try the applet,
and I also do not intend to create a Passport account.
It seems to me,
that if Harris is interested in folks seeing his works,
that he should make them more accessible.

Tom Potter http://tompotter.us
====
> I don't think it works for p = 3 either.
<2> = {2, 1} Z modulo 3.
GREG
Lurch
> Conjecture:
> For every prime p, the multiplicative group Z(modulo p) contains at
> least
> one prime q
It fails for p=2 :)
>
====
Consider a number p less than n!
If p is not divisible by the integers 2 .. n, then say that x is
nprime
(note that 1 will be in this set). x is not necessarily prime but is
potentially prime in that it cannot be divided by 2 through n.
Let N be the set of all p that are nprime.
All prime numbers less than n! will be in the set of N. This makes sense
because all primes less than N will also be nprime  that is not
divisible by 2 .. n.
There is a simple symmetry about n! Construct a new set M by adding n! to
each member of N. The set of primes between n! and 2n! will be in M.
Again, not all the numbers in M will be prime but M will contain all the
primes. In fact the number of primes in M will be less than the number of
primes in N. Even so, in many cases a prime less than n! will have a
mirror
prime between n! and 2n!.
For example, consider 4! = 24. The following numbers are nprime:
1, 3, 5, 7, 11,13,17,19,23
and form the set N.
The set M will be these numbers plus 24 or
25, 27, 29, 31, 37, 41, 43, 47
All of these are the prime except for 25 and all the primes between 24 and
48 fall in the set M.
N and M can be fairly easily constructed for 5!.
In short, if you know an nprime less than n! then you can find a candidate
prime greater than n! by simply adding n! to the nprime.
====
Consider a number p less than n!
If p is not divisible by the integers 2 .. n, then say that x is
nprime
>(note that 1 will be in this set). x is not necessarily prime but is
>potentially prime in that it cannot be divided by 2 through n.
Let N be the set of all p that are nprime.
All prime numbers less than n! will be in the set of N. This makes sense
>because all primes less than N will also be nprime  that is not
[SNIP]
God damn, I hate it when people start out a post with: Consider...
and then proceed to shit out some random math problem, with no
introduce yourself... and ONLY THEN give us your goddamned
homework problem.
FuckingA, this happens all the time. I don't give a shit about some
random math problem... but if you would like some help, then that's
a different story... but you have to ASK FOR IT.
..I don't know... sorry about the rant, but I just find it so annoying.
AS
====
> God damn, I hate it when people start out a post with: Consider...
> and then proceed to shit out some random math problem, with no
>
> introduce yourself... and ONLY THEN give us your goddamned
> homework problem.
Then again, this is a maths newgroup. :]

J K Haugland
http://www.neutreeko.com
====
> God damn, I hate it when people start out a post with: Consider...
>> and then proceed to shit out some random math problem, with no
>>
>> introduce yourself... and ONLY THEN give us your goddamned
>> homework problem.
Then again, this is a maths newgroup. :]
Maybe you are being a bit unfair... It wasn't a homework problem, it
was a result that he found, and overall not an unisnteresting one for
at least some people here. Before complaining, why not actually reas
the post.
====
>
> God damn, I hate it when people start out a post with: Consider...
> and then proceed to shit out some random math problem, with no
>
> introduce yourself... and ONLY THEN give us your goddamned
> homework problem.
>>Then again, this is a maths newgroup. :]
>
>
> Maybe you are being a bit unfair... It wasn't a homework problem, it
> was a result that he found, and overall not an unisnteresting one for
> at least some people here. Before complaining, why not actually reas
> the post.
I don't see Jan Kristian being unfair at all.
Before complaining, why not actually read his post?

Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html
The League of Gentlemen
====
>> God damn, I hate it when people start out a post with: Consider...
>> and then proceed to shit out some random math problem, with no
>> introduce yourself... and ONLY THEN give us your goddamned
>> homework problem.
Then again, this is a maths newgroup. :]
>
Good Point... I guess that's why I encluded the tags.
AS
====
>>Consider a number p less than n!
>>If p is not divisible by the integers 2 .. n, then say that x is
nprime
>>(note that 1 will be in this set). x is not necessarily prime but is
>>potentially prime in that it cannot be divided by 2 through n.
>>Let N be the set of all p that are nprime.
>>All prime numbers less than n! will be in the set of N. This makes sense
>>because all primes less than N will also be nprime  that is not
>>
>[SNIP]
and then proceed to shit out some random math problem, with no
>
>introduce yourself... and ONLY THEN give us your goddamned
>homework problem.
>
> FuckingA, this happens all the time. I don't give a shit about some
>random math problem... but if you would like some help, then that's
>a different story... but you have to ASK FOR IT.
>..I don't know... sorry about the rant, but I just find it so annoying.
>
This looks like a rant with
introduce yourself... and ONLY THEN give us your goddamned
homework problem.
FuckingA, this happens all the time.
But fortunately not that often on sci.math.
Jon Miller
====
>I don't see Jan Kristian being unfair at all.
>Before complaining, why not actually read his post?
>
>
Nah, Gartogg just replied to the wrong post. He was responding to as,
not Jan Kristian, even though he replied to JKs post.
Jon Miller
====
> Consider a number p less than n!
>
> If p is not divisible by the integers 2 .. n, then say that x is
nprime
> (note that 1 will be in this set). x is not necessarily prime but is
> potentially prime in that it cannot be divided by 2 through n.
>
> Let N be the set of all p that are nprime.
>
> All prime numbers less than n! will be in the set of N. This makes sense
> because all primes less than N will also be nprime  that is not
> divisible by 2 .. n.
>
> There is a simple symmetry about n! Construct a new set M by adding n!
to
> each member of N. The set of primes between n! and 2n! will be in M.
> Again, not all the numbers in M will be prime but M will contain all the
> primes. In fact the number of primes in M will be less than the number
of
> primes in N. Even so, in many cases a prime less than n! will have a
mirror
> prime between n! and 2n!.
>
> For example, consider 4! = 24. The following numbers are nprime:
>
> 1, 3, 5, 7, 11,13,17,19,23
>
> and form the set N.
>
> The set M will be these numbers plus 24 or
>
> 25, 27, 29, 31, 37, 41, 43, 47
>
> All of these are the prime except for 25 and all the primes between 24
and
> 48 fall in the set M.
>
> N and M can be fairly easily constructed for 5!.
>
> In short, if you know an nprime less than n! then you can find a
candidate
> prime greater than n! by simply adding n! to the nprime.
This is known as reinventing the wheel.
You cross 'em, I'll knock 'em in.
Phil
====
>
>>I don't see Jan Kristian being unfair at all.
>>Before complaining, why not actually read his post?
>>
> Nah, Gartogg just replied to the wrong post. He was responding to as,
> not Jan Kristian, even though he replied to JKs post.
He snipped as's name too.
Symptomatic though of someone who doesn't read carefully
before sounding off.

Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html
The League of Gentlemen
====
Conjecture 1:
If p is any odd prime & p1 contains at least 1 Quadratic nonresidue, then
at least one prime q which divides p1 is a primitive root.
Conjecture 2:
If p is any prime bigger than 3, then the multiplicative group Z(modulo
p1)
contains at least one primitive root of p.
Conjecture 3:
Suppose p is any odd prime and p1 is of the form 2q^x, where q is an odd
prime. If g is a quadratic non residue of both p and p1, then g is a
primitive root.

So, I'm looking for counterexamples, or reasons why these wouldn't be true.
So far, I'm thinking that both 1 & 3 are false, and 2 is true  although I
haven't found any counterexamples, or been able to prove them either way.
Any help would be great.
GREG
====
I have a function proportional to a probability distribution of interest
that is giving me fits.
y = x * I(1(x^2); y, 1/2)
where 'I' is the regularized beta function. What I need is the form of
this distribution as y>+oo and x>0. For large y, it looks awfully like
a gamma or beta distribution, and I'd really like to know if it *is* one
of those (or something similar). Can anyone help with this?
Zeus
====
Suppose X, Y, Z are positive random variable with the pdf f_X(t), f_Y(t),
f_Z(t), respectively. And F_X(t), F_Y(t), F_Z(t) are respective cdf
function. The quantity a is a positive real number. I need to evaluate the
following probabiltiy.
P( X
> Suppose X, Y, Z are positive random variable with the pdf f_X(t), f_Y(t),
> f_Z(t), respectively. And F_X(t), F_Y(t), F_Z(t) are respective cdf
> function. The quantity a is a positive real number. I need to evaluate
the
> following probabiltiy.
>
> P( X
> I develop the following expresion.
>
> P( X =int_{0}^{a} f_X(t) F_Y(t) [1F_Z(t) ]
dt
>
>
No. As you have stated it, you have definite values for Y and Z,
such that
0 < Y < X < min(a,Z)
Why the last inequality? Because if a < Z then X < a guarantees x <
Z,
and viceversa.
Of course the probability must be 0 if min(a,Z) < Y . Thus
P( Y < X < min(a,Z) )
= int _Y ^{min(a,Z)} {dt f_X (t) )
= [ F_X ( min(a,Z) )  F_X (Y) ] theta ( min(a,Z)
 Y )
You can then multiply by the pdf's f_Y (u) and f_Z (v), and
integrate
over u and v to get the probability of finding an X that satisfies
the
restrictions.

Julian V. Noble
Professor Emeritus of Physics
jvn@spamfree.virginia.edu
^^^^^^^^
http://galileo.phys.virginia.edu/~jvn/
Science knows only one commandment: contribute to science.
 Bertolt Brecht, Galileo.
====
Hey,
Im curious, what would you guys/gals say the probability of someone
entering a Ph.D. program in Math or Stats and not finishing it. i.e.
dropping out.
====
>Im curious, what would you guys/gals say the probability of someone
>entering a Ph.D. program in Math or Stats and not finishing it. i.e.
>dropping out.
Of not leaving with a Ph.D.? 75% would be my guess, based on the eight
years I've been at Colorado.
Doug
====
>Hey,
Im curious, what would you guys/gals say the probability of someone
>entering a Ph.D. program in Math or Stats and not finishing it. i.e.
>dropping out.
Wouldn't know about stat, but sad to say a large majority of the
people who enter the PhD program in math here at OSU end
up without a PhD, one way or another.
************************
David C. Ullrich
====
Im curious, what would you guys/gals say the probability of someone
>entering a Ph.D. program in Math or Stats and not finishing it. i.e.
>dropping out.
Of not leaving with a Ph.D.? 75% would be my guess, based on the eight
> years I've been at Colorado.
Doug
Depends a lot on the school, of course. If you want the highest
probability, I would guess in the states it might be University of Montana
or University of Idaho or Idaho State. Not disparaging, that's just how
they are.
====
> Hey,
>
> Im curious, what would you guys/gals say the probability of someone
> entering a Ph.D. program in Math or Stats and not finishing it. i.e.
> dropping out.
I just read that it was about 5050. Long ago, I heard that it is
another 5050 that one who finishes will do nothing after their
thesis. This suggests that a lot of theses are written by the
advisor.
====
> Hey,
Im curious, what would you guys/gals say the probability of someone
> entering a Ph.D. program in Math or Stats and not finishing it. i.e.
> dropping out.
I just read that it was about 5050. Long ago, I heard that it is
> another 5050 that one who finishes will do nothing after their
> thesis. This suggests that a lot of theses are written by the
> advisor.
Not in the least. In graduate school you are surrounded by excellent
mathematicians and the spirit of mathematics. Mathematics is everywhere;
it
is the whole world. Everybody around you thinks that it's the only thing
worth learning.
Then you get a job at Podunk, and discover that your newfound colleagues
think that knowing mathematics is knowing the difference between addition
and subtraction. Discussions in the faculty lounge are about football.
You teach 12 to 15 credits a week, same old stuff year after year. You get
numb and tired and disillusioned (Pirsig mentions this in Zen and the Art
of
Motorcycle Maintenance). You have no real contact with the living world of
mathematics and mathematicians; all you've got is your Calculus I textbook
and your colleagues. With great effort you can scare up money to go to the
occasional convention.
Some people overcome these obstacles, bless them.
====
...
>> I just read that it was about 5050. Long ago, I heard that it is
>> another 5050 that one who finishes will do nothing after their
>> thesis. This suggests that a lot of theses are written by the
>> advisor.
Not in the least. In graduate school you are surrounded by excellent
>mathematicians and the spirit of mathematics. Mathematics is everywhere;
it
>is the whole world. Everybody around you thinks that it's the only thing
>worth learning.
Then you get a job at Podunk, and discover that your newfound colleagues
>think that knowing mathematics is knowing the difference between addition
>and subtraction. Discussions in the faculty lounge are about football.
You teach 12 to 15 credits a week, same old stuff year after year. You
get
>numb and tired and disillusioned (Pirsig mentions this in Zen and the Art
of
>Motorcycle Maintenance). You have no real contact with the living world
of
>mathematics and mathematicians; all you've got is your Calculus I textbook
>and your colleagues. With great effort you can scare up money to go to
the
>occasional convention.
Some people overcome these obstacles, bless them.
I like to believe that the advent of Usenet, later the web,
arxiv.org, etc., are helping more people overcome those obstacles
more effectively.
Lee Rudolph
====
>> Hey,
>>
>> Im curious, what would you guys/gals say the probability of someone
>> entering a Ph.D. program in Math or Stats and not finishing it. i.e.
>> dropping out.
I just read that it was about 5050.
5050 for finishing or not finishing?
;)

Rouben Rostamian
====
> I just read that it was about 5050. Long ago, I heard that it is
> another 5050 that one who finishes will do nothing after their
> thesis. This suggests that a lot of theses are written by the
> advisor.
Most of those PhDs go to positions where research is not encouraged,
rather teaching and service are encouraged. 4year colleges, community
colleges, even high schools. That could also be a reason for not
writing more papers. The idea that writing no research papers equals
doing nothing shows a warped view of the world.

G. A. Edgar
http://www.math.ohiostate.edu/~edgar/
====
Not in the least. In graduate school you are surrounded by excellent
>mathematicians and the spirit of mathematics. Mathematics is everywhere;
it
>is the whole world. Everybody around you thinks that it's the only thing
>worth learning.
Then you get a job at Podunk, and discover that your newfound colleagues
>think that knowing mathematics is knowing the difference between addition
>and subtraction. Discussions in the faculty lounge are about football.
You teach 12 to 15 credits a week, same old stuff year after year. You
get
>numb and tired and disillusioned (Pirsig mentions this in Zen and the Art
of
>Motorcycle Maintenance). You have no real contact with the living world
of
>mathematics and mathematicians; all you've got is your Calculus I textbook
>and your colleagues.
At this point I would suggest: Guy's UPINT, GP/PARI, some decent coffee, a
supply of good Scotch Whiskey, and a great wife. Perhaps time on a trout
stream just outside Podunk two evenings a week may be of some benefit. A
summer working the wheat harvest might help also.
Clearly Podunk ain't MSRI. Southwest will, however, get you to Oakland for
$99. I don't know what AC Transit costs these days, but it can't be much.
Even the numb and tired and disillusioned have choices, I would think.
Rich
====
>
>Im curious, what would you guys/gals say the probability of someone
>entering a Ph.D. program in Math or Stats and not finishing it. i.e.
>dropping out.
>
> Of not leaving with a Ph.D.? 75% would be my guess, based on the eight
> years I've been at Colorado.
>
> Doug
Wow, i never would have thought it be that high. I would have guessed
maybe 30% drop out rate. I figured after someone got their Masters in
Mathematics, got straight A's in their graduate courses that it would
be sufficient to prepare them for the doctorate program in mathematics
/or statistics.
====
> I just read that [chance of completing Ph.D.] was about 5050. Long
> ago, I heard that it is another 5050 that one who finishes will do
> nothing after their thesis.
You mean 50% of mathematics Ph.D.s are unemployed, spending their
entire lives sitting in their room staring at the walls? I think
not. Perhaps there is a much more narrow (minded) interpretation of
the word nothing?
I'm going to speculate that nothing is interpreted along the lines
not inconsistent with the simple observation that something on the
order of 50% of math or science Ph.D. graduates enter careers other
than academics.
> This suggests that a lot of theses are written by the advisor.
I would suggest that one for whom this is suggested by the 5050
figure should consider investing some time in the study of logic or
statistics.
Kevin.
====
>
>Im curious, what would you guys/gals say the probability of someone
>entering a Ph.D. program in Math or Stats and not finishing it. i.e.
>dropping out.
>
> Of not leaving with a Ph.D.? 75% would be my guess, based on the eight
> years I've been at Colorado.
>
> Doug
>
> Wow, i never would have thought it be that high. I would have guessed
> maybe 30% drop out rate. I figured after someone got their Masters in
> Mathematics, got straight A's in their graduate courses that it would
> be sufficient to prepare them for the doctorate program in mathematics
> /or statistics.
One thing is that most people enter without a Masters degree in the
first place and switch to a Masters rather than finish the PhD. That
accounts for a big portion of the discrepancy you think is present. I
on the other hand was one of the few people that had passed most of
the hurdles of a math PhD program without actually getting such a
degree. I believe there was one other out of maybe two or three dozen
PhD graduates during the five year period I was trying for a PhD.
Karl Hallowell
====
>
think that knowing mathematics is knowing the difference between
addition
>and subtraction. Discussions in the faculty lounge are about football.
You teach 12 to 15 credits a week, same old stuff year after year. You
get
>numb and tired and disillusioned (Pirsig mentions this in Zen and the Art
of
>Motorcycle Maintenance). You have no real contact with the living world
of
>mathematics and mathematicians; all you've got is your Calculus I
textbook
>and your colleagues. With great effort you can scare up money to go to
the
>occasional convention.
Some people overcome these obstacles, bless them.
>
> I like to believe that the advent of Usenet, later the web,
> arxiv.org, etc., are helping more people overcome those obstacles
> more effectively.
I believe that is quite accurate. I currently (to be cured in a few
months) have no access to a nearby college library, community, etc.
The nearest college is more than fourty miles away and there I have
only a few informal contacts in the aerospace engineering community.
My real connections (as such) are online.
Any serious math or physics concept is available online. Ie, I can
google for Borel subgroups, Kaluza Klein models, the inverse Galois
problem, or the Eight Vertex model and quickly find relevant research
and expository material. The USENET might not be able to answer my
questions, but they never have failed to come up with some insight.
I'm still trying to figure out how to use arXiv.org (even after years
of playing with it), but it's proving to be an amazing research tool
even with my limited experience.
Karl Hallowell
====
>>
>>Im curious, what would you guys/gals say the probability of someone
>>entering a Ph.D. program in Math or Stats and not finishing it. i.e.
>>dropping out.
>>
>> Of not leaving with a Ph.D.? 75% would be my guess, based on the eight
>> years I've been at Colorado.
>>
>> Doug
Wow, i never would have thought it be that high. I would have guessed
>maybe 30% drop out rate. I figured after someone got their Masters in
>Mathematics, got straight A's in their graduate courses that it would
>be sufficient to prepare them for the doctorate program in mathematics
>/or statistics.
Again, I have no idea how things are in stat, but no that's not how
it is in math at all. A master's degree requires that you learn a
certain amount of mathematics  how much and how well you're
required to learn it varies from place to place. A PhD requires
_much_ more. First, it requires that you learn much more
mathematics  much deeper mathematics, and you're required
to understand it much better than a master's student (to
oversimplify that last point, a master's student gets credit
for knowing facts, while a PhD student only gets credit for
knowing how to _prove_ those facts).
And then there's the much more significant difference: A
PhD requires a thesis, which is supposed to be
significant original research. Of course some theses are
more significant and original than others, but regardless,
it's a totally different sort of requirement from anything
that's required in a typical master's degree  at least
theoretically, when you finish your PhD there's supposed
to be _something_ that you understand better than
anyone else on the planet.
************************
David C. Ullrich
====
...
> I just read that it was about 5050. Long ago, I heard that it is
> another 5050 that one who finishes will do nothing after their
> thesis. This suggests that a lot of theses are written by the
> advisor.
>>Not in the least. In graduate school you are surrounded by excellent
>>mathematicians and the spirit of mathematics. Mathematics is everywhere;
it
>>is the whole world. Everybody around you thinks that it's the only thing
>>worth learning.
>>Then you get a job at Podunk, and discover that your newfound colleagues
>>think that knowing mathematics is knowing the difference between addition
>>and subtraction. Discussions in the faculty lounge are about football.
>>You teach 12 to 15 credits a week, same old stuff year after year. You
get
>>numb and tired and disillusioned (Pirsig mentions this in Zen and the Art
of
>>Motorcycle Maintenance). You have no real contact with the living world
of
>>mathematics and mathematicians; all you've got is your Calculus I
textbook
>>and your colleagues. With great effort you can scare up money to go to
the
>>occasional convention.
>>Some people overcome these obstacles, bless them.
I like to believe that the advent of Usenet, later the web,
>arxiv.org, etc., are helping more people overcome those obstacles
>more effectively.
It can certainly help people stay in touch, or at least that seems
plausible. Hard to see how it can help with the huge teaching
loads at Podunk, though.
>Lee Rudolph
************************
David C. Ullrich
====
>>I like to believe that the advent of Usenet, later the web,
>>arxiv.org, etc., are helping more people overcome those obstacles
>>more effectively.
It can certainly help people stay in touch, or at least that seems
>plausible. Hard to see how it can help with the huge teaching
>loads at Podunk, though.
Why, by providing the students^Wclients^Wenrollees at Podunk
with sci.math to do their homework for them, of course.
And if you'd read Hyman Bass's report to the Carnegie Foundation
in the latest _Notices_, you'd realize that teaching load is
a doubleplusungood phrase. Time to talk about research burden
instead!
Lee Rudolph
====
>
> Most of those PhDs go to positions where research is not encouraged,
> rather teaching and service are encouraged. 4year colleges, community
> colleges, even high schools. That could also be a reason for not
> writing more papers. The idea that writing no research papers equals
> doing nothing shows a warped view of the world.
Adding to this ... A PhD program in mathematics that ONLY prepares the
participant for writing research papers is a seriously incomplete
program at best. Data shows that only about 20% of math PhDs in the US
will end up at PhDgranting universities.
====
>>
>> Most of those PhDs go to positions where research is not encouraged,
>> rather teaching and service are encouraged. 4year colleges, community
>> colleges, even high schools. That could also be a reason for not
>> writing more papers. The idea that writing no research papers
equals
>> doing nothing shows a warped view of the world.
Adding to this ... A PhD program in mathematics that ONLY prepares the
>participant for writing research papers is a seriously incomplete
>program at best. Data shows that only about 20% of math PhDs in the US
>will end up at PhDgranting universities.
There is neither a logical nor a pragmatic connection between your
last two sentences. Many universities and colleges which do not
grant PhDs (in mathematics) nonetheless have (however unreasonably
and/or unrealistically) a requirement that their (mathematics)
faculty members write and publish research papers, at least
if they expect to get tenure and/or merit raises.
Lee Rudolph
====
>
>Im curious, what would you guys/gals say the probability of someone
>entering a Ph.D. program in Math or Stats and not finishing it. i.e.
>dropping out.
>
> Of not leaving with a Ph.D.? 75% would be my guess, based on the eight
> years I've been at Colorado.
>
> Doug
Well, at my University you need an A average in your graduate courses
to be aloud entrance into the PH.D. program. Would it still be a 75%
failure rate you think ??
====
>> Hey,
>>
>> Im curious, what would you guys/gals say the probability of someone
>> entering a Ph.D. program in Math or Stats and not finishing it. i.e.
>> dropping out.
>I just read that it was about 5050. Long ago, I heard that it is
>another 5050 that one who finishes will do nothing after their
>thesis. This suggests that a lot of theses are written by the
>advisor.
How in the hell does it suggest that? If I go into industry after
finishing,
maybe I'm not writing papers, but that doesn't mean that my thesis was
written for me.
Doug
====
> Clearly Podunk ain't MSRI. Southwest will, however, get you to Oakland
for
> $99. I don't know what AC Transit costs these days, but it can't be much.
> Even the numb and tired and disillusioned have choices, I would think.
if you've got the patience to ride the bus from Oakland, kudos to you.
It's only $2.25, but it takes over an hour and a half just to get to
downtown Berkeley. if you take the BART, it's 4.25 total, and worth
the $2.
Ben
====
>> Most of those PhDs go to positions where research is not encouraged,
>> rather teaching and service are encouraged. 4year colleges, community
>> colleges, even high schools. That could also be a reason for not
>> writing more papers. The idea that writing no research papers
equals
>> doing nothing shows a warped view of the world.
>Adding to this ... A PhD program in mathematics that ONLY prepares the
>participant for writing research papers is a seriously incomplete
>program at best. Data shows that only about 20% of math PhDs in the US
>will end up at PhDgranting universities.
Such a program will only prepare the participant for doing
research in a narrow area. Unfortunately, these seem to be
most of what is being done now, especially in statistics.
The emphasis on interdisciplinary programs mainly produces
those who do not know the basics of anything, but these
programs have high rates of finishing.
Students are not getting the basics of set theory, algebra,
analysis, and topology these days. Learning how to compute
and how to solve certain types of problems fails if basic
material not covered in that is needed. Abstract concepts
are needed for understanding, even if the details of them
are not used.

This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Deptartment of Statistics, Purdue University
====
>Hey,
>Im curious, what would you guys/gals say the probability of someone
>entering a Ph.D. program in Math or Stats and not finishing it. i.e.
>dropping out.
No one seems to have mentioned it, but I think it may depend on the
specific university and what its criteria are for being admitted into
the program. Averaging over all U.S. schools would probably not provide
a meaningful statistic. My guess is that there is a significant variation.
Hard data could be obtained by asking the Director of Graduate Studies
for various programs. At my school I'll ask him the next time we're on
the tennis courts.

John E. Prussing
University of Illinois at UrbanaChampaign
Department of Aerospace Engineering
http://www.uiuc.edu/~prussing
====
available online in PDF format at http://www.ams.org/employment/asst.pdf
Everyone thinking about graduate school in mathematics should look at
this booklet.
For example, the first institution that I looked at in the booklet had
72 full time graduate students, 15 part time graduate students, and 15
full time first year graduate students. The department had graduated
18 MS students in the past year, and an average of 3 PhD's per year
gets an MS, but only about one in five entering graduate students goes
on to get a PhD. Of course, you should go back to previous years
booklets to see whether there have been any significant changes in
enrollment patterns.
Looking at about a dozen schools in this booklet, the ratio of
full time first year graduate students to PhD's per year (note
that PhD's for the last four years are given in the book, so this has
to be divided by four) runs from about 5to1 down to 2to1.
Of course, some students enter the graduate program intending to get
an MS degree. Unfortunately, I can't think of any way to distinguish
those students from students who were given an MS as a consolation
prize. In many cases, the total number of MS and PhD degrees per year
is very similar to the number of full time first year students, indicating
that most students get at least an MS. In other cases, far fewer degrees
are awarded than there are entering students.
For the big picture, it's worth pointing out that there are
approximately 15,000 graduate students in PhD granting departments of
math and statistics in the US, and that these departments produce
something like 1,0001,200 PhD graduates per year. These numbers
haven't changed dramatically in the last 10 years. (See the latest
AMS annual survey report for the numbers.)

Brian Borchers borchers@nmt.edu
Department of Mathematics http://www.nmt.edu/~borchers/
Socorro, NM 87801 FAX: 5058355366

Brian Borchers borchers@nmt.edu
Department of Mathematics http://www.nmt.edu/~borchers/
Socorro, NM 87801 FAX: 5058355366
====
> In <1eb18a7f.0307180852.4f4c5c6b@posting.google.com
>Hey,
Im curious, what would you guys/gals say the probability of someone
>entering a Ph.D. program in Math or Stats and not finishing it. i.e.
>dropping out.
No one seems to have mentioned it, but I think it may depend on the
> specific university and what its criteria are for being admitted into
> the program. Averaging over all U.S. schools would probably not provide
> a meaningful statistic. My guess is that there is a significant
variation.
Hard data could be obtained by asking the Director of Graduate Studies
> for various programs. At my school I'll ask him the next time we're on
> the tennis courts.

> John E. Prussing
> University of Illinois at UrbanaChampaign
> Department of Aerospace Engineering
> http://www.uiuc.edu/~prussing
I did mention it, when this thread started. I also suggested a few schools
where I guessed (I have no data) that the probability of finishing might be
highest.
====
>Wow, i never would have thought it be that high. I would have guessed
>maybe 30% drop out rate. I figured after someone got their Masters in
>Mathematics, got straight A's in their graduate courses that it would
>be sufficient to prepare them for the doctorate program in mathematics
>/or statistics.
But that ability to do well in classes is not particularly well correlated
with the ability to generate new mathematical ideas. You don't get a
PhD for taking a lot of classes, you know. (In some places, you don't
take _any_ classes to get a PhD.)
I would also object to a phrase like drop out. In secondary school
and below, there is a clear expectation that degree completion is the
necessary goal for everyone of that age. At the graduate level, and even
at the undergraduate level, leaving a program is not necessarily an
indication of some kind of failure. Students' eyes are opened in school
to the reality of the career choices for which they are preparing, and
they may well decide they don't like that image  even if they're doing
well and can continue to do well. Even if your mathematical skills are
superb, if what you want to do is make a lot of money, or to have time
to raise a family, or to work with some of the world's needy people,
then you would be making a mistake to complete a PhD in mathematics.
dave
====
This suggests that a lot of theses are written by the advisor.
Right. After all, why *wouldn't* a professor want to forego his or
her own research activities for a couple years to write an enormous
paper under a student's name?
Maybe you meant to say something less hilarious.

Kevin
====
> Hey,
> Im curious, what would you guys/gals say the probability of someone
> entering a Ph.D. program in Math or Stats and not finishing it. i.e.
> dropping out.
I just read that it was about 5050. Long ago, I heard that it is
> another 5050 that one who finishes will do nothing after their
> thesis. This suggests that a lot of theses are written by the
> advisor.
>
> Not in the least. In graduate school you are surrounded by excellent
> mathematicians and the spirit of mathematics. Mathematics is everywhere;
it
> is the whole world. Everybody around you thinks that it's the only thing
> worth learning.
>
> Then you get a job at Podunk, and discover that your newfound colleagues
> think that knowing mathematics is knowing the difference between addition
> and subtraction. Discussions in the faculty lounge are about football.
>
> You teach 12 to 15 credits a week, same old stuff year after year. You
get
> numb and tired and disillusioned (Pirsig mentions this in Zen and the Art
of
> Motorcycle Maintenance). You have no real contact with the living world
of
> mathematics and mathematicians; all you've got is your Calculus I
textbook
> and your colleagues. With great effort you can scare up money to go to
the
> occasional convention.
>
> Some people overcome these obstacles, bless them.
Ok, let me put it this way. I KNOW that a lot of PhD theses are
written by the advisors. Let me see, I have had 8 PhD students. Of
had only the most minimal help; four had a lot of help and explanation
described a PhD thesis as a work by the advisor under adverse
circumstances and I know for a fact that that was true in his case.
Wherever I have been there is always one supervisor who is known to
write all or nearly all of his students' theses. One once complained
that he didn't mind writing them, it was having to explain them that
he objected to.
But yes, there are other explanations for why people don't go on to do
their own work, but as I look at my students there is a strong
correlation between what they did in grad school and what they did
afterwards.
====
>> This suggests that a lot of theses are written by the advisor.
Right. After all, why *wouldn't* a professor want to forego his or
>her own research activities for a couple years to write an enormous
>paper under a student's name?
Maybe you meant to say something less hilarious.
A lot of people have pointed out that this does not necessarily
suggest that. Barr just posted a reply, saying let me put it
this way and then asserting that in _fact_ a lot of PhD theses
are written by advisors. That's not really putting it another
way, it's a separate assertion.
And whether you believe it or not, it's a _fact_ that a lot of
PhD theses are essentially written by the advisor. Barr
says he's seen a lot of this  so have I. Have you spent a
lot of time on the faculty in a PhDgranting math department,
or is your disbelief just motivated by your wonderment as
to why a professor would do such a thing?
(Regarding why a professor would do such a thing: First,
it doesn't mean he's putting his own research on hold for
those years. Anyway, there are all sorts of reasons: you
have a student who possibly should have been kicked
out years ago but wasn't  after the guy's been here for
five or six years, passed his exams and courses and
all, you really hate to kick him out just because he
can't do the thesis. Or in more cynical vein: If none
of the students get degrees then sooner or later the
bean counters will remove the PhD program from the
department, and then the professor will have to teach
trigonometry instead of advanced course. All sorts of
reasons it happens.
Not that _I_'ve ever done such a thing of course...)
************************
David C. Ullrich
====
>> Hey,
>>
>> Im curious, what would you guys/gals say the probability of someone
>> entering a Ph.D. program in Math or Stats and not finishing it. i.e.
>> dropping out.
I just read that it was about 5050.
>
> 5050 for finishing or not finishing?
>
> ;)
Yes.
;)
<877k6cyh1x.fsf@saurus.asaurus.invalid>
<6vrnhvsflefhciu879crdm26mc61pqabf6@4ax.com>
====
> This suggests that a lot of theses are written by the advisor.
>>Right. After all, why *wouldn't* a professor want to forego his or
>>her own research activities for a couple years to write an enormous
>>paper under a student's name?
>>Maybe you meant to say something less hilarious.
A lot of people have pointed out that this does not necessarily
> suggest that. Barr just posted a reply, saying let me put it
> this way and then asserting that in _fact_ a lot of PhD theses
> are written by advisors. That's not really putting it another
> way, it's a separate assertion.
And whether you believe it or not, it's a _fact_ that a lot of
> PhD theses are essentially written by the advisor. Barr
> says he's seen a lot of this  so have I. Have you spent a
> lot of time on the faculty in a PhDgranting math department,
> or is your disbelief just motivated by your wonderment as
> to why a professor would do such a thing?
What does it mean when you and he say that a thesis has been
(essentially) written by an advisor? You surely don't mean that the
advisor has contributed some nonnegligible portion of the LaTeX
source file, do you? Do you mean that the advisor has contributed
almost every original idea in the thesis? And also most (or
substantial parts) of the presentation decisions?
I suspect that I'm not alone in being unclear on what you and Michael
Barr mean.

[R]eality has a fascinating ability to check us when we get a little too
big for our britches... Make no mistake. There isn't a mathematician alive
today that I can't now touch, and not a mathematical career on the planet
that I can't now affect. James Harris, render of worlds
====
> Not that _I_'ve ever done such a thing of course...)
>
I believe Hans Zassenhaus was once quoted as saying he didn't mind
writing the thesis for the student, but he refused to then TEACH it to
the student so the student would be able to defend it.

G. A. Edgar
http://www.math.ohiostate.edu/~edgar/
====
>> This suggests that a lot of theses are written by the advisor.
>
> Right. After all, why *wouldn't* a professor want to forego his or
> her own research activities for a couple years to write an enormous
> paper under a student's name?
>
> Maybe you meant to say something less hilarious.
At some universities it's a written rule, and at others
it's an unwritten rule, that a criterion for tenure/promotion,
one must have produced Ph.D. students.
Besides that, it's personally embarrassing if one's students
don't make it, and it's egoboosting if one produces many
students.
Further, one hopes that if one feeds the student one idea
he'll get going. So one primes the pump. Then primes it
again. Then again. Then again. Then the student has
enough to call it dissertation.
It's easy to see how it happens. Especially if you're in
a department that produces few Ph.D.s and you'd like to
increase that number. How can you attract good students
to your program if you're record for number of Ph.D's
graduated per year is only 3 or 4? The pressure could be
tremendous to get the graduates out there, get the numbers
up, so that better students can be recruited.
Or maybe I'm the department chair working under a dean who
misunderstands what affirmative action is about and I'm under
the gun to produce quotas from underrepresented groups. So
by golly, this one advisee of mine is going to graduate if I
have to type the damn thing myself and walk across the stage
for him.
There are lots of motives, none very honest of course, for
sidelining one's own research a bit for pushing a slow student
through. Exasperation being the chief of these.
Bart
====
>>
>>Im curious, what would you guys/gals say the probability of someone
>>entering a Ph.D. program in Math or Stats and not finishing it. i.e.
>>dropping out.
>>
>> Of not leaving with a Ph.D.? 75% would be my guess, based on the
>> eight years I've been at Colorado.
>>
>> Doug
>
> Well, at my University you need an A average in your graduate courses
> to be aloud entrance into the PH.D. program. Would it still be a 75%
> failure rate you think ??
>
I think this points up a misunderstanding. By Entrance to the
Ph.D. program here, I think you originally meant someone who
has graduate courses, I suppose an MS, has passes qualifying
exams, and a certain amount of Ph.D. coursework done(?)
Every school has a difference set of hurdles for admission to
the program. Some have orals for qualifying, some have orals
as defense of the dissertaion. But I think you meant someone
postmasters degree.
In which case, 5050 might be the right answer.
But some people are taking entrance as right out of undergrad,
in which case the chances of finishing are much lower.
Bart
====
>But some people are taking entrance as right out of undergrad,
>in which case the chances of finishing are much lower.
I would say that the majority of students entering the Ph.D. program here
at Colorado have nothing more than a bachelor's degree. Not the vast
majority, but a majority nonetheless.
Doug
====
>
>>But some people are taking entrance as right out of undergrad,
>>in which case the chances of finishing are much lower.
>
> I would say that the majority of students entering the Ph.D. program here
> at Colorado have nothing more than a bachelor's degree. Not the vast
> majority, but a majority nonetheless.
My point was that does one mean by entering the Ph.D program
either A. starting grad school with the intent of getting a
Ph.D. or B. Being finally accepted into the program, having
passed qualifiers or the equivalent.
The answer to the OP's originial question depends on what
one means.
Bart
====
> This suggests that a lot of theses are written by the advisor.
Right. After all, why *wouldn't* a professor want to forego his or
>her own research activities for a couple years to write an enormous
>paper under a student's name?
Maybe you meant to say something less hilarious.
>> A lot of people have pointed out that this does not necessarily
>> suggest that. Barr just posted a reply, saying let me put it
>> this way and then asserting that in _fact_ a lot of PhD theses
>> are written by advisors. That's not really putting it another
>> way, it's a separate assertion.
>> And whether you believe it or not, it's a _fact_ that a lot of
>> PhD theses are essentially written by the advisor. Barr
>> says he's seen a lot of this  so have I. Have you spent a
>> lot of time on the faculty in a PhDgranting math department,
>> or is your disbelief just motivated by your wonderment as
>> to why a professor would do such a thing?
What does it mean when you and he say that a thesis has been
>(essentially) written by an advisor? You surely don't mean that the
>advisor has contributed some nonnegligible portion of the LaTeX
>source file, do you?
No.
>Do you mean that the advisor has contributed
>almost every original idea in the thesis?
Yes. Sometimes it appears to be somewhat more
than almost every. Honest, I've seen it happen
(much too close to naming names already, although
I don't think the people I have in mind were here
when you were here anyway... but I have two specific
students in mind, two different advisors.)
>And also most (or
>substantial parts) of the presentation decisions?
I suspect that I'm not alone in being unclear on what you and Michael
>Barr mean.
************************
David C. Ullrich
====
>
> suggest that. Barr just posted a reply, saying let me put it
> this way and then asserting that in _fact_ a lot of PhD theses
> are written by advisors. That's not really putting it another
> way, it's a separate assertion.
And whether you believe it or not, it's a _fact_ that a lot of
> PhD theses are essentially written by the advisor. Barr
> says he's seen a lot of this  so have I. Have you spent a
> lot of time on the faculty in a PhDgranting math department,
> or is your disbelief just motivated by your wonderment as
> to why a professor would do such a thing?
>>What does it mean when you and he say that a thesis has been
>>(essentially) written by an advisor? You surely don't mean that the
>>advisor has contributed some nonnegligible portion of the LaTeX
>>source file, do you?
>
> No.
>
>>Do you mean that the advisor has contributed
>>almost every original idea in the thesis?
>
> Yes. Sometimes it appears to be somewhat more
> than almost every. Honest, I've seen it happen
> (much too close to naming names already, although
> I don't think the people I have in mind were here
> when you were here anyway... but I have two specific
> students in mind, two different advisors.)
>
Well, how many original ideas are in an average thesis anyway? I would
bet no more than one, or the germ of one. And I wouldn't be too surprised
to learn if a given random thesis' one original idea was heavily hinted at
or suggested outright by the advisor. My impressions are that Ph.D.
theses are *usually* an indicator of hard work and persistence rather than
originality.
But all this raises an interesting point. Some people, on their own,
write terrific theses, and even soon after their theses are touted to
the community at large (or perhaps a smaller, more specialized community).
But some good mathematicians do not fall into this category. Their
theses problems are suggested by their advisors, who also occasionally
give hints of various kinds. The boundary between an advisor's
contributions and a student's can get quite blurred. In fact, I would
argue that a good advisor is defined by how blurred this boundary is.
It's not so easy to strike the right balance, and I think that's why a lot
of people end up suggesting too much to their students. The other extreme
of not suggesting anything is a luxury that only a minority of professors
can get away with.
====
>
...
>Do you mean that the advisor has contributed
>almost every original idea in the thesis?
>>
>> Yes. Sometimes it appears to be somewhat more
>> than almost every.
...
>Well, how many original ideas are in an average thesis anyway? I would
>bet no more than one, or the germ of one. And I wouldn't be too surprised
>to learn if a given random thesis' one original idea was heavily hinted at
>or suggested outright by the advisor. My impressions are that Ph.D.
>theses are *usually* an indicator of hard work and persistence rather than
>originality.
But, but...it says right there on my diploma that my Ph.D. is
in recognition of scientific attainments and the ability to
carry on original research as demonstrated by a thesis. And
it's *signed*, and everything. Are you calling Jerry Wiesner a
liar, sir? (For that matter, are you calling the me of 30 years
ago either hard working or persistent?)
>But all this raises an interesting point. Some people, on their own,
>write terrific theses, and even soon after their theses are touted to
>the community at large (or perhaps a smaller, more specialized community).
> But some good mathematicians do not fall into this category. Their
>theses problems are suggested by their advisors, who also occasionally
>give hints of various kinds. The boundary between an advisor's
>contributions and a student's can get quite blurred. In fact, I would
>argue that a good advisor is defined by how blurred this boundary is.
>It's not so easy to strike the right balance, and I think that's why a lot
>of people end up suggesting too much to their students. The other extreme
>of not suggesting anything is a luxury that only a minority of professors
>can get away with.
Not having been in a position to be an advisor myself, I have had
to rely on other people (some of whom I don't even know...) to give
their students hints of the form see if you can go any further with
this halfbaked idea of Rudolph's. I'll tell you, *that* is a luxury.
And it saves me the trouble of using my (nonexistent) pull to get
first jobs for my (nonexistent) advisees, too.
Lee Rudolph (what, me bitter?)
====
>
suggest that. Barr just posted a reply, saying let me put it
> this way and then asserting that in _fact_ a lot of PhD theses
> are written by advisors. That's not really putting it another
> way, it's a separate assertion.
And whether you believe it or not, it's a _fact_ that a lot of
> PhD theses are essentially written by the advisor. Barr
> says he's seen a lot of this  so have I. Have you spent a
> lot of time on the faculty in a PhDgranting math department,
> or is your disbelief just motivated by your wonderment as
> to why a professor would do such a thing?
What does it mean when you and he say that a thesis has been
>(essentially) written by an advisor? You surely don't mean that the
>advisor has contributed some nonnegligible portion of the LaTeX
>source file, do you?
>>
>> No.
>>
>Do you mean that the advisor has contributed
>almost every original idea in the thesis?
>>
>> Yes. Sometimes it appears to be somewhat more
>> than almost every. Honest, I've seen it happen
>> (much too close to naming names already, although
>> I don't think the people I have in mind were here
>> when you were here anyway... but I have two specific
>> students in mind, two different advisors.)
>>
Well, how many original ideas are in an average thesis anyway? I would
>bet no more than one, or the germ of one. And I wouldn't be too surprised
>to learn if a given random thesis' one original idea was heavily hinted at
>or suggested outright by the advisor. My impressions are that Ph.D.
>theses are *usually* an indicator of hard work and persistence rather than
>originality.
Of course. But after the advisor hints at or suggests the result the
student is supposed to have at least _something_ to do with
actually coming up with the proof. In the cases I have in mind
that was not so  instead there was an endless series of:
Ok, why not try this:
Ok, I'll look at that.
[week passes]
So did that work?
Don't know, couldn't figure anything out either way.
Hmm, let's see...
[delay of minutes or days]
No, that doesn't work [or does work]. Why not try this:
Ok...
(Or so the advisor claimed during endless bitch&moan sessions
during those years, and I believe him, because it's _exactly_ what
happened earlier when I was helping the same student, going
through all the exercises in some book one summer  he simply
never made any progress on any of them, with maybe two
exceptions, all the solutions were due to me the week after
the exercise was assigned.)
>But all this raises an interesting point. Some people, on their own,
>write terrific theses, and even soon after their theses are touted to
>the community at large (or perhaps a smaller, more specialized community).
> But some good mathematicians do not fall into this category. Their
>theses problems are suggested by their advisors, who also occasionally
>give hints of various kinds. The boundary between an advisor's
>contributions and a student's can get quite blurred.
Yes, no doubt it's quite blurry in the typical case. Why you think
it woud be blurry in the cases I'm referring to, given my
characterizations of them, is beyond me.
> In fact, I would
>argue that a good advisor is defined by how blurred this boundary is.
>It's not so easy to strike the right balance, and I think that's why a lot
>of people end up suggesting too much to their students. The other extreme
>of not suggesting anything is a luxury that only a minority of professors
>can get away with.
************************
David C. Ullrich
====
You guys make getting a math Ph.d sound as though it isn't even worth the
time! As a senior math student intending to go to grad school, I don't
find
anything in everyones' comments very encouraging.
It seems as though getting a math Ph.d boils down to the following:
1) Forego homeownership for student loans.
2) Work incredibly hard to understand something that nobody else cares to
understand because they are only interested in making lots of money.
3)Assiduously toil during your undergraduate years in order to attain a
near
perfect gpa that will get you into a respectable grad program.
4)Do the same as (3), but insert MA/MS and Ph.d program.
5)Enter a Ph.d program and have an advisor write a thesis for you.
6)Go to Podunk U. and attempt to do some original research.
7)When (6) fails, develop a drinking problem.
8)Retire
9)Die, but still in student loan debt and living in an offcampus apartment
unfulfilled and anonymous.
> Hey,
Im curious, what would you guys/gals say the probability of someone
> entering a Ph.D. program in Math or Stats and not finishing it. i.e.
> dropping out.
====
>You guys make getting a math Ph.d sound as though it isn't even worth the
>time! As a senior math student intending to go to grad school, I don't
find
>anything in everyones' comments very encouraging.
He asked a question  people tried to answer as accurately as they
could.
I don't think anyone's said it's not worth the time. People have said
it's not easy. It's not. When he asks what proportion of PhD students
get PhD's do you think we'd really be doing him or anyone else a
favor by _lying_, saying it's no problem for most students?
I mean really, by all means go to grad school in math! We did,
and we all think it would be great if you did too.
>It seems as though getting a math Ph.d boils down to the following:
1) Forego homeownership for student loans.
>2) Work incredibly hard to understand something that nobody else cares to
>understand because they are only interested in making lots of money.
>3)Assiduously toil during your undergraduate years in order to attain a
near
>perfect gpa that will get you into a respectable grad program.
>4)Do the same as (3), but insert MA/MS and Ph.d program.
>5)Enter a Ph.d program and have an advisor write a thesis for you.
>6)Go to Podunk U. and attempt to do some original research.
>7)When (6) fails, develop a drinking problem.
>8)Retire
>9)Die, but still in student loan debt and living in an offcampus
apartment
>unfulfilled and anonymous.
That's more or less the procedure, yes. Step 4 is optional  in a lot
of PhD programs they pay no attention to whether you have a
Master's dergree, many of the students are straight out of
undergrad. (At Wisconsin the Master's was more or less a
consolation prize for students who'd done the coursework
but didn't finish the PhD.)
>> Hey,
>> Im curious, what would you guys/gals say the probability of someone
>> entering a Ph.D. program in Math or Stats and not finishing it. i.e.
>> dropping out.
>
************************
David C. Ullrich
====
> You guys make getting a math Ph.d sound as though it isn't even worth
> the time! As a senior math student intending to go to grad school, I
> don't find anything in everyones' comments very encouraging.
>
> It seems as though getting a math Ph.d boils down to the following:
>
> 1) Forego homeownership for student loans.
> 2) Work incredibly hard to understand something that nobody else cares
> to understand because they are only interested in making lots of
> money.
> 3)Assiduously toil during your undergraduate years in order to
> attain a near perfect gpa that will get you into a respectable grad
> program.
> 4)Do the same as (3), but insert MA/MS and Ph.d program.
> 5)Enter a Ph.d program and have an advisor write a thesis for you.
> 6)Go to Podunk U. and attempt to do some original research.
> 7)When (6) fails, develop a drinking problem.
> 8)Retire
> 9)Die, but still in student loan debt and living in an offcampus
> apartment unfulfilled and anonymous.
You have Step 7 put off waaaaaaay to long. You'll want to get that
drinking problem going pretty much at the beginning of Step 3.
(It's a lot easier to get tenure if your colleagues percieve you
as a nonthreatening, harmless sot. And you'll make a better Dean
that way.)
Bart
====
>>But all this raises an interesting point. Some people, on their own,
>>write terrific theses, and even soon after their theses are touted to
>>the community at large (or perhaps a smaller, more specialized
community).
>> But some good mathematicians do not fall into this category. Their
>>theses problems are suggested by their advisors, who also occasionally
>>give hints of various kinds. The boundary between an advisor's
>>contributions and a student's can get quite blurred.
>
> Yes, no doubt it's quite blurry in the typical case. Why you think
> it woud be blurry in the cases I'm referring to, given my
> characterizations of them, is beyond me.
>
I didn't mean to imply that your cases were of that kind, but I only meant
to provide some kind of explanation of why that might happen, as I think I
did with the snippet below.
>> In fact, I would
>>argue that a good advisor is defined by how blurred this boundary is.
>>It's not so easy to strike the right balance, and I think that's why a
lot
>>of people end up suggesting too much to their students. The other
extreme
>>of not suggesting anything is a luxury that only a minority of professors
>>can get away with.
>
> ************************
>
> David C. Ullrich
====
>That's more or less the procedure, yes. Step 4 is optional  in a lot
>of PhD programs they pay no attention to whether you have a
>Master's dergree, many of the students are straight out of
>undergrad. (At Wisconsin the Master's was more or less a
>consolation prize for students who'd done the coursework
>but didn't finish the PhD.)
The same at Colorado  when I entered after completing my bachelor's
degree, I chose the Master's degree program (assuming that it had to be
done before the doctorate), and was convinced otherwise by the chair of
the department.
Doug
====
>It seems as though getting a math Ph.d boils down to the following:
1) Forego homeownership for student loans.
>2) Work incredibly hard to understand something that nobody else cares to
>understand because they are only interested in making lots of money.
>3)Assiduously toil during your undergraduate years in order to attain a
near
>perfect gpa that will get you into a respectable grad program.
>4)Do the same as (3), but insert MA/MS and Ph.d program.
>5)Enter a Ph.d program and have an advisor write a thesis for you.
>6)Go to Podunk U. and attempt to do some original research.
>7)When (6) fails, develop a drinking problem.
>8)Retire
>9)Die, but still in student loan debt and living in an offcampus
apartment
>unfulfilled and anonymous.
Oh, it's not that bad.
'Cuz if you post to sci.math then you won't be anonymous when you die.
Anon.
====
>Oh, it's not that bad.
>'Cuz if you post to sci.math then you won't be anonymous when you die.
Anon.
On the Internet, no one knows you're dead.
Lee Rudolph
====
> You guys make getting a math Ph.d sound as though it isn't even worth the
> time! As a senior math student intending to go to grad school, I don't
find
> anything in everyones' comments very encouraging.
>
> It seems as though getting a math Ph.d boils down to the following:
>
> 1) Forego homeownership for student loans.
> 2) Work incredibly hard to understand something that nobody else cares to
> understand because they are only interested in making lots of money.
> 3)Assiduously toil during your undergraduate years in order to attain a
near
> perfect gpa that will get you into a respectable grad program.
> 4)Do the same as (3), but insert MA/MS and Ph.d program.
> 5)Enter a Ph.d program and have an advisor write a thesis for you.
> 6)Go to Podunk U. and attempt to do some original research.
> 7)When (6) fails, develop a drinking problem.
> 8)Retire
> 9)Die, but still in student loan debt and living in an offcampus
apartment
> unfulfilled and anonymous.
#7 should occur during graduate school. Immediately after my
undergraduate
I went to graduate school drinking about the same amount as I did as
an
undergraduate (which seemed like a lot at the time). After 2 years, I
had
to quit graduate school because the material is so much harder than it
is
as an undergraduate. I don't think people have stressed that enough
on this
thread. Graduate level mathematics is much harder than undergraduate
level
mathematics. If you are the top student in your class, you will be
put in
your proper place quickly in graduate school. Herein lies the
importance
of developing a drinking problem. It is why I had to quit after 2
years.
Shortly before coming back to graduate school, I was talking to
someone
who gave me this piece of wisdom: The only thing that got me through
grad school was alchohol. As it stands now, I drink far more than I
ever did as an undergraduate. In my department, it seems to be a
general
rule that the ones who make it (or who are going to make it) have
drinking problems, and those who don't, don't. Is alchoholism worth
the opportunity
to pursue mathematical knowledge on a daily basis? No doubt its a
hard
question, but its one you have to ask.
On the more serious note of advisor's writing their student's thesis:
I have only attended one PhD defence and in that talk the student's
advisor
never once had to steer the student in the right direction during the
question/answer session. I do have to admit, though, that they did
ask
some fairly trivial questions  for example one could be done simply
by
using Fatou's lemma (which the candidate confidently answered).
Hugh
====
>Oh, it's not that bad.
>'Cuz if you post to sci.math then you won't be anonymous when you die.
Anon.
On the Internet, no one knows you're dead.
Lee Rudolph
Heard in an English pub, to the tune of Irish Washerwoman:
Oh, McTavish is dead and his brother don't know it,
They're both of them dead and they're in the same bed,
And neither one knows that the other is dead.
====
I have a Russian friend who was a Doctor in quantum physics.
I asked him about his education, and I was blown away about
what he told me about getting a PhD in Russia. (During the
cold war).
He said to get a PhD in Russia, you have to had 45 published
papers in a respectable journal of your field; that a PhD
is looked upon like a Masters in the States. There's
another level of education above PhD called Doctor. And,
of course, you need to write even more papers. And when you
become a Doctor, then you earn the title Doctor.
He said after Doctor, there's another step called Academic
but he said that such a title is mostly political.
I could be wrong about what he said, because I was just so shocked
to learn how hard it is there.
So I think the probability is determined by the country.
(Now for my very long aside. Sorry about it, but I'm just
so fond of my Russian friend.... My friend is a 'lowly' programmer
in the States, and back in Russia he was a respected quantum
physicist.
He is so eager to solve challenging math problems. I only know
a few Olympiad problems to give him. He pretty much solves them
in his head. So I went out one day, and bought an olympiad book
just to rattle off a problem when he would stop in my cube with
a happy smile asking for a math problem.
It was so entertaining/jaw dropping to see him usually solve
most of these olympiad problem in his head. I don't know,
perhaps these problems are easy for professional mathematicians.
I always thought it impressive.
I told him, Man, what are you doing as a corporate programmer?
You are too talented to be a programmer? He wasn't so sure himself.
But, it was along the lines of making ends meet. He was more
concerned about other issues like spending time with friends and
family, and how to live a good happy life.
This seems typical of most Russian emmigrants I've met so far.
Taxicab drivers, sofware testers, guitar bums all having an
extremely good education, and all happy to make some money,
and all more concerned about questions of living a meaningful
life. I contrast this with alot of my American friends who
seem to want a Masters, or an MBA just as an 'edge' in a rat
race.
Just the kinda thing that makes you go Hmmmm...
Well, those Russians, you gotta love them!)
> I don't think anyone's said it's not worth the time. People have said
> it's not easy. It's not.
====
>
I have found an elegant proof that the derivative of sin(x) is cos(x).
>I have studied two alevels in maths and read lots of math books but
>have not come across this particular proof before.
Obviously, this is not a groundbreaking proof, it is simply a
>different way of proving a fundamental result. (without using limits
>or infintesimals). However, for personal interest I would like to know
>if it is original.
Is there anywhere I can find a catalogue of existing proofs for the
>derivative of sin(x)?
If it turned out this proof was original should I consider getting it
>published or is it not worth it, since it is such a tiny proof.
Among these things, I am also working on some integration techniques.
>Again, I have a similiar problem: I do not know whether this stuff I
>am finding is original. I have done 6 modules of pure maths at school
>so I am not a complete novice, but on the other hand I am aware that
>there is many things that I do not know of pure maths since I have yet
>to start my maths degree. Can anyone suggest a website that provides
>information on advanced integration techniques, and for that matter
>information on higher level maths?
Any responses to the above would be gratefully received.
Flame.
>
1. There's no money in math. So if someone steals your idea, you
> haven't lost any money.
I made no reference to money. To quote myself: ...for personal
interest I would like to know if it is original
> 2. If you're doing original work, you'll continue to do original work.
> If someone steals an idea, just stay away from that person in the
> future and keep doing your original work.
If someone steals an idea as you put it, then i would lose priority
as the discoverer. I think that as unfair, strange though it may
sound.
> 3. The only fame you can expect from doing math is among other
> mathematicians.
I do not expect fame. I do not understand where you derived this idea
from my post.
> 4. There is money in applying mathematical ideas to other disciplines.
Please see my response to 1.
> Not deep mathematical ideas, but a little math and some logic and
> organization to business problems (and translating your results to
> English for your colleagues) will keep you steadily employed at quite
> reasonable rates. Thinking original thoughts is using time that could
> be spent thinking profitable thoughts.
Please see my response to 1.
(Of course, if you get paid well
> enough, you can afford to spend some time thinking original thoughts.
> It's much easier to take the lower salary and be a university
professor.)
>
> Jon Miller
I am currently making enquiries with other professional mathematicians
regarding my work so far. If any developments occur, I will post them
here along with my work.
Flame
====
Can someone give me a hint (not solution) on the following
problem. It is number 2.29 in Rotman's Introduction to
Homological Algebra.
Given:
g
A>B


f Prove the following diagram is a pushout:

V
C
g
A>B
 
 
f f'
 
V g' V
C>D
Where D=(C /osum B)/W, W={(fa,ga):a /in A}, f':b>(0,b)+W and
g':c>(c,0)+W.
What do I need to prove to prove the diagram is a pushout and
what is the significance of the set W?
====
>Can someone give me a hint (not solution) on the following
>problem. It is number 2.29 in Rotman's Introduction to
>Homological Algebra.
Given:
> g
> A>B
> 
> 
>f Prove the following diagram is a pushout:
> 
> V
> C
g
> A>B
>  
>  
>f f'
>  
> V g' V
> C>D
Where D=(C /osum B)/W, W={(fa,ga):a /in A}, f':b>(0,b)+W and
>g':c>(c,0)+W.
>What do I need to prove to prove the diagram is a pushout
The diagram is a pushout if it satisfies the following two conditions:
1. COMMUTATIVITY: f'g = g'f (I am applying functions on the left, so
they should be read righttoleft; f'g means g first, then f').
2. UNIVERSAL PROPERTY: Given any K and maps b:B>K, c:C>K such that
bg=cf, there exists a unique map d:D>K such that b=df' and c=dg'.
> and
>what is the significance of the set W?
You can think of a pushout as a coequalizer; you are finding the
largest object on which you can make f and g 'equal'. W is a measure
of how far they are on being 'equal' (not exactly, since we are
dealing with the dual notion, but maybe that makes some sense to
you?). In order for f'g(a) to be equal to g'f(a) for all a, you need
to make sure that (f(a),0) is the same as (0,g(a)); for them to be
the same, you need to mod out by (f(a),g(a)); so W is the closure of
all those identities in Cosum B; moding out by W is the same as
imposing those identities on Cosum B.
======================================================================
It's not denial. I'm just very selective about
what I accept as reality.
 Calvin (Calvin and Hobbes)
======================================================================
Arturo Magidin
magidin@math.berkeley.edu
====
[snip]
> The diagram is a pushout if it satisfies the following two conditions:
>
> 1. COMMUTATIVITY: f'g = g'f (I am applying functions on the left, so
> they should be read righttoleft; f'g means g first, then f').
>
> 2. UNIVERSAL PROPERTY: Given any K and maps b:B>K, c:C>K such that
> bg=cf, there exists a unique map d:D>K such that b=df' and c=dg'.
>
[snip]
Ok, I got the commutativity. That part was really obvious. The
universal property is the part giving me the most trouble. I am
assuming that one must use a previously proved universal property to
obtain the one in question. I am not seeing how to construct or prove
the existence of such a map d:D>K, for any K. I don't mind if you
spoil the problem now, unless you think you can give me a suitable
hint.
Chris
====
>[snip]
>> The diagram is a pushout if it satisfies the following two conditions:
>>
>> 1. COMMUTATIVITY: f'g = g'f (I am applying functions on the left, so
>> they should be read righttoleft; f'g means g first, then f').
>>
>> 2. UNIVERSAL PROPERTY: Given any K and maps b:B>K, c:C>K such that
>> bg=cf, there exists a unique map d:D>K such that b=df' and c=dg'.
>>
>[snip]
Ok, I got the commutativity. That part was really obvious. The
>universal property is the part giving me the most trouble. I am
>assuming that one must use a previously proved universal property to
>obtain the one in question.
I am not seeing how to construct or prove
>the existence of such a map d:D>K, for any K. I don't mind if you
>spoil the problem now, unless you think you can give me a suitable
>hint.
I don't follow what you mean. Assume you have an object K, and maps
b:B>K and c:C>K such that bg = cf.
f
A > C
 
g   g'
 
V V
B > D
f'
We know that D is defined as (Boplus C)/W; so to define a map from D
to K, we can define a map from Boplus C to K whose kernel contains W,
and factor it through the quotient. f' and g' are the obvious
inclusions, and W is the subgroup generated by all
pairs (g(a),f(a)) for a in A.
So let's consider what the d HAS to be. First, we want b=df' and
c=dg'. So given any x in B, we know what b(x) is (we are GIVEN the
maps b and c); and we know that f'(x) = (x,0) in Boplus C. So we map
(x,0) to b(x).
Likewise, we will need to map (0,y) to c(y) for all y in C.
That means that we need to map an element (x,y) in Boplus C to
b(x)+c(y).
That defines a map, call it e: B oplus C > K.
Now we need to verify that e factors through the quotient D, that is,
that W is contained in the kernel of e.
So let's take an element of W, which is of the form (g(a),f(a)) for a
in A. According to the definition of e, we map
e(g(a),f(a)) = b(g(a))+c(f(a))
= b(g(a))  c(f(a))
= bg(a)  cf(a).
But we are assuming futher that b and c are such that bg=cf; so
bg(a)cf(a)=0 for all a in A. Therefore, e takes W to 0, and so W is
contained in the kernel of e.
Therefore, e factors through the quotient
p:Boplus C > (Boplus C)/W = D.
So define d to be the unique map from (Boplus C)/W to K such that
commutativity condition, b=df' and c=dg'.
Moreover, since the definition of e was forced by the commutativity of
the diagram, the choice of d is also forced, so that d is the only
function that will fit in that diagram. Thus, d is unique.
In general, when you have a universal construction, IF you have an
>explicit< construction of the object, then the universal property
is easy to verify, because you will have no choice about how to define
the map in question. it should be obvious what the map has to be in
an object.
======================================================================
It's not denial. I'm just very selective about
what I accept as reality.
 Calvin (Calvin and Hobbes)
======================================================================
Arturo Magidin
magidin@math.berkeley.edu
====
====
to this question, which has been driving me a bit nuts. Some quick
background: as a roleplayer, I use a lot of dice. At times, the
rules call for one to roll multiple dice (say, five sixsided dice, or
5d6), then to drop the lowest two and total the other three. The
basic question is: is there a formula for determining the probability
of rolling a certain result, given these conditions?
Determining the probability of a particular outcome when just rolling
multiple dice is relatively straightforward (there's a brief
discussion here: http://mathforum.org/library/drmath/view/52207.html).
I can find a pattern to the summation needed when dropping a single
die from a set; but once I try to remove two dice from the set, the
pattern disappears and I find myself lost again. (The numbers can be
determined by brute force, of course, but that's neither practical nor
interesting.)
So, I guess the base question is: Is there a formula for calculating
the probability of achieving a result R on n dice with d sides,
dropping the k lowest dice?
====
How can the following WienerHopf operator W be bounded?
Define W = P M_f P, where M_f is multiplication by a function
f in C0(S^1),
P is the projection of L2(S^1) onto the subspace spanned by z^k, k >= 0,
when f is only assumed to be continuous (e.g what if f is not integrable?)
(This is an exercise in Booss: Topology and Analysis)
Andreas
====
>How can the following WienerHopf operator W be bounded?
Bounded on what space? (Or: Bounded in what norm?)
>Define W = P M_f P, where M_f is multiplication by a function
>f in C0(S^1),
P is the projection of L2(S^1) onto the subspace spanned by z^k, k >= 0,
If you're asking about boundedness in L2 this is obvious, because P
and M_f are both bounded. (M_f is bounded if and only if f is
bounded...)
>when f is only assumed to be continuous (e.g what if f is not integrable?)
??? A continuous function on S^1 is not integrable???
I must be missing what you mean by S^1  that's not the unit circle?
In any case, if S^1 is locally compact then f in C0(S^1) implies that
f is bounded.
>(This is an exercise in Booss: Topology and Analysis)
Andreas
>
************************
David C. Ullrich
====
> Bounded on what space? (Or: Bounded in what norm?)
With respect to the L2 norm.
>Define W = P M_f P, where M_f is multiplication by a function
>f in C0(S^1),
P is the projection of L2(S^1) onto the subspace spanned by z^k, k >= 0,
If you're asking about boundedness in L2 this is obvious, because P
> and M_f are both bounded. (M_f is bounded if and only if f is
> bounded...)
when f is only assumed to be continuous (e.g what if f is not
integrable?)
??? A continuous function on S^1 is not integrable???
It dawned on me later that f and M_f must be bounded because f is
continuous
and defined on a circle... it was my lack of experience in such things.
> I must be missing what you mean by S^1  that's not the unit circle?
> In any case, if S^1 is locally compact then f in C0(S^1) implies that
> f is bounded.
(9.4, page 97 of the German edition): If you know the book... I can't quite
see
the significance of the condition sup_S^1 fg  1 <1
does it mean that T_g is an inverse of P_n T_f , modulo a compact operator
because T_(fg 1) is compact?)
Not to worry  I'll figure it out or skip this (minor) point in the book.
Andreas
====
> Bounded on what space? (Or: Bounded in what norm?)
With respect to the L2 norm.
>Define W = P M_f P, where M_f is multiplication by a function
>>f in C0(S^1),
>>P is the projection of L2(S^1) onto the subspace spanned by z^k, k >=
0,
>> If you're asking about boundedness in L2 this is obvious, because P
>> and M_f are both bounded. (M_f is bounded if and only if f is
>> bounded...)
>>when f is only assumed to be continuous (e.g what if f is not
integrable?)
>> ??? A continuous function on S^1 is not integrable???
It dawned on me later that f and M_f must be bounded because f is
continuous
>and defined on a circle... it was my lack of experience in such things.
In fact, for future reference, if X is just locally compact and f is
in C0(X) then f is bounded  that's a large part of the difference
between C0 and C...
>> I must be missing what you mean by S^1  that's not the unit circle?
>> In any case, if S^1 is locally compact then f in C0(S^1) implies that
>> f is bounded.
(9.4, page 97 of the German edition): If you know the book...
Nope, sorry.
>I can't quite see
>the significance of the condition sup_S^1 fg  1 <1
>does it mean that T_g is an inverse of P_n T_f , modulo a compact
operator
>because T_(fg 1) is compact?)
Not to worry  I'll figure it out or skip this (minor) point in the book.
Andreas
************************
David C. Ullrich
====
I place a quarter (coin) on the table. Exactly how many quarters can I put
around this centered quarter?
====
>
> I place a quarter (coin) on the table. Exactly how many quarters can I
put
> around this centered quarter?
Six. Think honeycomb.

Ioannis
http://users.forthnet.gr/ath/jgal/
___________________________________________
Eventually, _everything_ is understandable.
====
> I place a quarter (coin) on the table. Exactly how many quarters can I
put
> around this centered quarter?
Define put.
====
>I place a quarter (coin) on the table. Exactly how many quarters can I put
>around this centered quarter?
That depends, of course, on the size of the table. :)

Stan Brown, Oak Road Systems, Cortland County, New York, USA
http://OakRoadSystems.com/
You find yourself amusing, Blackadder.
I try not to fly in the face of public opinion.
====
>>I place a quarter (coin) on the table. Exactly how many quarters can I
put
>>around this centered quarter?
> That depends, of course, on the size of the table. :)
It also depends on the definition of around. If we consider it in
a threedimensional sense and place noÊlimits on distance, then
every
quarter on Earth is around it.
And since quarter isn't defined, we could take it to mean onefourth
of anything, which raises the total a bit higher... :)

Wayne Brown  When your tail's in a crack, you improvise
fwbrown@bellsouth.net  if you're good enough. Otherwise you give
 your pelt to the trapper.
e^(i*pi) = 1  Euler   John Myers Myers,
Silverlock
====
hi all,
I am actually trying to understand this mathematical notion that is so
weird to me (I am far from being a god at maths...).
could someone drop the light onto the following for me ?
We will compute a rotation about the unit vector, u by an angle . The
quaternion that computes this rotation is
q = (s,v)
s = cos(teta/2)
v = u * sin(teta/2)
We will represent a point p in space by the quaternion P=(0,p) We
compute the desired rotation of that point by this formula:
P = (0,p)
Protated = qPq^1
The first thing I don't understand at all here is where the s and v
values come from ?!? It might sound stupid but I don't understand
this.
Any help ?
thanx
Sam
====
hi all,
I am actually trying to understand this mathematical notion that is so
weird to me (I am far from being a god at maths...).
could someone drop the light onto the following for me ?
We will compute a rotation about the unit vector, u by an angle . The
quaternion that computes this rotation is
q = (s,v)
s = cos(teta/2)
v = u * sin(teta/2)
We will represent a point p in space by the quaternion P=(0,p) We
compute the desired rotation of that point by this formula:
P = (0,p)
Protated = qPq^1
The first thing I don't understand at all here is where the s and v
values come from ?!? It might sound stupid but I don't understand
this.
Any help ?
thanx
Sam
====
> hi all,
> I am actually trying to understand this mathematical notion that is so
> weird to me (I am far from being a god at maths...).
> could someone drop the light onto the following for me ?
>
> We will compute a rotation about the unit vector, u by an angle . The
> quaternion that computes this rotation is
> q = (s,v)
> s = cos(teta/2)
> v = u * sin(teta/2)
>
> We will represent a point p in space by the quaternion P=(0,p) We
> compute the desired rotation of that point by this formula:
> P = (0,p)
> Protated = qPq^1
>
> The first thing I don't understand at all here is where the s and v
> values come from ?!? It might sound stupid but I don't understand
> this.
> Any help ?
> thanx
> Sam
expanding q = (s,v) gives a unit quaternion, which rotates R^3 in the
form you gave. rotation in R^3 requires a axis of rotation, which this
case is u, and an angle of rotation, theta. in general, the quaternion
which gives the rotation is q = cos (theta/2)  sin (theta/2) a
(i*j*k). it's much nicer to consider rotations in the clifford algebra
framework, where the unit ball of the even subalgebra rotates the
underlying quadratic space.
M.T.
====
ok thanx, but I still don't understand why we use cos(theta/2) and u *
sin(theta/2) as values for s and v...
besides does anyone could enlight me on this :
To rotate a vector v an angle of θ around about an arbitrary unit
axis w, you can use the formula:
v' = w(v.w) + (v  w(v.w))cos(θ) + (v^w)sin(θ)
how can we end up to this formula ?
> hi all,
> I am actually trying to understand this mathematical notion that is so
> weird to me (I am far from being a god at maths...).
> could someone drop the light onto the following for me ?
>
> We will compute a rotation about the unit vector, u by an angle . The
> quaternion that computes this rotation is
> q = (s,v)
> s = cos(teta/2)
> v = u * sin(teta/2)
>
> We will represent a point p in space by the quaternion P=(0,p) We
> compute the desired rotation of that point by this formula:
> P = (0,p)
> Protated = qPq^1
>
> The first thing I don't understand at all here is where the s and v
> values come from ?!? It might sound stupid but I don't understand
> this.
> Any help ?
> thanx
> Sam
>
> expanding q = (s,v) gives a unit quaternion, which rotates R^3 in the
> form you gave. rotation in R^3 requires a axis of rotation, which this
> case is u, and an angle of rotation, theta. in general, the quaternion
> which gives the rotation is q = cos (theta/2)  sin (theta/2) a
> (i*j*k). it's much nicer to consider rotations in the clifford algebra
> framework, where the unit ball of the even subalgebra rotates the
> underlying quadratic space.
> M.T.
====
>ok thanx, but I still don't understand why we use cos(theta/2) and u *
>sin(theta/2) as values for s and v...
The product of two quaternions s+v and t+w, where s and t are scalars
and v and w are vectors, is (s+v)(t+w) = (stv.w)+(sw+tv+v^w). It
follows that (s+v)(sv) = s**2+v.v, where s**2 denotes the square of s.
So if w is a unit vector, then (cos(theta/2)+sin(theta/2)w)^{1}
= cos(theta/2)sin(theta/2)w, and so
(cos(theta/2)+sin(theta/2)w) v (cos(theta/2)+sin(theta/2)w)^{1}
= (cos(theta/2)vsin(theta/2)w.v+sin(theta/2)w^v)
(cos(theta/2)sin(theta/2)w)
= cos(theta/2)**2 v + sin(theta/2) cos(theta/2) v.w
+ sin(theta/2) cos(theta/2) w^v  sin(theta/2) cos(theta/2) v.w
+ sin(theta/2)**2 (w.v)w + sin(theta/2) cos(theta/2) w^v
 sin(theta/2)**2 (w^v)^w
= cos(theta/2)**2 v + 2 sin(theta/2) cos(theta/2) w^v
+ sin(theta/2)**2 (w.v)w  sin(theta/2)**2 (w.w)v
+ sin(theta/2)**2 (w.v)w
= [cos(theta/2)**2  sin(theta/2)**2] v + 2 sin(theta/2) cos(theta/2) w^v
+ 2 sin(theta/2)**2 (w.v)w
= cos(theta) v + sin(theta) w^v + (1  cos(theta)) (w.v)w
= (w.v)w + (v  (w.v)w) cos(theta) + w^v sin(theta),
which is the result when v is rotated about w by an angle of theta
in the right handed sense.
>besides does anyone could enlight me on this :
>To rotate a vector v an angle of θ around about an arbitrary unit
>axis w, you can use the formula:
>v' = w(v.w) + (v  w(v.w))cos(θ) + (v^w)sin(θ)
The first thing to note is that this rotation looks like a left handed
rotation. For right handed rotations, which I will be dealing with below,
a right handed rotation of an angle theta about a unit vector w transforms
v to v' = (v.w)w + (v  (v.w)w) cos(theta) + w^v sin(theta). Note that
the only difference between this formula and the one you supplied above
is that the sign of the coefficient of v^w is reversed (recall that
w^v = v^w).
When you rotate about the unit vector w, then w and all its multiples
remain fixed, so in particular, for any vector v, (v.w)w remains fixed
under the rotation. The plane orthogonal to w turns about the origin,
and if a unit vector r is orthogonal to w, then the plane has basis
r and w^r, and a right handed rotation about w rotates the plane so that
r moves towards w^r. It follows that r is transformed by a right handed
rotation about u through an angle of theta to
r' = r cos(theta) + w^r sin(theta).
If v is a multiple of w, then (v.w)w = v and w^v = 0, so that
(v.w)w + (v  (v.w)w) cos(theta) + w^v sin(theta) = v,
which is the result of rotating v about w by any angle as a consequence
of the fact that v is a multiple of w.
If v is not a multiple of w, then v  (v.w)w is orthogonal to w, and
w^(v  (v.w)w) = v^w, with the result that under right handed rotation
about w through an angle of theta, v  (v.w)w transforms to
(v(v.w)w) cos(theta) + w^v sin(theta). Since (v.w)w transforms to
itself, then v = (v.w)w + (v(v.w)w) transforms to
(v.w)w + (v  (v.w)w) cos(theta) + w^v sin(theta).
>how can we end up to this formula ?
David McAnally
>> hi all,
>> I am actually trying to understand this mathematical notion that is so
>> weird to me (I am far from being a god at maths...).
>> could someone drop the light onto the following for me ?
>>
>> We will compute a rotation about the unit vector, u by an angle . The
>> quaternion that computes this rotation is
>> q = (s,v)
>> s = cos(teta/2)
>> v = u * sin(teta/2)
>>
>> We will represent a point p in space by the quaternion P=(0,p) We
>> compute the desired rotation of that point by this formula:
>> P = (0,p)
>> Protated = qPq^1
>>
>> The first thing I don't understand at all here is where the s and v
>> values come from ?!? It might sound stupid but I don't understand
>> this.
>> Any help ?
>> thanx
>> Sam
>>
>> expanding q = (s,v) gives a unit quaternion, which rotates R^3 in the
>> form you gave. rotation in R^3 requires a axis of rotation, which this
>> case is u, and an angle of rotation, theta. in general, the quaternion
>> which gives the rotation is q = cos (theta/2)  sin (theta/2) a
>> (i*j*k). it's much nicer to consider rotations in the clifford algebra
>> framework, where the unit ball of the even subalgebra rotates the
>> underlying quadratic space.
>> M.T.
====
> ok thanx, but I still don't understand why we use cos(theta/2) and u *
> sin(theta/2) as values for s and v...
> besides does anyone could enlight me on this :
>
> To rotate a vector v an angle of θ around about an arbitrary unit
> axis w, you can use the formula:
> v' = w(v.w) + (v  w(v.w))cos(θ) + (v^w)sin(θ)
>
> how can we end up to this formula ?
work out the geometry. in R^n, let n' be the greatest even number <=
n, the a rotation in R^n is a product of n' reflections. so work out a
formula for reflections and the result for rotation follows directly.
in doing so, one may want to note that successive reflections about
vectors a and b is a rotation about a X b by the angle between a and
b. again, it is more convenient to consider this in the clifford
algebra framework.
M.T.