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Primary Math Teacher Guide
Pri Math TEACHER'S Guide U.S. Ed 1A
The Rosenbaum Foundation, USA
Usually Ships Within One Business Day
Product Code: PMTG1A
Description Contents_Samples Extended Information
Primary Mathematics U.S. & 3rd Ed Teacher's Guides provide, in both flexibility and detail, a clear framework for the Primary Mathematics textbooks. Each lesson is accompanied by
numerous activities which expand and reinforce the concepts for that lesson and which are designed to fit both teachers who wish to adapt lessons to their own classroom situation, and Products in
teachers who desire easy-to-follow, effective teaching strategies. Through the notes to the teacher and the detailed objectives for each learning task in the text and activity in the same
guide, these Teacher Guides help teachers to fully understand the purpose and concept behind each set of problems, both within the context of the unit and the context of the overall category...
curriculum. The teacher's guide contains answers for the textbooks and workbooks.
This version of Teacher's Guide can be used with Primary Mathematics U.S. Edition and 3rd Edition textbooks and workbooks. It cannot be used with Primary Mathematics Standards Edition
Our recommendation:
Teacher's Guides are best suited for classroom use.
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two different values for angle A ?
May 20th 2010, 05:10 PM #1
May 2010
two different values for angle A ?
Okay so here is the question
Each angle of A is between 0 degrees and 180 degrees. Which equations result in two different values for angle A ?
-sin A = .7071
-cos A = -.5
-sin A = .9269
-cos A = -.7071
-sin A = .8660
-cos A = -1
-sin A = 3/4
-cos A = 3/4
-cos A = -3/4
I would like how you got there as well ! thanks !
Hello misspalmer
Welcome to Math Help Forum!
Okay so here is the question
Each angle of A is between 0 degrees and 180 degrees. Which equations result in two different values for angle A ?
-sin A = .7071
-cos A = -.5
-sin A = .9269
-cos A = -.7071
-sin A = .8660
-cos A = -1
-sin A = 3/4
-cos A = 3/4
-cos A = -3/4
I would like how you got there as well ! thanks !
First, I assume that the hyphens at the left-hand end of each equation are not minus signs. You do need to be careful in writing down the questions!
Then you need to know how sine and cosine behave between $0^o$ and $180^o$. So take a look at the diagram I've attached.
You'll see that $\sin x$ goes from $0$ to $1$ as $x$ goes from $0^o$ to $90^o$. Then, in a symmetrical way, it goes back down to $0$ as $x$ goes from $90^o$ to $180^o$.
So, for any number between $0$ and $1$, there will be two angles whose sine is this number: one between $0^o$ and $90^o$ and one between $90^o$ and $180^o$. But if the number lies outside the
range $0$ to $1$, there won't be any angles between $0^o$ and $180^o$ having this number as their sine.
On the other hand, $\cos x$ starts at $1$ and goes down to $0$ as $x$ goes from $0^o$ to $90^o$. Then it continues down to ${-1}$ as $x$ goes from $90^o$ to $180^o$.
So no value of $\cos x$ is repeated as $x$ takes values between $0^o$ and $180^o$. So none of the equations involving cosine will have two answers.
If you apply the rules I've just given you, it should be pretty clear what the answers are. For instance, the first equation, $\sin A=0.7071$, will have two solutions, but the second, $\cos A =
-0.5$, won't.
Can you do the rest of them now?
May 20th 2010, 10:55 PM #2
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Formerly Unknown Roots of Statway
The Statway began as an idea and a possibility: many community college students are interested in fields that require statistics. And math faculty to a surprising degree concur that elementary and
intermediate algebra are not really required for and do not actually prepare students for statistics. Would it be possible to create a one-year course that takes students who place into developmental
mathematics to and through a college level statistics course in one academic year? Could the course be designed so that the relevant mathematics is taught in the context of statistics, using the
power of contextual learning and the motivation of real world problems?
While we are building the first iteration of the Statway, it is delightful to learn of a one-year statistics course that had been organized in this spirit. With thanks Joe Malkevitch, a retired
mathematics professor from York College, for this description.
York College in Jamaica, N.Y., is one of nine CUNY four-year colleges. It opened in 1967 and served the same range of “at risk” students that community colleges regularly serve: single mothers,
military veterans (predominantly African-American and Hispanic), immigrants, first-generation college goers, and low-income students.
The goal—shared college wide—was to allow students to fulfill their dreams and not let deficiencies in their backgrounds get in the way. The college offered a range of mathematics options: Calculus,
Statistics, Finite Mathematics for Business, Computers in Modern Society, and a Mathematical Excursions course. Students could choose their mathematics based on their career plans. Students
interested in psychology could take statistics for a general education requirement, while students interested in nursing might also choose statistics.
The math department recognized that students could be highly motivated to take statistics, but didn’t necessarily have the background. Instead of requiring a prerequisite remedial class or an algebra
review, they designed a year-long four-credit statistics course. Each semester students signed up for a course that was four credit hours. If they succeeded in that course, they earned two hours of
college credit and two credits of developmental mathematics. However, for the purpose of being considered a full-time student, the course counted as four hours. Students had to complete the whole
year to get credit for the statistics course. Each semester’s content covered some college-level statistics, but also provided time to review background mathematics as needed.
This one-year statistics course wasn’t an experimental course, or the domain of one particular faculty member. It was a standard departmental offering from early in the 1970s to the late 1980s when
as a CUNY senior college they could no longer offer developmental courses. In recalling this class, Malkevitch said, “It seemed successful; students seemed motivated. Everyone got to start
statistics, and more than half went on to the second semester. Before, when algebra was required, fewer than half of the students would even get to start statistics. It certainly seemed possible to
teach the conceptual basis of statistics without a great deal of symbol manipulation.” This is an observation about learning statistics that Paul Nolting has also found in his work.
In education, good ideas have a way of appearing in different times and places. As we build the Statway, we are glad to know of a solid
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equation of the line AB
September 16th 2011, 06:05 AM #1
Super Member
Oct 2008
Bristol, England
equation of the line AB
i need to find the equation of the line AB from A(2,0) B(3,1)
for this would i do:
AB=(square root) 1^2 + 1^2= (square root)2?
is this correct? if not where am i going wrong?
Re: equation of the line AB
The equation of a line refers to an equation in x and y which all points on that line satisfy. You have tried to find out the length of the line-segment AB which is not required. The following
link should help.
Linear equation - Wikipedia, the free encyclopedia
Re: equation of the line AB
so would i do:
1=3+c 1/3=0.3
Re: equation of the line AB
would that be correct?
Re: equation of the line AB
Re: equation of the line AB
i did, 1/3=0.3
Re: equation of the line AB
That is not how it is done. In an equation you perform the same operation on both sides. For example you can subtract some quantity, or divide both sides by a non-zero number. Which of the above
would you do to get only c in a single side of 1 = 3 + c ?
Re: equation of the line AB
could i -1 from both sides to make c=2?
Re: equation of the line AB
That way you will get 0 = 2 + c. What is the next step?
Re: equation of the line AB
-2 from each side so c=-2 and the final answer would be y=1x-2???
Re: equation of the line AB
Re: equation of the line AB
thanks alot mate!!
September 16th 2011, 06:10 AM #2
September 16th 2011, 06:19 AM #3
Super Member
Oct 2008
Bristol, England
September 16th 2011, 06:27 AM #4
Super Member
Oct 2008
Bristol, England
September 16th 2011, 06:34 AM #5
September 16th 2011, 06:35 AM #6
Super Member
Oct 2008
Bristol, England
September 16th 2011, 06:38 AM #7
September 16th 2011, 06:39 AM #8
Super Member
Oct 2008
Bristol, England
September 16th 2011, 06:44 AM #9
September 16th 2011, 06:45 AM #10
Super Member
Oct 2008
Bristol, England
September 16th 2011, 06:56 AM #11
September 16th 2011, 06:57 AM #12
Super Member
Oct 2008
Bristol, England
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Math Forum - Problems Library - Primary, Operations with Numbers - Division
This page:
About Levels
of Difficulty
operations with numbers
number theory
algebraic thinking
data analysis
logical reasoning
Browse all
About the
PoW Library
The concept of division is central to solving these problems, though it may not be the only required operation.
Related Resources
Interactive resources from our Math Tools project:
Math 1: Operations with Numbers
NCTM Standards:
Number and Operations Standard for Grades Pre-K-3
Access to these problems requires a Membership.
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3D Math Basics Part 2 - 3D Math - Megabyte Softworks
Current series: 3D Math
(Return to list of 3D Math articles)
Welcome to the second part of 3D Math Basics articles. Here I will try to explain two very important vector operations - the cross and dot product.
Cross product
Cross product is very important in 3D graphics. With it, we can calculate the normal of polygon. For those, who doesn't know what is the normal, it is a directional vector, which is perpendicular to
the plane (there is a right angle between normal and plane), in which polygon lies, so it is also perpendicular to this polygon. It is used for thousands of things, for example lighting calculations
or back-face culling. And for those, who doesn't understand what is the plane, it's just a flat area in some 3D space. It can be defined by 3 points, which are not collinear - they don't lie on the
same line. Look at the picture:
So this is the plane (with yellow color):
It goes on forever, so actually this is just part of a plane. I hope you got the idea of what is the plane. When we have 3 (or more) points there, they form a triangle (or polygon). But in order to
set back-face culling and achieve nice lighting effects, we need to define a normal vector. How to do it? We need to take cross product. It's always taken from two directional vectors and resulting
vector will be normal. This normal is perpendicular both to first and second directional vector. Look at this:
We've got two directions - green is P2 - P1 and red is P3 - P2. Now, here is how is the cross product calculated:
CVector3 vecCross(CVector3 vVector1, CVector3 vVector2)
CVector3 vCross; // Here we will store it // Get the X value
vCross.x = ((vVector1.y * vVector2.z) - (vVector1.z * vVector2.y)); // X value
vCross.y = ((vVector1.z * vVector2.x) - (vVector1.x * vVector2.z)); // Y value
vCross.z = ((vVector1.x * vVector2.y) - (vVector1.y * vVector2.x)); // Z value
return vCross;
Green vector is (0, 0, 1) and red is (1, 0, 0). If we put these vectors as parameters, the resulting vector is (0, 1, 0) - vector pointing straight up! And it is perpendicular to both vectors: Look:
[S: I hope you now got it. But there is one problem. What if the resulting vector was (0, -1, 0)? It would be perpendicular too. How to know which direction will be vector pointing? Well, I heard
something about right-hand rule, but I didn't get the idea of it. But I found out that it depends on whether vectors are in clockwise or anticlockwise order. The best way to explain it is an example:
Edit on 21.01.2012: I was a n00b back then, and I'm not much of a pro right now, but I should be less n00b vVector1, your index finger as vVector2, and middle finger shows direction of resulting
vector from cross product. Take a look at wikipedia for a picture http://en.wikipedia.org/wiki/Right-hand_rule.
It's a cube. Now, we want to calculate its normals. For lighting effects and back-face culling. We will take as example the red side (front):
We want its normal to be pointing to the front (so the resulting vector should have direction (0, 0, 1) ). Look at this code:
CVector3 fourNormals[4];
fourNormals[0] = vecCross(P2 - P1, P3 - P2);
fourNormals[1] = vecCross(P3 - P2, P4 - P3);
fourNormals[2] = vecCross(P4 - P3, P1 - P4);
fourNormals[3] = vecCross(P1 - P4, P2 - P1);
for(int i = 0; i < 4; i++)vecNormalize(fourNormals[i]);
Now all of the four normals are the same and are pointing to the front (to us). Of course, you don't need to calculate all 4 (or more, if polygon has more vertices), you just need to calculate one
normal. You cannot forget to normalize it, because its length probably won't be 1. Normal's length must always be 1. As you can see, it's pointing where we wanted it to point. I used
counter-clockwise order. Look (green vector is normal):
Edit on 21.01.2012: Directions put here correspond with right hand rule, ignore clockwise and counter-clockwise stuff
If we would have put them in clockwise direction, normal would be (0, 0, -1). The clockwise direction would be such:
Dot Product
Another important operation is dot product. It can be used to find angle between two directional vectors. It can be used when you want to find angle to rotate some object to face another. The dot
product doesn't return a vector, it returns only a number (scalar). The dot product is calculated this way:
float vecDot(CVector3 vVector1, CVector3 vVector2)
return (vVector1.x * vVector2.x) +
(vVector1.y * vVector2.y) +
(vVector1.z * vVector2.z);
If we have only 2 dimensions, we would remove z from this formula (the dot product calculation is same for any number of dimensions, but this isn't much important here). So this is how we calculate
it. In any book that deals with math, you will find something like this about dot product:
A.B = |A|*|B|*cos theta
It means that dot product of vector A and B equals to length of A * length of B * cosine of angle between them. We want to find the angle. From that:
cos theta = (A.B) / (|A|*|B|)
So now we have the cosine of angle. To get actual angle, we need to use arc cosine. Following code does it all:
double vecAngle(CVector3 vVector1, CVector3 vVector2)
float fDotProduct = vecDot(vVector1, vVector2);
float fVectorsMagnitude = vecMagnitude(vVector1) * vecMagnitude(vVector2) ;
if(fVectorsMagnitude == 0.0)return 0.0; // Avoid division by zero
double angle = acos(fDotProduct / fVectorsMagnitude);
// The angle is in radians, so convert it to degrees
return angle * 180 / PI;
It returns angle from 0 to PI (0 to 180).
If you want to see usage of dot and cross products, check the codes of Ruined city of Verion (OpenGL Misc).
Now that's all I know about this. And I think it's enough. Yet. Maybe later it won't be enough and I will have to dive deeper into cross and dot products. I hope you learned something from this
article. If you've got any questions about this article, put them either into Message Board or e-mail them to michalbb1@gmail.com. And try to play around with these cross and dot products in order to
fully understand them.
No comments. Be the first to comment!.
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Virginia Tech killing: more than 30 dead
post #441 of 524
4/20/07 at 1:49am
Originally Posted by
A 15-minute lecture?
My JHS spent two hours lecturing us on the dangers of smoking and half my classmates now smoke. Go figure.
Drivers' Ed classes give lots of air time to driving the speed-limit. How many drivers really seem to have listened?
Hot air ain't gonna do nuthin.
OK, then. Nevermind. Fine by me. I know what the hell I am doing.
Hot air ain't gonna do nuthin when negotiating with a hell-bent criminal, either.
Everybody wear orange or maroon today.
Originally Posted by Jubelum
Balderdash. Poppycock.
Again, you have misrepresented my argument. The person on the bus carrying in Texas has been through 8 hours of classroom instruction, at least 2 hours supervised and tested at the range, and is
personally registered, fingerprinted, and all that with the State of Texas. The State sets the mandatory standards. I have confidence because I have met hundreds of instructors, and we know that one
clown can cause an accident and thus a PR nightmare that could cost us our rights. I've even seen CHL instructors throw people out of their classes because it was not being taken seriously. Not one
person in Texas, in decade plus since we got CHL, has been even indicted for unlawful use of their firearm while engaging in concealed carry. Not one. That in itself is a thumb in the eye of all the
liberals who said we'd have "the wild west" and "shootouts over parking spaces."
If they are carrying WITHOUT a license, then they should be put away because it is a state jail felony. At that point, they are acting criminally.
You seem to be putting forth that I want everyone to carry without a license and with no restrictions whatsoever. That's not it at all. Licensing someone to carry a concealed weapon is one thing.
Registering what is in my gun safe is quite another. One is insuring standards and public safety. The other has no purpose other than to eventually confiscate my firearms.
You keep saying this, but it appears to me IMHO to be a classic slippery slope argument;
Arguers also often link the slippery slope fallacy to the straw man fallacy in order to attack the initial position:
A has occurred (or will or might occur); therefore
B will inevitably happen. (slippery slope)
B is wrong; therefore
A is wrong. (straw man)
This form of argument often provides evaluative judgments on social change: once an exception is made to some rule, nothing will hold back further, more egregious exceptions to that rule.
Note that these arguments may indeed have validity, but they require some independent justification of the connection between their terms: otherwise the argument (as a logical tool) remains
The "slippery slope" approach may also relate to the conjunction fallacy: with a long string of steps leading to an undesirable conclusion, the chance of all the steps actually occurring is actually
less than the chance of any one of the individual steps occurring alone.
The last statement, suggests p (outcome) = p (1) * p(2) * p (3) * ... * p(n), where n in an arbitrary large number, and at least several p(i) are vanishingly small, if not in fact that (at least) one
p (i) = 0.
IMHO, in your oft repeated scenario, p (outcome) = 0, because we have the 2nd amendment (p (1) = 0 (thus only one term needs to be stated)).
And as you have already suggested, repeal the 2nd, watch out, because then we truly would be heading towards a police state. Not that IS a SCARY thought!
Every eye fixed itself upon him; with parted lips and bated breath the audience hung upon his words, taking no note of time, rapt in the ghastly fascinations of the tale. NOT!
Every eye fixed itself upon him; with parted lips and bated breath the audience hung upon his words, taking no note of time, rapt in the ghastly fascinations of the tale. NOT!
Originally Posted by franksargent
You keep saying this, but it appears to me IMHO to be a classic slippery slope argument;
Since we are speaking in Socrates... :
"H*P=C" where
H= The number of times registration has led to confiscation before
P= The chance it could happen in the US and
C= The (currently better than average) chance that it will happen again in the future.
A silly equation, but just trying to speak your language.
In short... look at history, frank.
Read Here about nearly 75 million dead in the 20th century because of registration, bans, and confiscation. And the Jews ought to know more than anyone...
Let's just look in the last 20 years or so...
Cambodia, 1975-1979 \t
2 million dead \t
•Licenses for guns, owners, ammunition & transactions •Photo ID with fingerprints •License inspected quarterly
Rwanda, 1994 \t
800,000 dead and counting
•Register guns, owners, ammunition •Owners must justify need •Concealable guns illegal •Confiscating powers
Originally Posted by Jubelum
Since we are speaking in Socrates... :
"H*P=C" where
H= The number of times registration has led to confiscation before
P= The chance it could happen in the US and
C= The (currently better than average) chance that it will happen again in the future.
A silly equation, but just trying to speak your language.
In short... look at history, frank.
p (outcome that ALL weapons are confiscated) = 0
p (outcome of total weapons confiscated > 0) = 1
Null set + infinite set + all finite sets = all possible sets, or p =1
But in the real world, the null and infinite sets are, for all intents and purposes, physical impossibilities.
Weapons are already confiscated all the time, by those who chose to break the law, duh.
Weapons will not be confiscated, by those who follow the laws.
And as I see it, given our 2nd amendment rights, that law itself guarantees access of weaponry (e. g. most certainly including a large array of firearms) to the citizenry.
Sorry, but I don't see a loophole in the 2nd amendment. To do that would mean an end to our form of government, pretty much the end of the USA as we currently know it.
So, like I originally stated, your confiscation argument (slippery slope + straw man), does not make logical sense!
Every eye fixed itself upon him; with parted lips and bated breath the audience hung upon his words, taking no note of time, rapt in the ghastly fascinations of the tale. NOT!
Every eye fixed itself upon him; with parted lips and bated breath the audience hung upon his words, taking no note of time, rapt in the ghastly fascinations of the tale. NOT!
Originally Posted by franksargent
p (outcome that ALL weapons are confiscated) = 0
p (outcome of total weapons confiscated > 0) = 1
Null set + infinite set + all finite sets = all possible sets, or p =1
But in the real world, the null and infinite sets are, for all intents and purposes, physical impossibilities.
Weapons are already confiscated all the time, by those who chose to break the law, duh.
Weapons will not be confiscated, by those who follow the laws.
And as I see it, given our 2nd amendment rights, that law itself guarantees access of weaponry (e. g. most certainly including a large array of firearms) to the citizenry.
Sorry, but I don't see a loophole in the 2nd amendment. To do that would mean an end to our form of government, pretty much the end of the USA as we currently know it.
So, like I originally stated, your confiscation argument (slippery slope + straw man), does not make logical sense!
And since when does civilian disarmament make sense? You can go through all the mental masturbation you want, just like I am sure it was done in Germany in the 30s. It's been done. Absent vigilant
people, it will happen again. Just look at the zeitgeist on this board that wants to take guns away from the populace. Those that keep telling me why guns should be illegal are making my point for
And you see, that is the rub. Gun grabbers want to make the law such that no one will have access to guns. See how neat and tidy that is?
Originally Posted by Jubelum
And since when does civilian disarmament make sense? You can go through all the mental masturbation you want, just like I am sure it was done in Germany in the 30s. It's been done. Absent vigilant
people, it will happen again. Just look at the zeitgeist on this board that wants to take guns away from the populace. Those that keep telling me why guns should be illegal are making my point for
Jubelum (very interesting screen name BTW),
You exist (at least I think you do
I exist (at least I think I do
That is all.
Every eye fixed itself upon him; with parted lips and bated breath the audience hung upon his words, taking no note of time, rapt in the ghastly fascinations of the tale. NOT!
Every eye fixed itself upon him; with parted lips and bated breath the audience hung upon his words, taking no note of time, rapt in the ghastly fascinations of the tale. NOT!
We definitely are going around in circles on this one!
The 2nd Amendment exists, we simply can't hit the delete button, on that one (or the 9 others for that matter).
But we also can't have carte blanche either WRT the USBOR, now can we?
Look most of what you have suggested makes perfect sense to me (enforce the existing laws, perhaps toughen them up a bit, more police wouldn't hurt IMHO, more enforcement (and surveillance/
Every eye fixed itself upon him; with parted lips and bated breath the audience hung upon his words, taking no note of time, rapt in the ghastly fascinations of the tale. NOT!
Every eye fixed itself upon him; with parted lips and bated breath the audience hung upon his words, taking no note of time, rapt in the ghastly fascinations of the tale. NOT!
Originally Posted by Jubelum
Read Here about nearly 75 million dead in the 20th century because of registration, bans, and confiscation. And the Jews ought to know more than anyone...
Let's just look in the last 20 years or so...
Cambodia, 1975-1979 \t
2 million dead \t
•Licenses for guns, owners, ammunition & transactions •Photo ID with fingerprints •License inspected quarterly
Rwanda, 1994 \t
800,000 dead and counting
•Register guns, owners, ammunition •Owners must justify need •Concealable guns illegal •Confiscating powers
Cambodia and Rwanda could have been stopped by arming the populace...? You're taking the piss.
Rwanda was the populace against the populace.
Perhaps the 4 million Vietnamese would have survived US bombing. OR those at Hiroshima would have been OK, and the 6m Jews would have been fine in Nazi Germany all for the love of hand guns - Wackos
Originally Posted by Jubelum
And since when does civilian disarmament make sense? You can go through all the mental masturbation you want, just like I am sure it was done in Germany in the 30s. It's been done. Absent vigilant
people, it will happen again. Just look at the zeitgeist on this board that wants to take guns away from the populace. Those that keep telling me why guns should be illegal are making my point for
1st sentence: I never said that, I never suggested that.
2nd sentence: Please no name calling OK, besides your falling into the exact same trap (i. e. mm) you accuse others of doing. Your comparisons to Nazi Germany, etceteras, does not make sense! I don't
see whatever myopic parallels that you appear to see in governments that are drastically different (structurally, culturally, etceteras) from ours. Again, that initiates your logical fallacy.
3rd sentence: OK, no problem there.
4th sentence: They are the few, we are the many.
5th sentence: No kidding!
Every eye fixed itself upon him; with parted lips and bated breath the audience hung upon his words, taking no note of time, rapt in the ghastly fascinations of the tale. NOT!
Every eye fixed itself upon him; with parted lips and bated breath the audience hung upon his words, taking no note of time, rapt in the ghastly fascinations of the tale. NOT!
Originally Posted by vinea
It is this kind of paranoia that makes gun proponents sound like nutjobs and why the rest of the gun owning populations stay the heck away.
Registration is part of responsible gun ownership. Training is part of responsible gun ownership. Gun locks and gun safes are part of responsible gun ownership. Being liable when your gun is used in
a crime is part of responsible gun ownership at both the individual and corporate level because really...the only way a criminal should get ahold of any of your guns is to pry it from your cold dead
Gun proponents fight ALL of these things and their only justification is paranoia. Well, screw that...you're behaving like children who won't agree to boundries on their toys. Which means you're more
likely to get all the toys completely taken away and ruin it for everyone.
So Jub...just STFU. I'm for repealing of all Bill of Rights protections from Texans and shipping the whole lot to Iraq. And I'm only half kidding.
Or heck, you guys can just secede from the Union. You have oil, I'm sure you'll do fine. You can keep W. and Cheney along with Iraq. Your mess. You deal. Heck, the roots of Vietnam can be found in
the Eisnehower administration and certainly well fertilized by Johnson.
This makes sense.
MA700LL/A arrived.
Latitude D600, PowerEdge 1600SC, OptiPlex GX520
MA700LL/A arrived.
Latitude D600, PowerEdge 1600SC, OptiPlex GX520
I guess we're still in introspective wash mode!
Wait for the true spin cycle, when our fearless DC/VA leaders have all those feel good and head bashing hearings!
Every eye fixed itself upon him; with parted lips and bated breath the audience hung upon his words, taking no note of time, rapt in the ghastly fascinations of the tale. NOT!
Every eye fixed itself upon him; with parted lips and bated breath the audience hung upon his words, taking no note of time, rapt in the ghastly fascinations of the tale. NOT!
Originally Posted by OfficerDigby
Cambodia and Rwanda could have been stopped by arming the populace...? You're taking the piss.
Rwanda was the populace against the populace.
Perhaps the 4 million Vietnamese would have survived US bombing. OR those at Hiroshima would have been OK, and the 6m Jews would have been fine in Nazi Germany all for the love of hand guns - Wackos
One group in Rwanda took guns from the other group, then slaughtered them.
Do you think people with guns ended up on cattle cars?
Absolutely perfect. Condenses the entire discussion into nine panels. Well done.
Originally Posted by franksargent
2nd sentence: Please no name calling OK, besides your falling into the exact same trap (i. e. mm) you accuse others of doing. Your comparisons to Nazi Germany, etceteras, does not make sense! I don't
see whatever myopic parallels that you appear to see in governments that are drastically different (structurally, culturally, etceteras) from ours. Again, that initiates your logical fallacy.
Was Germany not a republic when the guns were outlawed through the legal process? Like one we have here that Sarah Brady would like to employ? Remember, Bush and Hitler were both elected by the
people. A nation, (Germany) in the last 60 years, voted to make laws that disarmed their people, and then proceeded to butcher 6-8 million of their people, despite what Prez Tom says. Until the
moment it really began, the road to extermination took the LEGAL route to power. It can happen anywhere, under almost any type of government.
This tinfoil hat is getting hot in the Texas heat. BRB...
Originally Posted by Jubelum
Was Germany not a republic when the guns were outlawed through the legal process? Like one we have here that Sarah Brady would like to employ? Remember, Bush and Hitler were both elected by the
people. A nation, in the last 60 years, voted to make laws that disarmed their people, and then proceeded to butcher 6-8 million of their people, despite what Prez Tom says. Until the moment it
really began, the road to extermination took the LEGAL route to power. It can happen anywhere, under almost any type of government.
I think I've made my point, not to you obviously, but to others. BTW, last time I checked, Bush != Hitler, contrary to other people's opinions.
Committing a logical fallacy, does not absolve one from the inherent flawed logic therein.
Now, if you are attempting to make an emotional plea (as is apparent in your thought process), then by all means, do so (or should I say you will continue to do so).
PS - You might want to add a few years to your point above, or more explicitly what government are you referring to post 1947? And who/what is Prez Tom?
Every eye fixed itself upon him; with parted lips and bated breath the audience hung upon his words, taking no note of time, rapt in the ghastly fascinations of the tale. NOT!
Every eye fixed itself upon him; with parted lips and bated breath the audience hung upon his words, taking no note of time, rapt in the ghastly fascinations of the tale. NOT!
Originally Posted by franksargent
I think I've made my point, not to you obviously, but to others.
Committing a logical fallacy, does not absolve one from the inherent logic therein.
Now, if you are attempting to make an emotional plea (as is apparent in your thought process), then by all means, do so (or should I say you will continue to do so).
I'd like to hear more about why you do not think that gun confiscation can happen here as it has across the world, with dire consequences. Then go share that argument with IANSA, and have them tell
you why it CAN and SHOULD be done.
Originally Posted by franksargent
I think I've made my point, not to you obviously, but to others. BTW, last time I checked, Bush != Hitler, contrary to other people's opinions.
Committing a logical fallacy, does not absolve one from the inherent flawed logic therein.
Now, if you are attempting to make an emotional plea (as is apparent in your thought process), then by all means, do so (or should I say you will continue to do so).
PS - You might want to add a few years to your point above, or more explicitly what government are you referring to post 1947? And who/what is Prez Tom?
The Bush-Hitler reference is there to maybe make people think. What if Bush, in all his evil, got with Tom Tancredo, Pat Robertson, and Tom Delay and all of them decided, while in power
democratically, that political activism on the left, free speech, privacy rights, and the rest of the things that might "interfere with the state" are illegal and dissidents should be imprisoned or
killed. If the Killing Fields starting happening on AMERICAN university campuses, I think we'd see one hell of a change of heart among professors concerning gun rights.
Do free people deserve to have a recourse in the event of extremism from left or right?
Free thought for the day: things only happen three places- the ballot box, the jury box, and the cartridge box.
Originally Posted by franksargent
I think I've made my point, not to you obviously, but to others. BTW, last time I checked, Bush != Hitler, contrary to other people's opinions.
Committing a logical fallacy, does not absolve one from the inherent flawed logic therein.
Now, if you are attempting to make an emotional plea (as is apparent in your thought process), then by all means, do so (or should I say you will continue to do so).
PS - You might want to add a few years to your point above, or more explicitly what government are you referring to post 1947? And who/what is Prez Tom?
Prez Tom is the President of Iran. And I am speaking of Germany.
One for the ages... Ted gets it right again!....
Thirty-two people dead on a U.S. college campus pursuing their American Dream, mowed-down over an extended period of time by a lone, non-American gunman in illegal possession of a firearm on campus
in defiance of a zero-tolerance gun law. Feel better yet? Didn't think so.
Who doesn't get this? Who has the audacity to demand unarmed helplessness? Who likes dead good guys?
I'll tell you who. People who tramp on the Second Amendment, that's who. People who refuse to accept the self-evident truth that free people have the God-given right to keep and bear arms, to defend
themselves and their loved ones. People who are so desperate in their drive to control others, so mindless in their denial that they pretend access to gas causes arson, Ryder trucks and fertilizer
cause terrorism, water causes drowning, forks and spoons cause obesity, dialing 911 will somehow save your life, and that their greedy clamoring to "feel good" is more important than admitting that
armed citizens are much better equipped to stop evil than unarmed, helpless ones.
Pray for the families of victims everywhere, America. Study the methodology of evil. It has a profile, a system, a preferred environment where victims cannot fight back. Embrace the facts, demand
upgrade and be certain that your children's school has a better plan than Virginia Tech or Columbine. Eliminate the insanity of gun-free zones, which will never, ever be gun-free zones. They will
only be good guy gun-free zones, and that is a recipe for disaster written in blood on the altar of denial. I, for one, refuse to genuflect there.
You obviously know the players on both sides much better than I, go figure! Besides you (and yours). make more than a perfect counterweight, then them (and theirs).
I simply don't buy either end of the firearm policy spectrum position "extremes."
Your scenario could never possibly play out here in the USA. Suggesting such is 100% FUD! You would pretty much have to subvert all three branches of our government, declare 24/7 martial law,
We do have a 2nd Amendment you know? Which I believe in very, Very, VERY strongly!
IMHO, they tear that one down (or any of the first 10), say bye bye to the good old USA!
Imagine Civil War! Part Deux, with the weapon holders on one side, and the weaponless on the other, now which side do you suppose I would stand behind (figuratively and literally)? No, not the
weaponless side (no matter how far back behind them I could stand)!
Every eye fixed itself upon him; with parted lips and bated breath the audience hung upon his words, taking no note of time, rapt in the ghastly fascinations of the tale. NOT!
Every eye fixed itself upon him; with parted lips and bated breath the audience hung upon his words, taking no note of time, rapt in the ghastly fascinations of the tale. NOT!
Originally Posted by Jubelum
One for the ages... Ted gets it right again!....
I know you don't believe in statistics (of course we should all know that all statistical measures are way, way beyond lies, and way beyond damn lies!
But, here are three anyway;
US total annual death rate = 8.26 deaths/1,000 population (BTW, the world average is 8.67)
US total annual homicide rate = 5.6 deaths/100,000 population (FBI Excel spreadsheet for 2005 (filename 05tbl04.xls (but I can't find actual link at the moment))
So basically ~ 0.68% of the first statistic.
VT massacre (annualized) ~ 10.7 deaths/100,000,000 (32 deaths/300,000,000+ estimated US population)
So basically ~ 0.0013% of the first statistic or ~ 0.19% of the second statistic. And that assumes a mass murder event of that average magnitude occurs annually! So divide these by an ~ order of
magnitude, so, for example p = 0.2, for a 1/5 return period (a single event equal to this magnitude occurs on average 1 out of 5 years). Mind you this is an all time high senseless mass murder event
here in the US.
AND NO, I AM IN NO WAY TRYING TO TRIVIALIZE THE VT MASSACRE!
So as Artman's cartoon suggests, should we look at this as a "freak" occurrence, a crazed lone gunman, a very rare event that happens once every 10 years or so, or should we look at this as a 24/7
happens all the time, every minute of the day type event?
How about standard issue firearms to ALL students, along with standard issue IBA's (ESAPI plates included) or Dragon Skin armor vests (if you prefer)?
Sounds like a weiner to me!
PS - "The Nuge" (most famous for his quote: "You have to kill it, before you can grill it.") is way over the top on this one.
BTW - I went to a 1995 "Spirit of the Wild" concert in Memphis, TN, IMHO he's a likable kind of guy, and he ROCKS!
Every eye fixed itself upon him; with parted lips and bated breath the audience hung upon his words, taking no note of time, rapt in the ghastly fascinations of the tale. NOT!
Every eye fixed itself upon him; with parted lips and bated breath the audience hung upon his words, taking no note of time, rapt in the ghastly fascinations of the tale. NOT!
It took you a while, I expected it somewhat sooner!
Every eye fixed itself upon him; with parted lips and bated breath the audience hung upon his words, taking no note of time, rapt in the ghastly fascinations of the tale. NOT!
Every eye fixed itself upon him; with parted lips and bated breath the audience hung upon his words, taking no note of time, rapt in the ghastly fascinations of the tale. NOT!
Originally Posted by franksargent
You obviously know the players on both sides much better than I, go figure! Besides you (and yours). make more than a perfect counterweight, then them (and theirs).
I simply don't buy either end of the firearm policy spectrum position "extremes."
Your scenario could never possibly play out here in the USA. Suggesting such is 100% FUD! You would pretty much have to subvert all three branches of our government, declare 24/7 martial law,
We do have a 2nd Amendment you know? Which I believe in very, Very, VERY strongly!
IMHO, they tear that one down (or any of the first 10), say bye bye to the good old USA!
Imagine Civil War! Part Deux, with the weapon holders on one side, and the weaponless on the other, now which side do you suppose I would stand behind (figuratively and literally)? No, not the
weaponless side (no matter how far back behind them I could stand)!
Dude, you are REALLY killing my buzz. Don't be such a flat-liner. You're screwing up my fantasy life...
Isn't that what all of us gun right loving bubbas are all about anyway? What if the Rooskies invade and no one does anything but the Wolverines?
I have been an instructor of collegiate firearms programs for about six years, and I can tell you that about one in twenty people at that age I bounce so fast that they wonder if a keg-stand was
involved. The vast majority of students who are interested in having a CHL are very responsible people. The balance get washed out by those of us who know what bullshit looks like.
I think a college senior, age ~22 or so, who can carry legally almost everywhere else in the state, should have the right to in places he or she have a legal right to be. There is only one thing that
would have stopped Cho... a student or professor with the right tool. And, that tool is, take a deep breath... a concealed firearm in the hands of a capable, trained, and licensed CHL holder.
My daughter will have a firearm at college. Single, female, attractive, living alone, in an urban setting.
She's starting to beat me at the range these days. Girls CAN handle firearms...
Originally Posted by Jubelum
I have been an instructor of collegiate firearms programs for about six years, and I can tell you that about one in twenty people at that age I bounce so fast that they wonder if a keg-stand was
involved. The vast majority of students who are interested in having a CHL are very responsible people. The balance get washed out by those of us who know what bullshit looks like.
I think a college senior, age ~22 or so, who can carry legally almost everywhere else in the state, should have the right to in places he or she have a legal right to be. There is only one thing that
would have stopped Cho... a student or professor with the right tool. And, that tool is, take a deep breath... a concealed firearm in the hands of a capable, trained, and licensed CHL holder.
My daughter will have a firearm at college. Single, female, attractive, living alone, in an urban setting.
She's starting to beat me at the range these days. Girls CAN handle firearms...
I have NO problem with responsible firearm ownership! A lot of creeps out there, good for her.
Every eye fixed itself upon him; with parted lips and bated breath the audience hung upon his words, taking no note of time, rapt in the ghastly fascinations of the tale. NOT!
Every eye fixed itself upon him; with parted lips and bated breath the audience hung upon his words, taking no note of time, rapt in the ghastly fascinations of the tale. NOT!
Y'all can stop arguing now, @_@Artman wins this thread!!
You need skeptics, especially when the science gets very big and monolithic. -James Lovelock
The Story of Stuff
You need skeptics, especially when the science gets very big and monolithic. -James Lovelock
The Story of Stuff
Another shooting in the US.. this time at NASA (what is up with NASA these days?).
Your = the possessive of you, as in, "Your name is Tom, right?" or "What is your name?"
You're = a contraction of YOU + ARE as in, "You are right" --> "You're right."
Your = the possessive of you, as in, "Your name is Tom, right?" or "What is your name?"
You're = a contraction of YOU + ARE as in, "You are right" --> "You're right."
... I mean, if a bunch of rocket scientists can't handle guns, I'd say we're in trouble...
Your reading skills need a bit of work.
Your = the possessive of you, as in, "Your name is Tom, right?" or "What is your name?"
You're = a contraction of YOU + ARE as in, "You are right" --> "You're right."
Your = the possessive of you, as in, "Your name is Tom, right?" or "What is your name?"
You're = a contraction of YOU + ARE as in, "You are right" --> "You're right."
How can anybody promote the possession of guns by students in a college setting? Have you visited a college campus recently, particularly on party night? A college campus is not a place for guns,
period. I have been there and seen that and do not ever want to see it again.
Your = the possessive of you, as in, "Your name is Tom, right?" or "What is your name?"
You're = a contraction of YOU + ARE as in, "You are right" --> "You're right."
Your = the possessive of you, as in, "Your name is Tom, right?" or "What is your name?"
You're = a contraction of YOU + ARE as in, "You are right" --> "You're right."
Originally Posted by Bergermeister
How can anybody promote the possession of guns by students in a college setting? Have you visited a college campus recently, particularly on party night? A college campus is not a place for guns,
period. I have been there and done that and do not ever want to see it again.
Huh, you mean you've been to college, attended college, or had a firearm on a party night, intoxicated, etceteras???
Every eye fixed itself upon him; with parted lips and bated breath the audience hung upon his words, taking no note of time, rapt in the ghastly fascinations of the tale. NOT!
Every eye fixed itself upon him; with parted lips and bated breath the audience hung upon his words, taking no note of time, rapt in the ghastly fascinations of the tale. NOT!
Gunman knew engineer he killed at NASA, police say
Every eye fixed itself upon him; with parted lips and bated breath the audience hung upon his words, taking no note of time, rapt in the ghastly fascinations of the tale. NOT!
Every eye fixed itself upon him; with parted lips and bated breath the audience hung upon his words, taking no note of time, rapt in the ghastly fascinations of the tale. NOT!
Originally Posted by Bergermeister
How can anybody promote the possession of guns by students in a college setting? Have you visited a college campus recently, particularly on party night? A college campus is not a place for guns,
period. I have been there and done that and do not ever want to see it again.
Most college's ban guns on campus... My undergraduate institution had an immediate expulsion policy... It didn't stop some rich twerp from bringing his great-great grand father's loaded civil war
side arm...
"In a republic, voters may vote for the leaders they want, but they get the leaders they deserve."
"In a republic, voters may vote for the leaders they want, but they get the leaders they deserve."
The discussion is beginning in Virginia.
Seems there is no specific statue referring to colleges, but the "murky" laws are already being challengd by, you guessed it, those who prefer arming our campuses.
Your = the possessive of you, as in, "Your name is Tom, right?" or "What is your name?"
You're = a contraction of YOU + ARE as in, "You are right" --> "You're right."
Your = the possessive of you, as in, "Your name is Tom, right?" or "What is your name?"
You're = a contraction of YOU + ARE as in, "You are right" --> "You're right."
Originally Posted by Bergermeister
The discussion is beginning in Virginia.
Seems there is no specific statue referring to colleges, but the "murky" laws are already being challengd by, you guessed it, those who prefer arming our campuses.
I'd like to know a little more about concealed-carry permits (VA in particular), requirements, training, restrictions, and penalties for inappropriate use.
Every eye fixed itself upon him; with parted lips and bated breath the audience hung upon his words, taking no note of time, rapt in the ghastly fascinations of the tale. NOT!
Every eye fixed itself upon him; with parted lips and bated breath the audience hung upon his words, taking no note of time, rapt in the ghastly fascinations of the tale. NOT!
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the first resource for mathematics
Summary: The purpose of model selection algorithms such as All Subsets, Forward Selection and Backward Elimination is to choose a linear model on the basis of the same set of data to which the model
will be applied. Typically we have available a large collection of possible covariates from which we hope to select a parsimonious set for the efficient prediction of a response variable. Least Angle
Regression (LARS), a new model selection algorithm, is a useful and less greedy version of traditional forward selection methods. Three main properties are derived:
(1) A simple modification of the LARS algorithm implements the Lasso, an attractive version of ordinary least squares that constrains the sum of the absolute regression coefficients; the LARS
modification calculates all possible Lasso estimates for a given problem, using an order of magnitude less computer time than previous methods. (2) A different LARS modification efficiently
implements Forward Stagewise linear regression, another promising new model selection method; this connection explains the similar numerical results previously observed for the Lasso and Stagewise,
and helps us understand the properties of both methods, which are seen as constrained versions of the simpler LARS algorithm. (3) A simple approximation for the degrees of freedom of a LARS estimate
is available, from which we derive a Cp estimate of prediction error; this allows a principle choice among the range of possible LARS estimates. LARS and its variants are computationally efficient:
the paper describes a publicly available algorithm that requires only the same order of magnitude of computational effort as ordinary least squares applied to the full set of covariates.
62J05 Linear regression
62J07 Ridge regression; shrinkage estimators
65C60 Computational problems in statistics
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area of a spherical triangle
area of a spherical triangle
A spherical triangle is formed by connecting three points on the surface of a sphere with great arcs; these three points do not lie on a great circle of the sphere. The measurement of an angle of a
spherical triangle is intuitively obvious, since on a small scale the surface of a sphere looks flat. More precisely, the angle at each vertex is measured as the angle between the tangents to the
incident sides in the vertex tangent plane.
Theorem. The area of a spherical triangle $ABC$ on a sphere of radius $R$ is
$S_{{ABC}}=(\angle A+\angle B+\angle C-\pi)R^{2}.$ (1)
Incidentally, this formula shows that the sum of the angles of a spherical triangle must be greater than or equal to $\pi$, with equality holding in case the triangle has zero area.
Since the sphere is compact, there might be some ambiguity as to whether the area of the triangle or its complement is being considered. For the purposes of the above formula, we only consider
triangles with each angle smaller than $\pi$.
An illustration of a spherical triangle formed by points $A$, $B$, and $C$ is shown below.
Note that by continuing the sides of the original triangle into full great circles, another spherical triangle is formed. The triangle $A^{{\prime}}B^{{\prime}}C^{{\prime}}$ is antipodal to $ABC$
since it can be obtained by reflecting the original one through the center of the sphere. By symmetry, both triangles must have the same area.
For the proof of the above formula, the notion of a spherical diangle is helpful. As its name suggests, a diangle is formed by two great arcs that intersect in two points, which must lie on a
diameter. Two diangles with vertices on the diameter $AA^{{\prime}}$ are shown below.
At each vertex, these diangles form an angle of $\angle A$. Similarly, we can form diangles with vertices on the diameters $BB^{{\prime}}$ and $CC^{{\prime}}$ respectively.
Note that these diangles cover the entire sphere while overlapping only on the triangles $ABC$ and $A^{{\prime}}B^{{\prime}}C^{{\prime}}$. Hence, the total area of the sphere can be written as
$S_{{\mathrm{sphere}}}=2S_{{AA^{{\prime}}}}+2S_{{BB^{{\prime}}}}+2S_{{CC^{{% \prime}}}}-4S_{{ABC}}.$ (2)
Clearly, a diangle occupies an area that is proportional to the angle it forms. Since the area of the sphere is $4\pi R^{2}$, the area of a diangle of angle $\alpha$ must be $2\alpha R^{2}$.
Hence, we can rewrite equation (2) as
$\displaystyle 4\pi R^{2}=2R^{2}(2\angle A+2\angle B+2\angle C)-4S_{{ABC}},$
$\displaystyle\therefore~{}S_{{ABC}}=(\angle A+\angle B+\angle C-\pi)R^{2},$
which is the same as equation (1). ∎
AreaOfTheNSphere, Defect, SolidAngle, LimitingTriangle, SphericalTrigonometry
Mathematics Subject Classification
no label found
no label found
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Combination Lock with 281 Trillion Possible Combinations With a Trap! - Page 2
blackknight1337, on 27 November 2012 - 05:48 AM, said:
Nope, give someone enough time and it's straightforward to crack it. It's only near impossible if any single player can only have a single shot at it. So yeah, given enough time, anyone can crack it
pretty easily.
And like I said, it's only finite if the person attempting to crack it knows the combination length. Then it turns into n^n where n is the number of buttons. If the combination remains an unknown,
which it would, then it becomes n^x where n is the number of buttons and x is the guessed combination length. Which, because the x is an unknown, gives an infinite amount of possibilities. So yeah,
without prior knowledge (combination length or total possible combinations), the lock is infinite.
281000000000000/60/60/24/52/100 = 625445
Incase you don't know that means if you try a different combination every 1 second it will take you 625,445 CENTURIES! to try every combination.
Lets say you get lucky and do it within the first quarter it will still take you approx. 150,000 Centuries to crack it.
Average of lets say 100, you would need 150,000 people try every single second of the live to have a 1/4 chance of cracking it.
Still possible?? Still easy??
I dont think so
Also because of the redstone limits in minecarft x will be less than, lets say, 1,000,000 therefore not infinite but still near impossible to crack
|
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Mill Valley SAT Math Tutor
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Factorial calculation optimization
February 23rd, 2013, 11:08 AM
Factorial calculation optimization
For a problem I'm working on, I need to calculate some very, very large factorials; or at least the first five non-zero digits of said factorial. I've worked out the function:
F(n) = (^(n!)/[(10^S[5](n))])%10^5
S[5](n) = Attachment 1808 where p = 5 to get the trailing zeros of the factorial.
The Algorithm:
Code java:
public static void main(String[] args) {
// TODO Auto-generated method stub
Timer t = new Timer();//just a timer, can be replaced with long start = System.currentTimeMillis();
long fact = 1;//variable containing the needed value
long limit = (long) 100000.0;//the nth factorial
long pow10 = 10;//powers of 10
for(long i = 2; i <= limit; i++)//for loop to calculate the needed factorial
fact*=i; // (i-1)! * i = i! at that point
while(fact%10==0)//does the fact/10^(s_5(i)) function to strip the trailing zeros
fact %= 100000;// does the modulus to keep the numbers in bounds
if(i%pow10==0)//simply prints out the factorial of powers of ten
System.out.println(i + ": " + fact);
//System.out.println(i + ": " + fact);
System.out.println(fact);//print the result
t.end();//ends the timer and prints out the result
t.printElapsedTimeSeconds();//can be replaced with System.out.println("Elasped time: " + (System.currentTimeMillis()-start)/1000 + " seconds.");
I can't use De Polignac's formula because the required sieve would take too much memory and brute forcing every prime takes too long. This is the output of F(1000000000):
Code :
10: 36288
100: 16864
1000: 53472
10000: 79008
100000: 2496
1000000: 4544
10000000: 51552
100000000: 52032
1000000000: 64704
Elapsed time: 13.476 sec.
I need to calculcate F(1000000000000) or F of 1 trillion. This would take a very long time, there has to be some optimization or tweak that I'm missing somewhere.
February 23rd, 2013, 08:20 PM
Re: Factorial calculation optimization
Hmm, interesting problem...
So you want to find the last 5 digits of n! after all the trailing zeros?
The first thing to do is eliminate all the trailing zero.
Each trailing zero equates to a prime factor pair of (2,5).
So first order of business is to eliminate all of these pairs. An easy way to do this is count up how many times n! is divisible by 5. Since this is always going to be smaller than the number of
times n! is divisible by 2, we know that the factor 5 will be the limiting value of the (2,5) pairs.
Then as you're multiplying through, if a value is divisible by 2/5 and you haven't reached the limiting value for that number, divide the value by the appropriate value.
For example, let's take 10!:
There are two 5 prime factors of the result (5, 10).
Then as we multiply through, removing factors of 2 up to the limit:
num = 1
num = num * 2 / 2, two_count = 1
num = num * 3
num = num * 4 / 2, two_count = 2
num = num * 5 / 5, five_count = 1 (not actually necessary to keep track of five_count since every factor of 5 will be removed)
num = num * 6
num = num * 7
num = num * 8
num = num * 9
num = num * 10 / 5, five_count = 2 (not actually necessary to keep track of five_count since every factor of 5 will be removed)
Once you guarantee that the least significant digit is non-zero, you can use a simple math trick to calculate the last x digits of a*b:
You only need to multiply the last x digits from each number a, b to find the last x digits of the result.
For example, to find the last 2 digits of 123456 * 789012:
56 * 12 = 672
Last two digits are 72.
This last step isn't necessary for smaller factorials but is absolutely vital for larger numbers because there's the potential for a*b to overflow, especially for larger factorials.
The algorithm is O(1) space and ~O(n) runtime (possibly O(n log(n)), not sure about this).
Some run statistics (not comparable with your times because we likely have different hardware):
10: 36288
100: 16864
1000: 53472
10000: 79008
100000: 62496
1000000: 12544
10000000: 94688
100000000: 54176
1000000000: 38144
Time taken: 21.404 s
Interestingly, there are some discrepancies between my answers and your answers, especially for larger factorials (possibly due to numerical errors, or my mis-understanding of program
requirements). i checked using Wolfram Alpha and it looks like my answers are correct.
There might be some optimization for figuring out how many factors of 5 there are, I'm not sure.
You could also try optimizing by using divide and conquer and parallelizing, just make sure you ensure your counts are computed for each division (if you're going to try multi-threading).
February 23rd, 2013, 08:52 PM
Re: Factorial calculation optimization
Ok, well, you can use De Polignac's formula to figure out how many factors of a prime factor there are in n!.
Could you also explain this a different way?
So first order of business is to eliminate all of these pairs. An easy way to do this is count up how many times n! is divisible by 5. Since this is always going to be smaller than the number of
times n! is divisible by 2, we know that the factor 5 will be the limiting value of the (2,5) pairs.
Then as you're multiplying through, if a value is divisible by 2/5 and you haven't reached the limiting value for that number, divide the value by the appropriate value.
For example, let's take 10!:
There are two 5 prime factors of the result (5, 10).
Then as we multiply through, removing factors of 2 up to the limit:
num = 1
num = num * 2 / 2, two_count = 1
num = num * 3
num = num * 4 / 2, two_count = 2
num = num * 5 / 5, five_count = 1 (not actually necessary to keep track of five_count since every factor of 5 will be removed)
num = num * 6
num = num * 7
num = num * 8
num = num * 9
num = num * 10 / 5, five_count = 2 (not actually necessary to keep track of five_count since every factor of 5 will be removed)
This is a project Euler problem, so there is a solution possible that takes less than a few minutes possible. I noticed that there are a few thousand final five values under 100000! that have
quite a few repetitions, so that says to me that there may be a way to predict when and where those repetitious values come up.
February 23rd, 2013, 09:54 PM
Re: Factorial calculation optimization
Hmm, after close examination of your code you're right they are very similar. The only difference is that you use De Polignac's Formula which is faster, but you failed to properly handle the
factorial computation part.
In either case, much slower than should be expected for a Project Euler problem. Which problem number is it?
February 23rd, 2013, 10:07 PM
Re: Factorial calculation optimization
Problem 160 - Project Euler
Problem 160. I've been working on it for a while. First attempt was to brute force the factorial using JScience's LargeInteger class and huge multi-threading. That took too long. Then I moved to
trying De Polignac's algorithm specifically, but brute forcing prime numbers took too long and a large enough sieve is impossible to do in Java. This is the only method that's come close to
what's needed. I then noticed that there were last-five-digit-combos that never came up, and a whole lot that came up quite often. That's probably the key to figuring out the solution, but I
don't know how to apply it.
February 24th, 2013, 02:00 AM
Re: Factorial calculation optimization
Here's something I just thought of:
One of the key tricks we're taking advantage of is by only keeping track of the last 5 digits for multiplication. Even with 1 trillion factorial we're going to be repeatedly multiplying
essentially these same 5 digit numbers over and over again. You might have some luck with either a quick integer power function or look-up tables to quickly compute the repeated multiplications.
Dunno if it will work, but may be worth a shot.
|
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Geometry Festival
23rd Annual Geometry Festival
Previous Geometry Festival Speakers
1985 at Penn:
• Marcel Berger
• Pat Eberlein
• Jost Eschenburg
• Friedrich Hirzebruch
• Blaine Lawson
• Leon Simon
• Scott Wolpert
• Deane Yang
1986 at Maryland:
• Uwe Abresch, "Explicit constant mean curvature tori"
• Zhi-yong Gao, "The existence of negatively Ricci curved metrics"
• David Hoffman, "New results in the global theory of minimal surfaces"
• Jack Lee, "Conformal geometry and the Yamabe problem"
• Ngai-ming Mok, "Compact Kähler manifolds of non-negative curvature"
• John Morgan, "Self dual connections and the topology of 4-manifolds"
• Chuu-lian Terng, "Submanifolds with flat normal bundle"
1987 at Penn:
• Robert Bryant, "The construction of metrics with exceptional holonomy"
• Francis Bonahon, "Hyperbolic 3-manifolds with arbitrarily short geodesics"
• Keith Burns, "Geodesic flows on the 2-sphere"
• Andreas Floer, "Instantons and Casson's invariant"
• Hermann Karcher, "Embedded minimal surfaces in the 3-sphere"
• Jürgen Moser, "Minimal foliations of tori"
• Ed Witten. "Applications of quantum field theory to topology"
1988 at North Carolina:
• Detlef Gromoll, "On complete spaces of non-negative Ricci curvature"
• Nicolas Kapouleas, "Constant mean curvature surfaces in E^3"
• Robert Osserman, "Gauss map of complete minimal surfaces"
• Pierre Pansu, "L^p-cohomology of negatively curved manifolds"
• Peter Petersen, "Bounding homotopy types by geometry"
• Gang Tian, "Kähler-Einstein metrics on quasiprojective manifolds"
• DaGang Yang, "Some new examples of manifolds of positive Ricci curvature"
• Wolfgang Ziller, "Recent results on Einstein metrics"
1989 at Stony Brook:
• Eugenio Calabi, "Extremal singular metrics on surfaces"
• Harold Donnelly, "Nodal sets of eigenfunctions on Riemannian manifolds"
• Yasha Eliashberg, "Symplectic geometric methods in several complex variables"
• Thomas Farrell, "A topological analogue of Mostow's rigidity theorem"
• Lesley Sibner, "Solutions to Yang-Mills equations which are not self-dual"
• Carlos Simpson, "Moduli spaces of representations of fundamental groups"
1990 at Maryland:
• Michael T. Anderson, "Behavior of metrics under Ricci curvature bounds"
• Kevin Corlette, "Harmonic maps and geometric superrigidity"
• Kenji Fukaya, "Fundamental groups of almost non-negatively curved manifolds"
• Misha Gromov, "Recent progress in symplectic geometry"
• Werner Müller, "On spectral theory for locally symmetric manifolds with finite volume"
• Rick Schoen, "Least area problems for Lagrangian submanifolds"
• Gudlaugur Thorbergsson, "Isoparametric submanifolds and their Tits buildings"
• Shing-Tung Yau, "Some theorems in Kähler geometry"
1991 at Duke:
• Jeff Cheeger, "Transgressed Euler classes of SL(2n,Z)-bundles and adiabatic limits of eta-invariants"
• Chris Croke, "Volumes of balls in manifolds without conjugate points and rigidity of geodesic flows"
• Carolyn Gordon, "When you can't hear the shape of a manifold"
• Wu-Yi Hsiang, "Sphere packing and spherical geometry: The Kepler conjecture and beyond"
• Alan Nadel, "On the geometry of Fano varieties"
• Grigori Perelman, "Alexandrov's spaces with curvature bounded from below"
• Stefan Stolz, "On the space of positive curvature metrics modulo diffeomorphisms"
1992 at NYU/Courant:
• Jonathan Block, "Aperiodic tilings, positive scalar curvature and other homological phenomena"
• John Franks, "Infinitely many closed geodesics on the 2-sphere"
• Karsten Grove, "The inevitable presence of singular spaces in Riemannian geometry"
• Lisa Jeffrey, "Volumes of moduli spaces of flat connections on Riemannian surfaces"
• Jun Li, "Anti-self-dual connections on SU(2) bundles over algebraic surfaces"
• Dusa McDuff, "Symplectic 4-manifolds"
• Clifford Taubes, "Anti-self dual conformal structures in 4 dimensions"
1993 at Penn:
• Shing-Shen Chern, "Finsler geometry"
• Richard Hamilton, "An isoperimetric estimate for the curve-shrinking flow"
• Vaughan Jones, "Loop groups and operator algebras"
• Claude LeBrun, "Compact Kähler manifolds of constant scalar curvature"
• Louis Nirenberg, "The maximum principle and related things"
• Xiaochun Rong, "Collapsing in low dimensions and rationality of geometric invariants"
• I. M. Singer, "Geometry and quantum field theory"
1995 at Stony Brook:
• Dimitri Burago, "Asymptotic geometry of Z^n-periodic metrics"
• Tobias Colding, "Ricci curvature and convergence"
• Dominic Joyce, "Compact Riemannian manifolds with exceptional holonomy groups"
• Yael Karshon, "Hamiltonian torus actions"
• David Morrison, "Analogues of Seiberg-Witten invariants for counting curves on Calabi-Yau manifolds"
• Tom Mrowka, "The Seiberg-Witten equations and 4-manifold topology"
• Yongbin Ruan, "Higher genus pseudo-holomorphic curves"
• Edward Witten, "Monopoles and four-manifolds"
1996 at Maryland:
• J. Baez, "Quantum gravity and BF theory in 4 dimensions"
• Jean-Luc Brylinski, "Gauge groups and reciprocity laws on algebraic varieties"
• Bruce Kleiner, "Spaces of nonpositive curvature"
• G. Margulis, "Quantitative Oppenheim Conjecture"
• S. P. Novikov, "Laplace and Darboux transformations"
• Richard Schwartz, "The Devil's Pentagram"
• Guofong Wei, "Volume comparison with integral curvature bounds"
• Shmuel Weinberger, "Equivariant rigidity: For and against"
• Jeanne Nielsen Clelland , "Geometry of Conservation Laws for Parabolic PDE's"
• Anatole Katok , "Rigidity and invariant geometric structures for differentiable group actions"
• François Labourie , "Monge-Ampere problems, holomorphic curves and laminations"
• Gang Liu, "Floer Homology and the Arnold Conjecture"
• William Minicozzi, "Harmonic functions on manifolds"
• Lorenz Schwachhöfer, "The classification of irreducible holonomies of torsion free connections"
• Matthias Schwarz, "Symplectic fixed points and quantum cohomology"
• Stephen Semmes, "Geometry with little smoothness"
• Scott Axelrod, "Generalized Chern-Simons invariants as a generalized Lagrangian field theory"
• Jean-Michel Bismut, "Chern-Simons classes, Bott Chern classes and analytic torsion"
• Spencer Bloch, "Algebro-geometric Chern-Simons classes"
• Robert Bryant, "Recent progress on the holonomy classification problem"
• Robert Bryant (for S.-S. Chern), "Recent results and open problems in Finsler geometry"
• Jeff Cheeger and Blaine Lawson, "The mathematical work of James Simons"
• Jeff Cheeger, "Ricci Curvature"
• Jürg Fröhlich, "Physics and the Chern-Simons form (from anomalies to the quantum Hall effect to magnetic stars)"
• Mikhail Gromov, "Dynamics on function spaces"
• Maxim Kontsevich, "On regulators, critical values and q-factorials"
• Blaine Lawson, "Connections and singularities of maps"
• Robert MacPherson, "Spaces with torus actions"
• John Milnor, "Remarks on geometry and dynamics"
• I.M. Singer, TBA
• Dennis Sullivan "A combinatorial model for non-linearity"
• Clifford Taubes, "Seiberg-Witten invariants, harmonic forms, and their pseudo-holomorphic curves"
• Gang Tian, "Yang-Mills connections and calibration"
• C.-N. Yang, "Vector potentials and connections"
• S.-T. Yau, "Mirror symmetry and rational curves"
1999 at Penn:
• Peter Sarnak, "Some spectral problems on negatively curved manifolds"
• Zheng-xu He, "The gradient flow for the Möbius energy of knots"
• Curtis McMullen, "The moduli space of Riemann surfaces is Kähler-hyperbolic"
• Paul Biran, "Lagrange skeletons and symplectic rigidity"
• Helmut Hofer, Holomorphic curves and contact geometry"
• Werner Ballmann, "On negative curvature and the essential spectrum of geometric operators"
• Shlomo Sternberg, "Multiplets of representations and Kostant's Dirac operator"
• Samuel Ferguson, "The Kepler Conjecture"
• Robert Meyerhoff, "Rigorous computer-aided proofs in the theory of hyperbolic 3-manifolds"
• Herman Gluck, "Geometry, topology and plasma physics"
• Burkhard Wilking, "New examples of manifolds with positive sectional curvature almost everywhere"
• John Roe, "Amenability and assembly maps"
• Eleny Ionel, "Gromov invariants of symplectic sums"
• Mikhail Gromov, "Spaces of holomorphic maps"
• Robert Bryant, "Rigidity and quasirigidity of extremal cycles in Hermitian symmetric spaces"
• Tobias Colding, "Embedded minimal surfaces in 3-manifolds"
• Boris Dubrovin, "Normal forms of integrable PDE's"
• John Lott, "Heat equation methods in noncommutative geometry"
• Dusa McDuff, "Seminorms on the Hamiltonian group and the nonsqueezing theorem"
• Rick Schoen, "Variational approaches to the construction minimal lagrangian submanifolds"
• Shing-Tung Yau, "Mirror symmetry"
• Denis Auroux, "Singular plane curves and topological invariants of symplectic manifolds"
• Hugh Bray, "On the mass of higher dimensional black holes"
• Alice Chang, "Conformally invariant operators and the Gauss-Bonnet integrand"
• Xiuxiong Chen,"The space of Kähler metrics"
• George Daskalopoulos,"On the Yang-Mills flow in higher dimensions"
• Alex Eskin, "Billiards and lattices"
• Juha Heinonen, "On the existence of quasiregular mappings"
2004 at Courant:
• Jean-Michel Bismut
The Hypoelliptic Laplacian on the Cotangent Bundle
• Yasha Eliashberg
Positive Loops of Contact Transformations
• Blaine Lawson
Projective Hulls and the Projective Gelfand Transformation
• Dusa McDuff
Applications of J-holomorphic Curves
• Xiaochun Rong
Local splitting structures on nonpositively curved manifolds
• Dennis Sullivan
Algebraic topology in string backgrounds
• Gang Tian
Extremal Metrics and Holomorphic Discs
• Edward Witten
Gauge Theory Scattering From Curves In CP^3
• Juha Heinonen
On the existence of quasiregular mappings
• Nancy Hingston
Periodic solutions of Hamilton 's equations on tori
• Sergiu Klainerman
Null hypersurfaces and curvature estimates in general relativity
• Bruce Kleiner
Singular structure of mean curvature flow
• Frank Pacard
Blowing up Kahler manifolds with constant scalar curvature
• Rahul Pandharipande
A topological view of Gromov-Witten theory
• Igor Rodniansky
Non-linear waves and Einstein geometry
• Yum-Tong Siu
Methods of singular metrics in algebraic geometry
• Katrin Wehrheim
Floer theories in symplectic topology and gauge theory
• Jeff Cheeger (Courant)
Differentiation, bi-Lipschitz nonembedding and embedding
• Charles Fefferman (Princeton University)
Fitting a smooth function to data
• Helmut Hofer (Courant)
On the analytic and geometric foundations of symplectic field theory
• Ko Honda (University of Southern California)
Reeb vector fields and open book decompositions
• William Meeks (University of Massachusetts)
The Dynamics Theorem for embedded minimal surfaces
• Yair Minsky (Yale University)
Asymptotic geometry of the mapping class group
• Frank Morgan (Williams College)
Manifolds with Density
• Zoltan Szabo (Princeton University)
Link Floer homology and the Thurston norm
2008 and on: see here.
|
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Fraction Series Correlation to Texas Essential Knowledge and Skills
Grade 6: Products recommended: FraxKit, FraCards, FracMulti, FracDivide
TEKS # Description
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What is the composite score?
Print | Close
What is the composite score?
The composite score is used to determine the monthly ranking. It is calculated using a formula taking into account three things:
• The average score (the higher the better)
• The average time taken to complete a puzzle (the lower the better)
• The total number of puzzles completed during the month (the higher the better)
For those of you interested, here is the formula we are using:
CS = (AS + 100 * 10^(- AT/25000) + 100 * 10^(-(7*DM-NP)/3000))/3
• CS is the composite score
• AS is the average score
• AT is the average time in seconds
• DM is the number of days in the month
• NP is the number of puzzles completed during the month
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University of Rochester
Enter your name and a friend's email address in the fields below and click "Submit" to email this Press Release to a friend.
[YOUR NAME HERE] thought you might be interested in this story from the University of Rochester.
Alex Iosevich, professor of mathematics at the University of Rochester, has been named a Fellow of the American Mathematical Society (AMS).
The AMS awards fellowships to recognize "members who have made outstanding contributions to the creation, exposition, advancement, communication, and utilization of mathematics."
"This honor is a recognition of all the things done for me by many others, including my mentors, co-authors, and colleagues," said Iosevich.
One of those mentors is the chair of the Department of Mathematics at the University of Rochester, Allan Greenleaf, who calls the fellowship "a well-deserved honor" for Iosevich. "That he is one of
only 50 members named as an AMS Fellow this year indicates his stature in the national mathematical community."
Iosevich describes his work as "understanding the connections between different areas of mathematics." Specifically, he looks for ways to solve problems in one area of mathematics with knowledge
found in other areas, taking inspiration from the "giants in the field who were driven by problems, not labels."
One problem Iosevich is pursuing involves determining the maximum number of identical triangles that can be formed when connecting every three points in a grouping of dots. Specifically, he is trying
to show that the number of triangles repeated in any configuration of dots cannot be greater than the number of such triangles that are created from an orderly lattice of dots. He calls the problem
"simple-sounding, yet elegant."
While the triangle problem comes from the field of geometric combinatorics, Iosevich uses methods from harmonic analysis and geometric measure theory to work on a solution.
Iosevich earned his Ph.D. in pure mathematics from the University of California at Los Angeles. He joined the faculty at the University of Rochester in 2010, after five years as associate professor
at the University of Missouri-Columbia.
The American Mathematical Society, with headquarters in Providence, RI, was founded in 1888 to further the interests of mathematical research and scholarship.
The University of Rochester (www.rochester.edu) is one of the nation's leading private universities. Located in Rochester, N.Y., the University gives students exceptional opportunities for
interdisciplinary study and close collaboration with faculty through its unique cluster-based curriculum. Its College of Arts, Sciences, and Engineering is complemented by the Eastman School of
Music, Simon School of Business, Warner School of Education, Laboratory for Laser Energetics, Schools of Medicine and Nursing, and the Memorial Art Gallery.
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Worcester, PA Geometry Tutor
Find a Worcester, PA Geometry Tutor
...Someone who shows up on time.3. Someone who is prepared to teach the desired subject matter.4. Someone who is trained to change course if one approach isn't working to one that will.5.
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10 Subjects: including geometry, calculus, algebra 1, ASVAB
I've always been a technical-minded person and have enjoyed logic and math since I was a child. I'm an engineer and a homeschooling dad of two kiddos. I genuinely believe that math can be fun and
that as your child gains confidence, enjoyment of math will follow!
6 Subjects: including geometry, algebra 1, algebra 2, prealgebra
...I look forward to hearing from you and helping you learn some new things. I have been playing piano in a four person band for two years now. I took music lessons as a kid, but not piano
8 Subjects: including geometry, piano, algebra 1, ESL/ESOL
I am an Industrial Engineering major enrolled at the Pennsylvania State University. My education plan includes attending medical school after graduating. I come from a strong background in
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14 Subjects: including geometry, English, chemistry, calculus
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Mplus Short Courses
During the autumn of 2012, Örebro University in Sweden will offer a series of workshops on Structural Equation Modeling (SEM) using Mplus. SEM refers to a group of multidisciplinary methods for
studying causal relations. The courses are taught in English.
Content of the workshops
The workshops are multidisciplinary and are highly relevant for those interested in improving their knowledge and skills on a wide range of advanced methodology. The five workshops review the
theoretical background and application of data analysis with SEM techniques. Each workshop day will be composed of two integral parts: Theory and Application (using Mplus software).
During the first part of each workshop, a survey will be provided of the latest theoretical and statistical concepts and assumptions of SEM framework. The workshops will deal with recent developments
involving model building, model comparison, model specification and identification for several modeling techniques (e.g., path analysis, confirmatory factor analysis, structural equation models,
multiple group modeling). The conceptual overviews of each analytical technique will be illustrated by relevant examples.
During the second part of each workshop, the students will learn how to use Mplus software to specify and test the models discussed theoretically. Students are required to complete practice
assignments to enhance their understanding of the statistical techniques, and develop skills of independently applying analytical procedures.
The following workshops will be held on the following days:
SEM Workshop I: Introduction to Structural Equation Modeling (9th of October)
The aim of this workshop is to develop students' basic understanding of the underlying principles of Structural Equation Modeling (SEM). Additionally, students will acquire in-depth knowledge of the
conceptual basis of SEM and causal assumptions underlying different models. They will be introduced to Mplus and learn how to use the basic functions of the program.
SEM Workshop II: Path analysis and Model Testing (16th of October)
During this workshop, students acquire the latest and most recent techniques for model testing and path analysis. Students will gain knowledge of the ways in which one should formulate models, test
alternative models, and evaluate models with regard to statistical and practical significance. Particular attention will be aid to learning how to use Mplus and interpret the output when testing
different models.
SEM workshop III: Methods involving Latent Variables (18th of October)
Students will learn a sound and detailed understanding of the use of latent variabes in Structural Equation Modeling. New developments in SEM when using latent variables will be discussed according
to a wide range of examples. The program Mplus will be used to estimate and interpret models involving latent variables.
SEM workshop IV: Moderation Analysis and Interaction Effects (23rd of October)
During this workshop, students will acquire a basic understanding of moderation analyses. Furthermore, the students will learn the most recent developments on moderation analysis and interaction
effects. Mplus software will be used to estimate and interpret interaction effects and draw conclusions from the analysis with regard to moderation.
SEM workshop V: Mediation analysis and indirect effects (30th of October)
The last workshops deals with the use of SEM to model indirect effects and interpret mediation. Students will learn the basic conceptual basis underlying mediation analysis as well as more recent
advancements on this technique. Based on recent international publications on mediation and indirect effects in Mplus, students will learn how to use Mplus to examine research questions involving
Students applying for this course need to have a good understanding of regression analysis and Exploratory Factor Analysis. They need to bring their own laptop with SPSS and the SEM program Mplus.
Applying for the course:
The course is free of charge. For those of you interested to apply, please click here for registration information.
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Fermat's Last Theorem is Solved
PROOF OF FERMAT'S LAST THEOREM
James Constant
Fermat's Last Theorem is solved using the binomial series.
A new theorem determining the irrationality of a number using its infinite series expansion is presented.
For simple proofs see http://www.coolissues.com/mathematics/BealFermatPythagorasTriplets.htm
Proof of Fermat's Last Theorem Using the Exponential Series http://www.coolissues.com/mathematics/Fermat/fermatexp.htm
Fermat (1601-1665) claimed in 1637 to have discovered a marvelous proof of his last theorem.^1
(1) ...... ...........xyz 0 ............m>2 ........ x,y,z,m integers
Before seeking the solution to Fermat's last theorem, consider the binomial series^2
in which a is an arbitrary number, positive or negative, rational or irrational.
The exact conditions under which the series, equation (2), is convergent are as follows^3:
1. If index a is an integer the series terminates and is valid for all values of x and becomes the Binomial Theorem.
2. For all other values of a, the series is absolutely convergent for and divergent for
3. For x=+1 the series converges absolutely if a>0, converges conditionally if -1<a<0, and diverges if a Finally, at x=-1 the series is absolutely convergent if a>0, divergent if a<0.
Equation (2) can be written as the sum of its first n terms and a remainder function, in Lagrange's form^4
in which is Lagrange's (unspecified) number.
I will now prove Fermat's Last Theorem by invoking the binomial series (2) and the principle of the excluded middle of traditional logic. This principle holds that either A or not A, where A is
Fermat's Last Theorem, equation (1). I will also prove that x,y,z can be arbitrary numbers, positive or negative, rational or irrational. First, I assume not A, i.e., that the following equality
(4) ........ z^m = x^m + y^m........ xyz 0 ........ m>2........x,y,z,m integers
and note that the number z^m is an integer if z is so. I also note that equation (4) fails whenm=0 but succeeds when m=1 (addition of numbers) or m=2 (Pythagorean theorem). For all other numbers ,
(5) ........z = x (1 + ()^m)^1/m ........xyz 0 ........ m>2........ x,y,z,mintegers
which, when expressed as a binomial series, converges or diverges, I show that z cannot be an integer. More specifically, analogous to equation (2):
1. Since m>2, index 1/m is not an integer and the series cannot terminate becoming the Binomial Theorem.
2. The series is absolutely convergent for and divergent for .
It is easy to show that the parenthesis term in equation (5) is an irrational number, since
from which we immediately see that .
Equation (6) says that the left hand parenthesis term equals a rational number plus an irrational non-vanishing fraction since, if R[n] is assumed rational R[n]=p/q (p,q integers), we can multiply
each side by q and obtain qR[n]=p. This equation is impossible because for large n qR[n] is a non-vanishing fraction <1 but p is an integer 1. The assumption that R[n]=p/q is rational is wrong and,
since there are only two kinds of real numbers, rational and irrational, R[n] is irrational. The right hand sum of equation (6) is, therefore, an irrational number. Accordingly, the fraction z/x
obtained from equation (5) cannot be a rational number and, sincex is an integer, z is irrational and cannot be an integer when
3. For y/x=+1 the series converges absolutely since m>2 and thus 1/m>0. The same argument applies. In this case z=2^1/mis a non-integer.
Therefore, the assumed equality of equation (4) is wrong for the stated conditions x,y,z,m integers and, by the principle of the excluded middle of traditional logic, Fermat's Last Theorem, equation
(1), is proven. However, the assumed equality of equation (4) is correct for other values of x,y,z,m. In other words, the binomial series (2) confirms Fermat's inequality for the stated conditions
x,y,z,m integers but discomfirms Fermat's inequality for other values of x,y,z,m.
New Theorem
If a number is an irrational number. For if the contrary is true,i.e. if p,q integers), then since p',q' integers)
which says that a rational fraction p/q, equals a rational fraction p'/q' plus a non-vanishing fraction qq' we obtain
(8) ......
which says that an integer qq' is a fixed integer and thus
Pythagoras, Newton, and Fermat
The question why z is a real (rational or irrational) number when m=2, but is an irrational number when m>2 is beyond this paper. It is an issue between Pythagoras (geometry) and Newton (binomial
series). For example, if one setsx = x'(1 + ()^m)^1/m in equation (5), where x' is an integer, the resulting series for z terminates and the equality in equation (5) holds for m=1 and m=2. In other
words, the binomial series (2) confirms the addition of numbers when m=1 and Pythagora's theorem when m=2 for x and z absolutely converging infinite series and y any arbitrary number. These
conclusions fall short of the known universal applicability of the addition of numbers and Pythagoras' theorem for arbitrary numbers x,y,z. The question remains why the binomial series does not fully
explain the addition of numbers and Pythagoras' theorem. Nor can geometry prove Pythagoras' theorem when at least x or y is an absolutely converging infinite sum. Perhaps geometry and the binomial
series are complementary; the former contained in the domain of real numbers the latter in the domain of absolutely converging infinite sums.
The binomial series used in the present proof, equation (2), was invented by Newton (1642-1772) about 1676.^5 Fermat's claimed proof of equation (1) could not, therefore, have been based on the
results of equations (2) through (4). A recent book gives a detailed history of Fermat's Last Theorem and attempts to solve it.^6
^1 E. Bell, The Development of Mathematics, Dover Publications, Inc., New York 1972 page 157.
^2 e.g. see R. Courant, Differential and Integral Calculus, Interscience Publishers, Inc. New York 1947 Vol I pages 329-330.
^3 R. Courant note 2 above page 406.
^4 R. Courant note 2 above page 337.
^5 E. Bell note 1 above page 406.
^6 Simon Singh, Fermat's Last Theorem, Fourth Estate Ltd, 6 Salem Road, London W2 4BU,England.
By same author: http://www.coolissues.com/mathematics/sameauthor.htm
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irrational numbers - HELP!!
July 27th 2005, 07:17 AM
irrational numbers - HELP!!
:confused: HELP!! :confused:
What operations are available to irrational numbers?
Can you arrive at a nonzero answer for addition? subtraction? Examples?
Is sqrt of 3 times sqrt of 3 = 3 and a way to multiply irrational numbers?
Is sqrt of 3 divided by the sqrt of 3 = 1 and a way to divide irrational numbers?
Please help me if you can.
July 27th 2005, 07:46 AM
Math Help
irrational numbers have all mathematical operations available to them.
pi for example is irrational.
you can add and subtract pi
you can divide and multiply pi
you can squareroot pi and you can pi^x
any mathematical operation you can do to a rational number you can also do to an irrational.
sqrt(3) / sqrt(3) = 1
July 27th 2005, 08:24 AM
More Help!
Can you get a rational nonzero number with all operations? If so, do you have examples? :confused:
July 27th 2005, 10:30 AM
Can you get a rational nonzero number with all operations? If so, do you have examples?
Look man, u sound really upset -- be cool.
You can do everything you like with irrationals. They are artificially constructed, OK. They are hard to grasp, OK. But they are manageable, as much as rationals are.
Here are some facts you might like to ponder upon:
a) rational + rational = rational
b) rational + irrational = irrational
c) 1/(rational) = rational
d) 1/(irrational) = irrational
e) (rational)*(irrational) = irrational
f) If 1=2, then I am the King of France.
Because, the King and I are two, therefore we are one. :eek:
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Question about Lecture Answer and Problem
August 27th 2013, 10:01 PM #1
Jul 2011
Question about Lecture Answer and Problem
Hi, My instructor gave an example in class today about difference between probability and statistics. He used the following example: Suppose we flip a fair coin 10 times. What sis the probability
of fetting 9 0r more heads? He said the answer was 11/1024. He never showed us how to compute this and said it was simply the addition of two value to get 11/1024. How do you compute this? Thank
you for any advice.
Re: Question about Lecture Answer and Problem
Hey mathplease.
Hint: P(X >= 9) = P(X = 9) + P(X = 10) = (1/2)^10 + 10*(1/2)^10 = 1/1024 + 10/1024 = 11/1024.
Remember that in probability, the aim is to use the probability axioms. When you get used to the axioms (and when they become your intuition) then probability will get a lot easier.
August 27th 2013, 10:52 PM #2
MHF Contributor
Sep 2012
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Ramsey Numbers R(5,5) = 43
I set out a program containing a genetic algorithm I wanted to test to search R(5,5) = 43 for a counter example.
The algorithm rates each graph with a fitness value which is equal to the total number of 5 clique sub graphs.
After 3 days of running I got down from average values of 1,400 to 2,500 to a lowest value of 848 currently.
Does anyone know where I could find the lowest known fitness value for the specific problem of 5,5,43 that someone else may have found? (The goal is to find 0 to reach a valid counter example and
create a new lower bound of 44.)
If someone has found lower values, it might be nice to take the chromosome for those graphs and plug them into my algorithm.
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Nonnegative Matrix Factorization and Recommendor Systems
Albert Au Yeung provides a very nice tutorial on non-negative matrix factorization and an implementation in python. This is based very loosely on his approach. Suppose we have the following matrix of
users and ratings on movies:
If we use the information above to form a matrix R it can be decomposed into two matrices W and H such that R~ WH'
where R is an n x p matrix of users and ratings
W = n x r user feature matrix
H = r x p movie feature matrix
Similar to principle components analysis, the columns in W can be interpreted to represent latent user features while the columns in H’ can be interpreted as latent movie features. This factorization
allows us to classify or cluster user types and movie types based on these latent factors.
For example, using the nmf function in R, we can decompose the matrix R above and obtain the following column vectors from H.
We can see that the first column vector ‘loads’ heavily on ‘military’ movies while the second feature more heavily ‘loads’ onto the ‘western’ themed movies. These vectors form a ‘feature space’ for
movie types. Each movie can be visualized in this space as being a member of a cluster associated with its respective latent feature.
If a new user gives a high recommendation to a movie belonging to one of the clusters created by the matrix factorization, other movies belonging to the same cluster can be recommended.
Yehuda Koren, Yahoo Research Robert Bell and Chris Volinsky, AT&T Labs—Research
IEEE Computer Society 2009
Matrix Factorisation: A Simple Tutorial and Implementation in Python
R Code:
# ------------------------------------------------------------------
# | PROGRAM NAME: R nmf example
# | DATE: 10/20/12
# | CREATED BY: MATT BOGARD
# | PROJECT FILE: /Users/wkuuser/Desktop/Briefcase/R Programs
# |----------------------------------------------------------------
# | PURPOSE: very basic example of a recommendor system based on
# | non-negative matrix factorization
# |
# |
# |------------------------------------------------------------------
# X ~ WH'
# X is an n x p matrix
# W = n x r user feature matrix
# H = r x p movie feature matrix
# get ratings for 5 users on 4 movies
x1 <- c(5,4,1,1)
x2 <- c(4,5,1,1)
x3 <- c(1,1,5,5)
x4 <- c(1,1,4,5)
x5 <- c(1,1,5,4)
R <- as.matrix(rbind(x1,x2,x3,x4,x5)) # n = 5 rows p = 4 columns
res <- nmf(R, 4,"lee") # lee & seung method
V.hat <- fitted(res)
print(V.hat) # estimated target matrix
w <- basis(res) # W user feature matrix matrix
dim(w) # n x r (n= 5 r = 4)
h <- coef(res) # H movie feature matrix
dim(h) # r x p (r = 4 p = 4)
# recommendor system via clustering based on vectors in H
movies <- data.frame(t(h))
features <- cbind(movies$X1,movies$X2)
title("Movie Feature Plot")
Created by Pretty R at inside-R.org
5 comments:
1. Hi
I think that package 'nmf' is no longer in CRAN but there is 'NNMF'.
so that would be more or less the same:
> library(NMFN)
> res <- nnmf(R, 4)
> w<- res$W
> h<- res$H
1. I think you are correct. I was somehow able to download it on my Mac and use it. Not sure why I was able to do that and it still seems to work. I also found some documentation and examples
using it here: http://nmf.r-forge.r-project.org/vignettes/NMF-vignette.pdf Its good to know there is another package out there though. What I'd really like to do when I get time is implement
this in R ( or SAS/IML) similar to the way Albert Au Yeung did with Python, just to get my hands a little dirtier. Not sure when I will have time though. Thanks for your comment!
2. Hi Matt,
I programmed the same stuff for making a lab session for my students, but does this R package handle missing values in the matrix you feed the NMF with - and sparsity is typically the
situation for recommendation systems, and we can't just fill the holes with 0s, which NMF will interpret are "bullshit film" ? There are python packages that do, but I couln't get hold of a R
I also use it on mac.
Thanks if any clue !
3. I mean, with the factorization on your matrix you can identify the latent structure and perform the "understanding" task on the data set, but I guess not the prediction task - it that correct
2. Marc, you are correct. It's really about interpretation, & it does not seem to handle missing data. That is one reason I want to recode Yeung's python into R or SAS IML if I can find the time ,
because if you read his example that I link to, it appears to do the things we both are really interested in.
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Describe the behavior
October 17th 2007, 06:45 PM
Describe the behavior
Describe the behavior to the left ad right of any vertical asymptote.
I'm using a computer other than my own (no access to mine), so I can't graph anything online.
October 17th 2007, 06:54 PM
well, you actually don't need the graph to tell what the behavior at the ends are. but if you want a graph, here it is. i assume the function is $\frac {x - 1}{x^2(x + 2)}$ ...(you should use
parentheses more efficiently)
October 17th 2007, 07:46 PM
well, you actually don't need the graph to tell what the behavior at the ends are. but if you want a graph, here it is. i assume the function is http://www.mathhelpforum.com/
math-he...7f04ec6c-1.gif ...(you should use parentheses more efficiently)
You're right on both accounts. You got the equation right.
Why did I want a graph? Oh, now I know why. Still, I need to be able to do it without a graph.
Right hand: lim as x--> infinity f(x)/g(x) = 1
Left hand: lim as x--> - infinity f(x)/g(x) = 1
For the right, after expanding the equation
$\frac {x - 1}{x^3 + 2x^2)}$
We have $x/x^3 = \frac {1}{x^2}$ as the right,
Guess: and $x^3$ as the left, though no understanding how to prove, if even right.
Augh, this should be easy!!
Does it have to do with this?
Definition: A polynomial function (real) can be expressed in the form:
Our teacher, for some odd reason, didn't teach this, but it looks important since it was the first thing on the "Introduction to Calculus Tutorial" sticky on the Calculus board.
October 17th 2007, 08:07 PM
You're right on both accounts. You got the equation right.
Why did I want a graph? Oh, now I know why. Still, I need to be able to do it without a graph.
Right hand: lim as x--> infinity f(x)/g(x) = 1
Left hand: lim as x--> - infinity f(x)/g(x) = 1
For the right, after expanding the equation
$\frac {x - 1}{x^3 + 2x^2)}$
We have $x/x^3 = \frac {1}{x^2}$ as the right,
Guess: and $x^3$ as the left, though no understanding how to prove, if even right.
Augh, this should be easy!!
Does it have to do with this?
Definition: A polynomial function (real) can be expressed in the form:
Our teacher, for some odd reason, didn't teach this, but it looks important since it was the first thing on the "Introduction to Calculus Tutorial" sticky on the Calculus board.
ok...what are you talking about? what is f(x) and g(x)?
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MAT123J1 Introductory Mathematics
MAT517J2 Fractals, Chaos & Complex Systems
This is an optional module for students on the final year of the B.Sc. (Hons) Mathematics with Computing.
Fractals, chaos and complex systems are relatively ‘new kids on the block’ of science. If you want very rough definitions; a fractal is an object which no matter how closely you ‘zoom in’ on it looks
the same or similar, and chaos & complexity are about simple things which look like they should be predictable be aren’t (chaos) or look like they couldn’t possibly give rise to anything interesting
but do (complex systems).
One of the beauties of these topics is that they are highly visual. For example, the picture at the top of this web page is of the Mandelbrot set, whose infinite structure results from the iteration
of the simple map (where z & c are complex numbers).
There are many useful applets and pieces of freeware available over the net, a few of which are-
· Applets to go with one of the recommended texts, The Computational Beauty of Nature by GW Flake (MIT Press 2000).
· Visions of Chaos is an excellent package. The free trial version has an unlimited life which still gives much of the functionality of the full version. One of the many things you can do with this
package is generate chaotic music.
· An interactive Koch curve generator.
· An interactive logistic map generator.
Information last updated September 2007
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Random Drug testing Question
April 27th 2006, 03:52 AM #1
Mar 2006
Random Drug testing Question
Random drug testing is to be carried out at the Beijing Summer Olympics in 2008. It is known that the drug test for Human Growth Hormone (HGH) possesses the following properties: If an athlete
has taken HGH the test will be positive with probability 0.95. If an athlete has not taken HGH the test will be negative with probability 0.98. At the Athens Olympics it was estimated that 0.5 of
one percent of all athletes took HGH.
(a) Represent the above situation on a Venn diagram. Carefully define the sample
space when doing so.
(b) What is the probability that an athlete who is randomly tested for HGH at Beijing
has not taken HGH, given their test is positive?
(c) What is the probability that an athlete who is randomly tested for HGH at Beijing
has taken HGH, given their test is negative?
Last edited by Maitham; April 27th 2006 at 03:56 AM.
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Algorithms: Big-Oh Notation
How time and space grow as the amount of data increases
It's useful to estimate the cpu or memory resources an algorithm requires. This "complexity analysis" attempts to characterize the relationship between the number of data elements and resource usage
(time or space) with a simple formula approximation. Many programmers have had ugly surprises when they moved from small test data to large data sets. This analysis will make you aware of potential
Dominant Term
Big-Oh (the "O" stands for "order of") notation is concerned with what happens for very large values of N, therefore only the largest term in a polynomial is needed. All smaller terms are dropped.
For example, the number of operations in some sorts is N^2 - N. For large values of N, the single N term is insignificant compared to N^2, therefore one of these sorts would be described as an O(N^2)
Similarly, constant multipliers are ignored. So a O(4*N) algorithm is equivalent to O(N), which is how it should be written. Ultimately you want to pay attention to these multipliers in determining
the performance, but for the first round of analysis using Big-Oh, you simply ignore constant factors.
Why Size Matters
Here is a table of typical cases, showing how many "operations" would be performed for various values of N. Logarithms to base 2 (as used here) are proportional to logarithms in other base, so this
doesn't affect the big-oh formula.
│ │ constant │ logarithmic │ linear │ │ quadratic │ cubic │
│ n │ O(1) │ O(log N) │ O(N) │ O(N log N) │ O(N^2) │ O(N^3) │
│ 1 │ 1 │ 1 │ 1 │ 1 │ 1 │ 1 │
│ 2 │ 1 │ 1 │ 2 │ 2 │ 4 │ 8 │
│ 4 │ 1 │ 2 │ 4 │ 8 │ 16 │ 64 │
│ 8 │ 1 │ 3 │ 8 │ 24 │ 64 │ 512 │
│ 16 │ 1 │ 4 │ 16 │ 64 │ 256 │ 4,096 │
│ 1,024 │ 1 │ 10 │ 1,024 │ 10,240 │ 1,048,576 │ 1,073,741,824 │
│ 1,048,576 │ 1 │ 20 │ 1,048,576 │ 20,971,520 │ 10^12 │ 10^16 │
Does anyone really have that much data?
It's quite common. For example, it's hard to find a digital camera that that has fewer than a million pixels (1 mega-pixel). These images are processed and displayed on the screen. The algorithms
that do this had better not be O(N^2)! If it took one microsecond (1 millionth of a second) to process each pixel, an O(N^2) algorithm would take more than a week to finish processing a 1 megapixel
image, and more than three months to process a 3 megapixel image (note the rate of increase is definitely not linear).
Another example is sound. CD audio samples are 16 bits, sampled 44,100 times per second for each of two channels. A typical 3 minute song consists of about 8 million data points. You had better
choose the write algorithm to process this data.
A dictionary I've used for text analysis has about 125,000 entries. There's a big difference between a linear O(N), binary O(log N), or hash O(1) search.
Best, worst, and average cases
You should be clear about which cases big-oh notation describes. By default it usually refers to the average case, using random data. However, the characteristics for best, worst, and average cases
can be very different, and the use of non-random data (often more realistic) data can have a big effect on some algorithms.
Why big-oh notation isn't always useful
Complexity analysis can be very useful, but there are problems with it too.
• Too hard to analyze. Many algorithms are simply too hard to analyze mathematically.
• Average case unknown. There may not be sufficient information to know what the most important "average" case really is, therefore analysis is impossible.
• Unknown constant. Both walking and traveling at the speed of light have a time-as-function-of-distance big-oh complexity of O(N). Altho they have the same big-oh characteristics, one is rather
faster than the other. Big-oh analysis only tells you how it grows with the size of the problem, not how efficient it is.
• Small data sets. If there are no large amounts of data, algorithm efficiency may not be important.
Benchmarks are better
Big-oh notation can give very good ideas about performance for large amounts of data, but the only real way to know for sure is to actually try it with large data sets. There may be performance
issues that are not taken into account by big-oh notation, eg, the effect on paging as virtual memory usage grows. Although benchmarks are better, they aren't feasible during the design process, so
Big-Oh complexity analysis is the choice.
Typical big-oh values for common algorithms
Here is a table of typical cases.
│ Type of Search │ Big-Oh │ Comments │
│ Linear search array/ArrayList/LinkedList │ O(N) │ │
│ Binary search sorted array/ArrayList │ O(log N) │ Requires sorted data. │
│ Search balanced tree │ O(log N) │ │
│ Search hash table │ O(1) │ │
Other Typical Operations
│ Algorithm │ array │ LinkedList │
│ │ ArrayList │ │
│ access front │ O(1) │ O(1) │
│ access back │ O(1) │ O(1) │
│ access middle │ O(1) │ O(N) │
│ insert at front │ O(N) │ O(1) │
│ insert at back │ O(1) │ O(1) │
│ insert in middle │ O(N) │ O(1) │
Sorting arrays/ArrayLists
Some sorting algorithms show variability in their Big-Oh performance. It is therefore interesting to look at their best, worst, and average performance. For this description "average" is applied to
uniformly distributed values. The distribution of real values for any given application may be important in selecting a particular algorithm.
│ Type of Sort │ Best │ Worst │ Average │ Comments │
│ BubbleSort │ O(N) │ O(N^2) │ O(N^2) │ Not a good sort, except with ideal data. │
│ Selection sort │ O(N^2) │ O(N^2) │ O(N^2) │ Perhaps best of O(N^2) sorts │
│ QuickSort │ O(N log N) │ O(N^2) │ O(N log N) │ Good, but it worst case is O(N^2) │
│ HeapSort │ O(N log N) │ O(N log N) │ O(N log N) │ Typically slower than QuickSort, but worst case is much better. │
Example - choosing a non-optimal algorithm
I had to sort a large array of numbers. The values were almost always already in order, and even when they weren't in order there was typically only one number that was out of order. Only rarely were
the values completely disorganized. I used a bubble sort because it was O(1) for my "average" data. This was many years ago when CPUs were 1000 times slower. Today I would simply use the library sort
for the amount of data I had because the difference in execution time would probably be unnoticed. However, there are always data sets which are so large that a choice of algorithms really matters.
Example - O(N^3) surprise
I once wrote a text-processing program to solve some particular customer problem. After seeing how well it processed the test data, the customer produced real data, which I confidently ran the
program on. The program froze -- the problem was that I had inadvertently used an O(N^3) algorithm and there was no way it was going to finish in my lifetime. Fortunately, my reputation was restored
when I was able to rewrite the offending algorithm within an hour and process the real data in under a minute. Still, it was a sobering experience, illustrating dangers in ignoring complexity
analysis, using unrealistic test data, and giving customer demos.
Same Big-Oh, but big differences
Altho two algorithms have the same big-oh characteristics, they may differ by a factor of three (or more) in practical implementations. Remember that big-oh notation ignores constant overhead and
constant factors. These can be substantial and can't be ignored in practical implementations.
Time-space tradeoffs
Sometimes it's possible to reduce execution time by using more space, or reduce space requirements by using a more time-intensive algorithm.
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PH 310 Relativity Fall 2004
Midterm exam and final exam
Worksheets and homework
Homework for Monday September 6, 2004
• Text problem 1-4, p. 30
• Text problem 1-11, p. 31
HW for Thursday September 9, 2004 HW for Thursday September 16, 2004
• Problems P1, P2, P3 (on attachments in email)
HW for Thursday September 23, 2004
• Obtain equations (3-14) from equations (3-13)
• Problem 3-5
• Problem 3-9
HW for Thursday September 30, 2004
• Prob 3-8 . Use the Minkowski graph I handed out. For x'=1, ct'= 1, calculate x and ct from LTE and show they are consistent with x and ct values from your diagram.
• Emailed problem
HW for Thursday October 21, 2004
• In a universe where c = 100 m/s, the speed limit is 28 m/s (around 63 mph). A VW beetle is travelling at 25 m/s when it is passed by a Lincoln Continental. The police officers observing this
situation notice that the observed lengths of the two cars are the same, and they know that the Continental has a rest length twice that of the VW beetle. They write a ticket for the Continental
at $20 for every m/s over the limit. What is the amount of the ticket?
• An electron is travelling in the lab with vx = 0.5 c and vy = 0.3c. It is being overtaken by a spaceship travelling at v = 0.7 c. Find the velocity components of the electron as seen by the
• The doppler frequency formula for electromagnetic radiation of frequency fo and received at frequency f by an object moving directly away from the source of the light is givens by f = fo sqrt
((1-v/c)/(1+v/c)) as indicated in the text on p. 137, Eq. 5-15. Check this out for a light sosurce emitting a burst of photons once each second, and a rocket moving away from the source at v =
0.8c. Try starting the earth and rocket clocks and initializing the coordinate systems when one burst passes the rocket, then figure out how much time the rocket clock will elapse until the next
burst of photons passes it. This will give you the apparent frequency as seen on the rocket. [This is an overtaking problem, where a photon overtakes the moving rocket. When you figure out how
long it will take between overtakings in the lab, you can convert to rocket time.]
• Flash Gordon and Earth are stationary with respect to one another separated by a distance of 1.2 x 10^10 m.. Two rockets R1 and R2 are moving together, and at x=x' = 0 = t = t' = 0 rocket R1
passes Earth. At 1.00 minutes on Flash's clock, Rocket R1 passes Flash. a) What time does R1's clock read when it passes Flash? When Earth's clock reads 1.00 minutes, rocket R2 passes Earth. b)
What time does R2's clock read when it passes Earth?
• In class, we worked out the fringe shift for the Fizeau experiment. Use this fringe shift formula and the data given on p. 48 to calculate the expected fringe shift, based on Einstein's velocity
addition formula (it will be slightly smaller than the experimental value).
HW for Thursday October 28
• Text Prob. 5-11 Neon is an orange-red color, say, around 610 nm. Where is the visible limit of human vision?
• Text Prob 5-12
• Text Prob 5-14
• Text Prob. 6-4
HW for November 8, 2004
• A beam of protons is accelerated to a high enough energy to create a minimum of 10 additional protom mass equivalents when striking a stationary proton target. Find the kinetic energy needed in
the lab for the bombarding protons
• A particle of rest mass 2.50 x 10^-24 kg overtakes an identical particle and they stick together after the collision. The leading particle travels at 0.6c while the trailing particle travels at
0.8 c. Find a) the total energy after the collision, b) the total momentum after the collision, c) the velocity after the collision, and d) the rest mass after the collision.
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Quiz 11: Second Order Linear Differential Equations
Question 1
If $y={e}^{2t}$ is a solution to $\frac{{d}^{2}y}{d{t}^{2}}-5\frac{dy}{dt}+ky=0$, what is the value of $k$?
Your answer is correct
Not correct. You may try again.
Question 2
If $y={e}^{3x}cosx$ is a solution to $\frac{{d}^{2}y}{d{x}^{2}}-6\frac{dy}{dx}+ky=0$, what is the value of $k$?
Your answer is correct
Not correct. You may try again.
Question 3
Which of the following is the general solution to $\frac{{d}^{2}y}{d{x}^{2}}+3\frac{dy}{dx}-10y=0$?
In each case, $A$ and $B$ are arbitrary constants.
Your answer is correct.
Not correct. Choice (b) is false.
Note the roots of the auxiliary equation are 2 and $-5$.
Not correct. Choice (c) is false.
Note the roots of the auxiliary equation are 2 and $-5$.
Not correct. Choice (d) is false.
Note that the roots of the auxiliary equation are 2 and $-5$.
Question 4
Which of the following is the general solution to $\frac{{d}^{2}y}{d{t}^{2}}+4\frac{dy}{dt}-4y=0$?
In each case, $A$ and $B$ are arbitrary constants.
Not correct. Choice (a) is false.
Note that the roots of the auxiliary equation are $-2±2\sqrt{2}$.
Not correct. Choice (b) is false.
Note that the roots of the auxiliary equation are $-2±2\sqrt{2}$.
Not correct. Choice (c) is false.
Note that the roots of the auxiliary equation are $-2±2\sqrt{2}$.
Your answer is correct.
Question 5
Which of the following are general solutions to $\frac{{d}^{2}x}{d{t}^{2}}-4\frac{dx}{dt}+13x=0$?
In each case, $A$ and $B$ are arbitrary constants. More than one option may be correct.
There is at least one mistake.
For example, choice (a) should be false.
There is at least one mistake.
For example, choice (b) should be true.
There is at least one mistake.
For example, choice (c) should be true.
There is at least one mistake.
For example, choice (d) should be false.
Your answers are correct
1. False.
2. True.
3. True.
4. False.
Question 6
Which of the following is the general solution to $\frac{{d}^{2}y}{d{x}^{2}}+10\frac{dy}{dx}+25y=0$?
In each case, $A$ and $B$ are arbitrary constants.
Not correct. Choice (a) is false.
Note that $A{e}^{-5x}+B{e}^{-5x}=\left(A+B\right){e}^{-5x}$, so that there is really only one arbitrary constant involved in this option.
Your answer is correct.
Not correct. Choice (c) is false.
Note that $A{e}^{5x}+B{e}^{5x}=\left(A+B\right){e}^{5x}$, so that there is really only one arbitrary constant involved in this option.
Not correct. Choice (d) is false.
The auxiliary equation has a repeated root of $-5$.
Question 7
Consider the differential equation $\frac{{d}^{2}y}{d{x}^{2}}-7y=0$.
Which of the following options is correct?
Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Your answer is correct.
Question 8
The general solution to $\frac{{d}^{2}z}{d{t}^{2}}+6\frac{dz}{dt}+9z=0$ is
Which of the following options is correct?
Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Your answer is correct.
Not correct. Choice (d) is false.
Question 9
Find the particular solution to $\frac{{d}^{2}y}{d{t}^{2}}-3\frac{dy}{dt}-4y=0$ satisfying $y\left(0\right)=0$ and ${y}^{\prime }\left(0\right)=5$.
Not correct. Choice (a) is false.
This function satisfies the initial conditions, but not the differential equation.
Not correct. Choice (b) is false.
This function satisfies the differential equation, and $y\left(0\right)=0$, but ${y}^{\prime }\left(0\right)=-5$.
Not correct. Choice (c) is false.
This functionsatisfies neither the differential equation, nor the initial conditions.
Your answer is correct.
Question 10
Find a solution to $\frac{{d}^{2}x}{d{t}^{2}}+\frac{dx}{dt}-2x=0$ which satisfies $x\left(0\right)=3$ and does not tend to infinity (or minus infinity) as $t\to \infty$.
Not correct. Choice (a) is false.
Try again. This function does not satisfy the differential equation.
Not correct. Choice (b) is false.
Try again. This function tends to $-\infty$ as $t\to \infty$.
Your answer is correct.
Not correct. Choice (d) is false.
Try again. This function does not satisfy the differential equation.
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Math Help
January 27th 2009, 11:52 AM #1
Jan 2009
i have two questions.
so one problem is:
i don't remember if you apply the 2 to the (3x) or if it's like algebra and you just divide it by 2 to cancel out. oh and then you'd divide log 3x by log x right to combine them because they have
the same base?
the other one is:
15log base 1/3(1/27) + logbase 1/4(256)= u
so does the 15 go to the 1/3 or 1/27 or do you just divide it.
and you can't combine them because they have different bases right?
sorry if this is confusing..
i have two questions.
so one problem is:
i don't remember if you apply the 2 to the (3x) or if it's like algebra and you just divide it by 2 to cancel out. oh and then you'd divide log 3x by log x right to combine them because they have
the same base?
the other one is:
15log base 1/3(1/27) + logbase 1/4(256)= u
so does the 15 go to the 1/3 or 1/27 or do you just divide it.
and you can't combine them because they have different bases right?
sorry if this is confusing..
2 log(3x) - log x = 2
2 (log 3 + log x ) - log x = 2
can you do it from here?
$<br /> log_{10}1000 = 3<br />$
thats because 10^3 = 1000
can you work out your second equation now?
thanks for the help.
i understand what you meant for the first problem and i was able to solve it
and i understand the example you gave me, but i don't understand how it relates to the problem
$<br /> log_{(\frac{1}{3})}\frac{1}{27} = 3 <br />$
$<br /> log_{(\frac{1}{4})}256 = -4<br />$
January 27th 2009, 01:00 PM #2
Dec 2008
January 27th 2009, 02:55 PM #3
Jan 2009
January 27th 2009, 08:27 PM #4
Dec 2008
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How to find left or right matrix invese
September 19th 2005, 06:07 AM #1
Aug 2005
How to find left or right matrix invese
I know that in order for a matrix to have a left inverse (/right inverse)it must be full column rank (/row rank).
I know how to obtain an inverese for a non-singular square matrix (RREF of [A|I]) but I do not know how to obtain left or right invereses for an arbitrary mxn matrix that has either full column
or row rank.
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Maintainer diagrams-discuss@googlegroups.com
This module defines a type of names which can be used for referring to locations within diagrams, and related types.
data AName Source
An atomic name is either a number or a string. Numeric names are provided for convenience in naming lists of things, such as a row of ten squares, or the vertices of a path.
IName Integer
SName String
Eq AName Note that equality on names does not distinguish between integers and their String representations.
Ord AName
Show AName
newtype Name Source
A (qualified) name is a (possibly empty) sequence of atomic names. Atomic names can be either numbers or arbitrary strings. Numeric names are provided for convenience in naming lists of things, such
as a row of ten squares, or the vertices of a path.
Eq Name
Ord Name
Show Name
Monoid Name
Qualifiable Name Names can be qualified by prefixing them with other names.
IsName Name
Action Name a Names don't act on anything else.
Action Name (NameMap v) A name acts on a name map by qualifying every name in it.
class IsName n whereSource
Instaces of IsName are things which can be converted to names.
IsName Int
IsName Integer
IsName String
IsName Name
class Qualifiable a whereSource
Instances of Qualifiable are things which can be qualified by prefixing them with a name.
Qualifiable Name Names can be qualified by prefixing them with other names.
NameMaps are qualifiable: if ns is a NameMap, then n |> ns is the same NameMap except with every name
Qualifiable (NameMap v) qualified by n.
(HasLinearMap v, InnerSpace v, OrderedField (Scalar v), Monoid m) => Qualifiable ( Diagrams can be qualified so that all their named points can now be referred to using the qualification
AnnDiagram b v m) prefix.
(||>) :: (IsName n, IsName m) => n -> m -> NameSource
Convenient operator for writing complete names in the form a1 |> a2 |> a3 ||> a4. In particular, n1 ||> n2 is equivalent to n1 |> toName n2.
Name maps
newtype NameMap v Source
A NameMap is a map from names to points, possibly with multiple points associated with each name.
NameMap (Map Name [Point v])
Action Name (NameMap v) A name acts on a name map by qualifying every name in it.
NameMaps form a monoid with the empty map as the identity, and map union as the binary operation. No information is ever lost: if two maps have the same name in their
Monoid (NameMap v) domain, the resulting map will associate that name to the union of the two sets of points associated with that name.
VectorSpace v => HasOrigin
(NameMap v)
Qualifiable (NameMap v) NameMaps are qualifiable: if ns is a NameMap, then n |> ns is the same NameMap except with every name qualified by n.
HasLinearMap v =>
Transformable (NameMap v)
Constructing name maps
Searching within name maps
lookupN :: IsName n => n -> NameMap v -> Maybe [Point v]Source
Look for the given name in a name map, returning a list of points associated with that name. If no names match the given name exactly, return all the points associated with names of which the given
name is a suffix.
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Frequency Session
Frequency sessions can be used to calculate frequencies and trends in frequencies. If you are interested in other statistics, see Statistics Calculated by SEER*Stat for an overview of statistics
which can be calculated in other types of sessions.
It is recommended that you work through a Frequency session in the following order.
Tutorials involving Frequency sessions are available on the SEER*Stat Web site at:
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Physics Forums - View Single Post - Hausdorff topological space M of dimension m
Originally posted by meteor
I have printed a notes about differential geometry, and it says:
-A C^oo differentiable structure on a locally Euclidean, Hausdorff topological space M of dimension m is a collection of coordinate systems F
Then it specifies the conditions that F must satisfy, but I'm a little lazy and won't write it
Then it says:
-A C^00 differentiable structure F which is maximal is called an atlas.
Then the text do not specify what it means by maximal. this is my doubt, what is a maximal C^00 differentiable structure
a set of charts satisfying those requirements that you alluded to is called maximal if any other set of charts which satisfies the conditions is a subset of this one.
i find it a little more comfortable to call any set of charts that satisfies the conditions an atlas. then the above sentence is a little easier to read:
an atlas is maximal if any other atlas on the space is a subset.
by Zorn's Lemma, any space with an atlas has a maximal atlas.
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Frequency Density and Histograms. How to Calculate Frequensity Density (F.d)
Learn how to calculate the frequency density by dividing the frequency by the group width.
A histogram is usually drawn when you have continuous data and the groups in the frequency table are of unequal size. To draw a histogram you will need to work out the frequency density. The
frequency density can be calculated by using the following formula:
Frequency density = frequency ÷ class width
The class width is basically the width of the group.
The frequency density goes up the y-axis, and the area of each bar will represent the frequency.
Let’s look at a couple of examples on working out the frequency densities.
Example 1
Here is a frequency table on the weight of 17 people. Calculate the frequency densities needed to draw a histogram.
First, work out the frequency density for the 60kg up to 70kg group. The class width here is 10.
Frequency density = 11 ÷ 10 = 1.1
Next, work out the frequency density for the 70kg up to 75kg group. The class width here is 5.
Frequency density = 1 ÷ 5 = 0.2
Finally, work out the frequency density for the 75kg up to 95kg group. The class width here is 20.
Frequency density = 5 ÷ 20 = 0.25
So the frequency densities are 1.1, 0.2 and 0.25.
Example 2
Here is another frequency table this time on the heights of a group of people. Calculate the frequency densities needed to draw a histogram.
First, work out the frequency density for the 140cm up to 160cm group. The class width here is 20.
Frequency density = 60 ÷ 20 = 3
Next, work out the frequency density for the 160cm up to 170cm group. The class width here is 10.
Frequency density = 18 ÷ 10 = 1.8
Finally, work out the frequency density for the 170cm up to 200cm group. The class width here is 30.
Frequency density = 45 ÷ 30 = 1.5
So the frequency densities are 1.1, 0.2 and 0.25.
Frequency density = frequency ÷ class width.
Just make sure that you divide the numbers in the correct order!
4 Responses to “Frequency Density and Histograms. How to Calculate Frequensity Density (F.d)”
1. On April 7, 2011 at 3:40 am
it was really helpfull , was explianed really well.
2. On February 24, 2012 at 10:58 am
its quite batty
3. On April 3, 2012 at 11:03 am
4. On February 4, 2013 at 4:29 am
very useful to me
Post Comment
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Calculus: Calculus in 20 minutes Video | MindBites
Calculus: Calculus in 20 minutes
About this Lesson
• Type: Video Tutorial
• Length: 18:15
• Media: Video/mp4
• Use: Watch Online & Download
• Access Period: Unrestricted
• Download: MP4 (iPod compatible)
• Size: 197 MB
• Posted: 06/26/2009
Taught by Professor Edward Burger, this lesson comes from a comprehensive Calculus course. This course and others are available from Thinkwell, Inc. The full course can be found at http://
www.thinkwell.com/student/product/calculus. The full course covers limits, derivatives, implicit differentiation, integration or antidifferentiation, L'Hopital's Rule, functions and their inverses,
improper integrals, integral calculus, differential calculus, sequences, series, differential equations, parametric equations, polar coordinates, vector calculus and a variety of other AP Calculus,
College Calculus and Calculus II topics.
Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut
He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has
won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of
America. In 2006, Reader's Digest named him in the "100 Best of America".
Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, "Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas" and of the textbook "The
Heart of Mathematics: An Invitation to Effective Thinking". He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math
journals, including The "Journal of Number Theory" and "American Mathematical Monthly". His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of
numbers, and the theory of continued fractions.
Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.
About this Author
2174 lessons
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Thinkwell lessons feature a star-studded cast of outstanding university professors: Edward Burger (Pre-Algebra through...
Recent Reviews
~ JudyLee1234
This is a fantastic way to kick off my review of calculus for my students! A great reminder of concepts they "should" know.
~ JudyLee1234
This is a fantastic way to kick off my review of calculus for my students! A great reminder of concepts they "should" know.
Calculus in 20 Minutes
Calculus in 20 Minutes Page [1 of 4]
All right, so, now we're going to try to do the impossible. We're going to try to do all of calculus in under twenty minutes. So, we have to work really, really fast, go through the whole course.
Let's begin.
The first two basic issues of calculus, two big questions, what are they? How do you find instantaneous rate of change? How do you find how things have changed instantly? And, then completely
separate from that, how do you find areas under curves? Two completely different questions, turns out answers are completely related. Let's see how they go.
Question one, instantaneous rate of change, what are we going to do? Well, what you do there is you first remember, what does rate mean? Well, rate is just change in distance over change in time.
It's as easy as that, distance equals rate times time, not a big deal. Okay, now what do you do with that? Well, if you graph a function that represents, sort of, distance against time, then what do
you notice? If you want to look at the change in time and the change in distance, what do you got? You actually got a slope of a line, right? This is slope, rise over run, change in distance over
change in time. Distance over time, you've got a slope. So, automatically, we see a really neat thing. We see that the average rate of change between two points is equal to the slope of the lines
connecting them. Well, that's really cool. By the way, what about lines? Maybe you forgot about lines? We'll remind you about lines. Okay, no problem, we can do lines. We can do lines. Lines, y - y
[1] = m(x - x[1]). This is the point slope form. All you need to give me is a point on the line, x1y1 and a slope, m. You give me those two pieces of information, I can always write down the line
uniquely. Always, always, always. Never forget it. Okay, fine, now we're back to here. Now, we can find average rate and what do we see? Well, in fact, a line that touches the curve at two points, is
sometimes called a ces line. So in fact, we just discovered that the average rate change in distance over change in time is equal to the slope of the ces line, cool. So, all you have to do, want to
find average rate? Connect the two points of the line, find the slope, you got the average rate. No biggie. But, that's not what we want, we want instantaneous rate, so how do you do that? Well, what
would that be? Well, if I [inaudible] instantaneous rate here, what would I do? I'd make that ces line closer and closer and closer, bring those points together and look what I'm converging to? I'm
coming to a tangent line. Wow. Instantaneous rate of change equals the slope of the tangent line. So, what we want is the slope of tangent line. Well, what's the slope of the tangent line? It's
change in y over change in x, change in distance over change in time, but now, the change in time from that point to itself is 0. 0/0, so I'm getting 0/0, which is complete garbage and that is a big
problem. Our first major problem of the course. Okay, so, what do we do?
So, we can that. Distance equals rate times time, we're going to can that. Instantaneous rate of change, I really want that, so what do we do? Well, how do we get that 0/0 problem to go away? The
answer is, we inch up to it. We just approach the 0 point, and what would that look like? Well, I'm going to remind you what you've done when you were a little teeny kid. And when you were a little
teeny kid, this is what you were looking at, you were looking at a value of a function at a point. Value of a function at a point. Okay, not a big deal, there it is. F(a), it's that point. But you
don't know anything else but the function. You don't know what's going around around there, because all you're looking at is f(a), you open it up, whew hoo, the function could be quite interesting,
who knows? Okay, but, now what I invite us to do and what calculus invites us to do, is to look at the function this way, cover up that point and look at everything else. Open the window look
outside. That's what calculus is. And what you see here, is we can see what things are approaching, and we can actually determine the idea of a limit. The limit is what things are approaching. We
don't care about what actually happens at that point, only what things approaching. Armed with the idea of a limit, what can we do? Well, now we can return to the question and figure out, let's take
the limit as delta t goes to 0. What do we get? We get 0/0: that is called an indeterminate form when you get 0/0. So what do you do? You got to do some algebraic gymnastics. You got to factor the
top and cancel with the bottom. You can try to multiply by the cognizant. You can try to combine the fractions. There's always tricks of the trade to actually reduce this to something that you can
actually find. So, you find the limit, okay. Well, once you find the limit, and you take the limit as delta t goes to 0, what do you get? You get the answer to the question, how do you find
instantaneous rate of change, the answer is what we call, the derivative, and what's the derivative? It's the limit as delta x goes to 0, or f(x) plus delta x minus f(x)/delta x. Looks pretty
confusing, doesn't it? It's just rate, it's distance over time, but now I'm letting time go to 0. Not a big deal. So, that's the derivative, bingo, we're done with the first question.
And so now, what do we see? What we see now is we come back to here, and the derivative, in fact, gives us the slope of the tangent line. Well, that's really cool, if you want to find the slope of a
tangent line ever in life, you just take the derivative, and that gives you the slope of the tangent once you evaluate at the point you want. Great, but for free, we answer the first question,
because remember the derivative also represents the instantaneous rate of change. Do you want to find out how things are changing? No big deal, you take a derivative, plug in and that would tell you
how things are changing at that instant. Okay, great, well, now we know all about how to graph these things, how to look at these things, the derivative, velocity, boom, boom, boom, we got all the
way. Now, let's take a look at some applications. What can we do with this?
Well, how would you take derivatives of complicated functions? Well, if you've got a product, use the product rule. Remember the product rule, don't memorize the formula; memorize the chant. First,
times the derivative of the second, plus the second times the derivative of the first. So, the derivative of a product is this, it's not the product of derivatives, you got to use the product rule.
We've got five minutes, listen up here folks, I've got to move fast today. I'll move faster. What if you have a quotient, well, then use the quotient rule. So, what's the quotient rule say? If you
had a quotient, you take the bottom times the derivative of the top minus the top times the derivative of the bottom, over the bottom squared. That's the quotient rule, that's what you use when you
have a derivative of a quotient. Okay, great, no problem. Now, what about if I had a really complicated function? What if you got a function that looks like this? It's got insides; it's got guts
right in there. See, you got something this, and you want to take the derivative of that, what do you do? Well, you got to use the chain rule, folks. This is a thing that you can chain together,
there's an inside, there's a blop right here, there's a whole big blop there, and then you got an outside. So, take the derivative of the outside, the derivative of sin(blop) is actually cos(blop),
so it's cos(blop) and what's the blop? The blop is going to be 3x^3 + 1, and then you multiply that by the derivative of the inside, and the derivative of that turns out to be just, let's see, 9x^2 +
0. So, there's the derivative using the chain rule. The idea is to peel off, like an onion, just peel off, keep peeling off the outside until you get to the inside, always remember though, when you
take the derivative, if you have sin(blop), the derivative is cos of the blop. Don't put the derivative in there, put the blop and multiply by the derivative of the inside. That's the key to the
chain rule, now we got the chain rule.
Okay, so what about when you have functions that aren't functions? Like what if you have things that are relations? Like x^2 + y^2 = 1? Like a circle. How do you differentiate that? How do you find
dydx there? Well, the answer is we use something called implicit differentiation. Implicit differentiation, how does that work? Well, you've got to remember that dydx, that's an object, that's a
noun, and ddx is a verb, it's a commandment. Take the derivative with respect to x. So, you differentiate with respect to x, and what do you see? Well, you see something that looks like this. What
you do is, you say okay, I'll take dx of x^2 + y^2 = 1, the derivative of x^2, with respect to x, is just 2x, not a big deal. The derivative of y^2, remember how I think about this. I think about
this as clumping all this together, and I see this is a blop squared, so I actually used the chain rule, which we just developed, and the chain rule says the derivative of blop squared is 2 blop, and
then I multiply that by the derivative of the blop, which is the derivative of y with respect to x, that's called dydx, folks. The derivative of 1 is 0. And now, you can actually solve this for dydx,
by bringing this to the other side, that would be a -2x, you divide by the 2y, and you see dydx = -x/y, and there's the answer. That's implicit differentiation, just go right through and
differentiate implicitly, when you have a relationship, you can still find the derivative. And we're making progress here, folks. We are cutting through this stuff.
Well, now that you have derivatives, what can you do with that? Well, if you think about as velocity, you get instantaneous velocity. If you take the derivative of velocity, you actually get the
change in velocity, which is acceleration. So, we get acceleration now, we get velocity. Acceleration is just a second derivative of the position. So, then you have to take derivatives upon
derivatives upon derivatives as many derivatives as you want. So, it's great, we can do that, so how do I do derivatives? No problem. What can you use these things for? Okay, we know it's true for
velocity, we can use if for velocity, what else we use it for? Well, it turns out you can use it for linear approximation. Suppose you got some wacko function, you got the wacko function like this.
Woo, and you actually want to figure out the value right here. Right here and you don't know, you don't know what that value is. But you know near by there's a point that you can actually compute.
So, what do you do? You find the tangent line approximation, because you remember that the tangent line closely emulates the activity of the function. The tangent line closely emulates the activity
of what the function's doing, and it looks the same there, so you find the equation of the tangent line, and then plug in the mysterious point then you're all set because you can approximate the
value by plugging in the tangent line. So, how does that look? Well, that's call linear approximation, here's the formula, but don't bother memorizing it, just think about it, what you've got to do
is you've got to find the equation of the line, that's tangent at the known point which is x, here in this case. And this is going to be x + delta x, the known point plus a little teeny off set, so
it's approximately equal to the derivative times the change in x plus the function. So, that's it that's the whole thing right there, linear approximation. Allows you to actually compute things,
computers actually help this way, computers know calculus, everyone knows calculus, you guys know calculus.
Okay, now what else do you do? Well, suppose that a derivative were to be 0? Well, how could that possibly happen? That could possibly happen because maybe the function goes like this, and I see the
tangent over here has slope 0. Or maybe the function goes like this, and I see the tangent has slope 0. In particular, if the tangent equals 0, maybe we have a max or min. Also, maybe the slope or
the tangent doesn't exist? Like if we have a wave kind of thing. A cusp, very pretty cusps like a wave. Well, that might be a max, that might be a min. So, in particular you could find out when
objects are maximized or minimized. You can find the maxima or the minima very easily by using calculus. What do you do? You take a derivative and you see where equals 0, or where the derivative
doesn't exist, but the function does. Those, give you candidates for possible max and min. And what can you use that for? Well, you can do all sorts of max and min problems. You want to maximize
profits; you want to minimize costs? You got--oh my God! Five minutes, so there's five minutes left, they are trying to fool me here folks, but I'm not going to go for it. Five minutes left, okay, so
you want to maximize cost you want to minimize cost, you want to maximize profit? You want to maximize area; you want to minimize volume? Whatever it is, set up the problem really carefully, figure
exactly what you want to optimize, take the derivative, set it equal to 0, solve, find out where the derivative's undefined and you've got it made. Really, not, not, not a big deal. However, you
should always remember, and never forget the fundamental method of solving problems. So remember how you solve all of life's problems? The first thing you have to do is understand what you're being
asked. You can't answer a question that you don't understand. The next thing you do after you understand what you're suppose to find, what you're being asked, is figure what you know, list every
single thing that you know, every single fact; maybe it's frivolous, maybe you don't use it. Who cares, write it down, understand it make it your own. And the last thing is, take the information that
you know and see a relationship between that and the thing that you seek. Try to find a connection. Once you got the connection, then you're on the road to actually finding the solution. That is the
simple method to finding any single answer to any single problem.
Another application if you think about derivatives is the rate. We related rate. Suppose for example that you actually have a ladder, for example, it's falling down. The ladder is falling and you
don't want to be sued, but the only thing you do know is how fast the bottom is falling. You want to know how fast the top is falling. What you need to do there is if you know this rate, you can find
that rate by linking them up with a connection. And in this case, the connection would actually by the Pythagorean theorem. You can take the derivative with respect to time, because here you see the
variable, the thing that's independent that's always changing, is time. Time keeps on ticking into the future. So you can find this, you actually solve this, take the derivative using implicit
differentiation, differentiate with the respect to time, and plug in what you know, how this is changing and that tells you how this is changing. Pretty cool. That's called related rate. Suppose for
example, you drop a stone into a very, very still pool? You have a ripple effect, those make consensus circles, and they are getting larger, and you know how fast the radius is changing, you can find
how fast the area is changing because you have a connection between area and rate, area equals pr^2, so there you go, related rate. You know how one rate is changing; you can find how the related
rate is changing. Okay, what else can you use a derivative for?
Well, the other thing you can do is actually graph really, really accurate pictures of functions. Finally, you can figure out that a parabola looks really pretty and bowl like, like this, and it's
not something real exotic, that's just a nice pretty bowl. How do you do it? Well, you just start taking derivatives and analyzing things. First, you find the critical point. Those are the points
where the derivative either equals 0 or the derivatives is undefined but the function is defined. So, for example, we can do these examples right here, you'll notice that the derivative equals 0
here, the tangent is horizontal right there, the tangent is horizontal here, and this example, the tangent is horizontal here, and then where is the derivative of non-existence? The derivative
doesn't exist here and the derivative doesn't exist there. Those are tangents for max or min; those are called critical points. Then what do you do with those things? Well, you set them up on a
little number line on the x-axis and you look at the integrals all around it and you see whether the derivative is positive or negative. If the derivative positive, then that means slopes are
positive, so the function must be increasing. If a derivative is negative, then that means the function must be decreasing. So, you can see that the function is, for example here, is decreasing,
decreasing, decreasing, then increasing, then you can see it's increasing, increasing, increasing, then decreasing, then increasing. So, you can see where it's going up or it's going down by the sign
of the first derivative. That also determines whether you have max's or min's anywhere, and that's called the first derivative test. Now, how do you figure out the curvature? The curvature is given
by the rate of change of the derivatives. How the derivative is changing, so what you do there is you take the derivative of the derivative. So, you look at the second derivative, and with the second
derivative and with the second derivative of 0, are potential points of inflexion, points where the concavity changes. This is concave up; it's curving upward. This is concave down; this is concave
up. So, the cup is sitting up, the cup is sitting down, concave down. So, here you would see the second derivative is positive and the second derivative is negative, it changes here, now it's sitting
up, positive, positive, we're concave up. Here, we're concave up, the second derivative is positive. Now you here we see concave down, second derivative is negative and second derivative here's also
negative. This is a cusp point, the derivative doesn't exist there, this is a point of inflexion, this is a point of inflexion, this is a point of inflexion, and that's a minimum. So, just by taking
derivatives and second derivatives, you can actually figure out and graph a very accurate sketch of even very complicated almost scary looking functions. By the way that fractional exponent, expect
cusps. That's my warning for the day.
Okay, now, if you've got really exotic functions, that have denominators, then actually you may have asymptotes. So, don't forget that a vertical asymptotes is where the function after you simplify
it, the bottoms equal to 0. So, whatever the bottom equals 0, after you simplify and reduce, those are going to give you, your vertical asymptote. Horizontal asymptotes, you take the limit as the x
goes off to infinity, as you go off into the horizon, and you see what y value you are going to try to land to, if you are landing somewhere, then you know you've got a horizontal asymptote, and it's
y equals that value. So, you can put in the asymptotes and you can do all the other calculus, get the curvature, see exactly what the beautiful picture looks like. And that was the end of
differential calculus. Great, not problem, now what?
Well, now we move and look at the exact same thing we just did backwards. So, we look at math jeopardy. Oh, my goodness, we only have five minutes left. So, math jeopardy, here we go, so the idea is
here we go, if I tell you what the derivative is, how can you find the function whose derivative is that? Well, this is a notion of an anti-derivative. So, how do you find that anti-derivative? Well,
we set up formulas for that. If you want to find the anti derivative of x^n, it would be x^n + 1/n + 1. Take the derivative of that and see you get x^n, unless n = -1, then you're looking at the
integral of 1/x, and what's the integral of 1/x? Well, it's a natural log of the absolute value of x because the derivative of natural log, we already saw was 1/x, so great. Now, how can you find
exotic integrals? Well, remember that ddx represents differentiating with respect to x, and so therefore, the integral with respect to x, represents to integrate with respect to x. Now, if you were
very complicated thing there, with an inside and outside, you might be able to untangle that, which potentially was made by the chain rule, by using substitution. Let u equal some big blop and the
big blop derivative should appear somewhere else in your integral. If you got that, it sounds like a good candidate for udu substitution. And then you've got to change the dx to du by taking
derivatives and seeing what the u equals in terms of dx. So, that's the udu substitution. You got that going on here, and now what can you do? You can take that and we can study the motion again.
Now, I can give you acceleration. If you integrate, you get velocity, if you integrate again, you get position. So, vertical motion, not a big deal. Anything that moves, anything at all that moves,
we can now analyze. It's not a problem anymore folks. In this movement, we can anti-differentiate and figure out what it is. Not a problem. Now, where does this leave us, it brings us back to the
very first question of the course, which was how do you find the areas under curves? We have to answer that; we haven't done it yet. It turns out the surprising answer is that it's the fundamental
theorem of calculus. And the idea is if you want to find the area under this curve from a to b, right here, you want to find that area, then how do you do it? It turns out that if that function is
called, let's say f(x), then all you do is integrate from a to b, f(x)dx, because you are summing up little rectangles in here that are base, small change in x, base times height, which is a function
you use from a to b. And this equals F(b) - F(a), well, what's F? That's the anti-derivative. So, if you take the derivative of F, you actually get little f(x). So, just find the anti-derivative,
plug in the big point, plug in the small point, subtract that will always give you the area under the curve. You can look at areas and more exotic things. For example, if the thing actually goes like
this, then actually this is actually not very x-easy, the rectangles aren't very clear, the rectangles change from being going from green to green, over to green to orange, then orange to orange.
It's not very uniform, however, if you put the rectangles in this way, and stack them this way, now you're summing with respect to y. And so, here, you would actually sum this with respect to y. So,
you would integrate this, dy, and put the rectangles in this way, and you would put the rectangle in that way, and stack you would stack from low to high and you would stack the rectangle like this.
Okay, that is all of calculus; I did it under twenty minutes. That's what it is, go back think about it, have fun with it, congratulations folks, you just finished Calculus I. Have a good time.
Celebrate, good luck on the final, bye.
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Game theory
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Game theory is a branch of applied mathematics that studies strategic situations where players choose different actions in an attempt to maximize their returns. First developed as a tool for
understanding economic behavior, game theory is now used in many diverse academic fields, ranging from biology to philosophy. Game theory saw substantial growth and its first formalization by John
von Neumann before and during the Cold War, mainly due to its application to military strategy, most notably to the concept of mutual assured destruction. Beginning in the 1970s, game theory has been
applied to animal behavior, including species' development by natural selection. Because of interesting games like the prisoner's dilemma, in which mutual self-interest hurts everyone, game theory
has been used in political science, ethics and philosophy. Finally, game theory has recently drawn attention from computer scientists because of its use in artificial intelligence and cybernetics.
Although similar to decision theory, game theory studies decisions that are made in an environment where various players interact. In other words, game theory studies choice of optimal behavior when
costs and benefits of each option are not fixed, but depend upon the choices of other individuals.
Representation of games
The games studied by game theory are well-defined mathematical objects. A game consists of a set of players, a set of moves (or strategies) available to those players, and a specification of payoffs
for each combination of strategies. There are two ways of representing games that are common in the literature.
See also List of games in game theory.
Normal form
A normal form game
│ │ Player 2 chooses left │ Player 2 chooses right │
│ Player 1 chooses top │ 4, 3 │ -1, -1 │
│ Player 1 chooses bottom │ 0, 0 │ 3, 4 │
Main article: Normal form game
The normal (or strategic form) game is a matrix which shows the players, strategies, and payoffs (see the example to the right). Here there are two players; one chooses the row and the other chooses
the column. Each player has two strategies, which are specified by the number of rows and the number of columns. The payoffs are provided in the interior. The first number is the payoff received by
the row player (Player 1 in our example); the second is the payoff for the column player (Player 2 in our example). Suppose that Player 1 plays top and that Player 2 plays left. Then Player 1 gets 4,
and Player 2 gets 3.
When a game is presented in normal form, it is presumed that each player acts simultaneously or, at least, without knowing the actions of the other. If players have some information about the choices
of other players, the game is usually presented in extensive form.
Extensive form
Main article: Extensive form game
An extensive form game
Extensive form games attempt to capture games with some important order. Games here are presented as trees (as pictured to the left). Here each vertex (or node) represents a point of choice for a
player. The player is specified by a number listed by the vertex. The lines out of the vertex represent a possible action for that player. The payoffs are specified at the bottom of the tree.
In the game pictured here, there are two players. Player 1 moves first and chooses either F or U. Player 2 sees Player 1's move and then chooses A or R. Suppose that Player 1 chooses U and then
Player 2 chooses A, then Player 1 gets 8 and Player 2 gets 2.
Extensive form games can also capture simultaneous-move games as well. Either a dotted line or circle is drawn around two different vertices to represent them as being part of the same information
set (i.e., the players do not know at which point they are).
Types of games
Symmetric and asymmetric
Main article: Symmetric game
An asymmetric game
│ │ E │ F │
│ E │ 1, 2 │ 0, 0 │
│ F │ 0, 0 │ 1, 2 │
A symmetric game is a game where the payoffs for playing a particular strategy depend only on the other strategies employed, not on who is playing them. If the identities of the players can be
changed without changing the payoff to the strategies, then a game is symmetric. Many of the commonly studied 2×2 games are symmetric. The standard representations of chicken, the prisoner's dilemma,
and the stag hunt are all symmetric games. ^[1]
Most commonly studied asymmetric games are games where there are not identical strategy sets for both players. For instance, the ultimatum game and similarly the dictator game have different
strategies for each player. It is possible, however, for a game to have identical strategies for both players, yet be asymmetric. For example, the game pictured to the right is asymmetric despite
having identical strategy sets for both players.
Zero sum and non-zero sum
Main article: Zero-sum
A Zero-Sum Game
│ │ A │ B │
│ A │ 2, −2 │ −1, 1 │
│ B │ −1, 1 │ 3, −3 │
In zero-sum games the total benefit to all players in the game, for every combination of strategies, always adds to zero (or more informally put, a player benefits only at the expense of others).
Poker exemplifies a zero-sum game (ignoring the possibility of the house's cut), because one wins exactly the amount one's opponents lose. Other zero sum games include matching pennies and most
classical board games including go and chess. Many games studied by game theorists (including the famous prisoner's dilemma) are non-zero-sum games, because some outcomes have net results greater or
less than zero. Informally, in non-zero-sum games, a gain by one player does not necessarily correspond with a loss by another.
It is possible to transform any game into a zero-sum game by adding an additional dummy player (often called "the board"), whose losses compensate the players' net winnings.
Simultaneous and sequential
Main article: Sequential game
Simultaneous games are games where both players move simultaneously, or if they do not move simultaneously, the later players are unaware of the earlier players' actions (making them effectively
simultaneous). Sequential games (or dynamic games) are games where later players have some knowledge about earlier actions. This need not be perfect knowledge about every action of earlier players;
it might be very little information. For instance, a player may know that an earlier player did not perform one particular action, while she does not know which of the other available actions the
first player actually performed.
The difference between simultaneous and sequential games is captured in the different representations discussed above. Normal form is used to represent simultaneous games, and extensive form is used
to represent sequential ones.
Perfect information and imperfect information
A game of imperfect information (the dotted line represents ignorance on the part of player 2)
Main article: Perfect information
An important subset of sequential games consists of games of perfect information. A game is one of perfect information if all players know the moves previously made by all other players. Thus, only
sequential games can be games of perfect information, since in simultaneous games not every player knows the actions of the others. Most games studied in game theory are imperfect information games,
although some interesting games are games of perfect information, including the ultimatum game and centipede game. Many popular games are games of perfect information including chess, go, and
Perfect information is often confused with complete information, which is a similar concept. Complete information requires that every player know the strategies and payoffs of the other players but
not necessarily the actions.
Infinitely long games
Main article: Determinacy
For obvious reasons, games as studied by economists and real-world game players are generally finished in a finite number of moves. Pure mathematicians are not so constrained, and set theorists in
particular study games that last for infinitely many moves, with the winner (or other payoff) not known until after all those moves are completed.
The focus of attention is usually not so much on what is the best way to play such a game, but simply on whether one or the other player has a winning strategy. (It can be proved, using the axiom of
choice, that there are games—even with perfect information, and where the only outcomes are "win" or "lose"—for which neither player has a winning strategy.) The existence of such strategies, for
cleverly designed games, has important consequences in descriptive set theory.
Uses of game theory
Games in one form or another are widely used in many different academic disciplines.
Economics and business
Economists have used game theory to analyze a wide array of economic phenomena, including auctions, bargaining, duopolies and oligopolies, social network formation, and voting systems. This research
usually focuses on particular sets of strategies known as equilibria in games. These "solution concepts" are usually based on what is required by norms of rationality. The most famous of these is the
Nash equilibrium. A set of strategies is a Nash equilibrium if each represents a best response to the other strategies. So, if all the players are playing the strategies in a Nash equilibrium, they
have no incentive to deviate, since their strategy is the best they can do given what others are doing.
The payoffs of the game are generally taken to represent the utility of individual players. Often in modeling situations the payoffs represent money, which presumably corresponds to an individual's
utility. This assumption, however, can be faulty.
A prototypical paper on game theory in economics begins by presenting a game that is an abstraction of some particular economic situation. One or more solution concepts are chosen, and the author
demonstrates which strategy sets in the presented game are equilibria of the appropriate type. Naturally one might wonder to what use should this information be put. Economists and business
professors suggest two primary uses.
A three stage Centipede Game
The first use is to inform us about how actual human populations behave. Some scholars believe that by finding the equilibria of games they can predict how actual human populations will behave when
confronted with situations analogous to the game being studied. This particular view of game theory has come under recent criticism. First, it is criticized because the assumptions made by game
theorists are often violated. Game theorists may assume players always act rationally to maximize their wins (the Homo economicus model), but real humans often act either irrationally, or act
rationally to maximize the wins of some larger group of people (altruism). Game theorists respond by comparing their assumptions to those used in physics. Thus while their assumptions do not always
hold, they can treat game theory as a reasonable scientific ideal akin to the models used by physicists. However, additional criticism of this use of game theory has been levied because some
experiments have demonstrated that individuals do not play equilibrium strategies. For instance, in the Centipede game, Guess 2/3 of the average game, and the Dictator game, people regularly do not
play Nash equilibria. There is an ongoing debate regarding the importance of these experiments. ^[2]
Alternatively, some authors claim that Nash equilibria do not provide predictions for human populations, but rather provide an explanation for why populations that play Nash equilibria remain in that
state. However, the question of how populations reach those points remains open.
Some game theorists have turned to evolutionary game theory in order to resolve these worries. These models presume either no rationality or bounded rationality on the part of players. Despite the
name, evolutionary game theory does not necessarily presume natural selection in the biological sense. Evolutionary game theory includes both biological as well as cultural evolution and also models
of individual learning (for example, fictitious play dynamics).
The Prisoner's Dilemma
│ │ Cooperate │ Defect │
│ Cooperate │ 2, 2 │ 0, 3 │
│ Defect │ 3, 0 │ 1, 1 │
On the other hand, some scholars see game theory not as a predictive tool for the behavior of human beings, but as a suggestion for how people ought to behave. Since a Nash equilibrium of a game
constitutes one's best response to the actions of the other players, playing a strategy that is part of a Nash equilibrium seems appropriate. However, this use for game theory has also come under
criticism. First, in some cases it is appropriate to play a non-equilibrium strategy if one expects others to play non-equilibrium strategies as well. For an example, see Guess 2/3 of the average.
Second, the Prisoner's Dilemma presents another potential counterexample. In the Prisoner's Dilemma, each player pursuing his own self-interest leads both players to be worse off than had they not
pursued their own self-interests. Some scholars believe that this demonstrates the failure of game theory as a recommendation for behavior.
│ │ Hawk │ Dove │
│ Hawk │ (V-C)/2, (V-C)/2 │ V, 0 │
│ Dove │ 0, V │ V/2, V/2 │
Unlike economics, the payoffs for games in biology are often interpreted as corresponding to fitness. In addition, the focus has been less on equilibria that correspond to a notion of rationality,
but rather on ones that would be maintained by evolutionary forces. The most well-known equilibrium in biology is known as the Evolutionary stable strategy or (ESS), and was first introduced by John
Maynard Smith (described in his 1982 book). Although its initial motivation did not involve any of the mental requirements of the Nash equilibrium, every ESS is a Nash equilibrium.
In biology, game theory has been used to understand many different phenomena. It was first used to explain the evolution (and stability) of the approximate 1:1 sex ratios. Ronald Fisher (1930)
suggested that the 1:1 sex ratios are a result of evolutionary forces acting on individuals who could be seen as trying to maximize their number of grandchildren.
Additionally, biologists have used evolutionary game theory and the ESS to explain the emergence of animal communication (Maynard Smith & Harper, 2003). The analysis of signaling games and other
communication games has provided some insight into the evolution of communication among animals.
Finally, biologists have used the hawk-dove game (also known as chicken) to analyze fighting behavior and territoriality.
Computer science and logic
Game theory has come to play an increasingly important role in logic and in computer science. Several logical theories have a basis in game semantics. In addition, computer scientists have used games
to model interactive computations.
Political science
Research in political science has also used game theory. A game-theoretic explanation for the democratic peace is that the public and open debate in democracies send clear and reliable information
regarding the intentions to other states. In contrast, it is difficult to know the intentions of nondemocratic leaders, what effect concessions will have, and if promises will be kept. Thus there
will be mistrust and unwillingness to make concessions if at least one of the parties in a dispute is a nondemocracy.[3]
Game theory has been put to several uses in philosophy. Responding to two papers by W.V.O. Quine (1960, 1967), David Lewis (1969) used game theory to develop a philosophical account of convention. In
so doing, he provided the first analysis of common knowledge and employed it in analyzing play in coordination games. In addition, he first suggested that one can understand meaning in terms of
signaling games. This later suggestion has been pursued by several philosophers since Lewis (Skyrms 1996, Grim et al. 2004).
The Stag Hunt
│ │ Stag │ Hare │
│ Stag │ 3, 3 │ 0, 2 │
│ Hare │ 2, 0 │ 2, 2 │
In ethics, some authors have attempted to pursue the project, begun by Thomas Hobbes, of deriving morality from self-interest. Since games like the Prisoner's Dilemma present an apparent conflict
between morality and self-interest, explaining why cooperation is required by self-interest is an important component of this project. This general strategy is a component of the general social
contract view in political philosophy (for examples, see Gauthier 1987 and Kavka 1986). ^[4]
Finally, other authors have attempted to use evolutionary game theory in order to explain the emergence of human attitudes about morality and corresponding animal behaviors. These authors look at
several games including the Prisoner's Dilemma, Stag hunt, and the Nash bargaining game as providing an explanation for the emergence of attitudes about morality (see, e.g., Skyrms 1996, 2004; Sober
and Wilson 1999).
History of game theory
The first known discussion of game theory occurred in a letter written by James Waldegrave in 1713. In this letter, Waldegrave provides a minimax mixed strategy solution to a two-person version of
the card game le Her. It was not until the publication of Antoine Augustin Cournot's Researches into the Mathematical Principles of the Theory of Wealth in 1838 that a general game theoretic analysis
was pursued. In this work Cournot considers a duopoly and presents a solution that is a restricted version of the Nash equilibrium.
Although Cournot's analysis is more general than Waldegrave's, game theory did not really exist as a unique field until John von Neumann published a series of papers in 1928. These results were later
expanded in the 1944 book The Theory of Games and Economic Behavior by von Neumann and Oskar Morgenstern. This profound work contains the method for finding optimal solutions for two-person zero-sum
games. During this time period, work on game theory was primarily focused on cooperative game theory, which analyzes optimal strategies for groups of individuals, presuming that they can enforce
agreements between them about proper strategies.
In 1950, the first discussion of the Prisoner's dilemma appeared, and an experiment was undertaken on this game at the RAND corporation. Around this same time, John Nash developed a definition of an
"optimum" strategy for multiplayer games where no such optimum was previously defined, known as Nash equilibrium. This equilibrium is sufficiently general, allowing for the analysis of
non-cooperative games in addition to cooperative ones.
Game theory experienced a flurry of activity in the 1950s, during which time the concepts of the core, the extensive form game, fictitious play, repeated games, and the Shapley value were developed.
In addition, the first applications of Game theory to philosophy and political science occurred during this time.
In 1965, Reinhard Selten introduced his solution concept of subgame perfect equilibria, which further refined the Nash equilibrium (later he would introduce trembling hand perfection as well). In
1967, John Harsanyi developed the concepts of complete information and Bayesian games. He, along with John Nash and Reinhard Selten, won the Bank of Sweden Prize in Economic Sciences in Memory of
Alfred Nobel in 1994.
In the 1970s, game theory was extensively applied in biology, largely as a result of the work of John Maynard Smith and his evolutionary stable strategy. In addition, the concepts of correlated
equilibrium, trembling hand perfection, and common knowledge^[5] were introduced and analyzed.
In 2005, game theorists Thomas Schelling and Robert Aumann won the Bank of Sweden Prize in Economic Sciences. Schelling worked on dynamic models, early examples of evolutionary game theory. Aumann
contributed more to the equilibrium school, developing an equilibrium coarsening correlated equilibrium and developing extensive analysis of the assumption of common knowledge.
1. ^ Some scholars would consider certain asymmetric games as examples of these games as well. However, the most common payoffs for each of these games are symmetric.
2. ^ Experimental work in game theory goes by many names, experimental economics, behavioral economics, and behavioral game theory are several. For a recent discussion on this field see Camerer
3. ^ For a more detailed discussion of the use of Game Theory in ethics see the Stanford Encyclopedia of Philosophy's entry game theory and ethics.
4. ^ Although common knowledge was first discussed by the philosopher David Lewis in his dissertation (and later book) Convention in the late 1960s, it was not widely considered by economists until
Robert Aumann's work in the 1970s.
See also
Textbooks and general reference texts
• Bierman, H. S. and L. Fernandez, Game Theory with economic applications, Addison-Wesley, 1998.
• Fudenberg, Drew and Jean Tirole: Game Theory, MIT Press, 1991, ISBN 0262061414 (the definitive reference text)
• Gibbons, Robert (1992) Game Theory for Applied Economists, Princeton University Press ISBN 0691003955 (readable; suitable for advanced undergraduates. Published in Europe by Harvester Wheatsheaf
(London) with the title A primer in game theory)
• Ginits, Herbert (2000) Game Theory Evolving Princeton University Press ISBN 0691009430
• Osborne, Martin and Ariel Rubinstein: A Course in Game Theory, MIT Press, 1994, ISBN 0-262-65040-1 (modern introduction at the introductory graduate level)
• Rasmusen, Erik: Games and information, 4th edition, Blackwell, 2006. Available online [6].
Historically important texts
Other print references
• Camerer, Colin (2003) Behavioral Game Theory Princeton University Press ISBN 0691090394
• Gauthier, David (1987) Morals by Agreement Oxford University Press ISBN 0198249926
• Grim, Patrick, Trina Kokalis, Ali Alai-Tafti, Nicholas Kilb, and Paul St Denis (2004) "Making meaning happen." Journal of Experimental & Theoretical Artificial Intelligence 16(4): 209-243.
• Kavka, Gregory (1986) Hobbesian Moral and Political Theory Princeton University Press. ISBN 069102765X
• Lewis, David (1969) Convention: A Philosophical Study
• Maynard Smith, J. and Harper, D. (2003) Animal Signals. Oxford University Press. ISBN 0198526857
• Quine, W.v.O (1967) "Truth by Convention" in Philosophica Essays for A.N. Whitehead Russel and Russel Publishers. ISBN 0846209705
• Quine, W.v.O (1960) "Carnap and Logical Truth" Synthese 12(4):350-374.
• Skyrms, Brian (1996) Evolution of the Social Contract Cambridge University Press. ISBN 0521555833
• Skyrms, Brian (2004) The Stag Hunt and the Evolution of Social Structure Cambridge University Press. ISBN 0521533929.
• Sober, Elliot and David Sloan Wilson (1999) Unto Others: The Evolution and Psychology of Unselfish Behavior Harvard University Press. ISBN 0674930479
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Fractal questions: Weierstraß-Mandelbrot
up vote 2 down vote favorite
Coming from a specific field in algebraic geometry I am a total noob in Fractal Theory and I'd like to learn it a bit. I hope I am tolerated for my maybe-trivial questions. I just read about the
Weierstrass-Mandelbrot fractal (it's also simply called Weierstrass fractal using the Weierstrass function.. but there are dozens of Weierstrass functions so I'd rather call it
"Weierstrass-Mandelbrot" function). The definition of this fractal is found in wikipedia. I got easily impressed by it.
My question is whether there are nowhere differentiable continuous functions (between real numbers) whose graph are not fractals? Is the WM function the easiest example of a nowhere differentialbe
continuous function?
The other question is quite basic (for experts probably). I have seen the definition of fractal in wikipedia. This definition uses self-similarity. But in a reference of mine (from a lecture note) I
get a definition that makes use of an inequality with Hausdorff-dimension and inductive dimension. Are these definitions equivalent or are the precise definition still under debate (my reference
suggests that the suggested definition was former definition by Mandelbrot and then this definition was changed as Mandelbrot fractals don't follow this definition). A little enlightening would help
Weierstrass . – Qfwfq Jul 13 '11 at 10:57
thanks fixed- A common typo for me. – Jose Capco Jul 13 '11 at 11:00
1 there, now it is really fixed. =) – Willie Wong Jul 13 '11 at 11:12
Among the continuous functions on [0,1], the functions that are not differentiable at any point are topologically generic (classical result going back to Banach and Mazurkiewicz), and also
4 prevalent, which is a measure-theoretical genericity valid in infinite dimensional spaces (this is due to Hunt). Clearly, a generic continuous function won't have any self-similarity whatsoever.
When speaking of curves, though, nowhere differentiability is considered a hallmark of fractality. As others said, though, fractal is a vague concept which does not and cannot have a precise
definition. – Pablo Shmerkin Jul 13 '11 at 17:37
add comment
4 Answers
active oldest votes
My question is whether there are nowhere differentiable continuous functions (between real numbers) whose graph are not fractals?
Of course this depends on your definition of fractal. There are nowhere-differentiable functions with graph of Hausdorff dimension 1.
Is the WM function the easiest example of a nowhere differentialbe continuous function?
For example, a nowhere-differentiable function due to Kießwetter was designed to be used with high-school students in Germany. English translation in my book: Classics on Fractals
Are these definitions equivalent
No, the definition with self-similarity is not equivalent to Hausdorff dimension > topological dimension. [Using self-similarity as a definition of fractal should be considered something
up vote 9 to use for non-mathematicians who are curious about the subject, but have no hope to understand measures and such for the real definition.]
down vote
are the precise definition still under debate
Mandelbrot gave the definition: Hausdorff dimension strictly greater than topological dimension. He later wrote that he regretted this, and instead it should be left undefined. Others have
provided other definitions. For actual mathematical papers, the authors of course state what they are proving in real mathematical language, not using the word fractal or just using it for
the vague explanatory part of the paper.
Kiesswetter function, two figures from Classics on Fractals
Thanks for the detailed info. Do you happen to have a pictorial link to the Kießwetter fractal? I wasn't able to find it by doing a google image search. – Jose Capco Jul 14 '11 at 8:18
added to the answer – Gerald Edgar Jul 14 '11 at 14:48
add comment
A quick, partial answer to your second question about the definition of fractals. If a fractal is generated by an iterated function system with a scaling ratio less than one then you do
get a Hausdorff dimension less than the inductive dimension. However it is not particularly difficult to create a set with Hausdorff dimension less than inductive dimension that should be
up vote 4 a fractal that isn't self-similar. The idea is to choose between two iterated function systems aperiodically.
down vote
add comment
You can also create a continuous, non-differentiable function by restricting the height map produced by the midpoint displacement algorithm to a line: http://en.wikipedia.org/wiki/
up vote 1 down This will not be self-similar, each open segment will with probability 1 be unique (if I am not mistaken). The displacement factor can be tuned so that the fractal dimension is any
vote number between 1 and 2.
add comment
The question "are nowhere differentiable continuous functions (between real numbers) whose graph are not fractals?" has no answer, because there is no universally accepted
definition of fractal:
up vote 1 down http://mathoverflow.net/questions/56677#56779
But I'd say yes : Every nowhere differentiable C⁰ function has some "fractalness".
add comment
Not the answer you're looking for? Browse other questions tagged fractals or ask your own question.
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East Point, GA Precalculus Tutor
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On the heat equation subject to nonlocal constraints
Seminar Room 2, Newton Institute Gatehouse
I will consider a heat equation subject to integral constraints on the total mass and the barycenter, instead of more common boundary conditions. The natural operator theoretical setting is that of a
space of distributions on the torus. By variational methods I will show well-posedness and some relevant spectral properties of this problem. This is joint work with Serge Nicaise (Valenciennes,
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.
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Hebel-Slichter NMR Effect
Hebel-Slichter-Redfield field-cycling NMR experiment
This experiment was designed [11,12] to measure longitudinal relaxation rate R[1] (inverse of the relaxation time T[1 ]) of nuclides in both normal and superconducting metals at various applied
magnetic fields B (including zero field). Using current terminology, we recognize in it an almost standard [27] Fast Field-Cycling NMR Dispersion Relaxometer (FFC-NMRD). The original instrument
operated in continuous way (CW) as follows (Figure 1):
Figure 1. Timing diagram
of the Hebel-Slichter-Redfield experiment
The sample's nuclides are first polarized in a relatively large applied magnetic field (the thin blue line). This is then switched to the desired B[relaxation] (or just B[r ]) value at which it is
left to relax for a variable time τ. The field is then switched to a value which is somewhat higher than the resonance field corresponding to the continuously applied radiofrequency. As the magnetic
field flies across the resonance condition one detects a brief, ringing signal S(τ) whose intensity, of course, depends upon τ and also upon the relaxation field value and the absolute sample
temperature θ.
Plotting S(τ) as a function of τ, one obtains the relaxation curve for longitudinal magnetization which passes from its equilibrium value in the polarization field to that in the relaxation field. In
most metals, this curve is mono-exponential and its decay rate coefficient is the measured value of R[1](θ,B[r]).
The modern, pulsed version of the experiment is conceptually similar, but the transmitter and the receiver are both gated, the polarization field can be much higher (there is no interference with
nuclear magnetization when the field is switched down) and the signal is acquired at a constant field close to resonance (or even on resonance), giving a well-behaved free induction decay (FID). The
consequent gain in sensitivity and precision is obvious.
While this experimental setup is quite universal, in the case of metals the explanation really applies only to non-superconducting samples. Due to the Meissner effect [3,4], the internal magnetic
field in a superconductor is rigorously zero even in the presence of an external field (no matter whether static or dynamic). Consequently, whenever the applied field drops below a critical field
level, the effective internal field perceived by the sample's nuclides becomes abruptly zero, as indicated by the thick blue line in Figure 1.
Consequently, the only relaxation rate one can measure in a superconductor is that at zero magnetic field.
A few facts about relaxation in normal metals
Figure 2. NMRD profile of a normal metal
Dispersion profiles are best plotted on a log-log scale, as indicated by the labels of the axes. Following usual conventions, however, specific values are indicated without the log in front.
We want to avoid theory here but it is useful to recall at least the phenomenological aspects of NMR relaxation in normal metals which were known [9,10,11] already in 1957, the year when the
Hebel-Slichter effect was discovered.
First of all, the longitudinal relaxation rate R[1n](θ,B) in a normal metal is typically proportional to the absolute temperature θ and can be normalized by plotting the ratio r[1n] = R[1n]/θ as a
function of the applied magnetic field B. The normalized NMR dispersion curve looks typically as shown in Figure 2. There is a plateau at both low and high fields and a dispersion region around some
value B[d] which depends upon the particular metal. According to early theoretical models [10,12], the ratio r[1n](0)/r[1n](∞) should be equal to 2, but actual experimental values often deviate
In non-superconducting aluminum, for example, the center of the dispersion region is at B[d] = 1.5 mT, the temperature coefficients are r[1n](0) = 2.2 s^-1K^-1 and r[1n](∞) = 0.6 s^-1K^-1, their
ratio is 3.3, and the maximum log-log slope at the inflex point is -0.6 (the oblique dotted line).
In any case, it is relatively easy to measure NMRD profiles of powdered metal samples, determine their longitudinal-relaxation dispersion parameters, and use the latter to extrapolate the R[1n](θ,B)
values to any temperature θ and any magnetic field B.
A few facts about superconducting metals
Some metals become superconductors when their absolute temperature θ is lower than a critical value θ[c] and the external magnetic field B does not exceed some critical value B[c](θ) which decreases
with increasing temperature and becomes zero at θ[c]. In a [B,θ] diagram (Figure 3 on the right), the boundary of the region where a metal is in a superconducting state can be approximated quite well
by a formula containing just two parameters, the θ[c] and B[c] = B[c](0).
Figure 3. State diagram of a superconductor
Figure 3 is reminiscent of a phase diagram of a substance, especially when one considers that electric and magnetic field intensities are legitimate thermodynamic state variables. The question is
what is the substance, and it is not difficult to guess that it must be the electron gas of the delocalized metal electrons. Superconductivity therefore implies something akin to a phase-transition.
That much became clear relatively soon after its discovery by Heike Kamerlingh Onnes [1,2] in 1910, but the nature of the phenomenon remained unclear for nearly half a century. The subsequent history
up to - and including - the BCS theory [16-21] born in 1957 is well described in Wikipedia, so we can skip it here. Suffice it to say that the BCS theory is based on a microscopic phenomenon - the
formation of quasi-persistent and quasi-localized electron pairs (Cooper pairs). Since the spins of the two electrons in such a pair are anti-parallel, these pseudo-particles have zero total spin and
thus behave as doubly charged bosons - something totally different from normal electrons which, being fermions, must obey the rigors of Pauli's exclusion principle. I am of course oversimplifying a
complex phenomenon which involves interactions of electrons with phonons and second quantization of the electron field, but the general idea is just that.
Not all metals have a superconducting region and not all superconductors have such a neat diagram. But let us consider a well-behaved metal like aluminum (B[c]=9.84 mT, θ[c]=1.172 °K) and see what
its superconductivity diagram implies when it comes to the Hebel-Slichter-Redfield experimental setup of Fig.1. When the field is switched from a high value (well above B[c ]) to near zero, or vice
versa, it at some point crosses the critical value B[c](θ) indicated in Figure 1 by the dotted blue horizontal line. When changing the sample temperature θ (but staying below θ[c]), the B[c](θ) moves
vertically up or down. It is evident from the timing diagram, however, that this does not particularly affect the experiment. So, in principle, R[1] values can be measured in the superconductive
region at any absolute temperature between 0 and θ[c].
For completeness, let us say that the dynamics of the normal-to-superconducting transition is very fast and does not interfere with the experiment (at least, nobody has ever complained about it and I
don't know whether it has been ever measured).
Figure 4. Experimental Hebel-Slichter peak
The indices s and n refer to the superconducting and normal state, respectively.
Figure 5. Normalized Hebel-Slichter effect
Hebel-Slichter effect
Now we know how to measure the longitudinal relaxation rate R[1] of a metal's nuclides at any absolute temperature θ and at relaxation field B[r] = 0 (the latter value is imposed by the fact that in
the superconducting state we are not free to set it at will). We even know what to expect of the curve R[1](θ) = R[1](θ,0) in the normal state (a linear dependence). So what if we now measure R[1](θ)
at all temperature values, covering also the superconducting region?
The experimental result [11,12,13] is shown schematically in Figure 4. Just below the critical temperature, the relaxation rate abruptly increases above the expected 'normal' value. In the
superconducting state, the value of R[1s](θ) reaches its maximum at absolute temperature which is about 80% of θ[c] and then, at temperatures below about 10% of θ[c], drops even below the
extrapolated 'normal' value R[1s](θ) indicated by the dotted line. This R[1s] temperature profile is known as the Hebel-Slichter peak.
To account for and, to some extent, eliminate the 'normal' contributions to longitudinal relaxation, it is useful to normalize R[1](θ) dividing it by R[1n](θ) (either measured or extrapolated). When
the absolute temperature is also normalized dividing it by θ[c], one obtains a graph like that shown in Figure 5, where the deviation from the normal value of 1 is due entirely to the dynamics of the
Cooper pairs. A little bit below θ = θ[c] the peak reaches a height of almost 3. Notice also that at low values of the ratio θ/θ[c], the normalized curve becomes approximately linear.
In practice, the shape of the Hebel-Slichter peak may deviate considerably from the idealized one shown in Figure 4. In some semiconductors, typically the non-metallic ones with high critical
temperature, it may be even missing altogether. In all cases, however, the shape of the R[1](θ) curve, especially of its normalized version, tells us a lot about the formation and dynamics of the
Cooper pairs and about the phononic spectrum (lattice vibrations) of the superconducting material.
Historic notes
Longitudinal relaxation times T[1] and rates R[1] = 1/T[1] of nuclides in metals were of interest to physicists since the early days of NMR because they, together with Knight shifts [7,8,9], throw
light on the quantum states of electrons with energies lying close to the Fermi level which dominate most of every metal's properties. By late fifties, relaxation phenomena in normal metals were
phenomenologically relatively well known and there were reasonably sound theoretical models to describe them.
The Hebel-Slichter experiment was carried out in 1957 at the University of Illinois by Charles P.Slichter and his pre-graduate student L.C.Hebel (sponsored by General Electric). They resorted to the
field-cycling arrangement because that was (and still is) the only way to do NMR in a material which efficiently excludes any magnetic field from its interior. As they amply mention in their two key
papers [11,12], they were helped a lot by Alfred Redfield [10,14,15] who was at that time a promising young NMR relaxation expert at the IBM Laboratories, respected both for his theoretical work [10]
and for his experiments (he was already oriented towards variable-field NMR relaxometry). This is why I call the experiment Hebel-Slichter-Redfield's, while the effect itself is 'just' Hebel-Slichter
In any case, the actual experimental setup involved a specially build electromagnet with a relatively low-susceptibility yoke made of thin silicon steel plates, which permitted sufficiently fast,
actively driven field switching. They polarized their powder aluminum sample in a field of about 45-50 mT and their operating frequency was 400 kHz, corresponding for ^27Al to a field of about 36 mT.
Aluminum was chosen because both its NMR properties and its superconductivity had been already studied by a number of investigators (including Slichter and Redfield).
The outcome of the experiments was among the key factors in discarding previous theories of superconductivity [5,12] in favor of the new BCS theory which was being concurrently developed at the same
University by John Bardeen, Leon Cooper and John Schrieffer. The older theories were simply not able to explain the existence of a local extreme in the R[1](θ) curve, while it was possible to
slightly adapt the BCS theory to make it match the data (as acknowledged in the Hebel-Slichter papers [11,12], the two groups were in close contact).
It is interesting that the second set of experiments which cleared the way for the BCS theory regarded ultrasound attenuation [16,20]. Techniques like NMR relaxation, ultrasound absorption and
dielectric relaxation are in fact closely correlated since they all reflect the same stochastic processes at atomic (or molecular) level. Combining them should be a rule.
Continuing importance
The historic aspects of the Hebel-Slicher effect, interesting as they are, should not overshadow the fact that it continues to be an important tool in the study of superconductive materials. This is
particularly important for high-critical-temperature materials were the superconductivity mechanisms are still poorly understood.
It is also proper to mention here that curves remarkably similar to the Hebel-Slichter peak have been recently measured in light-scattering experiments on superfluids [26]. The parallelism, both
theoretical and experimental, is so close that it is not exaggerated to speak about a generalized Hebel-Slichter effect in many systems whose quantum behavior is due to the formation (condensation)
of quasi-particle pairs.
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Re: st: RE: dmexogxt questions
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Re: st: RE: dmexogxt questions
From "Mark Schaffer" <M.E.Schaffer@hw.ac.uk>
To statalist@hsphsun2.harvard.edu
Subject Re: st: RE: dmexogxt questions
Date Mon, 13 Sep 2004 14:43:10 +0100
Just a couple of footnotes to Steve's response:
Subject: st: RE: dmexogxt questions
Date sent: Tue, 14 Sep 2004 00:31:36 +1200
From: "Steve Stillman" <stillman@motu.org.nz>
To: <statalist@hsphsun2.harvard.edu>
Send reply to: statalist@hsphsun2.harvard.edu
> Hi Jean. The answers to your questions are below. Cheers, Steve
> -----Original Message-----
> From: owner-statalist@hsphsun2.harvard.edu
> [mailto:owner-statalist@hsphsun2.harvard.edu]On Behalf Of Salvati, Jean
> Sent: Saturday, September 11, 2004 8:49 AM
> To: statalist@hsphsun2.harvard.edu
> Subject: st: dmexogxt questions
> Hello,
> I have two questions about dmexogxt:
> 1) The joint test clearly rejects the null hypothesis that all
> regressors are exogenous, but the tests on individual regressors don't
> reject the null for any of the regressors (not even close).
> More precisely, let's say I estimate my model with the following
> command:
> xtivreg y x1 (x2 x3 = z2 z3), fe
> When I do "dmexogxt", the null hypothesis that all regressors are
> exogenous ism clearly rejected. However, when I do "dmexogxt x2" and
> "dmexogxt x3", I definitly can't reject the null for either x2 or x3 at
> the same level.
> How can I interpret these results?
> *** When you run the command dmexogxt x2, you are assuming that x3 is
> definitely endogenous and are only testing that x2 is exogenous given
> this assumption. For whatever reason, in your example, you cannot
> clearly distinguish between (x2 endog, x3 exog), (x2 exog, x3 endog), or
> (both endog). Since you do not seem to have a reason to assume either
> one is definitely endogenous (thus, leading to the reduced test), my
> instinct would be that you are best off treating both as being
> endogenous.
The text here can be interpreted in the same way as a Hausman test,
i.e., endogeneity/exogeneity is picked up by differences in the
coefficients between the two specifications. In effect, when you set
one or the other of x2 and x3 to be exogenous, the coeffs don't
change much compared to the benchmark case where both are
endogeneous. But when you set both to be exogenous, the coeffs
change a lot, again compared to the case of both being endogenous.
This doesn't sound very strange, at least to me.
> 2) After "xtivreg y x1 (x2 = z2 ), fe", both "dmexogxt" and "dmexogxt
> x2" yield F-statistics.
> *** with only one possible endogenous variable, "dmexogxt" and "dmexogxt
> x2" are identical tests and thus give identical results
> After "xtivreg y x1 (x2 x3 = z2 z3), fe", both "dmexogxt" still gives an
> F-statistic, but "dmexogxt x2" yields a chi2(1). Why is that? Is a Wald
> test used in the second case, and if so why?
> *** more generally, if "dmexogxt" is only run on a subset of endogenous
> variables you will end up with a chi2(number tested variables) instead
> of an f-test. This occurs because the auxiliary regression being run
> for the test is now an IV regression (we still need to instrument for
> the variables left out of the test) as opposed to an OLS regression (the
> case when all possible endogenous variables are being tested).
... and the Wu version of the test has an F-stat form in this case.
But if you're relying on asymptotics, it doesn't matter if it's an F
or chi-sq. If you want an F-stat instead of a chi-sq, you can always
get one by hand if you divide by the relevant dof.
> Thanks a lot.
> Jean Salvati
> *
> * 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/
Prof. Mark E. Schaffer
Centre for Economic Reform and Transformation
Department of Economics
School of Management & Languages
Heriot-Watt University, Edinburgh EH14 4AS UK
44-131-451-3494 direct
44-131-451-3008 fax
44-131-451-3485 CERT administrator
* For searches and help try:
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FUN math puzzles!
I have fun math puzzles
1.. If 1/2x +1/2(1/2x + 1/2(1/2x +1/2(1/2x + ... = y,
then x = ?
8. What is the area of a regular hexagon with sides 1 in. Long?
look on MIF for help on number 8
Re: FUN math puzzles!
Nehushtan wrote:
the awnser,not formula.
Re: FUN math puzzles!
Hi mathgogocart;
Last edited by bobbym (2013-03-18 12:01:39)
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
Re: FUN math puzzles!
5. What place in this world can have their temperatures Fahrenheit and Celsius equal? 10 points,5 if searched
Last edited by mathgogocart (2013-03-19 06:07:20)
Re: FUN math puzzles!
Yes, I had the formula wrong!
Last edited by bobbym (2013-03-18 16:59:29)
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
Re: FUN math puzzles!
Neshutan is partly correct
1.At a party, everyone shook hands with everybody else. There were 66 handshakes. How many people were at the party? 15 points
2.Using eight eights and addition only, can you make 1000? 20 points
Hint for 2:conbine like 2 8ts are 88
Last edited by mathgogocart (2013-03-19 06:11:18)
Re: FUN math puzzles!
Last edited by bobbym (2013-03-19 06:29:29)
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
Real Member
Re: FUN math puzzles!
Hi mathgogocart
The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
Re: FUN math puzzles!
Bobbym is right FORR BOTH!!!!!
1.6-4) Today is Friday. What day of the week is it 80 days from today?
7-1) The diameter of the circles is 2 centimeters. Calculate the area that is purple.
Last edited by mathgogocart (2013-03-19 06:38:03)
Re: FUN math puzzles!
Last edited by bobbym (2013-03-19 06:39:23)
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
Re: FUN math puzzles!
Re: FUN math puzzles!
Bozeman is the one I wanted but could not remember. It had a low of -43F in 1936.
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
Re: FUN math puzzles!
Neshutan:10-Junior Awnserer
Bobbin: 46-cool awnserer
Anononmistefy:35-cool awnserer
7-7) A half circle overlaps with a square. The diameter of the half circle is 12 inches. What is the area of the hatched/striped parts?-5 points
7-14) John went to the stores to buy Christmas presents for his kids. He spent 2 thirds of his money at a book store. He then spent 1 third of the remaining money at a craft store. At the end he had
$8 left. How much money did he have at the beginning?-8 points
Questions 1:
5.If Logx (1 / 8) = - 3 / 2, then x is equal to _________-5 points
A. - 4
B. 4
C. 1 / 4
D. 10
Re: FUN math puzzles!
Last edited by bobbym (2013-03-19 12:53:58)
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
Re: FUN math puzzles!
Neshutan:28-Junior Awnserer
Bobbin: 64-Awesome Awnserer
Anononmistefy:35-cool awnserer
1.A father in his will left all his money to his children in the following manner:
$1000 to the first born and 1/10 of what then remains, then
$2000 to the second born and 1/10 of what then remains, then
$3000 to the third born and 1/10 of what then remains, and so on.
When this was done each child had the same amount. How many children were ther?e-10 points
Question 2: Car A began a journey from a point at 9 am, traveling at 30 mph. At 10 am car B started traveling from the same point at 40 mph in the same direction as car A. At what time will car B
pass car A?No Explanation:12 points,Explantion 20 points
20-Junior Awnserer
30-Cool Awnserer
60-Awesome Awnserer
100-Epic Awnserer
more coming....
Last edited by mathgogocart (2013-03-20 10:01:44)
Re: FUN math puzzles!
Last edited by Nehushtan (2013-03-20 12:25:08)
Re: FUN math puzzles!
Neshutan:68-Awesome Awnserer-TOP AWNSERER!!
Bobbin: 84-Awesome Awnserer
Anononmistefy:35-cool awnserer
1.Question 2: Four times the first of three consecutive even integers is six more than the product of two and the third integer. Find the integers.
5 points,12 if explained
2.Question 1: Find the general solution for differential equation
(D4 - 5D3 + 5D2 + 5D - 6)y = 0
20 points,35 if explained
Last edited by mathgogocart (2013-03-21 10:51:51)
Re: FUN math puzzles!
I think the numbering of the questions should be more consistent.
Re: FUN math puzzles!
Last edited by bobbym (2013-03-20 13:37:13)
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
Re: FUN math puzzles!
New MATH RIDDLES!
1.If you take 3 apples from a group of 5, how many do you have?-3 points-Easy
2.What would be the seventh rung of the following pyramid?-6 points
3.So you think you're good at math? Complete the sequence: 1=3, 2=3, 3=5, 4=4, 5=4, 6=3, 7=5, 8=5, 9=4, 10=3, 11=?, 12=?-10 points
Re: FUN math puzzles!
You did not tally up the scores from posts #21 and #22.
In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.
Re: FUN math puzzles!
Can you please ddo it?
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Re: st: Teach an old dog new tricks
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Re: st: Teach an old dog new tricks
From "Joseph Coveney" <jcoveney@bigplanet.com>
To "Statalist" <statalist@hsphsun2.harvard.edu>
Subject Re: st: Teach an old dog new tricks
Date Wed, 14 May 2008 02:07:48 +0900
Austin Nichols <austinnichols@gmail.com>:
As for the sparse matrix problem in (A), you can generate a new
variable with all distinct concatenations of rowvar and colvar, then
cycle over the values of that, thereby ignoring the empty cells.
On Tue, May 13, 2008 at 10:18 AM, Sergiy Radyakin
<serjradyakin@gmail.com> wrote:
Thank you all, who responded to my request regarding obtaining a
matrix of means. Besides the answers posted in this thread I have
received a couple of suggestions privately. To summarize and close the
thread, the suggestions can be divided roughly into two groups:
A. Obtaining all possible levels of the by-variables, then cycling
through these values and computing means for each subgroup. This can
be quite slow, especially in case of "sparse" matrices, where only a
few non-empty cells exist (for a 50x50 matrix -summarize- must be
called 2500 times).
B. Using other Stata commands which can produce matrix of means as a
by-product. Unfortunately none of them is fast enough either. In
particular, Joseph Coveney suggested using xi to automatically create
all combinations of values and then estimating a univariate
regression. Although this is a very short code, it is perhaps the
slowest, and demands large amounts of memory.
Sergiy, it'll help us to help you better if you're more specific about the
scope of your problem up front; Austin's original reply's -tabmat- seemed
ideal to me given what you gave the list to go on; and my suggestion
works well for the example that you gave in your post, which I took to be
illustrative of scope of the individual summarization that you want to
repeat many times and therefore want to avoid -preserve-s, etc.
Austin's point above about concatenating applies to sparse matrix problems
in (B), too: see below for timing of a (B)-approach compared to -table ,
contents(mean )-, which is the benchmark you give in your original post.
Note that -anova , noconstant category()- is used in lieu of -xi: regress ,
noconstant-, because it's more efficient here.
Joseph Coveney
clear *
set matsize 800 // Nothing extraordinary
set memory 10M // Nothing extraordinary
set obs 250000 // I don't know how many you have--is this in the ballpark?
/* A 50 X 50 matrix */
generate byte a = mod(_n, 50)
sort a
generate byte b = mod(_n, 50)
generate float c = uniform()
/* Make that sparse */
foreach var of varlist a b {
replace `var' = 0 if !inrange(`var', 20, 30)
timer clear 1
quietly forvalues i = 1/10 {
timer on 1
table a b, contents(mean c)
timer off 1
timer clear 2
quietly forvalues i = 1/10 {
timer on 2
generate int ab = 100 *a + b // Concatenation
anova c ab, noconstant category(ab)
timer off 2
drop ab
timer list
. timer list
1: 24.29 / 10 = 2.4295
2: 7.62 / 10 = 0.7621
. exit
end of do-file
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Nonlinear Magnetization Dynamics in Thin-Films and Nanoparticles
d'Aquino, Massimiliano (2005) Nonlinear Magnetization Dynamics in Thin-Films and Nanoparticles. [Tesi di dottorato] (Inedito)
Full text disponibile come:
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In chapter 1 the micromagnetic model and the Landau-Lifshitz-Gilbert (LLG) equation is introduced to describe magnetization phenomena in ferromagnetic bodies. First, an approach in terms of the free
energy associated with the magnetic body is presented to derive the static equilibrium conditions for magnetization vector field. Then, the dynamic effects due to the gyromagnetic precession are
introduced. Both Landau-Lifshitz and Landau-Lifshitz-Gilbert equation are presented. Phenomenological Gilbert damping is analyzed in terms of Rayleigh dissipation function. In chapter 2 the study of
magnetization dynamics in uniformly magnetized particles is addressed. In particular, first the static Stoner-Wohlfarth model and then magnetization switching processes are analyzed. In addition,
novel analytical techniques to study magnetization dynamics under circularly polarized external fields and magnetization dynamics driven by spin-polarized currents are introduced and deeply
discussed. In this respect, it is shown how some behaviors indeed observed in experiments, can be explained in terms of bifurcations of fixed points and limit cycles of the LLG dynamical system. As a
further step, in chapter 3, the assumption of magnetization spatial uniformity is removed and the problem of studying thin-films reversal processes of technological interest is addressed. In this
respect, as preliminary step, the issue of the computation of magnetostatic fields, which is still the bottleneck of micromagnetic simulations, is illustrated together with the mostly used methods at
this time. Then, a comparison of damping and precessional switching processes in thin-films is performed, showing that fast precessional switching can be considered spatially quasi-uniform and,
therefore, its crucial aspects can be analyzed by means of uniform mode theory discussed in chapter 2. Finally, a uniform mode analysis is applied to the fast switching of granular tilted media which
represents one of the most promising solutions for high density magnetic storage in future hard disks. In chapter 4, the problem of the geometrical integration of LLG equation is considered. In
particular, the mid-point rule time-stepping is applied to the LLG equation. In fact, it is shown that the fundamental properties of magnetization dynamics, embedded in the continuous model, are
reproduced by the mid-point discretized LLG equation regardless of the time step. In addition, since the resulting numerical scheme is implicit, special and reasonably fast quasi-Newton technique is
developed to solve the nonlinear system of equations arising at each time step. The proposed mid-point technique is validated on the micromagnetic standard problem no. 4 which concerns with
thin-films reversal processes. Finally, numerical results and computational cost are discussed.
Tipologia di documento: Tesi di dottorato
Parole chiave: micromagnetics, Landau-Lifshitz-Gilbert equation, precessional switching, spin-polarized currents, perturbative tecniques, micromagnetic simulations, fast switching,
geometrical integration, micromagnetic standard problem n. 4
Settori Area 09 Ingegneria industriale e dell'informazione > ING-INF/02 CAMPI ELETTROMAGNETICI
Coordinatori della Scuola di ┌──────────────────────────────────────┬───────────────────┐
dottorato: │ Coordinatore del Corso di dottorato │ e-mail (se nota) │
│ Miano, Giovanni │ │
Tutor della Scuola di ┌───────────────────────────────┬───────────────────┐
dottorato: │ Tutor del Corso di dottorato │ e-mail (se nota) │
│ Serpico, Claudio │ │
Stato del full text: Accessibile
Data: 2005
Numero di pagine: 165
Istituzione: Università di Napoli Federico II
Dipartimento o Struttura: Dipartimento di Ingegneria Elettrica
Tipo di tesi: Dottorato
Stato dell'Eprint: Inedito
Ciclo di dottorato: XVII
Numero di sistema: 148
Depositato il: 02 Marzo 2007
Ultima modifica: 04 Febbraio 2009 09:38
Solo per gli Amministratori dell'archivio: edita il record
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Re: st: mfx algorithsm for standard error
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Re: st: mfx algorithsm for standard error
From rgutierrez@stata.com (Roberto G. Gutierrez, StataCorp.)
To statalist@hsphsun2.harvard.edu
Subject Re: st: mfx algorithsm for standard error
Date Tue, 03 Sep 2002 17:43:11 -0500
Mike Will <mystata@hotmail.com> has asked about the formula for the standard
errors calculated by -mfx-, and now writes:
> I have checked into mfx using findit mfx and couldn't get much out of it.
> Most of the explanations are too short and didn't even touch upon the
> standard errors of marginal effects. Any more inputs from those who wrote
> it or some other good references talking about the standard errors
> especially for the nonlinear options? Appreciate it.
The standard errors of the marginal effects are calculated using the "delta
method", which basically says that if you have a continuous function of some
estimators (like a marginal effect), then the standard error of this function
can be obtained from the derivatives of this function with respect to the
estimators in question and the variance-covariance matrix of the original
for Al Feiveson's excellent explanation of the delta method. In -mfx- the
derivatives are calculated numerically. Most of the time, the marginal effect
is such that it is a function of the whole linear predictor, x*beta, rather
than a function of each individual beta. When this occurs, this simplifies
the standard error calculation significantly since you can treat x*beta as a
scalar random quantity, rather than having to consider the marginal effect as
a function of each individual component of beta -- fewer derivatives to take.
* 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/
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Statistical Models Now Show Coleman as Slight Favorite
6:58 PM Dec 2, 2008
Statistical Models Now Show Coleman as Slight Favorite
With in excess of 90 percent of Minnesota’s votes now having been recounted, our statistical models now show Norm Coleman as the favorite to retain his senate seat, although with a high degree of
uncertainty and without accounting the effects of potential rejected absentee ballots.
The basic process behind our projections is as follows: using precinct-level returns available from the Minnesota Secretary of State, we use regression analysis attempt to predict the number of
ballots that a candidate has gained or lost in a given precinct based on the number of challenges issued by he and his opponent, and his share of the vote in the pre-recount stage of the process.
Then, we set the number of challenges to zero in the regression equation, which ideally represents the state that occurs once all ballot challenges have been considered by the state’s canvassing
board later this month.
I am now running eight separate versions of the model based on various permutations of assumptions that one can make about how to build the model. Specifically (and feel free to skip this description
if you don’t care about the technicalities):
‘Gross’ models evaluate each candidate’s results individually, e.g. how much Franken gains in the absolute count. ‘Net’ models evaluate how much Franken gains relative to Coleman, without
worrying about the absolute count. ‘Simple’ models include a maximum of three variables (plus a constant term): Franken’s share of the two-way vote in that precinct, the frequency of challenges
initiated by Coleman, and the frequency of challenges initiated by Franken. ‘Complex’ models account for two-way and three-way interactions (where statistically significant) between these
independent variables. The regression is weighted either based on the number of votes tabulated in that precinct (‘Straight’), or based on the square root of the number of votes in that precinct
(‘Root’). In all models, variables are dropped if not statistically significant at the 95 percent certainty level.
The reason I’m including these different versions is because the models are not especially precise, and so we want to get some sense for how robust they are. The margins of error on the models are
high — at least +/- 200 votes, and sometimes more depending on the complexity of the model.
Here are the results:
Type Depth Weight Franken Coleman Change Result
Gross Simple Straight +581 +454 F +127 C +88
Gross Simple Root +639 +544 F +95 C +120
Gross Complex Straight +545 +327 F +218 F +3
Gross Complex Root +584 +447 F +107 C +108
Net Simple Straight -- -- F +128 C +87
Net Simple Root -- -- F +80 C +135
Net Complex Straight -- -- F +209 C +6
Net Complex Root -- -- F +125 C +90
All eight versions of the model show Franken gaining significant ground in the recount, from a net of 80 votes to a net of 218. Since Coleman led by 215 votes in the state’s certified, pre-recount
tally, however, only one of the eight models now shows Franken gaining enough votes to overtake Coleman, and then only by 3 ballots.
This should not be interpreted to mean, however, that Franken only has a 1 in 8 chance of defeating Coleman. Given the high degrees of uncertainty and ambiguity implied by the models, they would
suggest that Franken has roughly speaking somewhere between a 25% chance and a 50% chance of overtaking Coleman depending on which model is selected.
In addition, the models do not consider the potential impact of rejected absentee ballots, which the Franken campaign is still attempting to get counted. If Franken is able to get such ballots
counted — and there is a strong chance that he will — they will likely be worth a net of somewhere between 25 and 100 votes to him. In this eventuality, the race should probably be considered a
UPDATE: Since several people have asked, the Daily Kos diary suggesting that Franken is “leading” the recount is grossly misleading. In the most literal sense, Franken has indeed won the plurality of
ballots counted so far in the re-count — but he also won a plurality of ballots from those same precincts on Election Day, because they tended to come from slightly bluer precincts than the state as
a whole. As the outstanding (mostly red-leaning) precincts are counted again, Coleman will gain ground and almost certainly overtake Franken in the Secretary of State’s total; the question then is
what will become of all the challenged ballots, which is what the statistical model is trying to address.
comments Add Comment
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When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.FSTTCS.2011.241
URN: urn:nbn:de:0030-drops-33535
URL: http://drops.dagstuhl.de/opus/volltexte/2011/3353/
Go to the corresponding Portal
Ananth, Prabhanjan ; Nasre, Meghana ; Sarpatwar, Kanthi K.
Rainbow Connectivity: Hardness and Tractability
A path in an edge colored graph is said to be a rainbow path if no two edges on the path have the same color. An edge colored graph is (strongly) rainbow connected if there exists a (geodesic)
rainbow path between every pair of vertices. The (strong) rainbow connectivity of a graph G, denoted by (src(G), respectively) rc(G) is the smallest number of colors required to edge color the graph
such that G is (strongly) rainbow connected. In this paper we study the rainbow connectivity problem and the strong rainbow connectivity problem from a computational point of view. Our main results
can be summarised as below: 1) For every fixed k >= 3, it is NP-Complete to decide whether src(G) <= k even when the graph G is bipartite. 2) For every fixed odd k >= 3, it is NP-Complete to decide
whether rc(G) <= k. This resolves one of the open problems posed by Chakraborty et al. (J. Comb. Opt., 2011) where they prove the hardness for the even case. 3) The following problem is fixed
parameter tractable: Given a graph G, determine the maximum number of pairs of vertices that can be rainbow connected using two colors. 4) For a directed graph G, it is NP-Complete to decide whether
rc(G) <= 2.
BibTeX - Entry
author = {Prabhanjan Ananth and Meghana Nasre and Kanthi K. Sarpatwar},
title = {{Rainbow Connectivity: Hardness and Tractability}},
booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011)},
pages = {241--251},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-34-7},
ISSN = {1868-8969},
year = {2011},
volume = {13},
editor = {Supratik Chakraborty and Amit Kumar},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2011/3353},
URN = {urn:nbn:de:0030-drops-33535},
doi = {http://dx.doi.org/10.4230/LIPIcs.FSTTCS.2011.241},
annote = {Keywords: Computational Complexity, Rainbow Connectivity, Graph Theory, Fixed Parameter Tractable Algorithms}
Keywords: Computational Complexity, Rainbow Connectivity, Graph Theory, Fixed Parameter Tractable Algorithms
Seminar: IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011)
Issue Date: 2011
Date of publication: 01.12.2011
DROPS-Home | Fulltext Search | Imprint
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Manhattan GMAT Challenge Problem of the Week – 22 Apr 10
Welcome back to this week’s Challenge Problem! As always, the problem and solution below were written by one of our fantastic instructors. Each challenge problem represents a 700+ level question. If
you are up for the challenge, however, set your timer for 2 mins and go!
An ant is clinging to one corner of a box in the shape of a cube. The ant wants to get to the most distant corner of the box by crawling only along the edges of the cube and without ever
revisiting a place it has been. How many different paths can the ant take to the most distant corner?
(A) 6
(B) 12
(C) 18
(D) 24
(E) 30
First, draw a picture of the situation.
Now, look for patterns as you construct and count paths that don’t retrace or touch themselves at all. One of the first patterns that is useful to spot has to do with the initial choice about which
edge to crawl along. The ant can take any one of three initial paths:
These three initial choices are equivalent, because the cube is symmetric. Thus, we can just consider one of the initial paths, count up the ways from that point forward, then multiply by 3.
Let’s take the middle choice:
First, try to go immediately to the goal. We get two similar paths:
If we stopped at this point, we would get 2 × 3 = 6 total paths. This is in fact the number of shortest paths to the goal. However, we can construct additional “zigzag” paths that also satisfy the
constraints of the problem.
There are no more possibilities without the path touching itself, so this initial edge leads to 6 possible paths. Since there are 3 initial edges, the total number of possible paths is 6 × 3 = 18.
Alternatively, we can map the corners by their “distance” away from the starting point, measured in edges:
Taking the leftmost edge to start, we can construct the 6 possible paths leading from that edge:
Again, there are 6 × 3 = 18 possible paths.
The correct answer is (C).
Special Announcement: Manhattan GMAT is now offering you a chance to win prep materials by solving the Challenge Problem. On our website, we will post a new question (without the answer) every week.
Submit a solution to the problem, and if we pick your name out of those who answer correctly, you could win free prep material from Manhattan GMAT. To view the current question, simply visit our
Challenge Problem Showdown.
If you liked this article, let Michael Dinerstein know by clicking Like.
4 comments
the question is ambigous as it says without ever revisiting a place it has been. what does place refer to? if it refers to an edge of the cube.
there are only 3 possible paths
This question is time consuming. Is there a short way to find the answer? or how would you apply an educated guess?
Long and tricky as well i guess....
It tricked me multiplying it by 3
As this question labels itself more than 700+ persons question ,i think you have to burrow some time from other short questions....
any way may be some other may be better idea...
i also dont see another than making diagram and using manual method
Ask a Question or Leave a Reply
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\[\large \text{Let} \quad f(x)=y\] \[\large y=2x-2\]Taking the inverse, we swap the x and y giving us,\[\large x=2y-2\]Solving for y gives us,\[\large y=\frac{x+2}{2} \qquad \rightarrow \qquad y=
\frac{x}{2}+1\] Pretty sure it's option number 3, unless I got confused about the order you posted them :o
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Rolling a convex body: Geodesics vs. rolling curves
up vote 5 down vote favorite
What are the curves of contact on a convex body $B$ rolling down an inclined plane? Assume $B$ is smooth, and there is sufficient friction to prevent slippage.
Certainly, one can develop a geodesic to a straight line on a plane by rolling $B$ so that the geodesic is the point of contact, but it doesn't appear that this in general would be the point of
contact in the physical situation of free rolling under gravity. It is for a sphere, which will roll along great circles. And an ellipsoid should roll along its three simple closed geodesics
(although there are significant instabilities with all but the shortest closed geodesic—I'd prefer to ignore stability issues). But for other shapes, I imagine that an off-center center of gravity
(and perhaps rotational momentum?) will cause the rolling to deviate from a geodesic. (But I am uncertain of this. Please correct me if I'm wrong!)
Assuming this is correct (that rolling curves are not always geodesics), what are the conditions that determine if a curve $\gamma$ is a rolling curve: $\gamma$ is the trace of the point of contact
between $B$ and an inclined plane as it rolls, from some initial position? Perhaps: If $p \in \gamma$ is a point of contact, then (a) the normal vector $N$ of $\gamma$ at $p$ must be perpendicular to
the plane, and (b) the center of gravity of $B$ must lie in the osculating plane of $\gamma$ at $p$ (the plane containing $N$ and the tangent $T$ vector at $p$)?
Have what I christened rolling curves been studied in the literature? If so, under what name? My searches have been unsuccessful. Can you think of shapes outside of {sphere, ellipsoid, cylinder}
where the rolling curves can be determined? Thanks for any thoughts or pointers!
mg.metric-geometry dg.differential-geometry reference-request classical-mechanics
springerlink.com/content/w2273555w336u976 – Steve Huntsman Jul 16 '10 at 12:58
@Steve: Thanks for reference to "On conjugation of solutions to two integrable problems: Rolling of pointed body in the plane." Definitely relevant, although he specifically attends to "the
horizontal plane." Haven't penetrated it entirely, and it gives a trail of related references in Russian journals. Thanks! – Joseph O'Rourke Jul 16 '10 at 13:20
Just for the record, a new paper appeared on this topic: "Rolling Manifolds: Intrinsic Formulation and Controllability," by Yacine Chitour, Petri Kokkonen, arxiv.org/abs/1011.2925 . – Joseph
O'Rourke Nov 15 '10 at 14:52
add comment
1 Answer
active oldest votes
The rolling motion of a convex symmetric body on a horizontal plane is a classical problem. In the symmetric case, Chaplygin was the first who showed that the full equations of motion
can be reduced to a linear integrable system of two ODEs. A modern exposition of Chaplygin's results can be found in the very recent book by Cushman, Śniatycki and Duistermaat.
The problem of rolling motion on an inclined plane is, in general, nonintegrable (this problem was studied, in particular, by V.V. Kozlov in 1990s).
up vote 3 down As for the tracing trajectories of the point of contact, you might be interested in this article (also available on arXiv) and the references therein. The authors discuss the case of
vote accepted a disc (i.e. a convex body of revolution) rolling on a horizontal plane. From the Introduction:
It appears that the point of contact performs the composite bounded motion: it periodically traces some closed curve which rotates as a rigid body with some constant angular
velocity about the fixed point...
@Andrey: Great! This is just what I need. I like this quote from the "Dynamics of Rolling Disk" paper: "We also present various types of trajectories which are traced by the point
of contact in the body-fixed and relative frames of references since they have curious forms which are difficult to predict"! Some of their "curious forms" are illustrated in
Fig.8--neat! Thanks! – Joseph O'Rourke Jul 16 '10 at 14:14
You're welcome. – Andrey Rekalo Jul 16 '10 at 14:19
Thanks for adding the Kozlov pointer on inclined planes. I see that Kozlov and Federov have a new book, "A Memoir on Integrable Systems," springer.com/mathematics/analysis/book/
978-3-540-59000-2 . – Joseph O'Rourke Jul 16 '10 at 15:45
Yeah, but it looks like it hasn't been published yet. – Andrey Rekalo Jul 16 '10 at 16:29
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PGP for secure e-mail
by Nat Queen
[Note. This article is adapted from an earlier version that originally appeared in the February 2000 issue of Archive magazine.]
Have you ever thought about the privacy of your e-mail? A message sent over the Internet usually passes through several relaying hosts before reaching its destination. Anyone with privileged access
to any of those computers can easily read it, just as a post office worker handling a postcard in transit can read its contents. There is also a risk of hackers.
In August 1999, hackers discovered a way to breach the security of Hotmail e-mail accounts, and the details were made public on the Internet, thus putting the privacy of 50 million subscribers at
risk. The entire Hotmail system was closed down for a short time, while steps were taken to fix the problem. Can you be confident that your e-mail is secure?
Encryption is often used to ensure privacy. In the more traditional type of cryptography, the same secret key is used to encrypt and decrypt a message. Such a key must be exchanged before the message
is sent. However, this is of little use if you want to send a one-off confidential message to someone in a different part of the world. If you have a secure means of transmitting a secret key, you
might as well send the message itself!
Even if you can somehow exchange secret keys with all your correspondents (say, by slow postal mail), you would still need to exchange different keys with all of them individually. This would be very
Public-key cryptography
Public-key cryptography overcomes these problems. As explained below, it enables you to communicate securely with people you have never met, over insecure channels, without first exchanging secret
Perhaps you think you have nothing to hide and don't need secure e-mail. Would you ever find it embarrassing if your e-mail is read by your sysadmin, your employer, your ISP, an unknown hacker, or
government intelligence agencies? Do you ever use e-mail to transmit confidential information like business plans, character references, credit card numbers, political strategies or love letters?
Would you like to use digital signatures to ensure that your e-mail is tamper-proof?
If you can answer "yes" to any of these questions, you will find public-key cryptography useful.
Key pairs
Each user of a public-key cryptosystem has a pair of keys: a public key and a secret key. The public key can be made available to anyone who wants it. It is even advantageous to publish it in an open
directory, like telephone numbers in a telephone directory.
The secret key, as its name suggests, is kept secret (in practice, it's strongly encrypted in the user's computer with a passphrase).
The two keys are mathematically related in such a way that any message encrypted with the public key can be decrypted only with the corresponding secret key, and vice versa. Anyone can send you a
secure message by encrypting it with your public key. Since you are the only person who has access to your own secret key, no one else will be able to decrypt the message.
For such a system to be secure, it must be designed so that it is computationally infeasible to discover a secret key from a knowledge of the corresponding public key.
Digital signatures
A byproduct of such a cryptosystem is the possibility of creating digital signatures. To see how this is possible, suppose that you encrypt a message with your own secret key. Since the public key
reverses the action of the secret key, anyone with access to your public key can decrypt the message. If the message decrypts correctly, this proves that it was created by you, since nobody else has
access to your secret key.
Thus, digital signatures can be used to authenticate messages and prevent forgeries or tampering. If a single byte of a message is changed in transmission, the digital signature would not be valid.
Digital signatures based on modern cryptosystems are virtually impossible to forge in practice - much more so than ordinary handwritten signatures.
How it works
Public-key cryptosystems are based on what mathematicians call "one-way functions". A one-way function is a relation between two objects A and B such that B can be readily calculated from A, while
there is no computationally feasible way of determining A from a knowledge of B.
As an example, consider the relation N = pq, where p and q are prime numbers. (A prime number p is a whole number which has no divisors except 1 and p itself.) Even if p and q have several hundred
digits each, a simple program can be written for any modern computer to calculate their product N in a negligible amount of time. However, if only N is given, the problem of finding its prime factors
p and q would require many millions of years of computation, using any known technology.
The one-way function described above is essentially the basis of one of the most popular public-key cryptosystems, the so-called RSA system, named after Rivest, Shamir and Adleman, who proposed it in
1978. The extreme difficulty of finding the prime factors of huge numbers explains why it is not feasible to determine a secret key if the corresponding public key is known.
The RSA system has been implemented in PGP, the standard program for secure e-mail. PGP, which stands for Pretty Good Privacy, was originally created by Philip Zimmermann, the first person to make
military-grade cryptography available to the masses. Since then, PGP has undergone numerous revisions. Freeware versions of PGP exist for all major operating systems. With certain limitations, the
different versions are interoperable.
PGP provides facilities for generating new key pairs, encrypting or decrypting messages, checking digital signatures, etc. The user need not be concerned with the mechanics of these processes. PGP
automatically takes care of all the "bookkeeping".
Readers can download the program itself for many different computer systems, and also further information about PGP for beginners, from my page Introduction to PGP.
for general information about PGP.
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A Reader's Solution
January 1999
Here is Tom Holden's solution to
puzzle number 6
When you get your first coin, you are guaranteed to get a coin you have not yet got.
When you get you second coin, there is 1/22 chance of it being one you already have, so there is a 21/22 chance of getting one you do not yet have.
When you have two different coins, there is a 2/22=1/11 chance of getting one you already have, so there is a 20/22=10/11 chance of getting a new one.
When you have n-1 different coins (or when if you get a new coin it will be your n'th coin), there is a (n-1)/22 chance of getting one you already have, so the re is a 1-(n-1)/22 chance of getting a
new one.
Now the average total number of coins you need to get before you get the full set is the sum (for i=1 to 22) of the average number of coins you have to get before you get your i'th different coin. So
to work out the solution of the problem, we must work out how the average number of coins needed to get the i'th different coin is affected by the number i.
This can be done easily since we know the probability of getting a coin we do not already have is 1-(i-1)/22 and so the expected number of coins we need to take before we get a new coin is 1/(1-(i-1)
/2 2)=22/(23-i) and so the total number of coins is the sum (for i=1 to 22) of 22/(23-i)=22 times the sum (for i=1 to 22) of 1/i since changing the order of a sum makes no difference to its result
Now the answer to the problem is given by:
22*(1+1/2+1/3+...+1/22)=81 to the nearest whole number of coins.
So on average, you would have to get 81 coins before you had a whole set of 22.
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Kurt Godel
He turned the lens of mathematics on itself and hit upon his famous "incompleteness theorem" — driving a stake through the heart of formalism By DOUGLAS HOFSTADTER
Source:ALFRED EISENSTAEDT/TIME LIFE PICTURES-Kurt Godel at the Institute of Advanced Study See:
The Time 100-Scientists and ThinkersKurt Gödel
(IPA: [kuɹtˈgøːdl]) (April 28, 1906 Brno (Brünn), Austria-Hungary (now Czech Republic) – January 14, 1978 Princeton, New Jersey) was an Austrian American mathematician and philosopher.
The Time 100-Scientists and Thinkers
The upshot of all this is that the cherished goal of formalization is revealed as chimerical. All formal systems — at least ones that are powerful enough to be of interest — turn out to be
incomplete because they are able to express statements that say of themselves that they are unprovable. And that, in a nutshell, is what is meant when it is said that Gödel in 1931 demonstrated
the "incompleteness of mathematics." It's not really math itself that is incomplete, but any formal system that attempts to capture all the truths of mathematics in its finite set of axioms and
rules. To you that may not come as a shock, but to mathematicians in the 1930s, it upended their entire world view, and math has never been the same since.
Gödel's 1931 article did something else: it invented the theory of recursive functions, which today is the basis of a powerful theory of computing. Indeed, at the heart of Gödel's article lies
what can be seen as an elaborate computer program for producing M.P. numbers, and this "program" is written in a formalism that strongly resembles the programming language Lisp, which wasn't
invented until nearly 30 years later.
In the late 1940s, Gödel demonstrated the existence of paradoxical solutions to Albert Einstein's field equations in general relativity. These "rotating universes" would allow time travel and
caused Einstein to have doubts about his own theory. His solutions are known as the Gödel metric.
Closed timelike curves
Because of the homogeneity of the spacetime and the mutual twisting of our family of timelike geodesics, it is more or less inevitable that the Gödel spacetime should have closed timelike curves
(CTC's). Indeed, there are CTCs through every event in the Gödel spacetime. This causal anomaly seems to have been secretly regarded as the whole point of the model by Gödel himself, who
allegedly spent the last two decades of his life searching for a proof that death could be cheated, and apparently felt that this solution provided the desired proof. This strange conviction came
to light decades after his death, when his personal papers were examined by a startled astronomer.[citation needed].
A more rational interpretation of Gödel's motives is that he was striving to (and arguably succeeded in) proving that Einstein's equations of spacetime are not consistent with what we intuitively
understand time to be (i.e. that it passes and the past no longer exists), much as he, conversely, succeeded with his Incompleteness Theorems in showing that intuitive mathematical concepts could
not be completely described by formal mathematical systems of proof. See the book A World Without Time (ISBN 0465092942).
General Relativity
CTCs have an unnerving habit of appearing in locally unobjectionable exact solutions to the Einstein field equation of general relativity, including some of the most important solutions. These
* the Kerr vacuum (which models a rotating uncharged black hole)
* the van Stockum dust (which models a cylindrically symmetric configuration of dust),
* the Gödel lambdadust (which models a dust with a carefully chosen cosmological constant term).
* J. Richard Gott has proposed a mechanism for creating CTCs using cosmic strings.
Some of these examples are, like the Tipler cylinder, rather artificial, but the exterior part of the Kerr solution is thought to be in some sense generic, so it is rather unnerving to learn that
its interior contains CTCs. Most physicists feel that CTCs in such solutions are artifacts.
Timelike topological feature
No closed timelike curve (CTC) on a Lorentzian manifold can be continuously deformed as a CTC to a point, because Lorentzian manifolds are locally causally well-behaved. Every CTC must pass
through some topological feature which prevents it from being deformed to a point. A test particle free falling along a closed timelike geodesic transits this feature; in the test particle's
frame, the feature propagates toward the test particle. This features resembles a glider in Conway's Game of Life, but in a continuous spatial automaton rather than a (discrete) cellular
Continuous spatial automaton
It is an important open question whether pseudo-photons can be created in an Einstein vacuum space-time, in the same way that a glider gun in Conway's Game of Life fires off a series of gliders.
If so, it is argued that pseudo-photons can be created and destroyed only in multiples of two, as a result of energy-momentum conservation.
2 comments:
1. Hi Plato,
Nice piece about Godel and his speculations about a rotating galaxy and how it relates to the nature of time. He truly was one of the great minds of the ages; and yet although he tried never did
see the world as he would have liked, in as he believed true as Plato and Descartes had. To serve perhaps as evidence of this, I leave you with a few quotes from Rebecca Goldstein's
book on Godel called [Incompleteness (The Proof and Paradox of Kurt Godel)-Atlas Books-2005-pages 259-260].
(what is contained in single quotation marks is what was said by Dr. Goldstein and in double by Dr. Wang)
“It is tempting to connect Godel’s attraction to these closed time loops with a passing remark that Hao Wang made, indicating how Godel had thought as his life as incomplete:”
“”In philosophy Godel has never arrived at what he looked for: to arrive at a new view of the world, it’s basic constituents, and the rules of their composition. Several philosophers, in
particular Plato and Descartes, claim to have had certain moments in their lives an intuitive view of this kind totally different from the everyday view of the world.””
“And, again Wang made reference to a transcendental experience that Godel had awaited all his life:”
“”He also looked for (but failed to obtain) an epiphany (a revelation or sudden illumination) that would enable him to see the world in a different light. (In his conversations with me, he
repeatedly said that Plato, Descartes, and Husserl all had such an experience).””
There is much more in this book that relates to your topic and our shared mutual concerns and if you haven’t already read it I would recommend it most highly. What stands out for me in this
regard is what I consider to be perhaps the reason for Godel’s failure as it appears. That is he was actually caught in the limitations of his methods of reason; for though he recognized and
proved that the axiom’s of mathematics although powerful could not provide the tools necessary to actual probe and then actually realize the depth to which it had. So like the axioms, Godel
himself was limited by his own incompleteness.
2. Hi Plato,
Sorry, I just noticed I have made an error, for I should have said a rotating universe not galaxy. This then stand as yet another proof that we all have our limitations; one of my many being not
to be able to proof read properly my own words. :-)
Editors of the world fear not, for a computer doesn't exist that has demonstrated the abilities required to replace you ;-)
P.S. The comment deleted above again serves as further testament to my claim :-<
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Zero Before the Decimal Point?
Date: 10/06/2000 at 09:22:14
From: Joe Bouril
Subject: Zero represented before the decimal
A fellow instructional designer and I are having a philosophical
dispute over the purpose and philosophy of textbooks and teachers
showing the zero in front of a decimal number. (I have a teaching
certificate and an M.A. in instructional design, while he has a Ph.D.
in math.)
"0.34 x 0.298 =" or "0.562 / 0.25 ="
As a former elementary teacher for 12 years, I contend that the zero
is redundant to the decimal, is a harmful crutch, and can even be
distracting. The prerequisite skills up to the decimal include mastery
of place values, and the comprehension of the decimal point. However,
I still see it used in classes past the fourth grade.
I agree with him that when used in a text sentence, it makes sense to
distinguish punctuation. I also agree that when the decimal point is
initially introduced, it can be used as a transition.
I disagree with him that when used in a stand-alone math problem, it
is still visually pleasing and, I guess, therefore better for the mind
to compute accurately.
Please explain and arbitrate.
Joe Bouril
Date: 10/06/2000 at 15:21:19
From: Doctor Peterson
Subject: Re: Zero represented before the decimal
Hi, Joe.
My main problem with your reasoning is that you evidently think that
redundancy is bad; I think it's good. Take another example, the use of
parentheses: There are many situations where we don't need parentheses
because the rules make the meaning clear; yet I recommend putting them
in, just to make it easier for the reader, and to ensure that if a
rule is forgotten, it will be hard to misread it. (This is commonly
recommended for good computer programming practice, where there are
extra rules that are easy to forget.) Sure, one could use extra
parentheses as a crutch, putting them in everywhere to avoid having to
learn the rules at all; but that's only in the extreme. Crutches are
useful to those who need them.
In the case of the zero before the decimal point, although it doesn't
affect the value of the number, it is almost universally considered
good practice to include it - not for mathematical reasons, but for
human reasons. It can be easy to miss the decimal point, and the zero
makes it stand out. It's not essential, but it's a good habit to
develop for those occasions when clear communication will be
Just to make sure this was not my own quirk, I did a quick search for
other people who might make the same - or opposite - recommendation.
Here are a few:
First, in writing drug prescriptions:
Good Prescribing Guidelines (Westmead Hospital Department of
Pharmacy - Khai Bui)
"Never leave a decimal point naked, such as .5 mL. When the decimal
point is not seen, a tenfold overdose may occur.
"When a decimal fraction must be prescribed, always write a zero
before the decimal point.
"Never put a decimal point and zero after a whole number such as
2.0 mg. This should be written as 2 mg. If the decimal point is not
seen, a tenfold overdose may result."
Medicine, Malpractice and the Law (a paper by Raymond Wacks)
"The expression of drug dose and units should be clear. For whole
numbers it is better to avoid following with a decimal point and a
zero which may be misinterpreted as ten or one hundred times the
appropriate dose. For numbers less than one it is essential to
place a zero before the decimal point."
Similarly, in product labels:
FDA Guidelines (U.S. Government FDA/ORA Compliance Policy Guides)
"A zero before the decimal point should be used in numbers between
1 and -1 to prevent the possibility that a faint decimal point will
be overlooked.
"Example: The oral expression "point seven five" is written 0.75."
Next, in the metric system:
Metrics the Right Way (George Sudikatus, ICF KH Metric Coordinator,
Pacific Northwest National Laboratory)
"In the United States, the standard decimal marker is a dot on the
line (i.e., a period or 'decimal point'). When writing numbers less
than one, add a zero before the decimal marker. For example, on a
drawing you might define a small length in English units as
.032 in., but write the metric length as 0.81 mm."
Also, some publishers and organizations include it as part of their
style guides:
Instructions for Authors (Publisher: Taylor and Francis Group)
"Use a zero before the decimal point for numbers less than one. For
t = 0.40
"However, do not use a zero before the decimal point when the
number cannot be greater than one. This occurs with correlations,
proportions and levels of statistical significance. For example:
r = .27, p < .01
On the other hand (!):
Preparing Manuscripts for Demography (Department of Demography,
Georgetown University)
"Decimal fractions should not include a zero before the decimal
point (e.g., .05 is correct; 0.05 is incorrect)."
These are not carefully chosen references, just those that I found in
a quick search. They should suggest that inclusion of the zero is a
common, though perhaps not universal, practice, and has good reasons
behind it. Therefore, I think it is appropriate for students to become
familiar with this style. We can let them become lazy later - if they
don't become pharmacists.
On the other hand, I don't think I would require them to always put in
the zero themselves; and I would make sure they saw numbers written
without the leading zero to make sure they knew it meant the same
- Doctor Peterson, The Math Forum
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Is Richard Dawson older than Diana Dors?
You asked:
Is Richard Dawson older than Diana Dors?
Diana Dors
Diana Dors (23 October 1931 - 4 May 1984), the English actress and sex symbol
Richard Dawson
Richard Dawson (born November 20, 1932), the English-American actor, comedian, game show panelist, and host
Assuming you meant
• Richard Dawson (born November 20, 1932), the English-American actor, comedian, game show panelist, and host
Did you mean?
Say hello to Evi
Evi is our best selling mobile app that can answer questions about local knowledge, weather, books, music, films, people and places, recipe ideas, shopping and much more. Over the next few months we
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[FOM] Hilbert's Vollstandigkeitsaxiom and Hilbert's Hotel
John Baldwin jbaldwin at uic.edu
Sat Jan 27 19:00:43 EST 2007
Note that the completeness is not of the axioms but of the model. This
is a formulation of what we now call the completeness of the real numbers.
It is equivalent to the least upper bound axiom. Such a second order
axiom is necessary to characterize the reals as an ordered set.
I hope someone can respond to Rodin's historical questions.
On Sat, 27 Jan 2007 Andre.Rodin at ens.fr wrote:
> Dear FOMers,
> I have few questions concernig Completeness axiom (Vollstandigkeitsaxiom) of
> Hilbert's "Grundlagen der Geometrie":
> "To a system of points, straight
> lines, and planes, it is impossible to add other elements in such a manner that
> the system thus generalized shall form a new geometry obeying all of the five
> groups of axioms."
> This axiom absent in the first edition of Grundlagen of 1899 appears in the
> second edition of this work. Earlier Hilbert used a similar axiom in his
> lecture "Ueber den Zahlbegriff" delivered in 1899 and published in 1900.
> My first group of question is historical: Are there historical evidences
> explaining Hilbert's motivation behind the introduction of this axiom? How
> exactly Hilbert formulated the problem the Vollstandigkeitsaxiom was supposed to
> treat? How he discovered this problem?
> The second group of questions concerns the axiom itself. Obviously Hilbert had
> in mind an infinite model of his geometry: points, straight lines and planes
> are infinitely many. Whatever is the cardinality of this model M, it is always
> possible to intruduce into M new elements - up to a countable number of new
> elements - and so obtain another set M' of the same cardinality. This property
> of infinite sets is often associated with Hilbert's name through the popular
> story of "Hilbert's Hotel". Hilbert's Vollstandigkeitsaxiom implies that
> although M' is isomorphic to M as a set M' unlike M is NOT a model of the
> theory in question. Hilbert provided no justification of why M with the
> required maximal property should exist (in any appropriate sense). Perhaps he
> didn't really think about models in set-theoretic terms. I wonder how the
> problem looks like from the today's viewpoint. Can Hilbert's intuition
> concerning the existence of maximal models of mathematical theories be
> justified? Was it plainly wrong? How "completeness" in Hilbert's sense relates
> to semantic completeness? How it relates to categoricity? (Hilbert's
> completeness implies categoricity but not the other way round. Can one say
> more?)
> I will be grateful for any hint or a reference.
> Thanks in advance
> Andrei Rodin
> _______________________________________________
> FOM mailing list
> FOM at cs.nyu.edu
> http://www.cs.nyu.edu/mailman/listinfo/fom
John T. Baldwin
Director, Office of Mathematics Education
Department of Mathematics, Statistics,
and Computer Science M/C 249
jbaldwin at uic.edu
Room 327 Science and Engineering Offices (SEO)
851 S. Morgan
Chicago, IL 60607
Assistant to the director
Jan Nekola: 312-413-3750
More information about the FOM mailing list
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Posts by
Total # Posts: 111
MgO (Magnesium Oxide)?
Is this some kind of survey and you're gathing data from people's responses? ---Well, you'll have to do some research on your career path. Ex: If you are a science major, you better take lot of
biology, chemistry, biochemistry, genetics classes and of course lot of...
Um... it uses dendrite to receive signals (inputs) from many other cells because dendrite has many branches of nerve endings. Axons are joined to the dendrite from the head body so the information
(message) can be facilitated to the head body. This is what I think, at least.
Translate these info into equations: y=3/5x x=5/3y y+140=2x Now you can use substitution method to solve for x easily.
crt 205
Okay? I believe I don't see any data or question anywhere here except your little introduction.
Ex: 1^(1/2) = square root of 2 of(1^1) = sqrt (1) = 1 Ex: 2^(3/2) = sqrt of 2 of (2^3) = sqrt (8) = 2 sqrt(2) Basically do the exponent normally for the numerator and do a square root of the
denominator number.
Don't know. Can I suggest you open up your book and look at the application problems section on the polynomial chapter? Those word problems should have real life cases.
Okay, this is what I think... it seems to be too easy. So the reaction produces 6 moles of CO2 and 6 moles of H2O, that mean they have a ratio of 1 to 1. In other word, when you have 174 moles of
H2O, then you'll also have 174 moles of CO2... so there isn't really any ...
In humans, melanin is the primary determinant of human skin color. So if you are tanned, maybe that's why.
You probably want to look at topics like the evolution of whales... good luck searching! I think their legs turn into the anal fins and the hands turn into the pectoral fins and they need fins to
adapt to their new oceanic environment. They probably go to water because they ha...
I haven't done this in years... but looks like you can do this using the formula PV = nRT if I remember correctly where P = pressure in atm, V = volume in L, n = number of moles, R = 0.0821 (atm*L*K)
/mol "I'm not so sure about the unit on R", and T = temperat...
The way you put it, there's no multiplication... so I don't know where you get 3p and 2p... maybe you want to revise your original problem by addiing in the parentheses. Anyway, by the way you put
it... it should be: p-p-1/2-2/3=1 (combine like-terms) 0-3/6-4/6=1 (comm...
Total cookies baked = 2*20 + 10 = 50 cookies 50x = 797.30 x= $15.95/cookie... wow that's expensive! Okay... this is actually the equation of how much each cookie cost and assuming that the bakery
sold all 50 cookies. This is not how many cookies sold... I don't think i...
Try to multiply first equation by 3 and second equation by 2. You will end up with 6x and -6x, you can eliminate those and then use the substitution method to find y.
First, convert your equations to y=mx+b form and then you can graph each of them into your graph individually. Where the two lines intersect, that's your answer.
9, 18 and 27
I believe all these properties can be found easily just by typing it in the google search box... Example, I pasted this exactly as it is from a site I found: Associative Property of Multiplication
The property which states that for all real numbers a, b, and c, their product i...
Algebra 1
1 and 31 because it is a prime #?
You sure you have it the right way (1 in = 0.01 in.) and not the other way around? The way you say it, Lin actually has to draw 100 times bigger than the actual size of the computer diode... usually
when you drawing a miniature, you are drawing with the picture smaller than th...
Ex: 1 2/3 Numerator = 1x3+2=5 Keep Denominator Answer: 5/3
6(y^2-4) 6(y-2)(y+2)...
LDH catalyzes the reaction between pyruvate and lactate. Pyruvate is important because it involves in glycolysis (glucose metabolism). Lactate is important because you can use lactate to quickly
generate the energy you need during intensive exercising. Hope this helps!
I just know that in order to be a ketone, this functional group looks like this C-C(O)-C so it must have at least 3 carbons to be a ketone... I don't know if you have to explain using
eletronegativity or dipole...
Um... I can't understand what you are trying to say. Please add the commas and periods to make it easier to read... maybe someone else can understand, but not me. Just pointing that out in case
someone else has the same problem as me, no offense.
I vote for bent.
1g = 100 cg --> 0.25g x 100cg/g = 25 cg 1mL = 1 cm^3, 1L = 1000mL, you figure the rest out. 1 m = 100 cm, 1 m^2 = (100 cm)^2 = 10,000 cm^2, you figure the rest out. Good luck!
D --> London's dispersion force.
Physical Science
1) Find the volume: V = (4/3)(pi)(r^3) 2) Density = mass/volume There you go... that's all, 2 steps!
I haven't done chemistry for a while, but here's my best guess: M = mol/L so 0.120 M means there is 0.12 moles of Ca(NO3)2 in a liter. The question ask for 500 mL, which is half of a litter, so I
would just divide 0.12 by two and get 0.06 moles of Ca(NO3)2 in 500 mL. A...
.. 3sqrt6 and sqrt2a^4? So you simplified the problem first as I said above and you get: 3(a^3)sqrt(2a) * 3(a^4)(b^3)sqrt(3) and then you multiply those 2 things together and you get: 9(a^7)(b^3)sqrt
Plug your 0 into the x in your equation and you get 1 = -1, which is false, which mean: no, the answer is not x = 0. Since you didn't show your work, I have no idea what you did wrong on your
process... or maybe you just guess 0 and didn't do any work at all. But I'...
... 3sqrt6 and sqrt2a^4? So you simplified the problem first as I said above and you get: 3(a^3)sqrt(2a) * 3(a^4)(b^3)sqrt(3) and then you multiply those 2 things together and you get: 9(a^7)(b^3)
18 = 9 * 2 then sqrt 18 = 3sqrt2 sqrt a^7 = a^3 * sqrt a 27 = 9 * 3 then sqrt 27 = 3 sqrt3 sqrt a^8 = a^4 sqrt b^6 = b^3 Now just multiply the like-terms together.
One to two
I meant, I was wondering that too, so thanks even though it isn't my question.
Alright thanks!
A base pairs with T, G with C. In RNA, A pairs with U and G pairs with C. I think you just need to do some more readings... most of these can be answered in your book or note for sure.
DNA is double stranded. It splited out into two single strand and produce another new strand and that created more DNA. Next, the DNA sequences are translated into amino acids. For every 3 base
sequences, that would represent a single amino acid. There are specific sequences f...
1. 10-99 are two digits # and that mean there are 99-10+1=90 possible numbers (and of course they're all less than 100). and so from 45-90, there are 46 numbers. 2. a and b looks right. There are 9
different letters in the word PROBABILITY so it should be 9/26. 3. Count th...
DNA is double strand, during transcription, they (the parents or original strands) separate into two single strands. Then the new strand will attach to each of the parent strand and form a new DNA
molecule and because this new molecule consist half the parent and half the new ...
Algebra II
1. y=kx plug in your # and solve for k 2. y=k/x solve for k and then plug in your y=4 to find x with the k value you found. 3. I forgot how to do this one.. it's been years lol.
Step one: combine like-term - put all the x on one side and all the whole number on the other side by addition/subtraction. Step two: get x on one side by division/multiplication. Ex: 3x+5=2x-3
Subtract 2x and -5 on each side of the equation 3x-2x + 5-5 = 2x-2x - 3-5 x = -8
I did this experiment in my science class while I was in middle school: Fill a glass with water and then put a piece of cardboard on top of it and then slowly turn the cup upside down and ta-da, the
cardboard will be able to hold the water from leaking out. Okay, empty the gla...
x-ray? Lol
On the step where you say y=-4y is where the problem is. Re-check that part and do it again. Everything else seems right except that you are not doing what the problem is trying to tell you to do.
You are supposed to solve by graphing, but here, you are doing substitution.
ummm... your body gives off heat and warm up the air and the warm air just circulate around your body inside your cloth to keep you warm?
advanced fuctions
I haven't done log for a while, but can't you just use subsitution and set up an equation like this? log(x-2)=1-log(x+1) and then solve for x? log(x-2)+log(x+1)=1 and I think there's a rule that let
you combine those... like multiplying them or something and go fro...
I thought constant and control are the same thing. Something that you keep constant, which mean you don't change it. The opposite of that would be a variable, which mean it changes and usually that's
what you're testing for.
1) Distribute ex: 2(5+x) = 2*5 + 2*x = 10 +2x 2)Combine like-term 3)Solve for h
(y-1)(y-6)/ (y+9)(y-1) No... you factor out and then the (y-1) cancel off and you'll end up with y-6/ y+9
algebra 2
A. Find the slope (rise/run). Plug x and y into your equation with the slope to find the y-intercept and then you'll have your y=mx+b equation. B. Should look like a "V" shape open upward starting at
the vertex at (0,0). Basically "x" can be any number ...
90 + (x-25) + (x+25) + (x-30) = 360 90 + x - 25 + x +25 + x - 30 = 360 60 + 3x = 360 3x = 300 x = 100 The concept of this problem is to recognize that ANY QUADRILATERAL (square, trapazoid, rhombus,
rectangle...etc) have a total angle of 360 degrees, so you add them all up and ...
16F=mg Solve for m. m*2.2 = weight in lb.
I always put the equation in the slope y-intercept form when I worked on these equations... some other people prefer other forms, which also works. However, I'll just show you my way since I know it
best. So the form we want to translate to looks like this: y = m*x + b Whe...
I suppose you are missing the "x" in your functions. Just graph it as you normally would, which mean graph y = (-3/2)x + 8. Then pick on any coordinate point that's NOT on the line for testing, just
FYI, I always use the point (0,0) just because it makes the math...
1) There's no way to do this without knowing the volume of the myocyte, so I would recommend you to search the book or your notes, should be somewhere in there. Basically, you would approach this
question by finding the volume of the actin molecule first. V = 4/3*pi*(1.8nm...
Good job! Sorry I posted the same thing twice because my net froze.
Well... as you said, you get v^16... so would it be 9+7=16 or 9*7=16?
Well... as you said, you get v^16... so would it be 9+7=16 or 9*7=16?
Is your submarine rectangular? If so, just multiply the two numbers... I can hardly imagine you would have to do calculus with the information given for a more accurate area of the submarine's actual
Looks like it's D, but B is quite goofy. If they mean the mass of the elements in the reactant side only, then I think it's true, of course that also including the number of moles of each element in
the balanced equation. Just a side note, in a real world, D is never t...
Let's see if this will help you visualize what Bob said: * = 1 valence electron H = Hydrogen There are 2 hydrogens atoms: H* and H* And they (the atoms) form a covalent bond by sharing the electrons:
H**H --> now each hydrogen is much happier with 2 valence electrons in...
8<w+3<10 8-3<w<10-3 5<w<7 Your answer should be (5,7) --> this mean any numbers between 5 to 7 except 5 to 7 themselves.
Algebra 2
Oh, and now you said you are at 56.7% I can give a rough prediction with the if...then statements for you, that way you don't have to ask your teacher and dig out all your tests to tell me the
results and stuffs. If you do not do anything different than what you did thus f...
Algebra 2
I don't think anyone of us can give you that answer without knowing the method of how your teacher keep tracks of score. For example, does your teacher treat all points weight equally? This is when 1
point in a test is equal to 1 point in the assignments and everything els...
Your question has already been answered... check your previous posting before you post another one please.
9 and 15 are the numerators 12 and ? are the denominators Start the problem by setting up an equation by multiplying the numerators with the denominators like this: 9 x ? = 12 x 15 Now just solve for
-6t=12 t= ?
Let me first convert them into the same y = mx + b form: -->y = -(3/4)x - 1 -->y = -(3/4)x + 5 Okay, slopes are the same so you know they're parallel to each other. Looking at the y-intercept you
know that one has a coordinate of (0,-1) and the other has a coordinate...
Where is the /x comes from on your x^2+2x/x... you were /2 up there on the above step... and even if the problem is f(2)/x... your f(2) = 0, that mean you get 0/x which equals 0, thus leaving you
only x^2+2x on the left side, which equals x(x+2). Basically, I disagree with wha...
Hmm... that should be the same answer because f(2) = 0 and 0 divide by any number is 0 and "x" cannot be zero.
=[(x+2)^2 - 2(x+2)] - [(2)^2 - 2(2)]/2 =x^2+4x+4 - 2x-4 - (4-4)/2 =x^2+2x - 0 =x(x+2) = 0 or -2 I have the x+2 part of your "correct answer", but I also got the "x" in there. That's the answer I came
up with, if you don't understand how I arrive my ...
Algebra- functions
When you put the function into your graphing calculator, just make sure you can see the whole window so you can see all the turns. Whenever you see your graph intersects with the x-axis, you have 1
zero... so just go ahead and count how many times it intersects the x-axis. For...
Math HELP please
sq24 = sq4 x sq6 = 2 x sq6 sq = square root of
Can you type out the problem? Maybe then people can actually help you. By reading what you said it sounds like it's sq (x^2/x-6) = 0... but I guess that's not what the problem looks like, be sure you
indicate where the parentheses are in your problem.
Yeah, half of 66 is 33.
Why Jiskha Is Full of Crap
Even I got my biology degree, I don't feel like I know most of the questions posted here. I just don't want to give a guess as an answer so I usually choose not to answer them. However, if I
encountered questions that I know how to deal with, I will answer them for you...
Um... I thought any non-whole number is considered a prime number so I say yes.
Lol! Capillaries?
...find the slope? I'm not sure what you don't understand here. I mean, the vertex coordinates are given on the diagram right? Then it's easy. Just find the slope on each side and if the slope on
both sides are the same then that proved they're parallel and thu...
Math is this right
Yeah, I think you get the idea. I just want to make sure your answer mean I/(Pr)=t not (I/P)r = t. Again, I think you got it right, but the way you typed your answer on the computer makes it look
like (I/P)r = t, which is wrong.
2=2x+4 -2=2x -1=x One solution?... if I read the question correctly.
Fe2O3 + 3CO --> 2Fe + 3CO2 This one is difficult, took me about 10 minutes, but it's pretty simple in a way that you just plug in number and do trial and errors...
The number of all the elements on the left side of the equation are equal to the number of the corresponding elements on the right side of the equation, thus no additional or less atoms are created
nor destroyed. For example, in this carbonic acid reaction where hemoglobin sto...
aqueous solution.
ap biology
Not sure... but if I have to guess, I'd say positive directional selection because 5 or 6 eggs seem like a big number. Positive directional selection favors those have higher trait values, in this
case, they produce more eggs, thus have higher fitness than other Starlings ...
Chemistry - Titrations
Distill water has pH = 7 and NaOH is a strong base (pH > 7)... if you mix NaOH with water, it will neutralize some of the base (decrease the pH to be closer to 7) and thus contaminate the NaOH
Introduction, hypothesis, data, result, conclusion.
ap biology again
Yeah, that's a bit confusing. Here is what I think they're trying to say: -Because unmethylated bacteriophages will be degraded by the restriction enzymes found in bacteria, natural selection will
favor those bacteriophages that have methylated DNA sequences (which ena...
Hehe, I didn't have to take ACT or SAT or anything while I was in high school. I don't know this is an option in any other states, but in Minnesota, we have a Post Secondary Program (PSEO) where you
can attend college during your junior and senior years so you can earn...
All organisms are constantly evolving. What they evolve to or what trait or characteristic they will develop in the future is uncertain and hard to predict in most cases. However, according to
Darwin, in short-term evolution, we can predict that organisms lower in fitness will...
ib math studies
I'm not sure what you meant by mathematical processes... but maybe measuring their velocities and times and then do a calculation on their accelerations? You can also apply math to find out their
gasoline utilization efficiency. Measure how far a car can go (distance) befo...
Small sedan will experience a larger force because the truck is heavier (more mass and more momentum if they both travel at equal velocity) so it's not going to move much from the collision.
x^2 - 4 = 2x-10 x^2-2x+6 = 0 There you go... solve by factor or that quadratic formula.
The word "line" is vague... you need to be specific. A line can be a segment, which has an end or a ray which never ends. Hope I answered your question.
0.15x=230 Transferring into words: 15% of the original price is equal to 230 dollars.
Not really for homework help, but just for curiousity. Is there a way to distinguish whether the graph is a parabola or a hyperbola if you do not know the equations and one side of the hyperbola is
missing (so there's just one U shape) by restricting the domains or ranges....
divide pi and r^2 on both sides of the equation.
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How I learned to stop worrying and love statistics
I've always hated statistics. Nothing smells more like accountancy, rimmed glasses, bookkeeping, and horrible little news reports than statistics. A career in statistics was the image of compiling
endless financial reports to a stony board of directors in an attempt to squeeze out those few more dollars from the public. It was the lowest. It was the selling of an mathematical mind to the
machine and the end of all beauty and expanse. There was no doubt in my mind that statistics was simply evil.
So it was a mysterious change when it happened, and it all began with a search engines module at University. This was easily one of the best courses I took in my time at University and from the
beginning of the course what became most clear was that making an effective search engine had nothing to do with understanding the English language, with extracting semantic meaning from queries or
documents, with logic, reason, or human experience. It was all to do with raw, unadulterated statistics.
And suddenly I saw the glint of gold. I saw a promise in statistics. Hiding beneath dusty logarithm lookup tables and hypothesis testing was the promise of an Oracle Machine. Something that could be
queried and provide answers in milliseconds. This was knowledge like had never been seen before and yet it was nothing to do with knowledge, logic, semantics, or meaning. It was just numbers, just
data and statistics and a query box. Ultimately the question in my mind was "how can this be?", and secondly, against my better judgement, "how can I get it?".
An internet search engine relies on the systematic de-construction and processing of text. The text is crippled; stripped of meaning until it is completely void and will fit into nice neat data
structures for processing. Only then would the data shine through. And once the numbers were ready, the statistical algorithms could roll along and process the data. Finally the questions we all had
could be answered in the blink of an eye. Building a search engine is not re-inventing the wheel, it is rediscovering the holy grail.
The first thing to go is syntax. The hierarchy of language, which structures and subjugates words into a towering tree, is unimportant under statistics. All web pages, documents, and queries are
reformed and stored as jumbled lists of words. Context is not truly lost. Those words which often are together are still in association via their combined presence in a list. Everything is just a
little more anarchic. The words have been freed of their sentences. There is no longer a primary verb, or a root pronoun. Under the statistical system all words are equal, and as you would expect,
some are more equal than others.
The important words are those which do not occur often. "The" is largely a useless citizen; syntactic glue. No room is left in our system for such common words and where possible they are removed.
The "aardvarks" and "armamentaria" are king, because you can be sure if they exist in a query then they must be key. So how are these statuses assigned? Not by some governing hand. We look toward the
Laws of Text, Zipf's law and Heaps' law. These laws tell you, in beautiful fairness and balance, the relative importance of words in a language. Even the numbers and numerals can be governed using
Benford's law. Nothing is left to chance, all is mathematical.
But all this begs the question. Do we really need words in the first place? Is this bureaucracy? Can something smaller suffice - say, a symbol, a letter? In languages such as Japanese, with no spaces
to separate words, we can simply assume that each overlapping pair of symbols, as well as each individual symbol, is a word in its own right. As we accumulate more and more web pages and documents,
the pairs which are actually words will continue to appear, while those which are not words will not. It soon becomes clear what is, and what isn't a word. This system, of taking N-grams, sets of
symbols is effective. Even more effective than just splitting via words. Even in European languages. The reason is we can match policeman with policemen, even with no idea of the semantic
relationship. Words are not required. A good search engine can use just symbols.
We note that in using N-Grams, the more documents you have the better. This is another devilish aspect of statistics. In statistics, more is more. You can never have too much data. The reason is
simple. Signal adds up and noise cancels out. More precisely, when you have more data, the probability of something becoming statistically significant via chance is lessened, while the probability of
something becoming statistically significant via actuality, is increased. In a logical formula, the smaller the formula the better. But in our search engine, the more websites scanned the better -
even if what they contain is largely junk.
So now our documents are simply jumbled lists of words and their relative importance. We have spiders crawling the web and accumulating more data for us, and we have an index slowly ticking over and
processing the document data. All that remains now is to design the statistical models via which we rate our documents for a given query. Because of our destruction of the text we can build effective
data structures and feed them into a huge database. The final step is just to turn it on.
A human-free system. A system of knowledge automated by the cold clicking hands of a computer.
The secret in statistics is rather simple. The power it provides is a concise mantra. In logic, deduction and mathematical proof one can divulge true answer to precise questions. Statistics, on the
other hand, can provide compelling answers to all questions. Statistics focuses on the question rather than the answer. As Douglas Adams revealed, if you wish to know the answer to the meaning of
life the universe and everything, you must first know the question.
The difference can be shown with a trick. When queried with the question...
"How many legs does the average person have?"
• A logical system will answer ~1.99
• A statistical system will answer 2
A good logical system will know the answer to a question.
A good statistical system will know what question you are asking.
This is both the beauty and the danger of statistics. Much like a search engine, the goal of a statistical system is to tell you exactly what you want to hear. A statistical system does not answer a
question with the precision and truth of a logical system, but it should capture the absolute intuition of what you are asking from it. It will know when "average" is translated to "mean" and when
one really intends for the "mode". When a CEO asks his statistician "is the company doing good?", a good statistician will formalise and calculate the exact notion the CEO holds of "company doing
good", and present it to the CEO.
The danger comes when the intuitive notion of "company doing good" differs from person to person. Perhaps the CEO is unconcerned with the variable counting toxic waste dumped, while a citizen rates
this variable highly on their intuition of that evaluation. The power of statistics comes from its subjectiveness and lack of true meaning, but it is also its heel.
What really is the "mean" or the "standard deviation" other than the formalisation of some human intuition? In exact terms the mean is not "the average" because as we discussed above, that is a
subjective and relative notion. The mean is only itself - that is the sum of all data points divided by the count of data points. The same is true for search engines. If I search for "The Best Page
In The Universe" Google does not return the best page in the universe. It returns the tf.idf weighted sum of my query terms against its index including user ranked weights, individual behaviour
weights and pagerank.
Statistics is not boring. Far from it. At its heart it is the beautiful and twisted cousin of logic and reason. It is deceptively powerful. It gives you the chance to throw your pennies in the well
and get an answer back. Most of all, statistics is agnostic, subjective and human. Unlike the godlike sentience of logic and reason, statistics is the devil inside. For that reason I love it.
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Approximation-solvability of Hammerstein equations
P. Milojevi\'c
Department of Mathematics and CAMS, New Jersey Institute of Technology, Newark, New Jersey, USA
Abstract: We study Hammerstein operator equations of the form $$ x-KFx=f \eqno (1.1) $$ where $K$ is linear and $F$ is a nonlinear map. We first study Eq. (1.1) in the operator form using the
(pseudo) $A$-proper mapping approach and the Brouwer degree theory. Then we apply the obtained results to Hammerstein integral equations.
Keywords: Approximation solvability, (pseudo) A-proper maps, surjectivity, elliptic, hyperbolic equations
Classification (MSC2000): 47H15, 35L70, 35L75; 35J40
Full text of the article:
Electronic fulltext finalized on: 1 Nov 2001. This page was last modified: 16 Nov 2001.
© 2001 Mathematical Institute of the Serbian Academy of Science and Arts
© 2001 ELibM for the EMIS Electronic Edition
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[Numpy-discussion] numpy.percentile multiple arrays
questions anon questions.anon@gmail....
Tue Jan 24 21:49:46 CST 2012
thanks for your responses,
because of the size of the dataset I will still end up with the memory
error if I calculate the median for each file, additionally the files are
not all the same size. I believe this memory problem will still arise with
the cumulative distribution calculation and not sure I understand how to
write the second suggestion about the iterative approach but will have a go.
Thanks again
On Wed, Jan 25, 2012 at 1:26 PM, Brett Olsen <brett.olsen@gmail.com> wrote:
> On Tue, Jan 24, 2012 at 6:22 PM, questions anon
> <questions.anon@gmail.com> wrote:
> > I need some help understanding how to loop through many arrays to
> calculate
> > the 95th percentile.
> > I can easily do this by using numpy.concatenate to make one big array and
> > then finding the 95th percentile using numpy.percentile but this causes a
> > memory error when I want to run this on 100's of netcdf files (see code
> > below).
> > Any alternative methods will be greatly appreciated.
> >
> >
> > all_TSFC=[]
> > for (path, dirs, files) in os.walk(MainFolder):
> > for dir in dirs:
> > print dir
> > path=path+'/'
> > for ncfile in files:
> > if ncfile[-3:]=='.nc':
> > print "dealing with ncfiles:", ncfile
> > ncfile=os.path.join(path,ncfile)
> > ncfile=Dataset(ncfile, 'r+', 'NETCDF4')
> > TSFC=ncfile.variables['T_SFC'][:]
> > ncfile.close()
> > all_TSFC.append(TSFC)
> >
> > big_array=N.ma.concatenate(all_TSFC)
> > Percentile95th=N.percentile(big_array, 95, axis=0)
> If the range of your data is known and limited (i.e., you have a
> comparatively small number of possible values, but a number of repeats
> of each value) then you could do this by keeping a running cumulative
> distribution function as you go through each of your files. For each
> file, calculate a cumulative distribution function --- at each
> possible value, record the fraction of that population strictly less
> than that value --- and then it's straightforward to combine the
> cumulative distribution functions from two separate files:
> cumdist_both = (cumdist1 * N1 + cumdist2 * N2) / (N1 + N2)
> Then once you've gone through all the files, look for the value where
> your cumulative distribution function is equal to 0.95. If your data
> isn't structured with repeated values, though, this won't work,
> because your cumulative distribution function will become too big to
> hold into memory. In that case, what I would probably do would be an
> iterative approach: make an approximation to the exact function by
> removing some fraction of the possible values, which will provide a
> limited range for the exact percentile you want, and then walk through
> the files again calculating the function more exactly within the
> limited range, repeating until you have the value to the desired
> precision.
> ~Brett
> _______________________________________________
> NumPy-Discussion mailing list
> NumPy-Discussion@scipy.org
> http://mail.scipy.org/mailman/listinfo/numpy-discussion
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An historical and critical study of the fundamental lemma in the calculus of variations ...
An historical and critical study of the fundamental lemma in the calculus of variations ...
Aline Huke
University of Chicago, 1930 - Calculus of variations - 110 pages
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admissible variation analogue analytic functions arbitrary function argument assume the lemma Bois Reymond Bois Reymond's lemma Bolza calculus of variations chapter class of functions consequence of
equation continuity properties continuous derivatives continuous function continuous on a,b continuous p th Crelle defined degree m-1 derivatives and vanish differential coefficients Dirksen's proof
double integrals end point problem equation 1*2 equation 3-1 Euler-Lagrange equation expression finite number finite polynominal fixed end point function n(x fundamental lemma funotion gives a proof
Haar Haar's theorem Heine Hence Hilbert Hobson holds integrand interior isoperimetric isoperimetric problem Kneser Kryloff Lagrange integration Legendre Legendre polynomials Mason's lemma method
minimizing arc Moigno mth derivative multiple integrals number of discontinuities obtained partial derivatives polynominal of degree positive integer proof given proves the following Sarrus Stegmann
Stegmann's proof sufficient surfaoe Todhunter triple integrals vanish separately variations problem involving Variationsreohnung whioh Zermelo
Bibliographic information
An historical and critical study of the fundamental lemma in the calculus of variations ...
Aline Huke
University of Chicago, 1930 - Calculus of variations - 110 pages
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Parametric and Polar Curves
Problem : Plot the polar curve given by r(θ) = cos(2θ) for θ = 0 to 2Π .
Figure %: Polar Plot of r(θ) = cos(2θ) for θ = 0 to 2Π
Problem : What is the area contained within the region bounded by r(θ) = cos(2θ) from θ = 0 to 2Π ? You may use that cos^2(θ) = (1 + cos(2θ))/2 .
We compute the area as follows:
exactly half the area of the unit circle in which it is contained!
Problem : Find the area bounded by the graph of the cardioid defined by r(θ) = sin(θ/2) for θ = 0 to 2Π , using the identity sin^2(θ) = (1 - cos(2θ))/2 .
The cardioid looks like
Figure %: Polar Plot of r(θ) = sin(θ/2) for θ = 0 to 2Π
The area is equal to
once again equal to half the area of the unit circle in which the region is contained!
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Math Forum - Problems Library - Pre-Algebra, Number Sense: Scientific Notation
This page:
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PoW Library
Scientific Notation
These problems require knowledge of the convention of representing a large or small number in scientific notation or basic operations of numbers written in scientific notation.
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Interactive resources from our Math Tools project:
Math 7: Number Sense
The closest match in our Ask Dr. Math archives:
Middle School: Exponents
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Access to these problems requires a Membership.
The Scientific Notation Game - Suzanne Alejandre
Pre-Algebra, difficulty level 3. Help Jay think about how points are calculated in the Scientific Notation Game. ... more>>
Scientific Sums - Judy Ann Brown
Pre-Algebra, difficulty level 2. Find the sum of a set of numbers written in scientific notation. ... more>>
Sizing Up Sequoias - Suzanne Alejandre
Pre-Algebra, difficulty level 2. Compared to a given seed weight, find out how much an average mature sequoia weighs. ... more>>
Page: 1
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Your abstract should be sent electronically as simple LaTeX or plain TeX. You may also use features from the amsmath, amsfonts, or amssymb packages; all other macros should be defined, please do not
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Let $S$ denote the set of all integers $n$ for which $n > 0$ and \[ \int_0^n x^n\, dx < 1/2 .\]
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Montebello, CA Trigonometry Tutor
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Hi folks! My name is Ryan. I'm a physicist/astronomer/educator with an excellent background in math, science, economics, business, and history.
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hi i really need help on this stuff
January 27th 2009, 08:41 PM #1
Jan 2009
hi i really need help on this stuff
i dont really suck at math...
i blame my teacher... he doesnt know how to teach...
he doesnt explain the problems very well..
im switching my teacher next week.. bleh....
please i really need help =3
2) standard form 9i - 3i^2
6)rational zero test
8) (1/2)^x=228
9) if logw = logx+1/2logp then w = ?
10) if f(x) = x^2-x+1 then f(2a) = ?
12) x^2+x-3=0 root = ? < --- not sure
Last edited by KyrieKaye; January 27th 2009 at 09:52 PM. Reason: done some of the questions
to #6:
Plug in the values $\pm1,\ \pm2,\ \pm5,\ \pm10$ and check if the term becomes zero.
to #8:
$\left(\frac12\right)^x=228~\implies~\left(2\right) ^{-x}=228~\implies~ -x=\log_2(228)~\implies~ x=-\log_2(228)$
But if the original problem reads:
$\left(\frac12\right)^x=128~\implies~\left(2\right) ^{-x}=2^7~\implies~ -x=7~\implies~ x=-7$
to #9:
Use the log rules:
$\log(a) + \log(b) = \log(a\cdot b)$
The final result should be: $w = \sqrt{px^2}$
to #10:
You are asked to plug in 2a instead of x:
$f(2a) = (2a)^2-(2a)+1$
to #12:
Solve $x^2+x-3=0$ . Use the formula to solve a quadratic equation.
The equation
$ax^2+bx+c=0$ has the solution $x=\dfrac{-b \pm\sqrt{b^2-4ac}}{2a}$
thank you soo much...
its much clearer to me now
January 28th 2009, 06:02 AM #2
January 28th 2009, 09:19 AM #3
Jan 2009
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Thomaston, NY SAT Tutor
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Mill Creek, Philadelphia, PA
Swarthmore, PA 19081
Professional physicist tutoring physics and math
My career in using physics and
has provided me with a boatload of examples of ways to use physics and
, solving real problems in the real world. These real life examples help me to share my love of learning
and science with the students I tutor. My...
Offering 10 subjects including algebra 1, algebra 2 and calculus
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Program to find out whether a given quadrilateral ABCD is rectangle. Assume that all the four angles and four sides of the quadrilateral are supplie
.model small
deg dw 90,90,90,60
Quad db 'Quadrilateral','$'
NotQuad db 'Not Quadrilteral','$'
mov ax,@data
mov ds,ax
mov ax,0
mov ax,deg
add ax,deg+2
add ax,deg+4
add ax,deg+6
cmp ax,360
je equal
mov ax,0900h
LEA DX,NotQuad
int 21h
jmp ext
mov ax,0900h
LEA DX,Quad
int 21h
mov ax,4C00h
int 21h
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Critical Points and Gröbner Bases: the Unmixed Case
Critical Points and Gröbner Bases: the Unmixed Case (2012)
Download Links
by Jean-charles Faugère , Mohab Safey El Din , Pierre-jean Spaenlehauer
author = {Jean-charles Faugère and Mohab Safey El Din and Pierre-jean Spaenlehauer},
title = {Critical Points and Gröbner Bases: the Unmixed Case},
year = {2012}
We consider the problem of computing critical points of the restriction of a polynomial map to an algebraic variety. This is of first importance since the global minimum of such a map is reached at a
critical point. Thus, these points appear naturally in non-convex polynomial optimization which occurs in a wide range of scientific applications (control theory, chemistry, economics,...). Critical
points also play a central role in recent algorithms ofeffectiverealalgebraicgeometry. Experimentally, it has been observed that Gröbner basis algorithms are efficient to compute such points.
Therefore, recent software based on the so-called Critical Point Method are built on Gröbner bases engines. Let f1,...,fp be polynomials in Q[x1,...,xn] of degree D, V ⊂ C n be their complex variety
and π1 be the projection map (x1,...,xn) ↦ → x1. Thecriticalpointsoftherestrictionofπ1to V are defined by the vanishing of f1,...,fp and some maximal minors of the Jacobian matrix associated to
f1,...,fp. Suchasystemisalgebraicallystructured:theidealitgenerates is the sum of a determinantal ideal and the ideal generated by f1,...,fp. We provide the first complexity estimates on the
computation of Gröbner bases of such systems defining critical points. We prove that under genericity assumptions on f1,...,fp, thecomplexityis polynomial in the generic number of critical points,
i.e. D p (D − 1) n−p () n−1.Moreparticularly,inthe p−1 quadratic case D =2,thecomplexityofsuchaGröbnerbasiscomputationispolynomial in the number of variables n and exponential in p. We also give
experimental evidence supporting these theoretical results.
1247 Commutative algebra; with view toward algebraic geometry, Graduate Texts - Eisenbud - 1994
907 Intersection Theory - Fulton - 1984
252 Basic algebraic geometry - Shafarevich - 1994
248 A new efficient algorithm for computing Gröbner bases (F4). Journal of pure and applied algebra - Faugere - 1999
206 O’Shea: Ideals, Varieties and Algorithms - Cox, Little, et al. - 1998
199 On the combinatorial and algebraic complexity of quantifier elimination - BASU, POLLACK, et al. - 1996
156 Efficient Computation of Zero-Dimensional Gröbner Bases by Change of Ordering - Faugère, Gianni, et al. - 1993
80 A Gröbner free alternative for polynomial system solving - Giusti, Lecerf, et al.
78 On the complexity of Gröbner basis computation of semi-regular overdetermined algebraic equations - Bardet, Faugère, et al. - 2004
61 Eagon: Cohen-Macaulay rings, invariant theory, and the generic perfection of determinantal loci - Hochster, A - 1971
51 Algorithms for Matrix Canonical Forms - Storjohann - 2000
40 On the complexity of semi-algebraic sets - Heintz, Roy, et al. - 1989
39 A new algorithm to find a point in every cell defined by a family of polynomials. In Quantifier elimination and cylindrical algebraic decomposition - Basu, Pollack, et al. - 1998
34 Solving Systems of Polynomial Inequalities in Subexponential Time - Grigoriev, Vorobjov - 1988
33 Polar varieties and computation of one point in each connected component of a smooth real algebraic set - Din, Schost - 2003
30 Feasibility testing for systems of real quadratic equations,” Discrete Comput - Barvinok - 1993
29 Polar varieties and efficient real elimination - Bank, Giusti, et al.
26 Generalized polar varieties: geometry and algorithms - Bank, Giusti, et al. - 2003
25 Safey El Din. Real solving for positive dimensional systems - Aubry, Rouillier, et al.
25 Polar varieties and efficient real equation solving: the hypersurface case - Bank, Giusti, et al. - 1997
24 Asymptotic expansion of the degree of regularity for semiregular systems of equations - Bardet, Faugère, et al. - 2005
23 On the Hilbert function of determinantal rings and their canonical module - Conca, Herzog - 1994
21 Generalized polar varieties and efficient real elimination procedure - Bank, Giusti, et al.
18 On the theoretical and practical complexity of the existential theory of the reals - Heintz, Roy, et al. - 1993
16 On the geometry of polar varieties - Bank, Giusti, et al.
14 Testing sign conditions on a multivariate polynomial and applications - Din
14 Properness defects of projections and computation of one point in each connected component of a real algebraic set - Din, Schost - 2004
13 The topology of the Voronoi diagram of three lines - Everett, Lazard, et al. - 2007
12 Introduction to Singularities and Deformations - Greuel, Lossen, et al. - 2007
11 Pasechnik: Polynomial-time computing over quadratic maps. I. Sampling in real algebraic sets. Comput. Complexity 14 - Grigoriev, D - 2005
10 Computing loci of rank defects of linear matrices using Gröbner bases and applications to cryptology - Faugère, Din, et al. - 2010
7 Variant quantifier elimination - Hong, Din - 2011
5 Resultants of determinantal varieties - Busé
5 Safey El Din. Classification of the perspective-three-point problem, discriminant variety and real solving polynomial systems of inequalities - Faugère, Moroz, et al. - 2008
5 Fast algorithm for change of ordering of zero-dimensional Gröbner bases with sparse multiplication matrices - Faugère, Mou - 2011
5 Variant real quantifier elimination: algorithm and application - Hong, Din, et al. - 2009
3 A class of perfect determinantal ideals - Hochster, Eagon - 1970
2 Algebraic degree of polynomial optimization - Nie, Ranestad
1 On the complexity of the Generalized Minrank Problem. arXiv:1112.4411 - Faugère, Din, et al. - 2011
1 Polar varieties and efficient real elimination. Mathematische Zeitschrift,238(1):115–144,2001 - Bank, Giusti, et al.
1 On the complexity ofGröbnerbasiscomputationofsemiregular overdetermined algebraic equations - Bardet, Faugère, et al.
1 A new algorithm to find apointineverycelldefinedbyafamily of polynomials. In Quantifier elimination and cylindrical algebraic decomposition.Springer-Verlag - Basu, Pollack, et al. - 1998
1 The voronoi diagram of three lines. Discrete &ComputationalGeometry,42(1):94–130,2009 - Everett, Lazard, et al.
1 ComputingLociofRankDefectsofLinear Matrices using Gröbner Bases and Applications to Cryptology - Faugère, Din, et al. - 2010
1 On the theoretical andpracticalcomplexityoftheexistential theory of the reals. The Computer Journal,36(5):427–431,1993 - Heintz, Roy, et al.
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Show work by using + and - on a number line
November 29th 2009, 12:32 PM
Show work by using + and - on a number line
I know how to find domains, but this instruction on the method to do it with confused me. I remember my teacher doing a long series of + and - symbols placed vertically below a number line, but
I'm not sure how she did it. The problem I am supposed to find the domain of is:
2x X > 3X
_____ + _ _________
X -4 X +X-6 X + 5X + 6
Any help would be appreciated. Thanks.
November 29th 2009, 06:22 PM
I know how to find domains, but this instruction on the method to do it with confused me. I remember my teacher doing a long series of + and - symbols placed vertically below a number line, but
I'm not sure how she did it. The problem I am supposed to find the domain of is:
2x X > 3X
_____ + _ _________
X -4 X +X-6 X + 5X + 6
Any help would be appreciated. Thanks.
The first thing you need to do is get that $\frac{3x}{x^2+ 5x- 6}$ on the left so you have $\frac{2x}{x^2- 4}+ \frac{x}{x^2+ x- 6}- \frac{3x}{x^2+ 5x+ 6}\ge 0$.
Now add the fractions on the left getting the common denominator. Fortunately, that's not too hard since those denominators have similar factors.
Now, determine where the numerators are 0 (that's easy: x= 0) and where the denominators are 0 (a little harder: solve that polynomial to 0 and factor it. Fortunately, you determined the factors
when you formed the common denominator!
The points where the numerator or denominator are 0 separate "+" from "-". The simplest way to figure which is correct for an integerval is to look at the individual factors. An odd number of -
factors, negative, and even number of - factors, positive. Also note that as you "step over" x= a, the term x- a is the only one that changes sign.
December 5th 2009, 08:05 AM
Thanks for the help! I did get that question right, but I have a couple more things to ask you. What did your last sentence mean? What's x=a? What's "stepping over?"
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Finite-Difference Discretization
Next: Treatment of Interfaces Up: Numerical Solution of Previous: Numerical Solution of
As illustrated in Fig. 3.1 we divide the interval 0<z<D into N equal intervals to construct a mesh of equally spaced points
where h is the mesh width given by N should be chosen large enough so that the modes are adequately sampled; usually 10 points per wavelength are sufficient.
Figure: Finite-difference mesh.
We shall assume for the moment that the density is constant, yielding the modal problem
where the primes denote differentiation with respect to z. Following a standard procedure for deriving finite difference equations, we use the Taylor series expansion to obtain
Rearranging terms, we obtain a forward-difference approximation for the first derivative,
An improved approximation is obtained by using the governing equation Eq. ( ) to evaluate the first term in the forward difference approximation. That is, we substitute
This yields the
Similarly, a backward-difference approximation is obtained starting with the Taylor series
yielding, the
and the
Finally, adding Eqs. ( ) and ( ) we obtain a centered-difference approximation to the second derivative:
With these finite-difference approximations in hand, we can proceed to replace the derivatives in the continuous problem with discrete analogues. Let us recall the continuous problem:
Using the centered, forward and backward difference approximations for the ODE, the top and bottom boundary conditions we obtain:
We next write the first of these equations as
Then collecting the difference equations together we obtain an algebraic eigenvalue problem of the form:
Here, ) evaluated at the mesh points. In addition, A is a symmetric tridiagonal matrix defined by
where the coefficients
We have consciously introduced a scaling factor of
Note that for a pressure-release surface the ratio k (as happens for the pressure release surface and rigid bottom conditions) then the above problem is a standard algebraic eigenvalue problem and
can be solved using standard routines. In general, only the lower-order modes will be sufficiently accurate: the higher order modes are undersampled by the finite-difference mesh. Thus, routines
which are designed to extract a subset of the eigenvectors and eigenvalues are desired.
The problem is a non-standard eigenvalue problem because the eigenvalue enters in a functional form through bottom boundary conditions. (For perfectly rigid or free boundary conditions the problem
reduces to a classical algebraic eigenvalue problem.) We shall discuss the solution of the eigenvalue problem in Sect. 3.3.
Next: Treatment of Interfaces Up: Numerical Solution of Previous: Numerical Solution of
Michael B. Porter
Tue Oct 28 13:27:38 PST 1997
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Kernel of a Character
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Why is the kernel of a character equal to the kernel of the representation that affords the character?
How do you define the kernel of a character?
Alain Valette May 14 '11 at 20:01
rep, I think your question would be more appropriate at math.stackexchange.com (although it seems that Geoff Robinson and David Ben-Zvi have given good answers to it).
S. Carnahan♦ May 16 '11 at 6:58
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closed as too localized by S. Carnahan♦ May 16 '11 at 6:56
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This is too elementary a question for MO. In the case of complex characters, the kernel of a character $\chi$ of a finite group $G$ is $\{g \in G: \chi(g) = \chi(1) \}.$ On the other hand, if
the character $\chi$ is afforded by a representation $\sigma$ (that is, $\chi(g) = {\rm trace}(g\sigma)$ for all $g \in G$), then for each $g \in G$, the eigenvalues of $g \sigma$ are all $o
(g)$-th roots of unity, where $o(g)$ is the order of $g$. Hence the triangle inequality yields $|\chi(g)| \leq \chi(1)$, and the only way we can have $\chi(g) = \chi(1)$ is if all eigenvalues
of $g\sigma$ are equal to 1. Since $g\sigma$ has finite order, in that case, $g \sigma$ must be the identity matrix. Thus ${\rm ker}\chi$ is precisely equal to ${\rm ker} \sigma$, as the
inclusion ${\rm ker}\sigma \leq {\rm ker}\chi$ is clear.
up vote Added later, in light of the discussion below: ``kernel" implicitly refers to the kernel of a group homomorphism here (actually, several group homomorphisms, as we will see). The (complex)
4 down character $\chi$ may be afforded by several different homomorphisms $\sigma: G \to {\rm GL}(n,\mathbb{C})$, where $n = \chi(1)$. But all such representations are equivalent( that is.
vote detemined up to conjugation by a matrix in ${\rm GL}(n,\mathbb{C})).$ Hence all such representations have the same kernel in the group-theoretic sense. Furthermore, the argument above shows
that the kernel can be seen directly from the character values. Hence the kernel of a character is really the kernel of an equivalence class of representations. One of the (many) advantages
of working with (complex) characters is that all normal subgroups of a finite group $G$ can be determined by inspection of the character table of $G$.
It would also be possible to speak of the kernel of an algebra homomorphim, but that would be the set of all elements of the group algebra $\mathbb{C}G$ sent to the zero matrix by a chosen
representation $\sigma$, and we would need to consider the algebra homomorphism as mapping into the full matrix ring $M_{n}(\mathbb{C})$. However, this kernel is not so easy to read from the
character values on group elements.
Yes, this question is too elementary for MO (and may well be a homework problem). This is standard textbook material: for instance, see pages 23-24 in I.M. Isaacs Character Theory of Finite
Groups (AMS Chelsea reprint) where it's also explained how to determine other normal subgroups than kernels from a character table. In some books the treatment is less thorough, e.g., Serre
Linear Representations of Finite Groups (Exercise 6.7).
Jim Humphreys May 15 '11 at 12:22
add comment
up vote 1 My question is why the kernel of a character $\chi$ of a finite group $G$ is $ \{g \in G: \chi(g) = \chi(1) \} $, instead of $ \{ g\in G: \chi(g) = 0 \} $?
down vote
Because the former matches up with the kernel of the representation while the latter doesn't!
Noah Snyder May 15 '11 at 5:20
I think Mark and Qiaochu's comments miss the point slightly. Given a complex-valued function, it is (as we all know) customary to use the word "kernel" for the set of elements in the
domain on which the function has the value zero. If, like me, you work mainly in analysis but dabble in algebra, then talk of the "kernel of a character $\chi$" causes momentary cognitive
dissonance, since I am used to thinking of characters as elements of $L^2(G)$.
Yemon Choi May 15 '11 at 6:22
Is it? I have only seen the term used for group or ring homomorphisms. In general there's no reason to single out the preimage of $0$.
Qiaochu Yuan May 15 '11 at 8:41
If the purpose of defining the kernel of a character is to match up with the kernel of the representation, why don't we simply call it "the kernel of the representation that affords it,
and write it as $ker (\Chi)$ instead of $ker (\chi)$?
rep May 15 '11 at 9:33
This discussion seems a little besides the point, since characters are MULTIPLICATIVE objects, so you would expect the kernel to be the inverse image of 1 not 0 (as indeed it is for
irreps of abelian groups, when the character is just a group homomorphism to the multiplicative group).
David Ben-Zvi May 15 '11 at 14:32
show 8 more comments
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One-dimensional model
2D effects: The mean-field model and variants
2D effects: Building on the mean-field models
Simple models yield accurate , and energy spread
All models that include profile shape evolution show the development of a linear self-similar profile on a time scale dictated by the plasma frequency
Two-dimensional profile evolution results in sharply defined turnaround points
Off-axis and inhomogeneous behaviors are more complex
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Towards Formal Power Series for Functors
Tue 13 May 2008
The post below will only compile on a version of GHC >= 6.9, since it uses type families.
There has been a lot of posting recently about automatic differentiation in Haskell, and I wanted to try the same thing with functors in the spirit of Conor McBride's Clowns to the Left of me, Jokers
to the Right and The derivative of a regular type is its type of one hole contexts, figuring that a Power Series could fully generalize Christophe Poucet's Higher Order Zippers, and might provide me
with a neat extension to the zipper comonadic automata I've been aluding to recently.
{-# OPTIONS -fglasgow-exts -fallow-undecidable-instances -fallow-overlapping-instances #-}
module Derivatives where
import Control.Monad.Identity
import Control.Arrow ((+++),(***),(&&&))
import Data.Monoid
infixl 9 :.:
infixl 7 :*:
infixl 6 :+:
To avoid importing category-extras and keep this post self-contained (modulo GHC 6.9!), we'll define some preliminaries such as Bifunctors:
class Bifunctor f where
bimap :: (a -> c) -> (b -> d) -> f a b -> f c d
instance Bifunctor (,) where
bimap f g ~(a,b) = (f a, g b)
instance Bifunctor Either where
bimap f _ (Left a) = Left (f a)
bimap _ g (Right b) = Right (g b)
Constant functors:
data Void
instance Show Void where show _ = "Void"
newtype Const k a = Const { runConst :: k } deriving (Show)
type Zero = Const Void
type One = Const ()
instance Functor (Const k) where
fmap f = Const . runConst
and functor products and coproducts:
newtype Lift p f g a = Lift { runLift :: p (f a) (g a) }
type (:+:) = Lift Either
type (:*:) = Lift (,)
instance Show (p (f a) (g a)) => Show (Lift p f g a) where
show (Lift x) = "(Lift (" ++ show x ++ "))"
instance (Bifunctor p, Functor f, Functor g) => Functor (Lift p f g) where
fmap f = Lift . bimap (fmap f) (fmap f) . runLift
and finally functor composition
newtype (f :.: g) a = Comp { runComp :: f (g a) } deriving (Show)
instance (Functor f, Functor g) => Functor (f :.: g) where
fmap f = Comp . fmap (fmap f) . runComp
So then, an ideal type for repeated differentiation would look something like the following, for some definition of D.
[Edit: sigfpe pointed out, quite rightly, that this is just repeated differentiation, and apfelmus pointed out that it not a power series, because I have no division!]
newtype AD f a = AD { runAD :: (f a, AD (D f) a) }
As a first crack at D, you might be tempted to just go with a type family:
type family D (f :: * -> *) :: * -> *
type instance D Identity = One
type instance D (Const k) = Zero
type instance D (f :+: g) = D f :+: D g
type instance D (f :*: g) = f :*: D g :+: D f :*: g
type instance D (f :.: g) = (D f :.: g) :*: D g
This could take you pretty far, but unfortunately doesn't adequately provide you with any constraints on the type so that we can treat AD f as a functor.
So, we'll go with:
class (Functor (D f), Functor f) => Derivable (f :: * -> *) where
type D f :: * -> *
and cherry pick the instances necessary to handle the above cases:
instance Derivable Identity where
type D Identity = One
instance Derivable (Const k) where
type D (Const k) = Zero
instance (Derivable f, Derivable g) => Derivable (f :+: g) where
type D (f :+: g) = D f :+: D g
instance (Derivable f, Derivable g) => Derivable (f :*: g) where
type D (f :*: g) = f :*: D g :+: D f :*: g
instance (Derivable f, Derivable g) => Derivable (f :.: g) where
type D (f :.: g) = (D f :.: g) :*: D g
With those instances in hand, we can define the definition of a Functor for the automatic differentiation of a Functor built out of these primitives:
instance (Derivable f, Functor (AD (D f))) => Functor (AD f) where
fmap f = Power . bimap (fmap f) (fmap f) . runPower
Unfortunately, here is where I run out of steam, because any attempt to actually use the construct in question blows the context stack because the recursion for Functor (AD f) isn't well founded and
my attempts to force it to be so through overlapping-instances have thus-far failed.
12 Responses to “Towards Formal Power Series for Functors”
1. apfelmus Says:
May 13th, 2008 at 7:38 am
I don’t see why you’d need overlapping or undecidable instances, all your class instances look like H98 (modulo the type family).
There seems to be a problem with the power series constructor, I fail to see how it’s different from
newtype Power _ a = P (a, Power _ a)
i.e. an infinite stream of a . The instance then simply becomes
instance Functor (Power f) where
fmap f = Power . bimap f (fmap f) . runPower
This is probably not what you want. In the case you want, the real obstruction is that for power series, the functor in question must be differentiable an infinite number of times. In other
words, something like
instance Differentiable f => Differentiable (D f) where …
Then, a context
instance Differentiable f => Functor (Power f)
should be enough.
2. Edward Kmett Says:
May 13th, 2008 at 10:04 am
Good point. OTOH:
instance Differentiable f => Differentiable (D f)
doesn’t work because D isn’t a type constructor for the compiler to grab onto, which is why I started beating my head against the other part.
What i was looking for was actually to say:
newtype Power f a = Power (f a, Power (D f) a)
instead of
newtype Power f a = Power (a, Power (D f) a)
[Edit: fixed]
3. apfelmus Says:
May 13th, 2008 at 11:08 am
Yeah, just
instance Differentiable f => Differentiable (D f)
doesn’t work.
And concerning the power series, I think what you really want is
newtype Power f a = Power (f () :+: (a :*: Power (D f) a))
but that’s wrong too, because you would have to divide by $latex \frac1{n!}$ somewhere. Power series are different from (but related to) automatic differentiation! Most likely, the impossible
division can be included by using bags of n elements instead of powers a :*: a :*: ... (= ordered pairs) of n elements. In any case, the functor instance for power series is no problem, since a
is used explicitly.
Not sure in the case of automatic differentiation; the main problem here is indeed to somehow tell the compiler that
instance Functor f => Functor (D f)
But since type families are open, I guess that’s not possible. I mean, a “mischievous” person could add a new case for this type family which violates this condition. So, the problem is not that
the compiler doesn’t grok it, the problem is that there is a good reason that the compiler doesn’t accept it.
4. Edward Kmett Says:
May 13th, 2008 at 11:32 am
True enough. I just started down this path because I was thinking in terms of inductive families of types, and realized with open type functions that you could put a more interesting operation in
than just a pair, etc.
And yeah, without a notion of type division (maybe using species?), I don’t see how to get from this form of AD to a power series.
5. Edward Kmett Says:
May 13th, 2008 at 11:43 am
Oh, maybe a more interesting instance for this would probably be the notion of a derivative for a list since that doesn’t zero out:
instance Derivable [] where
type D [] = [] :*: []
I also would like to find out a way to include a simplification step in the type function to go through and do the Zero :*: x = Zero step, etc to make the encoding a little less naive.
6. Dan Doel Says:
May 13th, 2008 at 3:21 pm
Although the context on Functor (AD f) is sufficient, it seems (to me) like the wrong place to specify it. Since
AD f a ~~ (f a, AD (D f) a)
With (Derivable f) => Functor (AD f) you only need (Derivable (D f)) to justify using bimap (fmap f) (fmap f). So, it seems like what you’d really want is:
class (Functor f, Derivable (D f)) => Derivable f where
type D f :: * -> *
instance (Derivable f) => Functor (AD f) where …
Where the context on Derivable enforces that all such functors are arbitrarily differentiable. But, of course, GHC complains about a cycle there (in 6.8, at least. I don’t have 6.9 handy).
Simplification should, I think, be possible with total type families once they go into 6.9. For instance:
type family (f :: * -> *) :+: (g :: * -> *) :: * -> * where
Zero :+: g = g
f :+: Zero = f
f :+: g = Lift Either f g
You’re allowed to overlap instances for total type families, because they’re closed.
7. apfelmus Says:
May 13th, 2008 at 3:30 pm
Concerning the division, we can appeal to the special case
a^n / n! = Bag n a
i.e. the multiset (bag) with n elements of type a. So, the power series expansion for a data type just counts how many different sets of elements it may contain. So, a :*: a contains two bags of
two a (one for each permutation in the tuple).
Btw, as the differential equation for the list type suggests, we have the power series expansion
[a] = 1 + a + a^2 + a^3 + ...
which “is” 1/(1-a).
8. sigfpe Says:
May 13th, 2008 at 7:10 pm
I’m confused. This doesn’t look like AD to me. The rules you give are the standard Leibniz rules for symbolic differentiation and then I think you’re trying to use those to get a (compile time)
list of all derivatives. This is a really neat thing to attempt (though I have a hunch it’s not possible in Haskell) but it seems quite different to AD to me. Anyway, I suspect a language like
Omega may be better suited for what you are doing.
9. sigfpe Says:
May 13th, 2008 at 7:14 pm
Oh yeah, I almost forgot. Check out Conor McBride et al’s handling of derivatives here. It includes the handling of fixed points.
10. Edward Kmett Says:
May 13th, 2008 at 8:04 pm
Yeah, mostly i was just trying to see what an inductive family of types built over an open type function would look like.
I left the fixed point cases off because I was still trying to get the finite cases to terminate and once I ran up against the wall of type-ability I didn’t check that what I had so far was what
I thought I had so far.
It has been a while since I did anything with classical analysis. Hrmm. Back to the drawing board I guess until I figure out how to do what I really want.
11. Edward Kmett Says:
May 13th, 2008 at 8:41 pm
So a simple encoding of the basic AD rules leaves your Dual class, just one level up.
data D f g a = D (f a) (g a)
type family (:+>) (f :: * -> *) (g :: * -> *) :: * -> *
type family (:*>) (f :: * -> *) (g :: * -> *) :: * -> *
type family DLift (f :: (* -> *) -> * -> *) (f’ :: (* -> *) -> * -> *) (g :: * -> *) :: * -> *
type instance (D x a :+> D y b) = D (x :+: y) (a :+: b)
type instance (D x a :*> D y b) = D (x :*: y) (x :*: b :+: a :*: y)
type instance DLift f f’ (D x a) = D (f x) (a :+: f’ x)
dConst :: a -> D Identity Zero a
dConst z = D (Identity z) zero
dVar :: a -> D Identity One a
dVar z = D (Identity z) (suck zero)
I think the general AD case requires polymorphic kinds, so yeah something like Omega is probably better suited. Alas.
12. The Comonad.Reader » Generatingfunctorology Says:
May 14th, 2008 at 2:45 am
[...] Ok, I decided to take a step back from my flawed approach in the last post and play with the idea of power series of functors from a different perspective. [...]
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Z-transform: what is Z{n * u[n - N]} ?
October 10th 2012, 03:08 AM #1
Z-transform: what is Z{n * u[n - N]} ?
These two Z-transform properties are relevant
$x[n-n_0] \Longleftrightarrow z^{-n_0}X(z)$
$nx[n] \Longleftrightarrow -z \frac{dX(z)}{dz}$
yet I don't know how (in which order) one should apply them on the following:
$Z(n u[n - N])$
Even using the z-transform definition, I cannot get a closed form solution:
$Z(n u[n - N]) = \sum_{n=-\infty}^{\infty} n u[n-N] z^{-n} = \sum_{n=N}^{\infty} n z^{-n} = \sum_{n=0}^{\infty} n z^{-n} - \sum_{n=0}^{N-1} n z^{-n} = \frac{z^{-1}}{(1-z^{-1})^2 } - \sum_{n=0}^
{N-1} n z^{-n} = ?$
Is there a closed form of $\sum_{n=0}^{N-1} n z^{-n} = z^{-1} + 2z^{-2} + ... + (N-1)z^{-(N-1)}$ ? What is it?
PS. Is there a more suitable sub-forum I should be posting these DSP questions? Or is this it?
Last edited by courteous; October 10th 2012 at 03:24 AM.
Re: Z-transform: what is Z{n * u[n - N]} ?
How silly of me, should've just take a derivative of $\sum_{n=0}^{\infty} z^n$ which gives (considering the linearity of a differentiation op.)
$\left(\sum_{n=0}^{\infty} z^n\right)' = \frac{1}{(1-z)^2} = \sum_{n=0}^{\infty} n z^{n-1}$. Therefore, $\sum_{n=0}^{\infty} n z^n = \frac{z}{(1-z)^2}$. And then some more paper-gymnastics for
the partial sum of the same thing.
I really need to take this "math side of physics" more seriously ... yes, I'm writing mostly to myself, but who cares anyway. Thanks for the help.
Re: Z-transform: what is Z{n * u[n - N]} ?
October 17th 2012, 08:16 AM #2
October 17th 2012, 08:24 AM #3
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Here's the question you clicked on:
The terms 5x+2, 7x-4, and 10x+6 are consecutive terms of an arithmetic sequence. Determine the value of x and state the three terms. I was just introduced to Tn=T1+(n-1)d so if I can answer using
this equation that would be great
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an arithmetical sequence has a common difference (d) so in this case d = (7x - 4) - (5x + 2) = 2x - 6 also d = (10x + 6) - (7x - 4) = 3x + 10 therefore 2x - 6 = 3x + 10 do you follow this?
Best Response
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ok but would the difference between the T1 T2 T3 be the same?
Best Response
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yes t2 - t1 and t3 - t2 are the same - thats what an arithmetical sequence is eg 2 , 4 , 6
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ok I follow so far
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now solve the equation to find x: 2x - 6 = 3x + 10 -6-10 = 3x - 2x x = -16 now you can find the first 3 terms by plugging x= -16 into the terms in x first one T1 = 5x+ 2 = 5(-16) + 2 = -78
Best Response
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now you can get the 2nd and third by plugging in x = -16 as above or you can use the formula for Tn which you quoted in your question
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awesome.Thanks for the help:)
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Part II. The case of independent random variables
• The TeX file of the article
• The dvi file of the article
• The pdf file of the article
The first part of this work
Part I. The general case
• The TeX file of the first part of the article
• The dvi file of the first part of the article
• The pdf file of the first part of the article
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Avogadro's law
Avogadro's law
Amedeo Avogadro
Avogadro's law (Avogadro's Hypothesis, or Avogadro's Principle) is a gas law named after Amedeo Avogadro, who in 1811 hypothesized that:
Equal volumes of ideal or perfect gases, at the same temperature and pressure, contain the same number of particles, or molecules.
Thus, the number of molecules in a specific volume of gas is independent of the size or mass of the gas molecules when related to an ideal gas approximate. It is very important to note that we apply
an ideal gas or perfect gas definition (a hypothetical gas consisting of identical particles of zero volume, with no intermolecular forces, but the ability to interchange momentum with identical gas
molecules) to a real gas such as hydrogen or nitrogen so we can statistically approximate the real gas behaviour.
As an example, equal volumes of molecular hydrogen and nitrogen would contain the same number of molecules, as long as they are at the same temperature and pressure and observe ideal or perfect gas
behaviour. Whilst this is not the real world case, it is statistically very close.
The minor aspect of the law can be stated mathematically as:
$\qquad {{V} \over {n}}= k$.
V is the volume of the gas.
n is the number of moles in the gas.
k is a proportionality constant.
However, this above equation is just a trivial one, which is valid for all homogeneous substances, including homogeneous liquids and solids. This relation is easy to deduce; its validity was assumed
before Avogadro's work.
The most important consequence of Avogadro's law is the following: The ideal gas constant has the same value for all gases. This means that the constant
$\frac{p_1\cdot V_1}{T_1\cdot n_1}=\frac{p_2\cdot V_2}{T_2 \cdot n_2} = const$
p is the pressure of the gas
T is the temperature of the gas
has the same value for all gases, independent of the size or mass of the gas molecules. This statement is nontrivial, and it embodies Avogadro's ingenious insight into the nature of ideal gases. It
took decades to prove Avogadro's law based on the kinetic theory of gases.
One mole of an ideal gas occupies 22.4 liters (dm³) at STP, and occupies 24.45 litres at SATP (Standard Ambient Temperature and Pressure = 25 degrees C and 1 atm/101.3kPa). This volume is often
referred to as the molar volume of an ideal gas. Real gases may deviate from this value.
The number of molecules in one mole is called Avogadro's number: approximately 6.022×10^23 particles per mole.
Avogadro's law, together with the combined gas law, forms the ideal gas law.
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Parametric and Polar Curves
Problem : Plot the polar curve given by r(θ) = cos(2θ) for θ = 0 to 2Π .
Figure %: Polar Plot of r(θ) = cos(2θ) for θ = 0 to 2Π
Problem : What is the area contained within the region bounded by r(θ) = cos(2θ) from θ = 0 to 2Π ? You may use that cos^2(θ) = (1 + cos(2θ))/2 .
We compute the area as follows:
exactly half the area of the unit circle in which it is contained!
Problem : Find the area bounded by the graph of the cardioid defined by r(θ) = sin(θ/2) for θ = 0 to 2Π , using the identity sin^2(θ) = (1 - cos(2θ))/2 .
The cardioid looks like
Figure %: Polar Plot of r(θ) = sin(θ/2) for θ = 0 to 2Π
The area is equal to
once again equal to half the area of the unit circle in which the region is contained!
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Search the 2007
Undergraduate Catalog
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Mathematical Sciences Course Descriptions
MATH 1306 College Algebra for the Non-Scientist (3 semester hours) This course is intended for students NOT continuing on to precalculus or calculus. The course is designed to develop both abstract
thinking and a practical approach to problem solving. The emphasis is on understanding rather than purely computational skills. Topics include logic, sets, the real numbers, linear equations and
their applications, functions, and graphs. Cannot be used to satisfy major requirements for majors in the Schools of Natural Sciences and Mathematics or Management, or degree requirements for the
School of Engineering and Computer Science. Credit given for only one of MATH 1306 or 1314. Prerequisite: High School Algebra II. (3-0) Y
MATH 1314 (MATH 1314) College Algebra (3 semester hours) Topics chosen from areas such as equations and inequalities, rational expressions, exponents, radicals and logarithms, functions, and graphs.
Cannot be used to satisfy major requirements for majors in the Schools of Natural Sciences and Mathematics or Management, or degree requirements for the School of Engineering and Computer Science.
Credit given for only one of MATH 1306 or 1314. Prerequisite: High School Algebra II. (3-0) S
MATH 1325 (MATH 1325) Applied Calculus I (3 semester hours) Functions and graphs, differentiation, maxima and minima, exponential and logarithmic functions, integration, applications of integrals.
Cannot be used to satisfy degree requirements or majors in the School of Engineering and Computer Science or major requirements in the School of Natural Sciences and Mathematics. Credit given for
only one of MATH 1325 or 2417. Prerequisite: A SAT II Mathematics Level IC Test score of at least 480 or a grade of at least C- in MATH 1314 or an equivalent course. (3-0) S
MATH 1326 Applied Calculus II (3 semester hours) Applications of differential equations, functions of several variables, least squares modeling, multiple integrals, infinite series. Cannot be used to
satisfy degree requirements for B.S. majors in Schools of Engineering and Computer Science or Natural Sciences and Mathematics. Credit given for only one of MATH 1326 or 2419. Prerequisite: A score
of at least 4 on the Advanced Placement Calculus AB exam, a score of at least 3 on the Advanced Placement Calculus BC exam, or a grade of at least C- in MATH 1325. (3-0) S
MATH 2017 Calculus I Problem Session (0 semester hours) Problem session required with Calculus I, MATH 2417. Corequisite: MATH 2417. (2-0) S
MATH 2018 Linear Algebra Problem Session (0 semester hours) Problem session required with Linear Algebra, MATH 2418. Corequisite: MATH 2418. (2-0) S
MATH 2019 Calculus II Problem Session (0 semester hours) Problem session required with Calculus II, MATH 2419. Corequisite: MATH 2419. (2-0) S
MATH 2020 Differential Equations Problem Session (0 semester hours) Problem session required with Differential Equations, MATH 2420. Corequisite: MATH 2420. (2-0) S
MATH 2051 Multivariable Calculus Problem Session (0 semester hours) Problem session required with Multivariable Calculus, MATH 2451. Corequisite: MATH 2451. (2-0) S
MATH 2312 (MATH 2312) Precalculus (3 semester hours) Trigonometric functions, rational functions, exponential and logarithmic functions and their graphs, analytic geometry, polynomial equations, and
linear system of equations will be covered. Cannot be used to satisfy degree requirements for majors in the School of Engineering and Computer Science, or major requirements for the Schools of
Management or Natural Sciences and Mathematics. Prerequisite: A SAT II Mathematics Level IC Test score of 480 or a grade of at least a C- in MATH 1314 or an equivalent course. (3-0) S
MATH 2333 Matrices, Vectors, and Their Application (3 semester hours) Matrices, vectors, determinants, inverses, systems of linear equations, and applications. Cannot be used to satisfy degree
requirements for majors in the School of Engineering and Computer Science, or major requirements in the School of Natural Sciences and Mathematics. Credit given for only one of MATH 2333 or 2418.
Prerequisite: MATH 1314 or equivalent. (3-0) S
MATH 2417 (MATH 2417) Calculus I (4 semester hours ) Functions, limits, continuity, differentiation; integration of function of one variable; logarithmic, exponential, and inverse trigonometric
functions; techniques of integration, and applications. Three lecture hours and two discussion hours (MATH 2017) a week. Credit given for only one of MATH 1325 or MATH 2417. Prerequisite: A SAT II
Mathematics Level IC Test score of 710, a Level II C Test score of 630, or a grade of at least C- in MATH 2312 or an equivalent course. Corequisite: MATH 2017. (4-0) S
MATH 2418 (MATH 2418) Linear Algebra (4 semester hours) Systems of linear equations, determinants, vectors and vector spaces, linear transformations, eigenvalues and eigenvectors, quadratic forms.
Three lecture hours and two discussion hours (MATH 2018) per week. Credit given for only one of MATH 2333 or 2418. Prerequisite: MATH 2419 or consent of instructor. Corequisite: MATH 2018. (4-0) S
MATH 2419 (MATH 2419) Calculus II (4 semester hours) Continuation of MATH 2417. Improper integrals, sequences, infinite series, power series, parametric equations and polar coordinates, vectors,
vector valued functions, functions of several variables, partial derivatives and applications, multiple integration. Three lecture hours and two discussion (MATH 2019) hours a week. Prerequisite: A
score of at least 4 on the Advanced Placement Calculus BC exam or a grade of at least C- in MATH 2417. Corequisite: MATH 2019. (4-0) S
MATH 2420 (MATH 2020) Differential Equations with Applications (4 semester hours) Topics covered will be drawn from the following list: First order differential equations, ordinary differential
equations, system of linear differential equations, stability, series solutions, special functions, Sturm Liouville problem, Laplace transforms and linear differential equations, and applications in
physical sciences and engineering using computers. Three lecture hours and two discussion hours (MATH 2020) per week. Prerequisite: MATH 2419. Corequisite: MATH 2020. (4-0) S
MATH 2451 Multivariable Calculus with Applications (4 semester hours) Vectors, matrices, vector functions, partial derivatives, divergence, curl, Laplacian, multiple integrals, line and surface
integrals, Green’s, Stoke’s, and Gauss’s theorems, and applications in physical sciences and engineering. Three lecture hours and two discussion hours (MATH 2051) per week. Prerequisite: MATH 2419.
Corequisite: MATH 2051. (4-0) S
MATH 2V90 Topics in Mathematics (1-6 semester hours) Special topics in mathematics outside the normal course of offerings. May be repeated for credit as topics vary (9 hours maximum). Consent of
instructor required. ([1-6]-0) S
MATH 3301 Mathematics for Elementary and Middle School Teachers (3 semester hours) This course is intended to develop future teachers' depth of mathematical understanding by examining concepts in
school mathematics from an advanced perspective. Topics include: numeration systems; arithmetic algorithms, prime factorization and other properties of the integers; proportional reasoning involving
fractions and decimals; counting methods; and basic ideas of geometry and measurement. Problem solving is stressed. Cannot be used to satisfy: [1] undergraduate mathematics core requirement, [2]
degree requirements by students in Mathematical Sciences, [3] the advanced electives sequence, or [4] certification requirements in 8-12 mathematics. Prerequisite: MATH 1306 or MATH 1314 or
equivalent course. (3-0) S
MATH 3303 Introduction to Mathematical Modeling (3 semester hours) An introduction to construction, use, and analysis of empirical and analytical mathematical models. Emphasis on using appropriate
technology with tools such as curve fitting, probability and simulation, difference and differential equations, and dimensional analysis. Cannot be used to satisfy mathematics requirements by
students in Mathematical Sciences and cannot be used to satisfy the advanced electives sequence. Prerequisites: MATH 2419 and 2418. (3-0) Y
MATH 3305 Foundations of Measurement and Informal Geometry (3 semester hours) An analysis, from an advanced perspective, of the basic concepts and methods of geometry and measurement. Topics include
visualization, geometric figures and their properties; transformations and symmetry; congruence and similarity; coordinate systems; measurement [especially length, area, and volume]; and geometry as
an axiomatic system. Emphasis on problem solving and logical reasoning. Cannot be used to satisfy: [1] undergraduate mathematics core requirement, [2] degree requirements by students in Mathematical
Sciences, [3] the advanced electives, or [4] certification requirements in 8-12 mathematics. Prerequisite: MATH 1312, MATH 3301 or equivalent course. (3-0) Y
MATH 3307 Mathematical Problem Solving for Teachers (3 semester hours) Development of the ability to solve mathematical problems and communicate their solutions throught the study of strategies and
heuristics. Practice in solving problems involving ideas from number theory, algebra, combinatorics and probability, etc. Communicating mathematics, logical reasoning, and connections between
mathematical topics will be emphasized. Cannot be used to satisfy degree requirements by students in Mathematical Sciences or the advanced electives. Prerequisites: MATH 2312 and MATH 3305 or MATH
3321. (3-0) Y
MATH 3310 Theoretical Concepts of Calculus (3 semester hours) Mathematical theory of calculus. Limits, types of convergence, power series, differentiation, and Riemann integration. Prerequisite: MATH
2419. (3-0) Y
MATH 3311 Abstract Algebra I (3 semester hours) Groups, rings, fields, vector spaces modules, linear transformations, and Galois theory. Prerequisite: MATH 2419. (3-0) Y
MATH 3312 Abstract Algebra II (3 semester hours) Continuation of Math 3311. Prerequisite: MATH 3311. (3-0) Y
MATH 3321 Geometry (3 semester hours) Elements of Euclidean, non-Euclidean, and projective geometry. Topics covered will be drawn from the following list: triangles and their distinguishing points,
Euler line, nine point circle, extremum problems, circles and spheres, inversions, the circles of Apollonius, projective geometry, axioms of the projective plane, Desargues’s theorem, conics,
elementary facts of the non Euclidean geometries. Prerequisite: MATH 2419. (3-0) Y
MATH 3379 Complex Variables (3 semester hours) Geometry and algebra of complex numbers, functions of a complex variable, power series, integration, calculus of residues, conformal mapping.
Prerequisites: MATH 2451 and 3310. (3-0) Y
MATH 4301 Mathematical Analysis I (3 semester hours) Sets, real number system, metric spaces, real functions of several variables. Riemann Stieltjes integration and other selected topics.
Prerequisites: MATH 2451 and 3310. (3-0) Y
MATH 4302 Mathematical Analysis II (3 semester hours) Continuation of Math 4301. Prerequisite: MATH 4301. (3-0) Y
MATH 4332 Scientific Math Computing (3 semester hours) Topics covered include introduction to Unix shells, basic and advanced use of Matlab for mathematical and scientific problem solving. Course is
conducted in a computer classroom and assignments include applications in numerical and statistical analysis, image processing, and signal processing. Prerequisites: MATH 2418 and MATH 2419 or
equivalent. (3-0) S
MATH 4334 Numerical Analysis (3 semester hours) Solution of linear equations, roots of polynomial equations, interpolation and approximation, numerical differentiation and integration, solution of
ordinary differential equations; computer arithmetic and error analysis. Prerequisites: MATH 2418, 2451, and CS 1337 or equivalent knowledge of a high level programming language. (Same as CS 4334)
(3-0) Y
MATH 4341 Topology (3 semester hours) Elements of general topology, topological spaces, continuous functions, connectedness, compactness, completeness, separation axioms, and metric spaces.
Prerequisites: MATH 2451 and 3310. (3-0) Y
MATH 4355 Methods of Applied Mathematics (3 semester hours) Topics include some frequently used tools in applied mathematics: Laplace and Fourier transforms, special functions, systems, signals, and
their applications in physical sciences and engineering. Prerequisites: MATH 2418 and 2420. (3-0) T
MATH 4362 Partial Differential Equations (3 semester hours) This course presents a survey of classical and numerical methods for the solution of linear and nonlinear boundary value problems governed
by partial differential equations. Modeling and application related issues are included throughout. Prerequisites: MATH 2420, 2451, and knowledge of a high level programming language. (3-0) T
MATH 4390 Senior Research and Advanced Writing (3 semester hours) For students conducting independent research and scientific writing. Individual instruction course designed to develop skills for
research and clear, precise and accurate scientific writing. Topics will vary from section to section depending upon the interests of the student, but will be selected from a specific area of
mathematics. Subject and scope to be determined on an individual basis. Satisfies the Advanced Writing Requirement. Prerequisite: Senior in Mathematics. (3-0) S
MATH 4398 Senior Honors in Mathematical Sciences (3 semester hours) For students conducting independent research for honors theses or projects. (3-0) S
MATH 4V03 Independent Study in Mathematics (1-6 semester hours) Independent study under a faculty member’s direction. Student must obtain approval from participating math sciences faculty member and
the undergraduate advisor. Can satisfy Communication elective (3 hours) if it has a major writing/report component. May be repeated for credit (9 hours maximum). Prerequisite: Consent of instructor.
([1-6]-0) S
MATH 4V91 Undergraduate Topics in Mathematics (1-9 semester hours) Subject matter will vary from semester to semester. May be repeated for credit as topics vary (9 hours maximum). Prerequisite:
Consent of instructor. ([1-9]-0) S
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Lakeside, CA Math Tutor
Find a Lakeside, CA Math Tutor
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[SciPy-user] [timeseries] Missing dates
Christiaan Putter ceputter@googlemail....
Sat Apr 4 12:41:42 CDT 2009
2009/4/4 Matt Knox <mattknox.ca@gmail.com>:
>> In the one plotting example (using yahoo finance) I saw that one can
>> fill missing dates before plotting so that the missing ones get
>> masked. Though when applying some moving windows functions that
>> caused all periods that were effected by the missing values to also
>> become masked, which isn't the behaviour I was expecting. It does
>> make sense to do it that way though.
>> Obviously it's simple enough to use the original timeseries to
>> calculate the moving window functions, or interpolate or something.
> You hit the nail on the head here. There is no way for the timeseries module to
> know what the user thinks is the proper way to handle the masked values here, so
> the sensible thing to do is mask the whole region. You can calculate the moving
> average on the original series (ie. before you call fill_missing_dates), or
> interpolate the data somehow first (eg. using forward_fill), etc.
>> The question I'm trying to get at though is if I'm going to store my
>> timeseries in hdf5 will I fill in the missing dates before I do so, or
>> only do that whenever I plot the timeseries? I'm working with stock
>> prices, so the "missing" dates over the weekends will increase file
>> size by more then 30%. Is there any other reason to fill in missing
>> dates besides for plotting?
> Note that in the example you are talking about, the series is a "BUSINESS"
> frequency series
> dates = ts.date_array([q[0] for q in quotes], freq='DAILY').asfreq('BUSINESS')
> so calling fill_missing_dates on this has the effect of adding masked values for
> the HOLIDAYS, but not Saturday and Sunday.
I'm trying to wrap my head around how the different frequencies
behave... Correct me if I'm wrong. Using the yahoo example as a
reference: A timeseries with daily frequency for one year does not
need to have a date value for every day in that year in its date
array. But when it gets plotted the index (x-axis) runs over the
entire year, and a line plot will simply connect all the dots
basically as if it were linearly interpolating the values for the
missing dates. Thus even though the frequency is expecting values for
every day, the process of masking the missing dates needs to be done
explicitly. Which is done by fill_missing_dates(), which then adds a
date to our array for every date that the frequency expects, and sets
the mask to true for those dates that weren't in the array initially.
So when I'm using a business frequency the index on plots doesn't
behave like a calender, but mondays immediately follow fridays. And
the only dates 'missing' are in fact the holidays as you pointed out,
which we need to add explicitly so that we can mask them.
> Now as to whether or not one should fill in the holidays for storage purposes is
> a judgement call, but I generally find it simpler to just forward fill all
> holidays (see the forward_fill function in the interpolation section of the
> docs) in a batch job overnight and that way any reports or models don't have to
> think about adding special logic to handle holidays which can be somewhat
> complicated, especially if you are talking about global data with different
> calendars and so forth. Yes, this can introduce inaccuracies to some degree, but
> for most use cases I have found the gains in simplicity more than outweigh those
> costs.
The way I've been working with my data up until now was purely looking
at trading days, ignoring weekends, holidays etc. So any analysis
that takes time into account basically has 'trading day' as its unit
of time. This makes some things simpler. close[-11] would always be
the closing price 10 trading days ago. Rate of change or other
indicators aren't effected by long holidays, though that's just a
minor issue.
The disadvantage is obviously that indexing the array with an actual
date becomes a bit harder and you always need to do a search.
What would really be the advantage if I were to use 'business day' as
my frequency?
And another question: Do you use the closing price from yahoo or the
adjusted closing price? It seems they use the adjusted prices
themselves, though I've come across one or two graphs where they
Thanks for all your advice Matt and Pierre.
> - Matt
> _______________________________________________
> SciPy-user mailing list
> SciPy-user@scipy.org
> http://mail.scipy.org/mailman/listinfo/scipy-user
More information about the SciPy-user mailing list
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An alternative method to crossing minimization on hierarchical graphs
Mutzel, Petra
MPI-I-97-1-008. March 1997, 15 pages. | Status: available - back from printing | Next --> Entry | Previous <-- Entry
Abstract in LaTeX format:
A common method for drawing directed graphs is, as a first step, to partition the
vertices into a set of $k$ levels and then, as a second step, to permute the verti
ces within the levels such that the number of crossings is minimized.
We suggest an alternative method for the second step, namely, removing the minimal
number of edges such that the resulting graph is $k$-level planar. For the final
diagram the removed edges are reinserted into a $k$-level planar drawing. Hence, i
nstead of considering the $k$-level crossing minimization problem, we suggest solv
ing the $k$-level planarization problem.
In this paper we address the case $k=2$. First, we give a motivation for our appro
Then, we address the problem of extracting a 2-level planar subgraph of maximum we
ight in a given 2-level graph. This problem is NP-hard. Based on a characterizatio
n of 2-level planar graphs, we give an integer linear programming formulation for
the 2-level planarization problem. Moreover, we define and investigate the polytop
e $\2LPS(G)$ associated with the set of all 2-level planar subgraphs of a given 2
-level graph $G$. We will see that this polytope has full dimension and that the i
nequalities occuring in the integer linear description are facet-defining for $\2L
The inequalities in the integer linear programming formulation can be separated in
polynomial time, hence they can be used efficiently in a branch-and-cut method fo
r solving practical instances of the 2-level planarization problem.
Furthermore, we derive new inequalities that substantially improve the quality of
the obtained
solution. We report on extensive computational results.
References to related material:
To download this research report, please select the type of document that fits best your needs. Attachement Size(s):
328 KBytes
Please note: If you don't have a viewer for PostScript on your platform, try to install GhostScript and GhostView
URL to this document: http://domino.mpi-inf.mpg.de/internet/reports.nsf/NumberView/1997-1-008
BibTeX @TECHREPORT{Mutzel97,
AUTHOR = {Mutzel, Petra},
TITLE = {An alternative method to crossing minimization on hierarchical graphs},
TYPE = {Research Report},
INSTITUTION = {Max-Planck-Institut f{\"u}r Informatik},
ADDRESS = {Im Stadtwald, D-66123 Saarbr{\"u}cken, Germany},
NUMBER = {MPI-I-97-1-008},
MONTH = {March},
YEAR = {1997},
ISSN = {0946-011X},
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Motivation for strong law of large numbers
up vote 18 down vote favorite
I always find the strong law of large numbers hard to motivate to students, especially non-mathematicians. The weak law (giving convergence in probability) is so much easier to prove; why is it worth
so much trouble to upgrade the conclusion to almost sure convergence?
I think it comes down to not having a good sense of why, practically speaking, a.s. convergence is better than convergence i.p. Sure, I can prove that one implies the other and not conversely, but
the counterexamples feel contrived. I understand the advantages of a.s. convergence on a technical level, but not on the level of everyday life.
So my question: how would you explain to, say, an engineer, the significance of having a.s. convergence as opposed to i.p.? Is there a "real-life" example of bad behavior that we're ruling out?
pr.probability ho.history-overview
3 en.wikipedia.org/wiki/… – Steve Huntsman Mar 24 '10 at 1:08
2 I think it is in Chung's undergraduate probability textbook, he juxtaposes two quotations. (I don't have it here now, so cannot check my recolloction.) Both are quoted from famous mathematicians,
one saying the weak law is useful and the strong law useless, and the other one saying the opposite. – Gerald Edgar Mar 24 '10 at 1:13
add comment
4 Answers
active oldest votes
Here is a nice post of T. Tao on SLLN. In the comments section he is asked a very similar question to which he answers the following: (I hope it's ok to reproduce it here, since it is
buried down in the comments)
Returning specifically to the question of finitary interpretations of the SLLN, these basically have to do with the situation in which one is simultaneously considering multiple
averages $\overline{X}_n$ of a single series of empirical samples, as opposed to considering just a single such average (which is basically the situation covered by the WLLN). For
up vote 15 instance, if one had some random intensity field of grayscale pixels, and wanted to compare the average intensities at 10 x 10 blocks, 100 x 100 blocks, and 1000 x 1000 blocks, then
down vote the SLLN suggests that these intensities would be likely to be simultaneously close to the average intensity. (The WLLN only suggests that each of these spatial averages are
accepted individually likely to be close to the average intensity, but does not preclude the possibility that when one considers multiple such spatial averages at once, that a few outlying
spatial averages will deviate from the average intensity. In my example with only three different averages, there isn’t much difference here, as the union bound only loses a factor of
three at most for the failure probability, but the SLLN begins to show its strength over the WLLN when one is considering a very large number of averages at once.)
updated link here toe the referenced comment terrytao.wordpress.com/2008/06/18/… – dan mackinlay Feb 6 at 16:12
add comment
Suppose you are in the context of collecting data and estimating the mean.
up vote 7 down vote Imagine a situation where SLLN does not hold. It means that with positive probability accumulating new data is useless.
add comment
You might use the three-series theorem to elaborate on a.s. convergence. This approach would also have the advantage of working towards the SLLN.
up vote 3 down vote
add comment
I think it is worth noting that even if real world systems are fundamentally finite (in which case the distinction between WLLN and SLLN gets a bit philosophical), history has shown that it
is extremely useful to approximate the discrete with the continuous. Thus we consider limit theorems to approximate statistics of large samples, we consider continuous distributions to
approximate complicated finite distributions, and we consider continuous stochastic processes in order to approximate finite ones (e.g. Donsker's invariance principle).
up vote The examples of sequences that converge in probability but not a.s. might seem a bit contrived, but then again most engineers seem to allow such philosophical absurdities as "let $X_n$ be an
3 down infinite sequence of coin tosses". In this regard, maybe it is best to phrase the distinction between convergence a.s. and convergence in probability in terms that seem more qualitative and
vote less analytic. For example, imagine you were presented a sequence of gambles, and you must take either all of them or none of them. There is a very significant distinction between knowing
that your wealth converges a.s. to some deterministic value vs knowing that it converges in probability (to that same value). In the former case, you expect in almost all states of the world
that if you play the game that your wealth eventually stabilizes. However, in the case of convergence in probability you could go bankrupt infinitely often. Yikes!
add comment
Not the answer you're looking for? Browse other questions tagged pr.probability ho.history-overview or ask your own question.
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Using lpsolve from R
R is a language and environment for statistical computing and graphics. It is a GNU project which is similar to the S language and environment which was developed at Bell Laboratories (formerly AT&T,
now Lucent Technologies) by John Chambers and colleagues. R can be considered as a different implementation of S. There are some important differences, but much code written for S runs unaltered
under R.
R provides a wide variety of statistical (linear and nonlinear modelling, classical statistical tests, time-series analysis, classification, clustering, ...) and graphical techniques, and is highly
extensible. The S language is often the vehicle of choice for research in statistical methodology, and R provides an Open Source route to participation in that activity.
One of R's strengths is the ease with which well-designed publication-quality plots can be produced, including mathematical symbols and formulae where needed. Great care has been taken over the
defaults for the minor design choices in graphics, but the user retains full control.
R is available as Free Software under the terms of the Free Software Foundation's GNU General Public License in source code form. It compiles and runs on a wide variety of UNIX platforms and similar
systems (including FreeBSD and Linux), Windows and MacOS.
The R environment
R is an integrated suite of software facilities for data manipulation, calculation and graphical display. It includes
• an effective data handling and storage facility,
• a suite of operators for calculations on arrays, in particular matrices,
• a large, coherent, integrated collection of intermediate tools for data analysis,
• graphical facilities for data analysis and display either on-screen or on hardcopy, and
• a well-developed, simple and effective programming language which includes conditionals, loops, user-defined recursive functions and input and output facilities.
The term "environment" is intended to characterize it as a fully planned and coherent system, rather than an incremental accretion of very specific and inflexible tools, as is frequently the case
with other data analysis software.
R, like S, is designed around a true computer language, and it allows users to add additional functionality by defining new functions. Much of the system is itself written in the R dialect of S,
which makes it easy for users to follow the algorithmic choices made. For computationally-intensive tasks, C, C++ and Fortran code can be linked and called at run time. Advanced users can write C
code to manipulate R objects directly.
Many users think of R as a statistics system. We prefer to think of it of an environment within which statistical techniques are implemented. R can be extended (easily) via packages. There are about
eight packages supplied with the R distribution and many more are available through the CRAN family of Internet sites covering a very wide range of modern statistics.
We will not discuss the specifics of R here but instead refer the reader to the R website. Also see An Introduction to R
R and lpsolve
lpsolve is callable from R via an extension or module. As such, it looks like lpsolve is fully integrated with R. Matrices can directly be transferred between R and lpsolve in both directions. The
complete interface is written in C so it has maximum performance.
There are currently two R packages based on lp_solve. Both packages are available from CRAN.
The lpSolve R package is the first implementation of an interface of lpsolve to R. It provides high-level functions for solving general linear/integer problems, assignment problems and transportation
problems. The following link contains the version of the driver: lpSolve: Interface to Lp_solve v. 5.5 to solve linear/integer programs. It does not contain the lpsolve API. Only the higher level
calls. Documentation for this interface can be found on: Interface to Lp_solve v. 5.5 to solve linear/integer programs
This driver is written and maintained by Sam Buttrey.
The lpSolveAPI R package is a second implementation of an interface of lpsolve to R. It provides an R API mirroring the lp_solve C API and hence provides a great deal more functionality but has a
steeper learning curve. The R interface to lpsolve contains its own documentation. See An R interface to the lp_solve library for the driver.
This driver is written and maintained by Kjell Konis.
Installing the lpsolve driver in R
How to install the driver depends on the environment.
In the RGui menu, there is a menu item 'Packages'. From there a package can be installed from a CRAN mirror or from a local zip file.
R command line
Packages can also be installed from the R command line. This is a more general approach that will work under all environments.
Installing the package takes a single command:
The lpSolve R package:
> install.packages("lpSolve")
and to install the lpSolveAPI package use the command:
> install.packages("lpSolveAPI")
The > shown before each R command is the R prompt. Only the text after > must be entered.
Loading the lpsolve driver in R
Installing the package is not enough. It must also loaded in the R memory space before it can be used. This can be done with the following command:
> library(lpSolveAPI)
> library("lpSolveAPI", character.only=TRUE)
Getting Help
Documentation is provided for each function in the lpSolve package using R's built-in help system. For example, the command
> ?add.constraint
will display the documentation for the add.constraint function.
Building and Solving Linear Programs Using the lpSolve R Package
This implementation provides the functions lp, lp.assign, lp.object, lp.transport and print.lp. These functions allow a linear program (and transport and assignment problems) to be defined and solved
using a single command.
For more information enter:
> ?lp
> ?lp.assign
> ?lp.object
> ?lp.transport
> ?print.lp
See also Interface to Lp_solve v. 5.5 to solve linear/integer programs
Building and Solving Linear Programs Using the lpSolveAPI R Package
This implementation provides an API for building and solving linear programs that mimics the lp_solve C API. This approach allows much greater flexibility but also has a few caveats. The most
important is that the lpSolve linear program model objects created by make.lp and read.lp are not actually R objects but external pointers to lp_solve 'lprec' structures. R does not know how to deal
with these structures. In particular, R cannot duplicate them. Thus one must never assign an existing lpSolve linear program model object in R code.
To load the library, enter:
> library(lpSolveAPI)
Consider the following example. First we create an empty model x.
> x <- make.lp(2, 2)
Then we assign x to y.
> y <- x
Next we set some columns in x.
> set.column(x, 1, c(1, 2))
> set.column(x, 2, c(3, 4))
And finally, take a look at y.
> y
Model name:
C1 C2
Minimize 0 0
R1 1 3 free 0
R2 2 4 free 0
Type Real Real
upbo Inf Inf
lowbo 0 0
The changes we made in x appear in y as well. Although x and y are two distinct objects in R, they both refer to the same lp_solve 'lprec' structure.
The safest way to use the lpSolve API is inside an R function - do not return the lpSolve linear program model object.
Learning by Example
> lprec <- make.lp(0, 4)
> set.objfn(lprec, c(1, 3, 6.24, 0.1))
> add.constraint(lprec, c(0, 78.26, 0, 2.9), ">=", 92.3)
> add.constraint(lprec, c(0.24, 0, 11.31, 0), "<=", 14.8)
> add.constraint(lprec, c(12.68, 0, 0.08, 0.9), ">=", 4)
> set.bounds(lprec, lower = c(28.6, 18), columns = c(1, 4))
> set.bounds(lprec, upper = 48.98, columns = 4)
> RowNames <- c("THISROW", "THATROW", "LASTROW")
> ColNames <- c("COLONE", "COLTWO", "COLTHREE", "COLFOUR")
> dimnames(lprec) <- list(RowNames, ColNames)
Lets take a look at what we have done so far.
> lprec # or equivalently print(lprec)
Model name:
COLONE COLTWO COLTHREE COLFOUR
Minimize 1 3 6.24 0.1
THISROW 0 78.26 0 2.9 >= 92.3
THATROW 0.24 0 11.31 0 <= 14.8
LASTROW 12.68 0 0.08 0.9 >= 4
Type Real Real Real Real
upbo Inf Inf Inf 48.98
lowbo 28.6 0 0 18
Now lets solve the model.
> solve(lprec)
[1] 0
> get.objective(lprec)
[1] 31.78276
> get.variables(lprec)
[1] 28.60000 0.00000 0.00000 31.82759
> get.constraints(lprec)
[1] 92.3000 6.8640 391.2928
Note that there are some commands that return an answer. For the accessor functions (generally named get.*) the output should be clear. For other functions (e.g., solve), the interpretation of the
returned value is described in the documentation. Since solve is generic in R, use the command
> ?solve.lpExtPtr
to view the appropriate documentation. The assignment functions (generally named set.*) also have a return value - often a logical value indicating whether the command was successful - that is
returned invisibly. Invisible values can be assigned but are not echoed to the console. For example,
> status <- add.constraint(lprec, c(12.68, 0, 0.08, 0.9), ">=", 4)
> status
[1] TRUE
indicates that the operation was successful. Invisible values can also be used in flow control.
Cleaning up
To free up resources and memory, the R command rm() must be used.
For example:
> rm(lprec)
See also Using lpsolve from MATLAB, Using lpsolve from O-Matrix, Using lpsolve from Sysquake, Using lpsolve from Octave, Using lpsolve from FreeMat, Using lpsolve from Euler, Using lpsolve from
Python, Using lpsolve from Sage, Using lpsolve from PHP, Using lpsolve from Scilab Using lpsolve from Microsoft Solver Foundation
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Creative cultural transmission as chaotic sampling
First, Chaos: Some formula produce unpredictable trajectories, for instance the Lorenz attractor. Here’s what part of a trajectory looks like:
You can play with the dynamics using this applet.
The trajectory will not pass through the same point twice, but is not completely random. Lorenz attractors have been used to re-sample sequences in the following way: Imagine you have a sequence
of musical notes. Pick a starting point on the Lorenz trajectory and associate each note with successive points. Now you have your notes laid out on the Lorenz attractor so that for any point in
the space you can find the closest associated note. If you start on the Lorenz trajectory from a different point, you can sample the notes in a different sequence. This sample will be different
from the original, but tends to preserve some of the structure. That is, the Lorenz attractor scrambles the sample, but in a chaotic way, not a random one.
related { Gilles Deleuze, Difference and Repetition, 1968 }
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Tangent plane to a surface
December 14th 2010, 05:40 AM #1
Tangent plane to a surface
Is it correct to assume that taking the dot product of the gradient and the derivative of a surface function will give me the equation of the tangent plane at any point on the surface? My notes
are not entirely clear about this. Thanks in advance.
Also, would a question of this nature be better suited in the calculus section?
To answer this question properly, I have be sure that you know how to write the equation of a plane given a point on the plane and its normal. Do you?
If so then the gradient to a surface at a point is the normal of the tangent plane at that point.
maybe i'm confusing the derivative of a vector function curve. I know that it represents the tangent at any point on a curve, and if I know the normal vector as well at that point (the
gradient?), then the dot product of the two equal to zero should give me a plane tangent to the curve at that point
I understand that when looking at surfaces, the tangent plane at a point (x1, y1, z1) of the function F(x,y,z) is going to be:
∇F(x,y,z) . <(x - x1), (y - y1), (z - z1)> = 0
and my visualization of this is the following:
It should be $abla F(x_1 ,y_1 ,z_1 ) \cdot \left\langle {x - x_1 ,y - y_1 ,z - z_1 } \right\rangle = 0$ for the tangent plane to the field $F(x,y,z)$ at $(x_1,y_1,z_1)$.
But remember that the gradient involves partial derivatives.
super, pretty much what the notes and readings say. Thanks for the help!
December 14th 2010, 06:43 AM #2
December 14th 2010, 07:23 AM #3
December 14th 2010, 07:48 AM #4
December 14th 2010, 12:53 PM #5
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9. Impedance and Phase Angle
The impedance of a circuit is the total effective resistance to the flow of current by a combination of the elements of the circuit.
Symbol: Z
Units: `Ω`
The total voltage across all 3 elements (resistors, capacitors and inductors) is written
To find this total voltage, we cannot just add the voltages V[R], V[L] and V[C].
Because V[L] and V[C] are considered to be imaginary quantities, we have:
Impedance V[RLC] = IZ
So `Z = R + j(X_L− X_C)`
Now, the magnitude (size, or absolute value) of Z is given by:
Phase angle
`tan\ theta=(X_L-X_C)/R`
Angle θ represents the phase angle between the current and the voltage.
Compare this to the Phase Angle that we met earlier in Graphs of y = a sin(bx + c).
Example 1
A circuit has a resistance of `5\ Ω` in series with a reactance across an inductor of `3\ Ω`. Represent the impedance by a complex number, in polar form.
Example 2(a)
A particular ac circuit has a resistor of `4\ Ω`, a reactance across an inductor of `8\ Ω` and a reactance across a capacitor of `11\ Ω`. Express the impedance of the circuit as a complex number in
polar form.
Interactive RLC graph
Below is an interactive graph to play with (it's not a static image). You can explore the effect of a resistor, capacitor and inductor on total impedance in an AC circuit.
Activities for this Interactive
1. First, just play with the sliders. Drag the red dot left or right to vary the impedance due to the resistor, `R`, the blue dot up or down to vary the impedance due to the inductor, `X_L`, and the
green dot up or down to vary the impedance due to the capacitor, `X_C`.
2. Observe the effects of different impedances on the values of X[L] − X[C] and Z.
3. Observe the effects of different impedances on θ, the angle the black line makes with the horizontal (in radians).
4. Consider the graphs of voltage and current on the right of the interactive. Observe the amount of lag or lead as you change the sliders.
5. What have you learned from playing with this interactive?
Example 2(b)
Referring to Example 2 (a) above, suppose we have a current of 10 A in the circuit. Find the magnitude of the voltage across
i) the resistor (V[R])
ii) the inductor (V[L])
iii) the capacitor (V[C])
iv) the combination (V[RLC])
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confused on this dielectric capacitance problem
The electric field between the plates of a paper-separated (K=3.75) capacitor is 9.41E4 V/m. The plates are 1.75mm apart and the charge on each plate is 0.775E-6C. Determine the capacitance of the
capacitor and the area of each plate?
I got the voltage by V=Ed. and that came out to 164.675
Where do i go from there because in order to get capacitance you need to know area, and finding the area of each plate is also a question in the problem. I thought about using C=Q/V, but that doesn't
apply for dielectrics. Can someone helpme out?
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