mm-729
===
Subject: Re: Mathematics and Powerpoint
...................
> well i can tell you're driven by emotion instead of logic
but i'll try
> to make one more appeal to that small shriveled section of
your brain
> used for rational thinking. when i said powerpoint slides
take less
> time. i meant over a period of 2 or 3 semesters in the ong
run it will
> save time. no writing on the chalkboard for all those 2 or
3 semesters,
> so more time to talk and answer questions. maybe instead
of trying to
> be imaginative you should go post in sci.idiot.. where you
might be
> respected for having realistic points and proofs.
That is assuming that you will give the same lectures in
the same order time after time. If you are planning to do
that, videotaping might be even easier. That is NOT the
way I teach, but this is what the students now want, to be
spoonfed into acting like robots.
>Let's take this to the extreme. Instead of every prof
preparing
>powerpoint slides and using 2-3 semesters to break even over
using the
>chalkboard. (I personally think it would take more than 2-3
semesters to
>break even, but nevermind that.) How about the textbook
company hiring
>one prof to prepare super power point slides. Then the
textbook company
>can distribute them to all the profs.
Are powerpoint slides any better than reading the textbook?
I can see animation for a FEW purposes as being better. We
do not need teachers to show how to go through the steps to
differentiate polynomials. But we do need teachers to get
the concepts across, to provide the understanding to be able
to use the material to the extent it can be used. Few now
coming out of calculus know anything more than how to compute
derivatives and antiderivatives, and this is useless unless
they understand the concepts. Also, computers can do this
better than all but the best, who will learn it anyhow, and
often even pocket calculators.
The same holds for theorems and proofs; understanding the
concepts is important, sometimes the ideas of the proofs
are as well, but rarely is the proof that important to
learn the gory details.
--
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, Department of Statistics, Purdue University
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
===
Subject: Re: Mathematics and Powerpoint
> That is assuming that you will give the same lectures in
> the same order time after time. If you are planning to do
> that, videotaping might be even easier. That is NOT the
> way I teach, but this is what the students now want, to be
> spoonfed into acting like robots.
It appears that the email I posted to begin this thread was
somewhat
inaccurate, or perhaps the people proposing Powerpoint have
retreated
in the face of responses they've received from teaching
faculty; in
any event, it doesn't look like anyone is going to force us
to use
Powerpoint to give our math lectures. At the very least, the
pdßatex
and other appraoches people have mentioned will be supported
by the
software & hardware we are getting; also, there is something
called
a Visualiser which is apparently capable of functioning as
the digital
equivalent of an overhead projector.
Also, at least for the time being, there will still be
overhead
projectors in the big theaters, and whiteboards in the smaller
rooms.
The longterm picture is still unclear.
--
Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email)
===
Subject: Re: Mathematics and Powerpoint
<cek5oi$526i@odds.stat.purdue.edu>
Discussion, linux)
> It appears that the email I posted to begin this thread was
somewhat
> inaccurate, or perhaps the people proposing Powerpoint have
retreated
> in the face of responses they've received from teaching
faculty; in
> any event, it doesn't look like anyone is going to force 
us
to use
> Powerpoint to give our math lectures. At the very least,
the pdßatex
> and other appraoches people have mentioned will be
supported by the
> software & hardware we are getting; also, there is
something called
> a Visualiser which is apparently capable of functioning as
the digital
> equivalent of an overhead projector.
In that case, if you're interested in using LaTeX to create
PDF slides
for your lectures, I recommend you look at prosper (see
http://prosper.sourceforge.net). Maybe you're already
familiar with
it, of course.
Note: I'm not saying that prosper is the best way to create
mathematics lectures. Whiteboards or handwritten slides are
still
preferable, as far as I'm concerned, but I could imagine
using prosper
to augment the whiteboard.
--
Jesse Hughes
Well, you know as soon as you have a new number I will be
happy to
add it to the list. Don't try those childish tit-for-tat
games with
me. -- Ross Finlayson on Cantor's theorem.
===
Subject: Re: Mathematics and Powerpoint
<cek5oi$526i@odds.stat.purdue.edu>
<87k6wiq71r.fsf@phiwumbda.org> <x5fz7599rj.fsf@lola.goethe.zz>
Discussion, linux)
> It appears that the email I posted to begin this thread
was somewhat
> inaccurate, or perhaps the people proposing Powerpoint
have retreated
> in the face of responses they've received from teaching
faculty; in
> any event, it doesn't look like anyone is going to force
us to use
> Powerpoint to give our math lectures. At the very least,
the pdßatex
> and other appraoches people have mentioned will be
supported by the
> software & hardware we are getting; also, there is
something called
> a Visualiser which is apparently capable of functioning as
the digital
> equivalent of an overhead projector.
> In that case, if you're interested in using LaTeX to
create PDF slides
> for your lectures, I recommend you look at prosper (see
> http://prosper.sourceforge.net). Maybe you're already
familiar with
> it, of course.
> Note: I'm not saying that prosper is the best way to 
create
> mathematics lectures.
> No. Beamer <URL:http://latex-beamer.sourceforge.net> would
be a
> better contender IMO: less visual distraction, better page
> utilization, much more convenient input syntax, better
navigation.
I wasn't familiar with that package, but the results do look
nice.
They don't look less distracting than my own prosper slides,
I think,
but I might have to give the project a try.
--
Jesse F. Hughes
If anything is true in general about Usenet, it's that 
people
can go
on and on about just about anything. -- James Harris speaks
the
truth.
===
Subject: Re: Probability & Statistics
> I'm looking for graduate books in probability and
statistics (not
> necessarily both subjects in the same book) which would be a
systematic -and
> sufficiently exhaustive- treatment of the subjects for
graduate
students
> (probability with measure theory, mathematical statistics).
By
sufficiently
> exhaustive I mean it should cover a level of Master 1st
year/2nd year
(or
> advanced 1st year, I'm not intending to specialize in 
those
subjects ).
> What are the top references in those subjects ?
> --
> Julien Santini
You might try Mood, Graybill, and Boes, Introduction to the
Theory of
Statistics, 3rd Edn.
===
Subject: Re: Internal Set Theory Uniqueness Principle
> In Edward Nelson's original paper on Internal Set Theory,
he gave a proof
> that if
> (1) P(x) is a formula in Internal Set Theory with one
> free variable x,
> (2) P(x) is relativized to a standard set V (i.e. all
> quantifiers are of the form for all x in V, for
> some x in V, for all standard x in V, for some
> standard x in V,
> (3) there is a unique value of x in V for which P(x) holds,
> then the unique value of x in V, such that P(x) holds, is
standard.
Yes, it's in V if that's what you mean by 
standard.
> My question is that if
> (1) P(u,v_1,...,v_k) is a formula in Internal Set Theory,
> (2) y_1, ..., y_k are standard,
> (3) there exists exactly one x such that P(x,y_1,..,y_k)
holds,
> then is it necessarily true that the unique value of x
specified in (3)
> is standard?
Yes, because u is standarized to V.
> Nelson answered the question in the case where there is
only one free
> variable and the formula is relativized to a standard set
V, with the
> answer being in the affirmative. Nelson's 
proof is easily
extendible
> to the case where there is more than one free variable. I
am curious
> about the more general case (i.e. when the formula is not
relativized
> to a standard set).
> If there is a proof in the general case that x must be
standard, I would
> be grateful to see it. If it is not true in general that x
must be
No, if u isn't standarized to V, then the unique x may not 
be
in V,
not standarized.
===
Subject: Re: Internal Set Theory Uniqueness Principle
> In Edward Nelson's original paper on Internal Set Theory,
he gave a
proof
> that if
> (1) P(x) is a formula in Internal Set Theory with one
> free variable x,
> (2) P(x) is relativized to a standard set V (i.e. all
> quantifiers are of the form for all x in V, for
> some x in V, for all standard x in V, for some
> standard x in V,
> (3) there is a unique value of x in V for which P(x) holds,
> then the unique value of x in V, such that P(x) holds, is
standard.
>Yes, it's in V if that's what you mean by 
standard.
That is not what I mean by standard. Internal set theory is a
theory
with two undefined predicates. The first is the 
binary
predicate is an
element of. The second is the unary predicate standard.
> My question is that if
> (1) P(u,v_1,...,v_k) is a formula in Internal Set Theory,
> (2) y_1, ..., y_k are standard,
> (3) there exists exactly one x such that P(x,y_1,..,y_k)
holds,
> then is it necessarily true that the unique value of x
specified in (3)
> is standard?
>Yes, because u is standarized to V.
I did not state that u was an element of V. And no connection
with V
has anything to do with whether a set is standard. Also,
infinite
standard sets have nonstandard elements as well as standard
elements.
> Nelson answered the question in the case where there is
only one free
> variable and the formula is relativized to a standard set
V, with the
> answer being in the affirmative. Nelson's 
proof is easily
extendible
> to the case where there is more than one free variable. I
am curious
> about the more general case (i.e. when the formula is not
relativized
> to a standard set).
> If there is a proof in the general case that x must be
standard, I would
> be grateful to see it. If it is not true in general that x
must be
>No, if u isn't standarized to V, then the unique x may not
be in V,
>not standarized.
That is not the definition of standard. Standard does not have
a
definition.
David
And all dared to brave unknown terrors, to do mighty deeds,
to boldly split infinitives that no man had split before -
and thus was the Empire forged.
-----
===
Subject: how to calculate sigma
Hi Group
is the sigma calculation formula same as StdDev ?
--
===
Subject: Re: how to calculate sigma
This sigma is for the Cp, Cpk calclulation
So its the same as StdDev?
Cp = (USL-LSL)/(6 x sigma).
Cpk = either (USL-Mean)/(3 x sigma) or (Mean-LSL)/(3 x sigma)
whichever is the smaller (i.e. depending on whether the shift
is up or
down).
I have also came across this on internet
stdDev = square root [sum( x - xbar)^2 / (N-1)]
sigma = square root [sum( x - mu)^2 / N]
what is mu ?
===
Subject: Re: how to calculate sigma
> This sigma is for the Cp, Cpk calclulation So its the same
as StdDev?
> Cp = (USL-LSL)/(6 x sigma).
> Cpk = either (USL-Mean)/(3 x sigma) or (Mean-LSL)/(3 x
sigma)
> whichever is the smaller (i.e. depending on whether the
shift is up or
> down).
> I have also came across this on internet stdDev = square
root [sum( x -
> xbar)^2 / (N-1)] sigma = square root [sum( x - mu)^2 / N]
> what is mu ?
I think in this particular case mu is a population mean and
xbar is a
sample mean. Thus stdDev is a sample standard deviation and
sigma is a
population standard deviation.
--
Lance Lamboy
Go F*ck Yourself ~ Dick Cheney
===
Subject: Re: Functions, the complex plane and Riemann
surfaces.
Right. So tell me, what's conesetter's real 
name?
George Beaman (almost certainly).
> troublemaker...
> Lee Rudolph gives me a name, David Ullrich gives me a
description. I
>am left wondering how and why Lee Rudolph carried out his
detective
>work.
Why: I was establishing that your posts, though pseudonymous,
were easily tied to an apparently genuine identity (which
seemed
to be relevant to some of the amateur copyright lawyering that
was going on). How: you've advertised your website more than 
a
few times; verb. sap.
> Perhaps in time we shall get back to mathematics.
Amateur lawyering and amateur detection are easier, especially
when the weather is as hot and humid as it has been here
lately.
Lee Rudolph
===
Subject: Vector rotation question
charset=US-ASCII
This will probably seem like an unbelievably simple question,
but I'm
bonking my head against the way:
Let's say I have two unit vectors, u and v. I want to
transform (rotate?
Translate?) v, such that instead of referring to the z-axis
as the origin
vector = {0,0,1}, it now uses u as the new origin. What is
computationally
--j
--
Jonathan Greenberg
Graduate Group in Ecology, U.C. Davis
http://www.cstars.ucdavis.edu/~jongreen
http://www.cstars.ucdavis.edu
AIM: jgrn307 or jgrn3007
MSN: jgrn307@msn.com or jgrn3007@msn.com
===
Subject: Re: Vector rotation question
> This will probably seem like an unbelievably simple
question, but I'm
> bonking my head against the way:
> Let's say I have two unit vectors, u and v. I want to
transform (rotate?
> Translate?) v, such that instead of referring to the z-axis
as the origin
> vector = {0,0,1}, it now uses u as the new origin. What is
computationally
> --j
As daniel noted, you need to clarify exactly what you're
trying to do here!
I
think you mean z is the Ôbasis vector' {0, 0, 
1}.
The origin is the point at {0, 0, 0}; the Ôorigin 
vector'
doesn't make much
sense.
(FYI, Ôposition vector' is the coordinate of a 
point relative
to the origin
(in
other words, just the plain coordinates of a point.))
What do you need to do this? What is the problem at hand? Is
it a test
question, homework, your own project, or what? If you can
give me the
original
problem/source it would help..
alex
===
Subject: How does Maths work?
I asked this question to a year 9 maths class, and the
responses were
very interesting. Suffice to say that they answered it as best
they
could with the limited tools and ideas at their disposal,
each with a
unique perspective, it begs the question to be asked to the
wider
community.
How does Maths work?
===
Subject: Re: How does Maths work?
>I asked this question to a year 9 maths class, and the
responses were
>very interesting. Suffice to say that they answered it as
best they
>could with the limited tools and ideas at their disposal,
each with a
>unique perspective, it begs the question to be asked to the
wider
>community.
>How does Maths work?
Define the question more clearly. One can easily state how
mathematics works as a subject by itself, and one can state
how it works as applied to the real world. The second
requires the first.
A major problem is that few know the first. Mathematics is
a formal system, in which one sets up axioms and proves
theorems. It is more than this, in that there are purely
abstract concepts created, which are in some sense useful
or important or just aesthetic. The same structure can
have many concepts related to it; the integers, for example.
The way that mathematics is used in the real world, or in
some other subject, is to model the other field in terms
of the mathematical concepts and structures. Then one
can apply the mathematical results to the model, and get
further results. This then translates back to the real
world or other subject.
It is important to keep these steps separate, although the
adept may combine them. By teaching how to get results,
without realizing the problem, understanding is lost.
It is unclear that the one who has had this can make the
necessary separation later.
--
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, Department of Statistics, Purdue University
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
===
Subject: Re: How does Maths work?
> I asked this question to a year 9 maths class, and the
responses were
> very interesting. Suffice to say that they answered it as
best they
> could with the limited tools and ideas at their disposal,
each with a
> unique perspective, it begs the question to be asked to the
wider
> community.
> How does Maths work?
seems like a dumb question.. a better question would be what
is
math? how does maths work sounds more like a poor illiterate
foreigner trying to pose a seemingly-deep and profound
question but
really just trying to sound thoughtful when he/she is an
idiot.
===
Subject: Re: How does Maths work?
What is math?
A poor abbreviation of Mathematics perchance?
Same as Physic might be for Physics.
An Ôs' on the end does not always imply plural.
Illiterate Johnny foreigner
> I asked this question to a year 9 maths class, and the
responses were
> very interesting. Suffice to say that they answered it as
best they
> could with the limited tools and ideas at their disposal,
each with a
> unique perspective, it begs the question to be asked to the
wider
> community.
> How does Maths work?
> seems like a dumb question.. a better question would be
what is
> math? how does maths work sounds more like a poor illiterate
> foreigner trying to pose a seemingly-deep and profound
question but
> really just trying to sound thoughtful when he/she is an
idiot.
===
Subject: Re: How does Maths work?
>What is math?
>A poor abbreviation of Mathematics perchance?
The standard abbreviation for it in North America. Please
don't
look down on us poor colonials.
>Same as Physic might be for Physics.
No, that would by phys.
>An Ôs' on the end does not always imply 
plural.
That much is obvious.
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia
Vancouver, BC, Canada V6T 1Z2
===
Subject: Re: How does Maths work?
> I asked this question to a year 9 maths class, and the
responses were
> very interesting. Suffice to say that they answered it as
best they
> could with the limited tools and ideas at their disposal,
each with a
> unique perspective, it begs the question to be asked to the
wider
> community.
> How does Maths work?
I've always felt that math was the study of abstraction. We
observe a
pattern in nature or in the math itself and begin to come up
with a
way of studying that pattern without the context. It's this
sort of
thinking that allows us to rely so heavily on logic. The goal
is to
make sure we don't change the fundemental structure of the
pattern
since we don't have a specific context - so this 
only allows
us to use
widely agreed upon rules for sound reasoning. So math works
by finding
a pattern, representing it in symbols and discovering what
you can
from it using logic. This pattern can take a variety of forms
from
compass and straight edge drawings to set theory to graphs
theory and
sometimes the same pattern can be represented many ways. This
is
essentially what I think math is about and the basic
structure it's
has historically.
===
Subject: Re: How does Maths work?
> I asked this question to a year 9 maths class, and the
responses were
> very interesting. Suffice to say that they answered it as
best they
> could with the limited tools and ideas at their disposal,
each with a
> unique perspective, it begs the question to be asked to the
wider
> community.
> How does Maths work?
Does year 9 mean they are 9 years old, or that this is their
ninth
year since starting to school?
--
G. A. Edgar
http://www.math.ohio-state.edu/~edgar/
===
Subject: Re: How does Maths work?
> I asked this question to a year 9 maths class, and the
responses were
> very interesting. Suffice to say that they answered it as
best they
> could with the limited tools and ideas at their disposal,
each with a
> unique perspective, it begs the question to be asked to the
wider
> community.
> How does Maths work?
This thread could become either pedant, or funny.
Ok; i'll give it a try:
Maths doesn't work. It's us mathematicians 
that do the work.
--
Herman Jurjus
===
Subject: Re: How does Maths work?
> I asked this question to a year 9 maths class, and the
responses were
> very interesting. Suffice to say that they answered it as
best they
> could with the limited tools and ideas at their disposal,
each with a
> unique perspective, it begs the question to be asked to the
wider
> community.
> How does Maths work?
Ôhow does maths work' is a little vague. 
(although maybe a
question that 9
years could Ôunderstand' more.)
To me, that vague question could be asking either Ôwhat are
the rules of
maths', or it could be asking Ôhow does maths 
maintain its
consistency and
cohesion throughout?', which may lead to Ôdoes 
mathematics
exist out there
somehow, or is it merely a human construction?', and so on.
alex
===
Subject: Re: How does Maths work?
Start over a bit: ask that same group of year 9 students:
How does arithmetic work?
How do patterns work?
How does a square work?
How does a rectangle work?
How does a triangle work?
How does a circle work?
How does a fraction work?
Mathematics is several things while the various more specific
topics
are,
naturally, more specific things. The year 9 students may be
able to give
meaningful responses more easily to the more specific
questions. (Maybe not
to
all of the questions, but to a few of the questions.)
G C
>I asked this question to a year 9 maths class, and the
responses were
>very interesting. Suffice to say that they answered it as
best they
>could with the limited tools and ideas at their disposal,
each with a
>unique perspective, it begs the question to be asked to the
wider
>community.
>How does Maths work?
===
Subject: Re: How does Maths work?
> How does Maths work?
A bunch of 9-year olds will try heroically to answer any
question
put by their teacher, whether or not it makes any sense. What
on
earth do you mean?
===
Subject: Identity question.
Is this identity correct for these 2 algebraic numbers?
-a = 2/10^(1/2)-4 = -2.387425886722..
b = 1/9 + 1/9 *10^(1/2) = .4624752955742..
Then --
b = (((-x*-1*2+2)^2 -2) /4) - (-x*-a)
Where -x = 1 - ((-a*-1)/2)
I think this is correct!
Dan
===
Subject: Re: Identity question.
>Is this identity correct for these 2 algebraic numbers?
>-a = 2/10^(1/2)-4 = -2.387425886722..
Not even close. Oh, you mean 2/(10^(1/2)-4). Please learn
about the precedence rules for operations. When in doubt, use
parentheses.
> b = 1/9 + 1/9 *10^(1/2) = .4624752955742..
>Then --
>b = (((-x*-1*2+2)^2 -2) /4) - (-x*-a)
What's with all these extra - signs? Surely you can
simplify...
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia
Vancouver, BC, Canada V6T 1Z2
===
Subject: Re: Vector rotation question
It's hard to understand what you're asking 
here, because an
origin is a
point, not an axis. Here is some information you may find
useful;
please clarify your question if you still need help.
To translate means to move the origin from one point to
another, while
leaving the axes pointing the same direction. To rotate means
to leave
the origin fixed and change the direction the axes point.
Rotation and
translation are transformations on coordinate systems; that
is, a
rotation takes one coordinate system (S) and transforms it
into a
different coordinate system (S').
A (rectilinear) coordinate system is effectively an ordered
basis of
R^3. Since you probably haven't had any linear algebra, that
means
roughly, this:
Let S be a list consisting of the three vectors i, j, and k,
corresponding to unit vectors in the direction of the x, y,
and z axes,
respectively. Then in the coordinate system S, the
coordinates (3, 5,
2) represent the vector 3i + 5j + 2k, i.e. the vector 3 units
to the
right, 5 units back, and 2 units above the origin.
But the vectors i, j, and k are not particularly special; we
can
replace them with any vectors we want. Say that we have three
(noncoplanar) unit vectors r, s, and t. We can make these
into a
coordinate system S'. If we have the vector (3, 5, 2) in S
(that is,
3i + 5j + 2s), we can extract the coordinates ar + bs + ct of
u in S'
as follows:
a = u (dot) r
b = u (dot) s
c = u (dot) t
A dot product of two vectors in the same is given by {X,Y,Z}
(dot)
{D,E,F} = XD + YE + ZF. (A proof of this fact can be found in
any
linear algebra text.)
Translations are much easier than rotations; you simply add
the
distances in the obvious way.
Hope this helps,
Daniel McLaury
===
Subject: Re: Vector rotation question
charset=US-ASCII
of a photon exiting the surface of a leaf. There are two
algorithms that
describe this behavior:
1) Determination of the leaf surface normal (a unit vector,
say {p,q,r})
2) Determination of the scattering direction with respect to
a standard
coordinate system (e.g. With respect to leaf normal being
{x,y,z}={0,0,1}),
which we'll say is {a,b,c}.
What I want to know is how to rotate the scattering direction
{a,b,c} into
leaf normal space {p,q,r}. Is it simply {a,b,c}.{p,q,r}? For
the
simple
case of the scattering direction being {0,0,1}, the new
scattering
direction
in leaf normal space would be {p,q,r}.
context...
--j
> It's hard to understand what you're asking 
here, because an
origin is a
> point, not an axis. Here is some information you may find
useful;
> please clarify your question if you still need help.
> To translate means to move the origin from one point to
another, while
> leaving the axes pointing the same direction. To rotate
means to leave
> the origin fixed and change the direction the axes point.
Rotation and
> translation are transformations on coordinate systems; that
is, a
> rotation takes one coordinate system (S) and transforms it
into a
> different coordinate system (S').
> A (rectilinear) coordinate system is effectively an ordered
basis of
> R^3. Since you probably haven't had any linear algebra,
that means
> roughly, this:
> Let S be a list consisting of the three vectors i, j, and k,
> corresponding to unit vectors in the direction of the x, y,
and z axes,
> respectively. Then in the coordinate system S, the
coordinates (3, 5,
> 2) represent the vector 3i + 5j + 2k, i.e. the vector 3
units to the
> right, 5 units back, and 2 units above the origin.
> But the vectors i, j, and k are not particularly special;
we can
> replace them with any vectors we want. Say that we have
three
> (noncoplanar) unit vectors r, s, and t. We can make these
into a
> coordinate system S'. If we have the vector (3, 5, 2) in S
(that is,
> 3i + 5j + 2s), we can extract the coordinates ar + bs + ct
of u in S'
> as follows:
> a = u (dot) r
> b = u (dot) s
> c = u (dot) t
> A dot product of two vectors in the same is given by
{X,Y,Z} (dot)
> {D,E,F} = XD + YE + ZF. (A proof of this fact can be found
in any
> linear algebra text.)
> Translations are much easier than rotations; you simply add
the
> distances in the obvious way.
> Hope this helps,
> Daniel McLaury
===
Subject: Re: Vector rotation question
> of a photon exiting the surface of a leaf. There are two
algorithms that
> describe this behavior:
> 1) Determination of the leaf surface normal (a unit vector,
say {p,q,r})
> 2) Determination of the scattering direction with respect
to a
standard
> coordinate system (e.g. With respect to leaf normal being
{x,y,z}={0,0,1}),
> which we'll say is {a,b,c}.
> What I want to know is how to rotate the scattering
direction {a,b,c}
into
> leaf normal space {p,q,r}. Is it simply {a,b,c}.{p,q,r}?
For the
simple
> case of the scattering direction being {0,0,1}, the new
scattering
direction
> in leaf normal space would be {p,q,r}.
> context...
Probably the easiest way to approach this, numerically and
conceptually, is in terms of basis vectors. You want to
express
v in terms of an orthonormal basis {u1, u2, u3} such that
u3 is the leaf normal and u1 and u2 define the plane of the
leaf.
There are two choices of u3 (up and down) but presumably you
have an unambiguous way of defining which one you want. There
are infinitely many choice of u1 and u2, consisting of any
two orthogonal vectors in the plane. Pick any that are
convenient
for you.
So given {u1,u2,u3} you want to write v = v1*u1 + v2*u2 +
v3*u3
where v1, v2 and v3 are scalars. Those are the components of v
in the {u1,u2,u3} coordinate system. Written that way it's
easy
to see that v1 = u1.v (vector dot product), since u1.u2 =
u1.u3 = 0.
Similary v2 = u2.v, v3 = u3.v.
So once you have your {u1,u2,u3} defined in your original
coordinate
system, all you have to do is take the three dot products to
get the components of v.
By the way if you have u3 and a choice of u1 (for instance the
main axis of the leaf) then you can get u2 with the vector
cross
product: u2 = u3 x u1 for a right-handed coordinate system.
Similarly, u1 = u2 x u3.
- Randy
===
Subject: Re: Representation of function over finite set
> Is it true that:
> Any function f of n variables defined on finite 
set can be
represented
> as follows f(x_1, ..., x_n) = f_1( x_1, f_2( x_2, ..., x_n
)).
if f_2 defined as f(x_2, ..., x_n) = (x_2, ..., x_n) 
(identity)
then f_1(x_1, f_2(x_2, ...,x_n)) = f_1(x_1, ..., x_n)
===
Subject: Re: Identity question.
Not so far as I can tell, but the incredibly strange way that
you've
written your equations suggests maybe you mean something
else. Why all
the things of the form (-x*-a) instead of just ax?
===
Subject: Re: Identity question.
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i724Kow00824;
>Is this identity correct for these 2 algebraic numbers?
>-a = 2/10^(1/2)-4 = -2.387425886722..
> b = 1/9 + 1/9 *10^(1/2) = .4624752955742..
>Then --
>b = (((-x*-1*2+2)^2 -2) /4) - (-x*-a)
>Where -x = 1 - ((-a*-1)/2)
>I think this is correct!
>Dan
Also another --
b = (-a*-1)/sqrt(10)+2
===
Subject: Re: Identity question.
>Is this identity correct for these 2 algebraic numbers?
>-a = 2/10^(1/2)-4 = -2.387425886722..
>b = 1/9 + 1/9 *10^(1/2) = .4624752955742..
>Then --
>b = (((-x*-1*2+2)^2 -2) /4) - (-x*-a)
>Where -x = 1 - ((-a*-1)/2)
>I think this is correct!
>Dan
> Also another --
> b = (-a*-1)/sqrt(10)+2
Hey, Dan,
Perhaps you haven't had the time to read the current
responses to your
to wield the sqrt operator must be accustomed to collecting
minus signs
in product expressions so as to leave the multiplication
operator
unadorned by superßuous material?
By which, I mean, do you really write this:
-a*-1
when you're in the privacy of your own room? If so, then
please leave
that sort of behavior aside, and learn to write its
equivalent:
a
You'll be glad you did. Once you've grasped 
that little
handful of
simplification, you may wish to try your hand at this:
(-x*-1*2+2)
or even this:
- (-x*-a)
Go ahead, you can do it!
A better life awaits:
Improved ability to wade through your own class notes and
homework!
Better grades on exams, since the instructor won't be
suffering from such fatigue at figuring out
what on earth you mean!!
What're you waiting for?
Dale
===
Subject: Enriched Category Theory: Weighted limit as a right
adjoint
Content-Length: 491
Originator: rusin@vesuvius
I am learning Enriched Category Theory, and I have a basic
question about
weighted limits:
Can a weighted limit be seen as a right adjoint? If yes,
which is its
left adjoint?
More generally, is it the case that all fundamental enriched
categorical
notions can be characterized by enriched adjunctions.
I guess answers to all my questions are available in GM
Kelly's book but
unfortunately I cannot find it. Is there an electronic version
available
somewhere?
David
===
Subject: Re: Identity question.
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i725kN106997;
>Not so far as I can tell, but the incredibly strange way
that you've
>written your equations suggests maybe you mean something
else. Why all
>the things of the form (-x*-a) instead of just ax?
Sorry for the confusion, in this instance I should have
written just
ax but either way it works. I Just wanted to show that 2 like
signs
(-) multiplied = (+)value.
Dan
===
Subject: Re: ALGABRIC SOLUTION
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i725kNq07001;
>Is there any example(s) where cube roots appear in general
solution of
>quintics
>x^5 + a3*x^3 + a2*x^2 + a1*x + a0 =0
>where a3,a2,a1 and a0 are rational numbers
No, a solvable but IRREDUCIBLE quintic with rational
coefficients
employs only square roots and fifth roots. You can read
Dummit's
paper for more details.
Titus (tpiezasIII@uap.edu.ph -> remove III for email)
===
Subject: Graphs: eccentricity, diameter, radius and tree
height
G=(V,E) a simple graph.
d(x,y): distance from node x to node y
eccentricity: e(v) = max{ d(u,v) / u in V and u!=v }
diameter: D(G) = max{ e(v) / v in V } (max distance between
to nodes)
radius: R(G) = min{ e(v) / v in V }
Let T a tree and h its height generated by BFS (may be not
important*),
then R(G) <= h <= D(G), what happened if T were generated by
DFS (*).
advance!
First, h <= D(G).
h
= max distance from r (root node) to a leaf
= max{d(u,r)/ u in V and deg(u) = 1}
<= max{ d(u,r)/ u in V }
= e(r)
<= max{e(v)/v in V}
= D(G)
Second, h >= R(G)
R(G) = min{e(v)/ v in V}
<= e(r)
[ Note: e(r) must be greater or equeal than min{e(v)/v in V},
otherwise e(r) < min{e(v)/v in V} is a contradiction ]
= max{ d(u,r)/ u in V }
<= h
[ because the leafs are the most distant nodes from the root ]
Then, R(G) <= h <= D(G). Doesn't matter if T is generated by
BFS, DFS or
any other algorithm.
Harry.
===
Subject: Re: chess
>But does there exist an unbeatable algorithm (ties OK)? If
one exists
>in theory, it might be something like: given the list of
all possible
>chess games, don't play any loss games. Could there be a
practical
>algorithm?
>is there an algorithm that ensures winning in chess in any
case...
>a set of moves...
> Please don't top-post.
> It is not known whether there is a winning strategy for
black. It does
not
> seem probable since in practical experience white seems to
have the
> advantage.
> Even if there were a theoretical guarantee that white had a
non-losing
> strategy, this does not guarantee the existence of a
practical
> algorithm that implements that strategy.
> The number of chess positions is large but finite(*). So in
theoretical
> terms, it would be possible to exhaustively list all the
non-losing moves
> from all possible positions. To play the game, you'd 
simply
> look up the current position in a [huge] database and read
out
> an acceptable move.
> As a practical matter, the number of positions is way too
large to
> allow such a database to be stored, to say nothing of
constructing
> it in the first place.
> (*) Assume that white will declare a draw at the first
opportunity.
> That way we don't get into the possibility of two players
playing
> forever, neither willing to accept an available draw.
Assumption (*) is quite appealing for these theoretical
discussions.
For example, there is a rule:
9.3
The game is drawn, upon a correct claim by the player having
the
move, if
(a)
intention to make a move which shall result in the last 50
moves having
been made by each player without the movement of any pawn and
without
the capture of any piece, or
(b)
the last 50 consecutive moves have been made by each player
without the movement of any pawn and without the capture of
any piece.
So under 9.3, a draw occurs only if the player having the
move makes
a claim, etc.
Using the assumption (*), it's interesting to me that some
positions
are known which would be a win for white without the 50-move
rule,
but are draws if we abide by the 50-move rule.
An example end-game position with King + Rook + Knight vs.
King + Knight + Knight where White safely captures a
http://plus.maths.org/issue28/features/dartnell/
David Bernier
===
Subject: Re: Calculus is EVIL, the DEVIL created it !
> /
> / EVIL dEVIL = 1/2 EVIL ^2 ??
 Since EVIL is negative, the result is always positive,
> and since the positive is GOOD we can write
> = 1/2 GOOD
No, two wrongs don't make a right!
===
Subject: Re: Strange relations, number theory tidbit
<10ggsocgg1a988e@corp.supernews.com>
Discussion, linux)
> Apparently, no one can figure out how I derived [(N-4)/6]
> No wonder, since
> [(N-3)/6]
> would be correct for both even and odd N, contrary to
[(N-4)/6] which
> gives incorrect results for N = 3, 9, 15 etc.
> Why do I have to keep repeating for even N?
You missed his point.
He *knows* your formula is correct for even N, but it is
correct
*only* for even N.
He gives a different formula which is just as simple as yours
but is
correct for all N[1].
This indicates that your formula is not exactly deep,
insightful or
useful.
Footnotes:
[1] I take Christian's word for this. I didn't 
check either
his or
James's formulas.
--
Mathematicians are rather important in the infrastructures of
many
organizations that protect civilization. I've determined 
that
they
are a consistent security risk, and seem to have other
agendas, other
loyalties beyond loyalty to their respective nations. --
James Harris
===
Subject: Re: Researching sensitive issues dilemma
[...]
>Please can folks tell me:-
I could but I won't. If you cannot deal with such an
elementary
problem you are not qualified to design and carry out such a
study.
Find a good course on ÔStatistic Methods for Research
Workers'.
J.9frgen
===
Subject: Re: Researching sensitive issues dilemma
===
Subject: Re: Researching sensitive issues dilemma
G'day, G'day.
>A researcher wishes to research the incidence of burn out
in a group
>of professionals. There is however a law which makes
withholding
>information about incompetence within this group from
authorities an
>offence. Put simply, if the researcher collects damning
information
>about individuals and doesn't report it, he/she could face
a jail
>term. Being aware of the penalties for both researcher and
>participants, participants are unlikely be forthcoming
with truthful
>information and the research becomes a white wash.
There's a rash of legislation city, state and national and
also corporate
maneuvers, that's making it more difficult for 
US to be an
informed
participative democracy. That law strikes me as another such.
Oh sigh,
is this beginning to happen New Zeland?
>Why should anybody participate,
>Good question. I might have thought since factual
information is
>vital for the health of their profession was an answer
however I
>must admit that is only an opinion.
If I collected damning information about some group of
people, I'd
disclose it anonymously, publish with pseudonym, lest I
suffer the fate of
whistle blowers. Researchers with damming information about
lead
poisoning have been marauded by the lead producers, college
research funds
slashed, results publicly discredited, tenure put into doubt,
etc.
>Perhaps there is no need to know the level of mental and
emotional
>competence of a group of professionals with considerable
power over
>the lives of others when weighed against an individual's
demands for
>privacy.
There is. Ronald RayGun's last two years of his term had
dementia which,
to keep him in office, was not told the public. Bush is taking
three
psychiatric medications to keep him from lashing out at
others.
>Knowing of the law they would not participate or they would
>blatantly lie unless they fully realised that statistical
information
>obtained using the technique suggested above would not
provide
>individual information.
Even if it doesn't, for any group to give personal or other
information
about themselves, is providing material for action to be
taken against the
group. I have seen this with anonymous surveys about drug
usage among
students. Most answered honestly and as a result the school
got cracked
down upon.
Social surveys are designed for the benefit of the inquirer.
They are not designed for the benefit of the participants.
>The statistics thrown up by the research will undoubtedly be
>published.
In addition, if any conclusions indicate damning
possibilities, the
inquirer is required to notified the government.
>The inquirer CAN'T report what he/she doesn't 
know. Put
simply,
>he/she doesn't know anything about or even suspect anything
about an
>individual if the number of such questions is small.
Then the inquirer has learned nothing and has no results to
show.
William Elliot
Email address upon request.
Have you an off topic topic to discuss?
How's it going down under? But just a nanosec.
Does that expression also apply to New Zeland?
----
===
Subject: Re: Researching sensitive issues dilemma
This post not CC'd by email
> Please can folks tell me:-
> A. Of web sites the give fully discussions of this and
other
> techniques that would be useful in such situations.
> B. The correct way to extract means and standard
deviations from the
> data collected.
> C. How much the masking procedure degrades the power of
the experiment
> or how much larger the sample has to be to get the same
power.
> Best wishes,
> What you are after is a scheme called Ôrandomized response.
>Search for that phrase in Google and you will find plenty of
>references. There has been a fair amount of research
published
>on it since the early 1980s which addresses your questions.
> Nora B.
G'day G'day Nora,
sensitive issues.
Guess I had better also Google randomised response to allow
for
spelling differences.
Best wishes and thank you,
--
Quentin Grady ^ ^ /
New Zealand, >#,#< [
/ /
... and the blind dog was leading.
http://homepages.paradise.net.nz/quentin
===
Subject: Re: Researching sensitive issues dilemma
This post not CC'd by email
>Why not just have the information submitted anonymously, so
the
>researcher can't match the responses with the participants?
G'day G'day Robert,
helping me with the statistical issues involved. Since your
suggestion here deals with a separate issue, I'd like to
explore it
separately from the stats.
My understanding of the situation is that the number of such
professionals to be surveyed in NZ is small. Anonymity is a
wonderful
... when it can be guaranteed. My guess is the researcher
doesn't
feel that anonymity can be guaranteed given the risk that
some answers
might unwittingly identify some participants.
Best wishes,
--
Quentin Grady ^ ^ /
New Zealand, >#,#< [
/ /
... and the blind dog was leading.
http://homepages.paradise.net.nz/quentin
===
Subject: Re: Job application strategies
>G'day G'day Folks,
> After thirty five years of teaching I found myself for the
first
>time unemployed. Each time a new position appears I
restructure my CV
>to suit and promptly post it in.
Keep Ôrestructuring' your CV long enough and 
you'll end up in
jail,
although a lunatic asylum is probably more appropriate. Are
you sure
you are quite well?
>Now I am wondering if this is a strategic/tactical error.
>While I have limited data there appears to be a pattern. At
first the
>recipients are delighted to have an application from a well
qualified,
>experienced applicant. This induces a false sense of
euphoria. Then
>there is a long wait till the closing date ... and nothing.
They probably called your previous employer and found out
what a
complete idiot you are.
===
Subject: Re: Job application strategies
This post not CC'd by email
>G'day G'day Folks,
> After thirty five years of teaching I found myself for the
first
>time unemployed. Each time a new position appears I
restructure my CV
>to suit and promptly post it in.
>Now I am wondering if this is a strategic/tactical error.
>While I have limited data there appears to be a pattern. At
first the
>recipients are delighted to have an application from a well
qualified,
>experienced applicant. This induces a false sense of
euphoria. Then
>there is a long wait till the closing date ... and nothing.
>My hypothesis is that the early application benefits the
employer.
>It relieves their anxiety about getting a suitable
applicant.
>However, IMHO it disadvantages the early applicant. Reasons
for
>rejection of one offer or another appear to be pretty
idiosyncratic
>and based on a few randomly occurring criteria. In my
hypothesis the
>early applicants run the risk of rejection by comparison
with every
>applicant that follows them. The last applicant risk
rejection only
>by comparison with the one that has survived the comparison
processes
>that have gone on before. Since selection/rejection is
essentially
>random (or idiosyncratic) the winning strategy is to
minimize the
>number of comparisons.
>Has anyone researched job applications and used as the null
hypothesis
>The rate of acceptance approximates (1/x)^n+1
>where x most cynically is 2, or more realistically 1 < x <=2
>where n = # applications following yours?
>Put simply, taking the most cynical case,
>the last applicant has a 1 in 2 chance of success.
>the one before that has only a 1 in 4 chance.
>the one before that has only a 1 in 8 chance.
>Best wishes,
>In the commercial (real) world, the employer will interview
and
>hire the first qualified candidate. In the 
academic and
government
>worlds, maybe not.
>phil
G'day G'day Phil,
Is this a conjecture based on your own personal experience or
is it a
result found in published in peer reviewed research?
I'm not saying or implying you should be quoting peer 
reviewed
research. I simply am looking for which strategy will give me
the best
chances. Whatever.
There could am important difference between the hiring
practices of
commercial employers and academic and government worlds ie a
closing
date for applications. In the academic/government situation
employers
cannot employ before the closing date. This doesn't stop 
them
reading
the CVs as they come in. IMHO this means that while they
might have
employed an early applicant immediately if they could, by the
time the
closing date comes around, their perception of them is stale.
More
recent applicants appear dynamic, alive, fresh etc etc by
comparison.
Best wishes,
--
Quentin Grady ^ ^ /
New Zealand, >#,#< [
/ /
... and the blind dog was leading.
http://homepages.paradise.net.nz/quentin
===
Subject: Re: Job application strategies
This post not CC'd by email
>these theories of yours seem quite irrelevant in the real
world.
G'day G'day,
Quite likely.
Nevertheless a large employer expressed the opinion that they
tended
to dismiss early applicants out of hand on the basis that
they were
probably,
a. desperately applying for every job going.
b. not taking the time to research the company and its
business before
applying.
>if teaching employment prospects in new zealand, or wherever
is
>that you live, behave similarly as in the us, rest assured
that
>with rare exception the strategy that works best is to ass
kiss
>the bureaucrat or manager in charge, and by all means sell
your
>self as one who shares their silly and often bogus
philosophies.
Unfortunately this appears to be more often true than one
would like
it to be.
Best wishes,
--
Quentin Grady ^ ^ /
New Zealand, >#,#< [
/ /
... and the blind dog was leading.
http://homepages.paradise.net.nz/quentin
===
Subject: Re: Job application strategies
> This post not CC'd by email
>these theories of yours seem quite irrelevant in the real
world.
> G'day G'day,
> Quite likely.
> Nevertheless a large employer expressed the opinion that
they tended
> to dismiss early applicants out of hand on the basis that
they were
> probably,
> a. desperately applying for every job going.
> b. not taking the time to research the company and its
business before
> applying.
>if teaching employment prospects in new zealand, or wherever
is
>that you live, behave similarly as in the us, rest assured
that
>with rare exception the strategy that works best is to ass
kiss
>the bureaucrat or manager in charge, and by all means sell
your
>self as one who shares their silly and often bogus
philosophies.
> Unfortunately this appears to be more often true than one
would like
> it to be.
> Best wishes,
maybe you say g'day too much during the interview and the
employer is
too polite to tell you it makes you look and sound like an
idiot?
===
Subject: Re: Gyroscopes - Usenet Physics FAQ - Reference
frames
JM Albuquerque <jm.aREMOV.E@sapo.pt> schreef in bericht
> Around these newsgroups there are current discussions about
> inertia, mass, centrifugal force, equivalence principle,
gravity, etc.
> I've looked at the Physics FAQ:
> http://math.ucr.edu/home/baez/physics/index.html
> And I've notice the following:
> 1 - Gyroscopes don't belong to Physics.
> 2 - Centrifugal force doesn't even exist.
> 3 - Gravity also is a fictitious force like the centrifugal
force.
>
http://math.ucr.edu/home/baez/physics/General/Centrifugal/
centri.html
> (See at bottom: Could gravity be a fictitious force too?
YES.)
> Why gyroscopes don't belong to the actual Physics is the
point.
> It looks like that the actual Physics don't like the
reference frame of
> gyroscopes, their 3 rotating motions, 3 force/torque at
right angles,
> and 3 non inertial reference frames (a triple problem with
mass inertia).
> Physics goes for inertial reference frames (those non
rotating) and
simply
> discard the rotating frames of reference.
> Looking at the universe, where are those non rotating
frames (inertial)?
> Since all features that one could see in the Universe is
rotating and
> working like a gyroscope (all of them are ßat, like a disk,
and all them
> most likely precess if an external force is applied)?
> In fact there are no true inertial frame of reference,
since everything
is
> rotating at large scale.
> If so why inertial reference frames and the equivalence
principle were
> both chosen?
> Experts, please explain why a bicycle wheel don't fall
under gravity, if
a
> precession motion exists?
> Please explain this:
> http://physics.nad.ru/Physics/English/gyro_tmp.htm
> http://physics.nad.ru/Physics/English/gyro_txt.htm
> And with 4Mb size the bicycle wheel:
> http://www.rci.rutgers.edu/~williebo/zzzgyrovideo.MPG
> (The usual bicycle wheel precessing around a vertical axis
which doesn't
> fall and turn around if allowed to do so).
> What happened to the space-time curvature, near the bicycle
wheel, in
order
> not to fall to the ground, like every other non rotating
mass will do?
> Is there any reasonable explanation why the bicycle wheel
doesn't fall?
> At such a small speed, how does GRT explain that the
bicycle wheel
doesn't
> fall? Gyroscopes show that locally the space-time curvature
clear depends
on
> the angular speed of the solid body.
> No rotating speed and it will fall.
> Just spin the body a little and it will not fall. Why?
I have read this whole thread,
when I have a bicycle weel and I hold it at both ends of the
axle and spin
the weel,and then i let it loose it will all spinning fall to
the ground
,what you probably mean is it won't fall over,yes?,
and when you try to bring it from vertical to f.i 45 degrees
it resist
you,therefore,if you ride a bicycly you won't fall over to
the ground,but
as
soon as you stop,you will fall over,why is that?that is,what
we are talking
about here,yes?
well,I have been thinking a long time about this problem,but
lets assume we
drive in a car and go around a corner,if you take that corner
very slow,you
dont need much force to let it take the corner,but now you go
very fast,you
have to pull your steering very hard,(therefore we have power
steering,we
don't need that for going slow around the corner,do we?),
so in fact ,you have to apply a force to let the car make that
corner,because you have to make an accelleration in a
different
direction,(that force for the acceleration comes out of your
motor,by
turning your steering wheel),
now,when you go slow,you need a small force,because you
spread the force
over a long time to make the corner,
but when you go fast,you have to apply a lot more force to do
that same
work
i.e.the force needed for the requiered acceleration,to bring
that car
around
the corner)in a lot shorter time,
,they call that anoloog?
change of direction is acceleration in that other changed
direction,and
that
takes a force,which over time cost energy,it just depends how
quick you
want
to accomplish that,
so actually it is a matter of time and f=ma ,
the faster the wheel spins the more force you need to bring
all the
the corner(also more force on your steering wheel is
requiered?
then I have a question,as gravity and acceleration is
equivalent,when you
have a balloon (with light gas in it,i.e. the balloon would
go in the air
when not being in the car)
in your car and you go around the corner,we people are pushed
the other way
than the corner goes,but what does the balloon do?he should
go the other
way
as we do? same as the ballon would do outside the car(going
against the
gravity),
it should do ,but I am not quite sure
marten
> Once upon a time Physics came to a problem about which
frame of
> reference is the best:
> 1 - Inertial reference frames;
> 2 - Rotating reference frames.
> Physics took the first and forget the second.
> And Physics failed to explain most of the basic working
mechanisms
> (those that everybody knows).
> I'm not confortable with the fact that Physics 
doesn't
explain the
> mechanism by which gyroscopic motion opposes gravity.
> It is a clear violation of the equivalence principle, but
nobody cares.
> But I do care!
> Without a good explanation I propose that Usenet Physics
FAQ clear say
that
> gyroscopic motion of matter cannot be explained by any
actual theory, at:
>
http://math.ucr.edu/home/baez/physics/General/open_
questions.html
> Gyroscopic motion is an open question in Physics.
> Or else please explain it.
> -------------------------------
> I've been studying and analysing gyroscopes.
> Also the mechanical to electromagnetic energy conversion
(the 3-phase
> synchronous generators that power the World, also the
homopolar
> generator, and so on).
> The conclusion I've found is that electromagnetic 
fields
forces and
> mechanical gyroscopic forces are very similar. Both apply
the right hand
> rule and the force field equations are the same in a local
reference
frame
> (at any laboratory here on Earth).
> The equations of circular motion, spiral motion,
electromagnetism,
> gyroscopic torque, etc. etc. could be of the form:
> x(t) = K t (cos(wt))^2
> y (t) = K t sin(wt)cos(wt)
> z(t) = K t sin(wt)cos(wt)
> Being:
> t = time
> K = constant
> w = angular speed
> The equations:
> x(t) = K (cos(wt))^2
> y (t) = K sin(wt)cos(wt)
> Physically meant a solid disk rotating simultaneously
around two
> perpendicular axis, being the x,y the positions of a point
of the disk
> surface projected in a plane.
> A circle is:
> x(t) = K cos(wt)
> y (t) = K sin(wt)
> multiplying both by cos(wt) produces another rotation going
in and out of
> the paper plane in the perpendicular direction.
> The first equation x(t) produces all the work (double
frequency, always
> positive and looks like power in electricity, could be
gravity and is the
> actual gyroscopic force applied).
> The other two equations don't produce any work since the
average is zero.
> Plotting those x(t) and y(t) equation in a computer
software enables one
to
> see many lovely things, like all the Physics passing in
front of your
eyes.
> (the first t refers to linear motion and could be made
constant at
the
> speed of light in one axis direction. The t close to w is
always
time
> from zero to infinite.)
> Them try:
> (cos(t))^2 - sin(t)cos(t) for the Doppler shift of light.
> Combinations of those circular terms:
> cos(t)^2
> cos(t)sin(t)
> produce many evidence that all the Universe could be a
large spinning
> device and spin is all that matters. Linear motions are
against mother
> nature and that's why all the know forces come to day 
light
when linear
> motion exists.
> A rotating Mach like principle perhaps could be the right
approach.
> But I don't have any theory, nor I want to have.
> The above is just a curiosity.
> I've been inspired here:
> http://www.rci.rutgers.edu/~williebo/
> And the best is here:
> http://www.rci.rutgers.edu/~williebo/g7Gyroscopicnalysis.pdf
> and predictions:
> http://www.rci.rutgers.edu/~williebo/za27Predictions.pdf
> Some strange things are:
> http://www.keelynet.com/gravity/gyroag.htm
> There are many other like the one above
> True of False is the question ???
> I guess that circular motion cannot be discarded form
Physics.
> Orbits and circular motion is not free fall, it is circular
motion.
> Remember Pioneer 10 going out of the solar system?
> There is so many evidence that Physics must have taken the
wrong
> road some where that it almost stinks.
===
Subject: Re: Gyroscopes - Usenet Physics FAQ - Reference
frames
> when I have a bicycle weel and I hold it at both ends of
the axle and
spin
> the weel,and then i let it loose it will all spinning fall
to the ground
> ,what you probably mean is it won't fall over,yes?,
> and when you try to bring it from vertical to f.i 45
degrees it resist
> you,therefore,if you ride a bicycly you won't fall over to
the ground,but
as
> soon as you stop,you will fall over,why is that?that
is,what we are
talking
> about here,yes?
Not really. A bicycle does NOT stay upright due to gyroscopic
action, it
does so because (a) the human rider continuously steers the
wheels under
the center of gravity, and (b) the geometry of the front fork
is
designed to do that automatically if left alone.
> well,I have been thinking a long time about this
problem,but lets assume
we
> drive in a car and go around a corner,if you take that
corner very
slow,you
> dont need much force to let it take the corner,but now you
go very
fast,you
> have to pull your steering very hard,(therefore we have
power steering,we
> don't need that for going slow around the corner,do we?),
> so in fact ,you have to apply a force to let the car make
that
> corner,because you have to make an accelleration in a
different
> direction,(that force for the acceleration comes out of
your motor,by
> turning your steering wheel),
Most of the force comes from the tires' friction on the 
road.
In fact,
that where it all comes from if you put the car in neutral
(so the
engine does not pull the car through the corner). Note that
the turning
force must be perpendicular to the car's motion, and a
rear-wheel-drive
car's engine can only exert a force parallel to the 
car's
motion (and
evn that requires friction of tires on road).
> now,when you go slow,you need a small force,because you
spread the force
> over a long time to make the corner,
No. For a circular turn the centripetal force of the turn is
proportional to your mass, proportional to the square of your
speed and
inversely proportional to your turning radius.
> but when you go fast,you have to apply a lot more force to
do that same
work
For a circular turn no work is done -- the force is
perpendicular to the
direction of motion.
> [...]
> then I have a question,as gravity and acceleration is
equivalent,when you
> have a balloon (with light gas in it,i.e. the balloon would
go in the
air
> when not being in the car)
> in your car and you go around the corner,we people are
pushed the other
way
> than the corner goes,but what does the balloon do?he should
go the other
way
> as we do? same as the ballon would do outside the car(going
against the
> gravity),
> it should do ,but I am not quite sure
This is a simple experiment to do, just get a helium-filled
balloon and
take it for a ride in your car. As expected, the helium
balloon goes to
the front of the car when accelerating, to the rear when
braking, and
toward the center of any turn. That's because the air is 
more
dense than
the balloon, and is pushed harder than the baloon (all those
forces are
proportional to the mass of the object, so there is more
force on the
air in the same volumes as the baloon than on the balloon
itself).
Tom Roberts tjroberts@lucent.com
===
Subject: Re: Gyroscopes - Usenet Physics FAQ - Reference
frames
> when I have a bicycle weel and I hold it at both ends of
the axle and
> spin
> the weel,and then i let it loose it will all spinning fall
to the
ground
> ,what you probably mean is it won't fall over,yes?,
> and when you try to bring it from vertical to f.i 45
degrees it resist
> you,therefore,if you ride a bicycly you won't fall over to
the
ground,but
> as soon as you stop,you will fall over,why is that?that
is,what we are
> talking about here,yes?
> Not really. A bicycle does NOT stay upright due to
gyroscopic action, it
> does so because (a) the human rider continuously steers the
wheels under
> the center of gravity, and (b) the geometry of the front
fork is
> designed to do that automatically if left alone.
In fact the bicycle does stay upright due to gyroscopic
action. When you
lean, due to the geometry of the front fork, a torque is
applied to the
gyroscopic fron wheel. That torque is in a vertical plane
perpenduclar to
the direction of travel. When you lean right, the torque
causes the front
of the wheel to turn to the right. The action isn't great,
but it does make
bikes (and Harleys) easier to ride.
> well,I have been thinking a long time about this
problem,but lets assume
> we drive in a car and go around a corner,if you take that
corner very
> slow,you dont need much force to let it take the
corner,but now you go
> very fast,you have to pull your steering very
hard,(therefore we have
> power steering,we don't need that for going slow around
the corner,do
> we?), so in fact ,you have to apply a force to let the car
make that
> corner,because you have to make an accelleration in a
different
> direction,(that force for the acceleration comes out of
your motor,by
> turning your steering wheel),
> Most of the force comes from the tires' friction on the
road. In fact,
> that where it all comes from if you put the car in neutral
(so the
> engine does not pull the car through the corner). Note that
the turning
> force must be perpendicular to the car's motion, and a
rear-wheel-drive
> car's engine can only exert a force parallel to the 
car's
motion (and
> evn that requires friction of tires on road).
> now,when you go slow,you need a small force,because you
spread the force
> over a long time to make the corner,
> No. For a circular turn the centripetal force of the turn is
> proportional to your mass, proportional to the square of
your speed and
> inversely proportional to your turning radius.
> but when you go fast,you have to apply a lot more force to
do that same
> work
> For a circular turn no work is done -- the force is
perpendicular to the
> direction of motion.
> [...]
> then I have a question,as gravity and acceleration is
equivalent,when
you
> have a balloon (with light gas in it,i.e. the balloon
would go in the
> air when not being in the car)
> in your car and you go around the corner,we people are
pushed the other
> way than the corner goes,but what does the balloon do?he
should go the
> other way as we do? same as the ballon would do outside
the car(going
> against the gravity),
> it should do ,but I am not quite sure
> This is a simple experiment to do, just get a helium-filled
balloon and
> take it for a ride in your car. As expected, the helium
balloon goes to
> the front of the car when accelerating, to the rear when
braking, and
> toward the center of any turn. That's because the air is
more dense than
> the balloon, and is pushed harder than the baloon (all
those forces are
> proportional to the mass of the object, so there is more
force on the
> air in the same volumes as the baloon than on the balloon
itself).
> Tom Roberts tjroberts@lucent.com
--
Russ Lyttle
Not Powered by ActiveX
http://home.earthlink.net/~lyttlec/philosophy/logos.html
===
Subject: Re: Gyroscopes - Usenet Physics FAQ - Reference
frames
> then I have a question,as gravity and acceleration is
equivalent,when you
> have a balloon (with light gas in it,i.e. the balloon would
go in the
air
> when not being in the car)
> in your car and you go around the corner,we people are
pushed the other
> way than the corner goes,but what does the balloon do?he
should go the
> other way as we do? same as the ballon would do outside the
car
> (going against the gravity), it should do ,but I am not
quite sure
> marten
The balloon is another story, I guess.
You cannot spin a balloon, like a rock and string does.
The spinning air, that goes along with the balloon, will have
more inertia
then the balloon it self, so the balloon is pushed inside and
the air
outside.
Again this is a centrifugal problem, not a gyroscopic problem.
I would like very much to discuss the gyroscope... but the
gyroscope looks
to be beyond any possible Physical explanation. So far I got
about 3
objective and different explanations for the gyroscopic
antigravity like
behaviour (none here in this thread). All of the said
explanations are very
different and all of then have some new physical concepts
involved.
There's only one thing clear to me. Gyroscopes cannot be
explained by the
actual Physics based on a space-time geometry. So far I never
seen and
explanation based on pure space-time geometry, which clear
shows that
space-time geometry is a big balloon that will blow up one
day.
===
Subject: Re: Heron's theorem in 3D and up
> Is there a higher-dimensional analog of Heron's theorem?
 Heron's theorem:
> If the sides of a triangle are a,b,c, then the area of the
triangle is
> sqrt(s(s-a)(s-b)(s-c))
> where s is the semi-perimeter of the triangle:
> s = (a+b+c) / 2
 MQ
> Here is a simple formula for simplices of any dimension:
> http://mathworld.wolfram.com/Cayley-MengerDeterminant.html
That's a really nice result. And I learned a new thing -
Ôcontent' is
the generalisation of Ôvolume' to higher 
dimensions. I would
have been
happy with Ôhypervolume', but there you go.
--
Larry Lard
Replies to group please
===
Subject: Penrose's Road to Reality - how penetrable is the
maths for
non-maths
types?
My dad has just bought a copy of The Road to Reality,
Penrose's new book.
Looks
very interesting and I am getting a copy myself.
My question, however, is... how penetrable is the book to
non-maths types? I
know that the jacket text says the book expects no specialist
knowledge,
and
that the first part of the book is giving the reader some
mathematical
knowledge, but it seems a bit maths-mad later on in the book.
I wonder if
anyone can give any feedback on what non-maths oriented
people they know
make
of the book, how much is readable to them, etc.?
Amazon page on the book for any interested people:
http://makeashorterlink.com/?H2D1263F8
alex
===
Subject: Re: Penrose's Road to Reality - how penetrable is
the maths for
non-maths types?
I've just brought it myself.
Firstly, I appreciate you ask for non-maths type opinions.
Nevertheless,
this is almost the other side of the fence, so hopefully you
will get some
balance.
One of the book's virtues is that, in my opinion, it is
written for those
that know roughly what most of the maths concepts are, its a
good refresher
in a lot of mathematical physics subjects, but I wouldn't 
say
Ôno
specialist
knowledge' - you will need to read around a lot. It is
mathematically a lot
more detailed than The Emperor's New Mind. For example, a
particular
section
I launched directly to was 16.1 on Finite Fields. He throws
around terms
like ÔCommutative Division Ring', Equivalence 
Classes,
Project Spaces Over
Fields'. Not in themsleves too bad for a student number
theorist/algebraist,
but I wouldn't think the book is, in itself, enough to
comprehende the
subject, let alone understand why he wants to introduce them.
The section
is
short and terse and, if rusty in these subjects, you'll need
to have an
Algebraic text to hand. Then there is Lie Groups, Derivatives
- Penrose
almost goes beserk (joke) on P315, section 14.6, take a
shufty. Not sure he
really needs to go that far, nevertheless.
Concluding, I don't think, IN MY OPINION, it is accessible.
But if you are
prepared to do a lot of background work, it will be very
rewarding. I'm
loving it, but I do have a mathermatical physics degree! I
wouldn't want to
put anyone off who is interested, just let them know it
requires some good
old fashioned hard graft
Enjoy
> My dad has just bought a copy of The Road to Reality,
Penrose's new book.
Looks
> very interesting and I am getting a copy myself.
> My question, however, is... how penetrable is the book to
non-maths
types?
I
> know that the jacket text says the book expects no
specialist
knowledge,
and
> that the first part of the book is giving the reader some
mathematical
> knowledge, but it seems a bit maths-mad later on in the
book. I wonder if
> anyone can give any feedback on what non-maths oriented
people they know
make
> of the book, how much is readable to them, etc.?
> Amazon page on the book for any interested people:
> http://makeashorterlink.com/?H2D1263F8
> alex
===
Subject: Re: Penrose's Road to Reality - how penetrable is
the maths for
non-maths types?
> My dad has just bought a copy of The Road to Reality,
Penrose's new book.
> Looks very interesting and I am getting a copy myself.
> My question, however, is... how penetrable is the book to
non-maths
types?
I haven't seen this book, but in one of his earlier books
he mentions that someone told him that each equation halves
the audience,
so at least he is aware of the problem!
--
Timothy Murphy
e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie
tel: +353-86-2336090, +353-1-2842366
s-mail: School of Mathematics, Trinity College, Dublin 2,
Ireland
Cc: tim@birdsnest.maths.tcd.ie
===
Subject: Re: Penrose's Road to Reality - how penetrable is
the maths for
non-maths types?
> I haven't seen this book, but in one of his earlier books
> he mentions that someone told him that each equation halves
the audience,
> so at least he is aware of the problem!
This anecdote is usually told of Hawking:
After he pulled off, with his Brief History of Time, the
biggest
publishing coup since the Bible, Stephen Hawking revealed
that he
owed it all to a sage injunction from the publisher: no
equations,
for every equation would halve the sales. E = mc2 alone was
permitted.
===
Subject: Re: 3 Questions on Z[sqrt(n)]
> I am searching a bit easy material on the question whether
Z[sqrt(n)] is
> factorial (=admits essentially unique decomposition into
primes).
> In the books I consulted there are long discussions on the
> properties of the algebraic integers in a quadratic number
field but
> very little on the rings Z[sqrt(n)]. The two look somewhat
related but
> they do differ, don't they?
The long discussions I alluded to are lists of integers of
which it
is known that the ring of algebraic integers in Q(sqrt(n))
are or are
not UFDs, or even PIDs, or even Euclidean, or even Euclidean
with the
norm in that field. For instance, Peter 
Bundschuh's German
number
theory book contains three such lists with a total of about 30
squarefree n. These lists are informative enough for the
purpose for
which I asked, even though they leave some questions open.
So, the case remains where the ring of algebraic integers in
> If n is square-free and n = 1 mod 4
should that be if n is not square-free or n = 1 mod 4?
> then Z[sqrt(n)] is not integrally closed in its
quotient-field,
> in which case I don't think it could be a UFD.
Maybe that's trivial, but I do not yet see why.
Helmut Richter
===
Subject: Re: 3 Questions on Z[sqrt(n)]
> So, the case remains where the ring of algebraic integers in
> If n is square-free and n = 1 mod 4
> then Z[sqrt(n)] is not integrally closed in its
quotient-field,
> in which case I don't think it could be a UFD.
> Maybe that's trivial, but I do not yet see why.
I think the result is true as I stated it.
Suppose n is square-free and n = 1 mod 4.
What I am saying is that the algebraic integers in Q(sqrt(n))
consist of the numbers (a + b sqrt(n))/2,
where a,b are integers and a = b mod 2,
ie a,b are both odd or both even.
In other words, 1 and (1 + sqrt(n))/2 form a Z-basis
for these algebraic integers.
To see this, suppose x = u + v sqrt(n),
where u,v are in Q (the rationals).
Then x satisfies the equation (x-u)^2 = nv^2, ie
x^2 - 2ux + (u^2 - nv^2) = 0.
Assuming v is not 0, this is the minimal equation for x.
Hence x is an algebraic integer if and only if
2u and u^2 - nv^2 are integers.
Hence u = a/2, where a is an integer;
and then 4nv^2 is an integer.
Since n is square-free, this implies that 4v^2 is an integer,
ie v = b/2, where b is an integer.
Since u^2 - nv^2 is an integer,
a^2 - nb^2 = 0 mod 4.
The result follows since a^2,b^2 = 0 or 1 mod 4.
Alternatively, if u + v sqrt(n) is an algebraic integer,
then so is u - v sqrt(n),
since it satisfies the same polynomial equations over Q (or 
Z).
Hence so are 2u and 2v sqrt(n).
But if an algebraic integer is rational it is an ordinary
integer.
Hence 2u is an integer, ie u = a/2.
By the same argument, 4v^2n is an integer;
hence so is 4v^2 and so v = b/2.
...
--
Timothy Murphy
e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie
tel: +353-86-2336090, +353-1-2842366
s-mail: School of Mathematics, Trinity College, Dublin 2,
Ireland
===
Subject: Re: Computational complexity, number theory tidbits
>Oh, is it just a matter to make the shift of using
>[N/5] - [N/10] - [N/15] + [N/30] - 1 = [(N-15)/10] -
[(N-15)/30]
>for N>6, so that you don't have to have even N?
Why don't you check yourself???
Anyway,
[N/5] - [N/10] = [(N+5)/10]
[N/15] - [N/30] = [(N+15)/30]
[N/5] - [N/10] - [N/15] + [N/30] - 1 = [(N+5)/10] -
[(N+15)/30] - 1
[(N+5)/10] = [(N-15+20)/10] = [(N-15)/10] + 2
[(N+15)/30] = [(N-15+30)/30] = [(N-15)/30] + 1
[(N+5)/10] - [(N+15)/30] - 1 = [(N-15)/10]+2-([(N-15)/30]+1)-1
so,
[N/5] - [N/10] - [N/15] + [N/30] - 1 = [(N-15)/10] -
[(N-15)/30]
>Or are the offsets for that different?
>So what's the big deal? Well two years ago I was working on
>algorithms to count prime numbers, and I wanted them fast,
so I put in
>formulas that already knew that 2 is even, which as you can
see,
>reduced complexity, which meant faster algorithms.
>Can anyone out there figure out a way to reduce it still
further?
No, because that isn't possible. Just put the formula in a
graph and
you'll see why you need at least two ßoor-functions...
>Like is there a simpler, shorter formula than
>[(N-16)/10] - [(N-16)/30], with even N>6,
>for the count of odd composites that do not have 3 as a
factor that do
>have 5 as a factor?
>I think it's an interesting question for bragging rights. 
If
I'm
>finding the most compact formulas possible then that might
mean
>something, especially about the people claiming my research
is
>useless.
You cannot even prove something as simple as this (you don't
appear to
be able to anyway) which is only using the observation that:
[N/k] = [N/(2*k)] + [(N+k)/(2*k)]
What do you think that says about the usefulness of your
research?
(BTW the correct answer is nothing. You simply might be lucky
once and
find something useful, but the problem is that even if you do,
someone
else needs to come along to prove that it is indeed correct
and
useful, because your guesses can't be the foundation of
anything. In
other words, the results of your research are in no way
directly
usable.)
>If they are not lying and ARE experts then it seems
reasonable that
>they can produce shorter relations, with even less
computational
>complexity.
This is a weird statement. It is like saying that an expert
must be
able to produce a shorter expression for the relation 2+2=4,
and if
they can't they are lying.
===
Subject: Re: Computational complexity, number theory tidbits
>Oh, is it just a matter to make the shift of using
>[N/5] - [N/10] - [N/15] + [N/30] - 1 = [(N-15)/10] -
[(N-15)/30]
>for N>6, so that you don't have to have even N?
> Why don't you check yourself???
I wasn't that interested.
> Anyway,
> [N/5] - [N/10] = [(N+5)/10]
> [N/15] - [N/30] = [(N+15)/30]
> [N/5] - [N/10] - [N/15] + [N/30] - 1 = [(N+5)/10] -
[(N+15)/30] - 1
> [(N+5)/10] = [(N-15+20)/10] = [(N-15)/10] + 2
> [(N+15)/30] = [(N-15+30)/30] = [(N-15)/30] + 1
> [(N+5)/10] - [(N+15)/30] - 1 =
[(N-15)/10]+2-([(N-15)/30]+1)-1
> so,
> [N/5] - [N/10] - [N/15] + [N/30] - 1 = [(N-15)/10] -
[(N-15)/30]
>Or are the offsets for that different?
>So what's the big deal? Well two years ago I was working on
>algorithms to count prime numbers, and I wanted them fast,
so I put in
>formulas that already knew that 2 is even, which as you can
see,
>reduced complexity, which meant faster algorithms.
>Can anyone out there figure out a way to reduce it still
further?
> No, because that isn't possible. Just put the formula in a
graph and
> you'll see why you need at least two ßoor-functions...
That's kind of interesting, but it's not 
surprising to me.
You see, I used my prime counting function to derive it.
>Like is there a simpler, shorter formula than
>[(N-16)/10] - [(N-16)/30], with even N>6,
>for the count of odd composites that do not have 3 as a
factor that do
>have 5 as a factor?
>I think it's an interesting question for bragging rights. 
If
I'm
>finding the most compact formulas possible then that might
mean
>something, especially about the people claiming my research
is
>useless.
> You cannot even prove something as simple as this (you
don't appear to
> be able to anyway) which is only using the observation that:
I'm not THAT interested. Besides I know how I derived the
formula,
which I did over two years ago, so there's even more reason
to be
bored on the subject.
I used [(N-4)/6] with my prime counting function. I figured
it'd give
the most compact result possible.
> [N/k] = [N/(2*k)] + [(N+k)/(2*k)]
> What do you think that says about the usefulness of your
research?
> (BTW the correct answer is nothing. You simply might be
lucky once and
> find something useful, but the problem is that even if you
do, someone
> else needs to come along to prove that it is indeed correct
and
> useful, because your guesses can't be the foundation of
anything. In
> other words, the results of your research are in no way
directly
> usable.)
Um, you're just some poster on sci.math who 
doesn't even give
his or
her name, so what makes you think I care about your asides?
If you post math, fine. Your opinions aren't of 
interest.
>If they are not lying and ARE experts then it seems
reasonable that
>they can produce shorter relations, with even less
computational
>complexity.
> This is a weird statement. It is like saying that an expert
must be
> able to produce a shorter expression for the relation
2+2=4, and if
> they can't they are lying.
I used my prime counting function, which allows me to do
things like
derive that formula from [(N-4)/6], while now I wonder what
mathematical tools you have available which are of use to
you, so I
have a simple challenge for you.
Give the formula that counts odd composites not divisible by
3 and not
divisible by 5 that are divisible by 7.
Can you manage a little more math, or are you done?
James Harris
http://mathforprofit.blogspot.com/
===
Subject: Re: Computational complexity, number theory tidbits
> What do you think that says about the usefulness of your
research?
> (BTW the correct answer is nothing. You simply might be
lucky once and
> find something useful, but the problem is that even if you
do, someone
> else needs to come along to prove that it is indeed
correct and
> useful, because your guesses can't be the foundation of
anything. In
> other words, the results of your research are in no way
directly
> usable.)
>Um, you're just some poster on sci.math who 
doesn't even
give his or
>her name, so what makes you think I care about your asides?
You didn't even consider the possibility that Way is my last
name and
that N.O. are my initials, now did you?
BTW you started about the usefulness of your research, I
merely
replied to what was stated.
>I used my prime counting function, which allows me to do
things like
>derive that formula from [(N-4)/6], while now I wonder what
>mathematical tools you have available which are of use to
you, so I
>have a simple challenge for you.
>Give the formula that counts odd composites not divisible by
3 and not
>divisible by 5 that are divisible by 7.
I can't give THE formula. I can give A formula:
[(N-7)/14] - [(N+21)/42] - [(N+35)/70] + [(N+105)/210]
>Can you manage a little more math, or are you done?
It appears I can manage...
===
Subject: Re: Computational complexity, number theory tidbits
> What do you think that says about the usefulness of your
research?
> (BTW the correct answer is nothing. You simply might be
lucky once and
> find something useful, but the problem is that even if you
do, someone
> else needs to come along to prove that it is indeed
correct and
> useful, because your guesses can't be the foundation of
anything. In
> other words, the results of your research are in no way
directly
> usable.)
>Um, you're just some poster on sci.math who 
doesn't even
give his or
>her name, so what makes you think I care about your asides?
> You didn't even consider the possibility that Way is my
last name and
> that N.O. are my initials, now did you?
Yeah right. All that's relevant is that you're 
not using your
actual
name, which means your opinions are irrelevant.
Now your math though is actually interesting.
> BTW you started about the usefulness of your research, I
merely
> replied to what was stated.
Whatever...
>I used my prime counting function, which allows me to do
things like
>derive that formula from [(N-4)/6], while now I wonder what
>mathematical tools you have available which are of use to
you, so I
>have a simple challenge for you.
>Give the formula that counts odd composites not divisible by
3 and not
>divisible by 5 that are divisible by 7.
> I can't give THE formula. I can give A formula:
> [(N-7)/14] - [(N+21)/42] - [(N+35)/70] + [(N+105)/210]
>Can you manage a little more math, or are you done?
> It appears I can manage...
Neat. It looks like there's a pattern.
It looks like inclusion-exclusion with an easy to determine
offset.
Can you prove that the pattern holds across all odd primes?
Given that Legendre's Method, which also uses
inclusion-exclusion, is
far less efficient, can those terms be compressed in some way
along
the lines of what has been used with Legendre's?
And hey, I'll admit it, I didn't know that 
pattern was there,
and I'm
finally a bit curious here.
What, did you find it in a textbook or something?
James Harris
http://mathforprofit.blogspot.com/
===
Subject: Re: Computational complexity, number theory tidbits
>Oh, is it just a matter to make the shift of using
>[N/5] - [N/10] - [N/15] + [N/30] - 1 = [(N-15)/10] -
[(N-15)/30]
>for N>6, so that you don't have to have even N?
> Why don't you check yourself???
> Anyway,
> [N/5] - [N/10] = [(N+5)/10]
> [N/15] - [N/30] = [(N+15)/30]
> [N/5] - [N/10] - [N/15] + [N/30] - 1 = [(N+5)/10] -
[(N+15)/30] - 1
> [(N+5)/10] = [(N-15+20)/10] = [(N-15)/10] + 2
> [(N+15)/30] = [(N-15+30)/30] = [(N-15)/30] + 1
> [(N+5)/10] - [(N+15)/30] - 1 =
[(N-15)/10]+2-([(N-15)/30]+1)-1
> so,
> [N/5] - [N/10] - [N/15] + [N/30] - 1 = [(N-15)/10] -
[(N-15)/30]
...
>Can anyone out there figure out a way to reduce it still
further?
> No, because that isn't possible. Just put the formula in a
graph and
> you'll see why you need at least two ßoor-functions...
I agree I think you need two ßoors but you can remove
something:
[(N+5)/10] - [(N+15)/30] - 1 = [2 [(N-5)/10]/3]
Proof: look at the similar
[(N+1)/2] - [(N+3)/6] - 1 = [2 [(N-1)/2]/3]
(this reduces the # of cases)
check for N = 6n+0,1,2,3,4,5
LHS RHS
6n+0: 2n-1,2n-1
6n+1: 2n, 2n
6n+2: 2n, 2n
6n+3: 2n, 2n
6n+4: 2n, 2n
6n+5: 2n+1,2n+1
Simpler but still 2 ßoors. How would a proof that you
-cannot- do it in 1 ßoor look?
--
Mitch Harris
(remove q to reply)
===
Subject: Re: Computational complexity, number theory tidbits
Oh, is it just a matter to make the shift of using
[N/5] - [N/10] - [N/15] + [N/30] - 1 = [(N-15)/10] -
[(N-15)/30]
for N>6, so that you don't have to have even N?
 Why don't you check yourself???
 Anyway,
 [N/5] - [N/10] = [(N+5)/10]
> [N/15] - [N/30] = [(N+15)/30]
 [N/5] - [N/10] - [N/15] + [N/30] - 1 = [(N+5)/10] -
[(N+15)/30] - 1
 [(N+5)/10] = [(N-15+20)/10] = [(N-15)/10] + 2
> [(N+15)/30] = [(N-15+30)/30] = [(N-15)/30] + 1
 [(N+5)/10] - [(N+15)/30] - 1 =
[(N-15)/10]+2-([(N-15)/30]+1)-1
 so,
 [N/5] - [N/10] - [N/15] + [N/30] - 1 = [(N-15)/10] -
[(N-15)/30]
 ...
>Can anyone out there figure out a way to reduce it still
further?
 No, because that isn't possible. Just put the formula in a
graph and
> you'll see why you need at least two ßoor-functions...
> I agree I think you need two ßoors but you can remove
> something:
> [(N+5)/10] - [(N+15)/30] - 1 = [2 [(N-5)/10]/3]
When I was deriving these formulas over two years ago, for
some reason
at the time I tried to get as few terms as possible, so I
rolled an
integer back into the ßoors.
However, though I did so again with the next in the series,
for 7, I
then thought better of it, and rolled it back out, and 
here's
what you
get for 7:
[(N-8)/14] - [(N-22)/42] - [(N-106)/70] + [(N-106)/210] - 2
with even N>36
where not surprisingly that's the count of odd composites 
not
divisible by 3 or 5 that are divisible by 7.
I find it interesting that you noticed I rolled an integer in
with the
other expression.
> Proof: look at the similar
> [(N+1)/2] - [(N+3)/6] - 1 = [2 [(N-1)/2]/3]
> (this reduces the # of cases)
> check for N = 6n+0,1,2,3,4,5
> LHS RHS
> 6n+0: 2n-1,2n-1
> 6n+1: 2n, 2n
> 6n+2: 2n, 2n
> 6n+3: 2n, 2n
> 6n+4: 2n, 2n
> 6n+5: 2n+1,2n+1
> Simpler but still 2 ßoors. How would a proof that you
> -cannot- do it in 1 ßoor look?
That poster said something about graphing a step function or
something.
I noticed that said posters subsequent reply to you was kind
of coy as
mention was made of Fourier series.
Maybe someone else is willing to give details?
Oh, and can anyone check to see if
[(N-8)/14] - [(N-22)/42] - [(N-106)/70] + [(N-106)/210] - 2
with even N>36
is the smallest possible for the count of odd composites not
divisible
by 3 or 5 that are divisible by 7?
Of course, I say it is the smallest (besides rolling that -2
back in)
which is kind of out now as a general position of mine that
these
expressions are fundamental in terms of smallest computational
complexity for what they do.
James Harris
http://mathforprofit.blogspot.com/
===
Subject: Re: Computational complexity, number theory tidbits
> When I was deriving these formulas over two years ago, for
some reason
> at the time I tried to get as few terms as possible, so I
rolled an
> integer back into the ßoors.
> However, though I did so again with the next in the series,
for 7, I
> then thought better of it, and rolled it back out, and
here's what you
> get for 7:
> [(N-8)/14] - [(N-22)/42] - [(N-106)/70] + [(N-106)/210] - 2
> with even N>36
> where not surprisingly that's the count of odd composites
not
> divisible by 3 or 5 that are divisible by 7.
Oh? This is, in fact, *very surprising* -- especially since
it is false. If
you had bothered to checkthe
work you posted, you would find that the above equation does
not perform as
advertised. On the other
hand, the equation:
[(N-7)/14] - [N+21)/42] - [(N+35)/70] + [(N+105)/210]
posted by No Way works just fine. Too bad you 
aren't
interested in his
opinions.
> Oh, and can anyone check to see if
> [(N-8)/14] - [(N-22)/42] - [(N-106)/70] + [(N-106)/210] - 2
> with even N>36
> is the smallest possible for the count of odd composites
not divisible
> by 3 or 5 that are divisible by 7?
It doesn't even work at all!! No bragging rights, no cigar.
> James Often in error, but never in doubt. Harris
--
There are two things you must never attempt to prove: the
unprovable -- and
the obvious.
--
Democracy: The triumph of popularity over principle.
--
http://www.crbond.com
===
Subject: Re: Computational complexity, number theory tidbits
> However, though I did so again with the next in the series,
for 7, I
> then thought better of it, and rolled it back out, and
here's what you
> get for 7:
 [(N-8)/14] - [(N-22)/42] - [(N-106)/70] + [(N-106)/210] - 2
 with even N>36
 where not surprisingly that's the count of odd composites
not
> divisible by 3 or 5 that are divisible by 7.
> Oh? This is, in fact, *very surprising* -- especially since
it is false.
If
> you had bothered to check the work you posted, you would
find that the
> above equation does not perform as advertised. On the other
hand, the
> equation:
> [(N-7)/14] - [N+21)/42] - [(N+35)/70] + [(N+105)/210]
> posted by No Way works just fine. Too bad you 
aren't
interested in
his
> opinions.
Actually both work as advertised. There is an off-by-one
difference.
James calculates the numbers less than N, No Way calculates
the numbers
less than or equal to N. If we adjust No Way's formula to
what we need
(less than N), we get (substitue N-1 for N):
[(N-8)/14] - [(N+20)/42] - [(N+34)/70] + [(N+104)/210].
It is easy to verify that the two are equivalent. (It may be
noted that
both do not work for many odd numbers, but both *do* work for
even numbers
larger than, or equal to, 8.
> Oh, and can anyone check to see if
 [(N-8)/14] - [(N-22)/42] - [(N-106)/70] + [(N-106)/210] - 2
 with even N>36
 is the smallest possible for the count of odd composites
not divisible
> by 3 or 5 that are divisible by 7?
> It doesn't even work at all!! No bragging rights, no 
cigar.
> James Often in error, but never in doubt. Harris
> --
> There are two things you must never attempt to prove: the
unprovable --
and the obvious.
> --
> Democracy: The triumph of popularity over principle.
> --
> http://www.crbond.com
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam,
nederland,
+31205924131
home: bovenover 215, 1025 jn amsterdam, nederland;
http://www.cwi.nl/~dik/
===
Subject: Re: Computational complexity, number theory tidbits
Originator: harris@tcs.inf.tu-dresden.de (Mitchell Harris)
> I agree I think you need two ßoors but you can remove
> something:
> [(N+5)/10] - [(N+15)/30] - 1 = [2 [(N-5)/10]/3]
...
> Proof: look at the similar
> [(N+1)/2] - [(N+3)/6] - 1 = [2 [(N-1)/2]/3]
> (this reduces the # of cases)
> check for N = 6n+0,1,2,3,4,5
> LHS RHS
> 6n+0: 2n-1,2n-1
> 6n+1: 2n, 2n
> 6n+2: 2n, 2n
> 6n+3: 2n, 2n
> 6n+4: 2n, 2n
> 6n+5: 2n+1,2n+1
> Simpler but still 2 ßoors. How would a proof that you
> -cannot- do it in 1 ßoor look?
>That poster said something about graphing a step function or
>something.
pictures are nice, and are often intuitively convincing. for
myself,
the graph did not convince me so I had to do the symbolic
work.
>I noticed that said posters subsequent reply to you was kind
of coy as
>mention was made of Fourier series.
>Maybe someone else is willing to give details?
Coy, ironic, knowing. (I think his point was that) a function
has all
sorts of representations. One way would be to have the
function's
domain include the reals (actually even the complexes). For
example,
[n/2] = n/2 - (1-(-1)^n)/2 which works over the complexes,
too. and
the latter representation doesn't use any ßoors. So maybe 
one
might
consider the integer ßoor operation expensive but
exponentiation and
real division cheap.
Mitch
===
Subject: Re: Computational complexity, number theory tidbits
>Simpler but still 2 ßoors. How would a proof that you
>-cannot- do it in 1 ßoor look?
Uhm... who said it couldn't be done with less than two 
ßoosr?
Ah,
crap! that was me... You could do the following:
f(N) = [(N+5)/10] - [(N+15)/30] - 1
g(N) = f(N) - (2*N/30)
g(N) is a periodic Ôsignal'. It can be described 
by a Fourier
series.
So g(N) has no ßoor function in it.
since g(N) can be written without a ßoor function, f(N) can be
written without a ßoor function as well. However, I 
wouldn't
call
this new form simpler.
===
Subject: Re: Computational complexity, number theory tidbits
[...]
>Can anyone out there figure out a way to reduce it still
further?
Eratosthenes of Cyrene (276-194 BC) could.
===
Subject: Re: Computational complexity, number theory tidbits
confuse everyone with this message:
> So finally some people came forward with some comments about
> derivation of ßoor((N-4)/6) which as has been noted works
for even
> N>2, while there also exists (and I admit I didn't know 
it)
> ßoor((N-3)/6) which works for N>2, as both give the count
of odd
> composites that have 3 as a factor.
> [x] = ßoor(x)
> as it's easier than repeatedly typing in ßoor.
>The usual convention would be
.89x.89 though
this requires a
>Unicode-enabled Newsreader.
In Russia we always use [x] for integer part aka ßoor. There
is no
such thing as ceiling though...
--
|Here you can see, how awful I'm at Quake speedrunning|
Grue(at)mail.ru|
|E1M2 :36 E1M6 :11 E2M3 :27 E4M1 :30 E4M6
:24|->grue3.tripod.com<--|
===
Subject: Re: Computational complexity, number theory tidbits
> confuse everyone with this message:
> So finally some people came forward with some comments about
> derivation of ßoor((N-4)/6) which as has been noted works
for even
> N>2, while there also exists (and I admit I didn't know 
it)
> ßoor((N-3)/6) which works for N>2, as both give the count
of odd
> composites that have 3 as a factor.

> [x] = ßoor(x)
 as it's easier than repeatedly typing in ßoor.
>The usual convention would be .89??x.89?? though this
requires a
>Unicode-enabled Newsreader.
> In Russia we always use [x] for integer part aka ßoor.
There is no
> such thing as ceiling though...
Ok, so I figure I can use [x] for ßoor(x), which is good 
as
that's a
lot easier.
So, the count of odd *composites* that do not have 3 as a
factor that
do have 5 as a factor is given by
[(N-16)/10] - [(N-16)/30] where you need even N>6
though I guess that my use of [(N-4)/6] for the count of odd
composites with 3 as a factor versus using [(N-3)/6] is the
reason the
formula only works for even N, still it does work, and
represents--using even N--a reduction in computational
complexity from
the formula using the inclusion-exclusion method, as
[N/5] - [N/10] - [N/15] + [N/30] - 1 = [(N-16)/10] -
[(N-16)/30]
with even N>6
and I'm curious about whether or not a shorter formula that
counts odd
composites that don't have 3 as a factor than
[(N-16)/10] - [(N-16)/30] with even N>6
can be found.
Here I think I managed the compression by using the
information that 2
is prime, which I think works because any odd can be
represented by
2k+1 where k is natural.
Here's how I derived [(N-4)/6], as given an *even* N, so I
know N
itself is not to be counted, the count of *composites* up to
N is
given by
3C <= N-1, where C is that count,
with the requirement that C is greater than 1 to keep out 3
itself
from the count.
Well then each odd in that count can be represented as 2k+1,
where k
is a counting number, and the largest odd that fits is 2k_max
+ 1, so
3(2k_max + 1)<=N-1, so
6k_max + 3 <= N-1, so k_max <= (N-4)/3, which is k_max =
[(N-4)/6]
and k_max is also the count of those composites.
So if I hadn't been using the requirement that N is even, 
and
deliberately skipped N itself, I would have used
3(2k_max + 1)<=N, and gotten [(N-3)/6]
but since it works with N even, it doesn't really matter a
lot from
what I've seen.
But that's the direct derivation which I recently 
remembered.
My thinking for some time, as I found these relations over
two years
ago, is that you get just the one compression, as the other
primes
aren't as easily handled as 2, but if someone can post a 
more
compact
formula that might mean there is more compression, and
reduction of
computational complexity possible.
I'm curious to know if there is some other compression
possible.
James Harris
http://mathforprofit.blogspot.com/
===
Subject: Re: Computational complexity, number theory tidbits
> [...]
>Look, it is trivial that
>[...]
>Similarly, it is trivial that
>[...]
>So it is quite trivial that
so your point is that james should write another paper
on his latest discovery, eh?
i mean really, a discovery that's so basic that anyone
with half a brain can prove it in a few minutes must be
a lot more important than that stuff you see in journals
where you need a friggin degree just to be able to -read-
the thing!
[which of course is not intended to say you have half
a brain...]
************************
David C. Ullrich
sorry about the inelegant formatting - typing
one-handed for a few weeks...
===
Subject: Re: Computational complexity, number theory tidbits
>confuse everyone with this message:
> So finally some people came forward with some comments
about
> derivation of ßoor((N-4)/6) which as has been noted works
for even
> N>2, while there also exists (and I admit I didn't know
it)
> ßoor((N-3)/6) which works for N>2, as both give the count
of odd
> composites that have 3 as a factor.

> [x] = ßoor(x)
 as it's easier than repeatedly typing in ßoor.
>The usual convention would be
.89x.89 though
this requires a
>Unicode-enabled Newsreader.
>In Russia we always use [x] for integer part aka ßoor. There
is no
>such thing as ceiling though...
that's funny, heh-heh: no ceilings in Russia. like the 
sky's
the limit...
************************
David C. Ullrich
sorry about the inelegant formatting - typing
one-handed for a few weeks...
===
Subject: F(x) as an F-vector space
In a recent thread over in <alt.math.undergrad>, the question
came up as to a basis for the field of rational functions F(x)
as a vector space over the field F.
I managed to convince myself that one such basis is {1, x,
x^2,
...} U {x^m/p(x)^n | p(x) a monic irreducible polynomial in
F[x], 0 <= m < deg(p(x)), 0 < n}, where the first set in the
union is of course a basis for the polynomial ring F[x].
The one thing that gives me pause is that this basis for F(x)
is
uncountable if F is uncountable, while the basis for F[x] is
obviously always countable. I wouldn't have expected that 
the
ring and its quotient field could have dimensions of different
infinite cardinalities. Am I worrying needlessly? Or worse,
have
I screwed up the basis for F(x)?
--
Jim Heckman
===
Subject: Re: F(x) as an F-vector space
> In a recent thread over in <alt.math.undergrad>, the
question
> came up as to a basis for the field of rational functions
F(x)
> as a vector space over the field F.
> I managed to convince myself that one such basis is {1, x,
x^2,
> ...} U {x^m/p(x)^n | p(x) a monic irreducible polynomial in
> F[x], 0 <= m < deg(p(x)), 0 < n}, where the first set in the
> union is of course a basis for the polynomial ring F[x].
So... you want the proof that partial fractions works
> The one thing that gives me pause is that this basis for
F(x) is
> uncountable if F is uncountable, while the basis for F[x] is
> obviously always countable. I wouldn't have expected that
the
> ring and its quotient field could have dimensions of
different
> infinite cardinalities. Am I worrying needlessly?
Are these functions linearly independent over the reals...
1/(x-a)
where a ranges over the reals? If so, there is no countable
basis
for R(x) over R.
> Or worse, have
> I screwed up the basis for F(x)?
--
G. A. Edgar
http://www.math.ohio-state.edu/~edgar/
===
Subject: Re: F(x) as an F-vector space
> In a recent thread over in <alt.math.undergrad>, the
question
> came up as to a basis for the field of rational functions
F(x)
> as a vector space over the field F.
> I managed to convince myself that one such basis is {1, x,
x^2,
> ...} U {x^m/p(x)^n | p(x) a monic irreducible polynomial in
> F[x], 0 <= m < deg(p(x)), 0 < n}, where the first set in the
> union is of course a basis for the polynomial ring F[x].
>So... you want the proof that partial fractions works
> The one thing that gives me pause is that this basis for
F(x) is
> uncountable if F is uncountable, while the basis for F[x]
is
> obviously always countable. I wouldn't have expected that
the
> ring and its quotient field could have dimensions of
different
> infinite cardinalities. Am I worrying needlessly?
>Are these functions linearly independent over the reals...
>1/(x-a)
>where a ranges over the reals?
i imagine that was a rhetorical question...
>If so, there is no countable basis
>for R(x) over R.
and hence R[x] and R(x) have different dimensions, although
they certainly have the same cardinality. [which i find
interesting because it's a simple example showing that
what i thought for 20 years was a proof that 
Ôdimension'
is well-defined is wrong. [realized some time ago that
the proof was wrong, didn't know such a simple 
counterexample
til today.]]
> Or worse, have
> I screwed up the basis for F(x)?
************************
David C. Ullrich
sorry about the inelegant formatting - typing
one-handed for a few weeks...
===
Subject: This Week's Thoughts on Fermat's Last 
Theorem: 3
Here's a history of Fermat's last theorem in 
Usenet:
n+biquadrates
Search Result 1
===
Subject: Fermat's Last Theorem Talk
Original Format
Now, this is post--Wiles's proof.
It would be interesting to find the earliest statement of
Fermat's Last
Theorem, excuse me, I mean The Problem, in Usenet in any
archive available
internet, and post a reference in this thread.
We're looking for the earliest one including Case One and
Case Two,
coprimality, and odd prime n, and the reasons for these
conditions. Of
course
such a post will just reiterate what is known. The quest is
to find the
first
one in Usenet.
Won't this be fun?
Should a judgment be called for, the basis for completeness
will be The
Problem in _Fermat's Last Theorem for Amateurs_ by Paulo
Ribenboim,
published by Springer-Verlag New York, Inc., and copyrighted
1999. Judgment
will be by highest vote. You may vote in next week's This
Week's Thoughts
on
Fermat's Last Theorem: 4 beginning 0600 Eastern Daylight
Time, August
9th,
votes and post timely.
If a summary of candidate statement posts exists in Usenet,
you may bring it
to
my attention and I will cancel this call.
To express my love of The Problem, I am looking at (a^n mod c
+ b^n mod c) =
c,
a sufficient condition for a^n + b^n = k*c and a necessary
condition for a^n
+
b^n = c^n. The period of c^n mod c is 1 and the value v is
always 0. The
period
P(a,c) with a and c prime is related to c-1 since the value 0
is forbidden.
I
think this period with a and c distinct primes is c-1, the
maximum
repeating
pseudorandom number generator of this type. With a and c
coprime, but
composite, P(a,c) is often less than c-1. Each period is
related to a
sequence,
a subset of the numbers 1 to c-1.
Starting with a^0 = 1, 0 < n, A.n = A.(n-1) * a (mod c).
Let's say c = 21
and a
= 5, and suppose that P(a,c) = 4. Starting with any number in
this sequence
as
A.0 provides the same sequence in the same order.
Now, what happens when we start with positive numbers not
included in the
sequence but less than c?
I hypothesize that c-1 / P sequences obtain, each with length
P, and am
doing
exhaustive searches, formatting the results, and looking for
patterns to
verify
this to limits like c = 21 or 41 or 100. I like patterns and
just want to
understand The Problem more. My goal in life is to implement a
self-reproducing
universal machine shop composed of self-reproducing machine
tools, and FLT
is
not related. However, that problem is also discrete. Proving
FLT is not my
object. I just like patterns.
Yours,
Doug Goncz ( ftp://users.aol.com/DGoncz/ )
Student member SAE for one year.
I love: Dona, Jeff, Kim, Mom, Neelix, Tasha, and Teri,
alphabetically.
I drive: A double-step Thunderbolt with 657% range.
===
Subject: Re: More math texts online
> here is a URL with online math texts that I somehow missed
in
> my first search:
> http://www.gotmath.com/notes.html
Hi Van Jacques
I can't find the original thread which it would 
appear you are
referring to,
can you help me find it by providing the subject line?
I'm not sure if you've seen (or listed) this 
page, but it has
loads of links
to
texts online or for downloading:
http://www.geocities.com/alex_stef/mylist.html
alex
===
Subject: Re: More math texts online
> here is a URL with online math texts that I somehow missed
in
> my first search:
> http://www.gotmath.com/notes.html
> Hi Van Jacques
> I can't find the original thread which it 
would appear you
are
referring to,
> can you help me find it by providing the subject line?
> I'm not sure if you've seen (or listed) this 
page, but it
has loads
of links to
> texts online or for downloading:
> http://www.geocities.com/alex_stef/mylist.html
> alex
This is the thread. My message (quoted in yours), was the
only one in
for posting it.
Van
===
Subject: Re: More math texts online
>here is a URL with online math texts that I somehow missed
in
>my first search:
>http://www.gotmath.com/notes.html
>Hi Van Jacques
>I can't find the original thread which it would 
appear you
are
> referring to,
>can you help me find it by providing the subject line?
>I'm not sure if you've seen (or listed) this 
page, but it
has loads
> of links to
>texts online or for downloading:
>http://www.geocities.com/alex_stef/mylist.html
>alex
> This is the thread. My message (quoted in yours), was the
only one in
> for posting it.
> Van
Oh, I see.
alex
===
Subject: Re: Identity question.
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i72BdxF01564;
>Is this identity correct for these 2 algebraic numbers?
>-a = 2/10^(1/2)-4 = -2.387425886722..
>Not even close. Oh, you mean 2/(10^(1/2)-4). Please learn
>about the precedence rules for operations. When in doubt, use
>parentheses.
I checked and it was right in the reverse symbolic calc.
table,
I did leave out the outer parentheses.
Sorry for the confusion.
Dan
> b = 1/9 + 1/9 *10^(1/2) = .4624752955742..
>Then --
>b = (((-x*-1*2+2)^2 -2) /4) - (-x*-a)
>What's with all these extra - signs? Surely you can
simplify...
>Robert Israel israel@math.ubc.ca
>Department of Mathematics <a
href=http://www.math.ubc.ca/~israel>http://www.math.ubc.ca/~
israel</aUniversity of British Columbia
>Vancouver, BC, Canada V6T 1Z2
===
Subject: Re: Identity question correction.
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i72Be0N01612;
>Is this identity correct for these 2 algebraic numbers?
>-a = 2/10^(1/2)-4 = -2.387425886722..
>Not even close. Oh, you mean 2/(10^(1/2)-4). Please learn
>about the precedence rules for operations. When in doubt, use
>parentheses.
> b = 1/9 + 1/9 *10^(1/2) = .4624752955742..
>Then --
>b = (((-x*-1*2+2)^2 -2) /4) - (-x*-a)
>What's with all these extra - signs? Surely you can
simplify...
>Robert Israel israel@math.ubc.ca
>Department of Mathematics <a
href=http://www.math.ubc.ca/~israel>http://www.math.ubc.ca/~
israel</aUniversity of British Columbia
>Vancouver, BC, Canada V6T 1Z2
A correction to my original post!
a = (2/(10^(1/2)-4))*-1
b = 1/9 + 1/9 *10^(1/2)
Then --
b = (((x*2)+2)^2 -2)/4 - (x*a)
Where x = (a/2)-1
This I hope is right?
Dan
Also another --
b = a/(sqrt(10) + 2)
Ps. The (a) value in the reverse symbolic calcualtor table is
negative, but I made it positive here to avoid confusion.
===
Subject: Re: Identity question correction.
> A correction to my original post!
> a = (2/(10^(1/2)-4))*-1
> b = 1/9 + 1/9 *10^(1/2)
> Then --
> b = (((x*2)+2)^2 -2)/4 - (x*a)
> Where x = (a/2)-1
> This I hope is right?
> Dan
> Also another --
> b = a/(sqrt(10) + 2)
> Ps. The (a) value in the reverse symbolic calcualtor table
is
> negative, but I made it positive here to avoid confusion.
Maple says:
> a := (2/(10^(1/2)-4))*(-1);
> b := 1/9 + 1/9 *10^(1/2);
> x := (a/2)-1;
2
a := - -----------
(1/2)
10 - 4
1 1 (1/2)
b := - + - 10
9 9
1
x := - ----------- - 1
(1/2)
10 - 4
> is(b = (((x*2)+2)^2 -2)/4 - (x*a));
true
===
Subject: Re: Identity question.
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i72Be1K01663;
>Is this identity correct for these 2 algebraic numbers?
>-a = 2/10^(1/2)-4 = -2.387425886722..
>Not even close. Oh, you mean 2/(10^(1/2)-4). Please learn
>about the precedence rules for operations. When in doubt, use
>parentheses.
Sorry about that Robert, but I coppied it from the reverse
symbolic
calcualtor table for that particular number. As I did with
the other
(b) value.
> b = 1/9 + 1/9 *10^(1/2) = .4624752955742..
>Then --
>b = (((-x*-1*2+2)^2 -2) /4) - (-x*-a)
>What's with all these extra - signs? Surely you can
simplify...
Yes where both are negative they can be xa or the other two,
(-a*-1) and -x*-1 I was just changing -a to +a and -x to +x.
Dan
>Robert Israel israel@math.ubc.ca
>Department of Mathematics <a
href=http://www.math.ubc.ca/~israel>http://www.math.ubc.ca/~
israel</aUniversity of British Columbia
>Vancouver, BC, Canada V6T 1Z2
===
Subject: Re: Identity question.
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i72Be3101862;
>Not so far as I can tell, but the incredibly strange way
that you've
>written your equations suggests maybe you mean something
else. Why all
>the things of the form (-x*-a) instead of just ax?
>Sorry for the confusion, in this instance I should have
written just
>ax but either way it works. I Just wanted to show that 2
like signs
>(-) multiplied = (+)value.
>Dan
You wanted to SHOW that? Were you under the impression that
we didn't
already know it?
===
Subject: How to calculate a useful size for a given pool
Scenario:
1. A firm receives 100,000 orders per week as electronic
documents.
2. All documents are archived on CDROM.
3. A percentage of orders (average around 10%) will go wrong
at some
stage and access to the original order will be required by
Customer
Services to resolve the problem.
4. Retrieval from CDROM is time-consuming but a limited
amount of space is
available to cache some of the documents on networked storage
for more
immediate retrieval. There is not enough space to cache all
the documents
so the older cached documents are deleted regularly.
5. It is not possible to determine up front which orders will
fail -
whilst the failure rate is fairly constant over time, just
about any
document could end up being requested by Customer Services.
Problem:
The firm would like to cache N documents (from the weekly pool
of
100,000 orders) such that they can satisfy X percent of
Customer
Service
requests from the networked storage. Customer Services are
prepared to
put up with [100-X]% of requests that would still need to be
retrieved
from the CDROM store.
In short, how do we calculate N?
This reminds me a little of the Cookie Jar or Sock Drawer
problem,
but it bugs me that I can't puzzle it out. I'm 
not a hard-core
statistician and would be grateful for any help. This is a
genuine (i.e.
non-homework) request, by the way. I've posted to other NGs
with no luck
so far.
--
===
Subject: Re: How to calculate a useful size for a given pool
X-RFC2646:  Original
> Scenario:
> 1. A firm receives 100,000 orders per week as electronic
documents.
> 2. All documents are archived on CDROM.
> 3. A percentage of orders (average around 10%) will go
wrong at some
> stage and access to the original order will be required by
Customer
> Services to resolve the problem.
> 4. Retrieval from CDROM is time-consuming but a limited
amount of space
is
> available to cache some of the documents on networked
storage for more
> immediate retrieval. There is not enough space to cache all
the
documents
> so the older cached documents are deleted regularly.
> 5. It is not possible to determine up front which orders
will fail -
> whilst the failure rate is fairly constant over time, just
about any
> document could end up being requested by Customer Services.
> Problem:
> The firm would like to cache N documents (from the weekly
pool of
> 100,000 orders) such that they can satisfy X percent of
Customer
Service
> requests from the networked storage. Customer Services are
prepared to
> put up with [100-X]% of requests that would still need to
be retrieved
> from the CDROM store.
> In short, how do we calculate N?
> This reminds me a little of the Cookie Jar or Sock Drawer
problem,
> but it bugs me that I can't puzzle it out. 
I'm not a
hard-core
> statistician and would be grateful for any help. This is a
genuine (i.e.
> non-homework) request, by the way. I've posted to other 
NGs
with no luck
> so far.
#4 mostly likely doesn't hold true anymore.
Hardrives have increased in capacity making caching all the
data easy. A
single 250gb drive caches 384 CDs worth at 650mb per CD. A
fully
decked out $10,999 Apple XServe RAID would cache 5384 CDs.
Thats 35/mb of electronic documents per order if 100,000 are
stored on the XServer at any one time.
===
Subject: Re: How to calculate a useful size for a given pool
> 1. A firm receives 100,000 orders per week as electronic
documents.
<snip 4. Retrieval from CDROM is time-consuming but a limited
amount of space
is
> available to cache some of the documents on networked
storage for more
> immediate retrieval. There is not enough space to cache
all the
documents
> so the older cached documents are deleted regularly.
<snip #4 mostly likely doesn't hold true anymore.
> Hardrives have increased in capacity making caching all the
data easy. A
> single 250gb drive caches 384 CDs worth at 650mb per CD. A
fully
> decked out $10,999 Apple XServe RAID would cache 5384 CDs.
> Thats 35/mb of electronic documents per order if 100,000 are
> stored on the XServer at any one time.
I was simplifying the figures for illustrative purposes - in
reality,
the volume is very much bigger. Sure, we could throw lots of
storage at
the issue, but I am grappling with the problem of what would
be the most
sensible amount of storage without having to cache everything
for an
over-long period of time. Having said that, a rough
calculation shows that
an XServe would hold over 100 days' worth of documents, 
which
may be quite
the way - sounds a steal at that price.
--
===
Subject: Re: How to calculate a useful size for a given pool
>Scenario:
>1. A firm receives 100,000 orders per week as electronic
documents.
>2. All documents are archived on CDROM.
>3. A percentage of orders (average around 10%) will go wrong
at some
>stage and access to the original order will be required by
Customer
>Services to resolve the problem.
>4. Retrieval from CDROM is time-consuming but a limited
amount of space is
>available to cache some of the documents on networked
storage for more
>immediate retrieval. There is not enough space to cache all
the documents
>so the older cached documents are deleted regularly.
>5. It is not possible to determine up front which orders
will fail -
>whilst the failure rate is fairly constant over time, just
about any
>document could end up being requested by Customer Services.
>Problem:
>The firm would like to cache N documents (from the weekly
pool of
>100,000 orders) such that they can satisfy X percent of
Customer
Service
>requests from the networked storage. Customer Services are
prepared to
>put up with [100-X]% of requests that would still need to be
retrieved
>from the CDROM store.
>In short, how do we calculate N?
You could do it based on document age, but you need more info.
Specifically, what is the pdf or cdf that the document will
ever be needed
based on the parameters (how many times it was accessed in
the past, how
long ago)?
I'm assuming that the pdf decays over time, that orders from
last week are
more likely to be needed than orders from last year.
If you want X to be 99%, look for the point (on the time
axis) where the
cdf
hits 99% of the eventual limit (10%).
--Keith Lewis klewis {at} mitre.org
The above may not (yet) represent the opinions of my employer.
===
Subject: Re: How to calculate a useful size for a given pool
>Problem:
>[A] firm would like to cache N documents (from the weekly
pool of
>100,000 orders) such that they can satisfy X percent of
Customer
Service
>requests from the networked storage.
>In short, how do we calculate N?
> You could do it based on document age, but you need more
info.
> Specifically, what is the pdf or cdf that the document will
ever be
needed
> based on the parameters (how many times it was accessed in
the past, how
> long ago)?
> I'm assuming that the pdf decays over time, that orders
from last week
are
> more likely to be needed than orders from last year.
Correct - There's an extreme +ve skew for most documents.
Last year's
orders are only ever resurrected rarely, e.g. if a significant
problem or
unusual customer query is submitted. Some distributions (e.g.
invoices
rather than orders) are more-or-less platykurtic, but most
look roughly
like this (hope the ASCII graphics turn out OK)
N|
U| x
M| x
| x
R| x
E| x
C| x
A| x
L| x
L| x
E| x x x
D|_________________________________________
30 60 90 120
DAYS BETWEEN RECEIPT & RECALL OF DOC
As I said, only a small percentage of orders fail (or are
otherwise
queried). Grateful for any pointers and/or good Web resources
to show me
how to calculate the PDF/CDF for this sort of distribution as
you suggest.
--
===
Subject: Re: How to calculate a useful size for a given pool
<celqdj$ls4$1@newslocal.mitre.orgProblem:
>[A] firm would like to cache N documents (from the weekly
pool of
>100,000 orders) such that they can satisfy X percent of
Customer
Service
>requests from the networked storage.
>In short, how do we calculate N?
>Correct - There's an extreme +ve skew for most documents.
Last year's
>orders are only ever resurrected rarely, e.g. if a
significant problem
or
>unusual customer query is submitted. Some distributions
(e.g. invoices
>rather than orders) are more-or-less platykurtic, but most
look roughly
>like this (hope the ASCII graphics turn out OK)
 N|
> U| x
> M| x
> | x
> R| x
> E| x
> C| x
> A| x
> L| x
> L| x
> E| x x x
> D|_________________________________________
> 30 60 90 120
> DAYS BETWEEN RECEIPT & RECALL OF DOC
>As I said, only a small percentage of orders fail (or are
otherwise
>queried). Grateful for any pointers and/or good Web
resources to show me
>how to calculate the PDF/CDF for this sort of distribution
as you suggest.
The graph that you have is the same shape as your pdf, only
the vertical
axis would be fraction of total recalled (unitless) instead
of number
of
orders recalled. You can work with it either way.
The cdf is the integral of the pdf, so for each point in time
you add up
the
numbers for that day and all previous days. You'll get a
graph that
approaches the total fraction which are ever recalled, which
you estimated
at 10% in your original post. I should point out that you
can't ever know
for sure what this value is based on historical data until
you have *all*
of
the historical data (which you can't possibly have until the
company goes
out of business). But you can estimate it.
If you want a cache hit rate of X%, look at the point on the
cdf where
you
hit X% of what you estimate to be the maximum end value. That
value is
associated with a time. Keep the (most recent) data from that
length of
time. If your order rate stays constant at 150,000/week, you
can calculate
N = time * 150,000 orders/week
but if your business grows N will also grow.
--Keith Lewis klewis {at} mitre.org
The above may not (yet) represent the opinions of my employer.
===
Subject: Re: How to calculate a useful size for a given pool
>[A] firm would like to cache N documents (from the weekly
pool of
>100,000 orders) such that they can satisfy X percent of
Customer
Service
>requests from the networked storage.
>Some distributions (e.g. invoices rather than orders) are
more-or-less
>platykurtic, but most look roughly like this (hope the
ASCII graphics
>turn out OK)
 N|
> U| x
> M| x
> | x
> R| x
> E| x
> C| x
> A| x
> L| x
> L| x
> E| x x x
> D|_________________________________________
> 30 60 90 120
> DAYS BETWEEN RECEIPT & RECALL OF DOC
> The graph that you have is the same shape as your pdf, only
the vertical
> axis would be fraction of total recalled (unitless) instead
of
number
> of orders recalled. You can work with it either way.
<snip>
--
===
Subject: Re: How to calculate a useful size for a given pool
> Scenario:
> 1. A firm receives 100,000 orders per week as electronic
documents.
> 2. All documents are archived on CDROM.
> 3. A percentage of orders (average around 10%) will go
wrong at some
> stage and access to the original order will be required by
Customer
> Services to resolve the problem.
> 4. Retrieval from CDROM is time-consuming but a limited
amount of space
is
> available to cache some of the documents on networked
storage for more
> immediate retrieval. There is not enough space to cache all
the
documents
> so the older cached documents are deleted regularly.
> 5. It is not possible to determine up front which orders
will fail -
> whilst the failure rate is fairly constant over time, just
about any
> document could end up being requested by Customer Services.
> Problem:
> The firm would like to cache N documents (from the weekly
pool of
> 100,000 orders) such that they can satisfy X percent of
Customer
Service
> requests from the networked storage. Customer Services are
prepared to
> put up with [100-X]% of requests that would still need to
be retrieved
> from the CDROM store.
> In short, how do we calculate N?
> This reminds me a little of the Cookie Jar or Sock Drawer
problem,
> but it bugs me that I can't puzzle it out. 
I'm not a
hard-core
> statistician and would be grateful for any help. This is a
genuine (i.e.
> non-homework) request, by the way. I've posted to other 
NGs
with no luck
> so far.
> --
N = X * 100,000
So if X percent is 15%
N = 0.15 * 100,000 = 15,000
Or if X percent is 50%
N = 0.50*100,000 = 50,000
Its really that simple. Nothing to do with combinations,
permutations, any
of that stuff. Just simple percentages.
===
Subject: Re: How to calculate a useful size for a given pool
> N = X * 100,000
> So if X percent is 15%
> N = 0.15 * 100,000 = 15,000
> Or if X percent is 50%
> N = 0.50*100,000 = 50,000
> Its really that simple. Nothing to do with combinations,
permutations,
any
> of that stuff. Just simple percentages.
--
===
Subject: Re: real numbers
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i72Be4q01910;
>Can u plz state a couple of properties of real that are not
satisfied
>by complex numbers.
The set of real numbers is orderable. That is, it is possible
to
define a linear order: x< y such that if x< y then x+z< y+ z
and if
x> 0 then x*x> 0.
Equivalently, there exist a set P (positives) that is closed
under
both multiplication and addition, and such that, given any
non-zero real
number, x, either x is in P or -x is in P, but not both.
Neither of those is true for the complex numbers.
And, of course, every polynomial, with complex coefficients,
has at least
one complex 0. Replacing complex with real, that statement is
not
true.
===
Subject: Re: limits, continuity, and differntiability
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i72Be3V01889;
>Can u plz tell me about the concepts of limits, conitnuity
and
>differentiability in descending order of their
inter-dependence.
Descending order?
Limits are used to define both continuity and
differentiability.
If a function is differentiable (at a point) then it must be
continuous
(at that point).
It is, of course, possible for a function to be continuous
without being
differentiable. Is that what it means?
===
Subject: Re: how to calculate sigma
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i72CIrL05035;
>Hi Group
>is the sigma calculation formula same as StdDev ?
Assuming that you are talking about probability and
statistics, yes, the
small sigma (as opposed to capital sigma used to mean sum) is
just the
symbol for standard deviation.
===
Subject: counting posets,dim=2
I was trying to enumerate the isomorphism classes of posets
of dimension <=2 that is all posets ({1,2,...,n},<) for which
there
is a permutation p in S_n such that i<j <=> p(i)<p(j)
This gives :
1,2,5,16,63,315,1956,14794,131526,1331854,
for n=1,2,..,10,
and then it is getting slow. Maybe there is a fast algorithm
to do this ?!
dividing by n! gives :
1.000,1.000,0.833,0.667,0.525,0.437,0.388,0.367,0.362,0.367
and I'm wondering, will it go up towards 1.000 again or will
it converge ?
--Guenter.
===
Subject: Re: counting posets,dim=2
> I was trying to enumerate the isomorphism classes of posets
> of dimension <=2 that is all posets ({1,2,...,n},<) for
which there
> is a permutation p in S_n such that i<j <=> p(i)<p(j)
> This gives :
> 1,2,5,16,63,315,1956,14794,131526,1331854,
> for n=1,2,..,10,
> and then it is getting slow. Maybe there is a fast
algorithm to do this
?!
FWIW, the first 6 terms of this sequence agrees with Sequence
A059685
at:
http://www.research.att.com/~njas/sequences/
which is (tantalizingly) described as Split interval orders
on n
points:
A poset (X,[pr]) is a split interval order (a.k.a. unit
bitolerance
order, proper bitolerance order) if a real interval and a
distinguished point in that interval can be assigned to each
x in X so
that x[pr]y precisely when x's distinguished point precedes
y's
interval, and x's interval precedes y's 
distinguished point.
This sounds suspiciously like saying that the Hasse diagram
of the
poset can be given a planar order.
At any rate the full sequence they give is:
1,2,5,16,63,315,1979,15576,151606
===
Subject: Re: counting posets,dim=2
Hi Guenter,
> I was trying to enumerate the isomorphism classes of posets
> of dimension <=2 that is all posets ({1,2,...,n},<) for
which there
> is a permutation p in S_n such that i<j <=> p(i)<p(j)
> This gives :
> 1,2,5,16,63,315,1956,14794,131526,1331854,
> for n=1,2,..,10,
> and then it is getting slow. Maybe there is a fast
algorithm to do this
?!
> dividing by n! gives :
> 1.000,1.000,0.833,0.667,0.525,0.437,0.388,0.367,0.362,0.367
> and I'm wondering, will it go up towards 1.000 again or 
will
> it converge ?
Are you sure the last 0.367 is not an error? I would
conjecture
that it converges to 0.
Stas
===
Subject: Re: Great-circle radius of ellipsoid
>BTW, here's another interesting mean radius. 
I've never seen
it
>mentioned before; nonetheless, the result must surely be
well known.
>An equation in polar coordinates for an ellipse centered at
the pole
>and with semiaxes of lengths a and b is
> r = a b / Sqrt( (a sin@)^2 + (b cos@)^2 )
>where @ is an ersatz theta. If we calculate the average of r
with
>respect to @ as it varies from 0 to 2 pi, we find that the
average is
> 2/pi b K(1 - (b/a)^2)
>where K is the complete elliptic integral of the first kind.
> It is the co-Gaussian radius, based on the co-elliptic
integral of
the
> second kind (co-EI2K).
Did you just make up those terms? If not, I would appreciate
your citing
some references in which they are used. I'm suspicious. Why
would anyone
want to call something easily expressed in terms of the
complete elliptic
integral of the _first_ kind the co-elliptic integral of the
second
kind?
It might also be noted that Google web searches for
co-Gaussian radius
and co-elliptic integral yield nothing.
David Cantrell
> EI2K: f(@) = sqrt(1-sin(@)^2*e^2)
> co-EI2K: g(@) = sqrt(1-cos(@)^2*e^2)
> Gauss rad: fr(@) = b/f(@)
> co-Gauss rad: gr(@) = b/g(@)
> BTW when @ is set to reduced latitude, g(@) is the integral
kernal
for
> the conformal sphere in geodetic formularies.
> Bud (just browsing the MF.org archives)
===
Subject: Probability fun...
I was reading the following:
http://en.wikipedia.org/wiki/Doomsday_argument
I stranded pretty quickly in something I didn't understand 
(I
almost
never strand on things I do understand :) ). It was the
following:
If you pick a number uniformly at random in a set of numbers
from 1
to N, where N is a finite number unknown to you, and if we
name that
number j, then:
Your unbiased estimate of N is 2 * j
Why is my unbiased estimate of N 2*j?
I tried to approach this from the following:
Suppose I don't know N, but I do know that N is not larger
than Nmax.
I can now calculate the chance for each N given j.
for k<j
P(N=k | given j) = 0
for j <= k<= Nmax
P(N=k | given j) = (1/k)/S with S = sum_{i=j to Nmax)(1/i)
It is easy to see that P(N=j) is greater than all the other
P(N=k).
This stays true for increasing Nmax (I think). So my guess
would be
that N=j... why is this incorrect?
===
Subject: Re: Probability fun...
> I was reading the following:
> http://en.wikipedia.org/wiki/Doomsday_argument
> I stranded pretty quickly in something I didn't understand
(I almost
> never strand on things I do understand :) ). It was the
following:
> If you pick a number uniformly at random in a set of
numbers from 1
> to N, where N is a finite number unknown to you, and if we
name that
> number j, then:
> Your unbiased estimate of N is 2 * j
> Why is my unbiased estimate of N 2*j?
> I tried to approach this from the following:
> Suppose I don't know N, but I do know that N is not larger
than Nmax.
> I can now calculate the chance for each N given j.
> for k<j
> P(N=k | given j) = 0
> for j <= k<= Nmax
> P(N=k | given j) = (1/k)/S with S = sum_{i=j to Nmax)(1/i)
> It is easy to see that P(N=j) is greater than all the other
P(N=k).
> This stays true for increasing Nmax (I think). So my guess
would be
> that N=j... why is this incorrect?
It's a long time ago since I had a serious course on this, 
so
it might have been covered somewhere. But I do remember
having seen (last year) this example as an application of the
method of maximum likelihood to estimate the parameters
of a density.
(I'm translating from Dutch and from memory now form the
notes of my son's course on probability and statistics so
I'm not sure whether the terminology is correct).
In the example n numbers ( j_1, j_2, ..., j_n) were drawn
at random (and put back) from the population with N
members (with N unknown).
So the total probability of the drawing ( j_1, j_2, ..., j_n)
is given by the function
P(N) = ( 1/N )^n
Then L(N) is maximized by the minimum of N that is
allowed by and compatible with the observed numbers j_n.
Therefore the maximum value of the (j_n) is the best
estimate for N:
N_est = j_max
(That was in the notes of that course)
Since in your example we have
n = 1,
and so we have
N_est = j_max = j
and there would be no reason to take
N_est = 2*j
This leaves me somewhat puzzled as well...
Dirk Vdm
===
Subject: Re: Probability fun...
> I was reading the following:
> http://en.wikipedia.org/wiki/Doomsday_argument
> I stranded pretty quickly in something I didn't understand
(I almost
> never strand on things I do understand :) ). It was the
following:
> If you pick a number uniformly at random in a set of
numbers from 1 to
> N, where N is a finite number unknown to you, and if we
name that
> number j, then:
> Your unbiased estimate of N is 2 * j
> Why is my unbiased estimate of N 2*j?
> I tried to approach this from the following:
> Suppose I don't know N, but I do know that N is not larger
than Nmax. I
> can now calculate the chance for each N given j.
> for k<j
> P(N=k | given j) = 0
> for j <= k<= Nmax
> P(N=k | given j) = (1/k)/S with S = sum_{i=j to Nmax)(1/i)
> It is easy to see that P(N=j) is greater than all the
other P(N=k).
> This stays true for increasing Nmax (I think). So my guess
would be
> that N=j... why is this incorrect?
> It's a long time ago since I had a serious course on this,
so it might
> have been covered somewhere. But I do remember having seen
(last year)
> this example as an application of the method of maximum
likelihood to
> estimate the parameters of a density.
> (I'm translating from Dutch and from memory now form the
notes of my
> son's course on probability and statistics so 
I'm not sure
whether the
> terminology is correct).
> In the example n numbers ( j_1, j_2, ..., j_n) were drawn
at random (and
> put back) from the population with N members (with N
unknown).
> So the total probability of the drawing ( j_1, j_2, ...,
j_n) is given
> by the function
> P(N) = ( 1/N )^n
> Then L(N) is maximized by the minimum of N that is allowed
by and
> compatible with the observed numbers j_n. Therefore the
maximum value of
> the (j_n) is the best estimate for N:
> N_est = j_max
> (That was in the notes of that course)
> Since in your example we have
> n = 1,
> and so we have
> N_est = j_max = j
> and there would be no reason to take
> N_est = 2*j
> This leaves me somewhat puzzled as well...
You are thinking of when there are more than one drawings and
you have
j_1, j_2, ... j_k. In this case an unbiased estimator of N is
a function
of the maximum j, max_i=1^k(j_i). In the OP's example there
was only one
j so the maximum is obvious. This simplifies things a lot.
Maximum likelihood is simply a way to derive estimators. It
doesn't
matter how you derive them, you still have to prove they are
unbiased.
> Dirk Vdm
--
Lance Lamboy
Go F*ck Yourself ~ Dick Cheney
===
Subject: Re: Probability fun...
> I was reading the following:
> http://en.wikipedia.org/wiki/Doomsday_argument
> I stranded pretty quickly in something I didn't understand
(I almost
> never strand on things I do understand :) ). It was the
following:
> If you pick a number uniformly at random in a set of
numbers from 1 to
> N, where N is a finite number unknown to you, and if we name
that number
> j, then:
> Your unbiased estimate of N is 2 * j
> Why is my unbiased estimate of N 2*j?
I would have guessed 2*j-1, but the two estimates are close.
> I tried to approach this from the following:
> Suppose I don't know N, but I do know that N is not larger
than Nmax. I
> can now calculate the chance for each N given j.
> for k<j
> P(N=k | given j) = 0
> for j <= k<= Nmax
> P(N=k | given j) = (1/k)/S with S = sum_{i=j to Nmax)(1/i)
> It is easy to see that P(N=j) is greater than all the other
P(N=k). This
> stays true for increasing Nmax (I think). So my guess would
be that
> N=j... why is this incorrect?
If there was a serious penalty to overestimating N then
you're approach
would make sense, but that doesn't mean that it is an
unbiased estimator.
To be an unbiased estimater, the expected value of the
estimate must be
the parameter.
E(j)=(N+1)/2
E(2*j-1)=N
--
Lance Lamboy
Go F*ck Yourself ~ Dick Cheney
===
Subject: Re: Ito integration
> The non-predictability of Z follows from the strong Markov
property.
I'd be grateful for a hint about that :)
Kuba
===
Subject: Re: Ito integration
> Intuitively, a predictable function (or process) is one
where it's
> always possible to tell where the process will be *at* time
t given
> information up to but not including t.
It sounds as if predictability was equivalent to
At=sidma(Fs,s<t)
adoptation. Is that true??
===
Subject: Re: Ito integration
> Intuitively, a predictable function (or process) is one
where it's
> always possible to tell where the process will be *at* time
t given
> information up to but not including t.
 It sounds as if predictability was equivalent to
At=sidma(Fs,s<t)
> adoptation. Is that true??
Not quite. The prototype predictable process is adapted and
*left-continuous*.
--
A.
===
Subject: Re: BHARAT'S (INDIA'S) CONTRIBUTIONS 
TO THE WORLD
Hanging on to the past is an escape from the current painful
reality.
Claims of historical glories serve no purpose outside the
country,they
are no substitute for a much needed urgent action to bring in
social
justice to a waiting billion people of an ancient
civilization.
> Following are a few highlights of some inventions by our
> ancient sage-scientists -- most of these are excerpted from
> the MICHIGAN DAILY of January 24, 1992:
===
Subject: Re: BHARAT'S (INDIA'S) CONTRIBUTIONS 
TO THE WORLD
mathma18@hotmail.com (Narasimham G.L.) posted:
> Hanging on to the past is an escape from the current
painful reality. . .
.
Is that how you utilize the past? Pity.
On the other hand, the wise learn from it.
Jai Maharaj
http://www.mantra.com/jai
Om Shanti
===
Subject: Re: BHARAT'S (INDIA'S) CONTRIBUTIONS 
TO THE WORLD
>Following are a few highlights of some inventions by our
>ancient sage-scientists -- most of these are excerpted from
>the MICHIGAN DAILY of January 24, 1992:
> ASTRONOMY: In the 5th century CE, Vedic scientist and
>astronomer Bhaskaracharya confirmed earlier calculations of
>the ancients of the time taken by earth to orbit the Sun to
>nine decimal places, 365.258756484 days -- a value that was
>later accepted by modern science.
That's ridiculous precision. 10^(-9) days = 86.4 
microseconds.
Since Bhaskaracharya had no clock better than a sundial,
there's
no way he could measure that. Anyway, he's way off: the 
right
answer is approximately 365.25636 days (assuming you're
talking
about the sidereal year).
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia
Vancouver, BC, Canada V6T 1Z2
===
Subject: Re: BHARAT'S (INDIA'S) CONTRIBUTIONS 
TO THE WORLD
Bhaskaracharya
http://www.geometry.net/detail/scientists/bhaskaracharya.html
Jai Maharaj
http://www.mantra.com/jai
Om Shanti
===
Subject: Re: BHARAT'S (INDIA'S) CONTRIBUTIONS 
TO THE WORLD
ASTRONOMY IN ANCIENT BHARAT
> Here is an extract from a book on the topic Astronomy in
> Ancient India: [...] Astronomy is one area which has
> fascinated all mankind from the beginnings of history. In
> India the first references to astronomy are to be found
> in the Rg Ved which is dated around 2000 BCE. Vedic
> Aryans in fact deified the Sun, Stars and Comets.
> Astronomy was then interwoven with astrology and
> since ancient times Indians have involved the planets
> (called Grahas) with the determination of human fortunes.
> The planets Shani, i.e. Saturn and Mangal i.e. Mars were
> considered inauspicious.
> In the working out of horoscopes (called
> Janmakundali), the position of the Navagrahas, nine
> planets plus Rahu and Ketu (mythical demons, evil forces)
> was considered. The Janmakundali was a complex mixture of
> science and dogma. But the concept was born out of
> astronomical observations and perception based on
> astronomical phenomenon.
> In ancient times personalities like Aryabhatta and
> Varahamihira were associated with Indian astronomy.
> It would be surprising for us to know today that this
> science had advanced to such an extent in ancient India
> that ancient Indian astronomers had recognised that stars
> are same as the sun, that the sun is center of the
> universe (solar system) and that the circumference of the
> earth is 5000 Yojanas. One Yojana being 7.2 kms., the
> ancient Indian estimates came close to the actual figure.
Count Louis Hamon, known to millions as Cheiro,
earned the highest appellations through thirty years of
diligent study of the science of prediction. He was an
expert in both astrology and numerology but was most
in 1949:
practiced this study of the hand, we find undisputed
proofs of their learning and knowledge. Long before Rome
or Greece or Israel was even heard of, the monuments of
India point back to an age of learning beyond, and still
beyond.
figures in their temples represent, it has been estimated
that the Hindus understood the precession of the
equinoxes centuries before the Christian era.
In some of the ancient cave temples, the
mystic figures of the [deities] silently tell that such
knowledge had been possessed and used in advance of all
those nations afterward so celebrated for their learning.
It has been demonstrated that to make a change
from one sign to another in the zodiacal course of the
sun must have occupied at least 2,140 years, and how many
centuries elapsed before such changes came to be observed
and noticed it is impossible even to estimate.
The intellectual power that was necessary to
make these observations speaks for itself; and yet it is
to such a people that we trace the origin of the study
under consideration. With the spread of the Hindu
teachings into other lands do we trace the spread of
knowledge of palmistry.
The Hindu Vedas are the oldest scriptures that
have been found, and according to some authorities they
have been the foundation of even the Greek schools of
learning.
Excerpts from Cheiro's Language Of The Hand by
Cheiro; Herbert Jenkins Limited, London, 1949.
Jai Maharaj
http://www.mantra.com/jai
Om Shanti
===
Subject: Re: BHARAT'S (INDIA'S) CONTRIBUTIONS 
TO THE WORLD
Why Indian science scores
By Shashi Tharoor
The Hindu
WORKING, as I have been for the last couple of years, on
a short biography of Jawaharlal Nehru, I became conscious
of the extent to which we have taken for granted one
vital legacy of his: the creation of an infrastructure
for excellence in science and technology, which has
become a source of great self-confidence and competitive
advantage for the country today. Nehru was always
fascinated by science and scientists. He made it a point
to attend the annual Indian Science Congress every year,
and he gave free rein (and taxpayers' money) to
scientists in whom he had confidence to build high-
quality institutions. Men like Homi Bhabha and Vikram
Sarabhai constructed the platform for Indian
accomplishments in the fields of atomic energy and space
research; they and their successors have given the
country a scientific establishment without peer in the
developing world. Jawaharlal's establishment of the
Indian Institutes of Technology (and the spur they
provided to other lesser institutions) have produced many
of the finest minds in America's Silicon Valley. 
Today,
an IIT degree is held in the same reverence in the U.S.
as one from MIT or Caltech, and India's extraordinary
leadership in the software industry is the indirect
result of Jawaharlal Nehru's faith in 
scientific
education. Nehru left India with the world's second-
largest pool of trained scientists and engineers,
integrated into the global intellectual system, to a
degree without parallel outside the developed West.
And yet the roots of Indian science and technology go far
deeper than Nehru. I was reminded of this yet again by a
remarkable new book, Lost Discoveries, by the American
writer Dick Teresi. Teresi's book studies the ancient
non-Western foundations of modern science, and while he
ranges from the Babylonians and Mayans to Egyptians and
other Africans, it is his references to India that caught
my eye. And how astonishing those are! The Rig Veda
asserted that gravitation held the universe together 24
centuries before the apple fell on Newton's head. The
Vedic civilisation subscribed to the idea of a spherical
earth at a time when everyone else, even the Greeks,
assumed the earth was ßat. By the Fifth Century A.D.
Indians had calculated that the age of the earth was 4.3
billion years; as late as the 19th Century, English
scientists believed the earth was a hundred million years
old, and it is only in the late 20th Century that Western
scientists have come to estimate the earth to be about
4.6 billion years old.
If I were to focus on just one field in this column, it
would be that of mathematics. India invented modern
numerals (known to the world as Arabic numerals because
the West got them from the Arabs, who learned them from
us!). It was an Indian who first conceived of the zero,
shunya; the concept of nothingness, shunyata, integral to
Hindu and Buddhist thinking, simply did not exist in the
in 1930, the invention of zero will always stand out as
one of the greatest single achievements of the human
race.) The concept of infinite sets of rational numbers
was understood by Jain thinkers in the Sixth Century B.C.
Our forefathers can take credit for geometry,
trigonometry, and calculus; the Bakhshali manuscript,
70 leaves of bark dating back to the early centuries of
the Christian era, reveals fractions, simultaneous
equations, quadratic equations, geometric progressions
and even calculations of profit and loss, with interest.
Indian mathematicians invented negative numbers: the
British mathematician Lancelot Hogben, grudgingly
acknowledging this, suggested ungraciously that perhaps
because the Hindus were in debt more often than not, it
occurred to them that it would also be useful to have a
number which represent the amount of money one owes.
(That theory would no doubt also explain why Indians were
the first to understand how to add, multiply and subtract
from zero  because zero was all, in Western eyes, we
ever had.)
The Sulba Sutras, composed between 800 and 500 B.C.,
demonstrate that India had Pythagoras' theorem before the
great Greek was born, and a way of getting the square
root of 2 correct to five decimal places. (Vedic Indians
solved square roots in order to build sacrificial altars
of the proper size.) The Kerala mathematician Nilakantha
pi before the West had heard of the concept. The
Vedanga Jyotisha, written around 500 B.C., declares:
Like the crest of a peacock, like the gem on the head of
a snake, so is mathematics at the head of all knowledge.
Our mathematicians were poets too! But one could go back
even earlier, to the Harappan civilisation, for evidence
of a highly sophisticated system of weights and measures
in use around 3000 B.C.
Archaeologists also found a ruler made with lines drawn
precisely 6.7 millimeters apart with an astonishing level
of accuracy. The Indus inch was a measure in consistent
use throughout the area. The Harappans also invented
kiln-fired bricks, less permeable to rain and ßoodwater
than the mud bricks used by other civilisations of the
time. The bricks contained no straw or other binding
material and so turned out to be usable 5, 000 years
later when a British contractor dug them up to construct
a railway line between Multan and Lahore. And while they
were made in 15 different sizes, the Harappan bricks were
amazingly consistent: their length, width and thickness
were invariably in the ratio of 4:2:1.
profound effect on neighbouring cultures. The greatest
impact was on Islamic culture, which borrowed heavily
from Indian numerals, trigonometry and analemma. Indian
numbers probably arrived in the Arab world in 773 A.D.
with the diplomatic mission sent by the Hindu ruler of
Sind to the court of the Caliph al-Mansur. This gave rise
to the famous arithmetical text of al-Khwarizmi, written
around 820 A.D., which contains a detailed exposition of
Indian mathematics, in particular the usefulness of the
zero. With Islamic civilisation's rise and spread,
knowledge of Indian mathematics reached as far afield as
Central Asia, North Africa and Spain. In serving as a
conduit for incoming ideas and a catalyst for inßuencing
others, Teresi adds, India played a pivotal role. His
research is such a rich lode that I intend to return to
ancient Indian science in a future column.
- - -
Shashi Tharoor is the United Nations Under Secretary-
General for Communications and Public Information and the
author of seven books, most recently Riot and (with M.F.
Husain) Kerala: God's Own Country.
End of forwarded message
Jai Maharaj
http://www.mantra.com/jai
Om Shanti
Hindu Holocaust Museum
http://www.mantra.com/holocaust
Hindu life, principles, spirituality and philosophy
http://www.hindu.org
http://www.hindunet.org
The truth about Islam and Muslims
http://www.ßex.com/~jai/satyamevajayate
The terrorist mission of Jesus stated in the Christian bible:
Think not that I am come to send peace on earth:
I came not so send peace, but a sword.
For I am come to set a man at variance against his
father, and the daughter against her mother, and the
daughter in law against her mother in law.
And a man's foes shall be they of his own
household.
- Matthew 10:34-36.
o Not for commercial use. Solely to be fairly used for the
educational purposes of research and open discussion. The
contents of
this post may not have been authored by, and do not
necessarily represent
the opinion of the poster. The contents are protected by
copyright law
and the exemption for fair use of copyrighted works.
o If you send private e-mail to me, it will likely not be
read,
considered or answered if it does not contain your full legal
name,
current e-mail and postal addresses, and live-voice telephone
number.
are not necessarily those of the poster.
===
Subject: Re: BHARAT'S (INDIA'S) CONTRIBUTIONS 
TO THE WORLD
Try replacing the word Hindu with White People, or Arian
Brotherhood
and see how bigoted you sound
???
===
Subject: Re: BHARAT'S (INDIA'S) CONTRIBUTIONS 
TO THE WORLD
> Edwin Clark <eclark@math.usf.edu> posted:
> GENERAL: It is already becoming clear that the chapter
>which had a Western beginning will have to have an Indian
>ending if it is not to end in self-destruction of the human
>race. - Dr. Arnold Toynbee, world historian.
>If we are all ONE, why do you . . .
> If we are all ONE, why the use of the term you above?
YOU are maya, a convenient fiction.
ONE should put it thusly:
If there is only ONE, why does ONE make dichotomies? In
paticular, why
does ONE distinguish between the East and the West? Why does
ONE desire
to distinguish between India and the rest of the world any
more than ONE
would distinguish between a rock and a bird?
The answer of course lies in the necessity for duality, for
the YIN that
requires the YANG, for the LIGHT that requires the DARK, the
GOOD that
requires EVIL, etc. ONE requires the tension of dichotomy
otherwise ONE
remembers that ONE is just ONE after all and the game
disappears.
Apparently ONE desires to play India is better than the rest.
---ONE (talking to ONESELF :-))
===
Subject: Re: BHARAT'S (INDIA'S) CONTRIBUTIONS 
TO THE WORLD
Edwin Clark <eclark@math.usf.edu> posted:
> Edwin Clark <eclark@math.usf.edu> posted:
> GENERAL: It is already becoming clear that the chapter
>which had a Western beginning will have to have an Indian
>ending if it is not to end in self-destruction of the
human
>race. - Dr. Arnold Toynbee, world historian.
>If we are all ONE, why do you . . .
> If we are all ONE, why the use of the term you above?
> YOU are maya, a convenient fiction.
What or who isn't a part of maya in this illusory existence?
> ONE should put it thusly:
> If there is only ONE, why does ONE make dichotomies? In
paticular, why
> does ONE distinguish between the East and the West? Why
does ONE desire
> to distinguish between India and the rest of the world any
more than ONE
> would distinguish between a rock and a bird?
> The answer of course lies in the necessity for duality, for
the YIN that
> requires the YANG, for the LIGHT that requires the DARK,
the GOOD that
> requires EVIL, etc. ONE requires the tension of dichotomy
otherwise ONE
> remembers that ONE is just ONE after all and the game
disappears.
> Apparently ONE desires to play India is better than the
rest.
> ---ONE (talking to ONESELF :-))
In a number of respects Bharat is indeed better than the rest.
Jai Maharaj
http://www.mantra.com/jai
Om Shanti
===
Subject: Re: BHARAT'S (INDIA'S) CONTRIBUTIONS 
TO THE WORLD
The Ancient World Had Only One Culture
By Ashok Singhal
Working President
Vishwa Hindu Parishad
THE ORGANISER
June 1, 1997
Hindus have a rich heritage. Not only the Hindu
philosophy Hindus have a long and distinguished tradition
in science from ancient times to the twentieth century.
It is indeed a happy augury that Hindu ethos has awakened
the hearts of Hindu youth of Europe. They are the
inheritors and carriers of ancient Hindu wisdom. The
home of the Vedas and the Upanishads is indeed the
fountainhead of world culture and civilisation.
Recently I came across a statement on a BBC serial about
the spread of Hinduism in Europe in ancient, pre-
Christian times. It was said that the ruins of a temple
found in the remote island nation of Iceland, or the
Egyptian pyramids, or the Tower of Babel all symbolise
the worship of Lord Shiva. It is now common knowledge
that the original inhabitants of the British isles, the
Celtics, were Vedic Aryans. The Celtic priests were Vedic
Brahmins known as Druids. The mythology of Scandinavian
countries closely resembles its Hindu counterpart in many
ways.
Hindu civilisation spread far and wide throughout Asia,
says Will Durant. It is noteworthy that the largest place
of religious worship in the world is an ancient Hindu
temple dedicated to Lord Vishnu in Cambodia at Angkor Wat
(Onkaar Vat).
Disillusioned by the 2000 years of their experience with
the lack of respect for life and nature on the part of
the Semitic religions and threatened by consumerism, many
in the West are today searching for their pre-Christian
roots. These roots were no other than the common culture
of mankind in the hoary past. It was the common heritage
of all mankind. Despite regional variations certain
characteristics were common to all which today, we
associate with Vedic culture. So we call this common
culture of mankind Vedic culture. Scholars, Western as
well as Hindu, have, therefore, unequivocally declared
that the ancient world had only one single culture which
they too name as Vedic culture.
Ignorance of the universal significance of holistic
approach of the Vedas is a tragedy. Our apathy in this
respect is causing loss of dignity to us even today. The
Vedas are the accumulated treasure of spiritual and
It is not a pumping-in from outside that gives wisdom;
it is the power and extent of your inner receptivity that
determines how much you can attain of true knowledge, and
how rapidly. Knowing and applying this truth, the
ancient rishis of India devoted themselves to practice of
yoga techniques for concentration and meditation that
enabled them to attain intuitive scientific understanding
of the universe-from the farthest reaches of theoretical
physics to the details of harmonious everyday living.
Their wisdom was recorded m the Vedas, which to this day
form the foundation of Indian spiritual and cultural
life. Ancient India played a pioneering role in shaping
human culture and civilisation throughout the world from
the Inca and Maya in South America to Shintoism in Japan.
The Buddhist monks, in fact, continued the tradition of
early sages of unfolding Sanatana Dharma throughout the
world.
The Science of Hindus
India was leading the world in scientific activity at
least till the seventh century AD. There was a downfall
of Indian science in the medieval ages owing to foreign
invasion, subjugation and exploitation. Science can
ßourish only in the proper socio-economic background
under optimum motivational conditions and with suitable
ideological environment. The faculty of speculation and
the capacity to accurately observe, infer, experiment and
interpolate were very well developed among the Hindus.
The wise men among them laid the foundations not only of
spiritual-cultural sciences but of scientific thinking in
numerous other disciplines. The observation of heavenly
bodies and nature inspired some of the thinkers towards
religious fervour, the more pragmatic ones however
utilised their experiments and observations for weather
prediction, navigation, agriculture and other utilitarian
purpose. Excavated ancient Hindus ruins reveal
remarkable feats achieved in drainage and other civil
engineering works and also systems of writing. The
ancient Hindus laid the foundations of mathematical and
scientific knowledge. They conceived and developed the
sciences of logic and grammar and made great advances in
fields so divergent as anatomy and astronomy, philosophy
and metaphysics, medicine and mathematics. Some of the
basic conclusions reached by the ancient Hindu insights
have become the fundamentals of modem science.
Professor of historical linguistics, George Cardona of
Pennsylvania, in a recent interview (Times of India 212-
97) said: Noam Chomsky is the giant of modem
linguistics. But Panini who lived no later than the 5th
century BC has even a greater stature. He was the first
to compose formal grammar with unbelievable theoretical
insight. His Ashtadhyayi deals with complex and
complicated issues about language, thought, relationship
of form and meaning and logic. He was the product of a
long history of thinking on every aspect of language. The
Upanishads, Aaranyakas.. and the Vedas speculate on
language, and all this Panini confided in crisp
statements on which long commentaries are written. He
was the culminating point and a brilliant scholar. he
Vedas: Rig, Yajur, Sama and Atharva (over 5000 BC)
represented the very best in speculative thinking apart
from containing numerous references on drugs, diseases
and stars. in recent decades the famous 16 formulae of
Vedic mathematics have been found from confided Vedic
mantras, which point to the fact that if we have the
capability to decipher, we can discover many other gems
in our ancient scriptures. Kanada and Kapila the Vedic
philosophers, referred to five 
Ôimmutable' elements which
were constituents of matter. They also developed the
concepts of molecule and atom. Kanad proposed the
undulatory theory for the propagation of sound and made
the seculation that light and sound waves are only
different manifestations of the same energy. Even in
that early age a statement was made in the Atharva
Samhita to the effect that though solar eclipse popularly
believed to be due to a monster called Rahu, the more
likely cause is the interposition Of the moon between the
sun and the earth. Expressions of scientific opinion
contrary to popular or religious beliefs were common in
the Vedic ages; the fact that there were scarcely any
religious beliefs were common in the Vedic ages; the fact
that there were scarcely any religious persecution bears
testimony to the spirit of tolerance in Hindu culture. In
Surva-siddhanta, the Indian meridian was fixed to be at
Ujjain, the centre of astronomical research. The most
fascinating topic in Surya-siddhanta is the sine-table,
which is the oldest in the world.
The discovery of the geometrical theorem stating that the
square on the diagonal of a rectangle equals the sum of
the squares on the two sides m area, is usually
attributed, without any justification, to the Greek
savant, Pithagorus. However, the following statement is
found in several places in Shatapatha Brahmana and Sulya-
sutra: The transverse chord of a rectangle produces, by
the construction of a square on itself, what the length
and the breadth produce separately. The decimal system
of number, perfected later by Aryabhatta was in vogue
during the Vedic ages. Large numbers up to 108 are
referred to in many ancient Hindu scriptures. Aryabhatta
his famous book on astronomy, Arvabhattiyan) that the
earth rotates round its axis and also theorised that the
earth revolves round the sun. His other scientific
contributions include: discovery of trigonometric sine,
accurate evaluation of pi, perfection of decimal system
of numbers, evaluation of AP and GP series and square and
cube roots of numbers, accurate calculation of number of
days in a solar year (up to seventh decimal point which
is better than Piolemy's figure), discovery of 
the causes
of solar and lunar eclipses, prediction of duration and
angular extent of eclipses, etc. - a long list indeed.
Aryabhatta was the pioneer to introduce the concept of
algebra. He had been the first and the foremost
mathematical genius, produced by the human race and a
forerunner of Newton and other ruropean mathematicians
who were to be born more than thousand years later.
Bhaskara advanced the concept of algebra, He dealt on
algorithm, zero and its use, unknown quantities, surds,
the pulverizer, solution of quadratic equations and of
certain equations of the third and the fourth degree.
Arabic numerals are still called ÔHindusa' as 
they were
borrowed from Hindustan.
Some of the comments found in Charak Samhita and Susruta
Samhita reveal an amazing degress of objective and
scientific thinking: to gain experience in surgery, one
has to properly dress and dissect a corpse and minutely
observe the anatomical details. Charaka and Susruta were
not only experts in medicinal chemistry but also authors
of several treatises on organic and inorganic compounds.
Some biased western scholars have claimed that all
scientific thinking emerged from Greece and that what
science the Hindu had, they borrowed. Jean Filliozat, the
eminent French indologist has pinpricked the buddle of
this fantasy.
Ancient Hindus were highly knowledgeable about the
intricate functioning of various phenomena of nature.
They appear to have not only made attempts to understand
these processes but have studied the interaction between
man and nature in much detail. Evidently, they recognised
the dangers and risks to nature and the environment by
activities of human beings and have, therefore, evolved
some code of conduct to control these activities. Also to
practically implement these rules, they developed several
traditions and customs. We need to analyse these
traditions and customs in the modem perspective. A
systematic of this field can provide several new ideas
and practical methods to preserve this planet from
further degradation and Hindus can show a way to the rest
of the world. The holistic approach of Hindus appreciates
humanism and not homocentricism of semitism, capitalism
and communism.
This presentation is illustrative, not exhaustive.
Constraints of space do not allow us to discuss further.
Many scholars are doing research into several such
things, not for an chauvinistic purpose, but to learn
from ancient insight for the benefit of mankind.
Research on ancient as well as modem ideas, development
and reconstruction can go on. Our youth could do these.
Let us learn from past experience
Let us, the Hindus of today know our past in its true
perspective to build a great future. How many of us
really know about the great exploits of Samudragupta or
of Shailendera whose military might was far superior to
that of Alexander or Napoleon. Just 500 years ago,
Hinduism was the faith of entire South-East Asia. The
evidence is still there in far-ßung places like Bali,
Cambodia and Sri Lanka. We all belong to a religion which
has taught the world both tolerance and universal
acceptance.
It is gratifying that Hindus the world over are now
realising their great heritage. Let the Hindu Youth of
Europe take a lead in worlwide revival of pride in our
heritage. The world has lived with the trauma of semitic
religions for too long a time. The time has come for
humanity to imbibe the universal wisdom of Hinduism. A
grea challenge and equally great opportunity are
beckoning us. Let us rise to the occasion and help the
world usher in a new era of peace and tranquillity.
Courtesy of the Hindu Vivek Kendra
Jai Maharaj
http://www.mantra.com/jai
Om Shanti
===
Subject: Re: BHARAT'S (INDIA'S) CONTRIBUTIONS 
TO THE WORLD
Try replacing the word Hindu with White People, or Arian
Brotherhood
and see how bigoted you sound
<snip>
===
Subject: Re: BHARAT'S (INDIA'S) CONTRIBUTIONS 
TO THE WORLD
The Ancient World Had Only One Culture
By Ashok Singhal
Working President
Vishwa Hindu Parishad
THE ORGANISER
June 1, 1997
Hindus have a rich heritage. Not only the Hindu
philosophy Hindus have a long and distinguished tradition
in science from ancient times to the twentieth century.
It is indeed a happy augury that Hindu ethos has awakened
the hearts of Hindu youth of Europe. They are the
inheritors and carriers of ancient Hindu wisdom. The
home of the Vedas and the Upanishads is indeed the
fountainhead of world culture and civilisation.
Recently I came across a statement on a BBC serial about
the spread of Hinduism in Europe in ancient, pre-
Christian times. It was said that the ruins of a temple
found in the remote island nation of Iceland, or the
Egyptian pyramids, or the Tower of Babel all symbolise
the worship of Lord Shiva. It is now common knowledge
that the original inhabitants of the British isles, the
Celtics, were Vedic Aryans. The Celtic priests were Vedic
Brahmins known as Druids. The mythology of Scandinavian
countries closely resembles its Hindu counterpart in many
ways.
Hindu civilisation spread far and wide throughout Asia,
says Will Durant. It is noteworthy that the largest place
of religious worship in the world is an ancient Hindu
temple dedicated to Lord Vishnu in Cambodia at Angkor Wat
(Onkaar Vat).
Disillusioned by the 2000 years of their experience with
the lack of respect for life and nature on the part of
the Semitic religions and threatened by consumerism, many
in the West are today searching for their pre-Christian
roots. These roots were no other than the common culture
of mankind in the hoary past. It was the common heritage
of all mankind. Despite regional variations certain
characteristics were common to all which today, we
associate with Vedic culture. So we call this common
culture of mankind Vedic culture. Scholars, Western as
well as Hindu, have, therefore, unequivocally declared
that the ancient world had only one single culture which
they too name as Vedic culture.
Ignorance of the universal significance of holistic
approach of the Vedas is a tragedy. Our apathy in this
respect is causing loss of dignity to us even today. The
Vedas are the accumulated treasure of spiritual and
It is not a pumping-in from outside that gives wisdom;
it is the power and extent of your inner receptivity that
determines how much you can attain of true knowledge, and
how rapidly. Knowing and applying this truth, the
ancient rishis of India devoted themselves to practice of
yoga techniques for concentration and meditation that
enabled them to attain intuitive scientific understanding
of the universe-from the farthest reaches of theoretical
physics to the details of harmonious everyday living.
Their wisdom was recorded m the Vedas, which to this day
form the foundation of Indian spiritual and cultural
life. Ancient India played a pioneering role in shaping
human culture and civilisation throughout the world from
the Inca and Maya in South America to Shintoism in Japan.
The Buddhist monks, in fact, continued the tradition of
early sages of unfolding Sanatana Dharma throughout the
world.
The Science of Hindus
India was leading the world in scientific activity at
least till the seventh century AD. There was a downfall
of Indian science in the medieval ages owing to foreign
invasion, subjugation and exploitation. Science can
ßourish only in the proper socio-economic background
under optimum motivational conditions and with suitable
ideological environment. The faculty of speculation and
the capacity to accurately observe, infer, experiment and
interpolate were very well developed among the Hindus.
The wise men among them laid the foundations not only of
spiritual-cultural sciences but of scientific thinking in
numerous other disciplines. The observation of heavenly
bodies and nature inspired some of the thinkers towards
religious fervour, the more pragmatic ones however
utilised their experiments and observations for weather
prediction, navigation, agriculture and other utilitarian
purpose. Excavated ancient Hindus ruins reveal
remarkable feats achieved in drainage and other civil
engineering works and also systems of writing. The
ancient Hindus laid the foundations of mathematical and
scientific knowledge. They conceived and developed the
sciences of logic and grammar and made great advances in
fields so divergent as anatomy and astronomy, philosophy
and metaphysics, medicine and mathematics. Some of the
basic conclusions reached by the ancient Hindu insights
have become the fundamentals of modem science.
Professor of historical linguistics, George Cardona of
Pennsylvania, in a recent interview (Times of India 212-
97) said: Noam Chomsky is the giant of modem
linguistics. But Panini who lived no later than the 5th
century BC has even a greater stature. He was the first
to compose formal grammar with unbelievable theoretical
insight. His Ashtadhyayi deals with complex and
complicated issues about language, thought, relationship
of form and meaning and logic. He was the product of a
long history of thinking on every aspect of language. The
Upanishads, Aaranyakas.. and the Vedas speculate on
language, and all this Panini confided in crisp
statements on which long commentaries are written. He
was the culminating point and a brilliant scholar. he
Vedas: Rig, Yajur, Sama and Atharva (over 5000 BC)
represented the very best in speculative thinking apart
from containing numerous references on drugs, diseases
and stars. in recent decades the famous 16 formulae of
Vedic mathematics have been found from confided Vedic
mantras, which point to the fact that if we have the
capability to decipher, we can discover many other gems
in our ancient scriptures. Kanada and Kapila the Vedic
philosophers, referred to five 
Ôimmutable' elements which
were constituents of matter. They also developed the
concepts of molecule and atom. Kanad proposed the
undulatory theory for the propagation of sound and made
the seculation that light and sound waves are only
different manifestations of the same energy. Even in
that early age a statement was made in the Atharva
Samhita to the effect that though solar eclipse popularly
believed to be due to a monster called Rahu, the more
likely cause is the interposition Of the moon between the
sun and the earth. Expressions of scientific opinion
contrary to popular or religious beliefs were common in
the Vedic ages; the fact that there were scarcely any
religious beliefs were common in the Vedic ages; the fact
that there were scarcely any religious persecution bears
testimony to the spirit of tolerance in Hindu culture. In
Surva-siddhanta, the Indian meridian was fixed to be at
Ujjain, the centre of astronomical research. The most
fascinating topic in Surya-siddhanta is the sine-table,
which is the oldest in the world.
The discovery of the geometrical theorem stating that the
square on the diagonal of a rectangle equals the sum of
the squares on the two sides m area, is usually
attributed, without any justification, to the Greek
savant, Pithagorus. However, the following statement is
found in several places in Shatapatha Brahmana and Sulya-
sutra: The transverse chord of a rectangle produces, by
the construction of a square on itself, what the length
and the breadth produce separately. The decimal system
of number, perfected later by Aryabhatta was in vogue
during the Vedic ages. Large numbers up to 108 are
referred to in many ancient Hindu scriptures. Aryabhatta
his famous book on astronomy, Arvabhattiyan) that the
earth rotates round its axis and also theorised that the
earth revolves round the sun. His other scientific
contributions include: discovery of trigonometric sine,
accurate evaluation of pi, perfection of decimal system
of numbers, evaluation of AP and GP series and square and
cube roots of numbers, accurate calculation of number of
days in a solar year (up to seventh decimal point which
is better than Piolemy's figure), discovery of 
the causes
of solar and lunar eclipses, prediction of duration and
angular extent of eclipses, etc. - a long list indeed.
Aryabhatta was the pioneer to introduce the concept of
algebra. He had been the first and the foremost
mathematical genius, produced by the human race and a
forerunner of Newton and other ruropean mathematicians
who were to be born more than thousand years later.
Bhaskara advanced the concept of algebra, He dealt on
algorithm, zero and its use, unknown quantities, surds,
the pulverizer, solution of quadratic equations and of
certain equations of the third and the fourth degree.
Arabic numerals are still called ÔHindusa' as 
they were
borrowed from Hindustan.
Some of the comments found in Charak Samhita and Susruta
Samhita reveal an amazing degress of objective and
scientific thinking: to gain experience in surgery, one
has to properly dress and dissect a corpse and minutely
observe the anatomical details. Charaka and Susruta were
not only experts in medicinal chemistry but also authors
of several treatises on organic and inorganic compounds.
Some biased western scholars have claimed that all
scientific thinking emerged from Greece and that what
science the Hindu had, they borrowed. Jean Filliozat, the
eminent French indologist has pinpricked the buddle of
this fantasy.
Ancient Hindus were highly knowledgeable about the
intricate functioning of various phenomena of nature.
They appear to have not only made attempts to understand
these processes but have studied the interaction between
man and nature in much detail. Evidently, they recognised
the dangers and risks to nature and the environment by
activities of human beings and have, therefore, evolved
some code of conduct to control these activities. Also to
practically implement these rules, they developed several
traditions and customs. We need to analyse these
traditions and customs in the modem perspective. A
systematic of this field can provide several new ideas
and practical methods to preserve this planet from
further degradation and Hindus can show a way to the rest
of the world. The holistic approach of Hindus appreciates
humanism and not homocentricism of semitism, capitalism
and communism.
This presentation is illustrative, not exhaustive.
Constraints of space do not allow us to discuss further.
Many scholars are doing research into several such
things, not for an chauvinistic purpose, but to learn
from ancient insight for the benefit of mankind.
Research on ancient as well as modem ideas, development
and reconstruction can go on. Our youth could do these.
Let us learn from past experience
Let us, the Hindus of today know our past in its true
perspective to build a great future. How many of us
really know about the great exploits of Samudragupta or
of Shailendera whose military might was far superior to
that of Alexander or Napoleon. Just 500 years ago,
Hinduism was the faith of entire South-East Asia. The
evidence is still there in far-ßung places like Bali,
Cambodia and Sri Lanka. We all belong to a religion which
has taught the world both tolerance and universal
acceptance.
It is gratifying that Hindus the world over are now
realising their great heritage. Let the Hindu Youth of
Europe take a lead in worlwide revival of pride in our
heritage. The world has lived with the trauma of semitic
religions for too long a time. The time has come for
humanity to imbibe the universal wisdom of Hinduism. A
grea challenge and equally great opportunity are
beckoning us. Let us rise to the occasion and help the
world usher in a new era of peace and tranquillity.
Courtesy of the Hindu Vivek Kendra
Jai Maharaj
http://www.mantra.com/jai
Om Shanti
===
Subject: Re: BHARAT'S (INDIA'S) CONTRIBUTIONS 
TO THE WORLD
@zV458T4QveIqBQ:
> ÔLost Discoveries': The Non-Western Roots of 
Science
What about those Battles at Panipat? Have you forgotten those?
r
--
Nothing beats the bandwidth of a station wagon filled with DLT
tapes.
===
Subject: Re: BHARAT'S (INDIA'S) CONTRIBUTIONS 
TO THE WORLD
Rich.Andrews <spmaway@ylhoo.com> posted:
> Dr. Jai Maharaj posted:
> ÔLost Discoveries': The Non-Western Roots of 
Science
> What about those Battles at Panipat? Have you forgotten
those?
Don't forget Kurukshetr and the Mahabharat.
Jai Maharaj
http://www.mantra.com/jai
Om Shanti
===
Subject: Re: BHARAT'S (INDIA'S) CONTRIBUTIONS 
TO THE WORLD
> Rich.Andrews <spmaway@ylhoo.com> posted:
> Dr. Jai Maharaj posted:
> ÔLost Discoveries': The Non-Western Roots of 
Science
> What about those Battles at Panipat? Have you forgotten
those?
The first battle of Panipat took place in northern India, and
marked the
beginning of the Mogul Empire.
In 1526, the forces of Babur, the ruler of Kabul and of
Timurid descent,
defeated the much larger army of Ibrahim Lodi, the ruler of
the large North
Indian Delhi Sultanate. The battle was fought near the small
village of
Panipat, in the present day Indian state of Haryana, an area
that has been
the site of a number decisive battles for the control of
Northern India
since the twelfth century. It is estimated that Babur's
forces numbered
about 12,000 men and he had between 15 to 20 pieces of field
artillery.
These guns proved decisive in battle because Ibrahim Lodi's
lacked any
field
artillery. Babur was an inspirational leader of men and
commanded a well
disciplined army. In contrast, Ibrahim Lodhi was loathed by
many of his
commanders and feudatories because of his legendary cruelty
and avarice;
his
army, which really consisted of separate feudal contingents,
loosely held
together, started disintegrating under the bombardment of
Babur's forces.
Ibrahim Lodi died on the field of battle, abandoned by his
feudatories and
generals, most of whom would change their allegiance to the
new master of
Delhi. The battle marked the foundation of the so called
Mughal or Mogul
empire in India - the word means Mongol and alludes to the
Turko-Mongol
origins of Baburs and his officers, although the majority of
his troops
would probably have had been of mixed Central Asian descent.
The other
significance of the battle is that it marked the beginning of
large scale
use of fire arms in Indian warfare.
> Don't forget Kurukshetr and the Mahabharat.
OK here they are.............
Kurukshetr--- King Dhritaraashtr said: O Sanjay, assembled in
the holy
field
of Kurukshetr and eager to fight, what did my people and the
Paandavs do?
(1.01) Sanjay said: Seeing the battle formation of the
Paandav's army, King
Duryodhan approached his guru and spoke these words: (1.02) O
master,
behold
this mighty army of the sons of Paandu, arranged in battle
formation by
your
other talented disciple. There are many great warriors,
valiant men,
heroes,
and mighty archers. I shall name a few of them for you.
(1.03-06) Also
know,
O best among the men, the distinguished ones on our side.
Mahabharat---The Mahabharata is a massive work attributed to
the sage
Vedavyasa. The myth is that he dictated it to Ganesha, the
elephant-headed
deity of wisdom and success, who was the one to actually pen
it. Divided
into 18 parvans or chapters, it is a War Epic, woven round
the royal Kuru
family of the Chandra Vamsha (Lunar Dynasty), which reigned
over Northern
India from Hastinapur. The war in question is the Kurukshetra
War fought at
Kurukshetra in northern India, around 3,000 BC according to
the Indian
tradition, and around 1,000-1,500 BC according to modern
scholars. Its
original name was Jaya or Victory. The core story is as
follows: King
Santanu, father of Prince Devavrata, re-married at a late
stage in life and
the Kuru throne went, not to Devavrata, but to a succession
of defective
princes born because of the second marriage. This resulted
eventually in a
controversy about whether Prince Duryodhana or Prince
Yudhistthira would be
king. Yudhistthira had four brothers, Bhima, Arjuna, Nakula
and Sahadeva.
And together the five brothers were known as Pandavas because
their father
had been named Pandu. They shared all sorts of adventures
together, as well
as a wife- Droupadi who was married to them all. Krishna, an
incarnation
of
Vishnu, and a cousin, was friend, philosopher and guide to
the Pandavas. At
the battlefield of Kurukshetra, all the major warrior kings of
India were
united under either Prince Duryodhana's banner or Prince
Yudhistthira's.
After a gory, 18-day battle, Yudhistthira emerged victor. But
the epic does
not end there. It takes us to great spiritual heights as
after a benign
reign, King Yudhistthira relinquishes his throne and, with
his brothers and
his wife, goes up the Himalayas into heaven.
What have these got to do with Roots of Science?
===
Subject: Re: ~ Latex and math papers
> Even with more experience it could be much too difficult to
reproduce
> the look of the book without knowing what particular
packages or
> style the author has used.
> Never mind, the main advantage of using LaTeX is, that you
can happily
> think about the content without having to fine-tune the
layout.
> Let me explain a bit about what I am doing. I want to
create a PDF file
> definition, up to sets of functions, as well as providing
all proofs for
the
> theorems presented. The first source of information that I
used is Proofs
> and Fundamentals: A first cource in abstract mathematics. I
then located
> online resources concerning functions to learn how various
definitions are
> presented in those sources, as well as any theorems or
names of functions
> that I did not already know. So far, I have definitions up
to sets of
> functions and theorems that apply therein. In the past, I
proved each of
the
> now. I simply wish to have a document with a very simple
presentation. I'm
> currently writing the proofs in Latex since the style can
be changed with
I
> learn the commands without having to alter the content.
> 1) Definitions.
> These should begin with Definition. in bold text, followed 
by
> non-italicized text.
> 2) Theorems.
> These should begin with Theorem followed by a number, both
in bold
> text. The text of the theorem should be italicized.
> 3) Proofs.
> These should begin with Proof. in bold text, have normal
text for
the
> body, and be ended with an qed symbol.
> Here is an online math text that has the look I wish to use:
> http://www.maths.tcd.ie/~dwilkins/Courses/311/311Groups.pdf
> eliminate the section numbers from the definitions. There is
a way to make
> my own definition style, but that created further
difficulties for me.
If pressed, you could redefine the material that is typeset
when
a definition environment is started. For example, in your
original
file you have defined an environment for theorems 
with the name
thm.
Adding the line
renewcommandthethm{hskip-0.3em}
results in the counter to be suppressed (the hskip takes care
that
the final . is not too far away.
However, I am certainly not proficient at LaTeX and I doubt if
such
a workaround is really a good way to go.
You should perhaps consider, to have the definitions numbered
as well.
How would you want to reference to a previous definition? Of
course
you could always write the definition of <concept 
defined>, but
for
the reader it might still be easier to find, if also some
number can
be supplied.
> I've read parts of it, along with many others. 
I'd prefer
technical
> details instead of a walk-through, though.
The Kopka books might be a good idea. There should be
references in
the TeX-FAQ. For plain TeX, the TeXbook is great.
If you have specific questions which are not dealt with in the
FAQ,
Marc
===
Subject: Re: Topology of the discriminant set
(Sorry for the late answer.) Yes I was aware of the work of
Michael
===
Subject: Re: Extending a function
Originator: grubb@lola
> Let X and Y be topological spaces, let A be a closed
subset of X, and
> suppose f:A-->Y is continuous. Can f be extended to a
continuous
> function on X?
>OK with good conditions. Maybe X is normal, Y is completely
regular.
No. Not even then. Let X be the closed disk in R^2 and
A=Y=unit circle.
Let f be the identity. Then f has no extension to X. All of
these spaces
are about as nice as you want in terms of compactness,
connectedness
and separation properties.
--Dan Grubb
===
Subject: Re: Extending a function
> Let X and Y be topological spaces, let A be a closed
subset of X, and
> suppose f:A-->Y is continuous. Can f be extended to a
continuous
> function on X?
>OK with good conditions. Maybe X is normal, Y is completely
regular.
>No. Not even then. Let X be the closed disk in R^2 and
A=Y=unit circle.
>Let f be the identity. Then f has no extension to X. All of
these spaces
>are about as nice as you want in terms of compactness,
connectedness
>and separation properties.
huh. we all know the result is true under various conditions
on X if
Y = R. this raises the question of what's so special about 
R;
there's
-some- topological condition on Y that suffices...
>--Dan Grubb
************************
David C. Ullrich
sorry about the inelegant formatting - typing
one-handed for a few weeks...
===
Subject: Re: Extending a function
> Let X and Y be topological spaces, let A be a closed
subset of X, and
> suppose f:A-->Y is continuous. Can f be extended to a
continuous
> function on X?
>OK with good conditions. Maybe X is normal, Y is
completely regular.
>No. Not even then. Let X be the closed disk in R^2 and
A=Y=unit circle.
>Let f be the identity. Then f has no extension to X. All of
these spaces
>are about as nice as you want in terms of compactness,
connectedness
>and separation properties.
>huh. we all know the result is true under various conditions
on X if
>Y = R. this raises the question of what's so special about
R; there's
>-some- topological condition on Y that suffices...
You are at least going to need homotopically trivial;
otherwise, the cited example will hold.
--
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, Department of Statistics, Purdue University
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
===
Subject: Re: Extending a function
Originator: grubb@lola
> Let X and Y be topological spaces, let A be a closed
subset of X, and
> suppose f:A-->Y is continuous. Can f be extended to a
continuous
> function on X?
>OK with good conditions. Maybe X is normal, Y is
completely regular.
>No. Not even then. Let X be the closed disk in R^2 and
A=Y=unit circle.
>Let f be the identity. Then f has no extension to X. All of
these spaces
>are about as nice as you want in terms of compactness,
connectedness
>and separation properties.
>huh. we all know the result is true under various conditions
on X if
>Y = R. this raises the question of what's so special about
R; there's
>-some- topological condition on Y that suffices...
Define Y to be an Ôabsolute 
retract' if whenever A is a closed
subset
of a normal space X and f:A->Y is continuous, then f can be
extended
to X. The Tietze extension theorem says that [0,1] is an
absolute retract.
As you pointed out, so is the real line. Products of absolute
retracts
are again absolute retracts. Also, an absolute retract *is* a
retract
of every normal space it is embedded in as a closed subspace,
i.e. if
Y is homeomorphic to a closed subspace A of the normal space
X, then
there is a map p:X->A with p(a)=a for all a in A.
There are many variants of the idea of Ôabsolute 
retract'
such as
Ôabsolute neighborhood retract' and 
Ôabsolute Euclidean
neighborhood
retract'. All of these are defined similarly.
--Dan Grubb
===
Subject: Re: Extending a function
> Let X and Y be topological spaces, let A be a closed
subset of X, and
> suppose f:A-->Y is continuous. Can f be extended to a
continuous
> function on X?
>OK with good conditions. Maybe X is normal, Y is
completely regular.
>No. Not even then. Let X be the closed disk in R^2 and
A=Y=unit circle.
>Let f be the identity. Then f has no extension to X. All
of these spaces
>are about as nice as you want in terms of compactness,
connectedness
>and separation properties.
>huh. we all know the result is true under various
conditions on X if
>Y = R. this raises the question of what's so special about
R; there's
>-some- topological condition on Y that suffices...
>Define Y to be an Ôabsolute 
retract' if whenever A is a
closed subset
>of a normal space X and f:A->Y is continuous, then f can be
extended
>to X. The Tietze extension theorem says that [0,1] is an
absolute retract.
>As you pointed out, so is the real line. Products of
absolute retracts
>are again absolute retracts. Also, an absolute retract *is*
a retract
>of every normal space it is embedded in as a closed
subspace, i.e. if
>Y is homeomorphic to a closed subspace A of the normal space
X, then
>there is a map p:X->A with p(a)=a for all a in A.
and hmm, a retract of an absolute retract is an absolute
retract...
[but as far as i can see what this says about my question is
that
absolute retracts are absolute retracts.]
>There are many variants of the idea of Ôabsolute 
retract'
such as
>'absolute neighborhood retract' and 
Ôabsolute Euclidean
neighborhood
>retract'. All of these are defined similarly.
>--Dan Grubb
************************
David C. Ullrich
sorry about the inelegant formatting - typing
one-handed for a few weeks...
===
Subject: Re: Fast Binary-to-Decimal Conversion Algorithms?
> You'll be multiplying by constants much of the time - 
store
the FFTs
> for those halves, so that you can re-use them without
having to FFT
> again. (Doubles your space requirement, turns a cost-3
operation
> into a cost-2 one).
I haven't studied the FFT, which is something I really 
should
do, but
from a brief look at it it does look like there are some very
nice
optimisations that might be possible :-)
Simon
===
Subject: Re: Fast Binary-to-Decimal Conversion Algorithms?
> You'll be multiplying by constants much of the time - 
store
the FFTs
> for those halves, so that you can re-use them without
having to FFT
> again. (Doubles your space requirement, turns a cost-3
operation
> into a cost-2 one).
> I haven't studied the FFT, which is something I really
should do, but
> from a brief look at it it does look like there are some
very nice
> optimisations that might be possible :-)
In brief, for those who've not seen it before, and using
ad-hoc notation
F(a*b) = F(a)^F(b)
i.e.
a*b = Finv(F(a)^F(b))
where a, b, vectors representing the bignum, *=normal
multiplication
F=fourier transform, ^=pointwise multiplicaton,
Therefore, if multiplying by the same number b repeatedly,
you don't need to
compute F(b) each time, just do it once, and remember it.
You'll need one full-sized constant, one half-sized one, one
quarter-sized
one, etc. so the exrta storage is quite managable.
Compared to large FFTs, the cost of pointwise multiplication
^ is
negligible, and so the work is dominated by the fourier
transforms
and their inverses. So basically the caching of F(b) causes
the cost
to drop from 3 to 2.
I believe Cory Bloodwood and Joerg Arndt have interesting
pages on FFTs
that can be used for multiplication (I dragged those names
from my dim
distant memory, and may have them totally wrong!), as does
the book
/Prime Numbers, a Computational Perspective/ by Crandall &
Pomerance.
(Richard Crandall is one of the movers and shakers in the
field of FFT
multiplication.)
Phil
--
1st bug in MS win2k source code found after 20 minutes:
scanline.cpp
2nd and 3rd bug found after 10 more minutes: gethost.c
Both non-exploitable. (The 2nd/3rd ones might be, depending
on the CRTL)
===
Subject: Re: Bernoulli and Euler numbers
> I am looking for a good reference (book or paper) on the
properties of
> Bernoulli and Euler numbers, i.e., where relations like
those (possibly
> all known relations of this type)
>
http://functions.wolfram.com/IntegerFunctions/BernoulliB/23/
01/
>
http://functions.wolfram.com/IntegerFunctions/BernoulliB/27/
01/
> http://functions.wolfram.com/IntegerFunctions/EulerE/23/01/
> http://functions.wolfram.com/IntegerFunctions/EulerE/27/01/
> can be found with proofs.
> I have already checked Abramowitz Stegun, but only few of
them are there.
Here are two things that are only peripherally related, but
you may find interesting:
http://www.nemitz.net/vernon/powersum.txt
http://www.nemitz.net/vernon/SUMPOWER.txt
Because Bernoulli's and Euler's Numbers may 
not be the best
ones for all occasions. :)
===
===
Subject: Re: Euclid and textbooks
>I think I understand: Analytic proofs use techniques that
are
>general, whereas synthetic proofs start from the axioms and
requires
>a new developement for every problem.
> Both approaches use techniques that are general. Both
require a new
> development for every problem to the same degree. Which
approach is
> simpler depends on the problem at hand.
>For almost all problems, it is easier to do analysis than to
invent a
>synthetic proof. That is the reason for analysis to be done
at all.
>I believe that Euclid and the other Greeks would present
results in
>synthetic form because of its greater elegance.
> More likely because they lacked the machinery to do
otherwise.
>The Greeks never used algebra?
Not until much after Euclid, and even then, not that much
of it. It seems that the use of one numerical variable
was invented by Diophantus around 300 CE. The use of two
variables gets us already to the middle ages.
They did use words, and more analysis than one would think.
They had the concepts of limit and integral, although not
QUITE the modern ones, but could only express them in words.
They did use algebraic ARGUMENTS, even why they could not
express them in symbols. Euclid had unique factorization
of integers, infinitude of primes, and irrationality of
such numbers as sqrt(2). The Greeks also had numerical
approximation schemes, which required verbal algebra.
>Though analysis is the easier method with which to derive
results,
> Quite often it isn't.
>Synthetic derivations are often easier to understand, which
might be
>were easy to derive.
>Andrew Usher
--
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, Department of Statistics, Purdue University
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
===
Subject: Re: Euclid and textbooks
> Synthetic derivations are often easier to understand,
which might be
> were easy to derive.
> Archimede's method was analysis of the problem followed by
synthesis to
> establish the validity of the solution. In modern math
analysis and
> synthesis are combined. Algebra is as synthetic as it is
analytic.
> Bob Kolker
Can you give an example of what was Ôanalysis' 
for Archimedes,
and what was Ôsynthesis'? Analysis 
wasn't supposed to be
Ôhandwaving
heuristics' and Ôsynthesis' 
wasn't Ôchecking via
calculation', right?
--
Herman Jurjus
===
Subject: Re: Job application strategies
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i72FW7622836;
>This post not CC'd by email
> After thirty five years of teaching I found myself for the
first
>time unemployed. Each time a new position appears I
restructure
<snip>
35 years?
Is there a possibility that you are the victim of age
discrimination?
Now we all know that that could never happen..... (nudge,
nudge,
wink, wink, know what I mean....)
===
Subject: Re: Vernon's Prime Sieve
> // This overall prime-sieve algorithm was independently
created by
> Vernon Nemitz. Someone else
Your code is an ideal example of why writing code no longer
than 79 chars
pers
line is a Good Idea!
alex
===
Subject: Re: Vernon's Prime Sieve
> // This overall prime-sieve algorithm was independently
> created by Vernon Nemitz. Someone else
> Your code is an ideal example of why writing code no longer
> than 79 chars pers line is a Good Idea!
(Off-Topic)
That's actually pretty easy to do if one never comments the
code. Having spent years programming first on a hardware-
limited 32-column text screen, and later on a hardware-limited
80-column text screen, I relished the graphics-software-based
editors that let me get 100+ characters per line.
As far as UseNet posts are concerned, I've noticed that
about 71 or 72 characters is about as much as you can get
per line before it wraps. I could wish the underlying
code was updated to not do that. Modern browsers are
able to provide wide lines that wrap without needed to
be told by the server.
Anyway, in one of the later posts to this Thread, you will
find the code edited to fit with almost no 
wrapping.
You will also find evidence of its inefficiency, 
in
comparison to other sieve algorithms. So it goes.
Still, it was fun, and I don't regret it.
As an aside, I've figured out a way to take all 
the
primes that fit in 32 bits (203million+) and compress
those 812+million bytes down to about 130 million bytes.
Is that efficient or not?
===
Subject: Re: More number theory tidbits, paper?
> How many odd composites less than N are divisible by 3?
[followed by the proof that it's [(N-4)/6], where [x] =
ßoor(x)]
> But, think about it, how would you even know that these
form[ula?]s
> *might* work unless you just had some really good wild
guess or
> someone told you, like I did?
Why would I care, unless someone told me to derive it, as you
did?
It's not like there's some Deep Truth hiding 
in the fact that
six
equals two times three (which is really all you're saying,
there).
> Now consider the more complex:
> [N/5] - [N/10] - [N/15] + [N/30] - 1 = [(N-16)/10] -
[(N-16)/30], for
> even N>6
> Care to brute force that one?
Sure. It would of course be easier if we could remove all
those
ßoor notations, though. And let's break it down into two
parts:
R1 = [N/5] - [N/10] - [N/15] + [N/30] - 1
R2 = [(N-16)/10] - [(N-16)/30]
The first thing we notice (empirically) is that R1 not= R2 
only
for N=5 (mod 30) or N=25 (mod 30). So let's take that as a
starting
point.
[(N-16)/10] = [N/10]-2 for N < 6 (mod 10)
[N/10]-1 for N >= 6 (mod 10)
[(N-16)/30] = [N/30]-2 for N < 16 (mod 30)
[N/30]-1 for N >= 16 (mod 30)
Now we can cancel corresponding terms:
R1 =?= R2 (N not= 5, 25 (mod 30))
[N/5]-[N/10]-[N/15]+[N/30]-1 =?= [(N-16)/10]-[(N-16)/30]
/ [N/5]-[N/10]-[N/15]+[N/30]-1 =?= [N/10]-[N/30]-4
(N=0..4,10..15 m 30)
| [N/5]-[N/10]-[N/15]+[N/30]-1 =?= [N/10]-[N/30]-3
(N=6..9,20..24 m 30)
[N/5]-[N/10]-[N/15]+[N/30]-1 =?= [N/10]-[N/30]-2
(N=16..19,26..29 m
30)
/ [N/5] - 2[N/10] - [N/15] + 2[N/30] =?= -3 (N=0..4, 10..15 m
30)
| [N/5] - 2[N/10] - [N/15] + 2[N/30] =?= -2 (N=6..9, 20..24 m
30)
[N/5] - 2[N/10] - [N/15] + 2[N/30] =?= -1 (N=16..19, 26..29 m
30)
And of course it's easy to check this. At a glance, the
reader can
see that [N/5] ~ 2[N/10] and [N/15] ~ 2[N/30] (where ~
indicates is
approximately equal to), so it makes sense that the sum
should be
close to zero in all cases. (The slight discrepancies from
zero are
due to the mismatched steps in the various graphs of [N/k].)
I don't know why JSH thinks this is an interesting family of
equations. It's pure adhockery on the level of Algebra I.
Look,
here are five more:
[N/4] + [N/6] + [N/8] = 13*[N/24] for all N=0,1,2,3 (mod 24)
[N/100] = [(N-17)/100] for all N with the second-from-right
decimal digit >1
[(N+1)/2] = N/2 for all even N>42
[1] = N for all N=1
[[N]] = N for all integer N
-Arthur
===
Subject: Re: More number theory tidbits, paper?
> The result is that ßoor((N-4)/6) is the count of odd
composites that
have
> 3 as a factor given an even N that is greater than 2.
 It is possible to prove that it does in fact give that
count, but, how
did
> I get the formula?
 Well, here is how I would get it. I'd note that the odd
composites
that
> have three as a factor are 9, 15, 21, ..., so to count
them, I want a
step
> function that is 0 on [1,8], 1 on [9,14], 2 on [15, 20],
and so on.
 That is obviously ßoor((N-3)/6).
 If for some reason I was only interested in even N, I'd
note that I can
> shift the steps by one and still be correct for even N,
giving
> ßoor((N-4)/6).
 I'd probably stop at the more useful ßoor((N-3)/6), 
though,
since I
don't
> see any obvious reason why I'd prefer a formula that only
works for even
N
> when I can have one that also works for odd N.
 Well, to be fair, I'm not sure this is what 
I'd do, since
this is so
trivial
> I wouldn't be aware of the thought processes involved.
> Well it is easy enough as my thinking at the time was to
count primes
> but ignore evens.
> That is, I wanted the formula to be specifically only
interested in
> counting odd numbers, though I admit I missed at the time
that
> ßoor[(N-3)/6] would work, as I made a direct calculation
which gave
> me [(N-4)/6].
> (Using the notation [x] = ßoor(x).)
> Now then, from easy to still easy, as for 5, that is for a
relation
> that gives the count of odd primes NOT divisible by 3 that
are
Should be odd composites. Yeah, same mistake as earlier which
does
worry me, as I've more than once put primes where I should 
be
putting composites.
Then again maybe I'm just fixated on the point 
of the composite
counting exercise.
A bit over two years ago, I was counting primes. I derived
these
formulas back then in order to count prime numbers.
Basically when I set out deriving these formulas over two
years ago, I
figured it'd make sense to skip past evens in 
the count of
primes.
The formulas themselves are used to find remaining composites
to leave
a count of primes.
> divisible by 5, it's easy to get
> [(N-16)/10] - [(N-16)/30] for even N>6
> and my guess then is that the shift using [(N-3)/6] would
give
> [(N-15)/10] - [(N-15)/30] for N>6,
> correct?
I was a bit surprised that I missed [(N-3)/6] though in
figuring out
how I derived [(N-4)/6] I see that my focus on N being even
probably
is why.
James Harris
http://mathforprofit.blogspot.com/
===
Subject: Re: More number theory tidbits, paper?
Discussion, linux)
> I'm still waiting to see a derivation like my original.
James, you have said that you don't know what your original
derivation
was. When will you know when you see a derivation like it?
--
It seems to me that in wartime Americans shouldn't be
attacking each
other in this way on a *worldwide* forum. Then again, I know
I'm an
American, but I have no way of knowing that you are, which
would
explain a lot. --James Harris, on why Yanks should accept his
proof
===
Subject: Re: More number theory tidbits, paper?
> I'm still waiting to see a derivation like my original.
>James, you have said that you don't know what your original
derivation
>was. When will you know when you see a derivation like it?
evidently his derivation is like pornography in this regard...
************************
David C. Ullrich
sorry about the inelegant formatting - typing
one-handed for a few weeks...
===
Subject: Enumerative Graph theory
Is there a formula or asymptotic known for the number of
labeled
2-edge-connected graphs on n vertices with k edges?
2-edge-connected means at least 2 edges must be removed to
disconnect the
graph.
===
Subject: Re: All roots real for small degree polynomials
> What are the conditions for all the roots of a polynomial
(degree small)
> to be real?
> This must be well-established but I cannot find a reference.
> John McKay
Computing the Sturm sequences symbolically seems to work. On
quadratic
polynomials it gives the condition that the dicriminant is
positive.
For the polynomial x^3+a x^2 + b x + c it gives the conditions
that
a^2 > 3b
and
a^2 b^2 - 4 b^2 - 4 a^3 c + 18 a b c - 27 c^2 > 0
and those seem to work for cubic equations. A symbolic
algebra package
should be able to crank out the next few faster than I can
copy them,
but they seem to get unpleasantly complicated fairly quickly.
===
Subject: Re: Hillary Ann Taman - June 12th 1983
> Just post your full name and birthday and post your parents
and siblinks
> birthdays, let's have a look at your family for free!!!
DON'T DO IT - you will be haunted by this kook until the day
they finally
lock him up and take away his Internet access for good...
===
===
Subject: Re: Roman numerals (tombstones)
> What's the grammatically sense of ) that it gives
> I) 500 this doesn't make sense
> CI) 1000 in view of the this
> CCI)) 10,000 etc. continuing sequence
> I do noth think that there is a gramatical sense of )
> it is just a special font:
> (|) is a round M
Aah....
That would make a great deal of sense.
Then a date beginning CI)I) or (I)I), as many dates did,
should be read as MD or 1500. That would be in the right
ballpark for lots of these churches, which seem to have
had a big renovation craze in the 1600s in Renaissance style
but mostly date from the 13th or 12th centuries, and are
often built on churches even older.
- Randy
===
Subject: Re: Roman numerals (tombstones)
> According to my high school Latin text, the Romans counted
backwards,
> which is not quite subtraction, with their spoken
language. I never
> understood this and always meant to go back and figure it
out.
>Well, you have The Ides of March, but then you give a
certain number
>of days before the Ides or after the Ides. Until you get
close enough
>to one of the other markers to add or subtract from it.
Without getting and getting the book, I think the counting
backwards
had to do with which words were used to represent a number.
So if
17 was the number to be spoken, the base of the spoken word
would be the word for 20, either preceded or followed by the
word
for three. I can't recall. Note that this was the oral
language.
/BAH
Subtract a hundred and four for e-mail.
===
Subject: sum f(z^n) analytic on |z|<1 if f(0)=0 and f
analytic on |z|<1
I'm trying to prove that if f(z) is analytic on the open 
unit
disc D and
f(0) then
sum f(z^n) converges in D to an analytic function.
I've tried playing with the terms of the series of f(z^n),
trying to
show that you get absolute convergence for |z|<1. I think the
coefficient of z^j in the power series of f(z^n) is the sum of
terms c_j
where j|n, with repetiton(where f(z)= sum c_k z^k).
Is this the right way to go about this?
===
Subject: Re: sum f(z^n) analytic on |z|<1 if f(0)=0 and f
analytic on |z|<1
>I'm trying to prove that if f(z) is analytic on the open
unit disc D and
>f(0) then
>sum f(z^n) converges in D to an analytic function.
>I've tried playing with the terms of the series of f(z^n),
trying to
>show that you get absolute convergence for |z|<1. I think the
>coefficient of z^j in the power series of f(z^n) is the sum
of terms c_j
>where j|n, with repetiton(where f(z)= sum c_k z^k).
>Is this the right way to go about this?
i don't know which way is 
Ôright', but there's a simple 
proof
without using the power series: there exists a constant b such
that
|f(z)| <= b|z|
for all z with |z| <= 1/2...
David C. Ullrich
===
Subject: support of a finite borel measure
I'm trying to prove :
Let mu be a finite Borel measure in a separable metric space
X. Define
supp(mu)={x in X : mu(D)>0 for every open D containing x}. Let
G=X-supp(mu). Show that G is the largest open set such that
mu(G)=0.
I think its fairly easy to show that G is open and contains
all open
null sets.
I'm not sure of my proof that G is null:
By Urysohns Metrization theorem, X is second countable.
So take a countable basis {B_i}. Now G is the union of all
open null
sets D. This may be uncountable.
Can I just write each D as a union of B_i's and hence have G
as a
countable union of B_i's. And each of the 
B_i's used is
contained in
some open null set D and hence each B_i is null. Therefore G
is null.
One of my causes for apprehension is that the question was
set for a
course which i didnt think included Urysohn's metrization
thm. Is there
a better/more elementary way?
===
Subject: Re: support of a finite borel measure
>I'm trying to prove :
>Let mu be a finite Borel measure in a separable metric space
X. Define
>supp(mu)={x in X : mu(D)>0 for every open D containing x}.
Let
>G=X-supp(mu). Show that G is the largest open set such that
mu(G)=0.
>I think its fairly easy to show that G is open and contains
all open
>null sets.
>I'm not sure of my proof that G is null:
>By Urysohns Metrization theorem, X is second countable.
>So take a countable basis {B_i}. Now G is the union of all
open null
>sets D. This may be uncountable.
>Can I just write each D as a union of B_i's and hence have 
G
as a
>countable union of B_i's. And each of the 
B_i's used is
contained in
>some open null set D and hence each B_i is null. Therefore G
is null.
>One of my causes for apprehension is that the question was
set for a
>course which i didnt think included Urysohn's metrization
thm. Is there
>a better/more elementary way?
i never recall when one can show a measure is regular [because
where i come from it doesn't come up...]. if you can show 
that
mu is inner regular you're done, since any compact subset of
G is covered by finitely many open null sets.
************************
David C. Ullrich
sorry about the inelegant formatting - typing
one-handed for a few weeks...
===
Subject: Re: support of a finite borel measure
>I'm trying to prove :
>Let mu be a finite Borel measure in a separable metric space
X. Define
>supp(mu)={x in X : mu(D)>0 for every open D containing x}.
Let
>G=X-supp(mu). Show that G is the largest open set such that
mu(G)=0.
>I think its fairly easy to show that G is open and contains
all open
>null sets.
>I'm not sure of my proof that G is null:
>By Urysohns Metrization theorem, X is second countable.
>So take a countable basis {B_i}. Now G is the union of all
open null
>sets D. This may be uncountable.
>Can I just write each D as a union of B_i's and hence have 
G
as a
>countable union of B_i's. And each of the 
B_i's used is
contained in
>some open null set D and hence each B_i is null. Therefore G
is null.
>One of my causes for apprehension is that the question was
set for a
>course which i didnt think included Urysohn's metrization
thm. Is there
>a better/more elementary way?
> i never recall when one can show a measure is regular
[because
> where i come from it doesn't come up...]. if you can show
that
> mu is inner regular you're done, since any compact subset 
of
> G is covered by finitely many open null sets.
G separable => G second countable => G Lindelof (which is
easy). G is the
union of open null sets, hence is the union of countably many
open null
sets.
===
Subject: Re: About the evidence and relativity
 General Relativity is a Tower of Babel,
> that generates more heat than light,
> and wastes time, money and minds,
> on speculation about time travel, worm holes,
> warped space, gravity waves, etc.
 Potter fails to realize that this combination of quantum
mechanic
> (electronics, atomic clocks, etc.), special and general
relativity
> (dealing with gravitational and motional frequency shifts
which are
> so large that, without carefully accounting for numerous
relativistic
> effects, the system would not work), combined with rocket
technology
> have resulted in a new infrastructure that has already
spawned a 10+
> billion dollar industry... with expected growth for decades!
> What a marvelous application of the two great pillars of
physics!
Woacoaoua! I didn't know there are quantum electronic 
devices
around.
Mike
===
Subject: Re: About the evidence and relativity
 General Relativity is a Tower of Babel,
> that generates more heat than light,
> and wastes time, money and minds,
> on speculation about time travel, worm holes,
> warped space, gravity waves, etc.
 Potter fails to realize that this combination of quantum
mechanic
> (electronics, atomic clocks, etc.), special and general
relativity
> (dealing with gravitational and motional frequency shifts
which are
> so large that, without carefully accounting for numerous
relativistic
> effects, the system would not work), combined with rocket
technology
> have resulted in a new infrastructure that has already
spawned a 10+
> billion dollar industry... with expected growth for decades!
> What a marvelous application of the two great pillars of
physics!
> Woacoaoua! I didn't know there are quantum electronic
devices around.
> Mike
For example: Tunnel Diode
http://www.americanmicrosemi.com/tutorials/tunneldiode.htm
===
Subject: Re: About the evidence and relativity
Assume one has made a revolutionary intellectual discovery,
but has no
relevant academic or research merits at all, what should one
do?
- Just give it away, e-mailing it to (A) some experts, or (B)
to
usenet.
- Publish it oneself, making website for it.
- Try somehow get it published, in some form, in a science
journal.
- Other.
===
Subject: Re: About the evidence and relativity
> Assume one has made a revolutionary intellectual discovery,
but has no
> relevant academic or research merits at all, what should
one do?
> - Just give it away, e-mailing it to (A) some experts, or
(B) to usenet.
> - Publish it oneself, making website for it.
> - Try somehow get it published, in some form, in a science
journal.
> - Other.
It is hard to think of a revolutionary intellectual discovery
that has no academic or research merit. But if there was a way
to make money off the discovery, that is what I would do.
The closest example I can come up with are post-it notes.
While
brillant, they don't appear to have any academic or research
merit. Their inventer did make a pile of money off them. They
were revolutionary but it is arguable whether they were
intellectual.
--
Lance Lamboy
Go F*ck Yourself ~ Dick Cheney
===
Subject: Re: About the evidence and relativity
> General Relativity is a Tower of Babel,
> that generates more heat than light,
> and wastes time, money and minds,
> on speculation about time travel, worm holes,
> warped space, gravity waves, etc.
> Potter fails to realize that this combination of quantum
mechanic
> (electronics, atomic clocks, etc.), special and general
relativity
> (dealing with gravitational and motional frequency shifts
which are
> so large that, without carefully accounting for numerous
relativistic
> effects, the system would not work), combined with rocket
technology
> have resulted in a new infrastructure that has already
spawned a 10+
> billion dollar industry... with expected growth for decades!
> What a marvelous application of the two great pillars of
physics!
I am pleased to see that Sam Wormley
understands that General Relativity
is a vastly inßated pillar supported by
Kepler's Laws, Newton's Laws,
Ohm's Law, Maxwell's Laws, 
Kirchoff's Laws, and
hundreds of others Laws, and thousands of other technologies,
without which the overly inßated pillar could not stand.
It is interesting to see that G.R. cult/charlatans,
claim credit for all good things for their God,
just as other cults do.
No doubt quantum mechanics finds wide spread, and 
cost-efficient
usage, but as can be seen by its works,
General Relativity is a Tower of Babel,
that generates more heat than light,
and wastes time, money and minds on the pursuit
of time travel, worm holes, gravity waves, warped space, etc.
A mind is a terrible thing to waste.
--
Tom Potter http://home.earthlink.net/~tdp
===
Subject: Re: About the evidence and relativity
> Look!
 http://www.space.com/news/cosmic_shear_000512.html
 Don't be stupid.
> Oh god, scientific research that is 4 years old! THROW IT
OUT. ing
> hell.
>At the same time, astronomers admit that their new method
for finding
dark
>matter has not yet been tested enough to allow experts to
make a
definitive
>generalization about the fate of the universe. Since our
approach is
new,
>it's not very precise yet, said Wittman. Really strict 
tests
of the
theory
>will come in the next few years as astronomers measure the
[weak]
lensing
>more and more accurately.
> The topic was dark matter, not the fate of the universe.
Your
> inability to comment is noted.
http://www.bell-labs.com/org/physicalsciences/projects/
darkmatter/darkmatter
.html
> Learn something.
> Hey, who are these people? Have they done any work? Lets
take a look!
http://xxx.lanl.gov/find/astro-ph/1/AND+au:+tyson+abs:+lensing
/0/1/0/2000,19
99/0/1
> panties in a twist. Lensing is lensing.
>Don't be so emotionally attached to a model
>that you defend it with ad hominem like a religious zealot.
> Oh you wound me so with such harsh words!
>As can be seen, the General Relativity industry/cult
>latches on to any artifact, and tries to twist it to
>promote their religiously and economically bias agenda.
> It is called evidence, crackpotter. Obviously evidence does
not give
> any merit to a scientific theory in your 
compactified
dementia.
>General Relativity is a Tower of Babel,
>that generates more heat than light,
>and wastes time, money and minds,
>on speculation about time travel, worm holes,
>warped space, gravity waves, etc.
>hundreds of millions of dollars
>are wasted promoting the agenda of a small cult,
>and speculating about places and events
>that lie far, far beyond the reaches of man
>in time and spaces, great and small.
>A mind is a terrible thing to waste.
> ...and we have come full circle. You live on a Mobius strip
of
> stupidity, it seems. No matter where you start or which
path you take,
> you always end up where you began - as shown by your above
speil.
Eric Gisse makes a good point!
No matter where he starts out on his Mobius strip,
he ends up attacking the messenger,
rather than addressing the message in
a rational, intelligent, moral way.
I was pleased to see that Eric Gisse
admit that the evidence is that
General Relativity is a Tower of Babel,
generates more heat than light,
and wastes time, money and minds,
on speculation about time travel, worm holes,
warped space, gravity waves, etc.
A mind is a terrible thing to waste.
--
Tom Potter http://home.earthlink.net/~tdp
===
Subject: Re: About the evidence and relativity
> Look!
 http://www.space.com/news/cosmic_shear_000512.html
 Don't be stupid.
 Oh god, scientific research that is 4 years old! THROW IT
OUT. ing
> hell.
>At the same time, astronomers admit that their new method
for finding
>dark
>matter has not yet been tested enough to allow experts to
make a
>definitive
>generalization about the fate of the universe. Since our
approach is
>new,
>it's not very precise yet, said Wittman. Really strict
tests of the
>theory
>will come in the next few years as astronomers measure the
[weak]
lensing
>more and more accurately.
> The topic was dark matter, not the fate of the universe.
Your
> inability to comment is noted.
>http://www.bell-labs.com/org/physicalsciences/projects/
darkmatter/darkmatte
r
>.html
> Learn something.
> Hey, who are these people? Have they done any work? Lets
take a look!
>http://xxx.lanl.gov/find/astro-ph/1/AND+au:+tyson+abs:+
lensing/0/1/0/2000,1
9
>99/0/1
> panties in a twist. Lensing is lensing.
>Don't be so emotionally attached to a model
>that you defend it with ad hominem like a religious zealot.
> Oh you wound me so with such harsh words!
>As can be seen, the General Relativity industry/cult
>latches on to any artifact, and tries to twist it to
>promote their religiously and economically bias agenda.
> It is called evidence, crackpotter. Obviously evidence
does not give
> any merit to a scientific theory in your 
compactified
dementia.
>General Relativity is a Tower of Babel,
>that generates more heat than light,
>and wastes time, money and minds,
>on speculation about time travel, worm holes,
>warped space, gravity waves, etc.
hundreds of millions of dollars
>are wasted promoting the agenda of a small cult,
>and speculating about places and events
>that lie far, far beyond the reaches of man
>in time and spaces, great and small.
A mind is a terrible thing to waste.
> ...and we have come full circle. You live on a Mobius
strip of
> stupidity, it seems. No matter where you start or which
path you take,
> you always end up where you began - as shown by your above
speil.
>Eric Gisse makes a good point!
>No matter where he starts out on his Mobius strip,
>he ends up attacking the messenger,
>rather than addressing the message in
>a rational, intelligent, moral way.
Before I insulted you like you deserve, I did include address
your
message by including more detailed evidence since you were too
it, you wern't apparently interested in learning - just
bitching about
your personal pet peeve with GR.
>I was pleased to see that Eric Gisse
>admit that the evidence is that
>General Relativity is a Tower of Babel,
>generates more heat than light,
>and wastes time, money and minds,
>on speculation about time travel, worm holes,
>warped space, gravity waves, etc.
>A mind is a terrible thing to waste.
===
Subject: Re: Great-circle radius of ellipsoid
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i72HWfK01753;
> 2/pi b K(1 - (b/a)^2)
where K is the complete elliptic integral of the first kind.
> It is the co-Gaussian radius, based on the co-elliptic
integral
> of the second kind (co-EI2K).
>Did you just make up those terms? If not, I would appreciate
your
>citing some references in which they are used. I'm
suspicious. Why
>would anyone want to call something easily expressed in
terms of the
>complete elliptic integral of the _first_ kind the 
co-elliptic
>integral of the second kind?
>It might also be noted that Google web searches for
co-Gaussian
>radius and co-elliptic integral yield nothing.
>David Cantrell
> EI2K: f(@) = sqrt(1-sin(@)^2*e^2)
> co-EI2K: g(@) = sqrt(1-cos(@)^2*e^2)
> Gauss rad: fr(@) = b/f(@)
> co-Gauss rad: gr(@) = b/g(@)
Unfortunately, I don't remember where I read (or maybe
heard?) that
expression (it was either complementary or supplementary).
I'm not really into the calculus stuff. I just thought the
relationship was interesting, that's all. :|
Bud (I'll go crawl back into the woodwork now)
===
Subject: Re: publication problem
> I have not had this happen, but I once had a paper
accepted and
> then learned that the main result had been done previously
by
> someone else. I withdrew the paper immediately. Not to have
> done so could have destroyed my academic career. You have
put
> yourself at risk simply by posting your question.
> A wild exaggeration! After all, as a rule nobody reads
proofs
>anyway. We can safely put all sorts of nonsense into our
papers.
>Nobody's academic career was ever destroyed by an incorrect
proof,
>let alone by a paper replicating stuff proved by somebody
else.
This is not the case. It may be that not many reads
proofs, but enough can find errors in them that it is
likely to be noted, and peoples careers can be adversely
affected by instances of this. It depends on how many
and of what type.
As for replicating results proved by somebody else, unless
there is reason to believe the author knew about the
previous works, this is not likely to do much harm. The
discovery of the Fast Fourier Transform by Cooley and
Tukey is not at all diminished by the fact, found later,
that Gauss had done it a century or so earlier.
--
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, Department of Statistics, Purdue University
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
===
Subject: Re: publication problem
> proofs. Unfortunately, a lot of the results and conclusions
i made were
> based on that erroneous proof, so there's not much good
about the paper
> left.
> I'm not sure what to do... can I still make adjustments to
the paper once
> it has been published, perhaps to the online version?
Should i leave it
> the way it is, and hope that noone reads it? ...
> I need your advice
> Geoff
contact editor.. or do it like james harris and bust out the
champagne
and celibrate that you passed peer review (whatever the hell
that
stupid phrase means.. ask the idiot himself)
in short, since you're probably not anything like that
dooshwad..
contact editor
===
Subject: Re: Ampere and his cats
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i72HwUZ03662;
Hi to anyone who is looking in. I am trying to make a
promotional trailer
for children's TV that uses this fact. However before I can
proceed I need to
substantiate the claim and make sure that it isn't just an
urban myth.
Any solid information would be gratefully received.
Ian
Ian Woodisse
Acting Brand Manager BBC
BBC Television Centre,
London W12 7RJ
><pre Ampere had two cats, a big cat and a little cat. He wanted
the cats
> to be free to come and go, without his having to tend to
the door for
> them. He called a carpenter and instructed him to cut two
cat-doors
> in his door, a large one for the large cat and a small one
for the
> small cat. The carpenter asked, why not just make one
large door for
> both cats?
>That's a story I've heard told about Sir 
Isaac Newton, along
with
>another about how when his fire was too warm he told a
servant to move
>the fire away (instead of moving his chair back).
>According to another source, he actually invented the
cat-ßap.
>--
>hwyl/cheers,
>Philip Anderson
>Alenia Marconi Systems
>Cwmbr.89n, Cymru/Wales
></pre>
===
Subject: PBZ: nonstandard math
I, Professor Ben Zona, have invented something quite amazing
- I call
it nonstandard math. Nonstandard math is the same an regular
math,
except it's different:
We have the 2 axioms (a=b & b=c) -> a!=c.
and (a!=b & b!=c) -> a=c in nonstandard math.
Here are some amazing theorems. We all know that 2+1=1+2 and
1+2=3.
Therefore,
2+1!=3 in nonstandard math. Isn't that wierd? But what does
2+1 equal?
We can show that 2+1=4 in nonstandard math:
2+1!=3 (from before) and 3!=4. Therefore, 2+1=4 in
nonstandard math.
Can anyone prove any more interesting theorems in nonstandard
math?
It's sort of like analogous to Euclidean Geometry and
Non-Euclidean
Geometry. I predict that people will laugh at me now, but in
100 years
from now, everyone will be doing nonstandard math. There are
many
schools in America where nonstandard math is the norm, I am
told,
starting a trend.
Professor Ben Zona
===
Subject: Re: PBZ: nonstandard math
> I, Professor Ben Zona, have invented something quite
amazing - I call it
> nonstandard math. Nonstandard math is the same an regular
math, except
> it's different:
> We have the 2 axioms (a=b & b=c) -> a!=c. and (a!=b & b!=c)
-> a=c in
> nonstandard math.
> Here are some amazing theorems. We all know that 2+1=1+2
and 1+2=3.
> Therefore,
> 2+1!=3 in nonstandard math. Isn't that wierd? But what 
does
2+1 equal?
> We can show that 2+1=4 in nonstandard math:
> 2+1!=3 (from before) and 3!=4. Therefore, 2+1=4 in
nonstandard math. Can
> anyone prove any more interesting theorems in nonstandard
math?
Yes, you did not specify in your axioms the domains of a, b,
and c, so I
will feel free to use my own imagination.
Lance Lamboy != Craig Feinstein and Craig Feinstein !=
smart.
Therefore by Axiom 2, Lance Lamboy = smart.
Craig Feinstein != Lance Lamboy and Lance Lamboy != dumb.
Therfore by Axiom 2, Craig Feinstein = dumb.
> It's sort of like analogous to Euclidean Geometry and
Non-Euclidean
> Geometry. I predict that people will laugh at me now, but
in 100 years
> from now, everyone will be doing nonstandard math. There
are many
> schools in America where nonstandard math is the norm, I am
told,
> starting a trend.
> Professor Ben Zona
--
Lance Lamboy
Go F*ck Yourself ~ Dick Cheney
===
Subject: Re: fundamental unit in quadratic function fields
> You do *exactly* the same thing as in real quadratic fields.
> (OK to define remainder, one expands one's A + 
B sqrt(D)
> as a series a_n T^n + a_{n-1} T^{n-1} + ... + a_0 + a_{-1}
T^{-1} + ...
> and then takes the terms in nonnegative powers).
I found something similar in
[2] C.Friesen Continued fraction characterization and generic
ideals
in real quadratic function fields
But - maybe I'm a bit confused at the moment - how can I in
general
write any A+B sqrt(D) as a power series
a_n T^n + a_{n-1} T^{n-1} + ... + a_0 + a_{-1} T^{-1} + ...
as you recommended?
For instance if I set D = X^2 + X + 1 - how can I write
sqrt(D) as a power series?
Martin
===
Subject: Re: fundamental unit in quadratic function fields
> You do *exactly* the same thing as in real quadratic fields.
> (OK to define remainder, one expands one's A + 
B sqrt(D)
> as a series a_n T^n + a_{n-1} T^{n-1} + ... + a_0 + a_{-1}
T^{-1} + ...
> and then takes the terms in nonnegative powers).
> I found something similar in
> [2] C.Friesen Continued fraction characterization and
generic ideals
> in real quadratic function fields
> But - maybe I'm a bit confused at the moment - how can I 
in
general
> write any A+B sqrt(D) as a power series
> a_n T^n + a_{n-1} T^{n-1} + ... + a_0 + a_{-1} T^{-1} + ...
> as you recommended?
> For instance if I set D = X^2 + X + 1 - how can I write
> sqrt(D) as a power series?
sqrt(X^2+x+1) = X sqrt(1 + x^{-1} + x^{-2})
= x ( 1 + (1/2) x^{-1} + (3/8) x^{-2} + ...)
Robin Chapman
===
Subject: Re: Hartogs's aleph function, revisited
> An exercise from Chapter I of Kunen:
> Let A(x) = sup{a: a is an ordinal and there exists an
injection
> from a to x}. In ZF - Foundation, show that A(x) <
> A(P(P(P(x)))), where P denotes the power set.
>Fred Galvin and Herman Rubin replied with some helpful
hints. After
>thinking about this (albeit intermittently) for months, I
finally
>determined the solution. Is it really as easy as this?
(Kunen marks
>the problem as difficult. BTW, this is not homework.)
The approach is more general than the deleted one, and
certainly goes back well into the 19th century. A good
ordered pairs construction was not available at the
time, although it can be based upon the old one; I do
not know if this is ever pointed out. (If x and y are
elements, consider the relation xRx, and if x ~= y,
xRy and yRy. This gives the present ordered pair.)
Also, the idea of ordinal numbers as corresponding to
equivalence classes of well-orderings was known, but not
the current definition of ordinals, due to von Neumann
in his thesis.
Suppose one has a transitive reßexive antisymmetric
relation R. One can define for each t,
t* = {y: y R t}.
Then it is not difficult to show that u R v if and only if
u* is a subset of v*, and t can be identified as that u for
which u in t* and u* = t*.
So consider the set of all well-orderings of subsets of x.
Each such well-ordering is an element of P(P(x)) by the above
construction. So the equivalence classes are a subset of
P(P(P(x))). This is the argument which Hartogs 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, Department of Statistics, Purdue University
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
===
Subject: Re: sum f(z^n) analytic on |z|<1 if f(0)=0 and f
analytic on |z|<1
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i72IWS506979;
>I'm trying to prove that if f(z) is analytic on the open
unit disc D
> and f(0)
=0
>then sum f(z^n) converges in D to an analytic function.
>I've tried playing with the terms of the series of f(z^n),
trying to
>show that you get absolute convergence for |z|<1. I think the
>coefficient of z^j in the power series of f(z^n) is the sum
of terms
> c_j where j|n, with repetiton(where f(z)= sum c_k z^k).
>Is this the right way to go about this?
No, it's simpler than that. The mapping takes f(z) to
F(z)=sum f(z^n).
This is linear, so look what it does to monomials. It takes
z^k to
sum (z^n)^k = sum z^(nk) = z^k/(1-z^k). So it takes the power
series
sum a_n z^n to sum a_k z^k/(1-z^k), which clearly converges
to an analytic
function on D. And here's why we need f(0)=0:
the mapping takes the function f(z)=1 to F(z) = sum 1 =
infinity.
Nick
===
Subject: Re: PBZ: nonstandard math
===
>Subject: PBZ: nonstandard math
>I, Professor Ben Zona, have invented something quite amazing
- I call
>it nonstandard math. Nonstandard math is the same an regular
math,
>except it's different:
I tried that argument once and the teachers still marked my
test wrong.
>We have the 2 axioms (a=b & b=c) -> a!=c.
>and (a!=b & b!=c) -> a=c in nonstandard math.
So if a=c, then
(c=b & b=c) -> c!=c
How can something not equal itself?
>Here are some amazing theorems. We all know that 2+1=1+2 and
1+2=3.
>Therefore,
>2+1!=3 in nonstandard math. Isn't that wierd? But what does
2+1 equal?
>We can show that 2+1=4 in nonstandard math:
>2+1!=3 (from before) and 3!=4. Therefore, 2+1=4 in
nonstandard math.
>Can anyone prove any more interesting theorems in
nonstandard math?
>It's sort of like analogous to Euclidean Geometry and
Non-Euclidean
>Geometry.
Except for the fact that Euclidean Geometry and Non-Euclidean
Geometry
are consistent.
>I predict that people will laugh at me now,
Except for the fact that you're not funny.
>but in 100 years
>from now, everyone will be doing nonstandard math. There are
many
>schools in America where nonstandard math is the norm, I am
told,
>starting a trend.
And what schools would those be?
Is that a thinly disguised racist comment?
>Professor Ben Zona
===
Subject: Re: PBZ: nonstandard math
===
>Subject: PBZ: nonstandard math
>I, Professor Ben Zona, have invented something quite amazing
- I call
>it nonstandard math. Nonstandard math is the same an regular
math,
>except it's different:
> I tried that argument once and the teachers still marked my
test wrong.
>We have the 2 axioms (a=b & b=c) -> a!=c.
>and (a!=b & b!=c) -> a=c in nonstandard math.
> So if a=c, then
> (c=b & b=c) -> c!=c
> How can something not equal itself?
I have no problem with c!=c. Can you prove that they are
equal? For
instance, tomato!=tomato. You may say the word, tom-a-to,
where I
may say the word, tom-a-to. We see from this that they are
pronounced differently, yet they are spelled the same way and
represent the same object.
>Here are some amazing theorems. We all know that 2+1=1+2 and
1+2=3.
>Therefore,
>2+1!=3 in nonstandard math. Isn't that wierd? But what does
2+1 equal?
>We can show that 2+1=4 in nonstandard math:
>2+1!=3 (from before) and 3!=4. Therefore, 2+1=4 in
nonstandard math.
>Can anyone prove any more interesting theorems in
nonstandard math?
>It's sort of like analogous to Euclidean Geometry and
Non-Euclidean
>Geometry.
> Except for the fact that Euclidean Geometry and
Non-Euclidean Geometry
> are consistent.
>I predict that people will laugh at me now,
> Except for the fact that you're not funny.
>but in 100 years
>from now, everyone will be doing nonstandard math. There are
many
>schools in America where nonstandard math is the norm, I am
told,
>starting a trend.
> And what schools would those be?
> Is that a thinly disguised racist comment?
No. Only the schools that teach evolutionism.
Professor Ben Zona
===
Subject: Re: Widely used calculus books must be mediocre. --
Walter
Rudin
>I've been reading Walter Rudin's 
autobiography The Way I
Remember
>It (American Mathematical Society, 1997), and came across
this
>interesting statement on page 113:
> The Mathematics Department [of the University of Rochester
in
>1952] was small, not active researchwise, but a nice friendly
>group of people. Its chairman, John Randolph, had written a
>calculus book with Mark Kac which was too good to be widely
used.
>(Widely used calculus books must be mediocre.)
>What are the widely used calculus books of today? Are they
indeed
>all mediocre?
> Some are worse than mediocre, Thomas for example.
Thomas is basically a 50-year-old textbook, and I believe it
was well
regarded at that time. Surely it would be interesting to know
what
changed circumstances make the book mediocre now.
> What makes them mediocre? How do the mediocre
>calculus books differ from the good calculus books? What are
some
>good calculus books?
> By far the best Calculus book ever written is Courant's,
which dates
> from the 1920's. I don't think there ever 
was an actual
English
> translation, but Courant & John is derived from the
original and
> represents an unsuccessful attempt to make it an acceptable
U.S.
> college text. The level is too high for U.S. freshmen, and
the
> structure is different from the usual Ôadvanced 
calculus'
fare.
Courant has been published in several editions in English,
see for
example http://www.bookfinder.com
>share. I always appreciate recommendations of good books
about
>mathematics, and detailed criticisms of bad books.
>---
>Karl M. Bunday
>P.O. Box 1456
>Minnetonka, MN 55345
>http://learninfreedom.org
===
Subject: Re: Widely used calculus books must be mediocre. --
Walter
Rudin
> I've been reading Walter Rudin's 
autobiography The Way I
Remember
> It (American Mathematical Society, 1997), and came across
this
> interesting statement on page 113:
> The Mathematics Department [of the University of Rochester
in
> 1952] was small, not active researchwise, but a nice
friendly
> group of people. Its chairman, John Randolph, had written a
> calculus book with Mark Kac which was too good to be
widely used.
> (Widely used calculus books must be mediocre.)
> What are the widely used calculus books of today? Are they
indeed
> all mediocre? What makes them mediocre? How do the mediocre
> calculus books differ from the good calculus books? What
are some
> good calculus books?
> share. I always appreciate recommendations of good books
about
> mathematics, and detailed criticisms of bad books.
> Karl M. Bunday
> P.O. Box 1456
> Minnetonka, MN 55345
> http://learninfreedom.org
>Keep in mind that standards of mathematical rigor have
changed quite a
>bit in this country since 1952. Ah, but which way! Among the
>cognoscenti, for the better; among the average student?
Well, that
>depends on the college and the mathematics department. I
haven't read a
>lot of calculus texts (R.L. Moore taught the subject without
a textbook)
>but I certainly like Tom Apostol's books; they seem 
rigorous
enough for
>a first pass, and interestingly written.
The standards of mathematical rigor have not changed since
1952, and not much since Euclid. Euclid was, from the
modern standpoint, not quite rigorous, as he made assumptions
about properties of the real numbers, etc., and assumed that
rigid motions behaved precisely; one has to somewhat reduce
what can be done at the early stages. An axiom is no longer
a self-evident truth, but something assumed.
I understand that the publisher finally recouped the advance
made to Apostol for his excellent texts, which it was believed
would sell well. It was too difficult for most of the students
then, and they were better prepared than most are now. At
least
then they had a Euclid geometry course, and college algebra
with induction, instead of just facts, formulas, and
computation.
I do not believe it is ever easier to understand concepts
after
mastering computations, and this shows up in many places.
--
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, Department of Statistics, Purdue University
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
===
Subject: Re: Widely used calculus books must be mediocre. --
Walter
Rudin
> I've been reading Walter Rudin's 
autobiography The Way I
Remember
> It (American Mathematical Society, 1997), and came across
this
> interesting statement on page 113:
> The Mathematics Department [of the University of Rochester
in
> 1952] was small, not active researchwise, but a nice
friendly
> group of people. Its chairman, John Randolph, had written a
> calculus book with Mark Kac which was too good to be
widely used.
> (Widely used calculus books must be mediocre.)
> What are the widely used calculus books of today? Are they
indeed
> all mediocre? What makes them mediocre? How do the mediocre
> calculus books differ from the good calculus books? What
are some
> good calculus books?
>I bet by good he meant theoretical or abstract. That would be
good
>for math majors but difficult for the Lumpencalculat. :-)
>Michael
What is the use of teaching the Lumpencalculat how to
compute derivatives and antiderivatives, and such things
as l'Hopital's rule without understanding what 
limits and
derivatives are? What is the use of teaching how to
compute integrals without knowing what an integral is?
These can be done by computer programs, and often even
by pocket calculators.
The users of mathematics need to know what the concepts
are, so they can formulate problems which can be solved
by those who know how to compute, and which is more
important, formulating a problem correctly, or formulating
it incorrectly so the person knows how to get a closed
form solution of the incorrect problem?
It is not necessary for the non-mathematicians to be able
to produce proofs on demand, but they need to understand
what is a proof, and recognize the difference. They need
to understand the structure of the integers, not how to
do arithmetic, and especially proofs by induction. They
need to understand the structure of the real numbers, and
the geometric interpretations. These ARE abstract, and
should be so taught.
What now happens when an instructor tries to get the ideas
across. The students complain, Don't teach me the 
`theory'!
Just teach me how to do the problems on the exam. And if
the instructor persists, the students give bad ratings.
--
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, Department of Statistics, Purdue University
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
===
Subject: Re: which kind of problem is this ?
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i72JJIa10880;
>I am facing a problem like this.
>Given:
>y=f(x)
>y+y0=f(x+x0)
>(f is a real function, (x0,y0) are known
>(x,y) are both real, x is the indipendent variable).
>Then which x satisfy
>f(x+x0)-y0-f(x)=0
>That is given a change in the cartesian system,
>where will the curves intersect itself ?
>A suggestion about the solution method for
>trigonometric functions would be useful.
>Luciano
if x1 is solution then we have:f(x1+x0)=f(x1)+f(x0) these tree
values x0,x1,x0+x1 may be located on a same line c*x , from
y0=f(x0)
we've got f(x1)=x1*y0/x0 and f(x1+x0)= (x1+x0)*y0/x0 thus we
have
to solve:f(x)=x*(y0/x0) .
===
Subject: Fourier Transform Question
Suppose I have a signal x(t) with Fourier Transform X(jw). It
is known
that if one scales the time axis by a factor K, then the
frequency
axis is scaled by 1/K. I am curious about non-linear scalings
of the
time axis. Is there a way to derive the Fourier Transform of
x(exp(t)), given that one knows the Fourier Transform of
x(t)??
Bob Adams
===
Subject: Re: Fourier Transform Question
>Suppose I have a signal x(t) with Fourier Transform X(jw).
It is known
>that if one scales the time axis by a factor K, then the
frequency
>axis is scaled by 1/K. I am curious about non-linear
scalings of the
>time axis. Is there a way to derive the Fourier Transform of
>x(exp(t)), given that one knows the Fourier Transform of
x(t)??
With the change of variables exp(t)=s, and assuming
convergence
int_{-infinity}^infinity exp(-ikt) x(exp(t)) dt
= int_0^infinity s^(-ik-1) x(s) ds
I'd like to write this as
int_{-infinity}^infinity G(w) X(w) dw
where G(w) is the Fourier transform of Heaviside(s) s^(-ik-1).
That doesn't quite work when k is real because 
it's too
singular at s=0,
but you can either replace k by k+i epsilon or Heaviside(s) by
Heaviside(s-epsilon) where epsilon>0, and take the limit as
epsilon->0.
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia
Vancouver, BC, Canada V6T 1Z2
===
Subject: Re: Fourier Transform Question
> Suppose I have a signal x(t) with Fourier Transform X(jw).
It is known
> that if one scales the time axis by a factor K, then the
frequency
> axis is scaled by 1/K. I am curious about non-linear
scalings of the
> time axis. Is there a way to derive the Fourier Transform of
> x(exp(t)), given that one knows the Fourier Transform of
x(t)??
> Bob Adams
If I were a sadistic professor and you were an undergrad I'd
reply to
that by saying hey, do you want to get a Doctorate?
I don't think there'd be a general solution to 
that, but I'm
willing to
be surprised.
--
Tim Wescott
Wescott Design Services
http://www.wescottdesign.com
===
Subject: Re: continuity of norm function in R^n
>I am trying to study analysis on my own, using SA Douglass'
>Mathematical Analysis. Marvelous textbook, but a hard job
>On page 124 the text states that evidently f(x)=||x|| is
continuous
>for all c in R^n. That seems entirely plausible indeed to
me, but I
>tried to devise a proof based on the epsilon-delta definition
of
>continuity. That did not work out well, however. Anyone out
there who
>would care to get me back on the track?
> if you want to get back on track you should -read- the
replies
> to your -previous- posts!
> i mean really - because i'm typing one-handed right now it
actually
> took me several minutes to type the inequality in my
previous reply.
> you'd prefer i do that again, so you can ignore -this-
reply and
> post the question a fourth time?
Sorry about the triple posting. I am new to this discussion
group
thing and I could not find the message I posted, so I decided
to redo
it, and again. Since I found it now I'll save you that 
fourth
posting.
If in the mean time I could do something else to hasten your
recovery,
please tell me - I happen to be a physician (although my field
is
nuclear medicine, really).
this type of inequality, but I did not know it held true in
R^n. Now
that you mention it, I find no difficulty in 
proving it - but my
textbook did not mention it on the previous pages. Anyway, you
Frank
===
Subject: Re: continuity of norm function in R^n
I am trying to study analysis on my own, using SA Douglass'
>Mathematical Analysis. Marvelous textbook, but a hard job
>On page 124 the text states that evidently f(x)=||x|| is
continuous
>for all c in R^n. That seems entirely plausible indeed to
me, but I
>tried to devise a proof based on the epsilon-delta
definition of
>continuity. That did not work out well, however. Anyone out
there who
>would care to get me back on the track?
> if you want to get back on track you should -read- the
replies
> to your -previous- posts!
> i mean really - because i'm typing one-handed right now it
actually
> took me several minutes to type the inequality in my
previous reply.
> you'd prefer i do that again, so you can ignore -this-
reply and
> post the question a fourth time?
>Sorry about the triple posting. I am new to this discussion
group
>thing and I could not find the message I posted,
oh. i assumed that you just wanted someone to work out all the
details and intended to keep posting til you got a reply that
didn't require any work on your part. sorry.
dunno how you could be unable to find your post. i see
Ôreal' news reader - go to
http://www.forteinc.com/main/homepage.php
and get a free copy of Agent. then you also need a news
Ônews server' and find free ones 
and commercial ones
[if you want extreme reliability for a small fee
try giganews]
>so I decided to redo
>it, and again. Since I found it now I'll save you that
fourth posting.
>If in the mean time I could do something else to hasten your
recovery,
>please tell me - I happen to be a physician
it's just a cracked shoulder - the er guy said he 
didn't see
the point to anything other than just putting the arm in a
sling
and waiting for it to get better. first few days i decided he
was
a sadistic incompetent lunatic, but as of the last few days
it seems
he may be right, i doubt you can add much, unless your field 
is
like nuclear medicine or something...
>(although my field is nuclear medicine, really).
keen. i've read that if i take a few milligrams of plutonium
my
pain will go away in at most a few days. been wondering where
to go for a second opinion... [plutonium isn't addictive, is
it?]
>this type of inequality, but I did not know it held true in
R^n. Now
>that you mention it, I find no difficulty in 
proving it - but
my
>textbook did not mention it on the previous pages. Anyway,
you
>Frank
************************
David C. Ullrich
sorry about the inelegant formatting - typing
one-handed for a few weeks...
===
Subject: Re: continuity of norm function in R^n
[...]
> ...
> David C. Ullrich
> sorry about the inelegant formatting - typing
> one-handed for a few weeks...
>Dave,
>OT: Why?
cracked shoulder. only hurts when i try to use the arm [so i
don't...]
>Presuming you are recovering from an accident of some form,
get well
>soon.
> At least your keyboard's not broken.
>Also, poke out a statement of position on ubiquitous
naturals, and about
>proper classes being the empty set, and so on.
>Also, please hunt and peck a brief synopsis of functions
between the
>naturals and reals.
too hard, sorry. concepts like Ôubiquitous 
naturals' and
Ôfunctions between the naturals and reals' are 
beyond me...
>Ross F.
===
Subject: Probability for chaos
I have the system:
f(t + 1) = F(f(t))
where:
t - integer >= 0
F - everywhere continuous function
What is the probability that the above system is chaotic for
f(0) = a (a
real)?
Chris
===
Subject: Re: The Electoral College and combinatorics
> If you are a minority in some
> other state, you get no attention at all from either
candidate.
> Somehow that doesn't jibe with your claims about more 
value
> for more votes.
I already explained this. If you'd stop ranting for a while
and listen, you might learn something. (At the very least,
you wouldn't look so silly in arguing against something
which doesn't exist.)
> Bush could perhaps obtain more total votes by
> campaigning in Texas, courting various minorities and
special
> interests there, but he already has all the *electoral*
votes in
> Texas, so why bother?
If Bush already has all the electoral votes in Texas, _how
did that come about?_ Answer: The special interests,
minorities included, in Texas are satisfied with 
Bush's
platform. (Duh.)
> An excellent example, really, of how the Electoral College
> causes both candidates to ignore the interests of vast
numbers
> of voters and focus all their energies on the people who
just happen
> to live within certain state boundaries.
In order for this to be a good example, you need to explain
why all the people of Texas just accidentally vote for Bush.
Or are _you_ really the one who has contempt for the masses?
Bart
===
Subject: Re: The Electoral College and combinatorics
>A candidate can _legally_ buy an election of the
>popular vote, by advertizing. He can buy an elector
>only illegally.
> really? what federal statute makes it illegal to try
> to inßuence an elector's vote? and how do they enforce
> it? seems like they have to prevent electors from
> watching tv or reading the paper - i didn't realize
> they did that.
You were the one who was asking which was cheaper and talked
about corruption, (not inßuence.) I sure don't know what
the federal statute is, but I'd bet good money that there
is one that makes it a serious offense to bribe or coerce an
elector.
> ignoring the question of whether the way things -seem-
> to -you- has any place in an argument that iirc started
> with a discussion of whether something is 
Ôlogical':
> what evidence do you have that the typical elector
> arrives at his opinions for less idiotic reasons than
> the general public?
Let's not quite ignore the question. The OP said What
possible reason could there be for keeping the EC? in a
rhetorical sense, meaning that she believed firmly that
there could be no sensible reason. To gainsay her, I
don't have to prove that the EC is the best or even in
the top ten. All I have to do is present a reasonable
argument that there might well be advantages to the EC,
which makes her argument by rhetorical question pretty
ineffective.
To your question, the elector doesn't vote his opinions
in the same sense that a voter does. An elector is
_supposed_ to vote as directed, and in some states, it
appears, he is bound. And, lo, almost everytime, the
electors vote they way they are directed.
Bart
===
Subject: Re: The Electoral College and combinatorics
>A candidate can _legally_ buy an election of the
>popular vote, by advertizing. He can buy an elector
>only illegally.
> really? what federal statute makes it illegal to try
> to inßuence an elector's vote? and how do they enforce
> it? seems like they have to prevent electors from
> watching tv or reading the paper - i didn't realize
> they did that.
>You were the one who was asking which was cheaper and talked
>about corruption, (not inßuence.) I sure don't know 
what
>the federal statute is, but I'd bet good money that there
>is one that makes it a serious offense to bribe or coerce an
>elector.
the distinction between corruption and inßuence can be
pretty subtle. like various other distinctions here:
you're pretty certain it's illegal to bribe or 
coerce
an elector. i'm -very- certain it's also 
illegal to
bribe or coerce a voter.
the voters can be inßuenced through advertising.
the electors can be inßuenced through advertising.
not much of a distinction regarding what's legal or
possible as far as i can see. now, whether we're
talking about legal or illegal forms of inßuence,
which do you suppose is cheaper, inßuencing a few
hundred electors or a few million voters?
> ignoring the question of whether the way things -seem-
> to -you- has any place in an argument that iirc started
> with a discussion of whether something is 
Ôlogical':
> what evidence do you have that the typical elector
> arrives at his opinions for less idiotic reasons than
> the general public?
>Let's not quite ignore the question. The OP said What
>possible reason could there be for keeping the EC? in a
>rhetorical sense, meaning that she believed firmly that
>there could be no sensible reason. To gainsay her, I
>don't have to prove that the EC is the best or even in
>the top ten. All I have to do is present a reasonable
>argument that there might well be advantages to the EC,
>which makes her argument by rhetorical question pretty
>ineffective.
fair point. but you haven't yet come up with anything
that seems to -me- like an advantage to having an ec.
it would be more harder or more dangerous to buy the
ec than the general public? i don't see why - see above.
my favorite was the idea that the ec saves us from
bad choices. you seem to have missed my comments on
that:
in any case, the ec has not protected us from
getting drunk drivers as president - i imagine
dui is not a felony but it certainly seems to
me to be the sort of thing that -should-
disqualify a person.
is going awol from military service a felony?
i don't know, but considering that deserters
[in a slightly different context] used to be shot
it doesn't seem impossible.
>To your question, the elector doesn't vote his opinions
>in the same sense that a voter does. An elector is
>_supposed_ to vote as directed, and in some states, it
>appears, he is bound.
and in half the states he is not so bound - he's 
Ôsupposed'
to -only- in the sense that that's what's 
traditional.
which amounts to exactly what i said: at least in half
the states, and in -all- the states as far as the
constitution is concerned, a voter's vote has -zero-
legal inßuence on the outcome of the election.
yes, zero. absolutely none whatever: in half of the
country if -every- voter votes for A and then -every-
electoral vote goes to B the voters have no legal
cause to complain.
> And, lo, almost everytime, the
>electors vote they way they are directed.
almost every time.
if murder were legal almost nobody would kill
anyone. that doesn't seem to me to be a very
good argument for legalizing murder.
>Bart
************************
David C. Ullrich
sorry about the inelegant formatting - typing
one-handed for a few weeks...
===
Subject: Re: The Electoral College and combinatorics
<snip and in half the states he is not so bound - he's 
Ôsupposed'
to -only- in
> the sense that that's what's traditional. 
which amounts to
exactly what
> i said: at least in half the states, and in -all- the
states as far as
> the constitution is concerned, a voter's vote has -zero-
legal inßuence
> on the outcome of the election.
IIRC, Kerry picks some elector-candidates, Bush picks some
elector-candidates, and whoever else is running picks
elector-candidates.
They are listed on the ballot under the name of the person
who picked
them. If a voter votes for a Bush elector and the Bush
elector then votes
for Kerry that would imply that one of Bush's top loyalists
changed his
mind. That would suggest that Bush couldn't find 
50 or so hard
line
loyalists in each and every state.
> yes, zero. absolutely none whatever: in half of the country
if -every-
> voter votes for A and then -every- electoral vote goes to B
the voters
> have no legal cause to complain.
What if A dies and B was the second choice of most A
supporters. For
example assume most Kerry-electors win, but Kerry dies before
the
electoral college vote. It is not obvious what should be
done. But if
the electors vote for Edwards then they are probably
respecting the wishes
of their constituents.
> And, lo, almost everytime, the
>electors vote they way they are directed.
> almost every time.
> if murder were legal almost nobody would kill anyone. that
doesn't seem
> to me to be a very good argument for legalizing murder.
>Bart
> ************************
> David C. Ullrich
> sorry about the inelegant formatting - typing one-handed
for a few
> weeks...
--
Lance Lamboy
Go F*ck Yourself ~ Dick Cheney
===
Subject: Re: The Electoral College and combinatorics
|Actually Electoral College makes each vote less powerful.
For example:
|suppose a nation has 9 people, and divided into 3 states
with 3 peple
|in each state.
|Then, the chance the your vote will be important is: 8!/(4!
cdot 4!)
|times 0.5^8= 0.2734275 in a popular election (when each of
the two
|condidates have 4 supporters out 8 people, except you).
Hoever, chance
|that your vote is important in Electoral College is: 0.5
times 0.5
|= 0.25.
|It is good or not, fair or not fair, I really do not know.
perhaps it was Mathematics Magazine-- about this phenomenon.
The
author claimed this reasoning was weak because it assumes
each voter
is equally likely to vote either way. As I recall it
sometimes goes
the other way if the probability is not 1/2. Of course, if the
probability deviates very far from 1/2, the chance that the
result
will hinge on one person's vote is relatively small anyway.
problem. Assume a voting system for a population P with N
people.
Initially assume for simplicity that they are deciding
between two
alternatives. The system consists of a family W of subsets of
P that
is upward closed, and such that for each subset X of P, just
one of
X and P-X is in W. (When the vote is taken, the winner is
decided by
which side's voters form a set in W.) Also for fairness
assume that
the permutations of P that leave W invariant are transitive.
If we
assumed that W was invariant under S_N, this would single out
the
majority-rule system, but if the symmetries are transitive,
then there
is at least some sense in which each voter has an equivalent
role in
the vote.
Given all that, what system maximizes the probability that a
given
voter's vote will be critical? Initially assume for
simplicity that
each voter votes either way with probability 1/2.
I have long had some curiosity (although not enough to put in
the
effort to answer the question) about the Nash equilibria of
voting
systems, modelled as games. Part of the problem is that for
many of
the ways of modelling voting systems as games, there would be
way
too many Nash equilibria. The condition of Nash equilibrium
makes
no requirement for agents if (given the strategies of all
other
agents) each of their own strategies gives them the same
payoff. This
tends to be the case when voting (i.e., when there are enough
other
voters who have decided to vote deterministically to ensure
one option
wins).
One can partially fix this by assuming a small perturbation
then
letting it go to zero. For instance, we could imagine a small
probability of each voter waking up on the wrong side of bed
that
morning and voting randomly. That would result in the
probability of
one's vote mattering being small but nonzero. Then one could
let the
probability go to zero.
The simplest sort of equilibrium would be one without ties. It
would make one option A liable to win, and leave a second
option B
as the one most likely to accidentally become a winner due to
the
perturbation. As the probability of waking up on the wrong
side of
the bed goes to zero, the ratio between the probability of B
winning
and the probability of anything other than A and B winning
goes to
zero. Hence to have an equilibrium without ties, the voters
who
prefer B to A will vote as if B is their first choice (a
notion which
makes sense in most ordinary voting systems).
I think it's interesting how theory fails to guide us in 
such
situations. With M options, we get at least M choose 2
equilibria,
where any two of the options have become the options and
voters
have lined up on either side. The only option that can't be
chosen
by voters rational in this sense is one that would lose in
pairwise
contests against all the other options. Any other option can
in
principle win by voters winding up resigned to its being the
only
viable alternative to some other option they find worse.
Actually,
there seems to be some degree of realism in this model. :-)
Economists and game theorists have considered stronger forms
of
equilibrium. One would like, for example, to have an
equilibrium
such that no subset of the agents can get an outcome
preferable to
all of them by collectively switching their strategies. In a
voting
system, this means that for there to be one likely winner, A,
there
must not be a second option B that would win in a pairwise
contest
with A. In a majority rule system such an option is known as a
Condorcet winner. If one does not have majority rule, I 
don't
think
this condition is called being a Condorcet winner anymore,
although
it is analogous.
Our voting system is not designed to put a candidate into
office who
satisfies this condition, but there are systems that are. I
tend to
think it would be good. It's been criticized as possibly
selecting too
bland a candidate, someone who merely manages not to annoy
people.
I'm agreeable to this possibility, though. It would also 
open
the
door to third-party candidates who might be preferred by most
people
to either of the major party candidates. (One does not
always, however,
have a Condorcet winner, which prevents the kind of strong
equilibrium
I described from always existing.)
Concerning the electoral college....
|It gives them voting power beyond there population size.
Each state gets
|two electors simply because it's a state(in addition to 
what
they get
|because of population). When enough small states combine
they can keep
|the large states from running roughshod over them by
blocking them in
|the Senate or for that matter in an election.
I see a certain wisdom in having two legislative chambers, so
that
when nobody is able to get the kind of support needed from
each,
legislation just stalls.
I think giving voters additional power merely because 
they're
spread
out geographically is not a good thing, however. This idea
that your
vote (for national elections) shouldn't count differently
depending
on where you live may seem like a mere aesthetic preference,
but it's
relevant when, for example, much of the rural poor moves into
cities
to look for work and thereby loses inßuence.
It's true that there is a possibility of a majority composed
of voters
in larger states running roughshod over the complementary
minority.
This is a possibility for most natural ways of dividing the
country
into two nearly equal halves, U and V, where U is able to
prevail over
V in elections, however. Arranging the system to protect V by
making
it able to overcome U does protect V, but enhances the
possibility of
either V, or even a majority containing V, of running
roughshod
over its complement in U.
|> Sorry, we disagree on this. If you think in general it is a
|> good idea to go against the will of the majority, you are
|> abandoning the idea of a democracy in favor of some form of
|> elitism or oligarchy.
|
|You're still confusing two things. I've said 
nothing like
|it's a good principle to go against the will of the 
majority.
|It's a huge (and unfounded) leap of logic to go from 
support
|for EC to we should always oppose the will
|of the majority. The issue has two sides, but the two
|sides are not for/against the will the majority. The sides
|are how will we solve the problems inherent in a popular
voting
|system? One solution: don't solve them because the 
solutions
|are worse than the problem. Another solution: write the
|Constitution so as to avoid some of the serious problems
without
|infringing too much on people's will.
Most of the ways that the Constitution limits majority rule
are
based on some consistent reason to think that a majority
opinion
is wrong. For example, when a majority decides that a minority
should just plain shut up, and decides that the government
should
enforce this preference, they are reliably wrong, and the
system
is meant to prevent such injustices. We can't of course
hard-wire
preventatives against all such errors; the ones we can write
into
the constitution are just the ones where we can state a
general
principle that holds well without having to look at the
details.
I find it hard to think of a geographical pattern of voting 
for
president that reliably indicates that the majority is liable
to be
wrong (independent of the issue that might be causing the
split).
Suppose one has a candidate who has gotten 60% of the vote in
states having 60% of the electoral college votes, and 30%
elsewhere.
If electoral college votes were proportional to voting
population,
this would mean a slim majority was opposed to this
candidate. This
is the sort of candidate for whom the electoral college tips
the
scales. I don't find it at all convincing that 
we have good
reason
to think that the 40% in his base and the 70% elsewhere are
more
likely to be wrong, more likely to have been bought off by
pork, or
whatever. Oh sure, it *might* be, and maybe you're glad that
it works
out that way in specific instances, but is there any 
consistent
reason to think it's the majority making the mistake?
Americans at the revolution were much more likely to be
strongly
affiliated with one state, and to have 
state-specific interests
(aside from spending earmarked by Congress for them) that
might
need protection. Now, many of the state-specific interests
really
are just pork, and protecting them tends to involve keeping
military
bases in borderline states open and the like. One effect of
the
system has been that larger states tend to subsidize smaller
states.
This is interesting given that there's an inverse 
correlation
between
the size of a state and how conservative it is (due to states
like
Wyoming).
Even as a mechanism for protecting a state from being run
over, I
don't see the electoral college as good.
One can think of the problem of designing a voting system as
akin
to a statistical inference problem (although there are *of
course*
big differences). The system as it now exists behaves as
though the
evidence for a candidate being better than another, provided
by the
voters in a given state, were a step function with one big
step right
at the point where they would tie. My intuition is quite the
opposite.
The voters in a state seem usually to wander around easily in
the
40-60% range on many issues. I don't think 
that's very
informative.
Someone managing to start persuading part of that 40% who
usually are
automatically on the opposite side-- *that's* interesting.
If, for example, some state were genuinely about to be
trampled on,
say by the rest of the country deciding to use it as a waste
dump,
the best evidence for this I can imagine appearing in a
presidential
election is that they would vote in a big way against the
candidate
favoring doing the trampling. But 90% of Arizona getting up
in arms
is equivalent as things stand to 51% of Arizona having a mild
preference for the other candidate. In fact, if they were to
approach
the 90% mark, the farther they went, the less the candidates
would
consider it necessary to pay any attention to them.
People also have this way of pretending that 60%=100%. The
town where
I grew up gets made fun of by being called the People's
Republic of
Boulder, on account of its voters being only 40% Republican.
It
would be really obnoxious if it were treated, not just by
supercilious
pundits but by the political process itself, as if it were
100%
Democrat rather than just around 55% as it actually is.
Of course, for purposes of the presidential election, it gets
treated
as if it were 100% Republican, because it's in Colorado. 
(Its
newspaper
editorialized against Bush all through the 2000 campaign, and
then
endorsed him, because it's part of a big newspaper chain 
now,
that
requires it.)
This is why I think that, even more than eliminating the
electoral
college, it would do us good to switch to a proportional
representation
system. That way, minorities having common political interest
would
have more of a visible face in the deliberations.
|> For instance, in Texas, Kerry will probably get 40% of the
vote, but
|> these votes will count for nothing.
|
|In one sense all people who vote for the loser can claim
their
|vote counted for nothing, since their candidate was not
elected
|and so their will will not be carred out. But that's hardly 
a
|good sense.
True. There's a different sense here, though. With another
system,
whether your vote is a deciding vote might still depend on
what is
going on outside of your state. If the rest of the country
tipped
closer to balance, whether he annoyed Lipscomb county in some
way
might matter. Now, it would be okay, so long as the rest of
Texas
was happy with it.
|Bush will get to ignore Texas (and some other states). But
|he still has to not piss us off too much, so at least
passively
|the Kerry supporters still have clout. The votes still have
to
|be earned, even though in Texas earning means not screwing
|up so horribly that people notice.
Well, this is true as far as it goes, but it seems to me that
it
makes for a very irregular pattern of concern. The spotlight
can get
focussed very intensely on some interest in some state, e.g.
in
separating Elian Gonzales from his father. Sometimes of
course this
has a good effect. But it's haphazard, and has a 
compensating
way of
leaving minorities within states that generally oppose them
politically with less access to mediation from less biased
outsiders.
Mathematically speaking :-) the incentive is very
nonproportional to
the magnitude of the irritation.
When Bush Sr. said that he didn't consider atheists
Americans, I
doubt this was because he thought that it would please more
people
than it would displease. But the people it might please would
be
concentrated in part of the country where he was eager to tip
the
balance his way. The people he would displease would tend to
be in
states he was already ready to write off. Atheist-sympathetic
libertarians in Massachusetts could hardly matter less.
|It's not so much that California overpowers Nebraska, but
|that smaller groups within Nebraska now get some attention
|paid to them. The candidates have to be more micro in
|there platform construction since there has to be a strategy
|to win the state. If they don't, they lose the entire
|state. In a two state model, it probably doesn't matter.
|But with 50 states, the candidates must have 50 plans
|(one plan is give up, of course.) And each plan must
|be fairly local. _If_ the candidate wants the Nebraska
|electoral votes, then he has appeal to groups of farmers
|who want more irrigation rights and ranchers who want
|beef prices protected and the Omahans who are tired of
|the minority agri-business community trying to run
everything,
|and the religious mood of the state etc.
|
|As another poster said smaller groups are protected.
You're concentrating on the way small differences can be
amplified.
Some shifts of 2% are treated as being as if they were shifts
of 50%,
or the entire state. Super. But you can't just amplify the
inßuence
of everybody. The amplification of some inßuences only 
works
because
others are damped down.
As I see it, there should be an equal incentive to persuade
the
ranchers (by appealing to principle, one hopes) regardless of
how
the rest of Nebraska votes. Instead, the incentive is either
extra
big (if Nebraska becomes borderline) or extra small (if the
rest of
them are happy).
Some smaller groups are temporarily protected, others are
temporarily
made irrelevant.
|>1. Let's get a referendum going to raise minimum wage to
|>$100/hour. Such a thing would surely pass.
|>
|> It would not! In places that have referendum powers now it
|> has not happened or come within miles of happening.
|
|Put it on a ballot and see what happens.
The aforesaid People's Republic of Boulder voted down a
measure
to raise the minimum wage to something like $9.50. Really,
you're
just wrong about this. Plenty of people worry about the
potential
problems of it.
I think I understand why people seem dumb to you, but 
they're
not as dumb as you think. (I've read your postings at 
various
times in the past.) Don't confuse apathy with stupidity; the
effect is often similar, but apathy can sometimes be cured.
|Yep. It's one of the dangers of our lack of civics
instruction.
|The Constitution is all about States having control, not
Federal
|government. People keep ceding control to the Federal
government
|rather than minding their state business.
But the Constitution is *not* all about States having
control. The
Articles of Confederation left states much more in control,
and they
realized it wasn't working well. Even then, states righters
were
complaining that the Articles prevented states from being
fully
sovereign, e.g. having their own foreign policy.
Madison considered various confederations for clues as to
what made
them either hold together or break apart later. He was of the
opinion
that it was really important that Congress have a veto power
over the
passing of state laws! He was quite disappointed when this
was not
incorporated into the Constitution.
It seems as though lately the framers' talk about preventing
tyranny of the majority is being portrayed as if the only kind
of majority they were worried about was a national majority,
with
special emphasis on its oppressing a local (state) minority.
Madison
was however concerned with local majorities tyrannizing over
local
minorities. (And I think the Founders created a system which
made it
too easy to do that, anyway.)
For example, he was concerned with incidents where hotheads
from one
state decided that they needed to go to war with the Indians.
It's
easier to whip up the passions in one small area than it is
to get the
whole nation to feel the same way.
If, however, you have to traipse your way all the way to
Congress and
explain to your fellow contrymen from many different places,
people
with a sense of loyalty to you, but disinterested, why it is
a good
idea to have a war, just having to reason with them will tend
to weed
out some of the nonsense. That's the brilliant thing about
federalism;
it turns one of their biggest worries, that the country might
be
simply too big to be self-governing, (sometimes) into an
asset.
Now of course we are as a whole much more like the small
community
that is vulnerable to being manipulated into favoring war,
without
feeling the need to explain our reasons to anybody else.
States
rights just makes it easier to tell people to go Cheney
themselves.
|And this gets pointed out every time the issue comes up: The
|World Series winner is determined by who wins the most games,
|not who tallies up the most total runs for the seven games.
But isn't that just for the drama of it? :-) Does anybody
argue
that this is the optimal way to measure some intrinsic fitness
of the team?
Keith Ramsay
===
Subject: Re: A dead subject
> The quadratic formula has been around for thousands of
years. I am
> extremely surprised that no one has been able to derive a
better
> version of it to solve second degree equations, so, I will
give you a
> better version of the formula.
 Taking into consideration second degree equations in
general,
> ax^2 + bx + c,
> set the expression like so... ax^2 = bx + c then identify
> a,b,and c as you see them on the page. Next use

> x = (b +-sqrt(b^2 + 4ac))/(2a)
 This version will give you the correct roots every time.
Notice that
> this version has 2 less minus signs and does not require
setting the
> quadratic equal to zero. For these reasons it is a better
version. I
> will give you some time to digest this and let it sink in.
For those
> of you that are interested I have 3 distinct ways of
deriving this
> version mathematically and will do so for you in the near
future.
 Laserman
> Did you know that the thousands of years old formula was
> actually many formulas? Before they believed in
> negative numbers (or zero), they did quadratics in cases:
> a x^2 + b x = c
> a x^2 = b x + c
> a x^2 + c = bx
> a x^2 = b
> a x^2 = b x
> b x = c
> It seems you have done case 2 only...
No you are incorrect, I have used case 2 to make a formula
that is
designed to lend itself easily for computation, even easier!
I have
made the formula easier, do you understand? All those
mathematicians
for thousands of years solving quadratic equations in all the
forms
you mentioned above and yet not one of them clever enough to
see the
better formula. NOT A ONE. You people do not WANT to see it,
as I
lay it before you plain as day. We are like Gallileo against
the
inquisition where you try to convince me to recant and say my
version
of the facts is not good enough because you refuse to see the
light.
Anybody else except Herman Rubin interested in the derivation?
===
Subject: Simple synthetic proof of an obvious geometric
theorem?
Originator: tchow@lagrange.mit.edu.mit.edu (Timothy Chow)
This problem is on the mathematical side for rec.puzzles, but
it's
completely elementary, so I'm crossposting it.
Charles Silver asks for a simple, elegant proof of the
following
obvious
theorem in high-school Euclidean geometry. In his words:
< GIVEN: Triangle ABC, point D on side BC, and point E on
side AC.
< Imagine a line segment connecting A to D, and one
connecting B to E.
< AD=BE. Furthermore, the lines AD and BE are angle
bisectors, meaning
< that angle ABE= angle DBE; and similarly angle BAD = angle
EAD.
<
< TO PROVE: ABC isosceles.
<
< If you draw the lines, ABC should be obviously isosceles,
but it's
< difficult to prove. I'm interested in a 
_direct, elegant
proof_, not a
< reductio one (e.g., not one that assumes for instance that
the two sides
< are not equal and then shows that the lengths of the angle
bisectors
< could not be equal). Also, I'm _not_ interested in an
algebraic proof
< (e.g., not one using formulas for the lengths of angle
bisectors,
< or whatever). I'd like to see a so-called pure (direct)
geometric
< proof that a student taking elementary geometry might
arrive at using
< constructions, triangle congruences, and so forth.
<
< I once knew such a proof and perhaps with great
concentration I could
< reproduce it. However, I've given this problem to many
smart math
people,
< who to a person have presented me with very long purported
proofs where
< about 30 lines into their proofs they tacitly assume
something that's
< equivalent to angle CAB = angle CBA, which of course is
tantamount to
< assuming what is to be proved.
--
Tim Chow tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the
artillery---however great, will
never exceed four of those miles of which as many thousand
separate us from
the center of the earth. ---Galileo, Dialogues Concerning Two
New Sciences
===
Subject: Re: Simple synthetic proof of an obvious geometric
theorem?
>This problem is on the mathematical side for rec.puzzles,
but it's
>completely elementary, so I'm crossposting it.
>Charles Silver asks for a simple, elegant proof of the
following
obvious
>theorem in high-school Euclidean geometry. In his words:
>< GIVEN: Triangle ABC, point D on side BC, and point E on
side AC.
>< Imagine a line segment connecting A to D, and one
connecting B to E.
>< AD=BE. Furthermore, the lines AD and BE are angle
bisectors, meaning
>< that angle ABE= angle DBE; and similarly angle BAD = angle
EAD.
>< TO PROVE: ABC isosceles.
See the thread with subject, A plane geometry problem, which
I started
hard problem, and it is called the Steiner-Lehmus theorem.
--
Stephen J. Herschkorn herschko@rutcor.rutgers.edu
===
Subject: Re: Simple synthetic proof of an obvious geometric
theorem?
Originator: tchow@lagrange.mit.edu.mit.edu (Timothy Chow)
>See the thread with subject, A plane geometry problem, which
I started
>hard problem, and it is called the Steiner-Lehmus theorem.
Excellent! I had momentary trouble locating that thread
because the
subject
line had the word geomerty rather than geometry, but I
eventually
found
--
Tim Chow tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the
artillery---however great, will
never exceed four of those miles of which as many thousand
separate us from
the center of the earth. ---Galileo, Dialogues Concerning Two
New Sciences
===
Subject: Re: Self-teaching of integration methods
<x53c366gpl.fsf@lola.goethe.zz>
Discussion, linux)
[...]
> this is an excerpt of a post by will willis. i haven't
tried it out
> myself it's been years since i used integration by parts
but it
> looks well-developed and confident enough to probably be
> correct.. he expressed great confidence in it
> According to that metric, you should be a great fan of
James Harris
> and his achievements.
We should all be great fans of James Harris (if not his
achievements). Bandel's metric has nothing to do with it.
--
Jesse F. Hughes
Mathematicians don't fit in with a consistent 
view, unless you
accept
that to a strangely large extent they are acting under the
inßuence
of something very powerful, dark, and negative. -- James S.
Harris
===
Subject: Re: Self-teaching of integration methods
> When solving integrals using the integration by parts
method, i.e.
> int (udv)=uv-int(vdu)
> i have always used, (and had much luck with) the acronym
LIATE
 L ogarithmic
> I nverse Trig
> A lgebraic
> T rigonometric
> E xponential

> this is an excerpt of a post by will willis. i haven't
tried it out
> myself it's been years since i used integration by parts
but it
> looks well-developed and confident enough to probably be
> correct.. he expressed great confidence in it
Once you get the technique for integration by parts,
are any of you familiar with the tabular method for this?
(Yes, I am, but I thought it worth mentioning here.)
Ray Steiner
> According to that metric, you should be a great fan of
James Harris
> and his achievements.
===
Subject: Re: Identity question correction.
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i72Ksxn18955;
> A correction to my original post!
> a = (2/(10^(1/2)-4))*-1
> b = 1/9 + 1/9 *10^(1/2)
> Then --
> b = (((x*2)+2)^2 -2)/4 - (x*a)
> Where x = (a/2)-1
> This I hope is right?
> Dan
> Also another --
> b = a/(sqrt(10) + 2)
> Ps. The (a) value in the reverse symbolic calcualtor table
is
> negative, but I made it positive here to avoid confusion.
>Maple says:
> a := (2/(10^(1/2)-4))*(-1);
> b := 1/9 + 1/9 *10^(1/2);
> x := (a/2)-1;
> 2
> a := - -----------
> (1/2)
> 10 - 4
> 1 1 (1/2)
> b := - + - 10
> 9 9
> 1
> x := - ----------- - 1
> (1/2)
> 10 - 4
> is(b = (((x*2)+2)^2 -2)/4 - (x*a));
> true
A.N.Neil,
This all came about when I was experimenting with
completing the square in the 3rd quadrant where
both coordinates x,y are negative.
For the one value of (a),lower limit, and all
values > (a) then using a pseudo form of a quadratic
this lower limit was reached where (a),lowest limit,
and > (a) was part of the equation and (-b),highest limit,
and <(-b) was the product.
I am not going to explain it because, I had people
confused about my first post which was my fault.
I am not sure I can clean it up to be understood.
There is certain rules on sign changes before and after
square-root and *,+,- operators.
Dan
===
Subject: Re: UPANISHADS AND THE UNIVERSE
[India shining stuff...]
So is the long-term plan here to generate enough embarassment
for all Indians to match in negativity, whatever positives
India might have generated in the past?
At the rate you are going, very soon anyone (else) who
says things like India gave such-and-such gift
to the world will be seeing responses like
Oh, really? But what about Dr. Jai Maharaj, the
netiquette-less cross-posting and multi-posting
biased racist astrology advertiser? Isn't he associated
with India too? Which will be enough
for any self-respecting honest Indian to wither away
in deep embarassment and shame, and drop the
topic altogether...
===
Subject: Re: UPANISHADS AND THE UNIVERSE
bhanwaram@netscape.net (bhanwara) posted:
> Dr. Jai Maharaj posted:
> [India shining stuff...]
> So is the long-term plan here to generate enough
> embarassment for all Indians . . .
If you are embarrassed by a shining Bharat then you
are either Bharat's enemy or a Bharatiya who is a
slave of foreigners. It's no wonder that you do not
sign your posts with your name.
Jai Maharaj
http://www.mantra.com/jai
Om Shanti
===
Subject: Re: UPANISHADS AND THE UNIVERSE
> bhanwaram@netscape.net (bhanwara) posted:
> Dr. Jai Maharaj posted:
> [India shining stuff...]
> So is the long-term plan here to generate enough
> embarassment for all Indians . . .
>If you are embarrassed by a shining Bharat then you
>are either Bharat's enemy or a Bharatiya who is a
>slave of foreigners. It's no wonder that you do not
>sign your posts with your name.
>Jai Maharaj
STFU Jay. You're doing nothing but embarrassing yourself
further.
http://www.geocities.com/drjosemariachi/jay_faq.html
--
Dr.Postman USPS, MBMC, BsD; Disgruntled, But Unarmed
Member,Board of Directors of afa-b, SKEP-TI-CULT member
#15-51506-253.
You can email me at: DrJaiMaharajFraud(at)hotmail.com
Did the Venus transit occur during sunset, idiot?
- Grant,on the GLP web board, explains to us how
sunrise happens in NY and Asia at the same time.
===
Subject: Re: UPANISHADS AND THE UNIVERSE
>Upanishads and the Universe
Hindu mythology
===
Subject: Re: UPANISHADS AND THE UNIVERSE
vonroach <hadrainc@earthlink.net> posted:
> Upanishads and the Universe
> Hindu mythology
No, Vedic-Hindu history and principles.
Jai Maharaj
http://www.mantra.com/jai
Om Shanti
===
Subject: Re: UPANISHADS AND THE UNIVERSE
X2b5S?7at*2R/5vY{L[AI_LKLHh.E
> vonroach <hadrainc@earthlink.net> posted:
 Upanishads and the Universe
> Hindu mythology
> No, Vedic-Hindu history and principles.
You are about as Vedic-Hindu as a three-dollar bill, Stevens.
-=-=-=-=-
* Official AFA-B Bully, Pest, Antagonist, Government/Media
Disinformation Agent, Dr. Green Sockpuppet, and Lemming
* Chief AFA-B Vote Rustler
===
Subject: Re: UPANISHADS AND THE UNIVERSE
> vonroach <hadrainc@earthlink.net> posted:
 Upanishads and the Universe
> Hindu mythology
>No, Vedic-Hindu history and principles.
Here's your history:
http://www.geocities.com/drjosemariachi/jay_faq.html
--
Dr.Postman USPS, MBMC, BsD; Disgruntled, But Unarmed
Member,Board of Directors of afa-b, SKEP-TI-CULT member
#15-51506-253.
You can email me at: DrJaiMaharajFraud(at)hotmail.com
Did the Venus transit occur during sunset, idiot?
- Grant,on the GLP web board, explains to us how
sunrise happens in NY and Asia at the same time.
===
Subject: best line fit
Hello
I am trying to come up with a formula to calculate the line
that best
fits a x,y point set, and then get the slope of that line.
any help is appreciated.
===
Subject: Re: best line fit
> Hello
> I am trying to come up with a formula to calculate the line
that best
> fits a x,y point set, and then get the slope of that line.
> any help is appreciated.
http://mathworld.wolfram.com/LeastSquaresFitting.html
(explanation and formulae)
http://www.hostsrv.com/webmaa/app1/MSP/webm1010/leastsquares
(web page calculates line)
and many others.
I'm assuming that what you mean by best is the least squares
fit.
If you have some other idea in mind, or even if you're not
sure
if that's what you want, you'd better make 
that clear. The
most
usual meaning by far of best is least squares.
The specific formula you asked for:
For data = {(X1,Y1), ..., (Xn,Yn)}
and the least squares fit line y = m*x + b,
m = ( n[xy] - [x][y] )/( n[x^2] - [x]^2 )
b = ( [y][x^2] - [x][xy] )/( n[x^2] - [x]^2 )
where
[xy] = sum{i = 1...n}(Xi*Yi)
etc.
===
Subject: Re: best line fit
> Hello
> I am trying to come up with a formula to calculate the line
that best
> fits a x,y point set, and then get the slope of that line.
> any help is appreciated.
It depends of your standard of best fit.
A common standard is least squares, i.e., minimize the sum of
squares
of differences between predicted y's and actual 
y's for each
given x.
This is called linear regression, for historical reasons, and
Googling
linear regression will give you more information on the
subject than
you could possibly use (over a million hits).
===
Subject: Re: UPANISHADS AND THE UNIVERSE
Exactly, but he can't see it. One notices the almost 
complete
absence of
participation of Indians in his threads, notwithstanding
those which are
obviously him using another name.
>[India shining stuff...]
>So is the long-term plan here to generate enough embarassment
>for all Indians to match in negativity, whatever positives
>India might have generated in the past?
>At the rate you are going, very soon anyone (else) who
>says things like India gave such-and-such gift
>to the world will be seeing responses like
>Oh, really? But what about Dr. Jai Maharaj, the
>netiquette-less cross-posting and multi-posting
>biased racist astrology advertiser? Isn't he associated
>with India too? Which will be enough
>for any self-respecting honest Indian to wither away
>in deep embarassment and shame, and drop the
>topic altogether...
===
Subject: Suspace interpolation
Does anybody have a good pointer to solutions to the
following problem?
Assume I have two vector sets {x} and {y} which contain
vectors
corresponding to realizations of multidimensional Gaussian
processes.
Assume now that we build a new vector set as follows: {z}
where z = a x
+
b y is a linear combination of the original vector sets (a+b
= 1).
{x} and {y} can be both modelled by their mean vector and
covariance
matrix.
Is there any result that allows you to compute the mean
vector and
covariance matrix of {z} only based on the knowledge of a, b,
and mean and
covariance of x and y?
This can be seen as given a and b, and two subspaces defined
by the
eigenvectors of the respective covariance matrices, find the
subspace of
the
interpolation between the two subspaces.
Alternativelly, it could be interpreted as, given two
statistical gaussian
models corresponding to two variables, find the interpolated
statistical
model of the interpolation of the very same two variables.
Alex
===
Subject: Re: Suspace interpolation
> Does anybody have a good pointer to solutions to the
following problem?
Yes; this is very basic. You would learn much by researching
it
yourself. You don't even need normality.
> Assume I have two vector sets {x} and {y} which contain
vectors
>corresponding to realizations of multidimensional Gaussian
processes.
> Assume now that we build a new vector set as follows: {z}
where z = a x
+
>b y is a linear combination of the original vector sets (a+b
= 1).
> {x} and {y} can be both modelled by their mean vector and
covariance
>matrix.
> Is there any result that allows you to compute the mean
vector and
>covariance matrix of {z} only based on the knowledge of a,
b, and mean and
>covariance of x and y?
--
Stephen J. Herschkorn herschko@rutcor.rutgers.edu
===
Subject: group theory/algebraic topology question
I'm trying to find all of the isomorphism 
classes of 3-sheeted
covering
spaces of the klein bottle.
This is essentially the same as finding all conjugacy classes
of
homomorphisms from the fundamental group of the klein bottle
<a,b :
a^2b^2=1> to S_3.
Apparently something has gone wrong with either my group
theory or
algebraic topology.
I cant see why sending a to (12) and b to (13) is not a valid
homomorphism/ covering space. Its just occured to me that the
Substitution Test proves that this is a valid homomorphism.
Or am I wrong?
===
Subject: Re: group theory/algebraic topology question
I have a feeling I was told this was wrong by mistake. Since
I took for
the presentation <a,b: a^2b^2=1>, I have a feeling the grader
thought I
was using the presentation <a,b : aba^{-1}b> in which case
(12)(13)(12)(13)not= 1 and so it would not have been a valid
homomorphism.
> I'm trying to find all of the isomorphism 
classes of
3-sheeted covering
> spaces of the klein bottle.
> This is essentially the same as finding all conjugacy
classes of
> homomorphisms from the fundamental group of the klein
bottle <a,b :
> a^2b^2=1> to S_3.
> Apparently something has gone wrong with either my group
theory or
> algebraic topology.
> I cant see why sending a to (12) and b to (13) is not a
valid
> homomorphism/ covering space. Its just occured to me that
the
> Substitution Test proves that this is a valid homomorphism.
Or am I
wrong?
===
Subject: Tautologies Then and Now
Tautologies Then and Now
----------------
The tautology is a law whose rules are exhaustive and exclude
no
possibility. Thus it is considered useless by many for
providing
information regarding the so called real world. Mathematics
is held to
be tautological, for example, in providing exhaustive
information on
its subjects. However there is no way to tell to what the
information
applies and in the so called real world, scientists are left
to divine
the applicability of tautological information for themselves,
a
doctrine which, for want of a better name, I refer to as
positivism.
We might illustrate simple tautological information in the
following
way: a train is either blue or not blue. This gives us an
exhaustive
account of all things blue and not blue but yields no useful
insight
as to which things are blue or not blue of analytical
necessity.
However, the case for the tautology is by no means as bleak
as the
conventional perspective makes out. For there is exactly one
and only
one form of tautology that is useful in explaining things to
which
tautological information applies because it applies of
necessity to
everything. And that tautology is P differences because
alternatives
Q different from differences are universally self
contradictory.
In other words, a tautology applies universally and if one
alternative
is self contradictory, then the other must perforce apply
universally
to all things. It makes no difference if we rephrase the
tautology as
P not and Q not not or P contradiction and Q contradiction of
contradiction or P negation and Q negation of negation.
Thus instead of a useless instrument of exhaustive truth we
find in
the tautology an instrument of universal truth applicable to
everything from math to science to the analysis of sentient
behavior.
And furthermore this also means that everything from math to
science
to the analysis of sentient behavior can only be universally
defined
in terms of differences, differences between differences, and
so on.
===
Subject: Re: Tautologies Then and Now
> We might illustrate simple tautological information in the
following
> way: a train is either blue or not blue. This gives us an
exhaustive
> account of all things blue and not blue but yields no
useful insight
> as to which things are blue or not blue of analytical
necessity.
Andreas Feininger (one of the old Life Magazine Photographers,
among other things) in his books, particularly, Light and
Lighting
in Photography could photograph a subject from different
perspectives, in different light of the day or season and come
up with just about any color you like.
Your train can be made to appear blue under the right
circumstances
whether it is blue or not.
Things are not so simple and depend of the object, idea or
entity,
the context and surroundings and who is and how are the
looking.
Human sense are easily fooled and nature fools us all the
time!
===
Subject: Re: Tautologies Then and Now
Sam Wormley
> Human sense are easily fooled and nature
> fools us all the time!
But remember what Bill Taylor reminded us:
Don't anthropomorphize nature.
She hates that!
my sig-nature :-)
--
Rainer Rosenthal, r.rosenthal@web.de _____________________
| _ | |
| (_) | Given A, P and a circle. Find B, C on the |
| A P | circle with P on BC and area(ABC)=maximum. |
|__________|___(Ingmar Rubin in de.sci.mathematik) ________|
===
Subject: Re: Tautologies Then and Now
> We might illustrate simple tautological information in the
following
> way: a train is either blue or not blue.
Bull. Morning glory blue and photographic film vs.
eyeballs. Dochroic pigments as on currently issued money.
Alexandrite and didymium glass in sunlight or incandescent
light.
If you know nothing then your conclusions will be equally
valid.
That is why the Frist World has porcelain ßush toilets and
East
Indians with their 360 million gods and 3000 years of
civilization squat and crap in their streets. You know nothing
by empirical counterdemonstration.
More to the point, in all the years you have been trolling
Usenet
with your crapola, you have never learned anything. That is
the
difference between ignorance (fixable) and stupidity (which is
forever). You are despicable.
--
Uncle Al
http://www.mazepath.com/uncleal/
(Toxic URL! Unsafe for children and most mammals)
http://www.mazepath.com/uncleal/qz.pdf
===
Subject: 4 colors and tetrahedrons
Is it true that the minimum number of sides
to enclose a 3D volume is 4, such as a tetra-
hedron?
Is it true the maximum number of colors on this
surface is 4, so that no same colors border?
Coincidence or logic?
Is it possible to modify a tetrahedron so that
5 colors are required so that no same colors
border on the surface of said volume?
Is the tetrahedron and the 4-color problem the
same thing?
Fun+++++
Ken S. Tucker
===
Subject: Re: Excel Math Bug
Do any of you SCI.MATH whizes want to weigh in on this?
MS Excel calculates =-5^2 as 25, not as -25.
This is because Ônegation' is handled 
first in Excel. (!?)
If you put a zero in the equation,
as in =0-5^2, your answer changes to -25.
Is this in line with standard math rules?
Is negation different than subtraction?
I'm getting a lot of comments in the Excel NG
basically saying that it's in the help section, so too bad.
I've had lots of math and as far as I know
negation and subtraction are the same thing.
> That's clearly explained in the XL Help topic The order in
which
> Microsoft Excel performs operations in formulas.
> In all the math I've ever done, from grade school on,
negation and
> subtraction have been separate operations (often, but not
always, using
> separate symbols, such as a hyphen for negation and an
n-dash for
> subtraction), so that -5^2 has always been interpreted to
equal 25.
> Personally, I'd take it up with your consultant, assuming
that he/she
> was working on the Excel model. That problem should have
been a piece of
> cake for someone with even moderate expertise to identify,
from the sign
> change alone! There's no way you should have to pay for 20
hours of
> troubleshooting (at my rates, at least).
> I did in another sub-thread. Dana was familiar with it
already.
> If you lead off with a negative sign it uses the negative
value inside
the
> exponentiation.
> So, instead of =-5^2 equalling -25 it equals 25.
> but, =0-5^2 is calculated correctly as -25 even though 
it's
mathematically
> the same.
===
Subject: Re: Excel Math Bug
>Do any of you SCI.MATH whizes want to weigh in on this?
>MS Excel calculates =-5^2 as 25, not as -25.
>This is because Ônegation' is handled 
first in Excel. (!?)
>If you put a zero in the equation,
>as in =0-5^2, your answer changes to -25.
>Is this in line with standard math rules?
>Is negation different than subtraction?
>I'm getting a lot of comments in the Excel NG
>basically saying that it's in the help section, so too bad.
>I've had lots of math and as far as I know
>negation and subtraction are the same thing.
> Well, this is an Excel forum, so one should expect a
programming point
> of view. But if you search on mathematical notation
generally, I
> think negation is viewed as a unary operator, while
subtraction is
> viewed as a binary operator; and the discussions are not
much clearer
> in that context. My own view, not as a mathematician, is
that the
> issue revolving around how to evaluate -1^2 depends on some
*order of
> precedence*, and is totally conventional as to negation and
> exponientation.
BOULDERDASH!!! This is a horrible bug in Excel (whereof I was
previously unaware). It is very standard that exponentiaion
takes
precedence over negation. Ask any semi-decent high school
student to
draw a graph of y = -x^2, and what will you get?
Stating it is a documented convention is not a legitimate
argument.
What if Microsoft(R) buried in its documentation that
addition takes
precedence over multiplication? That the spell checker would
always
change word friend to freind? That the sum function adds only
every
other term? That using a q in one of its products would cause
the
system to reboot? These effects would be just as valid by
this logic.
I have sent this comment to Microsoft(R), though I expect no
good to
come of it.
--
Stephen J. Herschkorn herschko@rutcor.rutgers.edu
===
Subject: Re: Excel Math Bug
> BOULDERDASH!!! This is a horrible bug in Excel (whereof I
was
> previously unaware). It is very standard that exponentiaion
takes
> precedence over negation. Ask any semi-decent high school
student to
> draw a graph of y = -x^2, and what will you get?
As another poster noted, M.P.E.P is a programming forum, but
told that it
violates math convention, they still argue. They probably
just didn't
believe me.
Another said much ado about nothing, but I think this is a
horrible bug
too. Excel should at least follow regular math conventions.
What other
surprizes await?!
===
Subject: Re: Excel Math Bug
> BOULDERDASH!!! This is a horrible bug in Excel (whereof I
was
> previously unaware). It is very standard that
exponentiaion takes
> precedence over negation. Ask any semi-decent high school
student to
> draw a graph of y = -x^2, and what will you get?
> As another poster noted, M.P.E.P is a programming forum,
but told that
> it violates math convention, they still argue. They
probably just
> didn't believe me.
> Another said much ado about nothing, but I think this is a
horrible
> bug too. Excel should at least follow regular math
conventions. What
> other surprizes await?!
I checked my copy of oocalc. (I don't use M$ products.) Much
to my
chagrin it exhibited the same Excel bug.
--
Lance Lamboy
Go F*ck Yourself ~ Dick Cheney
===
Subject: Re: Excel Math Bug
> Do any of you SCI.MATH whizes want to weigh in on this?
> . . .
> I've had lots of math and as far as I know
> negation and subtraction are the same thing.
Well, this is an Excel forum, so one should expect a
programming point
of view. But if you search on mathematical notation
generally, I think
negation is viewed as a unary operator, while subtraction is
viewed as a
binary operator; and the discussions are not much clearer in
that
context. My own view, not as a mathematician, is that the
issue
revolving around how to evaluate -1^2 depends on some *order
of
precedence*, and is totally conventional as to negation and
exponientation.
Alan Beban
===
Subject: Re: Excel Math Bug
> MS Excel calculates =-5^2 as 25, not as -25.
> Is this in line with standard math rules?
No. If all were OK, they'd call this program
Excellent
but since it lacks something it's just called
Excel
--
Rainer Rosenthal, r.rosenthal@web.de _____________________
| _ | |
| (_) | Given A, P and a circle. Find B, C on the |
| A P | circle with P on BC and area(ABC)=maximum. |
|__________|___(Ingmar Rubin in de.sci.mathematik) ________|
===
Subject: Archived geometry puzzle
A working copy of the Diamond 16 Puzzle is now in the
permanent
Internet Archive at
The puzzle illustrates concepts of symmetry-- and, more
generally, of
invariance under a group of transformations, as in Klein's
Erlangen
program.
Keywords: diamond theorem, diamond theory, diamond puzzle,
finite
geometry, galois geometry, affine group, symmetry
-- Steven H. Cullinane
===
Subject: A question about solving equations with non-integer
powers
How can I solve for x at the following equation???
ax^(7/3) + bx^(5/3) + cx + d = 0
Can someone please give me the given equation rearranged and
solved
for x please?
===
Subject: Re: All roots real for small degree polynomials
OK, I am wasting bandwidth again, but this problem is more
intriguing the more I look at it. I suspect that the answer
has
been known since Newton, but I cannot find it.
Let S be a set of numbers {s1,...sn} and define
d(S)=Product_{i<j} (si-sj)^2.
Let P be a polynomial and let
Dk(P)=the sum over all k element subsets S of the roots of P
of d(s).
Dk(P) is a symmetric polynomial in the roots P and so can be
expressed
in terms of the coefficients. If P is an n'th 
degree
polynomial then
Dn(P)
is the discriminant.
Now obviously if all the roots of P are real then all the
Dk(P) 2<=k<=n are positive. Is the converse true?
The Dk(P) appear to be the leading coefficients of the Sturm
sequence for polynomials of degree 4 or less, if my all too
sloppy
algebra is correct,
I apologize for top posting, but the combination of Safari and
for all the roots of a polynomial (degree small) > to be
real?  This must be well-established but I cannot find a reference.
>
symbolically seems to work. On quadratic > polynomials it
gives the
condition that the dicriminant is positive. > For the
polynomial
x^3+a x^2 + b x + c it gives the conditions > that > a^2 > 3b
> and >
a^2 b^2 - 4 b^2 - 4 a^3 c + 18 a b c - 27 c^2 > 0 > and
those seem
to work for cubic equations. A symbolic algebra package >
should be
able to crank out the next few faster than I can copy them, >
but they
seem to get unpleasantly complicated fairly quickly.
===
Subject: Re: PBZ: nonstandard math
>I have no problem with c!=c. Can you prove that they are
equal? For
>instance, tomato!=tomato. You may say the word, tom-a-to,
where I
>may say the word, tom-a-to. We see from this that they are
>pronounced differently,
How do we see this? The quotes look the same to me.
>yet they are spelled the same way and
>represent the same object.
> And what schools would those be?
> Is that a thinly disguised racist comment?
>No. Only the schools that teach evolutionism.
Racism, theism, it's all the same thing.
===
Subject: Theoretical questions raised by tidbits
Hey, I'm a dabbler who actually gets excited about some
number theory
and I don't care to admit it when I'm just 
ignorant. Here
with these
tidbits there's an interesting development to me, and I 
don't
mind if
someone comes forward with a number theory reference or some
paper
citation to show that I'm just missing something obvious.
The start was with my directly derived [(N-4)/6] which gives
the count
of odd composites divisible by 3 for even N>2 and I learned
that a
better formula is [(N-3)/6] where the offset actually looks
related to
the denominator.
Another poster (forget his name at the moment) noticed that
with the
relation for 5 that gives the count of odd composites that
have 5 as a
factor that do not have 3 as a factor that while I gave, from
my own
derivation,
[(N-16)/10] - [(N-16)/30], with even N>6
he had
[(N+5)/10] - [(N-15)/30] - 2, with N>6
where you can see the pattern again, though there's that -2
stuck on
the end.
Next in the series from my calculation is
[(N-8)/14] - [(N-22)/42] - [(N-106)/70] + [(N-106)/210] - 2
with even N>36, where I made all my derivations using
[(N-4)/6] and my
prime counting function, over two years ago, so not using
[(N-3)/6]
apparently gives an offset where you can still see the
pattern if
you're looking for it.
An anonymous poster put up
[(N-7)/14] - [(N-21)/42] - [(N+35)/70] + [(N-105)/210]
and there's the same pattern, where underlying it you can 
see
the
inclusion-exclusion principle at work.
That like how to get a count of composites that have 3 as a
factor
that don't have 2 as a factor you have
[N/3] - [N/6] - 1
as the count of naturals with 3 as a factor is given by
[N/3], while
the count of composites with 6 as a factor is given by [N/6],
so
you're correcting the first count with the 
second and then
subtracting
1 to handle 3 itself to give a count of odd composites that
have 3 as
a factor.
That method is inefficient for counting primes, as, not
surprisingly,
counting composites is a way to count primes.
However, if
[(N-7)/14] - [(N-21)/42] - [(N+35)/70] + [(N-105)/210]
is the most compact expression possible for counting the
number of
composites that have 7 as a factor that do not have 2, 3, or
5 as a
factor, then you have a piece of the most efficient explicit
prime
counting function possible!
How does it work?
You put them together. The count of evens is given by [N/2],
while
the count of odds with a factor of 3 that are not even is
given by
[(N-3)/6], and so forth, and you subtract that from N - 1 to
get the
count of primes!
For instance, with the formulas given so far you have
N - [N/2] - [(N-3)/6] - [(N+5)/10] + [(N-15)/30] + 2 -
[(N-7)/14] +
[(N-21)/42] + [(N+35)/70] - [(N-105)/210] - 1
will give you the count of primes up to 120, notice you can
leave off
the last term and going ahead and subtracting 1, so
pi(N) = N - [N/2] - [(N-3)/6] - [(N+5)/10] + [(N-15)/30] + 1 -
[(N-7)/14] + [(N-21)/42] + [(N+35)/70] for N<121.
If the expressions given *are* indeed the most compact
possible then
that is a piece of THE explicit prime counting function. That
is,
that's a piece of the fastest possible prime counting
function.
What's interesting is the pattern!
Now then, some interesting theoretical points:
1. Legendre's Method uses inclusion-exclusion and 
it's
terribly
inefficient.
2. THE formula clearly shows inclusion-exclusion!
Is it possible that the major big deal is that compression
gained by
using the fact that every odd can be written as a 2k+1 where
k is a
natural travels forward?
Or can the formula be further compressed?
If it can't, can it really be that there's 
this upper speed
limit to
*any* prime counting technique out there which was just
missed?
Well I'm just a dabbler, and I'd be interested 
in posters
with more
information in this area. I have no problem being wrong, and
putting
out the big ideas. And I'm sure plenty of posters will
happily go on
the attack.
Trouble with ideas like this for me is that when they're new
then I
tend to be wrong, and while the original formulas I showed
before were
derived by me over two years ago, this current speculation is
new.
Right now, in what may just be my ignorance, I'm thinking
this is just
plain fascinating.
James Harris
http://mathforprofit.blogspot.com/