ..., 200000 and writing their string representations to a file, timed 'show' at 4.26s, 'fshow' at 1.08s and 'show' for the newtype wrapper Double7 (rounds to seven significant digits) around Double at 0.55s. A corresponding C programme ran in 0.26s with the default precision and in 0.32s with a precision of 17 digits.
Tip: Filter by directory path e.g. /media app.js to search for public/media/app.js.
Tip: Use camelCasing e.g. ProjME to search for ProjectModifiedEvent.java.
Tip: Filter by extension type e.g. /repo .js to search for all .js files in the /repo directory.
Tip: Separate your search with spaces e.g. /ssh pom.xml to search for src/ssh/pom.xml.
Tip: Use ↑ and ↓ arrow keys to navigate and return to view the file.
Tip: You can also navigate files with Ctrl+j (next) and Ctrl+k (previous) and view the file with Ctrl+o.
Tip: You can also navigate files with Alt+j (next) and Alt+k (previous) and view the file with Alt+o. | {"url":"https://bitbucket.org/dafis/floatshow/commits/be672718fdb8/","timestamp":"2014-04-18T03:36:15Z","content_type":null,"content_length":"74938","record_id":"<urn:uuid:65320198-4c8a-490c-b846-93fed33dae2b>","cc-path":"CC-MAIN-2014-15/segments/1397609532480.36/warc/CC-MAIN-20140416005212-00444-ip-10-147-4-33.ec2.internal.warc.gz"} |
- 2 -
Cosmic Computer
Where does the complexity of the Universe come from? A simple computer program is generating it!
There is something fascinating about science. One gets such wholesale returns of conjecture out of such a trifling investment of fact.
Mark Twain
(Life on the Mississippi, 1884)
God has chosen the world that is the most simple in hypotheses and the most rich in phenomena.
Gottfried Leibniz
(Discourse de metaphysique, 1686)
It's AD 2068 and the survey expedition from Earth is picking its way through the ruins of an alien civilisation, long departed its home world for who knows where. Ahead, bathed in the sombre light of
the twin red suns, is a great slab of a building - the planet's central library, repository of the civilisation's accumulated wisdom.
Struggling visibly in the strong gravity, the expedition members clamber up the giant steps and push open the creaking door. Boots reverberating in the thick atmosphere, they hurry through
an empty, echoing chamber - until, finally, they come to a single cabinet, displaying a lone tablet inscribed with arcane symbols. Everyone crowds around while someone scans it with a translator...
RECIPE FOR UNIVERSE:
RUN COMPUTER PROGRAM (BELOW)
REPEAT FOR 13.7 BILLION YEARS
One person laughs. Another gasps in disbelief. The cosmic computer program they are all staring at is only 4 lines long!
Could the recipe for making a Universe really be as simple as this? One present-day physicist is convinced of it. His name is Stephen Wolfram and he claims to have stumbled on nature's
"big" secret. The source of all its bewildering complexity - from spiral galaxies to rhododendrons to human beings - is the application of a few simple instructions, over and over again. "Our
Universe is being generated by a simple computer program," says Wolfram.
Wolfram, a child prodigy from London, began publishing papers in professional physics journals at the age of 15. What led him to his extraordinary conclusion is a discovery he made around
1980. Contrary to all expectations, he found that simple computer programs have the ability to generate extraordinarily complex outputs.
Wolfram's discovery came about when he became interested in problems like how galaxies like our Milky Way form and how our brains work. "The trouble was that none of these 'complex systems'
seemed explicable by conventional science," he says.
Conventional science is synonymous with maths-based science. In the 17th century, Isaac Newton discovered that the laws which govern the motion of a cannon ball through the air and a planet
round the Sun could be described by mathematical formulae, or "equations". Following Newton's lead, generations of physicists have found that mathematical equations exist that can perfectly describe
everything from the character of the light given out by a hot furnace to the warping of space and time by the concentrated mass of a black hole .
But, despite the tremendous successes of equation-based science in penetrating nature's secrets, it has an Achilles' heel. It cannot do "complexity". It is utterly incapable of capturing
the essence of what is going on in a whole range of complex phenomena, ranging from turbulence in fluids to biology itself.
Most scientists lose little sleep over this. Complex phenomena may be "hard", they say, but this does not mean that science will not eventually get around to tackling them. Wolfram,
however, emphatically disagrees. Controversially, he believes that mathematical science will never, ever penetrate the mystery of complex phenomena.
A streetlight illuminates merely what it can illuminate - the circle of ground immediately beneath. Similarly, Wolfram believes mathematical science illuminates merely what it is capable of
illuminating - those phenomena whose essence can be captured by mathematical equations. But such phenomena, he contends, are rare and unusual. In the same way that a streetlight fails to reveal the
subways and sports grounds and art galleries of its surrounding city, science as practised for the past three centuries is blind to the overwhelming majority of phenomena in the Universe - complex
phenomena. "Since such phenomena include living things, the human brain and the biosphere, we are talking about all the truly interesting things that are going on in the Universe," says Wolfram.
This is radical stuff. For centuries, physicists have wondered why nature obeys mathematical laws, which can be distilled into neat mathematical equations and which can then be scrawled
across blackboards. The Hungarian-American physicist Eugene Wigner famously drew attention to this when he talked of "the unreasonable effectiveness of mathematics in the physical sciences".
According to Wolfram, however, Wigner was wrong to believe that the Universe is essentially mathematical. Mathematics is no more effective in revealing the inner workings of nature than a
streetlight is in revealing the city that surrounds it. Nature may "appear" to follow mathematical laws, he says. However, that is hardly surprising when scientists specifically seek out the rare
natural phenomena that follow mathematical laws. "Wigner could equally well have remarked on the unreasonable effectiveness of streetlights in illuminating the ground beneath them," says Wolfram.
If Wolfram is right, science has a serious problem. After all, if mathematical equations are incapable of describing nature's most interesting phenomena - complex phenomena - how cane such
phenomena be described? In the early 1980s, Wolfram gave this question a great deal of thought.
It was clear to him that the Universe must obey rules of some kind. If it did not, after all, there would be no pattern or regularity in nature. The Universe would be a meaningless
maelstrom of unpredictable randomness and chaos. But, if the rules are not embodied in mathematical equations, what are they embodied in? It was clear to Wolfram that it had to be something more
general than a mathematical equation. After thinking about it, he could come up with only one thing that fitted the bill: a computer program.
Nature's big secret
Wolfram decided to find out what kind of science could be built starting with the more general kinds of rules embodied in computer programs. The first big question he needed to answer was: what are
such rules capable of? Or, to put in another way, typically what do simple programs do?
The simplest computer program Wolfram could think of is known as a "cellular automaton". The most basic of these is simply a long line of squares, or "cells", drawn across a page. A cell
can either be one of two colours - white or black. At regular intervals of time, a new line of cells is drawn on the page, immediately above the first. Whether a cell in this second line is black or
white depends on a rule applied to its two nearest neighbours in the first line. The rule might, for instance, say: "If a particular cell in the first line has a black square on either side of it, it
should turn black in the second line". A third line of cells, immediately above the second, is then created by applying the cellular automata "rule" to the second line, and so on.
What we are talking about here is the operation of simple computer program embodying the cellular automaton rule. The program takes an input - the pattern of black-and-white cells on one
line - and produces an output - the pattern of cells on the next line. The key thing is that the output is fed back in as the next input to the computer program rather like a snake swallowing its own
tail. Such tail-swallowing is commonly called "recursion". And, as a wit once said: "To understand recursion, you must first understand recursion!"
For a 1-dimensional, two-colour, adjacent-cell cellular automaton like this, it turns out there are 256 possible rules, 256 kinds of program . The question is: what happens when the
programs are run, starting, say, with a single black cell in the first line of cells? In true scientific fashion, Wolfram began experimenting to find out.
He soon discovered that some rules and some starting patterns led to nothing interesting. As new lines of cells were created, any pattern quickly fizzled out. Or a particular arrangement of
black-and-white cells began repeating endlessly. However, in some cases, something very much more interesting happened.
The early 1980s was the time of the first cheap desktop computers so Wolfram was able to watch his cellular automata perform on a computer rather than on a piece of paper. Seeing the new
lines of cells marching steadily up the screen was much like watching a movie. Occasionally, the patterns of black cells coalesced into discrete "objects". These persisted - as unchanging and stable
as a table or chair - despite the fact they were being continually destroyed and regenerated.
Wolfram played with is a his cellular automata for hours on end, mesmerised by the marching patterns. And then, one day, he stumbled on something extraordinary. "I found a pattern which
appeared never to repeat, no matter how long I stared at it," says Wolfram.
If you see a complex thing like a car or a computer, you know it must have been made by a complex process. Even in biology, where natural selection is blind, the complexity of organisms is
a result of a complex series of processes operating over billions of years of evolution. In the everyday world, simple things have simple causes and complex things have complex causes. What Wolfram
had found, however, was something that bucked the trend - a complex thing that had a simple cause.
For Wolfram it was a life-changing moment. As he stared at his computer screen and the never-ending novelty scrolling down it, he wondered: Is this the origin of the Universe's complexity?
"When nature creates a rose or a galaxy or a human brain, is it merely applying simple rules - over and over again?" he says. "Is this its big secret?"
A survey of all possible worlds
From that moment on Wolfram became obsessed with the origin of complexity. At the time of his epiphany, he was at the California Institute of Technology in Pasadena. However, in the mid-1980s, he
moved first to Princeton's Institute for Advanced Study - Einstein's old institute - then to the University of Illinois at Urbana-Champaign, where he founded the Center for Complex Systems Research.
Around the same time, he started the first scientific journal on complexity. He even created his own computer language - "Mathematica" - which helped him in his investigation of the origin of
Mathematica led him to start his own company, Wolfram Research, and attract scientists and mathematicians to help develop the software. The programming language turned out to be not only a
tool but an inspiration of his work. Although Wolfram assembled it from simple program "modules", it was nevertheless capable of carrying out enormously complex tasks. "It hammered home to me once
again my central discovery -that simple programs can have hugely complex outcomes," says Wolfram.
Wolfram's hope was that others would pile into the research area he had created and that this would lead to rapid progress in understanding complexity. To his disappointment and
frustration, however, few joined in and progress was slow. Wolfram became increasingly impatient. By early 1991, he decided there was only one thing to do - carry out the work himself.
With a few million users worldwide, "Mathematica" had made Wolfram a multimillionaire. He did not need to be employed by a university and he did not need to fight constantly for research
money. He was free to concentrate all of his time on creating a science of complexity.
Wolfram set himself a gargantuan task but even he did not realise it would take him a decade. During that time, he published not a single research paper. Although he certainly talked and
corresponded with other scientists, he pretty much vanished off the edge of the scientific radar screen.
Month after month, year after year, while the rest of the world slept, Wolfram laboured through the night, painstakingly laying the foundations of a new way of doing science. In essence, he
was carrying out a systematic computer search for simple rules with very complicated consequences. "He set out to survey all possible worlds - at least all the ones generated by simple rules," says
mathematician Gregory Chaitin of IBM in Yorktown Heights, New York. "The result was a treasure trove of small computer programs that, when repeated again and again, yield extremely rich, complicated
and interesting behaviour."
Among the many things Wolfram discovered is the remarkable property of cellular automaton rule 110. Starting with a single black cell, this simple rule turns out to be capable of generating
infinite complexity, infinite novelty, infinite surprise. Not only that but a cellular automaton following rule 110 is a "universal Turing machine" . Despite being amazingly simple, it is like a
modern-day computer that can carry out any imaginable computation, simulate any other conceivable machine.
The remarkable ability of cellular automaton rule 110 is highly suggestive. After all, if even a simple 1-dimensional cellular automaton can create never-ending complexity, it shows the
kind of power that nature potentially has at its disposal. And Wolfram is convinced that nature avails itself of that potential. "I believe that physical systems subject to simple rules applied
recursively - with the output fed back in as the input - can have created everything from the tip of your nose to most distant cluster of galaxies," he says.
So is the Universe a giant cellular automaton - a 3-dimensional version of the 1-dimensional ones Wolfram has been playing with on his computer? Surprisingly, Wolfram thinks not. "I think
the truth is actually much more strange and interesting," he says.
The universe-generating program
A serious shortcoming of a cellular automaton as a model of the Universe is that all the cells update themselves together. This kind of coordinated behaviour requires a built-in "clock", whose ticks
provide the all-important cue for the cells to "all change". Unfortunately, this kind of clock is impossible to implement in the real Universe. The reason is the existence of a cosmic speed limit, as
discovered by Einstein.
Nothing, it turns out, can travel faster than the speed of light. This constraint means that, wherever the cellular automaton clock happens to be located in the Universe, the signal
carrying news of its tick will take longer to travel to a cell that is far away from than to one that is nearby. A possible way round this might be to have lots of clocks distributed throughout the
Universe. However, this does not overcome the fundamental problem because there is no way to make sure the clocks are all telling the same time. If a "reference clock" is used, inevitably the signal
carrying news of its time will take longer to reach some clocks than it does to reach others.
The impossibility of implementing a "global" clock in our Universe means that at the very least the Universe cannot be a "standard" cellular automaton. However, that does not rule out it
being a cellular automaton of some non-standard type - one that somehow gets by without a global clock. This seems a bit of a tall order. But Wolfram can think of an ingenious way it can be achieved.
"Say that, rather than updating all of its cells together, a cellular automaton updates just one cell at a time," he says.
At first sight this seem crazy. But consider for a moment the advantage of such a scheme. If, at each step, only one cell is updated, the sticky problem of getting all the cells to update
at the same time clearly goes away.
Of course there remains the small matter of how such a cellular automaton could possibly mimic our reality. After all, we have the very strong impression that everything in the Universe is
travelling forward through time together, not that one detail of reality is being updated in turn while everything else remains doggedly rooted to the spot.
Say you are playing in a football game. You see all the other players running about the pitch simultaneously. You do not see first one player take a step, as their thought processes click
on one notch, while everyone else remains paralysed in mid-stride; then another player take a step, and so on. "But, just because you do not see this happening, does not mean that this is not exactly
what is going on," says Wolfram.
But surely you would notice? No, says Wolfram. The only time you notice the world about you is when it is your turn to be updated. And, when this happens, all you see is that all the other
players have moved on a fraction. Because your awareness is frozen between your own updatings, it is impossible for you to notice when any of the other players are updated. Despite the fact that only
one player on the pitch is moving at any one time, your perception is of everyone on the pitch running about simultaneously.
Between any two successive moments of time as perceived by you, there are very many updating events, none of which you have any awareness of. In fact, all you can ever really know about,
says Wolfram, is what updating event influences what other updating event. For instance, the updating event that moved the football one step closer to the opposing team's goal influenced the opposing
team's defenders and goal keeper, who altered their positions to intercept the ball. This, of course, is the familiar story of a football game. "But that's all it is - a 'story'," says Wolfram. "A
network of cause and effect we impose on the underlying reality to make some kind of sense of it."
Contrary to common sense expectations, then, it appears that it is possible to mimic our Universe with a cellular automaton in which only one cell at a time is updated. The passage of the
"time" in the Universe is marked by the regular ticking of the cellular automaton's clock. That only leaves "space" to worry about. Unfortunately, it is here, according to Wolfram, that the idea of
the Universe-as-a-cellular-automaton comes to grief.
Wolfram is convinced that the computer program generating our Universe is a simple one. Every scrap of evidence he has accumulated since his key discovery that simple programs can produce
unexpectedly complex outputs bolsters this belief. But, if the program generating the Universe is simple, it stands to reason there will not be room in it for much "stuff". In other words, very few
of the features of our Universe - from gravity to space and time to koala bears - will be visible in the program. Instead, they will "emerge" - like an inflatable raft unfolding from a canister -
only after the program has been running for a long while.
But Wolfram does not simply think the Universe-creating program is simple. He goes further than this. He believes the program may be among the simplest possible programs capable of
generating the Universe. This is a leap of faith. All Wolfram knows for sure is that the rule for the Universe is not really complicated. If it was, he argues, there would be no perceptible pattern
to nature, which there clearly is. Wolfram thinks it is possible our Universe is the very simplest universe that is not obviously a silly one - for instance, a universe with no notion of space or of
time. Consequently, he thinks it is worth first trying the simple rules for size because our Universe might be among them.
If the Universe-generating program is indeed among the simplest programs capable of generating the Universe, it will contain the absolute bare minimum of stuff. And it is this that
persuades Wolfram that the Universe cannot possibly be a cellular automaton. A cellular automaton, after all, is a rigid array of cells laid out in “space”. In other words, the very notion of “space”
is built into its very foundations. To Wolfram, this is already too much stuff.
Wolfram believes the Universe-generating program will be so simple, so pared down, that even something as apparently fundamental as space will not be built into it. Instead, it will emerge
along with everything else only as the program runs, conjured out of something even more basic than space.
Wolfram believes space is not a smooth, featureless backcloth to the drama of the Universe. Instead, it has an underlying structure. The analogy he uses is water. Although water looks
smooth and continuous, in fact it is made up of tiny motes of matter called molecules. Wolfram thinks space is similar. If it were possible to examine it with some kind of super-microscope, we would
see that it is made of a huge number of discrete points. The points, or "nodes", are connected together in a vast extended network.
But how can a mere network of points have the properties of familiar space? "Surprisingly easily," says Wolfram. "It simply depends on the way the nodes are connected to each other."
Imagine being at one particular node, then going to all the nodes that are 1 connection away, then 2 connections, then 3, and so on. After going, say, r connections, simply count how many
nodes you have visited. If there are roughly pi X r^2 nodes - the area of a circle - then the space is 2-dimensional like the surface of a piece of paper. If there are roughly 4/3 pi X r^3 - the
volume of a sphere - then the space is 3-dimensional, like the space we live in. It turns out that a simple network of nodes can mimic the essential properties of absolutely any space imaginable, be
it 1-dimensional, 2-dimensional or 279-dimensional.
According to Wolfram, space is nothing more than a bunch of nodes connected together. Of course, there is a little bit more to it than that.
Wolfram envisions a space network being updated in a similar way to a cellular automaton. After all, a constantly updated cellular automaton has a proven ability to generate complexity
reminiscent of our Universe. Recall how it was possible to get over the synchronisation problem of a cellular automaton by updating just one cell at a time. Well, Wolfram thinks that this elegant
solution can be carried right over to a space network. Instead of having a rule which says - if a cell is surrounded by a certain pattern of coloured cells - change its colour, Wolfram imagines a
rule saying - if there is a piece of network with a particular form, replace it with a piece of network with another form. Remarkably, Wolfram claims that everything in our world can emerge from such
a space network.
Take particles of matter. In a cellular automata - for instance, the one subject to rule 110 - the system may quickly organise itself into a few localised structures which are persistent
and appear to move through space just like fundamental particles - quarks and electrons and so on. What is actually happening is that, as fast as the structures are destroyed, they are refreshed
again. It is just like a TV image of a football game. We may perceive that a football in flight. But, in reality, what is happening is that a picture of the ball is being refreshed 30 times a second
and giving us the illusion of the ball moving through the air .
Sometimes, in a cellular automaton subject to rule 110, there is a collision between "particles". They slam into each and a whole bunch of other particles come out. This is just the kind of
thing physicists observe at atom smashers like the one at the European centre for particle physics at CERN in Geneva. And what happens in a cellular automaton subject to rule 110 can also happen in a
space network. Instead of being stubbornly persistent patterns of cells, however, the "particles" are stubbornly persistent tangles of connections.
Remarkably, Wolfram has found that, with a constantly updated network of nodes, it is possible to create both the space we live in and the matter we are made of. "Reality", as Einstein
remarked, "is merely an illusion, albeit a very persistent one."
A problem arises, however, if a rule applies to a particular pattern of nodes and there are several places in the network with the same pattern. Which place should be updated first?
Updating the places in a different order will in general lead to different networks of cause and effect. Rather than having a unique history, the Universe will have several possible histories. We
will not know why we are following the history we are and not another, which is a highly unsatisfactory state of affairs.
Fortunately, there is a way out of this difficulty, says Wolfram. By a stroke of luck it turns out that there are certain rules with the property that it in fact does not matter in which
order they are applied. Wolfram calls them "causally invariant" rules. "Whenever they are used, there is always just a single thread of time in the Universe," he says.
Wolfram's progression from the Universe-as-a-cellular- automaton to the Universe-as-a-constantly-updated-space-network is a good illustration of the way in which physicists grope their way
towards a true picture of nature. They start with a crude model which mimics an aspect of reality which they consider to be important. In this case, the model is a cellular automaton which can
generate complexity tantalisingly like the complexity we see in the world around us. Inevitably, the model falls short in some way. In the case of cellular automaton, it contains too much ready-made
stuff such as "space". Nevertheless, physicists use the crude model as a bridge to reach a better model that mimics more reality more faithfully. Lastly, they throw away the bridge.
Wolfram's talk of space and matter "emerging" from a network may seems rather woolly. However, he maintains that it can explain concrete things too such as the general theory of relativity,
Einstein's theory of gravity. In a nutshell, the theory says that matter distorts, or warps, space-time, and that warped space-time, is what matter reacts to when it moves. In fact, warped space-time
is all gravity is. We think that the Earth pulls on the Moon with invisible fingers of force which somehow reach out across 400 000 kilometres of empty space. But, according to Einstein, this is an
illusion. In reality, the Earth's mass warps space-time, creating a sort of valley in its vicinity. We cannot see it because space-time is 4-dimensional and we can experience only 3 dimensions. But
the Moon "sees" it. It skitters around the rim of the valley in space-time like a roulette ball round a roulette wheel.
Wolfram claims that his perpetually updated space network behaves exactly like Einstein's warped space-time. For simplicity imagine things in 2 dimensions. Also, imagine that the network is
a network of hexagons which can be laid out flat like a fishing net spread out on a beach. What happens if some of the connections are changed so that some heptagons and pentagons are mixed in with
the hexagons? The answer is the network bulges out or in. "This is warped space," says Wolfram.
In ordinary flat 2-dimensional space, recall that the number of nodes we get by going out r steps through the network goes up as r^2. Well, in a warped network, it is not quite the same.
There is what mathematicians call a "correction term". And it turns out that the correction term is basically the "Ricci tensor". It is not necessary to know exactly what the Ricci tensor is. But it
crops up in Einstein's equations which, in general relativity specify the warpage of space-time.
The story of how is quite complicated. But Wolfram maintains that, with just a few assumptions, he can work out the conditions which the Ricci tensor must obey. "And, guess what?" he says.
"They seem to be exactly Einstein's equations of gravity."
The ubiquity of biological complexity
Wolfram believes the computer program that nature is using to generate the Universe is very short. We are certainly not talking about the 10 million or so lines of a program like "Microsoft Windows".
Far from it. "Nature's program may be expressible in as few as 4 lines of Mathematica," he says.
If he is right, those lines are responsible for creating everything from chocolate doughnuts to TV game shows to the very thought processes that have led Wolfram to the audacious claim that
a mere 4 lines of computer code are generating reality.
Wolfram admits that his decade of investigation has not yet furnished him with the elusive cosmic computer program - the “one rule to bind them all”. But he is hopeful that he will one day
find it.
One of the most important discoveries to have come out of Wolfram's decade of toil is the recognition that a cellular automaton following rule 110 is far from unique. Wolfram has been
surprised to find that many other real systems in the Universe - from turbulent fluids to colliding subatomic particles - also behave as universal computers. In other words, they too have the
capacity to simulate any other machine, carry out any conceivable computation.
Because a universal computer can compute, or simulate, absolutely anything, it is trivial to deduce from this that all systems that behave as universal computers can compute as much as each
other. In other words, they are equivalent. "Since universal computers are so widespread in nature, this has far-reaching implications," says Wolfram. "It means that everything from the behaviour of
a cell to turbulence in a hydrogen cloud drifting in the depths of space to rain pattering on the pavement is equivalent in terms of the computational complexity required to generate it."
Until now, scientists have assumed that the kind of complexity which is seen in living things - from single cells to human brains - can arise only in a system of large molecules based on
carbon atoms. This, after all, is what we observe on Earth. But if, as Wolfram firmly believes, a large range of systems in nature have equivalent computational complexity, it means that the
complexity we associate with life is not the unique preserve of planet-bound, water-soluble, carbon-based chemistry. Many of the things we thought were special about life and intelligence can be
present in numerous other kinds of physical systems. "The Universe may contain lifeforms - including intelligent life forms - the like of which we cannot begin to imagine," says Wolfram.
Wolfram elevates his discovery that large numbers of natural systems have the same computational complexity to an over-arching natural principle. He calls it "The Principle of Computational
Equivalence". Put crudely, it says that systems of similar complexity are equivalent. Take, for instance, the Earth's atmosphere. According to Wolfram's Principle, because the atmosphere's
circulation is as complex as any living thing, it has exactly the same right be classed as a living thing as you or me! "People say 'The weather has a mind of its own and think they're just using a
metaphor," says Wolfram. "I think there's something much more literally true about it."
Wolfram his believes his Principle of Computational Equivalence is a revolutionary and fertile new idea in science. Moreover, he sees it as the next logical step along a road that science
first embarked on more than four centuries ago.
In the 16th century, the Polish astronomer Nicolaus Copernicus realised that the Sun and planets did not turn about the Earth, as had generally been believed, but that the Earth occupied no
special place in the Universe. Later, in the 19th century, Charles Darwin deduced that humans were just another product of evolution by the process of natural selection and so they occupied no
special place in Creation. Wolfram sees himself as completing the revolution begun by Copernicus and Darwin. There is nothing special, he maintains, about the kind of computation that leads to living
things and the thought processes of human brains. Life and intelligence could be implemented in a myriad different physical systems. One consequence of this is that there is no barrier preventing us
from creating artificial intelligence - a machine that thinks and behaves like a human being.
All this spells trouble for a kind of reasoning currently favoured by some cosmologists. According to the "anthropic principle", the reason the Universe has many of the features it has -
for instance, laws of physics which permit the formation of galaxies, stars and planets - is because, if it did not, it would not have been possible for human beings to have arisen to notice those
features. It is a curiously topsy-turvy logic. And an inevitable consequence is that biology is the ultimate determinant of the physics that we observe around us.
However, the anthropic principle is fatally undermined if, as Wolfram believes, life can be implemented in any number of different physical systems, some as far away from carbon-based
chemistry as it is possible to imagine. "Cosmologists have no right to use the conditions necessary for our existence on Earth to deduce anything about the laws of physics that govern our Universe,"
says Wolfram.
Cosmologists wonder why the Universe appears so hospitable for life. The answer, Wolfram believes, is because almost any physical system, almost any set of parameters, can exhibit the
complexity of a living thing.
Is God a programmer?
Everything Wolfram discovered during his decade of toil - the equivalent, he maintains, of hundreds, maybe even thousands, of scientific papers - he eventually distilled into an enormous, epic book.
A New Kind of Science was finished in January 2002. It was almost 1200 pages long with about 1000 black-and-white pictures and half a million words. On the first day of publication it sold 50,000
copies. And it annoyed the hell out the scientific community.
Absolutely everything about the self-published book seemed to make other scientists see red. Wolfram was accused of not crediting the contributions of others. Wolfram was accused of
breathtaking arrogance. After all, he was saying, "Here in my book is an entirely new way of doing science". And nobody had dared say that since Isaac Newton.
A striking feature of the venom directed at Wolfram was its swiftness. Within days of the book’s publication, some scientists had posted damning reviews on Amazon’s website. Yet the book's
1200 picture-filled pages were crammed with examples that had to be worked through by the reader. It was hard to believe that anyone could have digested enough to have dismissed it in just a few
Chaitin is philosophical about the knee-jerk reaction of the scientific community. "If you write a book that offends no one and make sure everything you write is absolutely, 100 per cent,
correct, then you end up writing nothing," he says.
One specific criticism is that, although Wolfram has produced a 1200-page book of pretty pictures of what simple computer programs can do, he has deduced very few universal laws of the kind
first discovered by Newton. This, however, misunderstands Wolfram. His new kind of science is not at all like the old type - which, of course, why he has called it "a new kind of science". In the
old, maths-based science, the motion of, say, a planet travelling around the Sun is distilled into an equation, which predicts its behaviour from now into the infinite past and future. In the new
science, the only way to discover how something behaves is to run a computer program. There is no such shortcut. Or, rather, all the shortcuts have already been found. They are conventional,
equation-based science!
It is Wolfram's view that much of what is going on in the Universe cannot be distilled into neat equations. You have to run the program to find out what happens. Some of the programs can be
run, and a result obtained, more quickly than the Universe. This is because, by some fortunate quirk, some of what the Universe is doing is "computationally reducible". "Almost all of what
traditional equation-based science has been doing is looking just at those computationally reducible parts," says Wolfram.
Wolfram suspects, however, that most of what is going on in the Universe is computationally irreducible. In other words, the only way to find out the outcome of the program the Universe is
running is to run it for 13.7 billion years! This raises a spooky possibility. Is the program of the Universe being run by someone or something simply because there is no other way to discover the
outcome?! In The Hitch Hiker's Guide to the Galaxy, the Earth turns out to be a computer run by mice to discover the answer to the ultimate question. Might author Douglas Adams' jest, by some
tremendous irony, actually be near to the truth?
The American physicist Ed Fredkin thinks so. He is convinced that the Universe is nothing more than a computer which is being used to solve a problem. As others have pointed out, this is
both good news and bad news. The good news is that there really is a purpose to our lives. The bad news is that purpose may be to help someone or something work out pi to countless zillion decimal
The idea that the most fundamental stuff in the Universe - more fundamental even than matter or energy - is information - digital information - is certainly an idea which is taking hold
among today's physicists. Those who subscribe to this "digital philosophy", such as Wolfram and Fredkin, are in absolutely no doubt that what the Universe is doing is computation, in the most general
sense of the word. One consequence is unavoidable. Like the insects burrowing in the topsoil of Adams' terrestrial computer, we are a part of the great cosmic computation. "We never perform a
computation ourselves," says Tomasso Toffoli of Boston University . "We just hitch a ride on the great Computation that is going on already."
Of course, if you want to go one mystical step farther and talk about a computation not in its most general sense but in the sense of something directed to some end, like a human
computation, then you come to the arena of religious speculation. "The Universe begins to look more like a great thought than a machine," wrote the British astronomer Sir James Jeans. And Jeans was
really only echoing Bishop George Berkeley, the Irish philosopher who in the 18th century declared: "We exist only in the mind of God." Chaitin likes to put it in more modern terms: "Is God a
If the idea of the Universe "computing" something is not mind-blowing enough, consider what it really means if Wolfram is right and the complexity of the Universe is generated merely by
applying a simple computer program - a simple rule - over and over again. Information cannot be created out of nothing. Common sense says that what comes out cannot be any more than what is put in.
If Wolfram is right, it means that the Universe can contain no more complexity than simple the program responsible for generating it. Consequently, the complexity we see around us cannot be real
complexity. It must be "pseudo-complexity". The Universe only looks complex because we are unaware of the simple underlying rule generating it.
Newton's worldview was one in which the laws of physics orchestrate a predictable world. The planets, for instance, circle the Sun with the regularity of clockwork. However, in such a
clockwork universe, where the future is always utterly predictable, scientists faced a conundrum: how can there be any free will?
Wolfram sidesteps this problem. In his clockwork universe, the future of the Universe is predictable - but only in principle. In practice, you can never finish the computations and discover
the outcome faster than the Universe does. Free will survives - as pseudo free will!
Chaitin puts Wolfram's worldview in purely mathematical terms. Pi, the ratio of a circle's circumference to its radius, is a number that appears to be extraordinarily complicated, its
digits never repeating, but it can in fact be generated by a short computer program. Chaitin, however, has invented a number which is truly complex. "Omega" requires an infinitely long computer
program to generate it . "Is the Universe like pi or like Omega?" says Chaitin. "Most people think it's like Omega. Wolfram thinks it's like pi."
The reason that most people think the Universe is like Omega - in other words, that it has unadulterated, infinite complexity - is that most people believe in quantum theory. And quantum
theory tells us that events in the microscopic world such as the disintegration of an atom or the absorption of a photon of light by a window pane are completely random, as unpredictable as a perfect
coin toss. Such events generate an infinite amount of complexity - which is the same as randomness . And this gets permanently imprinted on the Universe - for instance, when a high-energy photon
strikes a strand of DNA and causes a mutation, which echoes down the generations, frozen into the fabric of life for all time.
As a consequence of quantum theory, then, much of what we see around us is in the Universe inherently unpredictable. It is the result of countless quantum coin tosses, which have been
happening one after the other since the beginning of time. We will therefore never be able to comprehend the Universe in its entirety.
On this score, Wolfram is far more optimistic than the majority of physicists. Because he believes the Universe has finite complexity like pi, he believes that quantum theory as currently
practised is wrong. All the randomness that quantum theory generates is therefore really only pseudorandomness, like that in the digits of pi. If he is right, then we may eventually be able to
comprehend everything.
Who is right - Wolfram or the rest of the scientific community? Chaitin confesses to spending long hours at Wolfram's house near Boston arguing with him about his ideas. "In A new kind of
science, Wolfram develops an extremely interesting and provocative vision," says Chaitin. "The question is: Does the physical Universe share Wolfram's vision? Time alone will tell." | {"url":"http://www.marcuschown.com/neverendingdayssample.htm","timestamp":"2014-04-18T23:27:47Z","content_type":null,"content_length":"75207","record_id":"<urn:uuid:3c23d0cd-b009-4f53-906c-e2fe38ad60e3>","cc-path":"CC-MAIN-2014-15/segments/1398223203422.8/warc/CC-MAIN-20140423032003-00105-ip-10-147-4-33.ec2.internal.warc.gz"} |
A Stochastic Dynamic Model of Computer Viruses
Discrete Dynamics in Nature and Society
Volume 2012 (2012), Article ID 264874, 16 pages
Research Article
A Stochastic Dynamic Model of Computer Viruses
School of Information Engineering, Guangdong Medical College, Dongguan 523808, China
Received 8 June 2012; Accepted 16 July 2012
Academic Editor: Bimal Kumar Mishra
Copyright © 2012 Chunming Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any
medium, provided the original work is properly cited.
A stochastic computer virus spread model is proposed and its dynamic behavior is fully investigated. Specifically, we prove the existence and uniqueness of positive solutions, and the stability of
the virus-free equilibrium and viral equilibrium by constructing Lyapunov functions and applying Ito's formula. Some numerical simulations are finally given to illustrate our main results.
1. Introduction
A generalized computer virus, including the narrowly defined virus and the worm, is a kind of computer program that can replicate itself and spread from one computer to another. Viruses mainly attack
the file system and worms use system vulnerability to search and attack computers. As hardware and software technology developed and computer networks became widespread, computer virus has come to be
one major threat to our daily life. Consequently, in order to deal with the threat, the trial on better understanding the computer virus propagation dynamics is an important matter. Similar to the
biological virus, there are two ways to study this problem: microscopic and macroscopic. Following a macroscopic approach, since [1, 2] took the first step towards modeling the spread behavior of
computer virus, much effort has been done in the area of developing a mathematical model for the computer virus propagation [3–13]. These models provide a reasonable qualitative understanding of the
conditions under which viruses spread much faster than others.
In [13], the authors investigated a differential model by making the following assumptions.() The total population of computers is divided into four groups: susceptible, exposed, infected, and
recovered computers. Let , and denote the numbers of susceptible, exposed, infected, and recovered computers, respectively. denotes the total number of computers.() New computers are attached to the
computer network with rate .() Computers are disconnected to the computer network with constant rate .() computers become computers with rate , where denotes the averaged number of neighbor nodes
(with various states) that are directly connected; is the transition rate from to . computers become computers with rate .() computers become computers with constant rate ; computers become computers
with constant rate ; computers become computers with constant rate .
According to the above assumptions, the following model (see Figure 1) is derived:
Notably, the first three equations in (1.1) do not depend on the fourth equation, since . Therefore, the forth equation can be omitted and the model (1.1) can be rewritten as
In [13], authors have proved the virus-free equilibrium is globally asymptotically stable if , and the viral equilibrium is globally asymptotically stable if , where
However, in the real world, systems are inevitably affected by environmental noise. Hence the deterministic approach has some limitations in mathematically modeling the transmission of an infectious
disease, and it is quite difficult to predict the future dynamics of the system accurately. This happens due to the fact that deterministic models do not incorporate the effect of a fluctuating
environment. Stochastic differential equation models play a significant role in various branches of applied sciences, including infectious dynamics, as they provide some additional degree of realism
compared to their deterministic counterpart. In this paper, we introduce a noise into (1.2) and we transform the deterministic problem into a corresponding stochastic problem.
In this paper, we introduce randomness into the model by replacing the parameters and by , and , where , and are mutual independent standard Brownian motions with , and , and intensity of white noise
and , respectively. Then the stochastic system is
The organization of this paper is as follows. In Section 2, we prove the existence and the uniqueness of the nonnegative solution of (1.3). In Section 3, if , we show that the solution is oscillating
around the virus-free equilibrium of (1.3). Section 4 focuses on the persistence of the virus. By choosing appropriate Lyapunov function, we show that there is a stationary distribution for (1.3) and
that it is persistent if . Some numerical simulations are performed in Section 5. In Section 6, a brief conclusion is given.
Throughout this paper, consider the -dimensional stochastic differential equation with the initial value . denotes -dimensional standard Brownian motion defined on the above probability space. Define
the differential operator associated with (1.4) by If acts on a function , then where .
By Ito’s formula, if , then for (1.4), assume that for all . So is a solution of (1.4), called the trivial solution or equilibrium position.
2. Existence and Uniqueness of the Nonnegative Solution
To investigate the dynamical behavior of a population model, the first concern is whether the solution is positive or not and whether it has the global existence or not. Hence, in this section, we
mainly use the Lyapunov analysis method to show that the solution of system (1.3) is positive and global.
Theorem 2.1. Let , then the system (1.2) admits a unique solution on , and this solution remains in with probability 1.
Proof. Since the coefficients of the equation are locally Lipschitz continuous, for any given initial value there is a unique local solution on , where is the explosion time [2, 13]. To show this
solution is global, we need to show that a. s. Let be sufficiently large so that every component of lies within the interval . For each integer , define the stopping time, where throughout this paper
we set (as usual denotes the empty set). Clearly, is increasing as . Set , whence a. s. If we can show that a. s., then and a. s. for all . In other words, to complete the proof we need to show that
a. s. For if this statement is false, then there is a pair of constants and such that Hence, there is an integer such that Define a -function for by The nonnegativity of this function can be seen
from , forall . Using Ito’s formula we get where By choosing , then Therefore, Setting for , then by (2.3), we know that . Note that for every , there is at least one of , , and that equals either
or . Then where is the indicator function of . Let lead to the contradiction that . So is necessary. The proof of Theorem 2.1 is completed.
3. Stability of Virus-Free Equilibrium
It is clear that is the virus-free equilibrium of system (1.3), which has been mentioned above, and is globally stable if , which means that the virus will die out after some period of time. Since
there is no virus-free equilibrium of system (1.3), in this section, we show that the solution is oscillating in a small neighborhood of if the white noise is small.
Theorem 3.1. If and , then the solution of system (1.3) with initial value has the property where is positive constants, defined as in the proof.
Proof. For simplicity, let , system (1.3) can be written as Let then is positive constants to be determined later. By Ito’s formula, we compute
Choosing , then we get Integrating this from 0 to and taking the expectation, we have Hence,
Remark 3.2. Theorem 3.1 shows that the solution of system (1.3) would oscillate around the virus-free equilibrium of system (1.1) if some conditions are satisfied, and the intensity of fluctuation is
proportional to , which is the intensity of the white noise . In a biological interpretation, if the stochastic effect on is small, the solution of system (1.3) will be close to the virus-free
equilibrium of system (1.1) most of the time.
4. Permanence
When studying epidemic dynamical systems, we are interested in when the computer viruses will persist in network. For a deterministic model, this is usually solved by showing that the viral
equilibrium is a global attractor or is globally asymptotically stable. But, for system (1.3), there is no viral equilibrium. In this section, we show that there is a stationary distribution, which
reveals that the computer viruses will persist.
Lemma 4.1 (see [14, 15]). Assumption : there exists a bounded domain with regular boundary , having the following properties.() In the domain and some neighborhood thereof, the smallest eigenvalue of
the diffusion matrix is bounded away from zero.() If , the mean time at which a path issuing from reaches the set is finite, and for every compact subset . If holds, then the Markov process has a
stationary distribution . Let be a function integrable with respect to the measure . Then
Lemma 4.2 (see [14, 15]). Let be a regular temporally homogeneous Markov process in . If is recurrent relative to some bounded domain , then it is recurrent relative to any nonempty domain in .
Theorem 4.3. If, and , then, for any initial value , there is a stationary distribution for system (1.3), and it has an ergodic property, where are defined as in the proof, is the viral equilibrium
of system.
Proof. When , there is an viral equilibrium of system (1.3). Then Define where , are positive constants to be determined later. Then is positive definite. By Ito’s formula, we compute where Let and ,
for all
Choosing , then
Choosing , then
Then the ellipsoid lies entirely in . We can take to be a neighborhood of the ellipsoid with , so, for , ( is a positive constant), which implies that condition in Lemma 4.1 is satisfied. Hence, the
solution is recurrent in the domain , which, together with Lemma 4.2, implies that is recurrent in any bounded domain . Besides, forall , there is an such that for all which implies that condition
is also satisfied. Therefore, the stochastic system (1.3) has a stationary distribution and it is ergodic. This completes the proof.
5. Numerical Simulations
In this section, we have performed some numerical simulations to show the geometric impression of our results. To demonstrate the global stability of infection-free solution of system (1.3) we take
following set parameter values: , , , , , ,, , . In this case, we have . In Figures 2(a), 2(b), and 2(c), we have displayed, respectively, the susceptible, infected and recovered computer of system (
1.4) with initial conditions: and .
To demonstrate the permanence of system (1.4), we take the following set parameter values: , , , , , , , , , . In this case, we have . In Figures 3(a), 3(b), and 3(c), we have displayed,
respectively, the susceptible and infected population of system (1.4) with initial conditions: and .
6. Conclusion
In this paper, a stochastic computer virus spread model has been proposed and analyzed. First, we prove the existence and uniqueness of positive solutions. Then, by constructing Lyapunov functions
and applying Ito’s formula, the stability of the virus-free equilibrium and viral equilibrium is studied.
This paper is supported by the National Natural Science Foundation of China (no. 61170320), the Natural Science Foundation of Guangdong Province (no. S2011040002981) and the Scientific Research
Foundation of Guangdong Medical College (no. KY1048).
1. J. O. Kephart and S. R. White, “Directed-graph epidemiological models of computer viruses,” in Proceedings of the IEEE Computer Society Symposium on Research in Security and Privacy, pp. 343–359,
May 1991. View at Scopus
2. J. O. Kephart, S. R. White, and D. M. Chess, “Computers and epidemiology,” IEEE Spectrum, vol. 30, no. 5, pp. 20–26, 1993.
3. L. Billings, W. M. Spears, and I. B. Schwartz, “A unified prediction of computer virus spread in connected networks,” Physics Letters A, vol. 297, no. 3-4, pp. 261–266, 2002. View at Publisher ·
View at Google Scholar · View at Zentralblatt MATH
4. X. Han and Q. Tan, “Dynamical behavior of computer virus on Internet,” Applied Mathematics and Computation, vol. 217, no. 6, pp. 2520–2526, 2010. View at Publisher · View at Google Scholar · View
at Zentralblatt MATH
5. B. K. Mishra and N. Jha, “Fixed period of temporary immunity after run of anti-malicious software on computer nodes,” Applied Mathematics and Computation, vol. 190, no. 2, pp. 1207–1212, 2007.
View at Publisher · View at Google Scholar · View at Scopus
6. B. K. Mishra and D. Saini, “Mathematical models on computer viruses,” Applied Mathematics and Computation, vol. 187, no. 2, pp. 929–936, 2007. View at Publisher · View at Google Scholar · View at
Zentralblatt MATH
7. B. K. Mishra and S. K. Pandey, “Dynamic model of worms with vertical transmission in computer network,” Applied Mathematics and Computation, vol. 217, no. 21, pp. 8438–8446, 2011. View at
Publisher · View at Google Scholar · View at Zentralblatt MATH
8. J. R. C. Piqueira and V. O. Araujo, “A modified epidemiological model for computer viruses,” Applied Mathematics and Computation, vol. 213, no. 2, pp. 355–360, 2009. View at Publisher · View at
Google Scholar · View at Zentralblatt MATH
9. J. R. C. Piqueira, A. A. de Vasconcelos, C. E. C. J. Gabriel, and V. O. Araujo, “Dynamic models for computer viruses,” Computers and Security, vol. 27, no. 7-8, pp. 355–359, 2008. View at
Publisher · View at Google Scholar · View at Scopus
10. J. Ren, X. Yang, L.-X. Yang, Y. Xu, and F. Yang, “A delayed computer virus propagation model and its dynamics,” Chaos, Solitons & Fractals, vol. 45, no. 1, pp. 74–79, 2012. View at Publisher ·
View at Google Scholar
11. J. Ren, X. Yang, Q. Zhu, L.-X. Yang, and C. Zhang, “A novel computer virus model and its dynamics,” Nonlinear Analysis: Real World Applications, vol. 13, no. 1, pp. 376–384, 2012. View at
Publisher · View at Google Scholar · View at Zentralblatt MATH
12. J. C. Wierman and D. J. Marchette, “Modeling computer virus prevalence with a susceptible-infected-susceptible model with reintroduction,” Computational Statistics & Data Analysis, vol. 45, no.
1, pp. 3–23, 2004. View at Publisher · View at Google Scholar
13. H. Yuan and G. Chen, “Network virus-epidemic model with the point-to-group information propagation,” Applied Mathematics and Computation, vol. 206, no. 1, pp. 357–367, 2008. View at Publisher ·
View at Google Scholar · View at Zentralblatt MATH
14. R. Z. Hasminskii, Stochastic Stability of Differential Equations, vol. 7, Sijthoff and Noordhoff, Groningen, The Netherlands, 1980.
15. C. Ji, D. Jiang, Q. Yang, and N. Shi, “Dynamics of a multigroup SIR epidemic model with stochastic perturbation,” Automatica, vol. 48, no. 1, pp. 121–131, 2012. View at Publisher · View at Google | {"url":"http://www.hindawi.com/journals/ddns/2012/264874/","timestamp":"2014-04-19T03:09:39Z","content_type":null,"content_length":"740971","record_id":"<urn:uuid:c7b04959-c5bb-4263-9758-013960f78a07>","cc-path":"CC-MAIN-2014-15/segments/1397609535745.0/warc/CC-MAIN-20140416005215-00249-ip-10-147-4-33.ec2.internal.warc.gz"} |
Both positive and negative curvature?
What happens to the Reimann tensor at the event horizon of a black hole? Do some of the 24 components become zero or infinite?
The components of a tensor depend on the coordinate system used to express them. So the answer to this question depends on which coordinate system you are using.
In standard Schwarzschild coordinates there are many zero components throughout the spacetime and there are some infinite components in the limit as you approach the horizon.
In Kruskal Szekeres coordinates there are also many zero components throughout the spacetime but there are no infinite components in the limit as you approach the horizon.
All curvature invariants are finite at the horizon. Furthermore, the larger the black hole the smaller the curvature invariants at the horizon. For a supermassive black hole there could be less tidal
gravity than at the surface of the earth.
What happens to parallel transport of a vector on the surface of an event horizon that is different than on a surface outside the event horizon?
Nothing. The event horizon is merely an outgoing null surface, so you cannot have any outgoing timelike paths which cross it. | {"url":"http://www.physicsforums.com/showthread.php?p=4229003","timestamp":"2014-04-20T11:30:34Z","content_type":null,"content_length":"45957","record_id":"<urn:uuid:175ef643-4869-439d-b73e-630bca262eed>","cc-path":"CC-MAIN-2014-15/segments/1398223205375.6/warc/CC-MAIN-20140423032005-00550-ip-10-147-4-33.ec2.internal.warc.gz"} |
Sea Cliff Math Tutor
Find a Sea Cliff Math Tutor
...I am certified in math in both Connecticut and New York, plus I previously taught pre-calculus both as a homebound tutor and in the classroom as a substitute teacher. And FYI, I scored a
perfect 10 out of 10 on the WyzAnt.com pre-calculus certification test. You'll definitely want to contact me for your pre-calculus tutoring needs!
10 Subjects: including algebra 1, algebra 2, geometry, prealgebra
...I escorted the children to the nurse's, bathroom, etc. I also helped the children with their projects, for example the counting caterpillar. I also read out loud to the children and depending
on the grade level helped them with their reading skills.
13 Subjects: including prealgebra, English, algebra 1, algebra 2
...I have many strengths especially in math, and social studies. My teaching method is that I make learning memorable by using child-friendly techniques like songs and TV. I will do everything
possible to make sure that your kids get the best education.
6 Subjects: including algebra 1, prealgebra, elementary math, elementary (k-6th)
...My students like me, but they also realize that I hold them to a high degree of excellence: I expect their best. As for my background, I have a master's degree in psychology, and when I am not
dedicating myself toward my students' test day success, I am working toward a second master's in statis...
14 Subjects: including algebra 1, writing, statistics, SAT math
...I look forward to showing you the way, but the effort has to come from you. Whether you want to understand the nuances of a subject or learn the basics I can help. I tutored undergraduates and
high school students through out my graduate school days.
11 Subjects: including calculus, differential equations, GRE, GMAT
Nearby Cities With Math Tutor
Baxter Estates, NY Math Tutors
Bayville, NY Math Tutors
Brookville, NY Math Tutors
Glen Cove, NY Math Tutors
Glen Head Math Tutors
Great Neck Estates, NY Math Tutors
Great Neck Plaza, NY Math Tutors
Kensington, NY Math Tutors
Lattingtown, NY Math Tutors
Locust Valley Math Tutors
Matinecock, NY Math Tutors
Old Brookville, NY Math Tutors
Roslyn Harbor, NY Math Tutors
Roslyn, NY Math Tutors
Thomaston, NY Math Tutors | {"url":"http://www.purplemath.com/sea_cliff_math_tutors.php","timestamp":"2014-04-20T09:13:33Z","content_type":null,"content_length":"23771","record_id":"<urn:uuid:7ce5fa99-ead1-4f63-9719-662860ba4fe6>","cc-path":"CC-MAIN-2014-15/segments/1397609538110.1/warc/CC-MAIN-20140416005218-00247-ip-10-147-4-33.ec2.internal.warc.gz"} |
Infinite-dimensional normed division algebras
up vote 16 down vote favorite
Let's say a normed division algebra is a real vector space $A$ equipped with a bilinear product, an element $1$ such that $1a = a = a1$, and a norm obeying $|ab| = |a| |b|$.
There are only four finite-dimensional normed division algebras: the real numbers, the complex numbers, the quaternions and the octonions. This was proved by Hurwitz in 1898:
Adolf Hurwitz, Über die Composition der quadratischen Formen von beliebig vielen Variabeln, Nachr. Ges. Wiss. Göttingen (1898), 309-316.
Are there any infinite-dimensional normed division algebras? If so, how many are there?
ra.rings-and-algebras division-algebras
1 The infallible wikipedia states that there are only four normed division algebras over the reals, period. – David Roberts Nov 11 '10 at 6:21
Note that a normed division algebra is a composition algebra if and only if its norm arises from an inner product via $|a| = \sqrt{\langle a,a \rangle}$. This is in fact always true in the
finite-dimensional case, and I bet it's true in the infinite-dimensional case as well, but I haven't checked it, so there's a slight gap here, at least in my brain. – John Baez Nov 12 '10 at 7:31
add comment
3 Answers
active oldest votes
A MathSciNet search reveals a paper by Urbanik and Wright (Absolute-valued algebras. Proc. Amer. Math. Soc. 11 (1960), 861–866) where it is proved that an arbitrary real normed algebra
(with unit) is in fact a finite-dimensional division algebra, hence is one of the four mentioned in the OP. A key piece of the argument (Theorem 1) is to show that such an algebra $A$
up vote 11 is algebraic, in the sense that if $x \in A$, then the subalgebra of $A$ generated by $x$ is finite-dimensional. The authors then invoke a theorem of A. A. Albert stating that a unital
down vote algebraic algebra is a finite-dimensional division algebra.
Thanks very much! By "real normed algebra" (with unit)" you mean what I'd called "normed division algebra (with unit)". (The authors actually use the term "absolute-valued algebra",
which is nonstandard.) Also, when you write "unital algebraic algebra", I'd write "unital algebraic normed division algebra" - we need $|ab| = |a||b|$ in Albert's theorem. But anyway:
you gave me what I wanted! And it's a very cute skinny PDF file, too – John Baez Nov 12 '10 at 7:18
add comment
Hurwitz's theorem is stated here (section 2.6 of 'A taste of Jordan algebras' by Kevin McCrimmon) as:
Any composition algebra $C$ over a field $\Phi$ of characteristic not 2 has finite dimension $2^n$ for $n=0,1,2,3$ and is one of the following ...
it goes on then to describe generalisations of the usual normed division algebras.
The wikipedia page composition algebra tells us that one only has a 1-dimensional composition algebra when the characteristic of the base field is not 2, but otherwise you can start from
a 2-dimensional composition algebra over a characteristic 2 field and perform the usual Cayley-Dickson construction.
Edit: The following theorem was proved by Kaplansky (Proc AMS 1953), which finishes off the classification. A quadratic form $g$ on an algebra $A$ over a field $F$ in this context is a
up vote 6 function $g:A \to F$ such that $g(kx) = k^2g(x)$ for $k\in F$ and $x\in A$.
down vote
Theorem. Let $A$ be an algebra with unit element over a field $F$. Suppose that $A$ carries a nonsingular quadratic form $g$ satisfying $g(xy) = g(x)g(y)$ for all $x, y \in A$. Then:
(a) A is alternative,
(b) except for the case where $A$ has characteristic two and is a purely inseparable field over $F$, $A$ is finite dimensional and of dimension 1, 2, 4, or 8,
(c) $A$ is either simple or the direct sum of two copies of $F$,
(d) $g(x) = x^\ast x$ where $x\mapsto x\ast$ is an involution of $A$.
So unless your base field has characteristic 2, and your division algebra is a purely inseparable extension of the base field, your division algebra has to be finite dimensional.
Note that normed division algebras are special cases of composition algebras... – David Roberts Nov 11 '10 at 6:30
Thanks, David. In the finite-dimensional case I know how to show that normed division algebras must have a norm arising from an inner product, and are thus composition algebras. I would
need to check this in the infinite-dimensional case. But the result you state is a very nice purely algebraic result along the general lines I sought. – John Baez Nov 12 '10 at 7:37
add comment
The associative case follows from Mazur's Theorem (see here). He proved that there are up to isomorphism precisely three Banach division algebras, namely $\mathbb R,\mathbb C$ and $\
up vote 4 mathbb H$. This applies to the completion of any normed division algebra, which still verifies the identity $|ab|=|a||b|$, and hence is a division algebra.
down vote
The question isn't assuming associativity, so I'm not sure if Mazur's theorem still applies. – Faisal Nov 11 '10 at 7:08
Sorry, I did not notice. I changed the text accordingly. – Andreas Thom Nov 11 '10 at 7:14
add comment
Not the answer you're looking for? Browse other questions tagged ra.rings-and-algebras division-algebras or ask your own question. | {"url":"http://mathoverflow.net/questions/45653/infinite-dimensional-normed-division-algebras?sort=oldest","timestamp":"2014-04-18T11:14:49Z","content_type":null,"content_length":"68034","record_id":"<urn:uuid:31d21a4f-6476-4918-bca5-bd8280b98d2b>","cc-path":"CC-MAIN-2014-15/segments/1397609533308.11/warc/CC-MAIN-20140416005213-00066-ip-10-147-4-33.ec2.internal.warc.gz"} |
Jefferson Lab's Workbench Projects - The Ring Fling Machine - Background
Faraday's law of induction states that a changing magnetic flux creates an induced electromotive force. In equation form, Faraday's law of induction is:
where the minus sign indicates that the direction of the induced electromotive force opposes the change of magnetic flux. This opposition becomes apparent in a closed, conductive circuit, where the
induced electromotive force gives rise to an electric current whose magnetic field opposes the change of magnetic flux. This effect is stated as Lenz's Law: An induced electromotive force always
gives rise to a current whose magnetic field opposes the original change in magnetic flux. For example, if the magnetic flux within a certain region is increasing, an electric current will be
established in such a way as to reduce the magnetic flux, assuming a closed, conductive path exists through which the induced current can flow. | {"url":"http://education.jlab.org/workbench/lenz/background.html","timestamp":"2014-04-17T18:51:16Z","content_type":null,"content_length":"5396","record_id":"<urn:uuid:c3d149f5-8e0d-48eb-94bc-bcd6e516e7a4>","cc-path":"CC-MAIN-2014-15/segments/1397609530895.48/warc/CC-MAIN-20140416005210-00192-ip-10-147-4-33.ec2.internal.warc.gz"} |
Math Help
October 12th 2009, 01:07 AM #1
im supposed to eliminate A from the pair of equations
x= sin2A
y= sec4A
ive learnt most trig identities i think, up to the double angle identities etc.
ive done several problems like this but here im really stuck
Well Remember that $\sin(2A) = 2\cos(A)\sin(A)$, remember that $\sec(x) = \frac{1}{\cos(x)}$, and remember that $\cos(2A) = \cos^2(A) - \sin^2(A)$.
I think mentioning the double angle formula for sin(2A) is an unintentional red herring. I'd be inclined to merely note that $\cos(4x) = \cos(2[2x]) = 1 - 2 \sin^2(2x)$ (using one of the
well-known forms of the cosine double angle formula).
i passed through those already, including the sin2A double angle identity which i already figured might be totally irrelevant to this problem. but where do i go from cos4A? i really didnt need
someone to repeat these identities to me. im actually stuck.
Actually, I think you did need the identity being repeated to you. It was being brought to your attention because it is relevant to answering the question. The hope was that you would contemplate
how it was linked to the question, see
$x = {\color{red}\sin (2A)}$
$y = \sec (4A) = \frac{1}{\cos (4A)} = \frac{1}{1 - 2 {\color{red}\sin^2 (2A)}}$
and then join the dots ....
maybe i just overestimated the difficulty of this problem cos it was the last of the section
anyway i sure know to whom i'll be running when i need help next time!(ironic)
October 12th 2009, 02:40 AM #2
Super Member
Dec 2008
October 12th 2009, 02:47 AM #3
October 12th 2009, 04:07 AM #4
October 12th 2009, 04:17 AM #5
October 12th 2009, 04:21 AM #6 | {"url":"http://mathhelpforum.com/trigonometry/107508-parameter.html","timestamp":"2014-04-19T22:15:00Z","content_type":null,"content_length":"48931","record_id":"<urn:uuid:e4286d75-0b0e-4d7a-8312-7db4e8553ca9>","cc-path":"CC-MAIN-2014-15/segments/1398223202774.3/warc/CC-MAIN-20140423032002-00005-ip-10-147-4-33.ec2.internal.warc.gz"} |
Audubon Park, NJ Algebra 1 Tutor
Find an Audubon Park, NJ Algebra 1 Tutor
...So I try to break down each concept, mechanism and problem down to its bare parts and build an understanding so that each concept, mechanism and problem can be solved logically on their own
without memorization. I am a graduate student getting my Ph.D. in organic chemistry. I have taught both organic chemistry 1 and 2 recitation and lab.
6 Subjects: including algebra 1, chemistry, algebra 2, prealgebra
Hello! I am in my 7th year as a local high school physics teacher. I graduated from the University of Maryland in 2007 with a degree in physics and I have been teaching ever since.
4 Subjects: including algebra 1, physics, geometry, algebra 2
...For the SAT, I implement a results driven and rigorous 7 week strategy. PLEASE NOTE: I only take serious SAT students who have time, the drive, and a strong personal interest in learning the
tools and tricks to boost their score. Background: I graduated from UCLA, considered a New Ivy, with a B.S. in Integrative Biology and Physiology with an emphasis in physiology and human anatomy.
26 Subjects: including algebra 1, chemistry, reading, writing
...While tutoring French, I focus on drawing parallels between French and other languages, particularly English, to enhance the retention of meaning. I primarily focus increasing ability to
communicate and confidence in one's abilities to do so. I have taken Physics in both high school and college, primarily focused around Mechanics and Acoustics.
33 Subjects: including algebra 1, English, French, physics
Hello! Thank you for your interest. My name is Lawrence and I am currently a senior at Temple University majoring in Sociology/Pre-Medical Studies.
11 Subjects: including algebra 1, geometry, algebra 2, SAT math
Related Audubon Park, NJ Tutors
Audubon Park, NJ Accounting Tutors
Audubon Park, NJ ACT Tutors
Audubon Park, NJ Algebra Tutors
Audubon Park, NJ Algebra 2 Tutors
Audubon Park, NJ Calculus Tutors
Audubon Park, NJ Geometry Tutors
Audubon Park, NJ Math Tutors
Audubon Park, NJ Prealgebra Tutors
Audubon Park, NJ Precalculus Tutors
Audubon Park, NJ SAT Tutors
Audubon Park, NJ SAT Math Tutors
Audubon Park, NJ Science Tutors
Audubon Park, NJ Statistics Tutors
Audubon Park, NJ Trigonometry Tutors
Nearby Cities With algebra 1 Tutor
Audubon, NJ algebra 1 Tutors
Barrington, NJ algebra 1 Tutors
Brooklawn, NJ algebra 1 Tutors
Collingswood algebra 1 Tutors
Glendora, NJ algebra 1 Tutors
Gloucester City algebra 1 Tutors
Haddon Heights algebra 1 Tutors
Hi Nella, NJ algebra 1 Tutors
Laurel Springs, NJ algebra 1 Tutors
Mount Ephraim algebra 1 Tutors
Oaklyn algebra 1 Tutors
Philadelphia Ndc, PA algebra 1 Tutors
West Collingswood Heights, NJ algebra 1 Tutors
Westville, NJ algebra 1 Tutors
Woodlynne, NJ algebra 1 Tutors | {"url":"http://www.purplemath.com/Audubon_Park_NJ_algebra_1_tutors.php","timestamp":"2014-04-17T19:48:45Z","content_type":null,"content_length":"24400","record_id":"<urn:uuid:8c959c43-aa7f-4c33-9f3e-8080e5427970>","cc-path":"CC-MAIN-2014-15/segments/1398223206770.7/warc/CC-MAIN-20140423032006-00589-ip-10-147-4-33.ec2.internal.warc.gz"} |
Got Homework?
Connect with other students for help. It's a free community.
• across
MIT Grad Student
Online now
• laura*
Helped 1,000 students
Online now
• Hero
College Math Guru
Online now
Here's the question you clicked on:
Find a number whose product with 6 is the same as its sum with 60. (Hint: you may want to try guess and check or a table.) Show all steps or reasoning. Be sure to check your answer for
• one year ago
• one year ago
Best Response
You've already chosen the best response.
Let the number be x. Then 6x=x+60 Solve for x to find the number
Best Response
You've already chosen the best response.
doesn't work because \(10\times 6=60\) but \(60+10=70\) try 12
Best Response
You've already chosen the best response.
THE NUMBER IS 12! haha sorry i ment to put it in the geometry sub section and when did anyone say it was 10?
Your question is ready. Sign up for free to start getting answers.
is replying to Can someone tell me what button the professor is hitting...
• Teamwork 19 Teammate
• Problem Solving 19 Hero
• Engagement 19 Mad Hatter
• You have blocked this person.
• ✔ You're a fan Checking fan status...
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. | {"url":"http://openstudy.com/updates/5090132ee4b043900ad8fec4","timestamp":"2014-04-18T08:05:37Z","content_type":null,"content_length":"32690","record_id":"<urn:uuid:94cf9f51-3be5-48f2-822d-52e1fb7e832f>","cc-path":"CC-MAIN-2014-15/segments/1397609533121.28/warc/CC-MAIN-20140416005213-00634-ip-10-147-4-33.ec2.internal.warc.gz"} |
PUBLICATIONS - Report, "Consideration of total energy loss in theory of flow to wells,"
UNITED STATES DEPARTMENT OF THE INTERIOR
GEOLOGICAL SURVEY
RESTON, VA. 22092
In Reply Refer To: August 20, 1980
EGS-Mail Stop 411
GROUND WATER BRANCH TECHNICAL MEMORANDUM NO. 80.10
Subject: PUBLICATIONS - Report, "Consideration of total energy
loss in theory of flow to wells," by
R. L. Colley and A.B. Cunningham.
The analytical solutions developed for drawdown induced by pumping
wells assume that the flux entering the well is uniformly
distributed along the screened zones and that the head along the
well screen is constant. The Theis nonleaky solution, and the
Hantush leaky aquifer solution for fully penetrating wells, for
example, make those assumptions. The assumptions imply that all
of the head (or energy) loss takes place within the aquifer.
Actually the drawdown in a pumped well represents the "resistance"
of the aquifer and energy loss in flowing through the screen and
up the well bore to the intake. Jacob (1947) and Rorabaugh (1953)
attempted to represent these components of energy loss by using an
equation for drawdown in the pumped well of the form:
s(subscript w) = BQ + CQ (superscript n)
Jacob assume n to be 2 and Rorabaugh considered the more general
case of n as any constant. Their approach tends to couple an
aquifer head loss component associated with radial flow in the
aquifer to a lumped term that represents all the other head losses
associated with getting the flow to the pump intake.
In the attached paper, Cooley and Cunningham analyze the total
energy losses in an aquifer-well system in a general manner. They
develop a coupled numerical scheme for unsteady flow in single or
multiple confined or semi-confined aquifers and in the well
penetrating the system. The results of their numerical
experiments suggest that for value of aquifer hydraulic
conductivity greater than about 0.015 meter/min (about 530
gpd/square ft) and for pumping rates greater than about 1.2
meter/min (about 315 gpm) that a significant region of non-radial
flow can develop because of the head losses in the well. They
note that these numerical studies suggest that the non-radial flow
related to energy losses in the well can lead to significant
errors in estimates of aquifer transmissivity computed from
drawdown data from the pumped well if the aquifer hydraulic
conductivity is greater than about 0.03 m/min (about 1050
gpd/square ft).
Cooley and Cunningham offer a qualitative explanation for
development of a non-radial flow region. Water movement in the
aquifer tends to follow a path such that the total energy loss
(the sum of all energy losses in the well and in the aquifer) is
minimized. For relatively low hydraulic conductivity the flow
paths in the aquifer tend to be more nearly radial than for cases
involving relatively high hydraulic conductivity. This is because
for a fixed pumping rate the lower the value of hydraulic
conductivity, the greater the proportion of total head losses that
occur in the aquifer. Thus, total energy loss is dominated by
aquifer head losses, which are minimized by all water taking the
shortest possible flow path--the radial one. Conversely, the
larger the hydraulic conductivity, the less the relative
significance of the head losses in the aquifer and the flow paths
will tend to be those that minimize energy losses in the well.
Energy losses in the well are minimized if most of the slow into
the well is near the pump intake. Therefore, in this case, flow
paths in the aquifer are directed more toward the top of the well.
We are not aware of field confirmation of the phenomenon of non-
radial flow to fully penetrating wells in aquifers having uniform
hydraulic conductivity. We would appreciate being advised of
field data pertinent to this question.
Limited additional copies of the attached paper and of the
following references are available upon request to the Ground
Water Branch.
Jacob, C. E., 1947, Drawdown test to determine effective
radius of artesian well: Trans. Am. Soc. Civ. Eng.,
vol. 112, p. 1047-1070.
Rorabaugh, M. I., 1953, Graphical and theoretical
analysis of step drawdown test of artesian well:
Proceedings, Am. Soc. Civ. Eng., vol. 79, separate no.
362, 23 p.
(s) Charles A. Appel
for E. P. Patten
Acting Chief, Ground Water
WRD Distribution: A (Memo only), B (limited), S (Memo only), FOL | {"url":"http://water.usgs.gov/admin/memo/GW/gw80.10.html","timestamp":"2014-04-19T00:19:31Z","content_type":null,"content_length":"5649","record_id":"<urn:uuid:71634be0-30e3-4a3d-a7dc-5479edf5fa70>","cc-path":"CC-MAIN-2014-15/segments/1398223205137.4/warc/CC-MAIN-20140423032005-00179-ip-10-147-4-33.ec2.internal.warc.gz"} |
Morphing of Bistable Composite Laminates Using Distributed Piezoelectric Actuators
Smart Materials Research
Volume 2012 (2012), Article ID 695475, 8 pages
Research Article
Morphing of Bistable Composite Laminates Using Distributed Piezoelectric Actuators
Department of Mechanical Engineering, Université Laval, Quebec City, QC, Canada G1V 0A6
Received 15 December 2011; Accepted 20 February 2012
Academic Editor: Michael W. Hyer
Copyright © 2012 Marie-Laure Dano et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
The use of bistable unsymmetric cross-ply laminates for morphing application has received growing attention in the last few years. So far, most studies use large rectangular piezoelectric Macro Fiber
Composite (MFC) patches bonded at the center of the laminate to induce snap-through. However, the use of large rectangular MFC patches bonded in the center of the laminates significantly influences
the shape of the laminate by greatly reducing the curvature at the midsection of the laminate where the MFC patches are bonded. This paper presents a study where narrow cocured MFC strips distributed
over the entire surface are used to induce snap-through of unsymmetric cross-ply laminates. This MFC configuration allows having a more uniform curvature in the laminate. Since the strips are bonded
on both sides, reverse snap-through should be obtained. The study was both theoretical and experimental. A finite element nonlinear analysis was used to predict the two stable cylindrical
configurations and the snap-through induced by MFC actuation. For the experimental study, a laminate-MFC structure was manufactured and tested. The shapes were measured using a 3D image correlation
system as a function of applied voltage. Good correlations for the cylindrical shape and displacement field were observed.
1. Introduction
The use of bistable composites for morphing application has received growing attention the last few years. The key characteristic of bistable structures is that they possess two stable equilibrium
shapes and that they can be changed from one shape to the other by a simple snap-through action. Morphing can be obtained by alternating between these two shapes. Well-known bistable structures are
unsymmetrically laminated composite plates [1–4]. For example, a two-layer [0/90][T] laminate that is flat at its elevated cure temperature, as in Figure 1(a), cools from its cure temperature to have
two equilibrium configurations. One configuration is cylindrical with a large curvature in the -direction, Figure 1(b). The other configuration is cylindrical with a large curvature in the
-direction, Figure 1(c). The curvatures for the two configurations are equal but of opposite signs, and the laminate can be changed from one configuration to the other by a simple snap-through action
initiated by applying equal and opposite moments to the edges of the laminate. Analysis indicates that there is a third equilibrium configuration which is a saddle shape, Figure 1(d). However, this
shape is unstable and is not observed in practice. Due to their bi stability, unsymmetric laminates are good candidates for morphing applications and most of the work done in this field involves this
kind of laminates, in particular cross-ply unsymmetric laminates [4–12]. The main advantage of using bistable structures is that large shape change can be obtained with little power consumption since
the application of forces is needed only to induce snap-through, not to hold the structure into a deformed configuration. Several theoretical and experimental studies have been reported to understand
and predict actuator-induced snap-through of unsymmetric laminates [5–9]. Models based on Rayleigh-Ritz approach [5–8] are useful for design purposes to carry out parametric studies, and finite
element models [9] are useful to predict more accurately the behaviour of the laminate/actuator system but are much more time consuming. Theoretical and experimental studies showed that unsymmetric
laminates could be changed from one shape to the other using either Shape Memory Alloy (SMA) wires [6, 7] or piezoelectric Macro Fiber Composite (MFC) patches [5, 9] attached on one side and that it
could be predicted fairly well. However, reverse snap-through could not be achieved because the actuator was just located on one side. Lately, Schultz et al. [8] added an MFC patch on the other side
of the laminate and showed that reverse snap-through could be induced using the second actuator. So far, investigators have all used rectangular MFC patches bonded in the center of the laminates [5,
8, 9]. This kind of patches significantly influences the shape of the laminate by greatly reducing the overall curvature [5]. When used in pair as in [8], the midsection of the laminate becomes quite
flat and most of the curvature occurs along the edges and at the corners. It is thought that other types of MFC configuration, such as narrow MFC strips that could be distributed on the entire
surface, may be more suitable and may produce more interesting results. The work discussed here will investigate this idea.
This paper begins in the first section with a general description of the problem that will be investigated, that is, the change of shape of a two-layer unsymmetric cross-ply [0/90][T] laminate using
cocured MFC strips distributed over the top and bottom surfaces. In the following section, a finite element model is presented and used to predict, in a first step, the behaviour of the laminate-MFC
structure as it cools from cure temperature to room temperature and, in a second step, the snap-through of the laminate induced by actuating the MFC strips. After the model is presented, the results
of the analysis are presented and discussed. Next, the experimental study that was conducted to validate the model predictions is presented. Experiments that consisted in measuring the shape of the
laminate as voltage was applied to the MFC actuators are described. Finally, the paper concludes with suggestions for future work.
2. Problem Description
In this study, the behavior of a [0/90][T] laminate with MFC actuator strips bonded over the top and bottom surfaces is investigated. The composite structure is a square laminate with side length of
190mm and thickness of 0.3mm. The laminate is made of two layers of M46J (12K)/RS-36 prepreg. Twelve MFC strips with 160mm by 10mm side lengths and a 0.3mm thickness (M14003-P1) are bonded on
each side of the laminate during the lay-up process. The laminate/MFC assembly is put on a flat plate and cured. Figure 2 shows the plate configuration. The MFC strips bonded on the top surface are
aligned in the -direction whereas the MFC strips bonded on the bottom surface are aligned in the -direction. Only the active part of the MFC (3mm by 140mm) is represented in the figure. Spacing is
larger at the center to have enough room to clamp the structure during testing. Table 1 presents the materials properties.
The objective of the paper is to investigate if using MFC strips distributed over the top and bottom surfaces of a cross-ply laminate is suitable to obtain a reversible bistable structure with
cylindrical shapes. This is done in two parts. First, a finite element model is developed to predict (i) the shape of the laminate/MFC structure due to cooling from cure temperature and (ii) the
response of the laminate/MFC structure as voltage is applied to the MFC strips. Next, experimental work is conducted to validate the model predictions.
3. Finite Element Analysis
The laminate/MFC assembly was modeled using the ABAQUS finite element software. The square [0/90][T] plate was modeled using approximately 2300 general purpose shell elements (S4R). The active part
of the MFC strips was modeled using 47 S4R elements. Since the configuration studied has a total of twenty-four MFC strips, the finite element model used 3400 shell elements overall.
The shells elements used to model the MFC strips were offset from the midplane of the laminate shell elements by 0.3mm. The distance corresponds to half the laminate and MFC thickness. The strips
are linked to the plate using TIE constraint. This approach follows the one used by Portela et al. [9].
The assembly was clamped at its central node. In order to obtain the different equilibrium shapes, small geometric imperfections need to be included [3, 11, 12]. The simplest way is to change
slightly the side lengths of the laminate. In this case, the square laminate with a nominal side length of 190mm was modeled as a rectangular plate with 189mm by 191mm side lengths.
The objectives of the finite element analysis are to predict the cured shape of the laminate/MFC assembly and to simulate the snap-through induced by applying voltage to the MFC actuators. To achieve
this, geometrical nonlinearities have to be taken into account. The finite element analysis is performed in three steps.
In the first step, the cooling from cure temperature to room temperature is simulated. The initial temperature is equal to 177°C. At this temperature, the laminate/MFC assembly is flat and stress
free. A temperature change °C which represents the difference between room temperature and cure temperature is applied to all shell elements. The thermally induced deformation of the laminate/MFC
assembly is computed using a simple *STATIC nonlinear analysis. The two stable cylindrical shapes can be computed by using either (mm, mm) or (mm, mm) for the side lengths.
In the second step, voltage is applied and the snap-through of the laminate induced by MFC actuation is simulated. In the literature, two procedures have been successfully used to simulate
snap-though, namely, the RIKS and STABILIZE procedure [9, 12]. In this study, the STABILIZE procedure was used. Since general purpose shell elements are used to model the MFC actuators, the electric
field cannot be applied directly as boundary conditions. Therefore, voltage is applied through the temperature field using the thermal deformation capabilities of ABAQUS. The coefficient of thermal
expansion for the MFC actuators had to be modified to take into account the piezoelectric and thermal expansion effects. The two-dimensional modeling equation for an MFC pooled in the 1-direction
under a state of plane stress is given by where are the strain components, are the compliance matrix components, are the stress components, are the coefficients of thermal expansion, are the
effective piezoelectric constants, is the temperature change, and is the electric field. The electric field is equal to , where is the applied voltage and is the electrode spacing. The first two
lines of the equation can be rewritten as By introducing modified coefficients of thermal expansion, (2) can be expressed as where the modified coefficients of thermal expansion are equal to The
modified coefficients of thermal expansion account for the effects of thermal expansion caused by and the piezoelectric expansion caused by . In the equation, the electrode spacing is equal to
0.5mm, the applied temperature change is equal to −152°C (as it was the case in the first step) and the applied voltage varies from 0 to 1500V. The modified coefficients of thermal expansion are
entered in ABAQUS as temperature-dependant properties but with the temperature field replaced by voltage.
In the third and last step of the analysis, is still equal to −152°C and voltage is removed . The shape obtained is the second stable cylindrical configuration.
Details of the different steps used in the analysis are represented in Table 2. The results of the analysis are presented in the Section 4.
4. Numerical Results
As described above, step 1 of the finite element analysis predicts shapes of the laminate/MFC structure at room temperature. The results obtained with mm and mm are presented in Figure 3(a). The
configuration is a cylindrical shape with its generator parallel to the -axis. Figure 3(b) illustrates the shape obtained with mm and mm. It is a cylindrical shape with its generator parallel to
the -axis. In both cases, the out-of-plane displacements have the same magnitude. The out-of-plane displacement increases regularly from the center to the edges with no flat area in the center as
observed when using one single large MFC patch on each side [8].
In step 2, the actuation of the MFC strips placed on the bottom surface of the laminate is simulated. As voltage is applied, the laminate curvature and the out-of-plane displacement decrease until
snap-through occurs as can be observed in Figure 4. The figure shows the out-of-plane displacement at the corners as a function of applied voltage. At the beginning of the step, the laminate is in
configuration I of Figure 3(a) (point A). As voltage increases, the out-of-plane displacement decreases. Past point B, there is a sudden change in the magnitude of the out-of-plane displacement which
indicates that snap-though has occurred (jump from point B to point C). The jump corresponds to an applied voltage V. The laminate is now in configuration II of Figure 3(b). After snap-through,
voltage continues increasing up to 1500V which induces a slight decrease in the out-of-plane displacement (point D). In Figure 5, the laminate shapes at (point A), V (point B), and V (point D) are
presented. Just before snap-through occurs, the laminate is almost flat except at the edges (point B). The edge displacements are small but opposite in sign similar to the unstable saddle shape of
Figure 1(d).
Finally, in step 3, voltage is removed. The laminate stays in configuration II. Compared to the shape at point D in Figure 5, the out-of-plane displacement has increased by about 8% (see Figure 6).
5. Experimental Results
5.1. Laminate-MFC Structure Manufacturing
The [0/90][T] laminate was cured and the twenty-four MFC strips bonded in one single step. First, the two prepreg plies were cut and stacked. Then, twelve MFC were bonded on one surface using Loctite
(HP-E120) two-part epoxy adhesive and hold in position using flash tape. The laminate and MFC dimensions and configuration are indicated in Figure 2. A layer of release ply was put on the top. The
laminate was laid on a flat tool with the MFC strips against the flat tool. Next, the other twelve MFC were bonded on the top surface of the laminate and hold in place with flash tape. A layer of
release ply was put on the top. The laminate-MFC assembly was cured in a vacuum bag at 177°C following the prepreg cure cycle. The laminate-MFC assembly was removed from the oven and vacuum bag. As
predicted, the structure has two cylindrical shapes. The first cylindrical shape has an out-of-plane displacement of about 10mm and a nice and uniform curvature (see Figure 7). By a snap-through
action, the second cylindrical shape can be obtained but the laminate does not stay in this configuration. It seems that imperfections due to manufacturing, such as uneven MFC spacing, adhesive
excess, uneven ply thickness, and fiber misalignment, prevent the laminate from having a second stable cylindrical shape. Thus, the ability of the laminate-MFC structure to exhibit two stable shapes
is highly sensitive to manufacturing imperfections.
5.2. Experimental Setup and Test Procedure
The experimental setup to study the behavior of the laminate-MFC structure subject to applied voltage consists mainly in a voltage supply, Dasylab data acquisition system, a stand to hold the
laminate, and a 3D image correlation system to measure the shape of the laminate.
The image correlation equipment used to measure the displacement fields during the tests is the ARAMIS system from Trilion. The system is composed of two 4 Megapixels digital CCD cameras. Before
testing, a random pattern is applied to the specimen surface. During testing, images of the specimen are taken by the two CCD cameras. Then, the system combines the two images to determine the
coordinates of each facets on the specimen surface. From the coordinates, the -, -, -displacements are computed. The displacement fields at different applied voltage can be visualized. A mm
calibration volume was used for the measurements.
After the electrical wires were connected to the MFCs and the pattern was applied on the laminate surface, the structure was fixed at its center on the test fixture (see Figure 8). With the
laminate-MFC structure in configuration I, voltage was applied. The application of voltage was controlled by the software Dasylab. Voltage was increased from 0 to 1500V by 30V increments every five
seconds. Pictures of the laminate were taken at zero voltage and after each increment by the ARAMIS system.
5.3. Results
After the experiment, the pictures were processed to calculate the 3D coordinates of facets on the plate surface using photogrammetric principles. Based on these 3D coordinates, the 3D displacement
and the shape of the plate were calculated for different voltages. The results can then be displayed as graphics of the out-of-plane displacement that can be compared directly to the finite element
Figure 9 shows the out-of-plane displacements obtained for the laminate at zero voltage both experimentally (a) and numerically (b). As can be seen, the two shapes are very similar. The maximum
displacements are 10.24mm and 11.17mm, respectively, for the experimental and theoretical shapes. As can be noticed in the figure, measurement was not possible at a few places on the laminate
surface. Most of them correspond to the location of the electrical wires.
Figure 10 shows the laminate at different applied voltages. As expected, the laminate curvature and out-of-plane displacements decrease with applied voltage. At 1500V, the laminate is quite flat
except at the edges where the displacements are very small with a magnitude around 2.5mm and opposite in sign. It is very similar to the shape obtained by Figure 5(b) just before snap-through
occurs. However, the change of shape could not be obtained experimentally. It is probably due to manufacturing defects.
Figure 11 compares the predicted and measured out-of-plane displacements at the corners. The measured out-of-plane displacement is an average of the values at the four corners. As can be observed,
the comparison is quite good up to 1180V. The values and slopes are close. After 1180V, the finite element analysis predicts that snap-through occurs and the displacement jumps suddenly to 9mm.
However, experimentally, the laminate does not show this behavior. At that voltage, the out-of-plane displacement is larger and continues decreasing until voltage reaches the maximum operational
value of 1500V. At this voltage, the out-of-plane displacement at the corners has a really small value (−1.8mm) and snap-through should have been imminent.
6. Concluding Remarks
This study investigated the use of MFC strips distributed over the top and bottom surfaces of a cross-ply unsymmetric [0/90][T] laminate to obtain a reversible bistable structure. The MFC strips were
bonded and cocured with the laminate. Finite element analyses and experimental work have been performed.
The use of distributed MFC strips allows the laminate to have a smooth cylindrical shape. Of course, since the MFC strips add stiffness to the structure, the laminate curvature is smaller than
without MFC strips. The finite element analysis predicts that the laminate-MFC structure exhibits two cylindrical shapes and that snap-though can be induced by applying 1180V to the MFC actuators.
Reversible snap-though can of course be obtained. However, the laminate-MFC structure that was manufactured exhibits only one stable cylindrical shape. The other cylindrical shape exists but the
structure does not stay in this configuration. It is though that manufacturing defects such as excess glue, uneven MFC spacing, and uneven ply thickness are critical enough to lead to the loss of the
bi-stability of the laminate. Further work needs to be conducted to study which parameter affects the most the stability of the second configuration.
The experimental work provided very interesting results since measurements of the displacement field were taken and could be compared to the predictions. Good correlations for the shape and
out-of-plane displacement were obtained. The finite element model is able to capture quite well the behavior of the laminate-MFC structure subjected to applied voltage and temperature change.
In light of this study and past works results [5, 8, 9] it is felt that MFC or piezoceramic actuators may not be the best choice to induce snap-through in unsymmetric laminates. They add stiffness to
the laminate which induces a decrease in curvature, require high values of voltage, and can lead to the loss of bi-stability of the laminate. Future works should focus on investigating the use of
other types of actuators that would be less intrusive and more efficient. Shape memory alloys as used in [7] may be finally a more promising alternative since they can develop large actuation force
with small voltage, can be attached to the laminate without altering the shape or the bi-stability.
The authors would like to thank the National Sciences and Engineering Research Council of Canada (NSERC) for its financial support. | {"url":"http://www.hindawi.com/journals/smr/2012/695475/","timestamp":"2014-04-17T21:38:39Z","content_type":null,"content_length":"113221","record_id":"<urn:uuid:30d4fd7d-8328-47c0-b658-e65d670a85f9>","cc-path":"CC-MAIN-2014-15/segments/1397609532128.44/warc/CC-MAIN-20140416005212-00497-ip-10-147-4-33.ec2.internal.warc.gz"} |
help on simple accumulator
January 14th, 2010, 02:54 PM #1
Junior Member
Join Date
Jan 2010
My Mood
Thanked 0 Times in 0 Posts
Hello everyone, I am new here and I am looking for help on a small assignment I have. It is a simple calculator which performs basic arithmetic operations (+ - / * power and square root) and I am
having trouble with making so that once a user has performed a calculation, the calculator will continue and use the result of the last equation as the new first operand and prompt the user for a
new operator and second operand. This will continue in a loop until the user enters 'q' as an operator to quit. As well I need to make it so when the user enters 'c' the accumulator will clear
and the calculator resets asking the user to enter the first number.
I've got everything else in my code I just cant wrap my head around how to do those two things.
package simlpecalculator;
* FileName: simpleCalculator.java
* @author XXXXXXX
* Date: January 13, 2010
* Description: A simple calculator program used to perform basic arithmetic operations
import java.util.Scanner;
public class Main {
* @param args the command line arguments
public static void main(String[] args) {
// variables
double frstNum = 0, scndNum = 0, ans = 0;
String operator = " ";
// create input object
Scanner keyboard = new Scanner(System.in);
while (!operator.equals("q")){ // loop to continue application process until user quits.
// obtain the first number and operator
System.out.println("Enter the first number: ");
frstNum = keyboard.nextDouble();
System.out.println(); // white space
System.out.println("Enter an operator: ");
operator = keyboard.next();
System.out.println(); // white space
// quit the application if operator entered is 'q'
// if not continue application processes
if (operator.equals("q")){
System.out.println("End of program");
if (operator.equals("s")){ // check to see if user desires square root
ans = Math.sqrt(frstNum); // take the square root of the first number
else{ // if user does not need square root, prompt for second number
System.out.println("Enter the second number: ");
scndNum = keyboard.nextDouble();
System.out.println(); // white space
if (operator.equals("+")){ // perform addition
ans = frstNum + scndNum;
else if (operator.equals("-")){ // perform subtraction
ans = frstNum - scndNum;
else if (operator.equals("*")){ // perform multiplication
ans = frstNum * scndNum;
else if (operator.equals("/")){ // perform division
ans = frstNum / scndNum;
else if (operator.equals("p")){ // take the power of a number
ans = Math.pow(frstNum, scndNum);
System.out.println("Your result is: " + ans);
Last edited by Khalon; January 14th, 2010 at 10:48 PM.
You need to re-assign firstNum with ans at the end of your while loop. Implementation of 'c' can be done at the end by asking the user for a new firstNum and restarting your while-loop.
The Following User Says Thank You to helloworld922 For This Useful Post:
Khalon (January 14th, 2010)
January 14th, 2010, 08:06 PM #2
Super Moderator
Join Date
Jun 2009
Thanked 619 Times in 561 Posts
Blog Entries
January 14th, 2010, 10:39 PM #3
Junior Member
Join Date
Jan 2010
My Mood
Thanked 0 Times in 0 Posts | {"url":"http://www.javaprogrammingforums.com/loops-control-statements/3039-help-simple-accumulator.html","timestamp":"2014-04-17T10:27:26Z","content_type":null,"content_length":"55522","record_id":"<urn:uuid:8590f0d0-7e39-494f-8570-9b9af3a17f28>","cc-path":"CC-MAIN-2014-15/segments/1397609537864.21/warc/CC-MAIN-20140416005217-00082-ip-10-147-4-33.ec2.internal.warc.gz"} |
Question and Answer thread
The generic term is fixed point. Suppose that X is some set, it matters not at all what sort of set. Suppose f is a function such that for each x in the set X, f(x) is also some point of X. In
shorthand f:X->X. Then: If f(x)=x then x is called a fixed point for the function f.
For whatever interest it may hold, here is one of the more famous fixed point theorems (called the Brouwer Fixed Point Theorem although the pedigree is maybe more complicated): If X is any closed
ball of any (finite) dimension and f: X->X is continuous then f has a fixed point. "Closed" means that the boundary points of X are included in X.
A closed 1-dimensional "ball" is just a closed interval, in 2 dimensions it's a disk, in 3 dimensions it is what you really think of as a ball, and mathematically you can have any number of
As for usefulness, it can be in the eye of the beholder. If you take any one theorem from mathematics, by itself it is apt to have little use. As a collective body of knowledge and as a way of
dealing with very practical problems, it would be an error to dismiss even the most theoretical mathematics. For a well-known example, the success of the Google search engine was based in part on
theorems in linear algebra. See
Similarly, the Brouwer Fixed Point Theorem, and a flock of other fixed point theorems, have been very useful in the right hands, applied to the right problems. Think Nash Equilibrium, see | {"url":"http://www.bridgebase.com/forums/topic/44668-question-and-answer-thread/page__st__40__p__618915","timestamp":"2014-04-17T04:04:10Z","content_type":null,"content_length":"193590","record_id":"<urn:uuid:6ce9abf8-c551-4613-bf7e-e19f4f479c0f>","cc-path":"CC-MAIN-2014-15/segments/1397609526252.40/warc/CC-MAIN-20140416005206-00361-ip-10-147-4-33.ec2.internal.warc.gz"} |
[R] Help on simple problem with optim
Zhang,Yanwei Yanwei.Zhang at cna.com
Thu Sep 9 20:54:02 CEST 2010
Dear all,
I ran into problems with the function "optim" when I tried to do an mle estimation of a simple lognormal regression. Some warning message poped up saying NANs have been produced in the optimization process. But I could not figure out which part of my code has caused this. I wonder if anybody would help. The code is in the following and the data is in the attachment.
da <- read.table("da.txt",header=TRUE)
# fit with linear regression using log transformation of the response variable
fit <- lm(log(yp) ~ as.factor(ay)+as.factor(lag),data=da)
# define the log likelihood to be maximized over
llk.mar <- function(parm,y,x){
# parm is the vector of parameters
# the last element is sigma
# y is the response
# x is the design matrix
l <- length(parm)
beta <- parm[-l]
sigma <- parm[l]
x <- as.matrix(x)
mu <- x %*% beta
llk <- sum(dnorm(y, mu, sigma,log=TRUE))
# initial values
parm <- c(as.vector(coef(fit)),summary(fit)$sigma)
y <- log(da$yp)
x <- model.matrix(fit)
op <- optim(parm, llk.mar, y=y,x=x,control=list(fnscale=-1,maxit=100000))
After running the above code, I got the warning message:
Warning messages:
1: In dnorm(x, mean, sd, log) : NaNs produced
2: In dnorm(x, mean, sd, log) : NaNs produced
I would really appreciate if anybody would help to point out the problem with this code or tell me how to trace it down (using "trace"?)?
Many thanks in advance.
Wayne (Yanwei) Zhang
Statistical Research
NOTICE: This e-mail message, including any attachments and appended messages, is for the sole use of the intended recipients and may contain confidential and legally privileged information.
If you are not the intended recipient, any review, dissemination, distribution, copying, storage or other use of all or any portion of this message is strictly prohibited.
If you received this message in error, please immediately notify the sender by reply e-mail and delete this message in its entirety.
-------------- next part --------------
An embedded and charset-unspecified text was scrubbed...
Name: da.txt
URL: <https://stat.ethz.ch/pipermail/r-help/attachments/20100909/9427d224/attachment.txt>
More information about the R-help mailing list | {"url":"https://stat.ethz.ch/pipermail/r-help/2010-September/252178.html","timestamp":"2014-04-18T20:50:09Z","content_type":null,"content_length":"5183","record_id":"<urn:uuid:9a8b1c75-5f5f-4110-8221-c870530ff5c4>","cc-path":"CC-MAIN-2014-15/segments/1398223205375.6/warc/CC-MAIN-20140423032005-00280-ip-10-147-4-33.ec2.internal.warc.gz"} |
The Open Group Base Specifications Issue 6
IEEE Std 1003.1, 2004 Edition
Copyright © 2001-2004 The IEEE and The Open Group, All Rights reserved.
hypot, hypotf, hypotl - Euclidean distance function
#include <math.h>
double hypot(double x, double y);
float hypotf(float x, float y);
long double hypotl(long double x, long double y);
These functions shall compute the value of the square root of x^2+ y^2 without undue overflow or underflow.
An application wishing to check for error situations should set errno to zero and call feclearexcept(FE_ALL_EXCEPT) before calling these functions. On return, if errno is non-zero or fetestexcept
(FE_INVALID | FE_DIVBYZERO | FE_OVERFLOW | FE_UNDERFLOW) is non-zero, an error has occurred.
Upon successful completion, these functions shall return the length of the hypotenuse of a right-angled triangle with sides of length x and y.
If the correct value would cause overflow, a range error shall occur and hypot(), hypotf(), and hypotl() shall return the value of the macro HUGE_VAL, HUGE_VALF, and HUGE_VALL, respectively.
^[MX] x or y is ±Inf, +Inf shall be returned (even if one of x or y is NaN).
If x or y is NaN, and the other is not ±Inf, a NaN shall be returned.
If both arguments are subnormal and the correct result is subnormal, a range error may occur and the correct result is returned.
These functions shall fail if:
Range Error
The result overflows.
If the integer expression (math_errhandling & MATH_ERRNO) is non-zero, then errno shall be set to [ERANGE]. If the integer expression (math_errhandling & MATH_ERREXCEPT) is non-zero, then the
overflow floating-point exception shall be raised.
These functions may fail if:
Range Error
If the integer expression (math_errhandling & MATH_ERRNO) is non-zero, then errno shall be set to [ERANGE]. If the integer expression (math_errhandling & MATH_ERREXCEPT) is non-zero, then the
underflow floating-point exception shall be raised.
The following sections are informative.
See the EXAMPLES section in atan2().
hypot(x,y), hypot(y,x), and hypot(x, -y) are equivalent.
hypot(x, ±0) is equivalent to fabs(x).
Underflow only happens when both x and y are subnormal and the (inexact) result is also subnormal.
These functions take precautions against overflow during intermediate steps of the computation.
On error, the expressions (math_errhandling & MATH_ERRNO) and (math_errhandling & MATH_ERREXCEPT) are independent of each other, but at least one of them must be non-zero.
atan2(), feclearexcept(), fetestexcept(), isnan(), sqrt(), the Base Definitions volume of IEEE Std 1003.1-2001, Section 4.18, Treatment of Error Conditions for Mathematical Functions, <math.h>
First released in Issue 1. Derived from Issue 1 of the SVID.
Issue 5
The DESCRIPTION is updated to indicate how an application should check for an error. This text was previously published in the APPLICATION USAGE section.
Issue 6
The hypot() function is no longer marked as an extension.
The hypotf() and hypotl() functions are added for alignment with the ISO/IEC 9899:1999 standard.
The DESCRIPTION, RETURN VALUE, ERRORS, and APPLICATION USAGE sections are revised to align with the ISO/IEC 9899:1999 standard.
IEC 60559:1989 standard floating-point extensions over the ISO/IEC 9899:1999 standard are marked.
IEEE Std 1003.1-2001/Cor 2-2004, item XSH/TC2/D6/49 is applied, updating the EXAMPLES section.
End of informative text.
UNIX ® is a registered Trademark of The Open Group.
POSIX ® is a registered Trademark of The IEEE.
[ Main Index | XBD | XCU | XSH | XRAT ] | {"url":"http://pubs.opengroup.org/onlinepubs/009695399/functions/hypotl.html","timestamp":"2014-04-16T04:33:38Z","content_type":null,"content_length":"7954","record_id":"<urn:uuid:8f3c2d20-7b66-4e5c-8965-e9aa600c0868>","cc-path":"CC-MAIN-2014-15/segments/1398223211700.16/warc/CC-MAIN-20140423032011-00093-ip-10-147-4-33.ec2.internal.warc.gz"} |
Voorhees Kirkwood, NJ Math Tutor
Find a Voorhees Kirkwood, NJ Math Tutor
Born in Atlantic City to immigrant parents, I was the first generation to attend university in my family, and was able to attend Princeton University. I now work as a college admissions
consultant for a university prep firm and volunteer as a mentor to youth in Camden. After graduating Princeton I...
36 Subjects: including algebra 2, SAT math, prealgebra, geometry
...I believe strongly in helping students to conceptualize numbers, and not just understand the processes of various operations, but also the concepts. I often incorporate manipulatives into my
lessons or show children how to do addition, subtraction, etc. using a number line or other visual strate...
32 Subjects: including algebra 1, SAT math, English, ACT Math
...I am comfortable tutoring students from the 7th grade forward. Additionally, I was a leader and editor of my high school and college newspapers and I am skilled in editing written material and
helping to develop stronger writing and reading skills. As of this moment, I privately tutor one stude...
33 Subjects: including precalculus, philosophy, Adobe InDesign, art history
...Student success determined how many sessions were needed, and student feedback was an integral part of the program. An important part of the development and implementation of the program was
not only ensuring that students' needs were met, but that relationships were built between the students a...
19 Subjects: including algebra 2, ACT Math, prealgebra, geometry
...Generally, there are two kinds of computers available in the general public market. You can either buy a computer that is made specifically to use Windows/Microsoft technology or Macintosh/
Apple technology. Computers that use Windows technology are referred to as PCs (a term that I personally d...
30 Subjects: including calculus, ACT Math, TOEFL, GED
Related Voorhees Kirkwood, NJ Tutors
Voorhees Kirkwood, NJ Accounting Tutors
Voorhees Kirkwood, NJ ACT Tutors
Voorhees Kirkwood, NJ Algebra Tutors
Voorhees Kirkwood, NJ Algebra 2 Tutors
Voorhees Kirkwood, NJ Calculus Tutors
Voorhees Kirkwood, NJ Geometry Tutors
Voorhees Kirkwood, NJ Math Tutors
Voorhees Kirkwood, NJ Prealgebra Tutors
Voorhees Kirkwood, NJ Precalculus Tutors
Voorhees Kirkwood, NJ SAT Tutors
Voorhees Kirkwood, NJ SAT Math Tutors
Voorhees Kirkwood, NJ Science Tutors
Voorhees Kirkwood, NJ Statistics Tutors
Voorhees Kirkwood, NJ Trigonometry Tutors
Nearby Cities With Math Tutor
Collingswood Math Tutors
Deptford Township, NJ Math Tutors
Echelon, NJ Math Tutors
Evesham Twp, NJ Math Tutors
Gibbsboro Math Tutors
Haddonfield Math Tutors
Hi Nella, NJ Math Tutors
Laurel Springs, NJ Math Tutors
Lindenwold, NJ Math Tutors
Mount Laurel Math Tutors
Pine Hill, NJ Math Tutors
Somerdale, NJ Math Tutors
Stratford, NJ Math Tutors
Voorhees Math Tutors
Voorhees Township, NJ Math Tutors | {"url":"http://www.purplemath.com/Voorhees_Kirkwood_NJ_Math_tutors.php","timestamp":"2014-04-18T14:14:37Z","content_type":null,"content_length":"24399","record_id":"<urn:uuid:834bbfdf-3fef-445c-96d3-41054874d702>","cc-path":"CC-MAIN-2014-15/segments/1397609533689.29/warc/CC-MAIN-20140416005213-00262-ip-10-147-4-33.ec2.internal.warc.gz"} |
Figure 5
Resolution: standard / high
Figure 5.
Correlation of observed with actual fold changes for a representative expression summary dataset (Additional data file 2, using dataset 9e.b). (a) The fold change for each probe set with spiked-in
target RNA is depicted as a cross. Empty probe sets are not shown. For each actual fold-change level (on the x axis), a boxplot shows the distribution of the corresponding observed fold changes. A
linear fit of the data is shown in cyan. Fit parameters: R^2 = 0.508; slope = 0.505; y-intercept = -0.061. (b-d) Increasingly more of the low-intensity probe sets are filtered out of the plot. All
probe sets are ranked according to average signal level, and those in the lowest 25th (b), 50th (c), or 75th (d) percentile of signal level are eliminated from (a). Fit parameters: (b) R^2 = 0.870;
slope = 0.546; y-intercept = -0.008; (c) R^2 = 0.895; slope = 0.517; y-intercept = -0.015; (d) R^2 = 0.906; slope = 0.457; y-intercept = -0.017.
Choe et al. Genome Biology 2005 6:R16 doi:10.1186/gb-2005-6-2-r16
Download authors' original image | {"url":"http://genomebiology.com/2005/6/2/R16/figure/F5","timestamp":"2014-04-20T09:00:24Z","content_type":null,"content_length":"12392","record_id":"<urn:uuid:42aa922d-ca43-43bb-97ff-693829d12014>","cc-path":"CC-MAIN-2014-15/segments/1397609538110.1/warc/CC-MAIN-20140416005218-00031-ip-10-147-4-33.ec2.internal.warc.gz"} |
Quadratic Formula and Equations Summary
This section contains 1,498 words
(approx. 5 pages at 300 words per page)
Quadratic Formula and Equations
A quadratic equation is an equation of the second degree, meaning that for an equation in x, the greatest exponent on x is 2. Quadratics most commonly refer to vertically oriented parabolas—that is,
parabolas that open upward or downward. The graph of a vertically oriented parabola has the shape of a rounded "v," and the bottom-most (or top-most) point is called the vertex.
The equation for a parabola is usually written in either standard or vertex form; however, the standard form is more commonly used to solve for the x-intercepts, or roots. The standard form is y = ax
^2+ bx + c for any real numbers a, b, c where a ≠ 0. The vertex form is y - k = a(x - b)^2 with vertex (b, k) and where a ≠ 0.
Because x-intercepts are the points at which the...
This section contains 1,498 words
(approx. 5 pages at 300 words per page) | {"url":"http://www.bookrags.com/research/quadratic-formula-and-equations-mmat-03/","timestamp":"2014-04-16T19:50:48Z","content_type":null,"content_length":"33503","record_id":"<urn:uuid:160aa1eb-3e94-4128-bf89-1e016a2e1927>","cc-path":"CC-MAIN-2014-15/segments/1397609524644.38/warc/CC-MAIN-20140416005204-00054-ip-10-147-4-33.ec2.internal.warc.gz"} |
2007.106: Complex cobordism classes of homogeneous spaces
2007.106: Victor M Buchstaber and Svjetlana Terzić (2007) Complex cobordism classes of homogeneous spaces.
This is the latest version of this eprint.
Full text available as:
PDF - Requires a PDF viewer such as GSview, Xpdf or Adobe Acrobat Reader
262 Kb
We consider compact homogeneous spaces $G/H$ of positive Euler characteristic endowed with an invariant almost complex structure $J$ and the canonical action $\theta$ of the maximal torus $T ^{k}$ on
$G/H$. We obtain explicit formula for the cobordism class of such manifold through the weights of the action $\theta$ at the identity fixed point $eH$ by an action of the quotient group $W_G/W_H$ of
the Weyl groups for $G$ and $H$. In this way we show that the cobordism class for such manifolds can be computed explicitly without information on their cohomology. We also show that formula for
cobordism class provides an explicit way for computing the classical Chern numbers for $(G/H, J)$. As a consequence we obtain that the Chern numbers for $(G/H, J)$ can be computed without information
on cohomology for $G/H$. As an application we provide an explicit formula for cobordism classes and characteristic numbers of the flag manifolds $U(n)/T^n$, Grassmann manifolds $G_{n,k}=U(n)/(U(k)\
times U(n-k))$ and some particular interesting examples. This paper is going to have continuation in which will be considered the stable complex structures equivariant under given torus action on
homogeneous spaces of positive Euler characteristic.
Available Versions of this Item
• Complex cobordism classes of homogeneous spaces (deposited 05 September 2007) [Currently Displayed]
Download Statistics: last 4 weeks
Repository Staff Only: edit this item | {"url":"http://eprints.ma.man.ac.uk/843/","timestamp":"2014-04-18T15:57:16Z","content_type":null,"content_length":"9811","record_id":"<urn:uuid:64917470-0b1f-4f57-bed8-fd0425a5da30>","cc-path":"CC-MAIN-2014-15/segments/1398223210034.18/warc/CC-MAIN-20140423032010-00183-ip-10-147-4-33.ec2.internal.warc.gz"} |
[SciPy-User] Hcluster Negative Distance Error?
disappearedng disappearedng@gmail....
Tue Apr 6 15:04:28 CDT 2010
Dear Everyone,
I have a input file which are all floating point numbers to 4 decimal place.
i.e. 13359 0.0000 0.0000 0.0001 0.0001 0.0002 0.0003
0.0007 ... (the first is the id).
My class uses the loadVectorsFromFile method which multiplies it by 10000
and then int() these numbers. On top of that, I also loop through each
vector to ensure that there are no
negative values inside. However, when I perform _hclustering, I am
continually seeing the error, "Linkage `Z` contains negative values".
I seriously think this is a bug because: 1) I checked my values, 2) the
values are no where small enough or big enough to approach the limits of the
floating point numbers and 3) the
formula that I used to derive the values in the file uses absolute value (my
input is DEFINITELY right).
Can someone enligten me as to why I am seeing this weird error? What is
going on that is causing this negative distance error?
def loadVectorsFromFile(self, limit, loc, assertAllPositive=True,
"""Inflate to prevent "negative" distance, we use 4 decimal points,
so *10000
vectors = {}
self.winfo("Each vector is set to have %d limit in length" % limit)
with open( loc ) as inf:
for line in filter(None, inf.read().split('\n')):
l = line.split('\t')
if limit:
scores = map(float, l[1:limit+1])
scores = map(float, l[1:])
if inflate:
vectors[ l[0]] = map( lambda x: int(x*10000),
scores) #int might save space
vectors[ l[0]] = scores
if assertAllPositive:
#Assert that it has no negative value
for dirID, l in vectors.iteritems():
if reduce(operator.or_, map( lambda x: x < 0, l)):
self.werror( "Vector %s has negative values!" % dirID)
return vectors
def main( self, inputDir, outputDir, limit=0,
Loads vector from a file and start clustering
vectors is { featureID: tfidfVector (list), }
IDFeatureDic = loadIdFeatureGroupDicFromIntermediate(
pjoin(self.configDir, mappingFname))
if not os.path.exists(outputDir):
vectors = self.loadVectorsFromFile( limit, pjoin( inputDir,
for threshold in map( lambda x:float(x)/30, range(20,30)):
clusters = self._hclustering(threshold, vectors)
if clusters:
outputLoc = pjoin(outputDir, "threshold.%s.result" %
with open(outputLoc, 'w') as outf:
for clusterNo, cluster in clusters.iteritems():
outf.write('%s\n' % str(clusterNo))
for featureID in cluster:
feature, group = IDFeatureDic[featureID]
outline = "%s\t%s\n" % (feature, group)
def _hclustering(self, threshold, vectors):
"""function which you should call to vary the threshold
vectors: { featureID: [ tfidf scores, tfidf score, .. ]
clusters = defaultdict(list)
if len(vectors) > 1:
results = hierarchy.fclusterdata( vectors.values(),
threshold, metric='cosine')
except ValueError, e:
self.werror("_hclustering: %s" % str(e))
return False
for i, featureID in enumerate( vectors.keys()):
clusters[results[i]].append( featureID)
return clusters
return False
-------------- next part --------------
An HTML attachment was scrubbed...
URL: http://mail.scipy.org/pipermail/scipy-user/attachments/20100407/625a2b2e/attachment.html
More information about the SciPy-User mailing list | {"url":"http://mail.scipy.org/pipermail/scipy-user/2010-April/024949.html","timestamp":"2014-04-16T13:35:13Z","content_type":null,"content_length":"6841","record_id":"<urn:uuid:66e74072-1269-4e72-aeae-82f103e91418>","cc-path":"CC-MAIN-2014-15/segments/1397609523429.20/warc/CC-MAIN-20140416005203-00556-ip-10-147-4-33.ec2.internal.warc.gz"} |
An algorithm for fast recovery of sparse causal graphs
Results 1 - 10 of 62
- Machine Learning , 1992
"... Abstract. This paper presents a Bayesian method for constructing probabilistic networks from databases. In particular, we focus on constructing Bayesian belief networks. Potential applications
include computer-assisted hypothesis testing, automated scientific discovery, and automated construction of ..."
Cited by 1081 (27 self)
Add to MetaCart
Abstract. This paper presents a Bayesian method for constructing probabilistic networks from databases. In particular, we focus on constructing Bayesian belief networks. Potential applications
include computer-assisted hypothesis testing, automated scientific discovery, and automated construction of probabilistic expert systems. We extend the basic method to handle missing data and hidden
(latent) variables. We show how to perform probabilistic inference by averaging over the inferences of multiple belief networks. Results are presented of a preliminary evaluation of an algorithm for
constructing a belief network from a database of cases. Finally, we relate the methods in this paper to previous work, and we discuss open problems.
, 1991
"... This paper concerns the empirical basis of causation, and addresses the following issues: 1. the clues that might prompt people to perceive causal relationships in uncontrolled observations. 2.
the task of inferring causal models from these clues, and 3. whether the models inferred tell us anything ..."
Cited by 208 (34 self)
Add to MetaCart
This paper concerns the empirical basis of causation, and addresses the following issues: 1. the clues that might prompt people to perceive causal relationships in uncontrolled observations. 2. the
task of inferring causal models from these clues, and 3. whether the models inferred tell us anything useful about the causal mechanisms that underly the observations. We propose a minimal-model
semantics of causation, and show that, contrary to common folklore, genuine causal influences can be distinguished from spurious covariations following standard norms of inductive reasoning. We also
establish a sound characterization of the conditions under which such a distinction is possible. We provide an effective algorithm for inferred causation and show that, for a large class of data the
algorithm can uncover the direction of causal influences as defined above. Finally, we address the issue of non-temporal causation. 1 Introduction The study of causation is central to the
understanding of hum...
, 1991
"... Theory refinement is the task of updating a domain theory in the light of new cases, to be done automatically or with some expert assistance. The problem of theory refinement under uncertainty
is reviewed here in the context of Bayesian statistics, a theory of belief revision. The problem is reduced ..."
Cited by 184 (5 self)
Add to MetaCart
Theory refinement is the task of updating a domain theory in the light of new cases, to be done automatically or with some expert assistance. The problem of theory refinement under uncertainty is
reviewed here in the context of Bayesian statistics, a theory of belief revision. The problem is reduced to an incremental learning task as follows: the learning system is initially primed with a
partial theory supplied by a domain expert, and thereafter maintains its own internal representation of alternative theories which is able to be interrogated by the domain expert and able to be
incrementally refined from data. Algorithms for refinement of Bayesian networks are presented to illustrate what is meant by "partial theory", "alternative theory representation ", etc. The
algorithms are an incremental variant of batch learning algorithms from the literature so can work well in batch and incremental mode. 1 Introduction Theory refinement is the task of updating a
domain theory in the light of...
, 1996
"... This literature review discusses different methods under the general rubric of learning Bayesian networks from data, and includes some overlapping work on more general probabilistic networks.
Connections are drawn between the statistical, neural network, and uncertainty communities, and between the ..."
Cited by 172 (0 self)
Add to MetaCart
This literature review discusses different methods under the general rubric of learning Bayesian networks from data, and includes some overlapping work on more general probabilistic networks.
Connections are drawn between the statistical, neural network, and uncertainty communities, and between the different methodological communities, such as Bayesian, description length, and classical
statistics. Basic concepts for learning and Bayesian networks are introduced and methods are then reviewed. Methods are discussed for learning parameters of a probabilistic network, for learning the
structure, and for learning hidden variables. The presentation avoids formal definitions and theorems, as these are plentiful in the literature, and instead illustrates key concepts with simplified
examples. Keywords--- Bayesian networks, graphical models, hidden variables, learning, learning structure, probabilistic networks, knowledge discovery. I. Introduction Probabilistic networks or
probabilistic gra...
"... This paper provides algorithms that use an information-theoretic analysis to learn Bayesian network structures from data. Based on our three-phase learning framework, we develop efficient
algorithms that can effectively learn Bayesian networks, requiring only polynomial numbers of conditional indepe ..."
Cited by 93 (5 self)
Add to MetaCart
This paper provides algorithms that use an information-theoretic analysis to learn Bayesian network structures from data. Based on our three-phase learning framework, we develop efficient algorithms
that can effectively learn Bayesian networks, requiring only polynomial numbers of conditional independence (CI) tests in typical cases. We provide precise conditions that specify when these
algorithms are guaranteed to be correct as well as empirical evidence (from real world applications and simulation tests) that demonstrates that these systems work efficiently and reliably in
- Data Mining and Knowledge Discovery , 1998
"... Mining for association rules in market basket data has proved a fruitful area of research. Measures such as conditional probability (confidence) and correlation have been used to infer rules of
the form "the existence of item A implies the existence of item B." However, such rules indicate only a st ..."
Cited by 88 (1 self)
Add to MetaCart
Mining for association rules in market basket data has proved a fruitful area of research. Measures such as conditional probability (confidence) and correlation have been used to infer rules of the
form "the existence of item A implies the existence of item B." However, such rules indicate only a statistical relationship between A and B. They do not specify the nature of the relationship:
whether the presence of A causes the presence of B, or the converse, or some other attribute or phenomenon causes both to appear together. In applications, knowing such causal relationships is
extremely useful for enhancing understanding and effecting change. While distinguishing causality from correlation is a truly difficult problem, recent work in statistics and Bayesian learning
provide some avenues of attack. In these fields, the goal has generally been to learn complete causal models, which are essentially impossible to learn in large-scale data mining applications with a
large number of variab...
, 1995
"... Previous algorithms for the recovery of Bayesian belief network structures from data have been either highly dependent on conditional independence (CI) tests, or have required on ordering on the
nodes to be supplied by the user. We present an algorithm that integrates these two approaches: CI tests ..."
Cited by 77 (8 self)
Add to MetaCart
Previous algorithms for the recovery of Bayesian belief network structures from data have been either highly dependent on conditional independence (CI) tests, or have required on ordering on the
nodes to be supplied by the user. We present an algorithm that integrates these two approaches: CI tests are used to generate an ordering on the nodes from the database, which is then used to recover
the underlying Bayesian network structure using a non-Cl-test-based method. Results of the evaluation of the algorithm on a number of databases (e.g., ALARM, LED, and SOYBEAN) are presented. We also
discuss some algorithm performance issues and open problems.
- In Proceedings of the Sixth ACM International Conference on Information and Knowledge Management
"... This paper presents an efficient algorithm for learning Bayesian belief networks from databases. The algorithm takes a database as input and constructs the belief network structure as output.
The construction process is based on the computation of mutual information of attribute pairs. Given a data ..."
Cited by 65 (7 self)
Add to MetaCart
This paper presents an efficient algorithm for learning Bayesian belief networks from databases. The algorithm takes a database as input and constructs the belief network structure as output. The
construction process is based on the computation of mutual information of attribute pairs. Given a data set that is large enough, this algorithm can generate a belief network very close to the
underlying model, and at the same time, enjoys the time complexity of O N ( ) 4 on conditional independence (CI) tests. When the data set has a normal DAG-Faithful (see Section 3.2) probability
distribution, the algorithm guarantees that the structure of a perfect map [Pearl, 1988] of the underlying dependency model is generated. To evaluate this algorithm, we present the experimental
results on three versions of the wellknown ALARM network database, which has 37 attributes and 10,000 records. The results show that this algorithm is accurate and efficient. The proof of correctness
and the analysis of c...
- Proc. of the Eighth Conference on Uncertainty in Artificial Intelligence , 1992
"... In a previous paper [8] we presented an algorithm for extracting causal influences from independence information, where a causal influence was defined as the existence of a directed arc in all
minimal causal models consistent with the data. In this paper we address the question of deciding whether t ..."
Cited by 60 (1 self)
Add to MetaCart
In a previous paper [8] we presented an algorithm for extracting causal influences from independence information, where a causal influence was defined as the existence of a directed arc in all
minimal causal models consistent with the data. In this paper we address the question of deciding whether there exists a causal model that explains ALL the observed dependencies and independencies.
Formally, given a list M of conditional independence statements, it is required to decide whether there exists a directed acyclic graph D that is perfectly consistent with M, namely, every statement
in M, and no other, is reflected via d-separation in D. We present and analyze an effective algorithm that tests for the existence of such a dag, and produces one, if it exists. Key words: Causal
modeling, graphoids, conditional independence. 1 1 Introduction Directed acyclic graphs (dags) have been widely used for modeling statistical data. Starting with the pioneering work of Sewal Wright
- IN PROCEEDINGS OF AI & STAT’97 , 1997
"... This paper presents an efficient algorithm for constructing Bayesian belief networks from databases. The algorithm takes a database and an attributes ordering (i.e., the causal attributes of an
attribute should appear earlier in the order) as input and constructs a belief network structure as output ..."
Cited by 43 (6 self)
Add to MetaCart
This paper presents an efficient algorithm for constructing Bayesian belief networks from databases. The algorithm takes a database and an attributes ordering (i.e., the causal attributes of an
attribute should appear earlier in the order) as input and constructs a belief network structure as output. The construction process is based on the computation of mutual information of attribute
pairs. Given a data set which is large enough and has a DAGIsomorphic probability distribution, this algorithm guarantees that the perfect map [1] of the underlying dependency model is generated, and
at the same time, enjoys the time complexity of O N ( ) on conditional independence (CI) tests. To evaluate this algorithm, we present the experimental results on three versions of the well-known
ALARM network database, which has 37 attributes and 10,000 records. The correctness proof and the analysis of computational complexity are also presented. We also discuss the features of our work and
relate it to previous works. | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=433607","timestamp":"2014-04-21T16:21:18Z","content_type":null,"content_length":"39367","record_id":"<urn:uuid:bdf6febb-c137-4f4c-8a83-78b659458de8>","cc-path":"CC-MAIN-2014-15/segments/1397609540626.47/warc/CC-MAIN-20140416005220-00096-ip-10-147-4-33.ec2.internal.warc.gz"} |
Oak Island, NY Math Tutor
Find an Oak Island, NY Math Tutor
...Have vast working experience with all working levels of students from those with learning difficulties to honors. I can either come to your home or go to your local public library. I have
taught high school math in a public high school for over 14 years.
7 Subjects: including statistics, algebra 1, algebra 2, geometry
...Head Math Teacher for Algebra 2 at the Bay Shore High School and over 10 years experience teaching and preparing students for Algebra 2 course and Regents Exam. I have been teaching/tutoring
Geometry for a public school district for over 10 years as well as tutoring agencies across Long Island. ...
17 Subjects: including prealgebra, ACT Math, algebra 1, algebra 2
...At the United States Naval Academy, it was required by all students to take one swim class a semester on top of our other PE classes. I know how to swim in all weather, survival swimming, and
all strokes. I have played soccer my whole life.
49 Subjects: including probability, reading, geometry, prealgebra
...I have a long background in Finance, mathematics and statistics, including Fellowship status as a Chartered Certified Accountant. I qualified with KPMG, then working as Audit Manager in the
Caribbean. During my time as a mathematics and test preparation tutor, I have worked extensively with ADD students of all ages.
55 Subjects: including calculus, discrete math, differential equations, career development
...I have taught every elementary grade 1-6. My certifications are in elementary education and technology. I like to reinforce skills taught with hands-on methods.
14 Subjects: including prealgebra, reading, grammar, elementary (k-6th)
Related Oak Island, NY Tutors
Oak Island, NY Accounting Tutors
Oak Island, NY ACT Tutors
Oak Island, NY Algebra Tutors
Oak Island, NY Algebra 2 Tutors
Oak Island, NY Calculus Tutors
Oak Island, NY Geometry Tutors
Oak Island, NY Math Tutors
Oak Island, NY Prealgebra Tutors
Oak Island, NY Precalculus Tutors
Oak Island, NY SAT Tutors
Oak Island, NY SAT Math Tutors
Oak Island, NY Science Tutors
Oak Island, NY Statistics Tutors
Oak Island, NY Trigonometry Tutors
Nearby Cities With Math Tutor
Amity Harbor, NY Math Tutors
Canaan Lake, NY Math Tutors
Captree Island, NY Math Tutors
Fair Harbor, NY Math Tutors
Fire Island, NY Math Tutors
Heer Park, NY Math Tutors
Kismet, NY Math Tutors
Lake Swannanoa, NJ Math Tutors
Marconiville, NY Math Tutors
Oak Beach, NY Math Tutors
Pelican Island, NJ Math Tutors
Point O Woods, NY Math Tutors
Shady Lake, NJ Math Tutors
Silver Lake, NJ Math Tutors
Washington Park, NJ Math Tutors | {"url":"http://www.purplemath.com/oak_island_ny_math_tutors.php","timestamp":"2014-04-18T21:19:47Z","content_type":null,"content_length":"23957","record_id":"<urn:uuid:540055ef-bae8-45b9-ac77-27e453b184da>","cc-path":"CC-MAIN-2014-15/segments/1398223202774.3/warc/CC-MAIN-20140423032002-00278-ip-10-147-4-33.ec2.internal.warc.gz"} |
Calculating Variance
October 6th 2008, 11:59 AM #1
Oct 2008
Calculating Variance
The number of new houses being built depends on the 20 year mortgage rates offered by local banks. Let X be the random variable denoting the average 20 year mortgage rate measured in percentage
points for a specific month. Assume that the expected value for X is 8.75 percentage points with a variance of 0.50. Define the random variable Y to be the number of new houses being built in a
specific month. If Y = 85 - 2X, what is the variance of the random variable Y?
I am confused about how to calculate this?
It seems that there are three ways to calculate this, with the notes given to me.
X=Random Variable c=Constant
I've been thinking of ways to solve this, but i keep on thinking it is only 85 since V(X+c)=V(X)
The number of new houses being built depends on the 20 year mortgage rates offered by local banks. Let X be the random variable denoting the average 20 year mortgage rate measured in percentage
points for a specific month. Assume that the expected value for X is 8.75 percentage points with a variance of 0.50. Define the random variable Y to be the number of new houses being built in a
specific month. If Y = 85 - 2X, what is the variance of the random variable Y?
I am confused about how to calculate this?
It seems that there are three ways to calculate this, with the notes given to me.
X=Random Variable c=Constant
I've been thinking of ways to solve this, but i keep on thinking it is only 85 since V(X+c)=V(X)
Var(aX + b) = a^2 Var(X).
In your case a = -2 and b = 85.
October 6th 2008, 06:35 PM #2 | {"url":"http://mathhelpforum.com/statistics/52274-calculating-variance.html","timestamp":"2014-04-16T08:29:06Z","content_type":null,"content_length":"34642","record_id":"<urn:uuid:b4fb677c-e7a2-4f3a-bc17-4652b4cf266c>","cc-path":"CC-MAIN-2014-15/segments/1397609521558.37/warc/CC-MAIN-20140416005201-00596-ip-10-147-4-33.ec2.internal.warc.gz"} |
Results 11 - 20 of 63
, 1997
"... The theory behind the success of adaptive reweighting and combining algorithms (arcing) such as Adaboost (Freund and Schapire [1995].[1996]) and others in reducing generalization error has not
been well understood. By formulating prediction, both classification and regression, as a game where one pl ..."
Cited by 138 (0 self)
Add to MetaCart
The theory behind the success of adaptive reweighting and combining algorithms (arcing) such as Adaboost (Freund and Schapire [1995].[1996]) and others in reducing generalization error has not been
well understood. By formulating prediction, both classification and regression, as a game where one player makes a selection from instances in the training set and the other a convex linear
combination of predictors from a finite set, existing arcing algorithms are shown to be algorithms for finding good game strategies. An optimal game strategy finds a combined predictor that minimizes
the maximum of the error over the training set. A bound on the generalization error for the combined predictors in terms of their maximum error is proven that is sharper than bounds to date. Arcing
algorithms are described that converge to the optimal strategy. Schapire et.al. [1997] offered an explanation of why Adaboost works in terms of its ability to reduce the margin. Comparing Adaboost to
our optimal ar...
, 2000
"... Much recent attention, both experimental and theoretical, has been focussed on classification algorithms which produce voted combinations of classifiers. Recent theoretical work has shown that
the impressive generalization performance of algorithms like AdaBoost can be attributed to the classifier h ..."
Cited by 115 (2 self)
Add to MetaCart
Much recent attention, both experimental and theoretical, has been focussed on classification algorithms which produce voted combinations of classifiers. Recent theoretical work has shown that the
impressive generalization performance of algorithms like AdaBoost can be attributed to the classifier having large margins on the training data. We present an abstract algorithm for finding linear
combinations of functions that minimize arbitrary cost functionals (i.e functionals that do not necessarily depend on the margin). Many existing voting methods can be shown to be special cases of
this abstract algorithm. Then, following previous theoretical results bounding the generalization performance of convex combinations of classifiers in terms of general cost functions of the margin,
we present a new algorithm (DOOM II) for performing a gradient descent optimization of such cost functions. Experiments on
- In Proceedings of the Fourteenth National Conference on Artificial Intelligence , 1997
"... An ensemble consists of a set of independently trained classi ers (such as neural networks or decision trees) whose predictions are combined when classifying novel instances. Previous research
has shown that an ensemble as a whole is often more accurate than any of the single classiers in the ensemb ..."
Cited by 84 (6 self)
Add to MetaCart
An ensemble consists of a set of independently trained classi ers (such as neural networks or decision trees) whose predictions are combined when classifying novel instances. Previous research has
shown that an ensemble as a whole is often more accurate than any of the single classiers in the ensemble. Bagging (Breiman 1996a) and Boosting (Freund & Schapire 1996) are two relatively new but
popular methods for producing ensembles. In this paper we evaluate these methods using both neural networks and decision trees as our classi cation algorithms. Our results clearly showtwo important
facts. The rst is that even though Bagging almost always produces a better classi er than any of its individual component classi ers and is relatively impervious to over tting, it does not generalize
any better than a baseline neural-network ensemble method. The second is that Boosting is apowerful technique that can usually produce better ensembles than Bagging � however, it is more susceptible
to noise and can quickly over t a data set.
, 1996
"... To Mom, Dad, and Susan, for their support and encouragement. ..."
- Combining Artificial Neural Nets , 1999
"... Several researchers have experimentally shown that substantial improvements can be obtained in difficult pattern recognition problems by combining or integrating the outputs of multiple
classifiers. This chapter provides an analytical framework to quantify the improvements in classification resul ..."
Cited by 65 (7 self)
Add to MetaCart
Several researchers have experimentally shown that substantial improvements can be obtained in difficult pattern recognition problems by combining or integrating the outputs of multiple classifiers.
This chapter provides an analytical framework to quantify the improvements in classification results due to combining. The results apply to both linear combiners and order statistics combiners. We
first show that to a first order approximation, the error rate obtained over and above the Bayes error rate, is directly proportional to the variance of the actual decision boundaries around the
Bayes optimum boundary. Combining classifiers in output space reduces this variance, and hence reduces the "added" error. If N unbiased classifiers are combined by simple averaging, the added error
rate can be reduced by a factor of N if the individual errors in approximating the decision boundaries are uncorrelated. Expressions are then derived for linear combiners which are biased or
correlated, and the effect of output correlations on ensemble performance is quantified. For order statistics based non-linear combiners, we derive expressions that indicate how much the median, the
maximum and in general the ith order statistic can improve classifier performance. The analysis presented here facilitates the understanding of the relationships among error rates, classifier
boundary distributions, and combining in output space. Experimental results on several public domain data sets are provided to illustrate the benefits of combining and to support the analytical
- CONNECTION SCIENCE , 1996
"... A neural-network ensemble is a very successful technique where the outputs of a set of separately trained neural network are combined to form one unified prediction. An effective ensemble should
consist of a set of networks that are not only highly correct, but ones that make their errors on differe ..."
Cited by 57 (6 self)
Add to MetaCart
A neural-network ensemble is a very successful technique where the outputs of a set of separately trained neural network are combined to form one unified prediction. An effective ensemble should
consist of a set of networks that are not only highly correct, but ones that make their errors on different parts of the input space as well; however, most existing techniques only indirectly address
the problem of creating such a set. We present an algorithm called Addemup that uses genetic algorithms to explicitly search for a highly diverse set of accurate trained networks. Addemup works by
first creating an initial population, then uses genetic operators to continually create new networks, keeping the set of networks that are highly accurate while disagreeing with each other as much as
possible. Experiments on four real-world domains show that Addemup is able to generate a set of trained networks that is more accurate than several existing ensemble approaches. Experiments also show
that Ad...
, 1999
"... . Boosting is a general method for improving the accuracy of any given learning algorithm. Focusing primarily on the AdaBoost algorithm, we briefly survey theoretical work on boosting including
analyses of AdaBoost's training error and generalization error, connections between boosting and game theo ..."
Cited by 50 (2 self)
Add to MetaCart
. Boosting is a general method for improving the accuracy of any given learning algorithm. Focusing primarily on the AdaBoost algorithm, we briefly survey theoretical work on boosting including
analyses of AdaBoost's training error and generalization error, connections between boosting and game theory, methods of estimating probabilities using boosting, and extensions of AdaBoost for
multiclass classification problems. Some empirical work and applications are also described. Background Boosting is a general method which attempts to "boost" the accuracy of any given learning
algorithm. Kearns and Valiant [29, 30] were the first to pose the question of whether a "weak" learning algorithm which performs just slightly better than random guessing in Valiant's PAC model [44]
can be "boosted" into an arbitrarily accurate "strong" learning algorithm. Schapire [36] came up with the first provable polynomial-time boosting algorithm in 1989. A year later, Freund [16]
developed a much more effici...
, 1999
"... Much recent attention, both experimental and theoretical, has been focussed on classification algorithms which produce voted combinations of classifiers. Recent theoretical work has shown that
the impressive generalization performance of algorithms like AdaBoost can be attributed to the classifier h ..."
Cited by 48 (2 self)
Add to MetaCart
Much recent attention, both experimental and theoretical, has been focussed on classification algorithms which produce voted combinations of classifiers. Recent theoretical work has shown that the
impressive generalization performance of algorithms like AdaBoost can be attributed to the classifier having large margins on the training data. We present abstract algorithms for finding linear and
convex combinations of functions that minimize arbitrary cost functionals (i.e functionals that do not necessarily depend on the margin). Many existing voting methods can be shown to be special cases
of these abstract algorithms. Then, following previous theoretical results bounding the generalization performance of convex combinations of classifiers in terms of general cost functions of the
margin, we present a new algorithm (DOOM II) for performing a gradient descent optimization of such cost functions. Experiments on several data sets from the UC Irvine repository demonstrate that
DOOM II gener...
- In Proceedings of the Fortieth Annual Symposium on Foundations of Computer Science , 1999
"... This paper connects two fundamental ideas from theoretical computer science: hard-core set construction, a type of hardness amplification from computational complexity, and boosting, a technique
from computational learning theory. Using this connection we give fruitful applications of complexity-the ..."
Cited by 38 (8 self)
Add to MetaCart
This paper connects two fundamental ideas from theoretical computer science: hard-core set construction, a type of hardness amplification from computational complexity, and boosting, a technique from
computational learning theory. Using this connection we give fruitful applications of complexity-theoretic techniques to learning theory and vice versa. We show that the hard-core set construction of
Impagliazzo [15], which establishes the existence of distributions under which boolean functions are highly inapproximable, may be viewed as a boosting algorithm. Using alternate boosting methods we
give an improved bound for hard-core set construction which matches known lower bounds from boosting and thus is optimal within this class of techniques. We then show how to apply techniques from
[15] to give a new version of Jackson’s celebrated Harmonic Sieve algorithm for learning DNF formulae under the uniform distribution using membership queries. Our new version has a significant
asymptotic improvement in running time. Critical to our arguments is a careful analysis of the distributions which are employed in both boosting and hard-core set constructions. | {"url":"http://citeseerx.ist.psu.edu/showciting?cid=279438&sort=cite&start=10","timestamp":"2014-04-18T13:39:03Z","content_type":null,"content_length":"37966","record_id":"<urn:uuid:3945fcd0-08d2-4ef1-9c56-79df6ac5737d>","cc-path":"CC-MAIN-2014-15/segments/1397609533689.29/warc/CC-MAIN-20140416005213-00508-ip-10-147-4-33.ec2.internal.warc.gz"} |
The SFIT function determines a polynomial fit to a surface and returns a fitted array. The function fitted is:
This routine is written in the IDL language. Its source code can be found in the file sfit.pro in the lib subdirectory of the IDL distribution.
Create a grid from zero to six radians in the X and Y directions:
X = (FINDGEN(61)/10) # REPLICATE(1,61)
Evaluate a function at each point:
Compute a sixth-degree polynomial fit to the function data:
Display the original function on the left and the fitted function on the right, using
WINDOW, XSIZE = 800, YSIZE = 400
!P.MULTI = [0, 2, 1] ; Set up side-by-side plots.
!P.BACKGROUND = 255 ; Set background color to white.
!P.COLOR = 0 ; Set plot color to black.
SURFACE, F, X, Y, ZRANGE = [-3, 3], ZSTYLE = 1 | {"url":"http://www.astro.virginia.edu/class/oconnell/astr511/IDLresources/idl_5.1_html/idl197.htm","timestamp":"2014-04-16T10:54:18Z","content_type":null,"content_length":"6220","record_id":"<urn:uuid:e57deafa-ee78-4293-bb14-e88f181794e1>","cc-path":"CC-MAIN-2014-15/segments/1397609523265.25/warc/CC-MAIN-20140416005203-00378-ip-10-147-4-33.ec2.internal.warc.gz"} |
VLOOKUP Formula Excel – How to use Excel VLOOKUP Function
Syntax of VLOOKUP Formula
Example of VLOOKUP Formula
Possible Errors returned by the VLOOKUP Formula
VLOOKUP formula matches a string against the 1st column of a range and returns any cell value from the matched row.
VLOOKUP Formula Syntax
VLOOKUP Formula has three parts:
VLOOKUP (value_to_find, range_to_search_in, column_number_to_return, match_type)
value_to_find is the value that we would like to find. You can specify a string, a number or a cell address as the value you would like to search for. While using a string enclose it within quotes
(Ex. "Apple"). When searching for a cell provide the address of the cell (Ex. B12). Numbers can be entered as such. (Ex. 19)
The range_to_search_in is the range in which we would like to search for the value and return the match. When you specify this range, the VLOOKUP only searches the first column of the range (and NOT
all the columns) for the value which we would like to find.
match_type is specified as either TRUE or FALSE. If this value is specified as FALSE (or false in small-case), the VLOOKUP function will find and exact match. If it does not find a match, it will
return an error value. However if this value is specified as TRUE, then the VLOOKUP function tries to find and exact match but if it does not, the next largest value that is less than lookup_value is
returned. This parameter is optional and if not specified is taken a TRUE by default. (To make this work properly, the values in the column that we want to return from the function have to in an
ascending order – but you can ignore that for now. For most cases, a match_type value of FALSE is all that we will ever need.)
Example of a VLOOKUP Formula
Let’s look at an example of the VLOOKUP formula. Suppose we had a table as shown in the above example. We have a list of products in the first column and their sales values in the second. Suppose you
wanted to know how much were the sales for the product “Software”. We can write the VLOOKUP formula as:
=VLOOKUP(A13,A2:B9,2,FALSE) or
Both of the above formulas would return a value of 10. How does it work ? Let’s look at each part of the formula carefully. The first part of the formula, as we saw earlier, is the value that we
would like to find. In our case, we can either specify the cell (A13) or the string (“Software”) that we would like to find. The second part is the range in which we would like to search which in our
case is A2:B9 (the table in which we have stored the values). The third part is the column that we would like to return a value from in case the value we were trying to find was found in a particular
row. Specifying a value of 1 would return the first column (which was also the column we searched against). A column value of 2 would return the figures from the second column (which is what we want
in our example).
How to enter the VLOOKUP formula in an Excel Sheet
1. Select the cell in which you want to place the formula
2. Type the formula as =VLOOKUP(
3. Move the cursor using the up-down or left-right arrow keys and take it to the cell which contains the value that you are trying to find.
4. Press the comma key (,)
5. Again move the cursor using the up-down or left-right arrow keys and take it to the first cell (top-left) of the range that you would like to search.
6. Now keeping the SHIFT key pressed, move the cursor again and take it to the last cell (bottom-right) of the range that you would like to search.
7. Press the comma key (,) again
8. Type in the column number from which you would like to return the value from. (If you don’t get it at first, simply type in 1).
9. Press the comma key (,) again
10. Type in FALSE and then close the formula bracket by typing in ).
(Check out the clip above for knowing if the values you’ve entered are in the same order. In the end your formula should look something like this =VLOOKUP(“Software”,A2:B9,2,FALSE) )
Possible Errors with the VLOOKUP Formula
The VLOOKUP formula can result in the following error values:
VLOOKUP #N/A Error
The #N/A Error value in VLOOKUP is one of the most frequently occurring error value. This signifies that the value that you are trying to find does not exist in the range in which you are trying to
find it. If you get this error, try going back and check whether the value that you are trying to find exists and is in the first column of the range in which you are trying to find it. Carefully
check the range that you’ve specified. Foe example, if we were to specify =VLOOKUP(“Cake”,A2:B9,2,FALSE) in the above example, it would result in an #N/A error value simply because the value “Cake”
does not exist in the first column of range in which we are trying to find it.
VLOOKUP #REF! Error
#REF! error in VLOOKUP specifies that you have specified a match to be returned from a column that does not exist in the range in which you are trying to find the value. For example if we were to
write the formula as =VLOOKUP(“Software”,A2:B9,33,FALSE), it would give us the #REF! value because we have specified that if there’s is a match, it should return the 33rd column from the range.
However the range that we have specified (A2:B9) has only two columns (A and B) and hence trying to reference the 33rd column results in an error.
VLOOKUP #NAME? Error
#NAME? Error in VLOOKUP can result from wrongly specifying address the range in which to find the value. Say for example you wanted to write =VLOOKUP(“software”,A1:B9,2,FALSE) but instead erroneously
entered =VLOOKUP(“software”,A1:BBB9,2,FALSE) (two extra B’s). Now the cell BBB9 does not exist anywhere in the sheet and as a result the VLOOKUP function throws up the #NAME? error value.
You can know more about hiding errors in a worksheet here.
You can download an example of VLOOKUP formula here or click on the button below: | {"url":"http://www.databison.com/vlookup-formula-excel-how-to-use-excel-vlookup-function/","timestamp":"2014-04-17T01:33:45Z","content_type":null,"content_length":"62422","record_id":"<urn:uuid:42990c56-9ba4-4aa6-9623-209c87ab8f6b>","cc-path":"CC-MAIN-2014-15/segments/1397609532480.36/warc/CC-MAIN-20140416005212-00574-ip-10-147-4-33.ec2.internal.warc.gz"} |
[Numpy-discussion] convert between structure arrays with different record orderings?
[Numpy-discussion] convert between structure arrays with different record orderings?
roger peppe rogpeppe@gmail....
Thu May 28 09:39:01 CDT 2009
sorry, i'm new to the list, and if this is a frequently asked question, please
point me in the right direction.
say, for some reason i've got two numpy structure arrays
that both contain the same fields with the same types
but in a different order, is there a simple way to convert
one to the other type?
i'd have thought that astype() might do the job,
but it seems to ignore the field names entirely, thus
values are lost in the following transcript:
> t1 = np.dtype([('foo', np.float64), ('bar', '?')])
> t2 = np.dtype([('bar', '?'), ('foo', np.float64)])
> a1 = np.array([(123, False), (654, True)], dtype=t1)
> a1.astype(t2)
array([(True, 0.0), (True, 1.0)],
dtype=[('bar', '|b1'), ('foo', '<f8')])
thanks for any help,
More information about the Numpy-discussion mailing list | {"url":"http://mail.scipy.org/pipermail/numpy-discussion/2009-May/042829.html","timestamp":"2014-04-16T19:07:40Z","content_type":null,"content_length":"3555","record_id":"<urn:uuid:68700fcf-a266-4440-a240-6987a8cb81f9>","cc-path":"CC-MAIN-2014-15/segments/1397609535745.0/warc/CC-MAIN-20140416005215-00320-ip-10-147-4-33.ec2.internal.warc.gz"} |
Summary: Applications of Homological Algebra Introduction to Perverse Sheaves
Spring 2007 P. Achar
Basic Facts on Sheaves
Definition 1. A sheaf of abelian groups F on a topological space X is the following collection of data:
· for each open set U X, an abelian group F(U), with F() = 0
· if U V , a restriction map V U : F(V ) F(U), with UU = id
such that
(1) (restriction) if U V W, then W U = V U W V
(2) (gluing) if {Vi} is an open covering of U, and we have si F(Vi) such that for all i, j, Vi,ViVj (si) =
Vj ,ViVj (sj), then there exists a unique s F(U) such that U,Vi (s) = si for all i.
If one omits the gluing condition from the above definition, one has a presheaf of abelian groups.
Elements of F(U) are called sections of F over U, and elements of F(X) are called global sections.
One can also define (pre)sheaves of R-modules, vector spaces, etc.
Notation 2. (U, F) := F(U). s|V := V U (s).
Lemma 3. Let F be a sheaf. If {Vi} is an open cover of U, and s, t F(U) are sections such that s|Vi
= t|Vi
for all i, then s = t. In particular, if s|Vi
= 0 for all i, then s = 0.
Definition 4. Let F be a presheaf on X. The stalk of F at a point x X, denoted Fx, is the group whose
elements are equivalence classes of pairs | {"url":"http://www.osti.gov/eprints/topicpages/documents/record/510/4798734.html","timestamp":"2014-04-20T01:34:48Z","content_type":null,"content_length":"8333","record_id":"<urn:uuid:856300f8-c966-4db0-a399-464dc37ffed9>","cc-path":"CC-MAIN-2014-15/segments/1397609537804.4/warc/CC-MAIN-20140416005217-00613-ip-10-147-4-33.ec2.internal.warc.gz"} |
Optimal Design under Heteroscedasticity for Gaussian Process Emulators with replicated observations
Seminar Room 1, Newton Institute
Computer models, or simulators, are widely used in a range of scientific fields to aid understanding of the processes involved and make predictions. Such simulators are often computationally
demanding and are thus not amenable to statistical analysis. Emulators provide a statistical approximation, or surrogate, for the simulators accounting for the additional approximation uncertainty.
For random output, or stochastic, simulators the output dispersion, and thus variance, is typically a function of the inputs. This work extends the emulator framework to account for such
heteroscedasticity by constructing two new heteroscedastic Gaussian process representations and proposes an experimental design technique to optimally learn the model parameters. The design criterion
is an extension of Fisher information to heteroscedastic variance models. Replicated observations are efficiently handled in both the design and model inference stages. We examine the effect of such
optimal designs on both model parameter uncertainty and predictive variance through a series of simulation experiments on both synthetic and real world simulators.
The video for this talk should appear here if JavaScript is enabled.
If it doesn't, something may have gone wrong with our embedded player.
We'll get it fixed as soon as possible. | {"url":"http://www.newton.ac.uk/programmes/DAE/seminars/2011090212001.html","timestamp":"2014-04-21T14:44:55Z","content_type":null,"content_length":"6938","record_id":"<urn:uuid:622098b4-e6a3-46cd-8087-0d249ac204a7>","cc-path":"CC-MAIN-2014-15/segments/1397609540626.47/warc/CC-MAIN-20140416005220-00528-ip-10-147-4-33.ec2.internal.warc.gz"} |
New Toads and Frogs results
In Games of No Chance, Richard J. Nowakowski, editor, pages 299-310.
Mathematical Sciences Research Institute Publications 29, Cambridge University Press, 1996.
• Download code (and help file) for evaluating Toads and Frogs positions, based on David Wolfe's Gamesman's Toolkit
• See also: Sowing games
• Read a review of the book from MAA Online, which mentions three "colorful" theorems from this paper!
• Update (22 May 2000): Jesse Hull has proved that Toads and Frogs is NP-hard, using a program that automates simple induction proofs involving Toads and Frogs positions. Specifically, his program
found the following identity, which by results of Yedwab and Moews is enough to show NP-hardness:
T^n_TFT^m_TF = m+n-1||n|2 for all m>0 and n>0
We present a number of new results for the combinatorial game Toads and Frogs. We begin by presenting a set of simplification rules, which allow us to split positions into independent components
or replace them with easily computable numerical values. Using these simplication rules, we prove that there are Toads and Frogs positions with arbitrary numerical values and arbitrarily high
temperatures, and that any position in which all the pieces are contiguous has an integer value that can be computed quickly. We also give a closed form for the value of any starting position
with one frog, and derive some partial results for two-frog positions. Finally, using a computer implementation of the rules, we derive new values for a large number of starting positions.
Publications - Jeff Erickson (jeffe@cs.uiuc.edu) 22 May 2000 | {"url":"http://compgeom.cs.uiuc.edu/~jeffe/pubs/toads.html","timestamp":"2014-04-20T03:11:13Z","content_type":null,"content_length":"3257","record_id":"<urn:uuid:b35c3dc7-a0f5-4958-b804-2ef6c430243e>","cc-path":"CC-MAIN-2014-15/segments/1398223202774.3/warc/CC-MAIN-20140423032002-00643-ip-10-147-4-33.ec2.internal.warc.gz"} |
Guide Entry 00.05.04
Yale-New Haven Teachers Institute Home
Math and Science Objectives Taught Using Sound and Music Concepts, by Mary Elizabeth Jones
Guide Entry to 00.05.04:
This unit was designed to be taught by a middle school teacher of math and/or science. The unit explores sound from its origins to the present. Students will learn about the characteristics and types
of waves. Wave frequency, pitch, amplitude and wave velocity are covered. Included in the unit are many classroom activities and research activities. The unit will be especially beneficial to
students with a limited musical background.
The unit is divided into three sections. The first section covers wave characteristics, sound velocity, pitch and frequency and the Doppler effect. The second section explores amplitude, sound
pressure level, noise and musical sounds. The last section focuses on persons involved in the origin of sound.
Students will learn to apply appropriate formulas in order to make mathematical calculations using data such as the speed and velocity of sound. Algebraic expressions will be constructed and solved
using data collected in science class or supplied by the teacher.
Students will learn to collect and analyze scientific data. The date can be used to teach graphing skills.
There are several hands-on mini labs, which will allow students to make and test a hypothesis.
(Recommended for Math and Science, grades 5-7.)
Contents of 2000 Volume V | Directory of Volumes | Index | Yale-New Haven Teachers Institute | {"url":"http://www.yale.edu/ynhti/curriculum/guides/2000/5/00.05.04.x.html","timestamp":"2014-04-16T07:38:10Z","content_type":null,"content_length":"4606","record_id":"<urn:uuid:e46ddcd1-5d0b-4804-867a-1c41c2e65ea5>","cc-path":"CC-MAIN-2014-15/segments/1397609521558.37/warc/CC-MAIN-20140416005201-00534-ip-10-147-4-33.ec2.internal.warc.gz"} |
Of the Safari Zone (Or, curse you Chansey!): A strategy guide - Page 3 - Guide Tavern
OK lets say I threw a bait at one Chansey then threw 29 balls before it left. I now have one ball left and I find another Chansey what should I do? That's what I'm getting at. Say I have 10 balls or
5 balls left, in these cases which method is more likely to get me a Chansey? Surely the amount of balls remaining can effect your chances | {"url":"https://forums.pokemmo.eu/index.php?/topic/3711-of-the-safari-zone-or-curse-you-chansey-a-strategy-guide/page-3","timestamp":"2014-04-17T04:01:29Z","content_type":null,"content_length":"226632","record_id":"<urn:uuid:6515d669-ccaf-4f97-8fa1-b43981f19f82>","cc-path":"CC-MAIN-2014-15/segments/1398223210034.18/warc/CC-MAIN-20140423032010-00513-ip-10-147-4-33.ec2.internal.warc.gz"} |
How to leak on key updates
Leakage resilient cryptography is an exciting area of cryptography that aims to build cryptosystems that provide security against side channel attacks. In this post I will give a nontechnical
description of a common leakage resilient security model, as well as describe a recent paper in the area with Allison Lewko and Brent Waters, titled “How to Leak on Key Updates”.
Review of Public Key Encryption
Let us (informally) recall the definition of a public key cryptography system. Alice would like to send Bob a private message $M$ over an unsecured channel. Alice and Bob have never met before and we
assume they do not share any secret information. Ideally, we would like a procedure where 1) Alice and Bob engage in a series of communications resulting in Bob learning the message $M$ 2) an
eavesdropper, Eve, who intercepts all of the communications sent between Alice and Bob, should not learn any (nontrivial) information about the message $M$. As stated, the problem is information
theoretically impossible. However, this problem is classically solved under the heading of public key cryptography if we further assume that:
1) Eve has limited computational resources,
2) certain computational problems (such as factoring large integers or computing discrete logarithms in a finite group) are not efficiently solvable, and
3) we allow Alice and Bob to use randomization (and permit security to fail with very small probability).
More specifically, a public key protocol works as follows: Bob generates a private and public key, say $SK$ and $PK$ respectively. As indicated by the names, $PK$ is publicly known but Bob retains
$SK$ as secret information. When Alice wishes to send a message $M$ to Bob she generates an encrypted ciphertext $C$ using the message $M$, Bob’s public key $PK$ and some randomness. She then sends
this ciphertext to Bob via the public channel. When Bob receives the ciphertext he decrypts it using his secret key $SK$ and recovers $M$. While Eve has access to the ciphertext $C$ and the secret
key $SK$, she is unable to learn any nontrivial information about the message $M$ (assuming our assumptions are sound). In fact, we require a bit more: even if this is repeated many times (with fixed
keys), Eve’s ability to decrypt the ciphertext does not meaningfully improve.
Leakage Resilient Cryptography and our work
In practice, however, Eve may be able to learn information in addition to what she intercepts over Alice and Bob’s public communications via side channel attacks. Such attacks might include measuring
the amount of time or energy Bob uses to carry out computations. The field of leakage resilient cryptography aims to incorporate protection against such attacks into the the security model. In this
model, in addition to the ciphertext and public key, we let Eve select a (efficiently computable) function $F:\{0,1\}^{\ell}\rightarrow\{0,1\}^{\mu \ell}$ where $\ell$ is the bit length of $SK$ and
$0<\mu<1$ is a constant. We now assume, in addition to $C$ and $PK$, Eve also gets to see $F(SK)$. In other words, Eve gains a fair amount of information about the secret key, but not enough to
fully determine it.
Moreover, we allow Eve to specify a different function $F$ every time Alice sends Bob a message. There is an obvious problem now, however. If the secret key $SK$ remained static, then Eve could start
by choosing $F$ to output the first $\mu \ell$ bits, the second time she could choose $F$ to give the next $\mu \ell$ bits, and if she carries on like this, after $1/\mu$ messages she would have
recovered the entire secret key. To compensate for this we allow Bob to update his secret key between messages. The public key will remain the same.
There has been a lot of interesting work on this problem. In the works of Brakerski, Kalai, Katz, and Vaikuntanathan and Dodis, Haralambiev, Lopez-Alt, and Wichs many schemes are presented that are
provably secure against continual leakage. In these schemes, however, information about the secret key is permitted to be leaked between updates, but only a tiny amount is allowed to be leaked during
the update process itself.
In our current work, we offer the first scheme that allows a constant fraction of the information used in the update to be leaked. The proof is based on subgroup decision assumptions in composite
order bilinear groups. | {"url":"http://lewko.wordpress.com/2011/02/03/how-to-leak-on-key-updates/","timestamp":"2014-04-18T15:39:39Z","content_type":null,"content_length":"58841","record_id":"<urn:uuid:c889a0a3-d08c-4c94-a5ce-7bd14a24d139>","cc-path":"CC-MAIN-2014-15/segments/1397609533957.14/warc/CC-MAIN-20140416005213-00076-ip-10-147-4-33.ec2.internal.warc.gz"} |
Patent application title: AUTOMATIC ESTIMATION OF WELDGUN SIZE USING SECTION GEOMETRY
Sign up to receive free email alerts when patent applications with chosen keywords are published SIGN UP
A method for estimating a plurality of geometrical parameters defining the size of a weld gun that has particular application for automatically selecting a weld gun for a welding operation. The
method includes iteratively estimating a plurality of geometric parameters based on part section curves corresponding to a direction of approach of the weld gun to weld point of the plurality of weld
points. Thereafter, a set of valid weld gun sizes are calculated based on the estimated plurality of geometric parameters. Similarly all the valid weld gun sizes are calculated corresponding to each
of the weld gun approach direction. Further, each set of the valid gun sizes are estimated for each of the weld point for the welding operation. Finally, a weld gun for performing the welding
operation is selected based on the set of weld gun sizes corresponding to the welding operation.
A method for estimating final values of a plurality of geometric parameters of a weld gun, wherein the plurality of geometric parameters define the size of the weld gun, wherein the weld gun performs
a welding operation at a weld point, said method comprising:a. generating part section curves corresponding to a direction of approach of the weld gun to the weld points, wherein the part section
curves are generated by using a part section plane, and wherein the part section plane is a plane containing the weld gun approach direction and a normal to the weld point;b. storing the part section
curves as a link list of obstructions in an array of Y scan lines to analyze a given part geometry for available clearance space;c. determining values of a first geometric parameter of the plurality
of geometric parameters, wherein the value of the first geometric parameter is determined based on the presence of the part section curves in a pre-defined area;d. estimating a first maximum and a
first minimum limit for a second geometric parameter of the plurality of geometric parameters, wherein the first minimum limit for the second geometric parameter is determined based on the limit for
the first geometric parameter, and wherein the first maximum limit of the second geometric parameter is determined based on the type of weld gun used;e. estimating a maximum and a minimum limit for a
third geometric parameter of the plurality of geometric parameters, wherein the maximum limit for the third geometric parameter is calculated based on the value of the first geometric parameter and
the first maximum and the first minimum limit for the second geometric parameter, and wherein the minimum limit for the third geometric parameter is estimated based on the first maximum and the first
minimum limit for the second geometric parameter, and wherein the maximum and the minimum limit for the third geometric parameter are calculated based on the part section curves;f. estimating a valid
maximum and a valid minimum limit for the third geometric parameter based on a value of the third geometric parameter for a first standard weld gun;g. computing a second maximum and a second minimum
limit for the second geometric parameter corresponding to the valid maximum and the valid minimum limit for the third geometric parameter, wherein the second maximum and the second minimum limit for
the second geometric parameter are calculated based on the part section curves;h. estimating a valid maximum and a valid minimum limit for the second geometric parameter based on a limit for the
second geometric parameter of a second standard weld gun; andi. repeating the steps e, f, g and h until each of the valid maximum and the valid minimum limit for the second geometric parameter
converge and each of the valid maximum and the valid minimum limit for the third geometric parameter converge, wherein the first maximum and the first minimum limit for the second geometric parameter
are replaced with the valid maximum and the valid minimum limits for the second geometric parameter, respectively, and wherein the limit for the first geometric parameter, the converged valid minimum
limit for the second geometric parameter, the converged valid maximum limit for the second geometric parameter, the converged valid minimum limit for the third geometric parameter and the converged
valid maximum limit for the third geometric parameter are the estimated final limits for the plurality of geometric parameters.
The method according to claim 1 wherein the first maximum limit for the second geometrical parameter is taken as a sum of a minimum limit for the second geometrical parameter for the type of weld gun
used, length of the gun frame for the type of weld gun used, and a value of tolerance for the type of weld gun used.
The method according to claim 1 wherein the method is carried out in a UG-NX design software.
The method according to claim 1 wherein the type of weld gun is selected from a group comprising a P type weld gun, a C type weld gun, an X type weld gun and an S type weld gun.
The method according to claim 1 wherein the first geometric parameter is a tip angle of shanks for stationary and moving arms of the weld gun.
The method according to claim 5 wherein the second geometric parameter is an arm stick-out of the weld gun.
The method according to claim 6 wherein the third geometric parameter is an arm offset corresponding to a moving arm and stationary arm of the weld gun.
The method according to claim 1 wherein a number of weld gun approach directions to the weld point is defined by a user.
A method for estimating a set of valid weld gun sizes for an approach direction of a weld gun to a weld point, wherein final values of a plurality of geometric parameters define the weld gun size,
and wherein the plurality of geometric parameters include a tip angle of a shank for moving and stationary arms of the weld gun, an arm stick-out of the weld gun, and arm offset of moving and
stationary arms of the weld gun, and wherein the weld gun performs a welding operation at the weld point, said method comprising:a. generating part section curves corresponding to the weld gun
approach direction to the weld point, wherein the part section curve is generated by using a part section plane, and wherein the part section plane is the plane containing the weld gun approach
direction and a normal to the weld point;b. storing the part section curves as a link list of obstructions in an array of Y scan lines to analyze a given part geometry for available clearance space;
c. determining a value of the tip angle of the shank for moving and stationary arms of the weld gun, wherein the value of the tip angle of the shank is determined based on the presence of the part
section curve in a three-dimensional cylindrical zone around the weld point;d. estimating a first maximum and a first minimum limit for the arm stick-out, wherein the first minimum limit for the arm
stick-out is determined based on the value of the tip angle of the shank of the weld gun and the first maximum limit for the arm stick-out is determined based on the type of weld gun used;e.
estimating a maximum and a minimum limit for the arm offset, wherein the maximum limit for the arm offset is calculated based on the limit for the tip angle of the shank of the weld gun and the first
maximum and the first minimum limit for the arm stick-out, and wherein the minimum limit for the arm offset is estimated based on the first maximum and the first minimum limit for the arm stick-out,
and wherein the maximum and the minimum limit for the arm offset are calculated based on the part section curves;f. estimating a valid maximum and a valid minimum limit for the arm offset on the
basis of a value of the arm offset for a first standard weld gun;g. computing a second maximum and a second minimum limit for the arm stick-out based on the valid maximum and the valid minimum limit
for the arm offset, wherein the second maximum and the second minimum limit for the arm stick-out are calculated on the part section curves;h. estimating a valid maximum and a valid minimum limit for
the arm stick-out based on the a limit for the arm stick-out for a second standard weld gun;i. repeating the steps e, f, g and h until each of the valid maximum and the valid minimum values of the
arm stick-out converge and each of the valid maximum and the valid minimum values of the arm offset converge, wherein the first maximum and the first minimum values of the arm stick-out are replaced
with the valid maximum and the valid minimum values of the arm stick-out, respectively, and wherein the limit for the tip angle of the shank, the converged valid minimum limit for the arm stick-out,
the converged valid maximum value of the arm stick-out, the converged valid minimum value of the arm offset and the converged valid maximum limit for the arm offset are the final limits for the
plurality of geometric parameters; andj. estimating the set valid weld gun sizes for the approach direction of the weld gun to the weld point, wherein the set of valid weld gun sizes is estimated by
mapping the final values of the plurality of geometric parameters onto a set of standard weld guns of the type of weld gun used.
The method according to claim 9 wherein the first maximum value of the arm stick-out is taken as a sum of a minimum limit for the arm stick-out for the type of weld gun used, length of a weld gun
frame for the type of weld gun used and a limit for tolerance for the type of weld gun used.
The method according to claim 9 wherein the method is carried out in a UG-NX design software.
The method according to claim 9 wherein the type of weld gun is selected from a group comprising a P type weld gun, a C type weld gun, an X type weld gun and an S type weld gun.
The method according to claim 9 wherein a number of weld gun approach directions to the weld point is defined by a user.
A method for selecting a weld gun corresponding to a welding operation, wherein the weld gun is selected based on the final values of a plurality of geometric parameters of the weld gun, and wherein
the plurality of geometric parameters define the size of the weld gun, and wherein the plurality of geometric parameters comprises a tip angle of a shank of moving and stationary arms of the weld
gun, an arm stick-out of the weld gun, and an arm offset of the stationary and moving arms of the weld gun, and wherein a welding operation comprises a plurality of weld points, said method
comprising:a. generating part section curves corresponding to a direction of approach of the weld gun to a weld point of the plurality of weld points, wherein the part section curve is generated by
using a part section plane, and wherein the part section plane is a plane containing the weld gun approach direction and a normal to the weld point;b. storing the part section curves as a link list
of obstructions in an array of Y scan lines to analyze a given part geometry for available clearance space;c. determining values of the tip angle of the shank of stationary and moving arms of the
weldgun, wherein the values of the tip angle of the shank are determined based on the presence of the part section curves in a three-dimensional cylindrical zone around the weld point;d. estimating a
first maximum and a first minimum limit for the arm stick-out, wherein the first minimum limit for the arm stick-out is determined based on the limit for the tip angle of the shank of the weld gun
and wherein the first maximum value of the arm stick-out is determined based on the type of weld gun used;e. estimating a maximum and a minimum limit for the arm offset wherein the maximum limit for
the arm offset is calculated based on the limit for the tip angle of the shank of the weld gun and the first maximum and the first minimum value of the arm stick-out, and wherein the minimum limit
for the arm offset is estimated based on the first maximum and the first minimum limit for the arm stick-out, wherein the maximum and the minimum limit for the arm offset are calculated based on the
part section curves;f. estimating the valid maximum and valid minimum limit for the arm offset based on a value of the arm offset for a first standard weld gun;g. computing a second maximum and a
second minimum limit for the arm stick-out based on the valid maximum and the valid minimum limit for the arm offset, wherein the second maximum and the second minimum limit for the arm stick-out are
calculated based on the part section curves;h. estimating a valid maximum and a valid minimum limit for the arm stick-out based on a value of the arm stick-out of a second standard weld gun;i.
repeating the steps e, f, g and h until each of the valid maximum and the valid minimum values of the arm stick-out converge and each of the valid maximum and the valid minimum values of the arm
offset converge, wherein the first maximum and the first minimum values of the arm stick-out are replaced with the valid maximum and the valid minimum values of the arm stick-out, respectively, and
wherein the value of the tip angles of the shank, the converged valid minimum limit for the arm stick-out, the converged valid maximum limit for the arm stick-out, the converged valid minimum limit
for the arm offset and the converged valid maximum limit for the arm offset are the final values of the plurality of geometric parameters;j. selecting a set of valid weld gun sizes corresponding to a
weld gun approach direction to the weld point, wherein the set of valid weld gun sizes is estimated by mapping the final values of the plurality of geometric parameters onto a set of standard weld
guns of the type of weld gun used;k. performing the steps of a, b, c, d, e, f, g, h, i and j for each of the weld gun approach direction to the weld point of the plurality of weld points;l. obtaining
a set of weld gun sizes corresponding to the weld point based on the set of valid gun sizes corresponding to each of the weld approach direction to the weld point;m. performing the steps of a, b, c,
d, e, f, g, h, i, j, k and l for each of the plurality of welding points of the welding operation;n. obtaining a set of weld gun sizes corresponding to the welding operation based on the set of valid
gun sizes corresponding to each of the weld point; ando. selecting a weld gun for performing the weld operation, wherein the weld gun is selected based on the set of weld gun sizes corresponding to
the welding operation.
The method according to claim 14 wherein the first maximum limit for the arm stick-out is taken as a sum of a minimum limit for the arm stick-out for the type of weld gun used, length of a weld gun
frame for the type of weld gun used and a value of tolerance for the type of weld gun used.
The method according to claim 14 wherein the method is carried out in a UG-NX design software.
The method according to claim 14 wherein the type of weld gun used is selected from a group comprising a P type weld gun, a C type weld gun, an X type weld gun and an S type weld gun.
The method according to claim 14 wherein a number of weld gun approach directions to the weld point is defined by a user.
The method according to claim 14 wherein selecting a weld gun for performing the welding operation is done on the basis of a plurality of user defined parameters.
The method according to claim 19 wherein the user defined parameters are selected from a group comprising a weld gun cost, a weld gun size and a list of available weld guns.
BACKGROUND OF THE INVENTION [0001]
1. Field of the Invention
This invention relates generally to a method for automatically selecting a welding gun for a welding operation and, more particularly, to a method for estimating the geometrical parameters defining
the size of a welding gun suitable for a welding operation.
2. Description of the Related Art
Welding is sometimes performed to join two parts together. There are various types of welding technique, such as electric arc welding and resistance spot welding. Resistance spot welding is the most
common joining operation used in automotive assembly line. In resistance spot welding process, the welding is performed by placing the parts to be joined together between the electrodes of a weld gun
and an electric current is passed through the electrodes. The parts are welded due to the high temperature caused by the resistance to electric current flow of the electrodes. In an automotive
assembly line, parts are joined at various weld points with a variety of orientations during a weld operation.
In a typical vehicle, the number of weld points are several thousand. To balance work across assembly line stations, these weld points are grouped into sets of six to eight weld points each called
weld operations, where a single weldgun on a single robot perform welding at weld points belonging to a single weld operation. Selection of a weld gun for a specific welding operation depends on a
set of operational and geometric constraints. The process constraints depend on material stack up and the geometric constrains depend on the geometry of parts and tools/fixtures near the weld point.
Current methods of weld gun selection typically employs software (UG-NX) assisted manual process. An engineer selects weld guns iteratively based on previous experience and by taking measurements
from 2-D sections manually. However, this method is time consuming and also requires manual iterations for selecting the correct weld gun. Further, the weld gun selected using this method can fail
while validating the weld gun for all the weld points of a welding operation. The weld gun is validated by manually operating the selected weld gun at the weld point such that the weld gun does not
interfere With the geometry of any part. Moreover, the method doesn't provide information about all the directions in which the weld gun can approach the weld point.
Another way to perform weld gun selection is to use eM-Simulate's weld gun selection capability. But that too is necessarily a validation approach rather than a selection approach. The eM-Simulate
method is a brute force method of validating each of the weld guns in a library of weld guns against all of the weld operations by performing successive interference checks around the weld points.
This method is time consuming and does not capture the complete solution space available for the valid weld gun sizes for a given set of weld operations.
SUMMARY OF THE INVENTION [0008]
In accordance with the teachings of the present invention, a method for estimating geometrical parameters defining the size of a weld gun suitable for a welding operation is disclosed. The method
includes generating part section curves corresponding to a direction of approach of the weld gun to a weld point in the welding operation. Then, a value of the tip angle of a weld gun shank is
determined for both arms of the weld gun based on the presence of the part section curves in a three-dimensional cylindrical zone around the weld point. Further, minimum and maximum limits for the
arm stick-out are estimated based on the value of the tip angles required and the type of weld gun used. Thereafter, minimum and maximum limits for arm offset are calculated for both arms of the
weldgun based on the value of the tip angle of the shank of the weld gun, and the presence of part section curves between the maximum and the minimum value limits of the arm stick-out, estimated
above. A valid maximum and minimum value of the arm offset are obtained based on standard weld gun sizes available. Thereafter, the maximum and minimum limits for the arm stick-out are computed and
updated based on the part section curves between the valid maximum limits of the arm offset for the two weld gun arms. A valid maximum and minimum value of the arm stick-out are obtained based on
standard weld gun sizes available.
The above described procedure of calculation of the arm stick-out limits and arm offset limits is repeated with the modified values of each, until each of the valid values of the arm stick-out limits
and the arm offset limits converge to a corresponding single value. The value of the tip angle of the shank, the converged valid minimum value and the converged valid maximum value of the arm
stick-out and arm offset are the final values of the plurality of geometric parameters. A set of valid weld gun sizes corresponding to a weld gun approach direction to the weld point is then selected
by mapping the final values of the plurality of geometric parameters onto a set of standard weld guns of the type of weld gun being used.
The above described procedure is repeated for each of the weld gun approach directions to the weld point of the plurality of weld points and a set of weld gun sizes corresponding to the weld point is
obtained based on the set of valid gun sizes corresponding to each of the weld approach direction to the weld point. The procedure is repeated for each of the plurality of weld points of the weld
operation and a set of weld gun sizes corresponding to the weld operation is selected based on set of valid weld gun sizes corresponding to each of the weld points. Finally, a weld gun for performing
the weld operation is selected based on the set of weld gun sizes corresponding to the welding operation and user defined parameters.
Additional features of the present invention will become apparent from the following description and appended claims, taken in conjunction with the accompanying drawings.
BRIEF DESCRIPTION OF THE DRAWINGS [0012]FIG. 1
illustrates geometrical parameters of a throat area for a pinch (P) type weld gun;
[0013]FIG. 2
is a scan line representation of part section curves illustrating a tip clearance zone corresponding to a weld point;
[0014]FIG. 3
is a scan line representation of part section curves illustrating estimation of tip angle for a moving arm;
[0015]FIG. 4
is a scan line representation of part section curves illustrating the initial arm offset limits for a weld gun corresponding to a weld point;
[0016]FIG. 5
is a scan line representation of part section curves illustrating the iteratively calculated arm stick-out limits for a weld gun corresponding to a weld point;
[0017]FIG. 6
illustrates a pinch (P) type weld gun of feasible dimensions for a weld operation; and
[0018]FIG. 7
is an exemplary depiction of the various geometrical parameters of a weld gun and part section curves used in their calculation.
DETAILED DESCRIPTION OF THE EMBODIMENTS [0019]
The following discussion of the embodiments of the invention directed to a method for estimating the geometrical parameters defining the size of a weld gun suitable for a welding operation is merely
exemplary in nature, and is in no way intended to limit the invention or its applications or uses.
[0020]FIG. 1
depicts a throat area 10 of a weld gun illustrating a plurality of geometric parameters that define the throat area 10. For the purpose of this description, a pinch (P) type weld gun has been used.
The plurality of geometric parameters include the tip angles θ
for a stationary arm 14 and θ
for a moving arm 16 of the weld gun, an arm stick-out X of the weld gun, an arm offset Y2 of the moving arm 16 of the weld gun and an arm offset Y1 of the stationary arm 14 of the weld gun. The space
available around the various weld points in a welding operation largely decides the various dimensions of a weld gun that would be suitable for the particular welding operation. Hence, the allowable
values of the various geometric parameters defined above for a welding operation are used to select a suitable weld gun for that welding operation. A method for calculating the allowable values of
these geometric parameters for a weld point in a welding operation and then selecting a suitable weld gun for the welding operation corresponding to the values of these parameters is as described
In one embodiment, the method for estimating the values of the above-mentioned geometric parameters for one of the weld points of the plurality of weld points in a welding operation for one direction
of approach of the weld gun is described. In this embodiment, the method is said to be carried out in a UG-NX environment. However, it will be readily apparent to a person of ordinary skill in the
art that software other than UG-NX can also be used. While carrying out the welding operation, there should be no interference of the weld gun with the geometry of the parts to be welded in the
direction of approach of the weld gun towards the weld point. To ensure this, part section curves of the weld parts in a weld gun approach direction to a weld point are generated. The part section
curves give an outline of how a part to be welded extends around the weld point and are generated using a plane containing the weld gun approach direction to the weld point and a normal to the weld
point. The part section curves are then digitized by storing them as obstructions in the form of a link list in an array of Y scan lines, as shown in
FIG. 2
. The scan lines store the data of the part section curves in the form of pixel values and represent this data in a two-dimensional coordinate system, say (x, y), as shown in FIGS. 2, 3, 4 and 5.
In the coordinate system shown in these figures, the origin is taken to be at the weld point, the arm stick-out (X) is measured along the X-axis and the arm offset (Y1 and Y2) is measured along the
Y-axis. The values of the various geometrical parameters of the weld gun are determined using the part section curves and the process for the same is as follows.
The tip angle of the shank 12 of the weld gun is determined based on the presence of the part section curves in a three-dimensional cylindrical zone around the weld point. The three-dimensional
cylindrical zone represents the area around the weld point that is available for the movement of the tip of the weld gun. This area is termed as the tip clearance zone and is shown in
FIG. 2
. When there is no part section curves present in the tip clearance zone, then the weld gun with a zero tip angle of the shank 12 is required to perform the welding operation. The tip angle of the
shank 12 of a weld gun is said to be zero when it has a straight shank. In a scenario, when the part section curve is present in the tip clearance zone, a weld gun with a bent shank is required to
perform the welding at the weld point. The procedure for calculating tip angle for moving shank is shown in
FIG. 3
. Draw line from the weld point at different standard tip angle values (θ
, θ
, . . . ) and calculate the X and Y values for its intersection with part section curves. Choose the tip angle with maximum Y value of intersection. The above process is also used to estimate tip
angle of the shanks for stationary arm.
The tip angles obtained above are used to determine the first minimum limit on the arm stick-out (X-Min) of the weld gun. The first minimum limit on the arm stick-out (X-Min) is taken as zero when
the weld gun with a straight shank is selected. For the weld gun with a bent shank the first minimum value of the arm stick-out (X-Min) is taken to be equal to X-Min θ, where θ is the larger of the
tip angles for stationary and moving arms of the weld gun. The first maximum value of the arm stick-out X-Max is taken as sum of minimum possible value of X-ARM-STICKOUT or arm stick-out, hereinafter
used interchangeably, for a weld gun type being used, length of the weld gun frame of the weld gun type along the x-direction, and tolerance values for the weld gun type. Examples of weld gun types
that can be used in this invention include, but are not limited to, P-type guns, X type guns, C-type guns and S-type guns.
Further, a minimum and a maximum value of the arm offsets (Y1 and Y2) of the weld gun arms 14 and 16 are determined based on the part section curves between the first minimum and first maximum values
of the arm stick-out, that is, X-Min and X-Max, respectively, as shown in FIG. 4. The minimum values of Y1 and Y2 are determined on the basis of a first y scan line, which does not have any X-pixels
between X-Min and X-Max. The minimum values of Y1 and Y2 should be greater than the corresponding minimum values of the arm offset of the moving arm 16 and the stationary arm 14 of the specified weld
gun type with clearance space as required. The maximum values of Y1 is determined on the basis of a first y scan line after Y1-min, which has one or more X-pixels between X-Min and X-Max. Similarly
the maximum value of Y2 is determined. The maximum and minimum values of Y1 and Y2 define the clearance space available for the weld gun to access the weld point in the y direction. If the clearance
space obtained based on the calculated values of the arm offset limits is too small for the weld gun type, a valid weld gun of the selected type can not be selected. This combination of values will
give the initial clearance space for arm offset of the weld gun arms.
Now, for more accurate determination of the various geometrical parameters, the values of the arm stick-out limits for the weld gun are again calculated based on the arm offset limits obtained above.
The second minimum limit for the arm stick-out is determined as the maximum limit of x pixel value along each scan line between maximum values of Y1 and Y2. The second maximum limit for the arm
stick-out is calculated as the minimum x pixel value of the scan line between maximum values of Y1 and Y2 greater than second minimum limit for the arm stick-out. If the difference between second
minimum and maximum values of arm stick-out is less than the minimum clearance space required for the weld gun type, re-estimation of various arm stick-out limits is done based on a standard weld
gun, which can be similar or different to the standard weld gun used for the validation of the arm offset limits. This gives the valid combination of the arm stick-out values for the weld gun. The
above outlined process of iterative calculation of the arm stick-out limits has been graphically represented in
FIG. 5
The above described procedure of calculation of the arm stick-out limits and arm offset limits is repeated with the modified values of each, until each of the valid values of the arm stick-out limits
converges to a corresponding single value and each of the valid values of the arm offset limits converges to a corresponding single value. The converged valid values of the arm stick-out limits and
the arm offset limits are then finalized for use in the further gun selection procedure.
The finalized values of the of the arm stick-out limits and the arm offset limits and the tip angles of the weld gun gives the clearance space bounds around the weld point for the weld gun approach
direction. These bounds are mapped to the throat parameters of standard weld guns of a given weld gun type. While mapping the clearance space bounds on to the throat parameters of the standard weld
guns of the given weld gun type, there should be enough tolerances to accommodate for the shank and arm thickness, the weld gun frame, the weld gun opening and the weld gun closing. Moreover, the
clearance space bound should take care of any additional gap on account of the shape of throat area of the weld gun of the gun type. In an exemplary embodiment the method associated with the mapping
of the clearance space bounds with the throat parameters of a P-type gun is as illustrated in
FIG. 6
and as described below.
Let the tolerance to accommodate for shank/arm thickness and some extra clearance be ε.
The required tip angle (estimated separately for moving and stationary arm) of the weld gun is determined as described above. Along weld gun length the difference between a X-ARM-STICKOUT of the
P-type weld gun and a minimum value of the arm stick-out should be greater then the tolerance value ε. The difference between a maximum value of the arm stick-out and the combined value of the
X-ARM-STICKOUT and the weld gun frame length should be greater than the tolerance value ε. These conditions are as shown in the equations below.
(X-Max)-(X-ARM-STICKOUT+Weld Gun frame length)>ε
Along the moving arm of the gun the difference between the arm offset or a Y2-ARM-OFFSET of the P-type weld gun and a minimum value of Y2 should be greater than the tolerance value ε. The difference
between a maximum value of Y2 and the combined value of the Y2-ARM-OFFSET and weld gun stroke opening is greater than the tolerance value ε. These conditions are shown in the equations below.
(Y2-Max)-(Y2-ARM-OFFSET+Weld Gun stroke opening)>ε
Along the stationary arm 14 of the weld gun the difference between the arm offset or a Y1-ARM-OFFSET of the P-type weld gun and a minimum value of Y1 should be greater than the tolerance value ε. The
difference between a maximum value of Y1 and value of the Y1-ARM-OFFSET is greater than the tolerance value ε. These conditions are shown in the equations below.
Based on the above mapping, a set of valid weld gun sizes is determined corresponding to the weld gun approach direction to the weld point. Similarly, different sets of weld gun sizes are determined
corresponding to each of the weld gun approach direction to the weld point. The final set of weld gun sizes corresponding to the weld point contains all the unique weld gun sizes shortlisted for each
weld gun approach direction for the weld point. Similarly, a plurality of final sets of the weld gun sizes corresponding to each weld point of the weld operation is determined. The method for
selecting the final weld gun sizes from the various sets of weld gun sizes shortlisted above, suitable for a weld operation, is described below. Let an i
weld point be represented as w
and a j
weld gun approach direction be represented d
. Hence, the set of weld gun sizes for the i
weld point in the j
weld gun approach direction can be written as:
Now, valid weld gun sizes for weld point w
(for m weld gun approach directions) can be written as:
) ∪ G-Size(w
) . . . ∪ G-Size(w
Hence, valid weld gun sizes for a welding operation wo
(for n weld points) can be written as:
) . . . ∩G-Size(w
The final weld gun best suited to perform a weld operation is selected from the final valid gun sizes based on the user defined parameters. The user defined parameters are selected from a group
comprising, but not limited to, weld gun size, weld gun cost, and on a list of common weld guns or existing guns and combination thereof.
[0037]FIG. 7
is an exemplary depiction of the various geometrical parameters of the weld gun that are calculated using the present invention and are in turn used to select a suitable weld gun for a welding
operation. Also shown in the figure are the exemplary part section curves used for the calculation of the various geometrical parameters of the weld gun. The graph also illustrates an area depicting
the clearance zone provided around a weld point for the weld gun to exist and perform welding.
Various embodiments of the present invention offer one or more advantages. The present invention provides a method for automatically selecting a weld gun for a welding operation. The method of the
present invention provides a quick way to find the right size gun for a welding operation. Further, the method finds all the feasible gun sizes (range of each parameter value) for a welding
operation, and user can select a gun based on appropriate criteria, such as lightest or cheapest and existing stock. This reduces design iterations and also enables optimization in gun selection.
The foregoing discussion discloses and describes merely exemplary embodiments of the present invention. One skilled in the art will readily recognize from such discussion and from the accompanying
drawings and claims that various changes, modifications and variations can be made therein without departing from the spirit and scope of the invention as defined in the following claims.
Patent applications by Ashish Gupta, Bangalore IN
Patent applications by Gopalakrishna Shastry, Bangalore IN
Patent applications by Narahari K. Hunsur, Bangalore IN
Patent applications by GM GLOBAL TECHNOLOGY OPERATIONS, INC.
Patent applications in class Constraints or rules
Patent applications in all subclasses Constraints or rules
User Contributions:
Comment about this patent or add new information about this topic: | {"url":"http://www.faqs.org/patents/app/20100198384","timestamp":"2014-04-18T23:42:41Z","content_type":null,"content_length":"67143","record_id":"<urn:uuid:a4a8304b-7de6-4bbd-9800-9b5fba9200b4>","cc-path":"CC-MAIN-2014-15/segments/1397609535535.6/warc/CC-MAIN-20140416005215-00036-ip-10-147-4-33.ec2.internal.warc.gz"} |
RE: st: RE: Constrained Lowess
[Date Prev][Date Next][Thread Prev][Thread Next][Date index][Thread index]
RE: st: RE: Constrained Lowess
From "Nick Cox" <n.j.cox@durham.ac.uk>
To <statalist@hsphsun2.harvard.edu>
Subject RE: st: RE: Constrained Lowess
Date Fri, 2 May 2008 18:26:49 +0100
Good. Your variable isn't binary, but a proportion, and you are treating
age as numerical, so your problem is fit for -lowess- after all. (I
prefer restricted cubic splines, but a transformation is probably useful
there too.)
Sergiy Radyakin
Hello Nick,
I am sorry for being inprecise. Indeed, I smooth the rates (of e.g.
unemployment) by groups defined by age (which is truncated to
integers, and thus I concider it categorical).
So I start with a table like the following:
Age Unemployment rate
10 0.01
11 0.02
99 0.01
Here unemployment rate is naturally between 0 and 1. It is the average
of the 0/1-responses within the group, defined by age.
If I just run lowess, it produces the picture similar to the one here:
sysuse auto
generate z=1/headroom^16
lowess z mpg
Note that the tails go below zero, and this is what I am trying to
Your advice of logit transformation before/after smoothing worked.
On 5/2/08, Nick Cox <n.j.cox@durham.ac.uk> wrote:
> Quite how to get useful results from smoothing a binary response is
> clear to me.
> If the data were proportions on (0,1) or even [0,1] I would suggest
> some kind of transformation approach. -lowess, logit- is presumably
> intended to help.
> Otherwise consider something like an angular or folded root
> transformation, applying -lowess- and then transforming back.
> But for binary data any transformation just maps two distinct values
> two other distinct values and so cannot help, so far as I can see.
> In the case of unemployment data, presumably you are dealing with
> individuals? If they are aggregate data for lots of individuals I
> collapse by age to get proportion of unemployed, and then smooth if
> necessary. It sounds as if you want something quite different,
> Also, as you regard -age- as categorical I probably don't understand
> what you are trying to do.
Sergiy Radyakin
> I am plotting a smoothed graph (-lowess-) of a binary variable (e.g.
> unemployed) by categorical (e.g. age). However the smoothed values are
> not necessarily in the [0;1] range, where unemployment must be by
> definition. I can save the smoothed values into a new variable with
> the option -generate(newvar)- and then truncate the negatives and
> values larger than one, but I believe smoothing must look differently
> if I could tell -lowess- to look for such a constrained value in the
> first place. As it follows from the description of -lowess- it doesn't
> have such a feature. Is there any user-written command or simple
> algorithm for this purpose?
* For searches and help try:
* http://www.stata.com/support/faqs/res/findit.html
* http://www.stata.com/support/statalist/faq
* http://www.ats.ucla.edu/stat/stata/ | {"url":"http://www.stata.com/statalist/archive/2008-05/msg00110.html","timestamp":"2014-04-16T07:30:29Z","content_type":null,"content_length":"8291","record_id":"<urn:uuid:66600188-bf65-402a-8117-ec6b4921ac5b>","cc-path":"CC-MAIN-2014-15/segments/1398223205137.4/warc/CC-MAIN-20140423032005-00175-ip-10-147-4-33.ec2.internal.warc.gz"} |
Material Results
Search Materials
Return to What's new in MERLOT
Get more information on the MERLOT Editors' Choice Award in a new window.
Get more information on the MERLOT Classics Award in a new window.
Get more information on the JOLT Award in a new window.
Go to Search Page
View material results for all categories
Click here to go to your profile
Click to expand login or register menu
Select to go to your workspace
Click here to go to your Dashboard Report
Click here to go to your Content Builder
Click here to log out
Search Terms
Enter username
Enter password
Please give at least one keyword of at least three characters for the search to work with. The more keywords you give, the better the search will work for you.
select OK to launch help window
cancel help
You are now going to MERLOT Help. It will open a new window.
Take random steps left or right and see how far you get. You won't go far very often.
Material Type:
Paul Trunfio, Gary McGath
Date Added:
Jun 29, 1997
Date Modified:
Apr 13, 2009
TeXCAD is a CAD program that can be used to comfortably generate LaTeX picture environments. The original TeXCAD version...
see more
Material Type:
Reference Material
Georg Horn
Date Added:
Jul 30, 2002
Date Modified:
Nov 11, 2010
Discussion of a technique for estimating pi, by dropping a needle between parallel lines.
Material Type:
George Reese
Date Added:
Jun 29, 1997
Date Modified:
May 10, 2011
This site provides about 35 graphical applets on topics relative to Algebra, Precalculus, Calculus, and Statistics. These...
see more
Material Type:
Mathematics Department, The Lawrenceville School
Date Added:
Jul 01, 2008
Date Modified:
Oct 04, 2011
The site is dedicated to the Pascal Triangle and its connections to different areas of mathematics. It also contains a set of...
see more
Material Type:
Michael Frame
Date Added:
Jan 08, 2009
Date Modified:
Dec 09, 2010
From the popular Annenberg/CPB Channel workshop "Private Universe Project in Mathematics," this Shockwave simulation provides...
see more
Material Type:
Alison Reid
Date Added:
Feb 13, 2001
Date Modified:
Oct 04, 2011
This applet is a web based lab that explores the properties of rational functions. The purpose of this lab is to help the...
see more
Material Type:
Alan Cooper
Date Added:
Sep 23, 2005
Date Modified:
Nov 18, 2011
This videopaper by the Math Forum's Bridging Research and Practice Group (BRAP) of teacher practitioners and Math Forum Staff...
see more
Material Type:
Date Added:
Nov 30, 2000
Date Modified:
Jun 17, 2011
The following applet demonstrates the relationship between secants and tangents, average rate of change and instantaneous...
see more
Material Type:
Garrett Durand Heath
Date Added:
Mar 13, 2002
Date Modified:
Jun 29, 2011
This is a subsite associated with the parent site called IDEA (Internet Differential Equation Activities). The activity on...
see more
Material Type:
Ronald Poshusta, Ray Huffaker, Kevin Cooper, Tom LoFaro
Date Added:
Oct 17, 2006
Date Modified:
Mar 07, 2011 | {"url":"http://www.merlot.org/merlot/materials.htm?nosearchlanguage=&pageSize=&page=9&category=2513&keywords=mathematics","timestamp":"2014-04-16T22:16:44Z","content_type":null,"content_length":"185252","record_id":"<urn:uuid:6d0c1b6a-e5b6-433b-863f-0a7881a0199f>","cc-path":"CC-MAIN-2014-15/segments/1397609525991.2/warc/CC-MAIN-20140416005205-00231-ip-10-147-4-33.ec2.internal.warc.gz"} |
CLASSICAL GEOMETRY & PHYSICS REDUX
Bill Hammel
The modern "pictures" of Euclidean geometry are incongruous
with its modern analytical Cartesian understanding. Some
details of this are explored, along with historical paths
that led to this unfortunate situation which infects even
topology and logic, and the mathematics upon which these depend.
Spinors, in particular, are creatures of analytic geometry,
and do not arise intrinsically from either quantum theory
or relativity. There is a long way to go in understanding
the fundamental aspects of "simple" Euclidean geometry.
This is also a longwinded explanation of why and how,
spin-1/2 particles must be understood, in terms of
classical Euclidean geometry, to be pointlike, and even
classically irreducible: why supposed classical models
by angular momentum, though interesting, are doomed to
failure as genuine models of the mathematical physics
of spin-1/2. Spin-1/2 is already a matter of classical
geometry and does not need attached fairy tales to
explain it. Points of physical space are not well modeled
by Euclidean points, even in a completely classical physics
or Euclidean geometry.
This is not something I had wanted, much less expected
to show. Up to now, I had resisted the idea that an
electron could be pointlike; now, I understand better
the concept and picture of "point": it is not what I was
taught in my mathematics or physics classes. A physical
point is Q-smeared, and, even without any Q-smearing by
intrinsic uncertainty, posseses algeraic structure.
This should put an end to the vast number of physics papers
over many decades devoted to such projects of explaining
putatively quantum or relativistic particle spin in terms
of constipated pictures of Euclidean geometry. Spin is
already a matter of classical Euclidean geometry seen from
a Cartesian point of view. It requires no applications of
quantum or relativistic physics.
Classical physics and mathematics can be seen to have at least two different beginnings. Where perhaps most physicists' minds go is to the beginnings of Greek natural philosophy around 600 BCE, as in
the traditions of classicists.
The beginnings might equally be said to start with the Sumerians and Egyptians, around 3000 BCE, continued in the Egypto+Sumer-Akkad-Babylonian tradition which contains a good amount of the
mathematics and astronomy that influences our mathematics and time keeping even to this day (360 degrees in a circle, 60 seconds in a minute, 60 minutes in an hour and 24 hours in a day with ancient
calendars having 360 days in a canonical year). That older line also contains a good amount of mathematics and science that was either ignored or forgotten by the Greeks, much of which involves
number systems and their calculational techniques, many of which were lost, forgotten or ignored from Greek mathematics on. Archeologists (one might coin "archeoepistemologists") have begun to
recover some of them.
[Should you think, as a pure mathematician, that the representational system of numbers is somehow irrelevant to the abstract body of pure mathematics, I invite you to attempt algorithms for decimals
and long division using only Roman numerals. Mathematics always comes down to actually computing numbers that correspond to measurements, and that is even more so in physics. Proofs by construction
are far more fecund than proofs by Reductio ad Absurdum. It is completely cool to know that solutions to certain problems exist; it is better to have a specific solution in hand that can be
constructed by algorithm; such things inevitably involve numerical calculation.]
A true beginning to point out with the Greeks is that mathematical logic distinguishes propositions that are "provably true." There is nothing that I am aware of predating the ideas of logic from
Thales of Miletus (ca. 624 BCE - 547 BCE) and Pythagoras of Samos (ca. 572 BCE - 497 BCE). The famous Pythagorean Theorem was known and understood a millenium before Pathagoras. This is one of the
first connections known between geometry and numbers that Descartes deals with many centuries later.
What physics (astronomy) and mathematics may have originally arisen in other earlier cultures around or even before the Egypto-Sumerian line seems, at the moment to be lost to us through lack of
written records. On the basis of some Greek and Roman historical writings, there are possibilities in both Vedic and Keltic cultures. What after all were the circular megalithic structures of, e.g.,
Stonehenge all about? No written records are available that tell us anything about them, so many interesting and curious stories can be made of them, guessing by their current structures, and how
their structures might have been at various times in the past.
In the later Greek classical era, the Greeks later invented a formal philosophical system of manipulation of linguistic symbols that we call "logic", the grand exponent of which is Aristotle (384 BCE
- 322 BCE). , [Internet Encyclopedia of Philosophy], Aristotle [Wikipedia]. Aristotle. A great observer of the world, he attempted to organize and make sense of the world he saw, and in so doing laid
the foundations of logic, physics, psychology and literary criticism, among others. Continuing the ancient line of thought contained in Greek (γεομετρια), he understood geometry of space as a part of
the description of physical reality and not simply as an abstraction.
We now speak of deductive and inductive logic, the latter relying on a proper understanding of probability and statistics; but, it is not difficult to understand that the very creation of deductive
logic was had by inductive means: it did not somehow descend from heaven magically as Athena was born fully grown from the head of Zeus by a simple stroke of the axe of Hephaestus. Analogously, the
priest class of ancient Sumer developed the mythology that their written cuneiform writing was given by the gods, but its development over centuries belies that manipulative fantasy.
It is also not difficult to understand that deductive logic needed the sense of a spoken language that was evolved culturally: neither did language descend magically from heaven, nor did its written
forms by which one generation can communicate to the next with more accuracy and precision than oral tradition provides.
This is not to say that oral traditions are then made irrelevant; the opposite of that conclusion is an important point being made here. Precision and accuracy can also often be better propagated by
an oral and interactive process, teacher to student. Some understandings cannot, and are not written with precision and accuracy - as masters and good students of Zen Buddhism would understand
immediately. They are induced thought patterns outside of language, and certainly outside of written symbols of language.
The object is, of course, to grasp the language and the symbols by which it is represented in terms of a conceptual level of understanding.
The writing down of certain things was even forbidden in some cultures, perhaps because it degraded the art of memory. We are now so dependent on the writing of things because no human being being
can remember the volume of what is called knowledge, even in a relatively small rarified area, e.g., analytic number theory. If one thinks of areas cut out with a more broad sword, like, mathematics,
physics or philosophy, the memory requirements are almost beyond conception, much less human abilities.
But, the oral traditions have never disappeared, though they seem to be disappearing now when we need them desperately. Scientists and scholars tread narrow paths through forests of knowledge to get
to an unknown frontier, and it is taking longer and longer to do that. Elder guides are needed, and they are disappearing for too many various reasons.
Euclid ca. (325 BCE - 265 BCE), summarized and organized the subject of Greek geometry in his systematic "Elements" on the subject, which also happened to include the foundations of number theory,
and which became the foundation of the teaching of mathematics that persists to this day, but with about a 1000 year hiatus of the oral teacher-student tradition which can be traced historically as
being essential in all the arts and sciences. I was rather amazed at one time to realize that my musical composition pedigree in these relationships goes back to Haydn.
Over the centuries of this new, derivative teaching, there arose a set of "pictures" of "Euclidean geometry" that are purported faithful descriptors of Euclid's combined axioms of points and lines
(extended to surfaces) with an Aristotelean manipulation of the linguistics involved. This was not a mathematics that was as potently symbolic (and linguistically specialized) as is modern
Mathematics has become so much of a language in itself that modern mathematicians can make sense to each other using standard written mathematical language without being able to speak each other's
human language.
Mathematicians have long since extended the general ideas, axiomatics of geometrical pictures from 2 to n dimensions [Sommerville 1958] and even to the curved spaces which quite literally leave
Euclidean geometry specifically flat, beginning with Bolyai (1802-1860) and Lobachevsky (Lobachewsky) (1792-1856) who simply began questioning Euclid's 5th axiom and its logical independence. The
conceptual breakthrough to generally curved spaces reaches an apex in the work of Bernhard Riemann (1826-1866).
The important ideas of René Descartes (1596-1650), [Internet Encyclopedia of Philosophy], finally combined the notions of ancient Euclidean geometry, together with the developed pictures or
descriptors with the older line of numerical calculations leading to the Islamic developments in algebra. It is unfortunate that the novel and inventive Descartes became intertwined with both
continuing Western theism (spirit converted to law and logic) and the equally pernicious escape from it called "deism" (spirit converted to a machine). Neither supports reality, and both have the
same level of epistemological enlightenment - commensurate with that of the tooth fairy, a product of Scholastic "thinking", the ultimate perversion of Aristotelean logic as instigated in Western
theologized philosophy by Thomas Aquinas. There are those who would disagree with the perversion part; I do not care. See, perhaps Sketches in the History of Western Philosophy for an academic and
detailed statement of the historical realities.
The Islamic line of mathematics and science, rather probably stemming from the Sumerian line was introduced to the Western world during the time of the so called "Reconquista" (718-1492) [Wikipedia].
and the fall of Toledo in 1085 CE. (The Reconquista is one of those great lies of history: there was no taking back of Iberian lands by Christians since land was occupied by Vandals, whose name has
wrongly been taken pejoratively into English.) Islam Spain and the history of technology At the same time, many of the destroyed works of the ancient Greeks, perhaps especially those of Aristotle
were also rediscovered mostly in their Arabic translations.
The continuing scholarship of Islamic civilization from both the Greek and Sumerian lines were essential to the "catching up" as it should be said, of previously destroyed knowledge in the West.
Artistically and intellectually speaking, the translation from the mostly unknown (to the West) Greek into Latin of the Corpus Hermeticum, commissioned by Cosimo de Medici in 1440 is the seminal, and
mostly ignored, turning point of Western culture that together with the contents of the great library at Toledo leaves Western culture not still wallowing like pigs in mud holes. [There is a great
deal of nonsense now surrounding the Corpus; beware.] See Marsilio Ficino (1433-1499). The cultural turning point was not simply a matter of the decline of medieval sociology and the rise of an
economic middle class as many "history" textbooks would have one believe. Seek simplicity - and distrust it. The Renaissance was indeed, as the name says, a rebirth of and rediscovery of culture, art
and science that had been deliberately suppressed and destroyed. Why would anyone think that all had been reclaimed?
Modern cultures, regardless of mother tongue, have adopted the rather perverse notion that language, and its methodical manipulations, is somehow the faithful encapsulator of all thought; what
These materials learned through the Renaissance were "rediscoveries" of not only forgotten, but suppressed and destroyed knowledge of and from the ancient world. Much is still lost, unknown and most
probably irreclaimable.
"Dark Ages" are where the history has been destroyed; there are a number of them, in various cultures. These are eras of glossed over history, all too gently placed bookmarks that ignore the fact
that both history and knowledge is too frequently lost, and all too often by purposeful destruction.
During the ages from Euclid and Archimedes (287 BCE - 212 BCE) to the Renaissance and its intellectual origins in the library of Toledo, the languages, texts and translations of ancient texts had to
some extent been regained, but the thought patterns were lost, and remained lost. They are still lost, in the same sense that the arts, the art of science or the arts of engineering can be lost.
The continuing lines of understanding, thought and "tricks of the trade" that pass only from teacher to student had not been merely severed; that line had been forcibly annihilated. We can only guess
at what came before.
The teaching of geometry is the subject, and it was a resuscitated subject, raised from the dead many centuries after Euclid. (One could say the same of reason, logic and Aristotelean philosophy,
philosophy generally, optics, biology, and a scientific concept generally, not to mention simple curiosity.) There is no available record of how our cultured ancestors may have thought about their
formalisms, or no record of their results. We have documents on ancient Greek musical theory, but have little idea of what the practices were, and there is also little idea of what the nouns actually
symbolized. What was, really, what we call a "mode"? Only the destroyed line of student to teacher connections might have told. Words can have subtle, and not so subtle distinctions in their use,
depending on where and when they are used. The word "raga" has rather different meanings in Northern and Southern traditions of classical Indian musical theory: it does not signify by itself, a
unified and well defined concept, as it might appear to a westerner.
Possibly, if that line of connections had not been broken, it would not have taken 1000 years for Descartes to have caught up with the connection of geometric thought with the beginnings of algebraic
thought that is also, at least conceptually but not exclusively, connected with the schools of mathematics and astronomy that existed in India (Kerala) at least as far back as 310 BCE with Indian
mathematics - and I suspect the existence goes considerably further back. As a matter of lost connections, one might simply notice that Elamite is a Dravidian language [Wikipedia], as is Malayalam,
the language of Kerala, intimately related to Tamil.
A point of this small history is that when the ancient mathematics and science had to be reconstructed from manuscripts and translations of translations, the thought patterns that lay in back also
had to be reconstructed - which is almost to say fabricated from thin air (or fat air?). While many of the early writings can be read literally and translated, the corpus of interpretive thought in
back of them is lost to us.
There are two pertinent peculiar parts of this story. One is that the pictorial descriptors are fabricated, much later than the Greek mathematical texts and that we mistake them as truly representing
Greek thought. The second, which is truly rather amazing, is that these fabricated descriptors have for the most part dominated all mathematical and physical thought, even beyond the work of
Descartes, despite the fact that this combining of algebra and geometry opens up entirely new worlds of mathematics and actually does it in unequivocal terms lead to quite different descriptors of
It might be worth mentioning that while Descartes did indeed work out the utility of using algebra to solve geometric problems, in Le Geometrie, an appendix to his Discourse on Method, he did not
conceive of analytic geometry, or even the celebrated "Cartesian plane" as we know them today. Descartes' Life and Works Arguably, Fermat (1601-1665) had as much to do with such developments. They
were not exactly on friendly terms, Descartes being the arrogant prick that he was. Even well before this, Al-Mahani, ca. (820-880), was engaged in the solution of geometric problems by algebraic
means. Relationships between the algebra of Al-Khwarizmi ca. (780-850), number theory and geometry were in fact signal aspects generally of this much earlier Arabic mathematics that itself was well
aware of prior mathematical and astronomical works from India.
The idea that geometry was a theory of the physical world and not merely some cute mathematical structure was part of ancient Greek physics, and that idea remains in modern physics, without bothering
to explore the new worlds of physical geometry opened up by Descartes, et al. The word itself, (γεομετρια), meaning "earth measurement" should be clue enough.
This is not to say that mathematicians have not done the exploring, because they have; I simply mention algebraic geometry and geometric algebra along with the writings of W. K. Clifford (1845-1929)
[Wikipedia], The Work of W.K. Clifford B. Riemann (1826-1866) and A. Einstein (1879-1955) [Wikipedia] that lead to the Clifford-Riemann-Einstein program of geometrizing physics which is quite
impossible without understanding connections between algebra and geometry. We do not get a conceptual leg up on this program if we noodle about with, wrong and restricted pictures that ignore and
defy those very connections.
Physicists' notions of space and time geometry with the small exception of the relativistic spacetime, have not changed much (in operative consensus) to include the Cartesian relations in centuries.
Because of this, physicists perhaps, have strange and antiquated working pictures and notions of space and time that produce confusion and paradox where there should be none.
The clearest example of such confusion is with the idea of spinor, (which we devoutly wish should have the universal spelling "spinnor", so that it is truly pronounced as it is spelled) over which
much fuss and many incantations have been said repeatedly, both relativistically and quantum mechanically. This mystical nonsense and confusion continues in physics textbooks to this day.
Spinors, and indeed Clifford algebras [Wikipedia] (an article with incisive mathematical particularity) have absolutely nothing to do with relativity theory or quantum theory per se. They are purely
aspects of classical geometry as algebraically suggested in principle by Descartes, and which can be developed with no additional physical assumptions, through a little more thought and even less
formal algebra.
Clifford algebras can arise in several different ways, all of which have origins in classical Cartesian-Euclidean geometry. One way is by simply considering the factorization of a quadratic form,
which is after all exactly what Dirac (1902-1984) (Paul A.M. Dirac - Biography and Nobel Lecture) did to derive his equation for the electron with spin-½ from the Klein-Gordon equation. [Wikipedia]
However it may seem, there is nothing magical here. The idea is routinely propagated that the electron spin is somehow a relativistic phenomenon. [Dirac equation - Wikipedia] when very clearly, it is
not: the very same kind of factorization of the Laplace operator (a quadratic differential operator) [Laplace's equation - Wikipedia] in the nonrelativistic Schrödinger equation [Wikipedia] can be
done, and it yields the Pauli algebra, [Pauli matrices - Wikipedia] an irreducible representation of the Lie algebra Lie algebra [Wikipedia] su(2), [Special unitary group - Wikipedia] whose
irreducible representations describe quantum spin [Spin (physics) - Wikipedia] generally, as it is understood in an Euclidean space of three dimensions. One can, in fact, pull this very same
"factorization trick" in any inner product space over R or C in any dimension, and with any signature, and so discuss the spin representations of SO(p, q) and SU(p, q), and associated Clifford
algebras generally.
Physically, Dirac's equation which takes the 2x2 ad hoc Pauli algebra to the 4x4 Dirac algebra merely adds the difficulty of ± signs for E by virtue of taking a square root; both the relativistic
Klein-Gordon equation and the nonrelativistic Schrödinger equation factorize to expose spin-½.
Moreover, both these equations are equations expressing a conservation of energy, as are all the core equations of physics.
Regarding Dirac's factorization, we might just as well have written
p[x]² + p[y]² + p[z]² - E²/c² = - m²c^4
and factorized the quadratic form of the LHS of the equation. The appropriate Clifford algebra would still have materialized by purely algebraic means in a nonrelativistic context. Now figure out
what that means - *geometrically*. This is not difficult if you transcend the erroneous idea that mathematical points are structureless. (Physical "points" as irreducible geometric atomics are even
more complicated.)
There are many ways of conceptualizing spinors (a hint, perhaps to their necessary ubiquity), one is as "square root of a vector", which seems only to be reasonable in a 3-Dim space. Spinors, can
also be understood sturcturally as two sided ideals in Clifford algebras, becoming the elements of the carrier space of the representations of the Clifford algebras, and then of orthogonal Lie
groups. There is no magic; just simple algebra that has nothing to do with quantum mechanics, and has also nothing to do with relativity theory.
Returning to the idea of square root of a vector, and generalizing via axial vectors to bivectors, understand spinors as eigendirections of antisymmetric forms that represent bivectors. I thank R. M.
Kiehn for this understanding and connection.
Any student of quantum physics who has looked at the meanings of spinors has encountered the "belt trick", nicely explained by John Baez in week61 of his always enlightening "Finds in Mathematical
Physics". It is an explicitly macro topological phenomenon, not a matter of quantum fiddling.
There is nothing, either relativistic or quantum mechanical about this; it is a macroscopic, yet non global topological property of the physical E^3 in which we seem to exist. We seemed to have
developed, on the basis of our false descriptors, the idea that topology is only considered as local or global, and that there is no significant understanding inbetween. The belt trick actually puts
the lie to that assumption, since it exists, is demontrable and *is* in between.
With regard to the rotational symmetry of E^3 the appropriate connected Lie group, SO(3) is *doubly* connected, and this is true for any SO(n), n > 2. This may have something to do with the
mathematical existence of the spinor representations of rotations; on the other hand, the Lie groups SU(n) also generally have spin representations which are extentions by complexification from SO
(n). An important aspect of spinor representations is a confluent isomorphism between associated Lie algebras, e.g., the isomorphism between the Lie algebras so(3) and su(2). But, while the
associated spin group for SO(3) happens to have a nice confluence with a classical SU(2), this is not the state of affairs with SO(n) for n > 2. Spin groups are generally not classical Lie groups;
the case of SO(3) is an isolated accident, but given the stress that this relationship is given, it can lead to gross misunderstandings of the mathematical reality. [This description does not answer
all questions, nor does it pretend that all relevant questions can be answered by existing mathematics or by me; many geometric questions of the Cartesian viewpoint remain.]
There is nothing magical about any this; it is all "classical" (but unfinished and possibly lost) mathematical understanding, and also most interestingly, a new understanding of classical physics.
Physical reality does seem to take advantage of possibilities of the mathematical model that combines the logic of pure Euclidean geometry with the natural algebraic extensions by Descartes - which
is interesting, bemusing and almost annoying because of the simplicity. If you can get the mathematics understood, the physics is not far behind. This is a bit more Platonic than I was prepared for.
What is the problem, and why does this stuff seem mysterious? It *seems* that way, because we have paid entirely too much attention to and placed entirely too much stock in those long ago fabricated
descriptors, and authoritarian text books.
We have become bovinely wedded to various descriptorial notions, in a 1000 year disconnection of thought patterns, of Euclidean geometry that refuse the Cartesian understandings, e.g., the idea that
"physical" points have no structure - because that is how we have been taught to picture them, and *not* because the mathematics tells us otherwise, which, in fact, it does. It is also the model that
our vision supports.
The question is, what does the physics tell us about the mathematical models of geometry and the pictures that we "glue on" to them? The physics tells us that the developed Cartesian mathematics is
quite right, and that the simplistic Medieval geometrical pictures that do not faithfully represent the Cartesian-Euclidean formulation in algebraic mathematics are wrong; they are overly simplistic
to the point of serious mathematical error in physical theory.
Why should there be a problem? The only problem is that we have persisted in "picturing" Euclidean geometry wrongly while the mathematics has told us rightly that our pictures are wrong, and have
been, for well over a century. The correct ideas have been available all along; they have simply been ignored. They need to be refigured and reinstalled in the minds of young mathematicians and
mathematical/theoretical physicists.
These ideas are not at all new, and were essentially understood by W. R. Hamilton [Biography of Hamilton] and J. C. Maxwell (1831-1879), though they did not enunciate them so forcefully: it was an
embryonic time for the resurgence of these concepts, and they were unsure.
In the first edition of Maxwell's great treatise on electromagnetics (1873), he did indeed flirt with idea of formulating electromagnetic (EM) theory in terms of Hamilton's William Rowan Hamilton
(1805-1865) [Wikipedia] Hamilton quaternions, [Quaternion, Wikipedia] and did a number of translations of the equations into a quaternionic language. In the following edition of 1884, by Oliver
Heaviside (1850-1925), the quaternionic sections were removed, and this probably was a great infortuity. They were almost side notes in the general progression done in Gibbsian [Josiah Willard Gibbs
(1839-1903) Willard Gibbs, Wikipedia] vector language, and may have seemed to get in the way.
O. Heaviside edited like some modern movie editors who have the strange idea that "moving the action along" is somehow always more important than actually making sense in telling a cohesive story.
You can get Maxwell's first edition from good libraries (interlibrary loan) and see that I have described the historical and mathematical realities reasonably accurately.
It should be noted, however, that Maxwell did write a manuscript in 1870 on the application of quaternions to electromagnetism that is reprinted in Vol. II of Maxwell's collected works. Maxwell was
not thinking in field theoretic terms, and did assume a physically real aether with classically physical proprties, and had the mathematical machinery available for waves in an elastic medium, both
vector and scalar parts, as given in [Morse 1953], pp 142-144, where it is shown that superluminal scalar waves are possible only if the medium is compressible. If spacetime is actually compressible,
it should only be so at energies that we have not yet achieved, somewhere in the neighborhood of greater than 250 GeV.
True historical reality is much like the reality of physics in that we never get to perceive it directly, but only indirectly through constructed models and theory applied to them. Since historical
models are rarely deniable and often fabricated, they have an innate dubiousness, and do not suggest a convergence on any truth, and so should be taken only with the proverbial grain of salt.
There are those who have claimed that Maxwell originally cast EM theory in quaternions, that the equations were somehow more general, and that some sort of conspiratorial suppression of the "real
truth" was engaged in, particularly by Heaviside. They speak wrongly; the quaternionic equations were not more general, and Heaviside simply did not understand the purpose of quaternions; nor was
Maxwell particularly clear in their significance, supremely careful mathematician and scientist though he was.
Written history does have a clearly conspiratorial aspect, but this one is pure confabulated nonsense. The first edition of Maxwell's treatise on EM used Gibbsian vectors throughout, with
quaternionic afterthoughts, and, as Maxwell himself shows in this treatise, the quaternionic formulation is isomorphic to his own vector formulation - which to a large extent missed some of the very
interesting points of modern EM formulations by David Hestenes (1933-) [Wikipedia] in terms of Clifford algebras, or as Hestenes would put it in "geometric calculus". Further developments in
understanding classical EMT can be found in the works of [E. J. Post 1963] and R. M. Kiehn: Maxwell Theory and Differential Forms
Prof. Kiehn also gives another way of understanding spinors as eigendirections of an antisymmetric matrix that has implications for the understanding of the Maxwell equations and its spinorial
I seem to remember also an earlier work of Penrose on spinorial solutions to the Maxwell equations related to his twistor theory [Wikipedia], in the Journal of Mathematical Physics, but I have not
yet rediscovered the citation.
Maxwell's use of quaternions is considerably more messy and less elegant than modern formulations, showing, I would think, that Maxwell was greatly intrigued by quaternions, but that he was also not
exactly facile with them; on the other hand, neither was anybody else. Hamilton himself continued to search for their logic and meaning through to his death. Quaternions are not a part of the general
education of mathematicians, even today; a pitiable condition.
Maxwell was definitely not talking about functions of a quaternionic variable, as say, noncommutative extensions of the theory of analytic functions of a complex variable. This mathematical theory of
noncommutative analysis was not yet a developed subject at the time of Maxwell, and is only now a subject of research. See also Notes on Noncommutative Geometry
All of quaternions (for DIM=4) and Clifford algebras have now elucidated a generalized meaning of EM theory, as has also the approach to the Maxwell equations though differential forms. EM is clearly
a topological (not metric) theory that expresses itself in antisymmetric forms (not symmetric forms). You can express the same sorts of equations on discrete, simplicial [Notes on Simplicial
Homology], networks using coboundary operators [Sorkin 1975], or Kuratowski closure operators. RMK Articles: Specials and Freeform Index Page [R. M. Kiehn]
One of the extended meanings of the Cartesian-Euclidean understanding is that the geometry of physical space and time should most generally be understood in at least locally complex coordinates, in
addition to the addition of the noncommutative Cliffordian and nonlinear spinor structures associated with physical "points". Physical points do indeed have physical structure, and that structure
seems best described by both a complex and Cliffordian nature. This is an understanding that I believe Clifford himself had, but that has been only sporadically picked up on.
In building classical spinors on an E^3 e.g., by starting with Eli Cartan's map of E^3 to Hermitean 2x2 matrices, [Contexts for Spinor Algebra], we discover the Pauli matricies, and also that the
building of spinors as null vectors, that we must allow that the real vector coordinates of E^3 must embrace a complexification that must be meaningfully geometric. One might also simply recall
"Cayley-Klein" - Google Search in E^3 and their purely classical origins, and uses. This is mathematics that begins in the the 19th century, before the physical quantum and relativistic theories -
and has everything to do with spinors, and the Pauli algebra, su(2) of the covering group of so(3).
A similar pattern of necessary extensions from the old Euclidean descriptors of E^3 to their complexification and Cliffordization can happen in any E^(p,q), p+q=n. It might be worth noting that when
a real Clifford algebra Cl(p,q) associated to an E^(p,q) is complexified that the inner product signature (p,q) is essentially lost since the generating basis elements with suitable factors of 'i',
can all be legitimately made to square to ±1, uniformly, and that while real Clifford algebras have "periodicity structure" mod 8, complex Clifford algebras have periodicity structure mod 2, and so
have a simpler structure theory.
Independently of the physically meaningful and necessary complexification, also consider the local analysis of curvature forms in differential geometry, e.g., Petrov's classification of Einstein
spaces, [Petrov 1969] (in which there is a mathematical error in Type III spaces that I will get around to defining sometime, the meantime, see [Hammel 1974]), or the similar analysis of the
electromagnetic field tensor, which shows the complexification to be physically meaningful in perfectly classical senses that are not directly connected to any relativistic assumptions. See again,
the works of R. M. Kiehn, RMK Articles: Specials and Freeform Index Page.
Once again, in both these cases, the physics tells us that physical geometry cannot simply be described by the old Euclidean pictures, and that these pictures are entirely too simple and simpleminded
in thinking about the physical geometry of space, time and spacetime. There is much more there, physically; it is the case that we have not been looking at physical geometry with this necessary
mathematical understanding, and have become confused, making the seemingly mysterious mystical, when the mystery is rather a concocted illusion in the first place.
Instead, Wheeler's notions "pregeometry", and "geometry without geometry" have taken hold in physics, mostly because J. A. Wheeler (1911-2008) [Wikipedia] was a brilliant guy, and has shown the way
in many areas. Anybody can miss the obvious, and a conceptual elision does not detract from Wheeler's genius. The return and connection with Wheeler's ideas might be through the concepts of topology
and new understandings in differential forms that go beyond the usual language of tensor analysis of classical fields worked out by R. M. Kiehn. See the wealth of this at Cartan's Corner, and more
particularly at RMK Articles: Specials and Freeform Index Page.
The error of the old pictures can be, and likely is, besides theoretical and philosophical inertia, a continued error of language: After all this time, i=sqrt(-1), is still called "imaginary". The
complex field is the smallest algebraically closed field. There is nothing imaginary or mystical about this, though it seemed there was at the time of its invention or calling into being; yet, the
designation "imaginary" remains, and befuddles students even now. Why still, "imaginary"? History! Tradition! Toscanini described tradition as "the last bad performance". Enough said? We also
prejudice our thinking simply by saying that a point "has dimension 0", and conclude from that its alleged structurelessness, and triviality. This picture is based on erroneously reconstructed
Euclidean geometry.
Physical "points" are not the same as mathematical points.
The technomages of mathematics should already have had enough fun messing with the heads of the acolytes on that score, and equally so in the matter of "classical geometry". Both a classical
Euclidean point as understood through Descartes, and a classical *physical* point do have structure, and it is both complex and spinorial/Cliffordian.
This means that classical physical points, having spinorial structure are, in a language of quantum theory, Fermionic objects. This is not a trivial conclusion, or understanding. See, e.g.
Introduction to quantum set theory and its (incomplete) sequel Set theory, quantum set theory & Clifford algebras. In the last reference, the exploration suggests that even a classical point in a
space 3 dimensions is well described by the 8 dimensional Lie algebra gl(2, C), and that a classical point in 4 dimensional classical spacetime (of which we are not thoroughly convinced) should be
described (perhaps, within a topological neighborhood) by the 16 complex dimensional complex space of the Lie algebra gl(4, C).
Perhaps, to beat a dead horse in the matter of complex numbers being required, in any formulation of QM, which always needs an expression of interfering alternative outcomes, a complex structure
cannot be avoided. [Mackey 1968] One cannot, e.g., in Q statisical mechanics, willy-nilly, separate q-space from p-space; they must be considered together. Then the complex structure on phase space
is unavoidable, as it is also essentially unavoidable in QM. The essential noncommutativity of Q phase space is also unavoidable; so it presents itself as a prototypical noncommutative geometry.
It is often suggested in QM texts that the complex projective Hilbert space is somehow the essential element of QM; this is patently wrong on two counts.
First, all separable (all one needs for QM) infinite dimensional Hilbert spaces are isometrically isomorphic; it is so then also for the associated projective Hilbert spaces. This leaves no room for
distinuishing physically different systems.
Second, and complementarily, the physics is actually captured in the structure of the algebra of "observables" corresponding to a *-algebra of linear operators (some, necessarily unbounded, if you
believe in CCR) acting on the Hilbert space, or more likely on a common domain within the Hilbert space of both the q and p operators. In any case, the kinematics is defined by the operator algebra,
and the dynamics is defined by a semigroup of operators acting on the *-algebra.
The Hilbert space can actually be conceptually eliminated by hiking it up into the C^*-algebra, using projection operators, as the boundary (set of extremal points) of the forward cone of the
algebra, whose interior describes the density operators of states in quantum statistical mechanics. The algebra of operators containing both the observables and the states of the system is the thing,
not the Hilbert space, a point made long ago, and I think originally by I. Segal. This does not seem to have caught on.
Yet another clue from physics that the fundamental picture tools of Euclidean geometry need to be reconsidered is in the concept of Supersymmetry - Wikipedia].
The distinctions among Bosonic (symmetry of interchange) and Fermionic (antisymmetry of interchange) particles with their different statistical behaviors are separate symmetries. Though there is
little in the way of physical evidence to suggest it, it would be completely cool to have a theoretical framework which combines these two "particle types" into simply "particles". This did lead to
the idea and investigation of superalgebras; it turns out that Clifford algebras are, in fact, superalgebras. Viewing Euclidean geometry in proper Cartesian fashion then provides evidence on the
mathematical level for the essential validity of superalgebras which express supersymmetry. While physical evidence suggests the mathematical language in which physics is expressed, it is also true
that the language suggests how to view the meaning of the physical evidence and mathematical language. Implications in both directions are operative here.
Perhaps, it is no great shock to see that these extended Cartesian descriptors of physical geometry fit nicely into Klein's Erlanger program for geometry where a geometry is specified by the action
of a group on a space together with a set of geometric invariants of the group action.
One way of understanding "structure of a point" in a classical sense of pictures goes like this: In E^3, for example, any point can be considered a Euclidean ball with antipodal points identified.
Physicaly and classically, we would not observe this substructure directly, and space would appear to be conforming to the old picture. Passage of anything "through the ball" is not apparent from the
outside. Such a ball is topologically equivalent to an SO(3) manifold, or a "contracted toroid with a half twist": take a finite cylinder or radius r, score it with lines parallel to its centroid.
Holding one end fixed, twist the other by π radians. Now, glue the two disc ends together. There are "geodesics" on the surface of this solid toroid which must go around the central hole of the
"doughnut" twice before returning to the starting point. Take the radius of the circle that is now the centroid of the Urcylinder to be R. If we contract the toroid's major radiu R→0, allowing the
substance of the toroid to pass through itself, the result is a ball with its antipodal points identified. The manifold of the Lie group SO(3) can be understood as the same thing by allowing two of
its angular parameters to have range [-π/2, +π/2] and [0, 2π] covering a spherical surface, and a third with range [0, π] that as a radial coordinate with the two surface coordinates finally
determines a ball in E^3 of radius π with antipodal points identified. Think Cayley-Klein parameters.
If one thinks of a physical space being made up of such structured physical points, an idea resembling Bose-Einstein condensation of the space itself is not too far behind - given enough points.
Simply put two such projective spheres in contact, and consider the identifications; continue the process. How many such classical angelic points may dance on the surface of another? As many as want
to, within some finite limit?
Limitations on that arise when points acquire quantum bulk, and some principle (Pauli, e.g.) that holds them apart, and from this notion of dimension can arise in the way that dimension is related to
the "Kissing Problem", of how many unit spheres in n dimensions can be tangent to a central unit sphere. [Conway 1993] If the general concepts of geometry need some rethinking, then most probably do
the general concepts of its abstraction, topology.
Keep in mind that a unitary quantum theory is spoken of in terms of a projective Hilbert space, and its sphere where antipodal points are identified.
There are fundamental geometric problems with combining the Poincaré symmetry of a relativistic spacetime and the internal symmetries of the Standard Model [Wikipedia] in elementary particle physics.
The suggestion is, of course, that these problems stem from ignoring the necessities of the very classical geometry upon which the physical model is predicated.
This extended way of seeing E^3 involves to start, only an additional structure of points that is physically consistent with the primitive Scholastic descriptors. One only gets to "see" such
structure physically when the quantum (fine grained) nature of physical geometry is probed or taken into account, and the classical points are fuzzed according to the basic necessities of quantum
Within the Brouwer-Urysohn concept of dimension, [Hurewicz 1948] one passes hierarchically in three dimensions from point to line (path), to plane (surface) to volume. We have already discussed the
necessary complication of the concept of Euclidean "point" from a classical viewpoint.
From a bottom up viewpoint, the obvious suggestion is that a classical 0-dimensional geometrical point, regardless of the space of which it is a member, should be replaced mathematically and
conceptually with the smallest complex Clifford algebra. The question, of course, is what is the "smallest"? In one sense, the smallest is simply the complex numbers; but these are commutative, while
the fundamental conceptions of quantum theory are noncommutative. So, the next and only possible choice of smallest "quantum point" is the complex Clifford algebra of complex dimension 4, and real
dimension 8, which happens to be represented by the algebra of complex 2x2 matrices. This also happens to be the definining and lowest dimensional faithful IRREP of the Lie algebra gl(2, C).
This geometrical viewpoint implies that all fermionic spin 1/2 particles are essentially geometrically irreducible, and "quantum pointlike". Thus, explanations of their existence in terms of wrong
Euclidean pictures will fail.
If one seeks further explanation, it must be found within a quantum theoretical language within the constraint that this quantum point with structure is still essentially irreducible, from the
viewpoints of classical geometry and from quantum theory. The further subtleties are in what quantum theory may contribute beyond the classical geometry.
The new business is then the concept of line/path which has to do with either or both the fitting together of quantum points, or from an opposed direction, the boundaries of quantum surfaces. I will
come back to the "line problem" soon, I hope.
Whether or not I live to complete this concept related to terms of Brouwer dimensions greater than zero, this is not some sort of new "nutsy" stuff: it is rather the digging out of aspects of the
mathematical understanding of Euclidean spaces that "should" have been understood and well known centuries ago.
Seeing this now, after all this studious time has passed, I am quite a bit mortified, and feel like an idiot for not having seen the obvious years ago.
There is nothing at all peculiar here, even in modern days, and maybe even especially in modern days, about rediscoveries in mathematics, physics and other sciences. The nicely attributed "Ising
Model" of spontaneous magnetization as a phase transition in statistical mechanics comes easily to mind.
Ising model - Wikipedia.
Ising originally used the concept of nearest neighbor interactions of spin 1/2 entities in a lattice to explain spontaneous magnetization. He worked out the model in one dimension and found that
there was no such behavior. The model was generally abandoned for a while, but revived by Heisenberg. The first problem turned out to be that in one dimension, there are not enough (2) nearest
neighbors, but that in two dimensions, 4 nearest neighbors are enough, and so certainly in three dimensions with 8 nearest neighbors there are certainly enough. Onsager, a chemist, showed in a most
clever but tedious way that in two dimensions, spontaneous magnetisation does indeed exist; just solving the problem and showing this was very difficult, and no one had done this before Onsager.
[Huang 1963], and references therein.
The models of the physics of the universe in which we live is has been highly restricted by the barriers of concepts based on the illusions of our common perceptual neurology, so much so that our
intellectual concepts even neglect primitive errors that obviously contradict reality. We rely entirely too much on direct visual perception.
Because we are so physically large, we have no perception of the ultimately small. Our visual perception is only of molecular order. Discovery of the quantum regime has broken one barrier of what is
possible and what is also necessary in the small for our own existence. Were it not for the existence of quantum indeterminancies, this mysterious universe in which we live could not exist: it would
be a dead thing that would not allow the birth of anything new beyond its primordial existence. Time could not have any meaning, or exist in any sense.
We construct a 3+1 dimensional existence because of our size and our inherited neurological substructure, and base our mathematics and mathematical models of reality on those things; we have
"discovered" molecules, atoms and certain elementary particles. What else may be discovered?
Even logically and mathematically it is clear that neither infinitesimals nor infinities can exist in reality; yet, all current physics is based on these 19th century interpolational delusions. Why?
Because that monumental hierarchy of of mathematics and mathematical physics exists, and it is how we were instructed to think, despite the fact that it is clearly wrong.
All of physics has manifestly gone awry, begiginning with the incompatibilities of quantum theory and relativity theory. Those began about a century ago! Ptolomeic epicycles come to mind.
This a minimal reconstruction of geometrical pictures, but there is no reason to suspect that complex Clifford algebras are the end of story. Spinors in higher dimensions than 3 are more complicated,
though still connected with Clifford algebras. There are at least further algebraic and geometric connections with division algebras, Lie, Jordan and Malcev algebras. This is still an open door to
epiphanies in the physics of space.
In considering the various possible connections from complex Clifford algebras for space itself, one should be mindful of the "time" concept and its multiplicities of meanings, but also of the
tantalizing SU(3)xSU(2)xU(1) symmetry of the "standard theory" or elementary particle physics, and that while EMT is an essential topological expresion of space and time, concerning antisymmetric
tensorial entities, general relativity theory is a metric theory concerning symmetric tensorial entities. Symmetry and antisymmetry are algebraic qualia, independent of "pictures".
To interpolate a mathematical structure between the complex Clifford algebra at a physical point and the mathematical point of continuous manifold containing the metrical substance of GR, we will
need something akin to a pseudohermitean complex manifold, where the complex structures in the local tangent spaces are not necessarily integrable.
This pseudohermitean complex manifold need not even be a manifold as such: it could be discrete in the quantum sense. Both EMT and GR can be done on simplicial complexes, but the 0-dimensional
structurless mathematical point must be replaced with a physical point modeled on its most elementary level by a complex Clifford algebra.
"The truth points to itself" -- Kosh Naranek
Many thanks Prof. R. M. Kiehn for very helpful discussion, for asking pregnant questions, and for leading me to his work that made so many connections for me. Thanks also to Mitch Smith for prior
discussions on continua, models and ancient mathematics, to Richard B. Carter for encouraging me to read Descartes, and for old and seminal discussion, to Elihu Lubkin, to Leonard Parker and Dale
Snider for discussion and comment on the geometrical meanings of spinors, and last, and certainly least, to my old physics teacher Harvey Kramer for repeatedly angrily telling me that every
unorthodox thing I thought was perfectly insane, including my naïve reinvention and working out of fractional differential calculus that was a new (useless and irritating) idea to him. It is so
wonderful to have knowledgeable teachers; in total, I think there were five or six: two were in musical composition. Others were mentors of one kind or another, Don Gelman, for one who guided me as
an undergrad. Love on their heads.
Nothing here has been supported by governmental or criminal corporate pseudoscience; so, no results, conclusions or opinions have been paid for.
"We live for the one; we die for the one." -- Zathras
"Zathras is used to being beast of burden to other people's needs. Very sad life, ... probably have very sad death, but at least there is symmetry."
Top of Page
Email me, Bill Hammel at
© August 2006 by Bill Hammel (bhammel@graham.main.nc.us).
Permission to use for any noncommercial, educational purpose.
This copyright and permission notice must appear in all copies.
Permission is also granted to refer to or describe these
documents in commercial books, products, or online services.
These documents may be freely reproduced, copied and disseminated
by any electronic, digital or written means, but in no case may
such copying or dissemination be charged for. The idea is very
simple, no person or body has supported any of the original
works contained in this pages. They are works of love given
freely. I find repugnant the idea of someone expropriating,
for profit, what I give freely. If you have a problem with
this, ask; rules always have exceptions.
The URL for this document is:
Created: September 26, 2005
Last Updated: September 27, 2005
Last Updated: October 19, 2005
Last Updated: November 11, 2005
Last Updated: November 14, 2005
Last Updated: December 5, 2005
Last Updated: December 18, 2005
Last Updated: December 31, 2005
Last Updated: June 17, 2006
Last Updated: July 22, 2006
Last Updated: October 22, 2006
Last Updated: November 3, 2006
Last Updated: February 2, 2007
Last Updated: October 31, 2007
Last Updated: January 17, 2008
Last Updated: October 16, 2009
Last Updated: January 27, 2010
Last Updated: March 12, 2010
Last Updated: December 2, 2010
Last Updated: January 5, 2011
Last Updated: July 9, 2011
Last Updated: November 9, 2011
Last Updated: | {"url":"http://graham.main.nc.us/~bhammel/MATH/cgpredux.html","timestamp":"2014-04-21T02:01:03Z","content_type":null,"content_length":"63655","record_id":"<urn:uuid:89491a95-6b4d-45b8-b3cb-3a408bf121ea>","cc-path":"CC-MAIN-2014-15/segments/1398223202774.3/warc/CC-MAIN-20140423032002-00639-ip-10-147-4-33.ec2.internal.warc.gz"} |
Math Help
February 22nd 2010, 02:33 PM #1
Junior Member
Aug 2009
Hi everybody,
We consider the following differential equation:
$(E) : y''+2y'+2y=-(x+1)e^{-x}+2$
$g(x)=\int_0^{x} te^{-t} dt$ is a solution to this equation E
1)-I must deduct the solutions of the equation (E)
I know that the solutions are: $e^{-2x}(\alpha cos(x)+ \beta sin(x))$
But how to determine $\alpha$ and $\beta$??
Hi everybody,
We consider the following differential equation:
$(E) : y''+2y'+2y=-(x+1)e^{-x}+2$
$g(x)=\int_0^{x} te^{-t} dt$ is a solution to this equation E
1)-I must deduct the solutions of the equation (E)
I know that the solutions are: $e^{-2x}(\alpha cos(x)+ \beta sin(x))$
But how to determine $\alpha$ and $\beta$??
I'll need initial or boundary conditions to determine those.
The 'incomplete' DE is...
$y^{''} + 2\cdot y^{'} + 2\cdot y=0$ (1)
... and its 'characteristic equation' is...
$x^{2} + 2\cdot x + 2 =0$ (2)
... the solution of which are $x= -1 \pm i$, so that the general integral of the (1) is...
$y= e^{-x}\cdot (\alpha\cdot \cos x + \beta\cdot \sin x)$ (3)
... that is different from the solution of the starting post.
Now a particular solution of the 'complete' DE...
$y^{''} + 2\cdot y^{'} + 2\cdot y= -(x+1)\cdot e^{-x} + 2$ (4)
... You can verify is ...
$g(x)= \int_{0}^{x} t\cdot e^{-t}\cdot dt$ (5)
... so that the general integral of (4) is...
$y= e^{-x}\cdot (\alpha\cdot \cos x + \beta\cdot \sin x) + g(x)$ (6)
Kind regards
Last edited by chisigma; February 23rd 2010 at 01:11 AM. Reason: trivial error corrected...
February 22nd 2010, 03:39 PM #2
February 22nd 2010, 11:44 PM #3 | {"url":"http://mathhelpforum.com/differential-equations/130185-equation.html","timestamp":"2014-04-19T07:58:09Z","content_type":null,"content_length":"41080","record_id":"<urn:uuid:ee8ab148-64ff-4477-86cc-dd6e40ea2687>","cc-path":"CC-MAIN-2014-15/segments/1397609536300.49/warc/CC-MAIN-20140416005216-00013-ip-10-147-4-33.ec2.internal.warc.gz"} |
Combinatorics Question
November 10th 2008, 09:52 AM #1
Oct 2008
Combinatorics Question
I have two questions about probabilities when choosing things from a set.
Assume you have a bag of x red marbles, y blue marbles, and z green marbles.
1. If you choose 6 marbles from the bag, without replacement, what is the probability that you get exactly 2 of each type of marble?
2. If you choose 6 marbles from the bag, with replacement, what is the probability that you get exactly 2 of each type of marble?
I'm not really sure how to go about solving these.
Thanks for any help.
You must assume that each of x, y, & z is at least 2.
${ x \choose 2} = \frac {x!}{(x-2)!(2!)}$ is the number of ways of choosing two reds marbles.
The total number of ways of choosing six marbles is ${x+y+z \choose 6}$ without replacement.
So, $\frac {{ x \choose 2}{ y \choose 2}{ z \choose 2}}{{x+y+z \choose 6}}$.
Now you try the next one.
November 10th 2008, 10:21 AM #2 | {"url":"http://mathhelpforum.com/discrete-math/58762-combinatorics-question.html","timestamp":"2014-04-18T05:53:50Z","content_type":null,"content_length":"33673","record_id":"<urn:uuid:4743f841-d60f-4ea9-a772-dadeed804940>","cc-path":"CC-MAIN-2014-15/segments/1397609532573.41/warc/CC-MAIN-20140416005212-00630-ip-10-147-4-33.ec2.internal.warc.gz"} |
Sunspot number is the most important index for tracking the level of solar activity. It is calculated as
where n is the number of individual spots, g is the number of sunspot groups and k is a constant that is different for each station. Records of sunspot number go back to the mid-17th century. This
site lists values of the daily mean sunspot number from 1818 to the present, monthly sunspot counts from 1749 to the present, and yearly sunspot counts from 1700 to the present. The most obvious
period in these records is an 11.4 year period called the solar cycle.
Another measure of the solar activity is provided by the 10.7 centimeter radio flux. Since intense emission at radio wavelengths are produced in magnetically active regions, a proxy can be created
for sunspot number using measurements of the radio flux. Statistical correlations between the sunspot number and f10.7 flux have been made using 40 years of data and is given by
R = (1.14).S - 73.21
where S is the solar flux (density) value in solar flux units. Tabulated values of the f10.7 fluxfor use in this calculation are available from DRAO, National Research Council of Canada | {"url":"http://www.windows2universe.org/spaceweather/solar_cycle.html","timestamp":"2014-04-17T16:13:59Z","content_type":null,"content_length":"5246","record_id":"<urn:uuid:cba11ed9-aaf5-4c18-844c-a762f735a8b8>","cc-path":"CC-MAIN-2014-15/segments/1397609537754.12/warc/CC-MAIN-20140416005217-00261-ip-10-147-4-33.ec2.internal.warc.gz"} |
Need help to clarify
March 25th 2009, 02:48 AM #1
Oct 2008
Need help to clarify
f(n) = 2n^1.1 + 4nlogbase2(n)-8n
Determine a suitable function g(n) such that f(n) is teta(g(n)). Justify your answer.
Can i let g(n) = 2n^1.1 + 4nlogbase2(n)-8n, same function as f(n) and justify that f(n) grows at the same rate as g(n). Is it correct.
Follow Math Help Forum on Facebook and Google+ | {"url":"http://mathhelpforum.com/discrete-math/80562-need-help-clarify.html","timestamp":"2014-04-21T13:04:14Z","content_type":null,"content_length":"28351","record_id":"<urn:uuid:6f168ae9-8db8-4858-b0b8-0604042f6059>","cc-path":"CC-MAIN-2014-15/segments/1397609539776.45/warc/CC-MAIN-20140416005219-00656-ip-10-147-4-33.ec2.internal.warc.gz"} |
┃ │Kenneth M. Levasseur │┃
┃UMass Lowell - CyberEd │Department of Mathematical Sciences │┃
┃92.419 Introduction to Mathematica│University of Massachusetts Lowell │┃
┃ │Lowell, MA 01854 │┃
Projects for
Introduction to Mathematica
Prof. Kenneth Levasseur
Department of Mathematical Sciences
University of Massachusetts Lowell
Email: Kenneth_Levasseur@uml.edu
I believe that in order to really start learning anything as complex as Mathematica , you've got to work toward a significant goal. This is why a project is such a major part of this course.
Required Elements of Projects
On occasion I've had as student do a wonderful project that used very little Mathematica . To avoid this, all projects now have to contain the items on the following short list. If you find it
impossible to include any of these, let me know and we can substitute some other feature.
1. Your name and email address - This isn't a feature of Mathematica, but it's something that a surprising number of people forget to put on their work.
2. At least one initialization cell that contains expressions that must be evaluated to use your project.
3. A function with at least one option. You will learn how to do this in Week 5.
4. At least one hyperlink either to a part of your project or to the Web.
Sources of Project Ideas
So far, the best source of project Ideas that I've been able to come up with are mathematics journals. The Journals of the Mathematical Association of America are particularly good sources since they
have many articles that are accessible to students. Some of the articles are prepared using a computer algebra system (usually Mathematica or Maple) and occasionally you can find one that can be
extended. Note: A project shouldn't be just a matter of replicating a calculation or a graphic - I'm looking for you to make use of Mathematica to explore new ground or clarify difficult concepts.
Sometimes, articles written by authors who didn't use a computer algebra systems are also a good choice.
Employment-Related Projects
You are welcome to do a project that is related to your job. It doesn't have to be mathematics, but then again, it you listen to Tom Lehrer, everything is mathematics!
More ideas
Evaluation of Projects
I've put together some comments on how I evaluate projects .
Sample Projects from past semesters
┃Project Name │Author │Semester ┃
┃Encryption │Henry Drew │Spring 1996┃
┃Math-termind │Jill Garland │Fall 1998 ┃
┃Wheels on Wheels... │Mike Pawlechek │Fall 1996 ┃
┃The Phi Number System │Sue Robinson │Fall 1995 ┃
┃Set, the Game │Tracey Smith │Spring 1998┃
When you're done.....
You might want to present your work or have it published. Some options are
• Every spring, UMass Lowell has a student research day.
• Also every spring, UMass runs a system-wide conference on undergraduate research.
• The Furman University Electronic Journal of Undergraduate Mathematics may publish your work.
If you have questions about whether these options are for you, contact me and we can discuss it.
Click here to ask a general question about projects | {"url":"http://faculty.uml.edu/klevasseur/courses/m419/proj/MMProj.html","timestamp":"2014-04-20T08:14:25Z","content_type":null,"content_length":"6517","record_id":"<urn:uuid:49276644-733a-4672-8eb7-f05b3d987b9d>","cc-path":"CC-MAIN-2014-15/segments/1397609538110.1/warc/CC-MAIN-20140416005218-00245-ip-10-147-4-33.ec2.internal.warc.gz"} |
Problems from Another Time
Individual problems from throughout mathematics history, as well as articles that include problem sets for students.
Discussion of 15th century French manuscript, with translation of its problems, including one with negative solutions
I have two fields of grain. From the first field I harvest 2/3 a bushel of grain/unit area; from the second, 1/2 bushel/unit area.
A fish sees a heron looking at him from across a pool, so he quickly swims towards the south. When he reaches the south side of the pool, he has the unwelcome surprise of meeting the heron.
Prove that the area of a regular polygon can be given by the product of its perimeter and half the radius of the inscribed circle.
Two circles of radii 25 feet intersect so that the distance between their centers is 30 feet. What is the length of the side of the largest square inscribable within their intersecting arcs?
A round pond sits in a rectangular garden. Its center is inaccessible; however, you know the distances from each corner of the garden to the center of the pond: 60, 52, 28, and 40 yards. What is the
radius of the pond?
Two cog-wheels, one having 26 cogs, and the other 20 cogs, run together. In how many revolutions of the larger wheel will the smaller gain in 12 revolutions?
Having been given the sum of two numbers,a, and the difference of their squares,b, find the numbers.
A circle is inscribed in an isosceles trapezoid. Find the relationship of the radius to the sides.
A carpenter has undertaken to build a house in 20 days. He takes on another man and says; "If we build the house together, we can accomplish the work in 8 days!" How long would it take this other man
to build the house working alone? | {"url":"http://www.maa.org/publications/periodicals/convergence/problems-another-time?page=3&device=mobile","timestamp":"2014-04-18T04:53:10Z","content_type":null,"content_length":"26094","record_id":"<urn:uuid:79400928-df88-4bbe-889b-e4293fd7e90b>","cc-path":"CC-MAIN-2014-15/segments/1397609537097.26/warc/CC-MAIN-20140416005217-00115-ip-10-147-4-33.ec2.internal.warc.gz"} |
FLT DEMONSTRATION By Anthony.R.Brown
Re: FLT DEMONSTRATION By Anthony.R.Brown
To Sekky "ITS ABOUT TIME YOU SHOWED SOMETHING!"
Super Member
Re: FLT DEMONSTRATION By Anthony.R.Brown
tell me
By induction!!!
Super Member
Re: FLT DEMONSTRATION By Anthony.R.Brown
This is my old trick, but still informative.
Re: FLT DEMONSTRATION By Anthony.R.Brown
To Maelwys
Quote:" Okay, what's the 1 case in 1000 that you can't prove with this? "
you have a shorty memory! 1,2,3,4,5,6,7,8,9..Odd or Even?
Re: FLT DEMONSTRATION By Anthony.R.Brown
Anthony.R.Brown wrote:
To Maelwys
Quote:" Okay, what's the 1 case in 1000 that you can't prove with this? "
you have a shorty memory! 1,2,3,4,5,6,7,8,9..Odd or Even?
is WHAT odd or even?
Re: FLT DEMONSTRATION By Anthony.R.Brown
To Sekky!
You are just too.. for words!......have a look at the Posts and you may also Remember??
Re: FLT DEMONSTRATION By Anthony.R.Brown
Anthony.R.Brown wrote:
To Maelwys
Quote:" Okay, what's the 1 case in 1000 that you can't prove with this? "
you have a shorty memory! 1,2,3,4,5,6,7,8,9..Odd or Even?
You left off part of my quote. "I don't believe it's been wrong yet, as long as it's being properly applied."
Whatever your 1,2,3... example is, it's not a proper application of mathematical induction.
Re: FLT DEMONSTRATION By Anthony.R.Brown
Ok, Anthony, let's do it your way, by formal mathematical induction.
Define your proposition P(n), for your case.
Re: FLT DEMONSTRATION By Anthony.R.Brown
I think it's time for a word from one of my old lecturers.
Induction is an axiom: in other words, if you don't believe it, there is not much I can do except advise you to study astrology instead.
He later went on to talk about goldfish that needed the whole of history recited to them every morning at breakfast, and how elephants were more useful. He was so great.
Anyway, I'm interrupting your debate. Just ignore me.
Why did the vector cross the road?
It wanted to be normal.
Re: FLT DEMONSTRATION By Anthony.R.Brown
MATH PROBLEMS INDUCTION CAN'T PROVE 100%
(1) Math Problems that use ( Pi ) Calculations any where within the Problem.
(2) Any other Infinite Math Problems
Re: FLT DEMONSTRATION By Anthony.R.Brown
Anthony, define your proposition, P(n), of the example you seem to be using.
Oh, by the way, failure to do so will just invalidate your claim, so drop the ad-hominem and just do it.
Re: FLT DEMONSTRATION By Anthony.R.Brown
Induction is an axiom: in other words, if you don't believe it, there is not much I can do except advise you to study astrology instead.
That's not exactly true. We define the natural numbers with a 5th property, that of induction. We can prove that such a set, in which induction applies, exists. So it isn't an axiom, just a property.
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
Re: FLT DEMONSTRATION By Anthony.R.Brown
Oh boy. Anthony is now refuting π as a number.
You're right in that you can't give the full decimal answer of any problem whose solution involves π, because you'd have to make an approximation, but it's perfectly allowed to work out answers by
just leaving π as a symbol.
Ricky, I'm not sure what you mean. It seems like you're saying that induction works because it is defined that way, which doesn't make any sense. Also, didn't someone say earlier that it was a Peano
Why did the vector cross the road?
It wanted to be normal.
Re: FLT DEMONSTRATION By Anthony.R.Brown
Quote:" Anthony, define your proposition, P(n), of the example you seem to be using.
Oh, by the way, failure to do so will just invalidate your claim, so drop the ad-hominem and just do it. "
For the Sequence 1,2,3,4,5,6,7,8,9.......etc. It's Impossible for Induction to Prove if the Number Sequence will End as an Odd Number or an Even
Re: FLT DEMONSTRATION By Anthony.R.Brown
Anthony.R.Brown wrote:
For the Sequence 1,2,3,4,5,6,7,8,9.......etc. It's Impossible for Induction to Prove if the Number Sequence will End as an Odd Number or an Even
There's two reasons for that:
1/ There is no end to an infinite sequence
2/ Inductive proof fails (n=odd, n+1=even). Induction can't prove everything, and nobody is claiming that it can. All we're saying is that in cases that you CAN use mathematical induction, it IS a
valid proof that is 100% correct.
Re: FLT DEMONSTRATION By Anthony.R.Brown
To Maelwys!
Dont tell me! tell Sekky! ( The Mirror Man/Person? )
Re: FLT DEMONSTRATION By Anthony.R.Brown
Anthony.R.Brown wrote:
To Maelwys!
Dont tell me! tell Sekky! ( The Mirror Man/Person? )
Tell him what? I'm pretty sure he agrees with me. That's why he was asking you to show him a formula for a valid proof by induction [the P(n) he was asking about] that wasn't 100% proof. You were the
one saying
Anthony.R.Brown wrote:
for 1 is 0.5n(n+1) you can only prove it 99.9% of the time!
Which is why we were asking what 1/1000 cases of THAT induction are not provable.
Re: FLT DEMONSTRATION By Anthony.R.Brown
To Maelwys
Quote:" Which is why we were asking what 1/1000 cases of THAT induction are not provable. "
The 1/1000 + 1 + 1 etc.......................................................................................................
Re: FLT DEMONSTRATION By Anthony.R.Brown
Anthony.R.Brown wrote:
To Maelwys
Quote:" Which is why we were asking what 1/1000 cases of THAT induction are not provable. "
The 1/1000 + 1 + 1 etc.......................................................................................................
I'm not even sure what that's supposed to represent...
Oh, and by the way. An ellipsis generally only requires 3 dots, not 33. The extra copies don't make it any more or less significant.
Re: FLT DEMONSTRATION By Anthony.R.Brown
Anthony.R.Brown wrote:
For the Sequence 1,2,3,4,5,6,7,8,9.......etc. It's Impossible for Induction to Prove if the Number Sequence will End as an Odd Number or an Even
because it's not induced
And yes, I agree with Maelwys, and I can prove it by induction.
Let M(Maelwys) |-> True
P(n) : (M(Maelwys) = True) implies (Anthony gets argumentative and hormonal)
P(1) is the current thread, hence P(1) is true.
Anthony gets argumentative and hormonal is a universal constant, hence P(n) is invariant under n.
Therefore P(n) implies P(n+1).
Super Member
Re: FLT DEMONSTRATION By Anthony.R.Brown
oh you!
The Beginning Of All Things To End.
The End Of All Things To Come.
Re: FLT DEMONSTRATION By Anthony.R.Brown
Ricky, I'm not sure what you mean. It seems like you're saying that induction works because it is defined that way, which doesn't make any sense. Also, didn't someone say earlier that it was a
Peano axiom?
Not quite. We define induction to be the property:
Let N be a set. Let A be a set with the following property: if n is in A, then *S(n) is in A. Induction is the property that if 1 is in A, then A = N.
And we can prove there exists a set which such a property. It's because we are proving the existence of a set with this property which makes it not an axiom.
*S(n) is known as the successor function, simply put, 2 = S(1), 3 = S(2), and so on.
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
Re: FLT DEMONSTRATION By Anthony.R.Brown
The Induction Challenge!
Anyone like to try and put forward a 100% Proof for any Math Problem/Calculation
Where the Problem/Calculation contains ( Pi ) within it!
Re: FLT DEMONSTRATION By Anthony.R.Brown
I just happened upon this ridiculous thread while googling something.
Claim: n > π for all integers n >= 4.
n = 4: 4 > π.
n > 4: Assume n-1 > π. Then n > n-1 > π.
Or perhaps Anthony thinks there may be some mysterious value of n out there for which this proof fails?
Re: FLT DEMONSTRATION By Anthony.R.Brown
To ephemere
Quote:" I just happened upon this ridiculous thread while googling something.
Claim: n > π for all integers n >= 4.
n = 4: 4 > π.
n > 4: Assume n-1 > π. Then n > n-1 > π. "
Can you prove the Value of Pi | {"url":"http://www.mathisfunforum.com/viewtopic.php?id=5389&p=3","timestamp":"2014-04-16T10:37:38Z","content_type":null,"content_length":"37016","record_id":"<urn:uuid:dea429cd-2748-4dfb-a12a-d85924255a24>","cc-path":"CC-MAIN-2014-15/segments/1397609523265.25/warc/CC-MAIN-20140416005203-00414-ip-10-147-4-33.ec2.internal.warc.gz"} |
Problems from Another Time
Individual problems from throughout mathematics history, as well as articles that include problem sets for students.
Discussion of 15th century French manuscript, with translation of its problems, including one with negative solutions
I have two fields of grain. From the first field I harvest 2/3 a bushel of grain/unit area; from the second, 1/2 bushel/unit area.
A fish sees a heron looking at him from across a pool, so he quickly swims towards the south. When he reaches the south side of the pool, he has the unwelcome surprise of meeting the heron.
Prove that the area of a regular polygon can be given by the product of its perimeter and half the radius of the inscribed circle.
Two circles of radii 25 feet intersect so that the distance between their centers is 30 feet. What is the length of the side of the largest square inscribable within their intersecting arcs?
A round pond sits in a rectangular garden. Its center is inaccessible; however, you know the distances from each corner of the garden to the center of the pond: 60, 52, 28, and 40 yards. What is the
radius of the pond?
Two cog-wheels, one having 26 cogs, and the other 20 cogs, run together. In how many revolutions of the larger wheel will the smaller gain in 12 revolutions?
Having been given the sum of two numbers,a, and the difference of their squares,b, find the numbers.
A circle is inscribed in an isosceles trapezoid. Find the relationship of the radius to the sides.
A carpenter has undertaken to build a house in 20 days. He takes on another man and says; "If we build the house together, we can accomplish the work in 8 days!" How long would it take this other man
to build the house working alone? | {"url":"http://www.maa.org/publications/periodicals/convergence/problems-another-time?page=3&device=mobile","timestamp":"2014-04-18T04:53:10Z","content_type":null,"content_length":"26094","record_id":"<urn:uuid:79400928-df88-4bbe-889b-e4293fd7e90b>","cc-path":"CC-MAIN-2014-15/segments/1397609532480.36/warc/CC-MAIN-20140416005212-00115-ip-10-147-4-33.ec2.internal.warc.gz"} |
Math Forum Discussions - Re: Matheology § 224
Date: Apr 3, 2013 6:40 AM
Author: mueckenh@rz.fh-augsburg.de
Subject: Re: Matheology § 224
On 3 Apr., 09:26, Virgil <vir...@ligriv.com> wrote:
> In article
> <2569eb91-7037-483e-be2c-17fce8394...@j9g2000vbz.googlegroups.com>,
> WM <mueck...@rz.fh-augsburg.de> wrote:
> > On 3 Apr., 00:29, Virgil <vir...@ligriv.com> wrote:
> > > The point being that removing one object from an infinite set does not
> > > diminish the infinite number left in the set
> > That is a good point. Alas induction holds for every natural number.
> No!
Your no is wrong. Induction holds for every natural number.
> It only holds for inductive sets:
But you don't know what the natural numbers are.
> One valid form of induction is:
> There exists a set of objects, N,
In mathematics that kind of nonsense is not required.
> and a special object such that:
> 1. The special object is a member of N.
> 2. For every object in N there is a successor object also in N.
That is not induction, but the property of natural numbers that is
required as the foundation of induction.
> 3. The special object is not a successor object of any object in N.
The special object it 2^22 and the elements of your N are 1^33, 0^44,
(-1)^55 and so on.
> 4. If successors of two objects in N are the same,
> then the two original objects are the same.
> 5. If any set contains The special object and the successor
> object of every object in N, then that set contains N as a subset.
The elements of N are the humans starting from Adam in the sequence of
their birth?
Regards, WM | {"url":"http://mathforum.org/kb/plaintext.jspa?messageID=8813989","timestamp":"2014-04-16T05:41:30Z","content_type":null,"content_length":"3069","record_id":"<urn:uuid:ec627c4e-7e0f-4eaa-9cf2-17266d97f80c>","cc-path":"CC-MAIN-2014-15/segments/1397609521512.15/warc/CC-MAIN-20140416005201-00430-ip-10-147-4-33.ec2.internal.warc.gz"} |
[SOLVED] linear kinematics
January 25th 2009, 12:28 PM #1
Junior Member
Jul 2008
[SOLVED] linear kinematics
hi there, i need apointer on this question, i've completed a, b half of c and d. BUt i'm getting abit stuck proving c again. Iill show the question then what i've got so far
A man standing at the top ad near the edge of a cliff of height 30m throws a ball into the air almost vertically upwars at a velocity of 12m/s. The ball then passes close to the man on he way
down, hitting the ground at the foot of the cliff.
a) The maximum height above the cliff top
b) The time that elapses before the ball pases the man on the way down (choose another method of calculation and use it to check your answer)
c)the total time takedn for the ball to reach the ground. (as in part b choose another method to check your answer)
d) The velocity of the ball when it strikes the ground
a) using v^2=u^2 +2as i rearranged for S and got 7.34m
b)using v=u+at i got 1.22s uptime and the same for the down time total 2.44s - proved it again using s=(u+V)t /2
c) used s=ut+1/2xat^2 , rearranged into a quadratic and got the values of 3.98s and -1.54s (usedthe 3.98 secs + 2.44 = 6.42s) - however i couldn't see another way of proving this without doing
part d and then going back.
d) used v=u+at using u=0 a=9.81 t=5.21s and got 51.012m/s
I tried to go back using the final velocity to prove my answer in part c but all the equation give me different answers. can anyone spot where i've went wrong?
Please help, be looking for my mistake for about an hour now!
a, b, and c are fine
d ...
$v_f = 12 - g(3.98) = -27 \, m/s$
how can you have a negative velocity?
You are confusing speed with velocity.
Velocity is a vector - it has magnitude and direction.
In straight line motion the direction of velocity is specified by +ve or -ve. If you define up as the positive direction then -ve is the downwards direction. Hence -27 m/s means a speed of 27 m/s
January 25th 2009, 04:29 PM #2
January 25th 2009, 11:06 PM #3
Junior Member
Jul 2008
January 26th 2009, 01:38 AM #4 | {"url":"http://mathhelpforum.com/advanced-applied-math/69842-solved-linear-kinematics.html","timestamp":"2014-04-20T14:20:46Z","content_type":null,"content_length":"41103","record_id":"<urn:uuid:4a8e44c4-6dbd-4c66-9599-66a7631cefad>","cc-path":"CC-MAIN-2014-15/segments/1397609538787.31/warc/CC-MAIN-20140416005218-00600-ip-10-147-4-33.ec2.internal.warc.gz"} |
Matches for:
Return to List
The Finite Calculus Associated with Bessel Functions
             
Contemporary Although Bessel functions are among the most widely used functions in applied mathematics, this book is essentially the first to present a calculus associated with this class of
Mathematics functions. The author obtains a generalized umbral calculus associated with the Euler operator and its associated Bessel eigenfunctions for each positive value of an index
parameter. For one particular value of this parameter, the functions and operators can be associated with the radial parts of \(n\)-dimensional Euclidean space objects. Some of the
1988; 122 pp; results of this book are in part extensions of the work of Rota and his co-workers on the ordinary umbral calculus and binomial enumeration. The author also introduces a wide
softcover variety of new polynomial sequences together with their groups and semigroup compositional properties. Generalized Bernoulli, Euler, and Stirling numbers associated with Bessel
functions and the corresponding classes of polynomials are also studied. The book is intended for mathematicians and physicists at the research level in special function theory.
Volume: 75
• The \(\nu\)-umbral algebra
ISBN-10: • The \(\nu\)-umbral field
0-8218-5083-0 • The group of \(\nu\)-delta functionals under composition
• Generalized binomial polynomial sequences
ISBN-13: • The composition of polynomial sequences
978-0-8218-5083-1 • Compositions of Moebius delta functionals
• Generalized shift invariant operators
List Price: US$29 • The generalized derivative of \(\nu\)-shift invariant operators
• Generalized Sheffer polynomials
Member Price: • Cross sets of polynomials
US$23.20 • A class of Laguerre type polynomials
• The generalized heat polynomials
Order Code: CONM/ • A primitive integral for the Euler operator
75 • Bernoulli type polynomials and numbers
• Generalized Euler polynomials and numbers
• Generalized Stirling numbers and factor polynomials | {"url":"http://ams.org/bookstore?fn=20&arg1=conmseries&ikey=CONM-75","timestamp":"2014-04-16T23:54:27Z","content_type":null,"content_length":"15406","record_id":"<urn:uuid:bba72a4b-3247-49c0-8f75-c85ff5a6e981>","cc-path":"CC-MAIN-2014-15/segments/1397609525991.2/warc/CC-MAIN-20140416005205-00504-ip-10-147-4-33.ec2.internal.warc.gz"} |
Oak Lawn Algebra 2 Tutor
...I won the Botany award for my genetic research on plants as an undergraduate, and I have done extensive research in Computational Biology for my Ph.D. dissertation. I was a teaching assistant
for both undergraduate and graduate students for a variety of Biology classes. I am fluent in a range of Science and History disciplines.
41 Subjects: including algebra 2, chemistry, English, writing
...Having also worked as a computer consultant for my university, I have a lot of experience helping clients with computer and technical problems that they encounter on campus. I've dealt with
people ranging from the technically illiterate to the geniuses of the electronic age. Having played the piano for over 9 years now, I am quite familiar with the basic and intermediate skills of
the piano.
16 Subjects: including algebra 2, chemistry, English, geometry
...Prior to receiving my B.S.E in Engineering Physics from the University of Michigan College of Engineering in 2005 I spent 3 years tutoring fellow college students in precalculus, calculus I &
II, differential equations, as well as physics I & II. The tutee and I met one-on-one in an agreed upon ...
7 Subjects: including algebra 2, calculus, physics, geometry
...In college, I tutored all levels of students including elementary and middle school and college students. I tutored my two wonderful kids from elementary to high school and college years. They
are now both highly educated and successful in their respective fields.
23 Subjects: including algebra 2, chemistry, geometry, biology
...I am very good at algebra and can generally be helpful to those students who are highly motivated to improve their skills in this area. I have taken many math courses in my academic career, and
I have helped many students get through algebra related topics. However, I am at my best with highly motivated students who seriously want to learn the subject matter.
13 Subjects: including algebra 2, statistics, algebra 1, geometry | {"url":"http://www.purplemath.com/Oak_Lawn_Algebra_2_tutors.php","timestamp":"2014-04-20T02:05:28Z","content_type":null,"content_length":"24093","record_id":"<urn:uuid:7309903c-8499-4800-9265-3166d5b659b3>","cc-path":"CC-MAIN-2014-15/segments/1398223211700.16/warc/CC-MAIN-20140423032011-00048-ip-10-147-4-33.ec2.internal.warc.gz"} |
Posts by
Posts by anna
Total # Posts: 1,882
A tyre manufacturer has chosen 48 tyres at random from a production line and issued them to taxis for testing. Assume that the 48 numbers you have selected refer to the number of thousands of
kilometres travelled during the life of each tyre. (For example, a value of 32&...
When BaCl2 reacts with Na3PO4, Ba3(PO4)2 and NaCl are formed. Balance the equation and write the coefficients in the blanks.
61.54 g
Fluorine gas can react with ammonia gas to produce dinitrogen tetrafluoride gas and hydrogen fluoride gas. This reaction is described by the following balanced equation: 5 F2 (g) + 2 NH3 (g) N2F4 (g)
+ 6 HF (g) What mass of hydrogen fluoride gas is produced from 135 g of ammon...
Check my physics
Would you mind checking one of my problems from my physics study guide? I know the answer and just want to make sure I am using the right formula. :) A 78-kg skydiver has a speed of 62 m/s at an
altitude of 870 m above the ground. a. Determine the kinetic energy possessed by t...
Hi! I have some questions from a study guide in my physics class, and they give me the answer, but I have to show how to get the answer. I did most of them, but need help with these. Here is problem
#8: A 20 kg block of ice slides without friction down a long hill. The start o...
Hi! I have some questions from a study guide in my physics class, and they give me the answer, but I have to show how to get the answer. I did most of them, but need help with these. Here is problem
#7: A 30 kg ball is launched at an angle of 15 degrees and is moving 20 m/s wh...
Hi! I have some questions from a study guide in my physics class, and they give me the answer, but I have to show how to get the answer. I did most of them, but need help with these. Here is problem
#6: A 30 kg kid on rollerskates is moving 1 m/s. His brother pushes him with a...
Hi! I have some questions from a study guide in my physics class, and they give me the answer, but I have to show how to get the answer. I did most of them, but need help with these. Here is problem
#5: A roller coaster car starts at Point A at a height of 100 m. When it reach...
Hi! I have some questions from a study guide in my physics class, and they give me the answer, but I have to show how to get the answer. I did most of them, but need help with these. Here is problem
#4: What is the increase in thermal energy to a 2000 kg bus and its surroundin...
Hi! I have some questions from a study guide in my physics class, and they give me the answer, but I have to show how to get the answer. I did most of them, but need help with these. Here is problem
#3: A 30 kg kid on rollerskates is being pushed by his mom with a force of 10 ...
Hi! I have some questions from a study guide in my physics class, and they give me the answer, but I have to show how to get the answer. I did most of them, but need help with these. Here is problem
#2: Suppose the spring in a pinball machine has a spring contant of k=400 N/m....
Hi! I have some questions from a study guide in my physics class, and they give me the answer, but I have to show how to get the answer. I did most of them, but need help with these. Here is problem
#1: A skier with a mass of 60.0 kg is standing at the top of a snow covered hi...
Exam: 986109RR - CHOOSING YOUR BUSINESS
Um, the point of this website is not to give out ABCD answers. It's to help guide you. But we can't even help you do that since you posted so many answers. Sorry, can't help.
If the equation were x= 0.03x - 58.91, x would equal -60.7320.
What are you asking?
Cultural Anthrolpogy
The field of computer technology could be used to illustrate that culture is
For launching a satellite into orbit around earth, would the proper conservation of energy formula be: Epi + Eki + O work = Epf + Ekf? Would work = 0 for launching a satellite? And, would Epi =
-Gm1m2/radius of earth or would it just equal 0 (because we usually set Ep on earth...
Calculate the pH of a .50 M solution of NaCl.
The electric field strength @ distance = 1.0 m from a point charge is 4.0 x 10^4 N/C. What is the Electric field strength @ 2.0 m from the same charge?
On the moon's surface, the force of gravity exerted by the moon on a camera is 2.4 N. What is the force of gravity exerted by the moon on the camera when it is in orbit with a radius = to twice the
moon's radius?
Maths A
Select 48 values according to the following criteria. o The first thirty (30) numbers must lie in the range from 48 to 58 inclusive. o The next ten (10) numbers must lie in the range from 30 to 70
inclusive. o The remaining eight (8) numbers must lie in the range from 50 to 95...
Maths A
A tyre manufacturer has chosen 48 tyres at random from a production line and issued them to taxis for testing. Assume that the 48 numbers you have selected refer to the number of thousands of
kilometres travelled during the life of each tyre. (For example, a value of 32&...
maths a
The Bough and Bud Company combined an insect spray with a growth hormone to produce the MOP (More Oranges Production) Treatment, which was expected to increase the number of oranges produced by each
tree. The data below shows the number of oranges produced by each tree in two ...
A tyre manufacturer has chosen 48 tyres at random from a production line and issued them to taxis for testing. Assume that the 48 numbers you have selected refer to the number of thousands of
kilometres travelled during the life of each tyre. (For example, a value of 32&...
Maths A
For this question, use the copy of the Random Number table provided at the end of this AAT. (This table was copied from page 111 in the Textbook.) Select 48 values according to the following
criteria. o The first thirty (30) numbers must lie in the range from 48 to 58 inclusiv...
For this question, use the copy of the Random Number table provided at the end of this AAT. (This table was copied from page 111 in the Textbook.) Select 48 values according to the following
criteria. o The first thirty (30) numbers must lie in the range from 48 to 58 inclusiv...
Math f(x)
given f(x)= x^2+x and g(x)=2x+1 determine f[g(x)] i dont even know where to start with this question help is much needed
A factory that specializes in the refinement of transition metals such as titanium was on fire. Provide kinetic-based arguments (concentration and temperature) explaining why water would be dangerous
for the firefighters.
The formula for rust can be represented by Fe2O3. How many moles of Fe are present in 21.3g of the compound? Please help. I am SO confused on all of this!!
Estimate the cost of ending inventory based on the retail method using the following information: Cost Retail Beginning Inventory $ 600,000 $ 800,000 Purchases $ 450,000 $ 600,000 Net Sales
$1,000,000 A. $150,000 B. $262,500 C. $300,000 D. $750,000
Business Analysis
Mahalyk's Water Fun Shoppe specializes in jet skis. Mahalyk's inventory records showed the following for the past year. Purchase Date Number of Jet Skis Cost Per Jet Ski February 10 30 $4,000 May 12
80 $3,000 June 15 20 $3,500 July 20 15 $2,500 Use the FIFO method to d...
Business Analysis
A firm has the following inventory information for the first quarter: 01/01 Beginning Inventory 50 units @ $5 01/15 Purchases 80 units @ $5.50 02/15 Purchases 60 units @ $5.25 02/15 Purchases 40
units @ $5.75 Sales 170 units @ $10 Total operating expenses $500 What is ending i...
How many distinct numbers greater and equal to 100 and smaller than 3000 can be formed using digits 0,1,2,3,4, without repetition of digits?
12.50g of an unknown compound (i=2) are dissolved in 50.00g of water and this solution freezes at 7.5 degrees C. What is the molar mass of this compound?
its concentration, so it should be mol/L
its concentration, so it should be mol/L
True, however, I keep getting a negative value which is not possible.
Find the initial concentration of NO2 if the concentration after 2.00 min is 0.113. k = 0.331 1/(M * s) 2 NO2 (g) + N2 (g) --> 2 N2O (g) + O2 (g)
Dr. Gillian explained that research has identified over 25,000 genes which are responsible for the genetic information which progresses from generation to generation. These genes are:
Organic chemistry
We conducted a diels-Alder reaction of cyclopentadiene with Maleic Anhydride experiment. Though I was wondering why we added ligroin to the maleic anhydride and ethyl acetate. I know ligroin can be
used as a solvent, but I thought that ethyl acetate was the solvent in this exp...
If it had gone down then how would we calculate it
Freezing point of pure water = -2, Freezing point of Unknown solution = -1 Mass of unknown in 50g of water = 1.9964g. Calculate the delta T for the solution of the unknown solid and determine the
molecular weight of the unknown solid.
Where reading the poem "LEGAL ALIEN" by Pat Mora . We have to find the rhythm,meter, and assonance . I cant fimd any of that in the poem . Please tell me what each one of them are in the poem ..
Reply ASAP please
general chemistry
A reaction involves fictional reagents A & B is begun with the inital concentration of 0.05M A and 0.01M B. The reaction reaches completion in 16 seconds. What is the initial rate of the reaction in
respect to A? and what is the order in respect to B
The expression ax^3+2x^2+cx+1 is 5x^3-3 greater than 3x^3+bx^2+d-7x. Find a, b, c, and d.
An aqueous KNO3 solution is made using 73.4g of KNO3 diluted to a total solution volume of 1.86 L. Calculate the molarity of the solution. (Assume a density of 1.05 g/ml for the solution.)
elementary math
thank you..
elementary math
i did use my poof to prove. im not sure im doing right. here is exercises: 'Suppose a,b,c are integers if a|b and a|c, then a|(b+c).' my answer: Proof. Suppose a|b ans a|c. By Definition of
divisibility. we know a|b means there is an integer x with b= ax. Likewise, a|c...
How do you solve this problem: sin(90* - x) ----------------- cot^2(90* - x) + 1 The * is a degree symbol!! I haven't seen any problems with degrees in it, usually it's only a variable! If anyone
could give me guidelines or show me where to start, I'd really apprec...
Calculate the speed of a proton having a kinetic energy of 1.00 × 10 −19 J and a mass of 1.673 × 10 −27 kg. Answer in units of m/s
Les vas a dar muchos regalos a tus padres. Quiero comprarles unos guantes a mis sobrinos Clara va a venderle sus libros de literatura francesa a su amiga. Los clientes nos pueden pagar con tarjeta de
A box with a lid is to be cut out of a 12 inch by 24 inch sheet of thin cardboard by cutting out six x-inch squares and folding them. What are the dimensions to two decimal places of all possible
boxes that will have a volume of 100 cubic inches?
A triangular entraceway has walls with corner angles of 50, 70, 60. The designer wants to place a tall bronze sculpture on a round pedestal in a central location equidistant from the three walls. How
can the designer find where to place the sculpture?
I have a board 18" x 18" x 1/8" I need to support it in an upright position at a 15 degree angle. How big must the base of support be to keep it from falling over.
The coordinates for a rhombus are given as (2a, 0), (0, 2b), ( 2a, 0), and (0, 2b). Write a plan to prove that the midpoints of the sides of a rhombus determine a rectangle using coordinate
geometry. Be sure to include the formulas.
Does anyone know a Commensalism of the Baltimore Checkerspot? If you do know, I need the awnser ASAP. Please help. Thank you.
how do I correctly show four as an array?
A bobcat jumps with a speed of 5m/s at an angle of 55 degrees. How long does the bobcat stay in the air?
A ball is thrown(from 0') straight up at 50' per second, how long before it hits the ground? How high does it go?
Calculus BC
How does a body fossil form? Is it just bones and teeth?
explain necessary and sufficient conditions and how providing these is one approach to defining a term
The rim of the London Eye (a 135m diameter ferris wheel) moves 26 cm/sec, slow enough for passengers to safely get on the wheel from the platform (2 meters above ground level) without stopping the
wheel at the bottom of its rotation. What's the height at the bottom of the ...
poli sci
Which of the fundamental American political values does the "Tea Party" most fundamentally support? Do you believe the "Tea Party" is a legitimate heir of the Founding Father's protest against the
British? Do you believe the Tea Party will transform Ame...
I calculated tan and sin (theta) for a physics lab experiment -- two-slit interference I had to use this equation y = L tan (theta), with y = 0.25 and L = 2 and tan (theta) came out to be 7.12 in
degrees and 0.1243 in radians. I found sin (theta) by using the triangle for X^2 ...
I need a 1.00 W motor to push a garbage compactor so that it squishes some garbage. The garbage has a spring constant of 49.7 N/m. If I am going to squish it 30 cm, determine how much time the
compactor takes.
What is the wavelength of waves transmitted by an AM radio station operating at a frequency of 840 kilohertz? Calculate the energy of a photon whose frequency is 2450 megahertz What is the difference
in energy for a photon of ultra-violet radiation, wavelength 319 nm and a pho...
A wolf us chasing a rabbit.Graph the wolf's motion using the following data: 15 m\s at 0 s,10 m\s at 3 s,1 m\s at 4 s,and 0 m\s at 5 s.What does the graph tell you???
Could someone explain how to graph the following problem? I don't quite understand what "Graph H for 0 ≤ t ≤ 3 and 300 ≤ H ≤ 340" is asking me to do... The National Oceanic and Atmospheric
Administration (NOAA) has been measuring atmospher...
American History
true or false The Great Migration was a large-scale movement of hundreds of thousands of southern African Americans to Weswtern farms.
exactly. This is all I was given. Possibly SumOf t = T(exp)*length* sin(theta) - W(hang) * length * sin(phi) - W(boom) * L(exp) * sin(phi) = 0 but im not sure.
The torques acting on the crane boom can be computed about anaxis passing through the top of the boom. (this is the place wherethe spring scale and mass hanger connect to the boom) apply thesecond
condition for equilibrium to the crane boom and derivethe resulting equation for...
a clothes washer starts from rest and spins up with an angular accceleration of 10/rad/s^2. Determine the angle through which the washer has moved afer 6 seconds.
this problem is modelling daylight hours using the form y = d+a sin[b(t-c)], btw
How would I solve for t if y equals 720 minutes of daylight and t=the number of days in 2012 720 = 713.7+152.7 sin [(2pi/366)(t-80.75)]
i think i figured out my mistake. it should be 720 = 713.7+152.7 sin [(2pi/366)(t-80.75)] this problem is modelling minutes of daylight in a town using the form y = d+a sin[b(t-c)]. let y equal 720
minutes of daylight and let t=the number of days in 2012
How would I solve this problem for t? 720 = 11.895+2.545sin [2pi/366(t-80.5)]
spanish literature
Osvaldo Dragun in his story Historia del hombre que se convirtio un perro: what is the conflict that the protagonist feels strongly about?
stat and research
3. A researcher is interested in whether listening to music helps or hinders test-performance. To control for differences in cognitive level, this researcher decides to use a within-participants
design. He selects a random sample of participants and has them study different ma...
use any method to estimate 667-651
A plane is capable of flying 200 km/h. There is a wind of 90 km/h from the east. The pilot flies in a northwest direction relative to the ground for three hours from city A to city B. Find the speed
of the plane relative to the ground, the heading of the plane, and the distanc...
A plane is capable of flying 200 km/h. There is a wind of 90 km/h from the east. The pilot flies in a northwest direction relative to the ground for three hours from city A to city B. Find the speed
of the plane relative to the ground, the heading of the plane, and the distanc...
3rd math
Tessie multiplied 3x4, and then doubled it to find 6x8 .did she get the correct answer ? Explained
thank you
A technician has prepared three solutions from a 600 mL stock solution. In preparing the first solution ½ of the original volume of the stock solution was used. The second solution required ¼ of
the remaining stock solution. Preparation of th...
Thank you
I have the fraction 1/6. I need to express it as a decimal, fraction expressed with denominator 100 and percentage. Is it correct to have 0.16666; 16.66/100 and 16.66%?
Math - urgent
thank you
Math - urgent
2. A technician has prepared three solutions from a 600 mL stock solution. In preparing the first solution ½ of the original volume of the stock solution was used. The second solution required ¼
of the remaining stock solution. Preparation of...
Cultral Diversity
1.Reserch about the impact of cultral valueson peer social interaction in an early education classroom may lead one to conclude that this enviroment may create biases if the primary social
orientation favors:a teacher centered learning b.individual activities c.non- maintream ...
Cultual Diversity
1.A mother's eventual submission to her young child's repquests for the toy matchbox car in the hobby shop is representative of the ______ parenting style. a. laissez faire b.authoritarian
c.Victorian d. democratic 2.A Native American child spent 30 minutes watching, c...
chem 11
a gas has an empirical formula C6H7O5. When put in a 50mL bulb at 120 degrees celcius and 120kPa (pressure) it's mass is determined: the mass is 10.37 times as much as C6H6O. What is the molecular
formula of the gas (C6H7O5)?
Chemistry 11
a gas has an empirical formula C6H7O5. When put in a 50mL bulb at 120 degrees celcius and 120kPa (pressure) it's mass is determined: the mass is 10.37 times as much as C6H6O. What is the molecular
formula of the gas (C6H7O5)?
At a given moment, a plane passes directly above a radar station at an altitude of 7 miles. (a) If the plane's speed is 400 mph, how fast is the distance between the plane and the station changing
half an hour later? (b) How fast is the distance between the plane and the s...
chemistry 11
if I was asked to set up a procedure that will allow me to check the concentration of aqueous Co2+ ions in a reaction vessel every 15 mintues, what could I do to quickly and accurately determine the
concentration of colbalt ions in the solution?
Culturally Diverse Background
Will someone check my answers. 1.During a class fingerpainting activity,Doresha appeared perplexed by the teacher's question, What are you doing, Doresha did not respond quickly or with ease. Why did
Doresha respond to this eventcast the way she did? a. Doresha is from a n...
Pages: <<Prev | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | Next>> | {"url":"http://www.jiskha.com/members/profile/posts.cgi?name=anna&page=3","timestamp":"2014-04-20T04:28:20Z","content_type":null,"content_length":"32352","record_id":"<urn:uuid:791f7d5e-f2dc-44aa-ac77-ab93b6dbe98d>","cc-path":"CC-MAIN-2014-15/segments/1397609537864.21/warc/CC-MAIN-20140416005217-00102-ip-10-147-4-33.ec2.internal.warc.gz"} |
Mplus Discussion >> Variances of growth factors
Anonymous posted on Wednesday, October 30, 2002 - 12:25 pm
One important advantage of LGM is that the model allows one to examine inter-individual differences in intra-individual growth in longitudinal studies. And the inter-individual differences are
captured by the variances of the growth factors.
To my understanding, if the variance of a slope factor is not statistically significant, I would say that there is no significant inter-individual differences, in regard to changes in the outcome
measure under study.
In my recent data analysis, I found that the variance of a lope factor, without covariates, was not statistically significant. This indicates that the changes over time in the outcome measure were
not significant different across individuals. However, once I added covariates into the LGM, I found two covariates (ethnicity and age) had significant effects on the slope factor and the R-square
was 0.42 (i.e., about 42% of the variation in the slope factor was explained by the covariates). How should I interpret these results?
Your help will be appreciated!
Linda K. Muthen posted on Wednesday, October 30, 2002 - 4:06 pm
If the slope growth factor mean is significant, this means that it is significantly different from zero which means that there is development over time on average. If the slope growth factor variance
is significant, this means that not all individuals grow at the same rate, but that there is significant variability in their growth rates. If the slope growth factor variance is not significant,
this means that all individuals have the same growth rate. Even if the slope growth factor variance is not statistically significant without covariates, inclusion of covariates often shows that they
have significant influence on the slope so that the slope does vary (as a function of the covariates). These seemingly conflicting results may be due to higher power to detect slope variability when
covariates are included.
Anonymous posted on Tuesday, August 10, 2004 - 7:18 am
I have a similar situation in a simultaneous process model. I have a non-significant mean slope with non-significant variance. Yet, this slope factor is significantly correlated with the slope factor
of the second growth process. There are no regression parameters in the model, just the intercept and slope parameters for the two processes. How should I interpret this? Thank you,
bmuthen posted on Friday, August 13, 2004 - 4:57 pm
I think it could be possible that you cannot reject a zero variance, but be able to reject a zero covariance. It may just be a matter of power. On the other hand, the non-zero covariance between the
growth factors may mask a model misspecification where the outcomes of the 2 processes need to correlate - and the correlation between the slopes help them do that - but the correct way to represent
the outcome correlation may be between contemporaneous residuals.
Jungmeen Kim posted on Thursday, February 21, 2008 - 2:28 pm
Dear Linda,
We ran parallel process growth models (between attention and anger) using standardized scores (since we had different reporters and slightly different questions for the variables we had to
standardize the scores to make composites), thus we know that there is going to be no significant mean changes over time. Then, we found that the intercept of the attention variable was significantly
predictive of the slope of the anger variable. The slope of the anger had a significant variance. How can we interpret this significant and negative regression path? Since the mean of anger slope is
not significant, does this mean that the higher intercept of attention is predictive of smaller (since it is negative) "variances" of anger?
Thank you for your attention and guidance in advance!
Bengt O. Muthen posted on Thursday, February 21, 2008 - 6:04 pm
No, a negative influence on a slope implies that as the predictor increases the slope decreases; it has nothing to do with the slope variance. Because your anger slope mean is not significantly
different from zero, this would mean that it gets negative as the predictor value increases.
But you should not do growth modeling on standardized scores - see the dangers described in the Seltzer article "The Metric Matters".
Sylvana Robbers posted on Thursday, July 24, 2008 - 2:03 am
Dear Dr. Muthen,
I am doing a twin-singleton comparison (with grouping option) on latent growth curves of externalizing problem behavior. I want to investigate if the growth factor variances of the twins are
different from the growth factor variances of the singletons.
I tested this by using equality constraints, like: i (1).
Then, I did a chi-square diff test by comparing the chi-sq of this model with the chi-sq of the unconstrained model.
My question is: is this a correct approach to test for variance differences in growth factors? I believe I should do an F-test instead of a chi-square diff test, but how can I do this in Mplus?
A related question: is it okay to test for group differences in growth factor MEANS using chi-sq diff testing, or do I also need another test for this?
Thanks in advance for your time.
Sylvana Robbers posted on Thursday, July 24, 2008 - 7:26 am
In addition, the estimator used in my analyses is ML.
Linda K. Muthen posted on Thursday, July 24, 2008 - 9:13 am
Using a chi-square difference test should be fine for both variances and means. The variances are not on the border of the admissible parameter space.
Sylvana Robbers posted on Thursday, July 24, 2008 - 2:33 pm
Thank you very much for your swift reply.
I am wondering, could I also use loglikelihood difference testing instead? When should chi-square difference testing be applied and not loglikelihood difference testing, or vice versa?
Linda K. Muthen posted on Thursday, July 24, 2008 - 5:13 pm
You can use either one. They will yield the same results when both are available.
Sylvana Robbers posted on Friday, July 25, 2008 - 12:10 am
Thanks alot!
jemila seid posted on Monday, August 25, 2008 - 7:04 pm
I am new to Mplus and doing a simple growth curve modeling - but with four dependent variables simultaneously.
I am just wondering if it is possible to get confidence intervals and/or p-values for the estimated correlations of the latent variables. If so, I would appreciate if you could tell me in which
output command I can find these values.
Thanks a lot
Linda K. Muthen posted on Tuesday, August 26, 2008 - 9:25 am
You would need to use MODEL CONSTRAINT to define these correlations. Then you would be able to get confidence intervals and p-values. See the user's guide for further information about MODEL
jemila seid posted on Saturday, October 04, 2008 - 4:58 am
Thanks a lot, Linda - I appreciate it. Is it possible to get residual correlations? that is a correlation between latent variables left unexplained by the model? how do I get it in the out put?
Best regards
Linda K. Muthen posted on Saturday, October 04, 2008 - 11:56 am
If you ask for RESIDUAL in the OUTPUT command, you can get residual covariances.
jemila seid posted on Monday, December 01, 2008 - 11:40 am
Thanks again, Linda. I appreciate it.
Wayne deRuiter posted on Monday, June 14, 2010 - 1:38 pm
I am having a little bit of trouble calculating the variances for my intercept and slope in my latent growth curve. When examining the standardized values for the variances, I get an estimate of
1.00, 999 for the est/se and 999 for the p-value. My outcome is physical activity levels (continuous variable). I know that 70% of my sample is sedentary at some point during the seven waves of data
and the data is no where near normally distributed. Is a negative variance the problem that I am encountering. If so, would this be caused by nonnormal data or some type of floor effect? How can I
fix it?
Linda K. Muthen posted on Tuesday, June 15, 2010 - 9:20 am
All variances are standardized to the value of one. Given the preponderance of zeroes in your variable, you might consider two-part growth modeling as shown in Example 6.16. See also Two-Part Growth
Modeling under Papers on the website.
Wayne deRuiter posted on Tuesday, June 15, 2010 - 10:40 am
Thank you for the suggestion on two-part growth modelling. My research looks at how change in one variable influences change in another variable (parallel process model). Would I still be able to see
how change influences change with a two-part growth model?
Linda K. Muthen posted on Tuesday, June 15, 2010 - 11:37 am
Yes, you can two two-part models or one two-part and one regular.
Christoph Weber posted on Wednesday, June 16, 2010 - 3:42 am
Dear Dr. Muthen,
just a short question regarding slope variances and individual differences in change?
1.) Should I use a one-tailed p-value for the evaluation of the slope variance? Because the question is, whether or not the variance is greater than 0.
2.) The strategie proposed by Hertzog et al (2008) (comparing a "random-intercept, fixed slope model" with a "random intercept, random slope model") yields often other conclusions. The random slope
is more often objected.
What would you suggest? What is the best way to analyse individual differences, especially in connection with the second posting on top (beside non significant slope variances, covariates have
significant effects)?
Would it be adequate not to "overrate" the slope variance, in case that one is especially interested in effects of covariates?
best regards
Christoph Weber
Bengt O. Muthen posted on Wednesday, June 16, 2010 - 4:12 pm
I don't think you need to emphasize how precisely the slope variance is estimated in order to let the slope be either predicted by a covariate or predicting another slope. You are more interested in
how precisely those relationships are estimated.
Wayne deRuiter posted on Tuesday, July 06, 2010 - 7:52 pm
Hello Dr. Muthen
In a previous posting you mentioned that all variances are standardized to a value of one. Why is this?
Bengt O. Muthen posted on Wednesday, July 07, 2010 - 10:05 am
Standardization implies that variances are turned into ones. You don't need to consider the standardized solution if you don't want to and then the variances would not be one.
matteo giletta posted on Tuesday, May 22, 2012 - 5:28 am
Dear Linda & Bengt,
I'm running some unconditional two-part LGM and would like to add covariates to my models. I would have two questions in this regard:
1) the variances of my growth factors (linear and quadratic terms)are not significant. Can I still add (and interpret in a meaningful way)covariates?
2)the continuous part of the model fits the data better when I introduce a cubic term. Yet, I get a warning "THE LATENT VARIABLE COVARIANCE MATRIX (PSI) IS NOT POSITIVE DEFINITE. THIS COULD INDICATE
I've solved the problem by fixing the variances of the quadratic and cubic term to zero. However, again, can I still add and interpret covariates?
Many thanks!!!
Linda K. Muthen posted on Tuesday, May 22, 2012 - 8:43 am
1. Yes, you will have more power when you add covariates.
2. Yes for the same reason stated above.
matteo giletta posted on Tuesday, May 22, 2012 - 9:31 am
Thanks a lot Linda!
I would like to ask you one more question:
some of the covariates I add in my model are significantly related to the growth factors. Yet, the growth factors that in the unconditional model were significant now turn out to be non-significant.
Is this possible? can I interpret my results without problem?
Linda K. Muthen posted on Tuesday, May 22, 2012 - 9:38 am
I think you are saying the variances of the growth factors were significant in the unconditional model and that the residual variances of the growth factors are not significant in the conditional
model. These are not the same parameters. This is to be expected because you are explaining the variance by the covariates. The residual variance is what is still not explained.
matteo giletta posted on Tuesday, May 22, 2012 - 1:04 pm
Sorry, I was not clear enough. Actually I meant the means of the growth factors, not variances. In the unconditional model I obtain means for all my growth factors and they are all significant. When
I add predictors (even simply gender) I do not obtain estimated means for the growth factors anymore, but estimated intercepts which are all not significant. I thought that the estimated means and
intercepts for the growth factors were the same parameters but I assume I am making some confusion!
Thus, is it possible that I obtain significant estimated means for growth factors in the unconditional model and non-significant estimated intercepts for growth factors in the conditional model?
Can I interpret the effects of my covariates based on the means of growth factors I obtained in the unconditional model?
Thanks a lot for your help!!!
Linda K. Muthen posted on Tuesday, May 22, 2012 - 2:38 pm
The same situation holds. In an unconditional model, a mean is estimated. In a conditional model, an intercept is estimated.
matteo giletta posted on Tuesday, May 29, 2012 - 7:56 am
Dear Linda,
thanks you so much for your reply!
I still have a main issue with my two-part models. In the unconditional model I fixed at zero the variances for the the slope and quadratic terms of the continuous part of the model, as well as all
the covariances except the one between the two intercepts. This model worked well. Yet, when I introduce predictors the covariance between the intercepts is higher than 1. I tried to center the
intercept at a different time point but did not help. Also both the variances around the intercepts are significant. Do you have any suggestion about how to solve it? Can I fixed it at zero?
Thanks a lot!
Yuliya Kotelnikova posted on Tuesday, May 29, 2012 - 5:19 pm
Dr. Muthen,
I am writing up a paper explaining a situation in which the slope variance in an unconditional model was not significant, however, I have significant time-invariant covariates predicting slope
variation. From your previous posts I understand that there is more power to detect slope variability, when covariates are added. I was wondering, if you could point out a paper or a chapter that
discusses the above mentioned issue in more detail.
Best regards,
Linda K. Muthen posted on Wednesday, May 30, 2012 - 10:28 am
This may be discussed in the Raudenbush and Bryk book. You might find an answer by posting on multilevel net.
Linda K. Muthen posted on Wednesday, May 30, 2012 - 10:30 am
Please send the conditional and unconditional outputs and your license number to support@statmodel.com.
Reiko Hirai posted on Saturday, February 02, 2013 - 2:58 pm
Dear Dr. Muthen,
I am new in Mplus discussion board. I am looking for a reference to justify my analyses. Two of my outcome variables do not have significant slope variances. I have significant intercept variances
for both outcomes. From what I read, Nagin recommends not to run trajectory analyses if there is no slope variance. I heard that you take a different approach - it is still useful to run trajectory
analyses if there is significant intercept variance. I was unable to find the citation. Could you kindly point me to the appropriate article or book? I appreciate your help very much.
Bengt O. Muthen posted on Saturday, February 02, 2013 - 4:04 pm
When you say trajectory analysis, do you mean using latent trajectory classes?
And when you say significant slopes variance, are you referring to a regular growth model with random effects, so a 1-class model?
Reiko Hirai posted on Saturday, February 02, 2013 - 4:34 pm
Dear Dr. Muthen,
I am sorry that I was not clear. Yes, I believe it is latent trajectory class analysis. I ran a single class model without constraining the variance zero to test the significance of slope and
intercept variances. The output indicated significant variance in intercept but non-significant variance in slope. Hope I provided enough information. Thank you very much for your prompt response.
Bengt O. Muthen posted on Sunday, February 03, 2013 - 11:32 am
I think you can find trajectory classes even if only the intercept has significant variance in a single-class random effect growth model. For instance, groups of individuals may have distinctly
different starting points but develop approximately at the same rate. I know of no specific citation for this, but I don't think it needs it.
Reiko Hirai posted on Tuesday, February 05, 2013 - 8:12 pm
Dear Dr. Muthen,
Thank you very much for your answer. Hearing it from you made me feel good about all the analyses I have done.
Barry Wagner posted on Tuesday, March 19, 2013 - 2:07 pm
Dr. Muthen:
I have a question about setting a slope factor variance to 0 in a piecewise growth model. The observed dependent variables are five time points of suicidal ideation. There are 2 linear slope factors:
S1 (from time 1 to time 2), and S2 (time 2 through time 5). In order to get the model to run properly we set the variance of S1 to 0 (otherwise the standard errors could not be computed). Setting
S1@0 is fine, because the computed variance is very small and non-significant. In our model we examined the regressions of the intercept and both slope factors on our covariates. Mplus provides
answers for the regressions of S1 on the family predictors, a couple of which are significant. My question is: What do those findings for S1 mean? That is, how can covariates predict S1 when there is
no variance in S1? Are the answers interpretable? If not, should we omit regressions of S1 on predictors in the model, in other words, only look at regressions of I and S2 on the predictors? Omitting
S1 regressions changes the answers for regressions involving S2, so it is not an inconsequential decision. Thank you!
Bengt O. Muthen posted on Tuesday, March 19, 2013 - 2:27 pm
Without covariates the s1 variance is not identified, so any estimates you see are not trustworthy (as you say, the SE signals the non-ident). With covariates, fixing s1@0 implies that the s1
residual variance is zero. The s1 regression on covariates is still fine. For instance, a gender covariate says that the s1 mean is different for males and females.
matteo giletta posted on Wednesday, August 21, 2013 - 9:55 am
Dear Drs. Muthen,
My colleagues and I are working on a paper in which we have conducted a LGM and found a non-significant slope variance (p =.12). Based on the previous posts we have decided to still conducted a
conditional LGM and we found significant effects of predictors on the slope. Yet, a reviewer is now questioning our analyses stating that given the non-significant slope we should have never moved to
examine a conditional model. So, my question is: do you know any paper or book to which we could refer to support our decision? May we cite this post as a "personal communication"?
Thank you so much for your help!
Bengt O. Muthen posted on Wednesday, August 21, 2013 - 10:14 am
I would not go by the reviewer's rule. It is often found that adding covariates can tease out variance in a slope, presumably due to increased power. I can't put my finger on a reference right now -
others? Also, the usual z test (Wald test for one parameter) is known to have low power; see, e.g., Berkhof-Snijders (2001) in JEBS.
matteo giletta posted on Wednesday, August 21, 2013 - 1:10 pm
Thank you very much for your answer!
Sointu Leikas posted on Monday, September 09, 2013 - 5:00 am
Dear Drs Muthen,
I'm testing a simple growth model with 4 time points. The linear (a) and quadratic (b) models show poor fit, so I tested a "free growth model" (c) with two of the 4 time scores free (estimated). This
model has a good fit (and visually, the growth seems to be non-linear and non-quadratic, so this makes sense).
However, for each of the above models I get the "THE LATENT VARIABLE COVARIANCE MATRIX (PSI) IS NOT POSITIVE..." warning. In models a and c, the warning says "PROBLEM INVOLVING VARIABLE I" and in
model b, the problematic variable is S.
I found in the outputs that the variance of the problematic latent variable (I or S) is negative, causing the warning (correlations between latent variables are clearly below 1).
But I don't know why this happens and what to do with it. The data looks OK, kurtosis and skew are within acceptable limits for all variables etc.
My input for the free growth model:
Model: i s | LS1@0 LS2@2.5 LS3* LS4*;
I'm grateful for all suggestions!
Sointu Leikas posted on Monday, September 09, 2013 - 5:02 am
An addition: constraining the variance of I or S to zero causes a very poor fit.
Linda K. Muthen posted on Monday, September 09, 2013 - 9:44 am
It sounds like the negative residual variances are not small and insignificant. In this case, they should not be fixed at zero. Instead you need to find a better model for your data.
Katy Roche posted on Tuesday, January 07, 2014 - 10:07 am
I am having trouble finding the slope variances in my conditional model (and I did request Tech 4). Thus, although I can see them in my unconditional model output, I just do not see them in the
conditional model output. Can you let me know if there is a default supressing these or how I can call them out?
Linda K. Muthen posted on Tuesday, January 07, 2014 - 10:46 am
They will be residual variances in the conditional model.
Danyel A.Vargas posted on Saturday, March 08, 2014 - 4:34 pm
I have computed a growth model with four time points and results indicate significant intercept variance. I went to examine this +1sd and -1sd below the intercept and the value for +1sd was above my
scale. My scale is 1-5 and the value for +1sd above = 5.09. I have double checked my values and they are in the correct range (i.e., 1-5). Do you know what could be happening here? Also, the SRMR for
model fit is indicating poor fit (i.e., .14) but the chi-square and SRMR are not.
Can you please help me?
Thanks so much.
Bengt O. Muthen posted on Monday, March 10, 2014 - 5:26 pm
You would have to send the output and license number to Support.
Jason Bond posted on Tuesday, March 18, 2014 - 12:19 pm
I’m trying to estimate the effect of a grouping variable G in a 3-time point design using the random intercept model:
y(i,t) = a(i) + b(i)I(T=1) + c(i)I(T=2) + e(i,t)
a(i) = a1 + a2*G + u(i)
b(i)= b1 + b2*G
c(i)= c1 + c2*G
where I(T=x) is an indicator variable for data from wave x. When I ran the below syntax, it indicates that PSI is not positive definite. Any suggestions re the problem? Thanks much,
TYPE = RANDOM;
ESTIMATOR = ML;
i BY n28_hdd@1 fn28_hdd@1 gn28_hdd@1;
s1 BY n28_hdd@0 fn28_hdd@1 gn28_hdd@0;
s2 BY n28_hdd@0 fn28_hdd@0 gn28_hdd@1;
[n28_hdd@0 fn28_hdd@0 gn28_hdd@0 i s1 s2];
n28_hdd fn28_hdd gn28_hdd (1);
s1@0 s2@0;
i on group;
s1 on group;
s2 on group;
Bengt O. Muthen posted on Tuesday, March 18, 2014 - 2:13 pm
Are you sure that the 3 growth factors have fixed zero covariances? If that's not the problem, send output and license number to Support.
Jason Bond posted on Tuesday, March 18, 2014 - 4:24 pm
That was indeed the problem...thanks.
RuoShui posted on Sunday, March 23, 2014 - 7:45 pm
Dear Drs. Muthen,
I read that when doing LGCM, we need to compute pseudo R squared to understand the variance of I and S explained by the covariates. I am wondering does Mplus provide pseudo R squared statistics or
the R squared provided in the output serve the same function?
Thank you very much!
Linda K. Muthen posted on Monday, March 24, 2014 - 8:22 am
I believe Pseudo R-square is for logistic regression. The regression of one growth factor on another is a linear regression. Growth factors are continuous variables. Pseudo R-quare would not apply in
this situation.
RuoShui posted on Monday, March 24, 2014 - 3:49 pm
I see. Thank you very much Dr. Muthen.
On a related note, the variance of the slope growth factor is .003 in the unconditional LGCM model. When I included covariates, R square statistics indicate 4% of the variance in S was explained. But
the residual variance of the slope growth factor in the conditional model is .004. Is this even possible?
Thank you a lot!
Linda K. Muthen posted on Monday, March 24, 2014 - 4:24 pm
Please send the output and your license number to support@statmodel.com so I can take a look at it. Are you looking at the standardized residual variance?
Back to top | {"url":"http://www.statmodel.com/cgi-bin/discus/discus.cgi?pg=next&topic=14&page=224","timestamp":"2014-04-16T22:39:38Z","content_type":null,"content_length":"101347","record_id":"<urn:uuid:7fbb9ba0-1f49-4b52-9b9d-911e9a070419>","cc-path":"CC-MAIN-2014-15/segments/1398223211700.16/warc/CC-MAIN-20140423032011-00174-ip-10-147-4-33.ec2.internal.warc.gz"} |
Spring 2010
The two most important applications of General Relativity are, first, black holes, and second, the cosmic microwave background. The basis of the latter is general relativistic perturbation theory in
a uniform expanding background. The intention in this course is to give you a thorough understanding of black holes, and to get you to the point where you can begin to understand the cosmic
Unfortunately, general relativity is challenging mathematically, which poses barriers to understanding the interesting applications. We will cover the mathematics that is required, but with a clear
view that the purpose of the mathematics is to allow you to understand the applications.
I will hand out notes in advance for each section of the course, and I will lecture from those notes.
This is the planned path for the course:
• Special Relativity
• Coordinate approach to General Relativity
□ Ideal black holes (Schwarzschild, Reissner-Nordström, Kerr-Newman)
□ Homogeneous, isotropic cosmology (FRW metric)
• Tetrad approach to General Relativity
□ Black hole interiors
□ General relativistic perturbation theory
□ Cosmic microwave background
Grading will be weighted as follows:
│Problem Sets │60%│
│Midterm │20%│
│Final │20%│
Class and University Policies
Updated 2010 Jan 12 | {"url":"http://casa.colorado.edu/~ajsh/phys5770_10/syllabus.html","timestamp":"2014-04-19T01:47:19Z","content_type":null,"content_length":"3189","record_id":"<urn:uuid:529682a0-04af-4e44-bb6b-fbffcfe0681f>","cc-path":"CC-MAIN-2014-15/segments/1398223206647.11/warc/CC-MAIN-20140423032006-00054-ip-10-147-4-33.ec2.internal.warc.gz"} |
Reply to comment
Submitted by Anonymous on August 15, 2010.
Richard Elwes
Re the New Scientist article 'it doesn't add up', (New Scientist, Vol 207, No. 2773, pp34-38) is fascinating. The whole area of infinities and number theory is wonderful and, as a science trained
artist, has captured my imagination since childhood.
The Nature of the Infinity Boundary.
I would agree with Doron Zeilberger - in the sense that its a nonsense to consider that as we count up numbers we eventually continue until 'infinity'. Of course, there is no magnitude limit!. We can
always add one to whatever an 'infinite' number is. So what is the nature of the boundary of this infinity?
Let us consider that, in the extremely simple case of integer number, the infinity boundary in this case may be understood to describe a 'realm of quality' - a dimension-like entity with an
'attribute boundary'.
We can make sense of this boundary, which defines the reality of infinity (of integer number), if it describes the boundary of an unambiguous 'realm of all integer number' - an attribute boundary
describing a realm of quality of number - in this case the realm of 'All Integer Number'. The set of all integer number if you like. But in this case the 'set' is not merely a collection of numbers,
it is invested with dimension-like properties.
So the reality of the 'infinity', in the case of integer number, can be understood by observing it from the 'ouside', in which case we see a dimension-like attribute boundary, a phase boundary,
rather than any kind of simplistic, ambiguous and nonsensical 'magnitude limit' (which has no limit), which is of course, nonsense. As an integer number becomes increasing large it becomes more
dimension-like (cf limit condition as the dimension-like realm of quality of All Integer Number), whereas 'smaller' numbers are more 'particle' like with the quantum of 1 as the smallest integer
best wishes,
dallas simpson,
location performance environmental sound artist. | {"url":"http://plus.maths.org/content/comment/reply/2293/744","timestamp":"2014-04-20T08:18:58Z","content_type":null,"content_length":"21820","record_id":"<urn:uuid:6b0cc4a3-3f74-469b-8bdb-4326de6813a4>","cc-path":"CC-MAIN-2014-15/segments/1398223204388.12/warc/CC-MAIN-20140423032004-00086-ip-10-147-4-33.ec2.internal.warc.gz"} |
Regaining Lost Knowledge
A recent Python newsgroup query asked for an efficient solution to the problem of computing a running median as a large sliding window advances over a stream of data
One category of replies can be classified as clever. The respondents used their innate intelligence and knowledge of Python for a fresh look at the problem. Their solutions focused on the fact the
position of the median doesn’t move much between successive updates. Unfortunately, these solutions were catastrophically slow for large data windows.
Another category of reply relied on education. A respondent remembered that QuickSelect is a fast O(n) way of finding a median in unsorted data. I responded with an ASPN recipe implementing
QuickSelect (written by yours truly). These posts represented progress, a triumph of education over cleverness, but even that improved solution was unusably slow for large window sizes.
A more promising type of reply relied on research. Surely, this problem had been solved before. Indeed, there is a published paper: Efficient Algorithm for Computing a Running Median by Soymya D.
Mohanty with an O(sqrt(n)) solution. Score one for science!
However, that solution was trumped by respondents who characterized the solution mathematically, “the obvious way to compute a running median involves a tree structure so you can quickly insert and
delete elements, and find the median. That would be asymptotically O(log n) but messy to implement.” Fortunately, such an implementation exists using the blist Python extension. Alas, we had a good
solution but not a portable one. Without the extension module, the B+ tree structure is non-trivial to implement.
When I thought about the problem, the mathematical characterization suggested data structures that maintained sorted data with O(log n) updates, and previous education indicated a skiplist would fit
the bill, but it took cleverness to discover that indexing the skiplist to find the median could be reduced to O(log n) time by adding link widths to the structure. This thinking led to my solution
which is easily portable across languages and scales well to very large window sizes.
Had I discovered something new under the sun? Yes and no.
Yes, as far as I can tell the idea of using an indexable skiplist to solve the running median problem in O(log n) time had never been presented before anywhere else. The best published solution was
Mohanty’s O(sqrt n) solution. Score one for combining mathematical characterization with education and cleverness.
And no, the big inspiration of figuring out how to make a skiplist indexable was not a new result. Score a big failure for research. Everywhere I had looked for skiplist resources, only the basics
were presented (insertion and deletion in O(log n) time). No resource mentioned indexable skiplists. The previous work on the problem had effectively been lost. An entire generation of programmers
was learning about skiplists but not being taught that they could be made efficiently indexable.
To help the world regain this lost knowledge, I updated the wikipedia entry for skiplists to show how to make them indexable with my Python recipe and I’ve added a link to Pugh’s earlier research on
the problem.
Will that wikipedia entry really solve the problem of lost knowledge? The page view statistics suggest that it will. Only time will tell.
Explore posts in the same categories: Algorithms
Tags: Lost Knowledge, Running Median, Skiplists, Wikipedia
You can
comment below
, or
link to this permanent URL
from your own site.
7 Comments on “Regaining Lost Knowledge”
1. March 26, 2010 at 8:55 am
I came up with my own indexed skip list several years ago. I like that you updated the wikipedia page. Skip lists are great. My current version of skip lists is where you can operate on it via
both index and object. For example, if you want to search for a node both by ID or by index, this list will do it for you in O(log n) in both cases. Note that the objects can be inserted in
random order (and not sorted by name).
It’s great for implementing ordered (indexed) sets where you would also like to retrieve elements by name. Again, sorting by index and sorting by name would give two different lists.
2. March 31, 2010 at 7:40 am
(Sorry for double post…)
I think we can use 2 heaps (one for ), that’s a lot easier than skip lists.
When you add a value, just put the value in the right heap. If the median isn’t the median anymore, just put a value from a heap to the other.
To delete value in a heap, set “deleted” in a table. When the top of a heap is set “deleted”, delete it (that’s “lazy” heap).
3. April 29, 2011 at 7:16 pm
Maybe the reason you didn’t find any published literature is because you were searching for the wrong terms?
The CLRS algorithms book has a chapter dedicated to order statistics for binary search trees. They support insertion/deletion and finding the i’th item in sorted order all in O(log(n)). It’s
clear how to implement the sliding window median if you have one of these already built.
When you get a new item in the stream, delete the trailing part of the window, insert the new item into the window, and then ask for the median.
A lecture on order statistics for BSTs
And a homework problem assigned (I think for undergrads?) in 2004 to add order statistics to skip lists: Problem 5-3 here:
□ May 3, 2011 at 1:22 pm
The world of binary search trees seems to be very well documented and throughly covers searching and indexing.
In contrast, the world of skiplists is sparsely documented. Aside from the homework problem in some course handouts and a brief mention (without source code) in the original skiplist paper,
it seems to have been completely forgotten that skiplists can augmented for O(log n) indexing.
Both data structures can solve the running median problem. The purported advantage of skiplists over binary search trees is that balancing is accomplished trivially through randomization.
That makes the implementation comparatively simple.
5. May 6, 2011 at 2:51 am
I for one appreciated your addition to the Wikipedia article — without it Pugh’s article would have remained unknown to me!
6. December 31, 2011 at 6:50 pm
It’s a great addition to the article, and it’s great to have an easily portable one.
Incidentally, you are not the first. Redis (an in-memory data structure server written in C) has had an indexable skiplist as part of it’s available data structures for over 2 years. | {"url":"http://rhettinger.wordpress.com/2010/02/06/lost-knowledge/","timestamp":"2014-04-20T04:18:34Z","content_type":null,"content_length":"56240","record_id":"<urn:uuid:4c6d1648-1da2-4c99-805c-33b588f1f103>","cc-path":"CC-MAIN-2014-15/segments/1397609537864.21/warc/CC-MAIN-20140416005217-00092-ip-10-147-4-33.ec2.internal.warc.gz"} |
MathGroup Archive: August 2010 [00146]
[Date Index] [Thread Index] [Author Index]
Surface integral on a 3D region (Solution)
• To: mathgroup at smc.vnet.net
• Subject: [mg111565] Surface integral on a 3D region (Solution)
• From: dr DanW <dmaxwarren at gmail.com>
• Date: Thu, 5 Aug 2010 07:01:07 -0400 (EDT)
Here is the solution I worked out for the surface integral question I
posed to the group about a week ago. I hope someone else can make use
of this solution in the future.
These are the expressions that, when equal to zero, define the
surfaces of my solid region, which is a circular cylinder with one
flat end and one slanted end.
L = 15; R = 3.5; \[Theta] = 40.*Degree;
cylinder = -R^2 + $y^2 + $z^2;
piston = $x;
termination = $z*Cos[\[Theta]] + ($x - L)*Sin[\[Theta]];
xmax = L + R/Tan[\[Theta]];
So, the solid region is defined as
tubeRegion = cylinder <= 0 && piston >= 0 && termination <= 0
Out[6]= -12.25 + $y^2 + $z^2 <= 0 && $x >= 0 && 0.642788 (-15 + $x) +
0.766044 $z <= 0
My problem is that I want to be able to integrate a function over the
entire surface of this region. That was not covered in my calculus
class, since I have boundaries that are not on constant coordinate
planes. The first thing to do is to break the problem down to
integrating over each of the three surfaces that bound the region.
To work one surface, my first thought was to change one of the
inequalities to an equality and do a 3D NIntegrate, but that gave me
zero since the resulting region has zero volume (duh.) So, it was
evident that I had to reduce the problem to 2D. So, pick a variable
and solve the surface equation for that variable:
xform = Solve[cylinder == 0, $y]
Out[7]= {{$y -> -0.5 Sqrt[49. - 4. $z^2]}, {$y -> 0.5 Sqrt[49. - 4.
I will use this variable transformation to eliminate the $y variable
from the remaining expressions, essentially turning the 3D problem
into 2D problem in the parameters $x and $z. Looking up some formulas
for parametric surface integration in MathWorld, I calculate the
element of surface in the new coordinates
ds = (Norm[Cross[D[#1, $x], D[#1, $z]]] & ) /@ ({$x, $y, $z} /. xform)
Out[8]= {Sqrt[1. + 4. Abs[$z/Sqrt[49. - 4. $z^2]]^2], Sqrt[
1. + 4. Abs[$z/Sqrt[49. - 4. $z^2]]^2]}
I also use the transformation of the remaining two bounding
inequalities. In this case, I cheat since one of the bounding
conditions is trivial and I can put them in directly as limits of
select = Boole /@ (termination <= 0 /. xform)
Out[9]= {Boole[0.642788 (-15 + $x) + 0.766044 $z <= 0],
Boole[0.642788 (-15 + $x) + 0.766044 $z <= 0]}
To integrate a function f[x,y,z] over the one surface we have been
working with, the integral would be
(* NIntegrate[(f[x,y,z]/.xform).(ds select),{$x,0,xmax},
{$z,-3.35,3.35}] *)
For example, for the trivial case where we only want the surface area,
the function f[x,y,z] = 1, so the integral evaluates
NIntegrate[(1 /. xform) . (ds*select), {$x, 0, xmax}, {$z, -3.35,
Out[11]= 268.164
Now repeat for the other two surfaces.
Other solutions were suggested, such as giving the surfaces a small
thickness, performing a volume integration, and dividing by the
thickness, or using Dirac functions to define the surfaces. I have
not tried either of these. Thanks to everybody for their help. | {"url":"http://forums.wolfram.com/mathgroup/archive/2010/Aug/msg00146.html","timestamp":"2014-04-21T09:46:03Z","content_type":null,"content_length":"28089","record_id":"<urn:uuid:a34142d3-2be3-4276-9c66-cba07ca7e0fa>","cc-path":"CC-MAIN-2014-15/segments/1397609539705.42/warc/CC-MAIN-20140416005219-00457-ip-10-147-4-33.ec2.internal.warc.gz"} |
Mplus Discussion >> Multiple group approach with latent classes
Ravi Jasuja posted on Wednesday, November 08, 2006 - 8:28 am
Is it possible to use latent factors to define groups in a multiple group approach in SEM? Could you please direct me to articles which discuss this approach? My knowledge of statistics is at best
Any help would be appreciated.
Thank you.
Linda K. Muthen posted on Thursday, November 09, 2006 - 10:10 am
If you want to use a categorical latent variable is used to define latent classes based on unobserved heterogeneity in the data, use latent class analysis.
Ravi Jasuja posted on Thursday, November 09, 2006 - 10:33 am
Dear Dr. Muthen,
Thank you for your prompt response.
Does that also mean I can use the latent factor as a moderator with latent class analysis? Could you direct me to literature where latent class analysis is used to test the effects of a latent
variable as a moderator?
Thank you again.
Linda K. Muthen posted on Thursday, November 09, 2006 - 6:04 pm
It is not clear what you are trying to do. If you are interested in latent variable interactions, then you would not use latent class analysis. Mplus has a special option, XWITH to use for latent
variable interactions. You would not form groups using the continous latent variable. Latent class analysis use categorical latent variables.
Michael Spaeth posted on Wednesday, April 08, 2009 - 9:39 am
Dear all,
I would like to do a multiple group analysis and test the equality of parameters of a cross domain growth curve model (deviant peers/Delinquency). The groups, however, are not observed but should be
based on a growth mixture analysis (family climate). One approach would be to save the Cprb of the mixture analysis and use the most likely probability class membership as grouping variable
afterwards when doing multiple group analysis, right?
However, I prefer group memberships based on posterior probabilities due to class uncertainty. Is there a way to conduct the above mentioned multiple group analysis based on fractional class
membership? I guess one has to handle the cross domain model and the mixture model in ONE model to achieve this, but I don't want my growth mixture model to be influenced by the cross domain growth
Any help would be greatly appreciated!
Bengt O. Muthen posted on Wednesday, April 08, 2009 - 3:15 pm
Sounds like you have 3 processes and you want to keep the GMM for 2 (deviant, delinq) separate from the GMM of the third (family). You say you want to take a "analyze-classify-analyze" approach. The
quality of this depends on the entropy of the family GMM - see different approaches discussed in the newly posted paper:
Clark, S. & Muthén, B. (2009). Relating latent class analysis results to variables not included in the analysis. Submitted for publication. Submitted for publication.
which you find under Papers, LCA.
Jerry Cochran posted on Wednesday, February 01, 2012 - 12:02 pm
I have two datasets from different behavioral health RCTs that contain a similar measure of alcohol use consequences.
I have run LCAs seperately for each dataset to see if the models are similar. The dataset with 750 cases produces a model with five classes, and the other dataset with 250 cases produces a model with
four classes. Conditional item probabilities, covariate effects, and distal outcome posses several similarities, with the exception of the missing class in the second dataset.
My question is if there is a statistical test compare these two models directly to one other, maybe like a chi-square difference test for multiple groups (but, I am not sure if such an application
would be appropriate).
Any ideas would be appreciated. Thank you
Bengt O. Muthen posted on Wednesday, February 01, 2012 - 2:40 pm
You can analyze the two datasets jointly and test any parameter equality using Model Test.
Laura Marie Schons posted on Wednesday, February 29, 2012 - 7:33 am
I also would like to estimate a multiple group analysis for a given structural equation model using an unobserved group variable from a latent class analysis. I have read the paper that you
Clark, S. & Muthén, B. (2009). Relating latent class analysis results to variables not included in the analysis.
I understand that applying sigle step regressions delivers the best results. I have read the syntax at the end of the paper but I still do not understand how I would have to specify this type of
analysis for an MGA in MPlus in my given case?
Any help would be greatly appreciated.
Thank you very much in advance!
Bengt O. Muthen posted on Wednesday, February 29, 2012 - 8:39 am
Are you saying that you are interested in how to simultaneously, in one single analysis, do a combination of an LCA and an SEM?
Laura Marie Schons posted on Wednesday, February 29, 2012 - 1:13 pm
Yes, would that be possible?
Linda K. Muthen posted on Wednesday, February 29, 2012 - 6:09 pm
Yes, see Example 7.19 in the user's guide.
Maira Covre-Sussai posted on Thursday, June 07, 2012 - 10:25 am
I am estimating a multiple group mixture model using the KNOWNCLASS specification (7.21 in MPLUS user's guide).
I have two continuous indicators, collected over 8 regions (groups). I want to estimate model with 2 latent classes therefore, I use the following set-up:
Data: FILE IS LA_LCA.dat;
NAMES = LA AGECOH NCHILD;
MISSING = ALL (999);
CLASSES = cg (8) c(2);
KNOWNCLASS = cg (LA = 1 LA = 2 LA = 3 LA = 4 LA = 6 LA = 7 LA = 8 LA = 9);
Type is Mixture;
c ON cg;
Model c:
[AGECOH NCHILD];
[AGECOH NCHILD];
Model cg:
tech1 tech8;
(1) Is this the correct specification?
(2) As I understand it, this model implies complete homogeneity, right? I also want to verify if the associations between the latent variable and each indicator variable are identical across all of
the samples (structural equivalence, McCutcheon, 2002). How can I do this?
Thank you in advance.
Linda K. Muthen posted on Thursday, June 07, 2012 - 2:26 pm
To me your model looks like heterogeneity. You have means and variances free over all classes. The variances are equal as the default. The means are free. To hold them equal change all of your
variance statements to:
[AGECOH NCHILD] (1);
Maira Covre-Sussai posted on Friday, June 08, 2012 - 3:06 am
Dear Dr. Muthen,
Thank you for your answer.
My next step is to extend the model by including two categorical (binary) and one nominal variable with 4 categories.
I also have doubts about this:
(1) How can I include them in the model specification?
(2) How can I hold thresholds equal across classes of c?
(3) How can I hold the classes of c equal across groups (cg)?
Thank you in advance!
Linda K. Muthen posted on Friday, June 08, 2012 - 10:45 am
1. Add them to the USEVARIABLES list.
2. For variables on the CATEGORICAL list:
[u$1] (1);
For variables on the NOMINAL list:
[u#1] (1);
3. Remove c ON cg.
Maira Covre-Sussai posted on Thursday, June 21, 2012 - 2:21 am
Dear Dr. Muthen,
Thank you for your answer.
I am not sure if I express myself correctly or if I am doing something wrong, but I am not achieving the results that I am aiming at.
I am trying to fit a Multiple Group Latent Class analysis, with continuous, ordinal and nominal indicators. For this, I am trying to assess the Measurement Equivalence of my model by comparing three
models: (1) an unrestricted, heterogeneous model, which allows both intercept and slope parameters to vary across countries; (2) a partially homogeneous model in which slope parameters are
constrained to be equal across countries; and (3), a structurally homogeneous model that constrains both intercept and slope parameters to be equal across countries.
I am actually trying to replicate the study of Kankaras, Vermunt and Moors (2011)* using MPlus and I confess I am not succeeding at all!
Could you help me with the set up, please?
* Kankaras, Vermunt and Moors (2011). Measurement Equivalence of Ordinal Items: A Comparison of Factor Analytic, Item Response Theory, and Latent Class Approaches. Sociological Methods & Research
Linda K. Muthen posted on Thursday, June 21, 2012 - 2:57 pm
Please send your output and license number to support@statodel.com.
shaun goh posted on Tuesday, March 19, 2013 - 11:57 pm
Dear Dr Muthen,
I am interested in conducting multiple-group latent growth curve modelling. I was wondering if it is possible to utilise groups from a LCA/LPA to conduct multiple-group LGCM in Mplus?
To give more context, I would like to use LCA/LPA on language indicators to define children with different severity of language ability (i.e. impaired vs non-impaired), and then compare the growth
trajectories of their psychological outcomes and effects of covariates using multiple group LGCM.
From my understanding, there are two ways to proceed.
Approach A.) A 3 step proccedure to utilise LCA/LPA first to define the groups of interest, then do model fitting for each group independently to determine configural and measurement invariance, then
finally run a multiple-group model.
Approach B.) To run LCA/LPA and the final multiple-group LGCM simulatenously in one model.
However, it seems that approach(A) is statistically limited by having to treat LCA/LPA class probabilities as observed classes, and that approach (B) is limited by assuming that all groups have the
same underlying growth factors.
I was wondering if there's another way to model this proposal without running into these limitations, or a comment about which approach may be better (A or B, e.g. perhaps utilise A when posterior
probability values >.80?)
Thank you for all your help so far,
Bengt O. Muthen posted on Wednesday, March 20, 2013 - 9:19 am
I think the answer depends on what you want your classes to represent substantively. Let's call your language indicators U and your psychological outcomes for which you have repeated measures Y. Do
you want classes to reflect features of only U or both U and Y?
If you want the classes to be formed using only U information, you have 2 choices. If entropy is high you can simply use most likely class membership as a grouping variable in a multiple-group
analysis. If entropy is not high you can do a "manual 3-step" in line with the revised Web Note 15 where the 3rd step is the growth model; this takes care of classification error (check that the
class formation doesn't change).
In contrast, in a 1-step analysis of U and Y jointly, your classes will be reflect features of both U and Y. It is a substantive question whether or not you want that. In a 1-step analysis you can
actually have a latent class variable for U and another one for Y and see how they relate.
Regarding your objection to your Approach B of having the same growth factors for all groups, note that this model does allow for different growth factor means.
Maksim Rudnev posted on Thursday, January 09, 2014 - 4:14 am
Dear Dr.Muthen,
I have the following model with two latent classes and three known classes (a multiple group LCA) and I want to constrain and relax thresholds across groups.
CLASSES ARE cluster(2) known(3);
KNOWNCLASS is known (group = 1 2 3);
cluster on known;
MODEL cluster:
[u1$1 - u10$4] (c1_1-c1_40);
[u1$1 - u10$4] (c2_1-c2_40);
[u1$1 - u10$4] (k1_1-c1_40);
[u1$1 - u10$4] (k2_1-c2_40);
[u1$1 - u10$4] (k3_1-c3_40);
1. It seems that Mplus completely ignores section MODEL KNOWN since the output shows that the thresholds of classes CLUSTER are constrained between groups KNOWN. What should I do to relax these
2. I wish to constrain thresholds u1$1-u1$4 to be equal across groups known and all the other thresholds are allowed to differ across groups. Labels seem too be ambigious in my case and I am not sure
what should be specified in MODEL CONSTRAINT section.
--Maksim Rudnev
Bengt O. Muthen posted on Thursday, January 09, 2014 - 8:30 am
1. If you delete the Model cluster and Model known statements, all threshold equalities are relaxed.
2. You can obtain full flexibility in constraining the thresholds by using the dot (.) command, e.g. for the first class of both of the latent class variables:
Back to top | {"url":"http://www.statmodel.com/discussion/messages/13/1780.html?1389285035","timestamp":"2014-04-19T09:26:35Z","content_type":null,"content_length":"50883","record_id":"<urn:uuid:cb7968fb-d46d-4734-b62c-9057d36128e8>","cc-path":"CC-MAIN-2014-15/segments/1397609537097.26/warc/CC-MAIN-20140416005217-00412-ip-10-147-4-33.ec2.internal.warc.gz"} |
Coproducts of schemes ("gluing construction") ?
up vote 0 down vote favorite
In this MO question it was raised the topic of "gluing constructions" in the category of schemes. I understand the phrase "gluing two schemes along maps to them" as "there exists a coproduct of the
two schemes (with respect to the two given morphisms) in the category of schemes".
Let's consider the affines first. If $R'$, $R''$ and $R$ are rings, and $\phi': R' \to R$ and $\phi'':R''\to R$ are homomorphisms, then one can define the ring
$A=R'\times_{\phi',R,\phi''} R'':=\;${$(a,b)\in R'\times R''$ | $\phi'(a)=\phi''(b)$}.
The first question is:
Is the ring $A$ so constructed always the fibered product in the category of rings of $R'$ and $R''$ along the prescribed maps $\phi'$ and $\phi''$ ? (I guess this may be answered by abstract
nonsense alone)
In case the answer to the above question is "yes", then one automatically gets the existence of fibered co-products (i.e. verifying the dual universal property than fibered products) in the category
of affine schemes.
So one may ask:
Under which assumptions does it carry over to the non-affine case?
ag.algebraic-geometry schemes
Ya, i suspected it was some general thing about equalizers. – Qfwfq Mar 2 '11 at 22:12
2 Sorry for the barge-in-and-edit, but the typo in the title just screamed at me. – David Roberts Mar 2 '11 at 22:30
Not literally, of course. – David Roberts Mar 2 '11 at 22:30
Thank you ! – Qfwfq Mar 3 '11 at 12:35
add comment
2 Answers
active oldest votes
There is a pretty good account of how this works in Karl Schwede's paper:
MR2182775 (2006j:14003)
Karl Schwede, Gluing schemes and a scheme without closed points.
Recent progress in arithmetic and algebraic geometry, 157–172, Contemp. Math., 386, Amer. Math. Soc., Providence, RI, 2005
up vote 5 See also this paper:
down vote
MR2044495 (2005a:13016)
Ferrand, Daniel Conducteur, descente et pincement.
Bull. Soc. Math. France 131 (2003), no. 4, 553–585.
1 Beat me to it. Note that you're also interested in the coproduct in the category of ringed spaces (so the "gluing" is the topological gluing that you expect). The paper shows that if
R is a closed subscheme of one of the others then all three concepts coincide. – Lloyd Smith Mar 2 '11 at 19:21
add comment
Gluing produces a coproduct only when you glue along the empty subscheme. In general, if we glue $X$ and $Y$ along maps $U \to X , U \to Y$ together, we want to get the pushout (also
called amalgamated sum) of the diagramm $X \leftarrow U \to Y$.
The answer to the first question is, of course, yes. You can verify the universal property directly. More generally, every limit can be constructed via products and equalizers, which is a
up vote 3 fact from basic category theory. In particular, a fiber product of the diagram $A \to C \leftarrow B$ is equal to the equalizer of $A \times B \to A \to C$ and $A \times B \to B \to C$.
down vote
Also Sándor Kovács has already pointed you to Karl Schwede's paper in which the second question is answered, let me just say that this article shows that pushouts along closed immersions
exist and are given locally by the fiber product of rings. But in general, pushouts of schemes do not exist at all. See this question.
add comment
Not the answer you're looking for? Browse other questions tagged ag.algebraic-geometry schemes or ask your own question. | {"url":"http://mathoverflow.net/questions/57156/coproducts-of-schemes-gluing-construction","timestamp":"2014-04-16T16:41:58Z","content_type":null,"content_length":"61728","record_id":"<urn:uuid:ba7ffb00-6836-4ade-863d-dbbfe2dfc01e>","cc-path":"CC-MAIN-2014-15/segments/1398223206672.15/warc/CC-MAIN-20140423032006-00374-ip-10-147-4-33.ec2.internal.warc.gz"} |
Mount Ephraim Calculus Tutor
Find a Mount Ephraim Calculus Tutor
I have been a part time college instructor for over 10 years at a local university. While I have mostly taught all levels of calculus and statistics, I can also teach college algebra and pre-
calculus as well as contemporary math. My background is in engineering and business, so I use an applied math approach to teaching.
13 Subjects: including calculus, geometry, statistics, algebra 1
...I took a semester-long class that involved projects created in MatLAB, and learning the essentials of the programming language. My specialty is helping novice users become more comfortable with
the software. I've been using Microsoft Outlook in a professional setting since 2011, and have learned the ins and outs of the program.
21 Subjects: including calculus, reading, physics, geometry
...My name is Michael, and I am an experienced young professional conducting independent research at UPenn. I studied at West Chester University of Pennsylvania for Physics and am currently at
Philadelphia University for Mechanical Engineering. I have worked with students in math and science for over two years from middle school to college levels.
9 Subjects: including calculus, physics, geometry, algebra 1
...For the SAT, I implement a results driven and rigorous 7 week strategy. PLEASE NOTE: I only take serious SAT students who have time, the drive, and a strong personal interest in learning the
tools and tricks to boost their score. Background: I graduated from UCLA, considered a New Ivy, with a B.S. in Integrative Biology and Physiology with an emphasis in physiology and human anatomy.
26 Subjects: including calculus, chemistry, English, reading
...I am highly qualified in mathematics and physics, I am patient, and passionate to help anyone seeking assistance in a wide range of mathematics and physics. I am looking forward to helping you
understand and gain the confidence to achieve more in your education.I majored in Math, and as part of ...
27 Subjects: including calculus, chemistry, economics, elementary math | {"url":"http://www.purplemath.com/Mount_Ephraim_calculus_tutors.php","timestamp":"2014-04-21T02:28:15Z","content_type":null,"content_length":"24446","record_id":"<urn:uuid:f8ced737-c540-4567-b040-3a076d04e698>","cc-path":"CC-MAIN-2014-15/segments/1397609539447.23/warc/CC-MAIN-20140416005219-00231-ip-10-147-4-33.ec2.internal.warc.gz"} |
The Eilenberg Swindle
Recall from the last post that if $R$ is a commutative ring, we define $K_0(R)$ to be the Grothendieck group of the isomorphism classes of finitely generated projective $R$-modules. It is natural to
ask what happens if we replace finitely generated projective modules with countably generated projective modules. Let us write $\mathfrak{K}_0(R)$ for this group. It turns out that understanding $\
mathfrak{K}_0(R)$ is extremely easy.
Theorem: For any commutative ring $R$, $\mathfrak{K}_0(R)=0$.
Proof: We have to show that if $A$, $B$, $C$, and $D$ are countably generated projective $R$-modules, there is some countably generated projective $R$-module $E$ so that $A\oplus D\oplus E\cong B\
oplus C\oplus E$. Define $E=\bigoplus_{i=1}^\infty (A\oplus B\oplus C\oplus D)$. Hence $\mathfrak{K}_0(R)=0$.
A similar construction shows up in the theory of group rings. Here’s an exercise from T.Y. Lam’s Exercises in Classical Ring Theory:
Exercise 8.16: Let $G$ and $H$ be any two groups. Show that there is some ring $R$ so that $R[G]\cong R[H]$. (Here $R[G]$ is the ring of finite $R$-linear combinations of elements of $G$, and
multiplication is defined by the group multiplication of $G$.)
Solution: Let $K=(G\times H)\times(G\times H)\times\cdots$, and set $R=\mathbb{Z}[K]$. Then $R[G]\cong R[H]$.
Lam makes the comment that, although consideration of the group rings $\mathbb{Z}[G]$ and $K[G]$ are very useful for determining properties of $G$ (for instance, the modules over these rings are the
objects of study in group cohomology and representation theory, respectively), the group ring $R[G]$ for an arbitrary ring $R$ might not give us much information about $G$.
There’s an interesting article I found on more general Eilenberg swindles, but the authors don’t define progenerators, so I’ll include that here.
Let $R$ be a ring and $M$ a right $R$-module. Define $M^\ast=Hom_R(M,R)$ and $S=Hom_R(M,M)$. Then $M$ and $M^\ast$ are $S-R$ and $R-S$ bimodules, respectively. Furthermore, we can define
multiplications $M^\ast M\subseteq R$ by $m^\ast m=m^\ast(m)$ and $MM\ast\subseteq S$ by $mm^\ast(m')=m(m^\ast m')$. We say that $M$ is a progenerator if $MM^\ast=S$ and $M^\ast M=R$.
5 comments
Interesting, I was just reading about this in a book on K-theory.
Actually that’s the same book I am currently studying. It was an exercise though. I think Weibel probably mentioned the trick somewhere in his book on homological algebra, so I realized the
[...] The Eilenberg swindle can also be stated in a slightly different form. [...]
Great post, thank you. | {"url":"http://complexzeta.wordpress.com/2007/08/28/the-eilenberg-swindle/","timestamp":"2014-04-19T22:28:35Z","content_type":null,"content_length":"56644","record_id":"<urn:uuid:5f0620ae-3ae7-42ba-90d5-aa23adc6cbb0>","cc-path":"CC-MAIN-2014-15/segments/1397609537754.12/warc/CC-MAIN-20140416005217-00613-ip-10-147-4-33.ec2.internal.warc.gz"} |
Gamma functions
October 16th 2013, 10:18 PM
Gamma functions
Hi all,
I am reviewing a section of my device physics textbook which covers the effective density of states. In the books' derivation, they reach a point where they say this:
The integral is the gamma function, with a value of
$\int_0^\infty \eta ^\frac{1}{2} exp(-\eta ) d\eta = \frac {1}{2} \sqrt{\pi}$
...and then jump back into the full derivation of the effective density of states (using other factors not shown here).
My Question:
Can someone please show me (in a detailed way) how they arrived at $\frac {1}{2} \sqrt{\pi}$? Also, in laymen's terms... what is a gamma function, and how is it advantageous? I tried looking this
up, and could not make sense of what was being explained.
Thank you!
October 17th 2013, 12:25 AM
Re: Gamma functions
start from here:
Gamma function - Wikipedia, the free encyclopedia
then have alook here:
Gamma Function -- from Wolfram MathWorld
October 17th 2013, 06:29 AM
Re: Gamma functions
Thank you, but I came here because you need a PhD in math to understand either the Wikipedia entry or the Wolfram entry :)
October 17th 2013, 06:39 AM
Re: Gamma functions
yes you are right the gamma and beta functions are part of very advanced mathematics available to very few...
you may get some help from youtube videos like the one here:
Gamma Function - Part 1 - Functional Equation - YouTube | {"url":"http://mathhelpforum.com/advanced-applied-math/223137-gamma-functions-print.html","timestamp":"2014-04-19T00:21:15Z","content_type":null,"content_length":"5830","record_id":"<urn:uuid:a165fa35-d919-443c-8896-3f293edbc48c>","cc-path":"CC-MAIN-2014-15/segments/1397609535535.6/warc/CC-MAIN-20140416005215-00150-ip-10-147-4-33.ec2.internal.warc.gz"} |
The Open Group Base Specifications Issue 6
IEEE Std 1003.1, 2004 Edition
Copyright © 2001-2004 The IEEE and The Open Group, All Rights reserved.
cproj, cprojf, cprojl - complex projection functions
#include <complex.h>
double complex cproj(double complex z);
float complex cprojf(float complex z);
long double complex cprojl(long double complex z);
These functions shall compute a projection of z onto the Riemann sphere: z projects to z, except that all complex infinities (even those with one infinite part and one NaN part) project to
positive infinity on the real axis. If z has an infinite part, then cproj( z) shall be equivalent to:
INFINITY + I * copysign(0.0, cimag(z))
These functions shall return the value of the projection onto the Riemann sphere.
No errors are defined.
The following sections are informative.
Two topologies are commonly used in complex mathematics: the complex plane with its continuum of infinities, and the Riemann sphere with its single infinity. The complex plane is better suited
for transcendental functions, the Riemann sphere for algebraic functions. The complex types with their multiplicity of infinities provide a useful (though imperfect) model for the complex plane.
The cproj() function helps model the Riemann sphere by mapping all infinities to one, and should be used just before any operation, especially comparisons, that might give spurious results for
any of the other infinities. Note that a complex value with one infinite part and one NaN part is regarded as an infinity, not a NaN, because if one part is infinite, the complex value is
infinite independent of the value of the other part. For the same reason, cabs() returns an infinity if its argument has an infinite part and a NaN part.
carg(), cimag(), conj(), creal(), the Base Definitions volume of IEEE Std 1003.1-2001, <complex.h>
First released in Issue 6. Derived from the ISO/IEC 9899:1999 standard.
End of informative text.
UNIX ® is a registered Trademark of The Open Group.
POSIX ® is a registered Trademark of The IEEE.
[ Main Index | XBD | XCU | XSH | XRAT ] | {"url":"http://pubs.opengroup.org/onlinepubs/009695399/functions/cprojf.html","timestamp":"2014-04-16T22:35:22Z","content_type":null,"content_length":"5278","record_id":"<urn:uuid:454c45d1-a8b5-4a10-9b66-fcbb6adf73a0>","cc-path":"CC-MAIN-2014-15/segments/1398223202774.3/warc/CC-MAIN-20140423032002-00511-ip-10-147-4-33.ec2.internal.warc.gz"} |
The `%` operator
up vote 0 down vote favorite
I want to find all the numbers divisble by all the numbers between 1 and 5. how do I write the program so that if the remainder of 'start' divided by all the numbers that x goes through is equal to 0
that it will print start. Is there any syntax that will calculate what I'm looking for. thanks.
import math
def main():
one = 1
start = 1
while one == 1:
for x in range(1, 5):
if start % x == 0:
print start
start += 1
python math
is that your real problem? are you really just looking for all multiples of 60? – hop Feb 20 '09 at 23:09
almost exactly the same code was posted stackoverflow.com/questions/567222/… and what exactly "Is there any syntax that will calculate what I'm looking for" supposed to mean? – SilentGhost Feb 20
'09 at 23:15
First, "while True:" is much more concise. Second, the current code looks broken; for the iterations, x is increasing from 1 to 4 and start is increasing from 1 to 4, but simultaneously. You want
to loop independently for each one. – Nikhil Chelliah Feb 20 '09 at 23:52
I recognize this example - it is how I used to think about programming when I learned QuickBasic. (Using excessive variables to control status of a loop) – Marc Maxson Dec 21 '12 at 15:23
add comment
2 Answers
active oldest votes
if I understood correctly you want something like this:
start = 1
while (True):
flgEvenlyDiv = True
for x in range(1, 5):
if (start % x != 0):
up vote 0 down vote flgEvenlyDiv = False
if (flgEvenlyDiv == True):
print start
start += 1
add comment
First of all, you seem to ask for all multiples of 60. Those can be rendered easily like this (beware, this is an infinite loop):
from itertools import count
for i in count():
print i*60
If you just oversimplified your example, this is a more pythonic (and correct) solution of what you wrote (again an infinite loop):
from itertools import count
# put any test you like in this function
up vote 3 down vote def test(number):
return all((number % i) == 0 for i in range(1,6))
my_numbers = (number for number in count() if test(number))
for number in my_numbers:
print number
You had a grave bug in your original code: range(1,5) equals [1, 2, 3, 4], so it would not test whether a number is divisble by 5!
PS: You have used that insane one = 1 construct before, and we showd you how to code that in a better way. Please learn from our answers!
sorry ure completly right i forgot to change it. thanks – marc lincoln Feb 21 '09 at 12:47
add comment
Not the answer you're looking for? Browse other questions tagged python math or ask your own question. | {"url":"http://stackoverflow.com/questions/571538/the-operator?answertab=oldest","timestamp":"2014-04-20T21:28:11Z","content_type":null,"content_length":"72704","record_id":"<urn:uuid:cda16f52-7cbe-483b-8741-ab53b863fa27>","cc-path":"CC-MAIN-2014-15/segments/1397609539230.18/warc/CC-MAIN-20140416005219-00525-ip-10-147-4-33.ec2.internal.warc.gz"} |
Binary Coordinate Systems
by Steven H. Cullinane
(Article intended for American Mathematical Monthly readers, written July 1984)
Mathematics Subject Classification (MSC2000) -
20B25, Finite automorphism groups of algebraic, geometric, or combinatorial structures.
05B25, Finite geometries;
51E20, Combinatorial structures in finite projective spaces.
This article tells how to visualize some low-dimensional vector spaces over the finite field F[2] as square or cubical arrays of points in Euclidean 3-space. Simple geometric operations on the
point-arrays are shown to generate linear and affine groups over F[2] of orders up to approximately 1.3 trillion. Some properties of these groups are sketched.
The typical example of a finite group is GL(n,q), the general linear group of n dimensions over the field with q elements. The student who is introduced to the subject with other examples is
being completely misled. -- J. L. Alperin [AL]
Among the algebraic structures known as fields, clearly the most important for applications are the rationals, the reals, and the complex numbers, together with the fields of rational functions with
coefficients in one of these number fields. What is the next most important field? Combinatorialists, coding theorists, and computer scientists would be likely to name the ridiculously simple
structure F[2], consisting of the elements 0 and 1, with 1 + 1 = 0.
One reason for the importance of this field is that the 1's in a vector in an n-space over F[2] can be used to indicate membership in a subset of an n-set, with addition of vectors corresponding to
taking the symmetric difference of sets. A family of subsets corresponds to a set of vectors, which in turn generate a vector subspace that may be easier to work with than the original subsets.
Permutations on the parent n-set that leave the family of subsets invariant induce linear transformations on the corresponding subspaces. (J. H. Conway's analysis of M[24], sketched below,
exemplifies this approach to working with subsets.)
These considerations, and Alperin's remarks quoted above, indicate that linear and affine maps on vector spaces over F[2] are reasonable things to study. Just as in linear algebra over the reals, the
student working over F[2] may be helped by seeing low-dimensional examples of how linear and affine maps actually move points throughout a vector space. (Over the reals, the student can visualize the
global effects of such things as shears, dilations, rotations, and reflections.) We will describe how one can at least begin to visualize global actions of the linear and affine groups of 2, 3, 4,
and 6 dimensions over F[2], by constructing generators for these groups that have easily pictured actions on square and cubical arrays of points. (The awkward case of 5 dimensions we leave as an
Before delving into the construction of these low-dimensional groups, we first show that their sizes are nontrivial, and then note that one of them, GL(4,2), has several nice properties that make it
of particular interest to group theorists.
The notation GL(n,q) gives little idea of the size of these groups, which grow quite rapidly as n increases. As an aid to intuition, we list the orders of the first six general linear groups over F
[2], and of the corresponding affine groups AGL(n,2). Here AGL(n,2) is a semidirect product of the elementary abelian group V(n,2) of order 2^n (the translation group) by GL(n,2). (In fact, AGL(n,q)
is the holomorph of V(n,q). See [RO, pp. 136-139] and [CA, pp. 68, 112-113].)
│ Group │ Order of group │
│ GL(1, 2)│ 1│
│ AGL(1, 2)│ 2│
│ GL(2, 2) (=S[3])│ 6│
│ AGL(2, 2) (=S[4])│ 24│
│ GL(3, 2)│ 168│
│ AGL(3, 2)│ 1,344│
│ GL(4, 2)│ 20,160│
│ AGL(4, 2)│ 322,560│
│ GL(5, 2)│ 9,999,360│
│ AGL(5, 2)│ 319,979,520│
│ GL(6, 2)│ 20,158,709,760│
│ AGL(6, 2)│ 1,290,157,424,640│
To verify these numbers, note that the order of GL(n,q) is
(q^n - 1)(q^n - q)(q^n - q^2)...(q^n - q^n-1),
which is the number of ways to choose successive images of basis vectors, and that
|AGL(n,q)| = q^n|GL(n,q)|.
Among the groups above, GL(4,2) has several nice properties that make it of particular interest to students of group theory, combinatorics, or finite geometry.
1. GL(4, 2) is one of two nonisomorphic simple groups of the same order, the other being the projective unimodular group PSL(3, 4). (See AR, p. 171, or RO, p. 172.)
2. GL(4, 2) is isomorphic to A[8], and this isomorphism plays a role (described below) in the structure of the quintuply transitive Mathieu group M[24].
3. Underlying the isomorphism of GL(4, 2) with A[8] is a structure that by itself is of interest in combinatorics and finite geometry. This structure is a correspondence J between the 35 partitions
of an 8-set into two 4-sets and the 35 2-dimensional subspaces of V(4, 2) when the latter is regarded as a vector 4-space over F[2]. Under the correspondence J, two partitions into 4-sets have a
common refinement as a partition into four 2-sets if and only if the corresponding 2-dimensional subspaces have a nontrivial intersection. (See HA or CA, p. 60.)
4. J. H. Conway [CO] has shown that M[24] can be generated by combining three sorts of permutations, one sort being an action of GL(4, 2) on the 16 points of V(4, 2), combined with the corresponding
action of A[8] on a separate set E of 8 points, the second sort being a translation acting on V(4, 2), with all points of E fixed, and the third sort being an interchange of E with either of two
halves of V(4, 2). Conway shows that a family of 8-element subsets (octads) is left invariant by the action of M[24] on a 24-set, and also shows that the vectors over F[2] corresponding to these
subsets span a 12-dimensional space C over F[2] that is known as the binary Golay code, in which each vector has 0, 8, 12, 16, or 24 1's. In a related paper, R. T. Curtis [CU] displays the
correspondence J, described above, in his "miracle octad generator," which beautifully pictures the locations of octads within a 4x6 array.
5. GL(4, 2) is the collineation group of a finite projective geometry, PG(3, 2), that as the smallest projective space is a rich source of examples for geometers.
We now describe how to generate eight of the groups on our list above, of orders 6 through approximately 1.3 trillion, by letting the symmetric groups S[3] (for linear groups) or S[4] (for affine
groups) permute natural subdivisions of square or cubical arrays.
We first recall that a permutation group acting on spatially located objects has isomorphic "alias" and "alibi" interpretations, depending on whether the cycle (ab...) means that object a is replaced
by object b, etc. ("alias"), or whether it means that the object in location a is replaced by the object in location b, etc. ("alibi"). To avoid confusion we will use the alibi approach throughout,
and will apply labels (numbers or (0,1)-vectors) only to locations within square or cubical arrays, and not to the objects permuted, which are "points" represented by unlabeled square or cubical unit
cells. (The reader may, of course, label these cells as he pleases.)
Our groups will act on the following arrays of unit cells: a 2x2 square, a 2x2x2 cube, a 4x4 square, and a 4x4x4 cube. In each case, the same mathod of labeling locations is used:
• First, think of the locations in a square array, or in the top layer of a cubical array, as labeled by the integers 0, 1, 2, ... in normal reading order (left to right, top to bottom); for a
cubical array, continue this process on lower layers, with the numbers increasing as the layers get lower.
• Now change the numbers to binary notation.
• Finally, regard the string of 0's and 1's labeling a given location as a vector over F[2], rather than a binary number.
By a standard result of linear algebra [AR, ch. IV] the group of all unimodular linear transformations on an n-space over a field, i.e. the group of all nxn matrices with determinant 1, is generated
by transvections, i.e. by the matrices B[ij](t) obtained from the identity matrix by substituting the scalar t for one of the 0's. Since we are working over F[2], every nonsingular linear
transformation is unimodular, so the transvections B[ij] are the only generators for GL(n, 2) we need examine.
A quick check shows that for the 2x2 array as labeled above, the B[ij] for GL(2, 2) generate the action of S[3] on the three unit cells not located at the origin (0, 0).
Given this S[3]action, clearly AGL(2, 2) acts as S[4], since any nonzero translation moves the cell at the origin into the S[3] orbit.
For the 2x2x2 array, the transvections of GL(3, 2) are easily seen to generate copies of the S[3] action on the top, left, and back of the cube, with this action being extended through adjacent
layers. In other words, GL(3, 2) is generated by S[3] acting in each of 3 sets of 4 parallel 1x1x2 sections of the cube, with the fourth section in each set being fixed under S[3]'s action on that
set, since the fourth section contains the cell located at the origin. By the same argument as above, AGL(3, 2) is generated by S[4] acting on each of the 3 sets of 4 parallel 1x1x2 sections.
For the 4x4 array, 6 of the B[ij] generate all 12 transvections, and also generate the action of S[3] on the 3 rows, the 3 columns, and the 3 2x2 quadrants of cells that do not contain the cell
located at the origin. Again, it is clear that AGL(4, 2) is generated by the action of S[4] on rows, on columns, and on quadrants.
For GL(6, 2) acting on the 4x4x4 cube, consider the three 4x4 submatrices of the 6x6 identity matrix that correspond to the 3 sets of parallel 1x4x4 layers of the cube such that 4 of the 6
coordinates are identical throughout the four layers in a set. Together, these three submatrices cover all possible locations for a 1 located outside the diagonal in a 6x6 transvection matrix. It
follows that the actions of GL(6 2) on the 4x4x4 cube are generated by S[3] permuting any 3 parallel 1x4x4 layers not containing a cell at the origin and by S[3] permuting any 3 parallel 2x2x4
sections not containing a cell at the origin. Naturally, AGL(6, 2) is generated by arbitrary permutations of parallel layers and of parallel 2x2x4 sections.
We conclude by mentioning an application to symmetries of ornamental patterns. Consider a cubic cell that is colored white around each of two opposite corners, in an area bounded by diagonals of the
faces surrounding these corners, and black elsewhere, so that a band of black running around the cube separates the two white "caps," which are images of one another under a reflection in the center
of the cube. If the reader arranges such cubes into a diamond pattern within any of the arrays we have discussed, he is likely to find that the resulting pattern has either an ordinary or a
color-interchange symmetry under some rigid motion (rotation, reflection, or inversion) of the entire array. That such symmetry always occurs, given the right sort of starting pattern, is due to the
Euclidean symmetries of partitions by affine hyperplanes.
For a summary of these results, see Affine groups on small binary spaces.
For a look at AGL(4, 2) in action, see The Diamond 16 Puzzle.
AL.... J. L. Alperin, book review,
Bulletin (New Series) of the American Mathematical Society 10 (1984) 121.
AR.... E. Artin, Geometric Algebra,
Interscience, New York, 1957.
CA.... P. J. Cameron, Parallelisms of Complete Designs,
Cambridge University Presss, Cambridge, 1976.
CO.... J. H. Conway, "Three lectures on exceptional groups,"
in Finite Simple Groups, edited by M. B. Powelll and G. Higman,
Academic Press, New York, 1971.
CU.... R. T. Curtis, "A new combinatorial approach to M[24],"
Math. Proc. Camb. Phil. Soc. 79 (1976) 25-42.
HA.... J. I. Hall, "On identifying PG(3, 2) and the complete 3-design on seven points,"
Ann. Discrete Math. 7 (1980) 131-141.
RO.... J. J. Rotman, The Theory of Groups, 2nd ed.,
Allyn and Bacon, Boston, 1973.
Page last updated April 29, 2002.
Page created January 27, 2001.
Copyright © 2001-2002 by Steven H. Cullinane. All rights reserved. | {"url":"http://finitegeometry.org/sc/gen/coord.html","timestamp":"2014-04-21T02:22:49Z","content_type":null,"content_length":"16025","record_id":"<urn:uuid:ddfdcb9b-b805-4ef0-8a0e-04806d808e59>","cc-path":"CC-MAIN-2014-15/segments/1397609539447.23/warc/CC-MAIN-20140416005219-00475-ip-10-147-4-33.ec2.internal.warc.gz"} |
Posts by
Total # Posts: 109
Why does Cl- have a larger ionic radius than Ca2+ ?
Why does Cl- have a larger ionic radius than CA2+ ?
Geometry: Ms. Sue or Steve or someone help please?
Given a triangle ABC with A(6b,6c) B(0,0) and C (6a,0), prove that the medians of the triangle are concurrent at a point that is two thirds of the way from any vertex to the midpoint of the opposite
side. I'm not sure how to prove this. I tried and when I got to finding th...
Given a triangle ABC with A(6b,6c) B(0,0) and C (6a,0), prove that the medians of the triangle are concurrent at a point that is two thirds of the way from any vertex to the midpoint of the opposite
side. I'm not sure how to prove this. I tried and when I got to finding th...
What does a change in the color of precipitate over time when a mixture of two solutions is combined with a reagent indicate?
Whats Ñ
IDENTIFY AND DESCRIBE 3 Environmental health hdzards that cause ill health and disasters within your community
If a solid is stretched or compressed beyond the point that it can return to its original shape it has passed its
Grand Canyon University
Show that 480 W of power is expended by a weightlifter when lifting a 60-kg barbell a vertical distance of 1.2 m in a time interval of 1.5 s.
i need help in math can someone help me
just confused with part c
Suppose the U.S government decides that the incomes of dairy farmers should be maintained at a level that allows the traditional family dairy farm to survive. It therefore implements a price floor of
$1 per pint by buying surplus milk until the market price is $1 per pint. Use...
D because with a positive attitude you can help improve your physical health and also boosts your mental health
Heat gained = Heat loss Let final temperature be x, Specific heat of Iron: 0.12 cal/g°C = 502 J/kg °C Specific Heat of Water: 4186 J/kg °C Heat gained by water = mass x specific heat capacity x
change in temperature =100(4186)* (x-25) =(418600x-10465000)J Heat lost...
if angle aob=4x+15 and aoc=3x+25, find the degree measures of angle aob and angle aoc
It only brought me back up to this page.
What 'EXPERIMENT' did Craig Venter & Francis Collins do in the Human Genome Project? Any links would be great! I've found a lot of info. on what it is, but I'm looking for info on what they did to
get there.
Grand Canyon University
Show that 480 W of power is expended by a weightlifter when lifting a 60-kg barbell a vertical distance of 1.2 m in a time interval of 1.5 s.
In Great Expectations, Pip is talking to the 'mysterious sailor' that is at his house. He asks the man "What floor do you want?" and the man replies "The top, Mr. Pip." What does this mean???
You have been hired as the marketing manager. Research and identify major approaches of how innovations like these are adopted by consumers. Discuss how the marketing strategy for the car needs to
rapidly move through these processes. Who or what are the key groups that need t...
ehich of the following is an example of projectile motion?
cultural diversity
I need help with identifying the difference in the three main groups, muslim, arabs, and christians where can I go to get some extra help
a boy sledding down a hill accelerates at 2.4 miles per second squared. if he starts from rest, in what distance would he reach a speed of 8.0 miles per second?
8 boxes of 1000 counters is greater than less than, or egual to 1,000
9th grade biology
How do macromolecules carry out the function of life within a cell?
9th Grade Biology
How do macromolecules carry out the function of life?
A solid that has a high melting point and conducts electricity in solution is a(n) ______________ solid. Im not sure what the answer is it molecular?
Vapor pressure ___________ as the temperature of the liquid in equilibrium with it increases.
Vapor pressure ___________ as the temperature of the liquid in equilibrium with it decreases.
What concentration of aqueous CaCl2 solution freezes at -10.2ºC? The freezing point of pure water is 0.0ºC and Kf of pure water is -1.86ºC/m.
ALGEBRA :)
its it 44in and the little 2
ALGEBRA :)
nd what about triangle;b =8 in..,h=11 ???
ALGEBRA :)
oh sorry = 60?
ALGEBRA :)
ALGEBRA :)
GIVE THE AREA OF THE FIGURE DESCRIBED. RECTANGLE; L=12CM , W=5CM HOW SHOULD I GET THE RESULT?
What is the freezing point of an aqueous 2.25 m potassium nitrate (KNO3) solution? The freezing point of pure water is 0.0ºC and Kf of pure water is -1.86ºC/m.
What is the freezing point of a solution made by dissolving 450 g of ethylene glycol (C2H6O2) in 550 g of water? The freezing point of pure water is 0.0ºC and Kf of pure water is -1.86ºC/m.
The solute molecules mixed with the solvent molecules in a solution cause the freezing point of a solution to be ______ than it is for an equal volume of pure _________.
What are the 7 properties that all living things share?
Quantitative Research
What is the probability of 59 if the mean is 60 and the standard deviation is 7?
social studies
Where is Friday before Wenesday? this is a riddle, pls. help me with the answer.
social studies
Where is Friday before Wednesda?
Algebra 1
How would you solve the problem (w - 2)^2 ???? So far I've gotten w^2-2w*2-4 Where do I go from here? Have I even gotten what I have right? I understand when I'm with my teacher, but as soon as I got
home, I was a little lost. So any help would be great!! THANKS!! (:
Not sure if I'm doing this right Ln(3x-4)-Ln(x+1)=Ln(2) My solution: Ln 3x-4/x+1 =Ln2 3x-4/x+1=2/1 2x+2=3x-4 2=x-4 6=x
I need a good hook for an essay. The essay's about being a peer for the children at my school who are mantally disabled. My 1st paragraph is about what I do in the class, 2nd is about the kids, & 3rd
is about special olumpics. Help PLEASEE!!!??
How many moles of oxygen will occupy a volume of 2.5 liters at 1.2 atm and 25 degrees celcius
(x-1)/(2x+3) is the answer because the (2x+7) in the numerator and denominator cancels out.
Language Arts
What's a good way to teach the word "Accolade" to the class. I have to make a prsentation that's less than 2 minutes, and teach my classmates the word so that it sticks. Any help???
Algebra 1
Like.... You solve it and then change it to the opposite. It's hard to explain now, but Igot it! Thank you guys so much! You're life savers!
Algebra 1
Oh!! You don't do the absolute value part until the end!! I get it now! I was changing it at the beginning!! THank you soo much!
Algebra 1
I had a quiz in math. The problem is: |4.3+(-7.2)| I got the answer 11.5 and my teachr marked it wrong. When she gave us a chance to fix the errors for half credit, I tried it again. But I KEEP
getting the same answer. What am I doing wrong?
8th grade
Umm... maybe to remind them that what they did was wrong and that they shouldn't do it. I don't know.... that's a good question. (:
7th grade 3D math
the answer is lwh
Algebra II
Factor Completely: x^2-2x-99
if there are 2 cylindrical cans. A label on the small one says it has a volume of 513 grams. the large one was destroyed. the small can's diameter was the same as the large can. large was 1/3 taller
than the small can. what is the volume of the large can?
social studies; AP US
I agree with you! William Jennings Bryan's "Cross of Gold" speech had to do with the issue of bimetallism, which was delivered at the 1896 Democratic National Convention in Chicago. Rudyard Kipling's
ideas had to do with the "White Man's Burden"...
US History
Oh, yes you're right! I just remembered that Henry Cabot Lodge had traveled oversea's with Teddy Roosevelt so I just assumed he would be different from the others, as they had all been senators. But
I didn't make the connection about the war with Spain! And thanks ...
US History
1) Which man does not belong in the same group with the others? a. Carl Schurz b. James G. Blaine c. Henry Cabot Lodge d. Albert Beveridge 2) The American record in the Philippines includes all the
following, EXCEPT: a. Training Filipinos for self-government b. Development of ...
If a 90%, 95% and 99% confidence interval are found for a population mean, how are the three interval estimates related to each other?
What does "llanura costera del pacifico" mean?
yeah, I tryed to look these up but my computer wasn't loading well. When it did load, the website didn't show me what I was looking for.
Is there any possible way that someone can give me the answer, I can't find it anywhere!
What is the plain that extends through northern Aregentina, Paraguay, and southern Bloivia? What is the grassy plateau located in southern Argentina between the Andes mountains and the AAtlantic
ocean? What is the fertile plain that covers most of central Argentina?
torque problem!
A spring (k =819N/m) is hanging from the ceiling of an elevator, and a 5.1-kg object is attached to the lower end. By how much does the spring stretch (relative to its unstrained length) when the
elevator is accelerating upward at a = 0.46m/s^2?
What's the question?
Yes, you should!
It is too big to carry around, like you can a map. Can you imagine trying to carry a huge globe around in your backpack, when instead, you can carry around a map?
social studies
Alrighty...so it is b then! Thanks for helping me! :)
social studies
Wait...no...I was looking up what "bicameral" meant and now I'm confused...is the answer d because we have the senate and the house of representatives? Or is it still b? Sorry...I think I just
confused myself! ;)
social studies
Oh so is it b then? Thank you for helping me!
social studies
Sorry...one more question! A feature of the American political system NOT adopted from practice in England is: a)appeal of disputed court decisions to higher courts b) legislative control of
government finaces c) local government through town meetings d) the bicameral legislat...
social studies
Thank you so much! :)
social studies
Prior to 1763, the British government generally followed a policy of salutary neglect towards the American colonies because it: a) wished to secure the loyalty of the colonies b) lacked a
well-organized imperial system c) lacked the power to make laws for the colonies d) did l...
7th grade technology... help!
ohhhhhh! okay, that explains a lot! thank you!
7th grade technology... help!
Can anyone come up with an acronym for COPYRIGHT ??????????
technology... please help!
Okay, thank you!!
technology... please help!
I have to write a poem and a song about computer graphics and animation. I'm really bad at this so can someone give me an idea?
social studies
A phisical maps shows water and landforms. (Such as mountain, rivers, etc.)
social studies
Athens, Greece. =) I really hope this helped you!!!!
social studies
stars mark the capital of a place. dots mark a city.
social studies
Pacific, Atlantic, Indian, Arctic, Southern Oceans. I had this question before! =)
social studies
The Map Key or Legend. (I'm not sure if I spelled that right)
I need to write a song and a poem about computer graphics and animation. Does anyone have any ideas about that?
social studies
I had this question in Geography! It is north. If you need to remember that, my teacher taught us a way to: Cancer would come first in a dictionary, so it comes first on a map. I hope this helps! =)
erupt and explode????? are you studying volcanoes???? =)
7th grade math
divide 5 by 16.
7th grade advanced math
Not off the top of my head, I don't. I'm in 7th grade advanced math and we get to use calculators. Can you use a calculator? If so, you can enter this. I have a scientific calculator, it's really
nice to have aroud for probblems like that!
count the number of lines then divide by two... then you'll see how much half a page is. or you can just stop when it looks like you've written half a page!
social studies
Your flight takes off from the eastern United States at 10 AM. What time will you arrive on the west coast, if the flight takes six hours?
maths for science
use the quotient rule to differentiate the following function f(x)=(1+In(x))/(x)
differentiate the following function using the composite rule h(x)=e^x(4-x)/6
How can a person distinguish between the prejudicial and nonprejudicial use of rhetorical devices?
Thank you.
This problem has to do with exponential models. The question says, you deposit $1600 in a bank account. Find the balance after 3 years for each of the following situations. The first one says: 1. The
account pays 2.5% annual interest compounded monthly. 2. The account pays 1....
consider the sequence given by U1=-0.3 Un+1=Un+0.7 i)state what type of sequence this is. ii)write down the first 4 terms of the sequence. iii) find a closed form for sequnce. iv) use the closed form
to find the nth term of the sequnce when Un=36.1 (n=1,2,3 ETC)
yes i do now - thank you again - i can go to sleep now!! bye thanks
5 hours
Pages: 1 | 2 | Next>> | {"url":"http://www.jiskha.com/members/profile/posts.cgi?name=cecilia","timestamp":"2014-04-18T07:25:21Z","content_type":null,"content_length":"25428","record_id":"<urn:uuid:39a198b4-5ba9-4136-bcd7-08fe056ffc07>","cc-path":"CC-MAIN-2014-15/segments/1397609532573.41/warc/CC-MAIN-20140416005212-00351-ip-10-147-4-33.ec2.internal.warc.gz"} |
Math Forum Discussions - Re: Matheology � 224
Date: Mar 17, 2013 7:59 PM
Author: Virgil
Subject: Re: Matheology � 224
In article
WM <mueckenh@rz.fh-augsburg.de> wrote:
> On 17 Mrz., 21:48, Virgil <vir...@ligriv.com> wrote:
> > Mathematical truth is independent of time.
> In fact??? Amazing! After Cantor's list has been diagonalized, it is
> possible to include all diagonals into the list. But someone has
> forbidden to change the list after time t_0 when the diagonalizers
> start to do their work.
Why does WM claim that after what WM calls "Cantor's list" has been
diagonalized, he can include all anti-diagonals, when it is always
possible to find others that have been so far overlooked?
After each anti-dagonal of any list is found, prefix it to that list and
then the anti-diagonal to the new list is not in the new list or the old
This procedure always finds new lines which are non-members of any of
the prior lists of lines including all lines of any original list and
all previously found anti-diagonals of those prior lists.
WM is just not paying attention!
WM has frequently claimed that HIS mapping from the set of all infinite
binary sequences to the set of paths of a CIBT is a linear mapping.
In order to show that such a mapping is a linear mapping, WM would first
have to show that the set of all binary sequences is a linear space
(which he has not done and apparently cannot do) and that the set of
paths of a CIBT is also a vector space (which he also has not done and
apparently cannot do) and then show that his mapping, say f, satisfies
the linearity requirement that f(ax + by) = af(x) + bf(y),
where a and b are arbitrary members of the field of scalars and x and y
and f(x) and f(y) are arbitrary members of suitable linear spaces.
While this is possible, and fairly trivial for a competent mathematician
to do, WM has not yet been able to do it.
But frequently claims already to have done it. | {"url":"http://mathforum.org/kb/plaintext.jspa?messageID=8657980","timestamp":"2014-04-17T07:10:16Z","content_type":null,"content_length":"3235","record_id":"<urn:uuid:be34693b-bc84-4067-92f4-87a9fca1fd93>","cc-path":"CC-MAIN-2014-15/segments/1397609526311.33/warc/CC-MAIN-20140416005206-00063-ip-10-147-4-33.ec2.internal.warc.gz"} |
RE: st: RE: RE: A Question on Selecting a Sample from a Panel Data Set
Notice: On March 31, it was announced that Statalist is moving from an email list to a forum. The old list will shut down at the end of May, and its replacement, statalist.org is already up and
[Date Prev][Date Next][Thread Prev][Thread Next][Date Index][Thread Index]
RE: st: RE: RE: A Question on Selecting a Sample from a Panel Data Set
From "Nick Cox" <n.j.cox@durham.ac.uk>
To <statalist@hsphsun2.harvard.edu>
Subject RE: st: RE: RE: A Question on Selecting a Sample from a Panel Data Set
Date Wed, 16 Jun 2010 10:23:29 +0100
No, I don't think so. You need to change the code upstream, not
set obs 19
gen int year=_n+1986
//10 firms
expand 10
bys year: gen byte firm =_n
gen byte derivuse=runiform()>.25 if year==2000 | year == 2001
bysort firm (derivuse) : replace derivuse = derivuse[1] == 1
Martin Weiss
bysort firm (derivuse): /*
*/ replace derivuse = derivuse[1] == 1 | /*
*/ derivuse[2] == 1
Hong Nguyen
I have two years of derivative use data (2000 and 2001). I could select
a sample of firms based on derivative use in either 2000 or 2001 (and
spread that property to other observations for the same firms). How
would I modify your previous commands to fit this situation?
Nick Cox wrote:
> Not quite.
> -derivuse- is undefined, namely missing, for years not 2000. You need
> spread that property to other observations for the same firms.
> bysort firm (derivuse) : replace derivuse = derivuse[1] == 1
Martin Weiss
> If your dataset looks like this:
> ***********
> clear*
> set obs 19
> gen int year=_n+1986
> //10 firms
> expand 10
> bys year: gen byte firm =_n
> gen byte derivuse=runiform()>.25 if year==2000
> ***********
> you can simply condition on -if derivuse==1- in your analyses.
Hong Nguyen
> I have a panel data set consisting of financial data for 423 firms
> a 19-year period. For one of the years (2000) I have data on
> derivatives use (use or non-use) for these 423 firms. I need to
> a sample of firms that used derivatives in 2000 and use it to run
> regressions over the entire period 1987-2005. This is where I'm stuck.
* 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/ | {"url":"http://www.stata.com/statalist/archive/2010-06/msg00918.html","timestamp":"2014-04-18T23:56:02Z","content_type":null,"content_length":"10129","record_id":"<urn:uuid:3f8b78f4-73cc-4e66-922a-79568a6987a3>","cc-path":"CC-MAIN-2014-15/segments/1398223211700.16/warc/CC-MAIN-20140423032011-00497-ip-10-147-4-33.ec2.internal.warc.gz"} |
The Colony Algebra 1 Tutor
Find a The Colony Algebra 1 Tutor
...I am a degreed engineer (Mechanical) with an MBA. I offer the distinct ability to explain mathematical concepts in "layman's" terms, often through the use of non-technical analogies and
demonstrations. Being married to an educator, I have tutored numerous students in the past.
8 Subjects: including algebra 1, geometry, algebra 2, ASVAB
...I have successfully tutored several students in the PSAT exam. I tutor all levels of math, starting with basic math. As a father I learned how to tutor my daughter and three sons when they were
48 Subjects: including algebra 1, chemistry, physics, calculus
...I specialize in standardized testing, with intimate knowledge of the MCAT, and whether it is your desire to go to medical or dental school or you just need help with biology or math, I can
assist you in a number of different ways. I can communicate a number of different topics in plain English, ...
15 Subjects: including algebra 1, chemistry, physics, algebra 2
...I have more than 10 years of public school teaching experience, a master's degree in gifted education, and I am completing my dissertation for my PhD in educational psychology. I have completed
coursework for a PhD in cognition and instruction. I have done research in the fields of problem solving and the testing effect, and have extensive knowledge of learning and memory.
39 Subjects: including algebra 1, reading, English, chemistry
...I will work with you at the level you are beginning on and take you to where you would like to be to complete your program of study.This course is the foundation for high school mathematics
courses. It is the bridge from the concrete to the abstract study of mathematics. Topics include simplify...
8 Subjects: including algebra 1, grammar, study skills, elementary (k-6th) | {"url":"http://www.purplemath.com/the_colony_algebra_1_tutors.php","timestamp":"2014-04-18T00:51:41Z","content_type":null,"content_length":"23963","record_id":"<urn:uuid:2362768f-e252-4704-83e6-58d2746a3034>","cc-path":"CC-MAIN-2014-15/segments/1397609532374.24/warc/CC-MAIN-20140416005212-00531-ip-10-147-4-33.ec2.internal.warc.gz"} |
2012-2013 Math News 2011-2012 Math News
The Putnam Exam is the first Saturday in December (Putnam home).
If you are interested please contact Scott Beaver (beavers@wou.edu) or Matt Nabity (nabitym@wou.ed).
Mathematics Fall Alumni Get Together (3rd Annual), University House, Friday, October 11, 2013
Sonia Kovalevsky Day for High School Girls, Saturday, March 1, 2014
Mathematics Spring Picnic, Thursday, June 5, 2014, WOU's Gentle House
(Undergraduates in) Mathematics Scholarship Winners
Scholarship page and information
Charlie Dolezal and Ernie and LaVerne Cummins Scholarships
2013-2014 awardees: Lorena Avila Perez, Camarie Campfield, Andy Fry, and Jillian Johnson
Special note: Long time friend, colleague and supporter of the Mathematics Department, Charlie Dolezal is the winner of this year's Alumni Award of Excellence!
Π Μ Ε (The National Mathematics Honor Society)
To learn about how students qualify for membership for Π Μ Ε at Western Oregon University, click here.
To learn more about Π Μ Ε, visit the national Π Μ Ε home page.
Congratulations to Andy Fry who won the Pi Mu Epsilon Outstanding Scholarship second prize at the 2014 Northwest Undergraduate Mathematics Symposium for his talk on Abelian Sandpiles.
Student News
Academic Excellence Showcase 2014
Western Oregon University mathematics majors will present their senior projects and senior capstone projects at the 2014 WOU Academic Excellence Showcase. Details TBA
Conferences and Papers 2013-2014
Molly Stubbliefield'13, pursuing her Ph.D. in the Mathematics program at the University of Oklahoma in Norman, Oklahoma has had her paper "Nowhere-zero k-flows on graphs" written by Matthias Beck,
Alyssa Cuyjet, Gordon Kirby, Molly Stubblefield, and Michael Young accepted for publication in Annals of Combinatorics.
Employment News
Emily Trigg BS '10 MAT '11 is the math teacher at Woodburn Success Alternative High School.
Graduate School News
WOU students off to graduate school for fall term 2014:
Current graduate school news:
REUs Summer 2014 (check out the Summer Program for Undergraduates page)
Andy Fry '15 accepted an offer from the San Diego State University Mathematics Research Experience for Undergraduates to study Numerical Semigroups summer 2014.
Study Abroad | {"url":"http://www.wou.edu/las/natsci_math/math/news.php","timestamp":"2014-04-17T04:04:49Z","content_type":null,"content_length":"18262","record_id":"<urn:uuid:fe54024d-d143-4c0f-89cc-ca2ec3a0c1bf>","cc-path":"CC-MAIN-2014-15/segments/1397609526252.40/warc/CC-MAIN-20140416005206-00017-ip-10-147-4-33.ec2.internal.warc.gz"} |
Re: st: comparing 25th percentile survival time between two race groups
Notice: On March 31, it was announced that Statalist is moving from an email list to a forum. The old list will shut down at the end of May, and its replacement, statalist.org is already up and
[Date Prev][Date Next][Thread Prev][Thread Next][Date Index][Thread Index]
Re: st: comparing 25th percentile survival time between two race groups
From Steve Samuels <sjsamuels@gmail.com>
To statalist@hsphsun2.harvard.edu
Subject Re: st: comparing 25th percentile survival time between two race groups
Date Sat, 29 Sep 2012 18:50:40 -0400
I should clarify. When survival in one group is "better" that might not mean at all percentiles. Think of two survival curves are similar up to 80% survival (i.e. 20-th percentile for failure) and then one curve is higher than the other thereafter.
About -ranksum- :
1. It is not for use with censored survival data, so you should not have
used it. I apologize for not catching this mistake earlier.
2. The corresponding test for censored survival data is: -sts test- with
a "wilcoxon option". (The full name for the original rank sum test is
"Wilcoxon two-sample rank sum test")
3. I previously said that a rank-sum test "tests for any difference
where survival in one group is better, not just differences in medians."
That means that if one distribution is shifted to the right relative to
another, all percentiles, including 10th, 25th 40th, etc. will also be
shifted. There is no specific test for a 25th percentile. You will see
that the formulas for the Wilcoxon test for censored or
uncensored data do not single out a specific percentile.
4. You appear unfamiliar with elementary survival concepts. Before
going any further, I suggest that you consult the Cleves
book that I referenced earlier or another introductory text.
On Sep 28, 2012, at 7:04 PM, Haena Lee wrote:
Thank you Steve, That's really good point to look at the graph
especially for the failures.
According to ranksum, I was not able to find an option for 25th percentiles.
When I was comparing the differences of median survival time between
African American and White across 10 different regions in US, I used
this following command;
. sort UNOS
. by UNOS: ranksum d_enrollment, by(race_all)
In this case, I didn't need to let Stata know it is median survival
time because I guess median was default. However my question is how I
can particularly tell Stata that it is to test 25th percentiles this
time. Any help? Thanks much.
On Fri, Sep 28, 2012 at 4:26 PM, Steve Samuels <sjsamuels@gmail.com> wrote:
> Dear Haena:
>> On Sep 28, 2012, at 2:43 PM, Haena Lee wrote:
>> Hi Listers,
>> I have tried to produce the median survival time and its 95% CI with
>> renal transplant data. "Stci" command, however, did not produce any
>> median values. So I generated the graph of median survival time by
>> race (African American vs. White) and the graph showed it only reached
>> to the length of 25% in the course of follow-up, indicating both of
>> groups have died not even near 50% (median), but mostly near the
>> length of 25% tile in the course of follow-up.
> What you observe can be better phrased as saying that followup was not
> long enough to observe 50% of failures.
>> 1. Given this, I attempted to produce 25th percentile survival time.
>> However I am struggling with comparing the difference of 25th
>> percentile survival time between African American and White across 10
>> regions in US. The command that I used:
>> sort UNOS
>> by UNOS: stci , by(race_all) p(25)
>> I was trying to use "ranksum" but it doesn't have an option for
>> testing differences of other percentiles besides median. Which command
>> should I use?
> That's a correct command. It tests for any difference where survival
> in one group is better, not just differences in medians.
>> 2. In comparing the median to the 25 percentile, there are some
>> regions where the length to 25% tile is significantly longer than
>> median. In this case, what would you guys do? If you would choose to
>> use median survival time, then what else could I do in order not only
>> to produce the median survival time and 95% CI, but to test the
>> difference of median survival time between groups instead stci (since
>> it didn't produce in previous analysis)?
> Your observation is impossible in theory; to persuade us, you'd need to
> follow the advice in Statalist FAQ (3.3) and show us the commands and
> results that lead you to this observation.
> If I had to guess, I'd say that you looked at the Kaplan-Meier survival
> curve, not at -stci- results. A survival curve shows the proportion of
> people who haven't had the event at a time, not the proportion who have
> failed. Thus the point corresponding to the 25% mark on the y axis is where
> 25% of people haven't failed and 75% have. In other words it is the 75th
> percentile for failure. To see the failure time percentiles, the command
> is:
> *******************
> sts graph, failure
> *******************
> You might benefit from reading a good text on survival data, e.g. Cleves
> et al. (2010).
> Reference: Cleves, Mario, William Gould, Roberto Gutierrez, and Yulia
> Marchenko. 2010. An Introduction to Survival Analysis Using Stata, Third
> Edition. College Station, Tex: Stata Press.
> Steve
>> Thank you so much!
> *
> * For searches and help try:
> * http://www.stata.com/help.cgi?search
> * http://www.stata.com/support/faqs/resources/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/faqs/resources/statalist-faq/
> * http://www.ats.ucla.edu/stat/stata/
Haena Lee
Ph.D Student
Sociology Department
The University of Chicago
312 - 405 - 3223
* For searches and help try:
* http://www.stata.com/help.cgi?search
* http://www.stata.com/support/faqs/resources/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/faqs/resources/statalist-faq/
* http://www.ats.ucla.edu/stat/stata/ | {"url":"http://www.stata.com/statalist/archive/2012-09/msg01188.html","timestamp":"2014-04-19T00:20:27Z","content_type":null,"content_length":"14663","record_id":"<urn:uuid:b4130d5f-7739-4d34-915e-bd871506b340>","cc-path":"CC-MAIN-2014-15/segments/1397609539776.45/warc/CC-MAIN-20140416005219-00554-ip-10-147-4-33.ec2.internal.warc.gz"} |
Robert Bridson
Adjunct professor in the Imager and SCL labs, UBC Computer Science
Ph.D. Stanford '03, MMATH Waterloo '99, BMATH Waterloo '98
email rbridson @ cs.ubc.ca
phone 604-822-1993
office ICICS/CS X663
mail Dept. Computer Science, UBC
201-2366 Main Mall
Vancouver, V6T 1Z4, Canada
Current Position
As of August 2013 I am no longer a full-time professor at UBC, but retain adjunct status. I am now a Senior Principal Research Scientist for Visual Effects at Autodesk.
Symposium on Computer Animation
I helped organize SCA 2011, August 5-7 in Vancouver, just before SIGGRAPH: www.siggraph.org/sca2011
Past Industry Work
I cofounded Exotic Matter, a graphics company which made physical simulation software for the film industry. Our Naiad software, acquired by Autodesk in 2012, has featured in some of the most
impressive liquid effects to date, including films such as Avatar, Narnia: Voyage of the Dawntreader, X-Men First Class, Harry Potter and The Deathly Hallows Part 2, Pirates of the Caribbean 4, Rise
of the Planet of the Apes, and many others. It is currently used at studios around the world.
My official screen credits are for The Hobbit: An Unexpected Journey, The Adventures of Tintin, The Rise of the Planet of the Apes, and Inkheart, but I have helped write in-house software at studios
used in many other films. Most recently I spent ten months as part of R&D at Weta Digital in New Zealand following earlier visits. Before that I co-wrote the Squirt fluid simulator for Double
Negative Visual Effects, seen in many films for smoke, water, fire, clouds, ink-in-water, etc. including Harry Potter and the Half-Blood Prince, 2012, The Boat that Rocked/Pirate Radio, Inkheart,
Quantum of Solace, The Dark Knight, and Hell Boy II: The Golden Army. Even further back I helped out with cloth simulation code used for Star Wars Episode II: Attack of the Clones at Industrial Light
and Magic, as one of the original contributors to the PhysBAM project, under my Ph.D. supervisor Ron Fedkiw.
Animations and Images
Fluid Simulation for Computer Graphics, R. Bridson, A K Peters, 2008.
My website for the book is at: www.cs.ubc.ca/~rbridson/fluidbook.
I've supplied preprints in most cases, which may be missing figures or include typos etc. Please see the publishers' websites for the official versions.
• Synthesizing waves from animated height fields, M. B. Nielsen, A. Soderstrom, and R. Bridson, ACM Transactions on Graphics, 2012.
• Linear-time smoke animation with vortex sheet meshes, T. Brochu, T. Keeler, and R. Bridson, ACM SIGGRAPH/Eurographics Symposium on Computer Animation 2012. (movie)
• Multiphase flow of immiscible fluids on unstructured moving meshes, M. K. Misztal, K. Erleben, A. Bargteil, J. Fursund, B. Bunch Christensen, J. A. Bærentzen, and R. Bridson,ACM SIGGRAPH/
Eurographics Symposium on Computer Animation 2012. (best paper award)
• Ghost SPH for animating water, H. Schechter and R. Bridson, Proc. SIGGRAPH 2012. (movie)
• Efficient Geometrically Exact Continuous Collision Detection, T. Brochu, E. Edwards, and R. Bridson, Proc. SIGGRAPH 2012. (open-source code, project)
• Steady state Stokes flow interpolation for fluid control, H. Bhatacharya, M. Nielsen, and R. Bridson, to appear in Eurographics 2012, short papers.
• MultiFLIP for Energetic Two-Phase Fluid Simulation, L. Boyd and R. Bridson, ACM Trans. Graph. (movie)
• A high-order accurate particle-in-cell method, E. Edwards and R. Bridson, to appear in Intl. J. Num. Meth. Engr.
• Guide shapes for high resolution naturalistic liquid simulation, M. Nielsen and R. Bridson, Proc. ACM SIGGRAPH 2011.
• Computational physics in film, R. Bridson and C. Batty, Science 24 December 2010: Vol. 330 no. 6012 pp. 1756-1757.
• A simple finite difference method for time-dependent, variable coefficient Stokes flow on irregular domains, C. Batty and R. Bridson, 2010, arXiv preprint.
• Optimization-based fluid simulation on unstructured meshes, M. K. Misztal, R. Bridson, K. Erleben, A. Baerentzen, and F. Anton, in Proceedings of Virtual Reality Interaction and Physical
Simulation (VRIPHYS), 2010.
• Matching fluid simulation elements to surface geometry and topology, T. Brochu, C. Batty, and R. Bridson, Proc. ACM SIGGRAPH 2010, 9 pages. (project)
• SpikeNav: using stylus tilt in three-dimensional navigation, R. Bridson, ACM UIST 2009, poster session.
• Animating smoke as a surface, T. Brochu and R. Bridson, Symposium on Computer Animation 2009, poster session. (movie)
• Robust topological operations for dynamic explicit surfaces, T. Brochu and R. Bridson, accepted in SIAM Journal on Scientific Computing 2009. (for related software, see the El Topo project page.)
• Evolving sub-grid turbulence for smoke animation, H. Schechter and R. Bridson, Symposium on Computer Animation 2008. (full movie)
• Accurate viscous free surfaces for buckling, coiling, and rotating liquids, C. Batty and R. Bridson, Symposium on Computer Animation 2008. (movie, project page)
• Animating developable surfaces using nonconforming elements, E. English and R. Bridson, Proc. ACM SIGGRAPH 2008. (movie)
• Fast Poisson disk sampling in arbitrary dimensions, R. Bridson, ACM SIGGRAPH 2007 Sketches Program. (download the curl-noise example code for a public-domain implementation).
• A fast variational framework for accurate solid-fluid coupling, C. Batty, F. Bertails, and R. Bridson, Proc. ACM SIGGRAPH 2007. (movie, project page)
• Curl noise for procedural fluid flow, R. Bridson, J. Hourihan, and M. Nordenstam, Proc. ACM SIGGRAPH 2007. (movie and public domain example code available.) See also Ivan DeWolf's technical
report Divergence-free noise for a slightly different approach.
• Fluid animation with explicit surface meshes, T. Brochu and R. Bridson, Symposium on Computer Animation 2006, poster session.
• Cloth animation through unbiased strain limiting and physics-aware subdivision, D. Tsiknis and R. Bridson, Symposium on Computer Animation 2006, poster session.
• A multi-preconditioned conjugate gradient algorithm, R. Bridson and Chen Greif, SIAM Journal on Matrix Analysis and Applications, 2005, vol. 27, no. 4, pp. 1056-1068.
• Animating sand as a fluid, Y. Zhu and R. Bridson, ACM SIGGRAPH 2005. (see below for code)
• Adaptive physics based tetrahedral mesh generation using level sets, R. Bridson, J. Teran, N. Molino and R. Fedkiw, Engineering with Computers 2005.
• Nonconvex rigid bodies with stacking, E. Guendelman, R. Bridson, and R. Fedkiw, ACM Transaction on Graphics, vol. 22, no. 3, Proc. ACM SIGGRAPH 2003, pp. 871-878.
• Simulation of clothing with folds and wrinkles, R. Bridson, S. Marino, and R. Fedkiw, Proc. ACM/Eurographics Symposium on Computer Animation 2003, pp. 28-36.
• A crystalline, red green strategy for meshing highly deformable objects with tetrahedra, N. Molino, R. Bridson, J. Teran, and R. Fedkiw, Proc. International Meshing Roundtable 2003.
• Robust treatment of collisions, contact and friction for cloth animation, R. Bridson, R. Fedkiw and J. Anderson, ACM Transactions on Graphics, vol. 21, no. 3, Proc. ACM SIGGRAPH 2002, pp.
• Multiresolution approximate inverse preconditioners, R. Bridson and W.-P. Tang, SIAM Journal on Scientific Computing, vol. 23, no. 2, pp. 463-479.
• A structural diagnosis of some IC orderings, R. Bridson and W.-P. Tang, SIAM Journal on Scientific Computing, vol. 22, no. 5, pp. 1527-1532
• Refining an approximate inverse, R. Bridson and W.-P. Tang, Journal on Computational and Applied Math, 123 (2000), Numerical Analysis 2000 vol. III: Linear Algebra, pp. 293-306.
• Ordering, anisotropy and factored approximate inverses, R. Bridson and W.-P. Tang, SIAM Journal on Scientific Computing, vol. 21, no. 3, pp. 867-882.
• The asymptotic regimes of tilted Bianchi II cosmologies, C. G. Hewitt, R. Bridson, and J. Wainwright, General Relativity and Gravitation, vol. 33, no. 1, pp. 65-94.
Other Publications
My Group
• Athena Chang (BSc)
• Crawford Doran (MSc - NSERC)
• Essex Edwards (PhD - dept. scholarship)
• Todd Keeler (PhD)
• Xinxin Zhang (PhD)
• Yufeng Zhu (PhD)
All files in this section are in the public domain unless otherwise indicated.
Many of my or my group's projects use a variety of files from a "common" directory: not exactly a proper library, but a collection of occasionally very useful files for C++ hacking. Some of the more
stable members are available in the public domain here:
Here are some more specific projects, organized by topic:
• Linear Algebra and Optimization
• Geometry
• Graphics
• Miscellaneous
Current/Upcoming courses:
• CPSC 426, Computer Animation, September-December 2011
• CPSC 314, Computer Graphics, January-May 2012
I have taught in the past:
• CPSC 314, Computer Graphics, September-December 2009
• CPSC 542G, Scientific Computing (Graduate Breadth), September-December 2009
• CPSC 548, Directed Studies (physics-based animation)
• CPSC 314, Computer Graphics, September-December 2008
• CPSC 542G, Scientific Computing (Graduate Breadth), September-December 2008
• CPSC 426, Computer Animation, January-May 2008
• CPSC 542G, Scientific Computing (Graduate Breadth), September-December 2007
• CPSC 542G Scientific Computing (Graduate Breadth), September-December 2006
• CPSC 533D Animation Physics, September-Decemeber 2005
• CPSC 426 Computer Animation, September-December 2005
• CPSC 533D Animation Physics, January-April 2005
• CPSC 426 Computer Animation, September-December 2004
• CPSC 533B Animation Physics, January-April 2004
Here's some music I've written (more to come as I typeset it):
I also wrote a little duet part for the Bach "Minuet No. 3" in Suzuki Viola Book One: minuet3.pdf (and MIDI output from LilyPond too).
A little poem: Comfort.
A few tracks I recorded: Recovery, Marked.
My son's first movie is Atlantic Ocean (I got to have a supporting role, and served as technical consultant :-)).
Some other short animations, from a traditional animation class with Lorie Loeb:
• Surfacing (listen to this with Otis Redding's "Sittin' on the Dock of the Bay" playing)
Here's source code for a simple 2D fluid simulator I wrote for a course project once.
I did a hectic project involving digital sculpting, environment map acquisition from uncalibrated photos, fast ray-tracing of micro-facet textured level sets, and compositing (producing images like
this) for the Stanford cs348b rendering competition.
Here's source code for a 3D first-person video-game, Spaceman Spiff: Escape from Zorg, which I wrote for the Stanford cs248 video game competition.
A word-for-word translation I made of the sung parts of the Latin mass: mass.pdf.
Aliens apparently invaded my left ear.
And like the book says, we might be through with the past, but the past, it ain't through with us. | {"url":"http://www.cs.ubc.ca/~rbridson/index.html","timestamp":"2014-04-18T00:38:52Z","content_type":null,"content_length":"30963","record_id":"<urn:uuid:e0edba5e-1ebd-4992-9499-61d9a010c07f>","cc-path":"CC-MAIN-2014-15/segments/1397609532374.24/warc/CC-MAIN-20140416005212-00534-ip-10-147-4-33.ec2.internal.warc.gz"} |
by Robert fitz John, Master Gamesplayer
The Guildmaster and I were observed playing a rather obscure dice game at St Briavels Castle. The game was Hazard, which is mentioned in Chaucer's 'The Canterbury Tales': In Flanders whilom was a
company / Of younge folkes, that haunted folly, / As riot, hazard, stewes, and taverns. Here are the rules, which are not (as some suggested) "whoever has the rule book wins"!
You need two or more players, two ordinary dice, and gaming tokens of some sort. If you're the gambling type, the tokens could be sweetmeats, bangles or reproduction coins. Otherwise, anything will
do - pebbles, for example.
Start off by choosing the caster. The other players are then faders. One or more faders place a token on the table, and the caster matches each token with one of his or her own. These are the stakes.
(Non-participants are welcome to place side-bets if they wish.)
The caster begins by throwing the dice to determine the Main Point. This must be a score between 5 and 9. If anything else is rolled, the dice are thrown again until a Main Point is obtained.
Now the caster throws the dice again. If the score is the same as the Main Point, this is known as a nick and the caster wins. If a 2 or 3 was rolled, that's an out and the caster loses. 11 and 12
are also outs, except in certain cases: a roll of 11 after a Main Point of 7 is a nick, and so is a roll of 12 after a Main Point of 6 or 8. The table below should make this clearer.
│ Main Point: │ 5 │ 6 │ 7 │ 8 │ 9 │
│ Nicks (win): │ 5 │ 6, 12 │ 7, 11 │ 8, 12 │ 9 │
│ Outs (lose): │ 2, 3, 11, 12 │ 2, 3, 11 │ 2, 3, 12 │ 2, 3, 11 │ 2, 3, 11, 12 │
Assuming a nick or out hasn't been rolled, the result will have been a score between 4 and 10 that isn't the same as the Main Point. This is called the Chance Point. The caster now throws the dice
until either the Main Point or the Chance Point is rolled. (There are no nicks or outs at this stage.) The caster wins if the Chance Point comes up, but loses if the Main Point is thrown.
If the caster wins, he or she collects all the stakes and the process starts all over again. Otherwise, the stakes go to the faders, and the nearest fader on the caster's left becomes the new caster.
Play continues until somebody wins all the tokens, everyone's too drunk to carry on, or a fight breaks out.
The rules for nicks and outs sound rather complicated, but they ensure the odds are pretty much even, regardless of what was rolled for the Main Point. In each case, the caster has about a 49% chance
of winning. The precise odds aren't too important anyway, as the role of caster will tend to move round the table fairly rapidly.
Petty Hazard
In case you find these rules too complicated, I've invented a variant which I'll call Petty Hazard. In this version, you only get a nick if you roll the Main Point, and you only get an out if you
roll 2 or 3. If you roll 11 or 12, you throw the dice again. Once the Chance Point has been decided, the rules are the same as for Hazard. In this variant the odds are slightly in favour of the
caster, who has nearly a 51% probability of winning.
© 2003, 2005, Trevor Barker. Permission is given to reproduce and distribute this work, on the following conditions: this must not be done for profit, and this copyright notice must remain attached
and unaltered.
About the Author
Dr Trevor Barker studied Chemistry at Oxford University. He works for a leading software and systems integration company and is married with two sons.
Robert fitz John was born in Flanders, circa 1047. He joined the army of William, Duke of Normandy, as a mercenary archer, and participated in the invasion of England in October 1066. He was formerly
Sheriff of Blackwater and then Master Secretary to the High Council of the Kingdom of the Far Isles. He had to dictate this article to his scribe, as he is illiterate. | {"url":"http://homepage.ntlworld.com/trevor.barker/farisles/guilds/games/hazard.htm","timestamp":"2014-04-18T18:28:24Z","content_type":null,"content_length":"5883","record_id":"<urn:uuid:be6de7b5-26d7-4f10-919e-c7408abfad42>","cc-path":"CC-MAIN-2014-15/segments/1397609535095.7/warc/CC-MAIN-20140416005215-00603-ip-10-147-4-33.ec2.internal.warc.gz"} |
Convert kilopond/square centimetre to centibar - Conversion of Measurement Units
›› Convert kilopond/square centimetre to centibar
›› More information from the unit converter
How many kilopond/square centimetre in 1 centibar? The answer is 0.0101971621298.
We assume you are converting between kilopond/square centimetre and centibar.
You can view more details on each measurement unit:
kilopond/square centimetre or centibar
The SI derived unit for pressure is the pascal.
1 pascal is equal to 1.01971621298E-5 kilopond/square centimetre, or 0.001 centibar.
Note that rounding errors may occur, so always check the results.
Use this page to learn how to convert between kiloponds/square centimeter and centibars.
Type in your own numbers in the form to convert the units!
›› Definition: Centibar
The SI prefix "centi" represents a factor of 10^-2, or in exponential notation, 1E-2.
So 1 centibar = 10^-2 bars.
The definition of a bar is as follows:
The bar is a measurement unit of pressure, equal to 1,000,000 dynes per square centimetre (baryes), or 100,000 newtons per square metre (pascals). The word bar is of Greek origin, báros meaning
weight. Its official symbol is "bar"; the earlier "b" is now deprecated, but still often seen especially as "mb" rather than the proper "mbar" for millibars.
›› Metric conversions and more
ConvertUnits.com provides an online conversion calculator for all types of measurement units. You can find metric conversion tables for SI units, as well as English units, currency, and other data.
Type in unit symbols, abbreviations, or full names for units of length, area, mass, pressure, and other types. Examples include mm, inch, 100 kg, US fluid ounce, 6'3", 10 stone 4, cubic cm, metres
squared, grams, moles, feet per second, and many more!
This page was loaded in 0.0029 seconds. | {"url":"http://www.convertunits.com/from/kilopond/square+centimetre/to/centibar","timestamp":"2014-04-17T07:04:51Z","content_type":null,"content_length":"20717","record_id":"<urn:uuid:ec4b6717-605c-4173-b4f9-e9a77651dd68>","cc-path":"CC-MAIN-2014-15/segments/1398223203841.5/warc/CC-MAIN-20140423032003-00519-ip-10-147-4-33.ec2.internal.warc.gz"} |
solve for x - WyzAnt Answers
solve for x
Tutors, please
to answer this question.
9/2 log[3](x) = 1
log[3](x) = 2/9
now, to "undo" the log[3] on the left side of the equation, we raise 3 to the power of each side of the equation.
x = 3^(2/9)
x = 1.28, rounded to three digits. | {"url":"http://www.wyzant.com/resources/answers/16087/solve_for_x","timestamp":"2014-04-16T15:21:17Z","content_type":null,"content_length":"40546","record_id":"<urn:uuid:0eedaffd-d85e-4330-88b2-169b8a687fd8>","cc-path":"CC-MAIN-2014-15/segments/1397609523429.20/warc/CC-MAIN-20140416005203-00220-ip-10-147-4-33.ec2.internal.warc.gz"} |