mm-165
=
In sci.math, Alloran1<alloran1@aol.com>Where ^ is to the
power of:->>Is (-1)^(0.5*2) = (-1)^(2*0.5) ?>>But (-1)^0.5
=+i,-i and +-i^2 = -1>>Also (-1)^2 = 1 and 1^0.5 = -1, +1>>So
why are the solutions different?>>J.> i is not the square
root of negative 1!> i is a number that, when you happen to
square it you get negative one.> (-1)^0.5 therefore does not
make any sense.(-1)^0.5 = e^(ln(-1) * 0.5)Now ln is an
interesting periodic function (period is 2 * i *
pi).Therefore, (-1)^0.5 is ambigous, returning two values (as
it should).These roots are +i and -i.It makes perfect sense to
me but one has to be carefulhow one applies the mathematics (a
number of fallaciesare occasionally bantered about which
depend on pickingthe wrong sign for the square root). But
negative numbersdo have square roots, if one extends the
problem into thecomplex plane.It turns out ((-1)^2)^0.5 = +1
and -1, and ((-1)^0.5)^2 = -1.For the former, the same
0.5-power dichotomy applies, asone is either doing e^0 or e^(i
* pi); the former is 1,the latter -1.> --Daniel Briggs--
#191, ewill3@earthlink.netIt's still legal to go
.sigless.
 I was wondering about the set of positive
integers n where> (n!+1) AND (n!-1) are both composite.>
<snip> What is the lowest positive integer n such that c(n)
= 1/n, ie. what n> is the lowest such that (n!+1) AND (n!-1)
are both primes?3
 I was wondering about the set of
positive integers n where> (n!+1) AND (n!-1) are both
composite.>> For n's = positive integers:>> Let a(N) = 
{number
of n's all <= N such that (n!-1) is a composite}/N>> And let
b(N) = {number of n's all <= N such that (n!+1) is a>
composite}/N>> And let c(N) = {number of n's all <= N such
that (n!+1) AND (n!-1) are> both composites}/N>> First, the
preliminary question:>> Is a(N) ~ b(N) ?>> And the main
question:>> Is a(N)*b(N) ~ c(N)?>> And, in any case, what are
a(N), b(N), and c(N) asymptotic to?>> I cannot check this now
on my computer due to issues with my> math-software, but I was
wondering:>> What is the lowest positive integer n such that
c(n) = 1/n, ie. what n> is the lowest such that (n!+1) AND
(n!-1) are both primes?>> Leroy Quet2! +1 is prime3! +/-1 is
prime.4! -1 is prime.6! -1 is prime.7! -1 is prime.11! +1 is
prime.
=I was wondering about the set of positive integers n
where(n!+1) AND (n!-1) are both composite.For n's = positive
integers:Let a(N) = {number of n's all <= N such that (n!-1)
is a composite}/NAnd let b(N) = {number of n's all <= N such
that (n!+1) is acomposite}/NAnd let c(N) = {number of n's 
all
<= N such that (n!+1) AND (n!-1) areboth composites}/NFirst,
the preliminary question:Is a(N) ~ b(N) ?And the main
question:Is a(N)*b(N) ~ c(N)?And, in any case, what are a(N),
b(N), and c(N) asymptotic to?I cannot check this now on my
computer due to issues with mymath-software, but I was
wondering:What is the lowest positive integer n such that c(n)
= 1/n, ie. what nis the lowest such that (n!+1) AND (n!-1) are
both primes?Leroy Quet
>Leroy Quet (...)>> I was
wondering about the set of positive integers n where>> (n!+1)
AND (n!-1) are both composite.(...)>11! +1 is prime.>How did
you 'nd that out ? is there (or did you write) a piece
ofsoftware helping you to test this number for primality ->
one needsto divide 11!+1 by all primes <= 6317 (for the
elementary way) !
>Leroy Quet (...)>> I was
wondering about the set of positive integers n where>>
(n!+1) AND (n!-1) are both composite.> (...)>11! +1 is
prime.>> How did you 'nd that out ? is there (or did you
write) a piece of> software helping you to test this number
for primality -> one needs> to divide 11!+1 by all primes <=
6317 (for the elementary way) !Yes, there is a pretty nice
software for factoriztion:Try Dario Alperns ECM:
http://www.alpertron.com.ar/ECM.HTMFor primality testing look
at OpenPFGW:http://www.primeform.net/openpfgw/There are better
ways to solve Leroy's question:Do a lookup in the Online
Encyclopedia of Integer sequences(Leroy should be aware of
that ;-)n!-1 is
prime:http://www.research.att.com/projects/OEIS?Anum=
A0029823,4,6,7,12,14,30,32,33,38,94,166,324,379,469,546,974,196
3,3507,3610,6917,21480n! + 1 is
primehttp://www.research.att.com/projects/OEIS?Anum=
A0029810,1,2,3,11,27,37,41,73,77,116,154,320,340,399,427,872,14
77,6380,269513 seems to be the only common term and any future
'ndingof another term would be a sensation.Hugo Pfoertner
>Leroy Quet (...)>> I was wondering about the set of
positive integers n where>> (n!+1) AND (n!-1) are both
composite.> (...)>11! +1 is prime.>> How did you 'nd that
out ? is there (or did you write) a piece of> software helping
you to test this number for primality -> one needs> to divide
11!+1 by all primes <= 6317 (for the elementary way) !I
cheated! with software of course.http://indigo.ie/~mscott/The
MIRACL library comes with a few demo executables; among thema
very good factoring
program.
=http://home.att.net/~numericana/wilson.htmTapio> I
was wondering about the set of positive integers n where>
(n!+1) AND (n!-1) are both composite.>> For n's = positive
integers:>> Let a(N) = {number of n's all <= N such that
(n!-1) is a composite}/N>> And let b(N) = {number of n's all
<= N such that (n!+1) is a> composite}/N>> And let c(N) =
{number of n's all <= N such that (n!+1) AND (n!-1) are> 
both
composites}/N>> First, the preliminary question:>> Is a(N) ~
b(N) ?>> And the main question:>> Is a(N)*b(N) ~ c(N)?>> And,
in any case, what are a(N), b(N), and c(N) asymptotic to?>> I
cannot check this now on my computer due to issues with my>
math-software, but I was wondering:>> What is the lowest
positive integer n such that c(n) = 1/n, ie. what n> is the
lowest such that (n!+1) AND (n!-1) are both primes?>> Leroy
Quet
=(This is a repost with a primes replaces with the
intended COMPOSITES.)---- I was wondering about the set of
positive integers n where(n!+1) AND (n!-1) are both
composite.For n's = positive integers:Let a(N) = {number of
n's all <= N such that (n!-1) is a composite}/NAnd let b(N) 
=
{number of n's all <= N such that (n!+1) is acomposite}/NAnd
let c(N) = {number of n's all <= N such that (n!+1) AND 
(n!-1)
areboth composites}/NFirst, the preliminary question:Is a(N) ~
b(N) ?And the main question:Is a(N)*b(N) ~ c(N)?And, in any
case, what are a(N), b(N), and c(N) asymptotic to?I cannot
check this now on my computer due to issues with
mymath-software, but I was wondering:What is the lowest
positive integer n such that c(n) = 1/n, ie. what nis the
lowest such that (n!+1) AND (n!-1) are both COMPOSITES?Leroy
Quet
 (This is a repost with a primes replaces with
the intended COMPOSITES.)> ----> I was wondering about
the set of positive integers n where> (n!+1) AND (n!-1) are
both composite.> For n's = positive integers:> Let a(N) =
{number of n's all <= N such that (n!-1) is a composite}/N>
And let b(N) = {number of n's all <= N such that (n!+1) is 
a>
composite}/N> And let c(N) = {number of n's all <= N such
that (n!+1) AND (n!-1) are> both composites}/N> First, the
preliminary question:> Is a(N) ~ b(N) ?> And the main
question:> Is a(N)*b(N) ~ c(N)?> And, in any case, what
are a(N), b(N), and c(N) asymptotic to?> I cannot check this
now on my computer due to issues with my> math-software, but I
was wondering:> What is the lowest positive integer n such
that c(n) = 1/n, ie. what n> is the lowest such that (n!+1)
AND (n!-1) are both COMPOSITES?> Leroy QuetLeroy,check Paul
Leyland's Tables of factorials plus/minus 1folowing the 
links
given
athttp://research.microsoft.com/~pleyland/factorization/
main.htmcontinued for n>400 at Andrew Walker'sN!+-1 
Factoring
Pagehttp://www.uow.edu.au/~ajw01/ecm/curves.htmlHugo
Pfoertner
=The Dover classic, Three Pearls of Number
Theory gives a proof ofthe following theorem of Van der
Waerden:Let k, l, be 'xed integers >= 2. Then, for a
suf'ciently largepositive integer, N, any partition into k
disjoint subsets of theintegers in the interval [ 1 , N ]
results in one of the k subsetscontaining an arithmetic
progression of length l .My question is: For k and l 'xed,
what is known about the least suchN (which depends on k and
l)? For l = 2, N is clearly k + 1. Ilooked at the simplest
case which is not completely trivial -- k = 2,(for k = 2, l =
3) = 9. However, the Three Pearls method gives abound for
(2, 3) of 780 . (There tends to be a negative
correlationbetween the size of a bound N, and the ease of
demonstrating that Nworks as a bound.)I'd be interested to
learn about any results about the least N. If wecall the least
N (k, l), n (k, l), what is known about this function? I bet
that n is not a polynomial function of k and l, but has
anybodyproved it? It would not surprise me if it has been
proved that, in acertain sense, no such function can be
constructed. Does anyone knowany results in that
direction?Paul Epstein
 The Dover classic, Three Pearls
of Number Theory gives a proof of> the following theorem of
Van der Waerden:> Let k, l, be 'xed integers >= 2. Then, for 
a
suf'ciently large> positive integer, N, any partition into k
disjoint subsets of the> integers in the interval [ 1 , N ]
results in one of the k subsets> containing an arithmetic
progression of length l .> My question is: For k and l 'xed,
what is known about the least such> N (which depends on k and
l)? For l = 2, N is clearly k + 1. I> looked at the simplest
case which is not completely trivial -- k = 2,> (for k = 2, l
= 3) = 9. However, the Three Pearls method gives a> bound
for (2, 3) of 780 . (There tends to be a negative correlation>
between the size of a bound N, and the ease of demonstrating
that N> works as a bound.)> I'd be interested to learn about
any results about the least N. If we> call the least N (k, l),
n (k, l), what is known about this function? > I bet that n is
not a polynomial function of k and l, but has anybody> proved
it? It would not surprise me if it has been proved that, in a>
certain sense, no such function can be constructed. Does anyone
know> any results in that direction?> Paul EpsteinYou may
be able to 'nd more by searching on Ramsey theory or Ramsey
bounds.Nemo
The Dover classic, Three Pearls of Number
Theory gives a proof of>the following theorem of Van der
Waerden:>Let k, l, be 'xed integers >= 2. Then, for a
suf'ciently large>positive integer, N, any partition into k
disjoint subsets of the>integers in the interval [ 1 , N ]
results in one of the k subsets>containing an arithmetic
about any results about the least N. If we>call the least N
(k, l), n (k, l), what is known about this function? >I bet
that n is not a polynomial function of k and l, but has
anybody>proved it? It would not surprise me if it has been
proved that, in a>certain sense, no such function can be
constructed. Does anyone know>any results in that
direction?The Van der Waerden bound was originally quite
astronomical as you saw,in fact it grew faster than any
primitive-recursive function (just likeAckermann's function
does).It was markedly improved by Sharon Shelah (who in fact
gave a boundfor a stronger statement called the Hales-Jewett
theorem), which wasthe 'rst primitive-recursive upper bound.
More recently, there isTim Gowers' celebrated proof of
Szemeredi's theorem, which gives anupper bound on n(k,l) of
(if I'm interpreting the review correctly):
e^(k^(2^(2^(l+9)))).Berlekamp has shown by explicit
construction that n(2,p+1) > p*2^pfor p prime, so there is no
way that n(k,l) can be polynomial in l.I don't know whether,
say, n(k,3) is known to grow superpolynomiallywith k. Maybe
I'm missing an obvious argument?See also:
http://mathworld.wolfram.com/vanderWaerdenNumber.html --
Erick
(who in fact gave a boundThat's Saharon Shelah actually, in
case you want to look him up. Oneof the most fascinating
mathemticians of our time.- EM
A photo of Mr Harris
can be found here:>http://groups.msn.com/AmateurMath/memberspictures.msnw?
action=ShowPhoto&Pho>toID=2> And we should care
because .... ?> ...surely the fans want to have a picture of
their idol?But it doesn't show the clay feet!
=<snip> Now
you could ask me *why* to follow this approach and you may be>
aesthetically dissatis'ed with a *foundational* theory based
jointly> on two primitive concepts that in *other* theories
could be de'ned> one in terms of the other. But I 
don't see
why *a priori* this> possibility should be excluded: of course
there may well be a *good*> reason why it is indeed so!>I
cannot speak to your remarks concerning categories and sets
since I do nottheory without sets individuates on the basis
of one-to-one maps, orinclusions. In a certain way, I have
made that claim in set theory on thebasis of the sentences,Aa
Ab( a proper_subset b <-> ( Ac( b proper_subset c -> a
proper_subset c ) / Ec( a proper_subset c / ~(b
proper_subset c) ) ) )Aa Ab( a in b <-> ( Ac( b proper_subset
c -> a in c ) / Ec( a in c / ~(b proper_subset c) ) ) )It
took me a long time to learn how to think about the proper
subset relationas a language primitive distinct from the
membership relation. Of course, nowI no longer have to think
that way. About a year ago I learned of mereology.It is a
foundational discipline based on the part-whole relationship.
Whathas been labeled as the proper subset relation in the
above sentences I nowunderstand as the proper part relation in
mereology.But, the point is this. I have been trying to do what
you are suggesting,although in a slightly different context.
For the most part, a person isignored or villi'ed. Another
poster, Galathaea, even pointed this out withrespect to
established authorities and their intuitionistic leanings.It
seems that this is a question of belief revision more than
anything else.If the mathematics of belief interests you at
all, I found a nice summaryDempster-Shafer's Model by
Philippe Smets,
http://iridia.ulb.ac.be/~psmets/TBM_and_DST.pdfYou will 'nd a
dynamic component that seems to re§ect intuitionism (It
isbased on upper and lower probablities and the denominator is
measuretheoretic in that it has factors referencing positive
criteria and negativecriteria). You will also 'nd a dynamic
component that re§ects monotonicbelief revision. Both of
these, however, are rejected in formulating thetransferable
belief model of the title.If you wonder what belief has to do
with any of this, there is a nicediscussion of theories of
justi'cation at
http://www-philosophy.ucdavis.edu/phi102/tkch1.htmApparently,
philosophers are interested in logic, too.
:-):-)mitch
But, the point is this. I have been trying to
do what you are suggesting,>although in a slightly different
context. For the most part, a person isSorry for not taking
care of reading carefully the rest of your post.Overall it
seems interesting and, also, I thank you in any case
forHowever, with particular reference to the lines quoted
above, I'd liketo point out that the scenario 
I've depicted
(IMHO) is common tocompletly different contexts, and maybe the
actual circumstance thatcategory theory is a foundational one
tends to raise arti'cialissues with the approach I naively
sketched.More precisely, I'd like to make an example in a
context where such abias should not exist. Of course I'll be
rather vague here too: yousurely know that to extend the
concept of integral you can eitherde'ne a general notion of
measure and *build* an integral in terms o't in the usual 
way,
or you can abstractly de'ne a linear functionalwith certain
properties, that you will call integral, and de'ne ameasure 
in
terms of it (by means of charachteristic functions).Now the
point is that a priori nobody prevents you from formulating
a*third* theory in which both a mutilated measure (in the
sense thatit is not subject to all of the axioms of the 'rst
theory) and amutilated integral (in the obvious sense) are
primitive concepts andthe missing axioms are replaced by
other ones involving jointly bothof them (e.g.: there should
be one stating that the former is thelatter applied to
charachteristic functions).Most important, it must be possible
to show that the axioms of 'rstand/or second theory can be
proved as theorems, so that both thenewly de'ned measure and
integral are not mutilated in fact.Again, you might ask *why*
to follow this approach. The answer is thatthere's no
particular reason why: originally I was commenting onanother
poster who said they can't be *both* primitive concepts...As
an aside, however, it should be noted that there's the
remotepossibility that what I've called the third theory
might lend in anatural way to an higher order
generalization. But then I'm not anexpert in this 
'eld. (in a
trivial way, since I'm not an expert in*any* 
'eld!)Michele-- >
Comments should say _why_ something is being done.Oh? My
comments always say what _really_ should have happened. :)-
Tore Aursand on comp.lang.perl.misc
=What is the difference
between:- formal adjoint- the usual adjoint (as a map between
two normed spaces)?I know two of them should be the same, 
it's
a question about convention.
 Im taking a split
undergraduate/graduate Real Analysis course which> covers
Lebesgue measure and Lebesgue integral as well as
radon-nikodym> theorem. What book would you recommend ???>
Kolmogorov (trans by fomin?): Real Analysis> cdjThis is an
excellent book which starts out at set theory, moves
intotopology and goes into real analysis and integration. Its
4.5 starrated on Amazon and several of the respondents jummped
on the one guywho did not give it 5 stars. Its quite down to
earth with manyexamples. I personally used it to self teach
myself. I would verymuch recommend that you do every exercise
if you really want to learnthe stuff.
=[...] I would very>
much recommend that you do every exercise if you really want
to learn> the stuff.Well spoken.
=Could someone give an
example of a fourth degree polynomialP(x) with rational
coef'cients and with a real root r suchthat there is no 
second
degree extension of Q contained inQ(r)?Jose Carlos Santos
>
Could someone give an example of a fourth degree polynomial>
P(x) with rational coef'cients and with a real root r such>
that there is no second degree extension of Q contained in>
Q(r)?> Jose Carlos SantosI think this is not what you want,
but it answers your question as stated,so you will perhaps
want to restate the question.P(x)=(x-1)(x^3-2),
r=2^(1/3)
> Could someone give an example of a fourth
degree polynomial>> P(x) with rational coef'cients and with a
real root r such>> that there is no second degree extension of
Q contained in>> Q(r)?>> I think this is not what you want, but
it answers your> question as stated,> so you will perhaps want
to restate the question.>> P(x)=(x-1)(x^3-2), r=2^(1/3)the
subject: I want r to have degree 4 over the rationals.Jose
Carlos Santos-- Direct access to this group with
http://web2news.comhttp://web2news.com/?sci.mathTo contact in
private, remove noosp+5a9mm
>Could someone give an example
of a fourth degree polynomial>P(x) with rational coef'cients
and with a real root r such>that there is no second degree
extension of Q contained in>Q(r)?> Almost all degree-4
polynomials have galois group S_4(symmetric group of order 4).
Show that such polynomialslack the second-degree extensions you
want. Then 'nd sucha polynomial with a real root.
>>Could someone give an example of a fourth degree
polynomial>>P(x) with rational coef'cients and with a real
root r such>>that there is no second degree extension of Q
contained in>>Q(r)?> Almost all degree-4 polynomials have
galois group S_4> (symmetric group of order 4). Show that such
polynomials> lack the second-degree extensions you want.OK.
That took me a few minutes, but I got it. > Then 'nd such a
polynomial with a real root.Great! Now I have something to
think about during the weekend! :)Jose Carlos Santos
Could someone give an example of a fourth degree polynomial>
P(x) with rational coef'cients and with a real root r such>
that there is no second degree extension of Q contained in>
Q(r)?One fairly simple way to see if P(x) has this propertyis
to set out to solve the equation P(x) = 0using the standard
technique, getting an associated cubic.If the 'eld Q(r)
contains a quadratic 'eldthen this cubic must have a rational
root.-- Timothy Murphy tel: +353-86-233 6090
>Could
someone give an example of a fourth degree polynomial>>P(x)
with rational coef'cients and with a real root r such>>that
there is no second degree extension of Q contained in>>Q(r)? One fairly simple way to see if P(x) has this property> is
to set out to solve the equation P(x) = 0> using the standard
technique, getting an associated cubic.> If the 'eld Q(r)
contains a quadratic 'eld> then this cubic must have a
rational root.That's a rather simple way of 
'nding an aswer to
my question.ThnaksJose Carlos Santos
 some months ago I've
seen a web page with alternative methods to calculate>
factorial numbers.> There was Stirling's approximation but
also 6/7 other (not approximative)> methods to calculate n!>
other than 1 X 2 X 3 X .... X n> Unfortunately I didn't
bookmark the page!> Could you please help me in retrieve that
page of something similar.> Basically I'm looking for all 
the
knowns alternative methods to calculate> factorials.>
SandraNOT THAT PAGE, BUT A NEW ONE!I have just published a
piece on the Gamma function, and a new use
at:http://www.wehner.org/euler/I hope this is of use.Charles
In such a program, would one be able to rigorously de'ne the
measure of> an angle in a purely algebraic way, without an
appeal to analysis?> One could de'ne angles as isometry
classes of pairs of rays,> de'ne their ordering and
addition/subtraction but to show that> this gives a structure
isomorphic to the numbers in some interval> would require some
argument involving limits as far as I can see.If you choose
some constructible reference angle, say 90 degrees, andcall it
1, then by Euclidean (ruler and compass) constructions
tobisect angles, you can show that the set of angles at least
includesthe set {n/2^m : n,m integers} (intersect an
appropriate interval, ofcourse).Isn't (classical) Euclidean
geometry independent of the notion o§imits (i.e., can you not
construct Euclidean geometry over anon-complete space - a
missing axiom is that [suf'ciently close]circles
intersect)?There are (as far as I know - I am far from an
expert in thesubtleties of classical Euclidean geometry!) some
interestingquestions here.For example, 60 degrees is
constructible, but not included in the setI constructed above.
What is the set of constructible angles,assuming completeness?
The obvious generators that occur to meare arcsin(n/m) and
arctan(n/m) (n,m positive integers). However, I'mnot sure 
this
is a well-de'ned problem; if you can draw two linesegments of
arbitrary and independent length (can you do this?), thenof
course any angle is constructible (but then showing
isomorphism ofangles with an interval of real numbers is, as
Robin says, probablydif'cult without using limits).The same
question can be asked without the completeness
assumptions.Kevin.
> Now, just for the sake of argument, let
me run this by you. If you>> have two
instantaneously successive tangential velocities v1 and v2,> what is the vector difference between them? I think
it would have to>> be dr/dt centripetally directed
along r1.>>What do you mean by instantaneously
successive? Standard calculus>explanations draw
diagrams of two points/vectors, just a tiny, butde'nitely nonzero distance apart. If you mean this, 
it's
better (I>think) to avoid instantaneously. *In the
limit* we consider>properties derived from those two
points as they become inde'nitely>close together.>But if you mean two velocities a tiny time step apart,
then indeed the>difference between v1 and v2 is the
increment in velocity over that>time interval, and
indeed it points centripetally. But an increment invelocity is not dr/dt, because dr/dt means the derivative of
the>position vector r w.r.t. time; this is the
*instantaneous* velocity,>not a velocity increment over
a small time interval.> I'm not talking about a
velocity increment, which would be> acceleration.>I'm trying to be very careful (not just to nitpick), but a
velocity>>increment has units of velocity (as in a snail
moving at 3 ells per>>fortnight sees a lettuce and (perhaps
instantaneously) increases this>>by 0.5 ells per fortnight.
If the change is instantaneous, the>>acceleration is
inde'nitely large (by snail standards at least). But>>an
acceleration has units of velocity/time - it's not the
increment,>>but the increment per unit time. But ok, anyway,
you're not talking>>about this.> I'm not 
trying to
nitpick either. In fact we have far too much ground>> to cover
to spend a lot of time hyperventillating over small points.>>
The fact that the difference between velocities can be taken
as a>> velocity per unit time isn't at issue.>>But I think
that it's precisely the case that a *difference
between>velocities* and a *velocity per unit time*
(acceleration) are not the>same thing, nor in any sense that I
understand can they be taken as>the same. This might be at
the root of the confusion.Might be. I think when different
velocities are applied to the samemass point, they necessarily
imply a temporal succession with a forcerequired to explain the
difference and that the force operates in thedirection of the
difference or change in velocity.>> What is at issue is the
direction and magnitude of the difference>> between successive
tangential velocities in circular rotational>> motion.>>Apart
from your use of the odd term tangential velocities,
nothing>is in dispute here. (Velocity is a vector; you could
have the>tangential speed, being the component of the
velocity in the>tangential direction, but tangential
velocity is either tautological>if the velocity is tangential
[which it is in this case] or>contradictory if it isn't.)I
understand what you are saying and don't necessarily 
disagree
thatthe terminology is confusing. It's just that in the case
of circularrotation there are two categorically distinct
velocities - at leastthis is how the problem is de'ned in
celestial mechanics. There isone velocity associated with
tangential motion and one directedcentripetally. And rotation
results from the vector combination.The tangential component
is constant but the centripetal componentresults from some
kind of centripetal acceleration like gravitation.The latter
is the instantaneous integral of dr/dt dt = dr/dt.>> ... If
you want to consider that difference as divided by some>>
incremental time, the result is acceleration. But the
difference>> between tangential velocities remains a
centripetally directed>> velocity in any case.>>The difference
between velocities is a velocity increment over a>nonzero time
interval. It is not an instantaneous property of the>motion.
(Your argument is uncannily similar to Zeno's stuff
about>motion being contradictory, btw.)I think you may be
looking at the problem backward or upside down. Wedon't take
dr/dt as the ratio of dr divided by dt in situations
whereforces are present.Instead we integrate the force over dt
to produce dr/dt and ifnecessary dr/dt over dt to determine dr.
In the case of circularrotation we have a centripetally
directed force that produces dr/dtand the centripetally
directed dr/dt that produces dr.In the absence of a force we
integrate dr/dt over dt to produce dr butthen we don't have
circular rotation. Only in the case where we have aforce
directed normal to dr/dt in linear terms do we have rotation.I
think most people regard this linear dr/dt and its resultant dr
asthe appropriate measures. However, neither produces rotation.
Theyonly produce linear changes in displacement whereas what we
need to belooking for is some measure of what produces changes
in angulardisplacement. And I contend that the appropriate
measure needed to dothat is dr/dt directed normal to linear
dr/dt.In circular rotation there are actually two mutually
normal velocitiespresent: linear dr/dt which is constant and
represents what I calltangential velocity and radial dr/dt
which is the result ofcentripetal acceleration. We can analyze
these in different ways butthe net result is that the presence
and nature of rotation isdetermined by the nature of radial
dr/dt if linear dr/dt is constant.> I think I see
what you're getting at. But let me suggest that dr/dt>
represents a velocity corresponding to the difference between
v1 and> v2.>>But it doesn't, does it. dr/dt is the
differential w.r.t. time of the>>position; it's a vector
which _is_ the instantaneous velocity (which>>of course is
in the direction the object is moving). How could it>simultaneously represent a velocity in a different
direction? I>>can't avoid a suspicion that this goes back to
an intuition that when>>you apply a force to an object the
object must move in the direction>>of that force. Plainly
this can't be true, or a weight swung>>circularly on a piece
of string would spiral inexorably inwards.> Well, you know,
I think this is exactly what happens in Newtonian>> celestial
mechanics. A mass, weight, or planet spirals inexorably>>
inwards. But the spiral inwards is exactly offset by
tangential>> velocity. And the mass, weight, or planet
maintains its orbit.>>When I say spiral inwards, I mean
describe a spiral path, of which>the radius is monotonically
decreasing. When something is moving in a>circle, the radius
is constant.I understand and was really just being facetious.
If there is some'nite value of centripetal dr/dt then I would
agree. In point of factcentripetal dr/dt is equal in magnitude
to linear tangential dr/dt.But because centripetal dr/dt is
only applied for in'nitessimal dt'sits actual 
magnitude is
also in'nitessimal and its only effect is tochange the
direction of tangential v.The same is true of centripetally
directed dr. Its magnitude isin'nitessimally small so it has
no 'nite effect on the magnitude ofthe radius r. So we have
rotation rather than a spiral.>> ... dr/dt is what we
have in the cross product of dL/dt and it is> produced as
the result of the centripetally directed force.>>I'm a
bit lost here, since I don't actually know what 
ïL' is (nor
what>>a unary cross product is). Again, it seems to me,
produced as is a>>dangerously suggestive expression.>
The original post in the thread explained that the angular
momentum is>> L = r x p. Meron suggested we analyze the issue
of whether circular>> rotation represents constant angular
momentum - his and everyone>> else's position - or 
constantly
changing angular momentum - as I>> suggest - in terms of dL/dt
= dr/dt x v + r x dv/dt where x represents>> the cross
product.> Now, the conventional interpretation of dr/dt
takes dr in the>> direction of tangential velocity parallel to
tangential velocity v.>> Consequently, dr/dt x v would be zero
by the de'nition of a cross>> product.>>No, the conventional
interpretation of dr/dt is that it is the>derivative w.r.t.
time of the position vector. It doesn't take>anything in any
direction, it looks to see what direction dr/dr _is_>in. Take
two values of r (r1 and r2) at close, but separated times
t1>and t2 (difference delta_t); the difference vector between
r1 and r2>is a chord of the circle. As delta_t tends to zero,
this chord>approaches a tangent to the circle. In the limit,
this tangent is the>direction of the velocity vector, dr/dr
aka v; it's a relief to most>people to see that the velocity
vector does indeed point in the>direction of movement of the
body, i.e. around the circle. When you>look at any chord
approximation, it does have a centripetal component>w.r.t. r1,
but a centrifugal component w.r.t. r2. These both tend to>zero
in the limit. Motion around a circle does not spiral inwards.I
understand what everyone's getting at. If we have a constant
linearv we also have a constant rate of change in terms of
dr/dt. However,this does not de'ne any kind of rotation
because it does not de'neany difference in linear v: it only
de'nes constant motion in astraight line.If we want to 
de'ne
any kind of rotation we have to analyze thecentripetally
directed force and velocity normal to linear dr/dt toexplain
changes in linear dr/dt. So, if we're trying to
comprehendangular mechanics as opposed to linear mechanics I
contend we can onlydo so in terms of centripetally directed
dr/dt because this is whatproduces differences in linear dr/dt
and in fact the only thing thatdoes.So, if we're trying to
determine if the cross product L = r x p isconstant or changes
and if we do so by analyzing whether its timederivative is zero
or not, then the only thing pertinent to that is the measure of
centripetally directed dr/dt and not linearly directeddr/dt. We
already know that linearly directed dr/dt produces no
changebecause it is constant. So, the only relevant question
is reallywhether there are changes to linear dr/dt. And if
there are they canonly be produced by some non zero
centripetally directed dr/dt.>> On the other hand my
contention is that dr/dt represents a velocity.>>Yes, a
velocity.> So, that the direction of dr/dt has to correspond
to the difference>> between successive tangential velocities
and thus is directed>> centripetally along r rather than
parallel to tangential v. >>No, because the change in velocity
aka the acceleration does not have>to be in the same direction
as the velocity, and can and in this case>does have a zero
component in the direction of the velocity.>>OK, I give
up.Yeah, Brian, I have gone into further detail above that I
hopeclari'es the thinking for what I contend. 
I'm not trying
to beevasive in closing out the thread, and if there remain
signi'cantconceptual problems, please have at, although it
would be helpful ifwe could eliminate as much redundant
material as possible andconcentrate on just a couple of
critical points at a time.I'm still trying to get the post 
on
the analytical derivation forPlanck's constant ready for 
early
next week. So, I'm also trying toconcentrate on that.
 [.
. .]> This is frustrating because we're up against a
deadline. So, I'd like> to reply to just the tail end of 
your
post now and get to the> beginning later. I have a semi-free
day, so...>>... if your assertion is that a stable orbit
>>represents changing angular momentum, where does the energy
>>come from/go to?> You know, initially I thought there was
a problem in this regard. But> now I'm not so sure. Granted
that I think I'm talking about a> constantly changing 
measure
of angular momentum, the fact is that> we're ever in a
position to measure such changes in isolation. If you're
talking about spin, you can't measure it in isolation. RPM 
is
always WRT the external Universe, neh? Anyway energy is
conserved. If it isn't, where're the 
source/sink?> In other
words, circular rotational motion can only be mechanized>
through the interaction of point masses in opposition to one
another. Many cases to consider. Two co-orbiting objects that
aren't spinning WRT each other see each other as gravitating
masses at a constant distance.> And I think that the changes
offset one another. At least this is how> it seems to me at
present and is the reason I asked to address the> problem
later.> The whole issue relates to the difference between
constant angular> momentum considered in purely linear
contexts at constant velocity and> cases of circular rotation.
I think there has to be some analytical> and mechanical
difference between the two cases. I'm just not quite> sure 
yet
what that difference is and what the implications may be for>
angular mechanics in general. But I'm considering the 
problem.
When you step from galactic rotation through gyroscopes of
rotation to handle. Are you trying to say they're the 
same?>>
Are we getting into cosmological issues here?> Probably.
If not right away almost certainly in the next go round. I>
don't think I'm attacking the practical 
implications of macro
angular> and celestial mechanics. But I do think I'm 
attacking
the analytical> foundation for quantum mechanical theory in
pretty speci'c terms. An angular version of 
Mach's Principle?
I don't see that QM is going to be helpful. IIUC spin in
elementary way charge and magnetism transform; nobody (sane)
thinks electrons act like little tops. Then there're things
like precession and spin-spin quantized, but not in macro
systems.>> Trying to think about angular motion in linear
terms is >>trouble. Breaking angular motion into linear bits
works _if_ >>you remember to refer them to an inclusive
coordinate frame >>_and_ to use the same coordinate frames
for the bits and the >>system.>>I think the best
explanation I've been able to come up with is the>circular
celestial orbit with the centripetally directed
gravitational>force.>> Yes, and?> Well, to me it
pretty much seems to prove the case I'm making. Mainly>
because dr/dt is taken to be centripetally directed velocity
which> when combined vectorially with tangential v results in
rotation of> both vectors.>... I seem to recall that>on
the original Trivial Pursuit thread you remarked that the
center of>the galaxy was rotating faster than the amount of
matter would>indicate.>> Not sure that was me, but
evidence seems to say that, >>hence all the dark matter/energy
§ap.> Actually, pretty much sure it was.> I wonder if
that's true for other galaxies as well, such>as the
Andromeda.>> AFAIK yes.> Good. Then I may be on the
right track.>I've had a couple of thoughts on the subject
which may>explain the effect.>> So think out loud. You
can't be any more wrong than >>anyone else on the issue.>
I intend to. I'm just waiting for the right moment.
Unfortunately, I> have to establish one other unobvious
principle before I do. I just> wanted to make sure I'd have
someone to talk it over with. This other> principle has a well
established and easily accessible phenomenon in> experimental
terms. So, I'd rather establish it before trying to apply> 
the
principle to galactic rotation. But it should be coming up in
a> few weeks. Nothing to do with the present subject as far as
I can> tell. OK, we'll see. Mark L. Fergerson
 I'm still
trying to get the post on the analytical derivation for>
Planck's constant ready for early next week. So, 
I'm also
trying to> concentrate on that.> You're just trying to drive
us all crazy right? What arrogance! How galling!!
[sputter...gasp...gulp]Ahem.You still haven't got a handle 
on
the most basic ideas of circular motion (the above post is a
nice summary of the extent of your confusion) and now you're
onto an analytical derivation for Planck's constant?!?!The
interesting aspect of your communications has nothing to do
with modern physics. Alas, it is about cognitive science. Why
do otherwise intelligent people keep on responding to you? I
try to limit my own input but the urge is there. Sometimes
it's just irresistible.Self control. That's 
what I need.JUST
... SAY ... NO-- Joe Legris
>> I'm still trying to get
the post on the analytical derivation for>> Planck's 
constant
ready for early next week. So, I'm also trying to>>
concentrate on that.>You're just trying to drive us all
crazy right? What arrogance! How >galling!!
[sputter...gasp...gulp]>>Ahem.>>You still haven't got a 
handle
on the most basic ideas of circular >motion (the above post is
a nice summary of the extent of your >confusion) and now
you're onto an analytical derivation for 
Planck's
>constant?!?!>>The interesting aspect of your communications
has nothing to do with >modern physics. Alas, it is about
cognitive science. Why do otherwise >intelligent people keep
on responding to you? I try to limit my own >input but the
urge is there. Sometimes it's just irresistible.>>Self
control. That's what I need.>>JUST ... SAY ... NOI know, I
know. I used to experience the same thing in politics. Thenone
day I learned to curb the urge. Extraordinarily frustrating
butultimately rewarding.I would like to say that I've
sublimated politics by trolling physicsnewsgroups. Just 
isn't
the case unfortunately. Arrogance helps in theapproach but it
doesn't do anything for the analysis.As a simple bridge, let
me ask in general if you consider thateverything there is to
say on the subject of angular momentum andPlanck's constant
has already been said. Not in detail. Just whetheryou think
there are things remaining to be said.
=[. . .]>>There are
lots of things to be said. Start by learning a little calculus
>and saying: the integral of dr/dt dt is neither dr nor dr/dt.
It is r.>>Save Planck's constant for next year.I think the
in'nitessimal integral of dr/dt dt would be dr and
thein'nitessimal integral of dv/dt dt would be dv = dr/dt.
Byin'nitessimal I mean instantaneous. The 'nite 
integral of
dr/dt dtis r. But that only occurs over a 'nite interval of 
t.
Over aninstantaneous or in'nitessimal interval, the integral 
of
dr/dt dtwould be dr. It would lie in the same direction as r
but wouldrepresent no 'nite change to r.But I'm 
always happy
to hear what you have to say on the subject.By the way, as
long as we're talking math, I've been 
reconsidering thematter
of de'nition when it comes to taking the derivative of a
crossproduct.If we have a cross product like a = b x c and
take a derivative, theresult is a' = b' x c + 
b x c'. But the
b' always lies in thedirection of b and the c' 
always lies in
the direction of c byde'nition. Otherwise, there is no
derivative of the cross product abecause if b' lay in the
direction of c' or vice versa they would notrepresent cross
products by de'nition and would not re§ect anychange to a 
= b
x c which is what the derivative of a cross product isdesigned
to re§ect.Which means that for a cross product such as L = r x
p the timederivative dL/dt = dr/dt x p + r x dp/dt must re§ect
dr/dt in thesame direction as r by de'nition. 
= > [. . .]
>>There are lots of things to be said. Start by learning a
little calculus >>and saying: the integral of dr/dt dt is
neither dr nor dr/dt. It is r. >Save Planck's constant for
next year. > I think the in'nitessimal integral of dr/dt dt
would be dr and the > in'nitessimal integral of dv/dt dt 
would
be dv = dr/dt. By > in'nitessimal I mean instantaneous. The
'nite integral of dr/dt dt > is r. But that only occurs over 
a
'nite interval of t. Over an > instantaneous or 
in'nitessimal
interval, the integral of dr/dt dt > would be dr. It would lie
in the same direction as r but would > represent no 'nite
change to r. > But I'm always happy to hear what you have to
say on the subject. > By the way, as long as we're talking
math, I've been reconsidering the > matter of 
de'nition when
it comes to taking the derivative of a cross > product. > If
we have a cross product like a = b x c and take a derivative,
the > result is a' = b' x c + b x 
c'. But the b' always lies
in the > direction of b and the c' always lies in the
direction of c by > de'nition. Otherwise, there is no
derivative of the cross product a > because if b' lay in the
direction of c' or vice versa they would not > represent 
cross
products by de'nition and would not re§ect any > change to 
a =
b x c which is what the derivative of a cross product is >
designed to re§ect. > Which means that for a cross product
such as L = r x p the time > derivative dL/dt = dr/dt x p + r
x dp/dt must re§ect dr/dt in the > same direction as r by
de'nition. >Your 'rst paragraph is just 
nonsense. You
knew that r was of constantlength and that it is directed
outward radially from the initialassumptions of uniform
circular motion. Getting to an inwardpointing vector of
constant length is a simple matter of taking the second
derivative, i.e. 'nding the acceleration. There is no need to
fudge a differential, hoping to a achieve a what amounts to a
variable constant. Pure silliness.Your second paragraph is
just plain wrong. If a, b and c are arbitraryvector functions,
it is not true that b' always lies in the directionof b and
the c' always lies in the direction of c. For example,
asfunctions of time if b = (1,1,t) then b' = (0,0,1). They 
are
NEVER inthe same direction for 'nite t.-- Joe Legris
I
think the in'nitessimal integral of dr/dt dt would be dr and
the>in'nitessimal integral of dv/dt dt would be dv = dr/dt.
By>in'nitessimal I mean instantaneous.Lester, Lester - it
doesn't matter what you think. The integral of DR/DT is
Vbecause that's how an integral is de'ned.BTW, 
in'nitesimal
integral is a meaningless.-- Best Wishes,Wolf Kirchmeir,
Blind River ONNot that brains are everything --you'll also
need a skull to put them in. (Nancy Franklin, 1997)
>
[. . .]>>There are lots of things to be said. Start by
learning a little calculus>>and saying: the integral of
dr/dt dt is neither dr nor dr/dt. It is r.>>Save
Planck's constant for next year.> I think the
in'nitessimal integral of dr/dt dt would be dr and the>
in'nitessimal integral of dv/dt dt would be dv = dr/dt. By>
in'nitessimal I mean instantaneous. The 'nite 
integral of
dr/dt dt> is r. But that only occurs over a 'nite interval
of t. Over an> instantaneous or in'nitessimal interval, the
integral of dr/dt dt> would be dr. It would lie in the same
direction as r but would> represent no 'nite change to r.> But I'm always happy to hear what you have to say on the
subject.>> By the way, as long as we're talking math, 
I've
been reconsidering the> matter of de'nition when it comes to
taking the derivative of a cross> product.>> If we have
a cross product like a = b x c and take a derivative, the>
result is a' = b' x c + b x c'. 
But the b' always lies in the direction of b and the c' always lies in the direction of 
c
by> de'nition. Otherwise, there is no derivative of the
cross product a> because if b' lay in the direction of 
c' or
vice versa they would not> represent cross products by
de'nition and would not re§ect any> change to a = b x c
which is what the derivative of a cross product is> designed
to re§ect.>> Which means that for a cross product such as L
= r x p the time> derivative dL/dt = dr/dt x p + r x dp/dt
must re§ect dr/dt in the> same direction as r by 
de'nition.>Your 'rst paragraph is just nonsense. You knew that
r was of constant>length and that it is directed outward
radially from the initial>assumptions of uniform circular
motion. . . .So far I'm with you.> . . . Getting to an
inward>pointing vector of constant length is a simple matter
of taking the >second derivative, i.e. 'nding the
acceleration. There is no need to >fudge a differential,
hoping to a achieve a what amounts to a variable >constant.
Pure silliness.But here I've lost you. I just 
don't see what
you're getting at.>>Your second paragraph is just plain 
wrong.
If a, b and c are arbitrary>vector functions, it is not true
that b' always lies in the direction>of b and the 
c' always
lies in the direction of c. For example, as>functions of time
if b = (1,1,t) then b' = (0,0,1). They are NEVER in>the same
direction for 'nite t.Think it through, Joe. A cross product
such as a = b x c is alwaysproportional to area. So, the
derivative of a cross product isnecessarily de'ned in terms 
of
changes in area. Which means you canonly evaluate the
derivative of a cross product in terms of changes tothe area
subtended by b x c. Which means b' has to point normal to
c.That's by the de'nition of a cross product 
and the
derivative of across product.This is why b' has to be 
radially
directed. The product b' x c has tode'ne a 
change in area to b
x c even if the change is zero. But a b'in the direction of 
c
does not correspond to any area zero or notbecause they 
don't
intersect and don't subtend or de'ne an area 
ofany kind.
Consequently, b' x c with b' in the direction 
of c does
notcorrespond to the derivative of a cross product. It's as
simple asthat.>Oh, and by the way, I'll probably be running
I
think the in'nitessimal integral of dr/dt dt would be dr and
the>>in'nitessimal integral of dv/dt dt would be dv = dr/dt.
By>>in'nitessimal I mean instantaneous.>>Lester, Lester - it
doesn't matter what you think. The integral of DR/DT is
V>because that's how an integral is 
de'ned.>>BTW,
in'nitesimal integral is a meaningless.Ah, 
c'mon, Wolf. A
time derivative is just the ratio dr/dt as t --> 0.So, I think
that integral of v dt = dr/dt dt = dr over the sameinterval.
That's what I mean by in'nitessimal or
instanteousintegral.Likewise the in'nitessimal integral of a
dt = dv/dt dt = dv for thesame reason. After all it is called
in'nitessimal calculus for areason. So, if you can take a
derivative over an in'nitessimalinterval I 
don't see why you
can't take an antiderivative or integralover the same
interval.When you take a 'nite integral obviously the results
will be 'nite.This is the reason centripetally directed v and
r do not change by'nite amounts in circular rotation despite
the presence ofcentripetal force.And of course it doesn't
matter what I think unless of course I'mright.>>-- >Best
Wishes,>Wolf Kirchmeir, Blind River ON>Not that brains are
everything -->you'll also need a skull to put them in. 
(Nancy
Franklin, 1997)>>
Ah, c'mon, Wolf. A time derivative is
just the ratio dr/dt as t --> 0.No, it's as dt --> 0, which 
is
not quite the same thing.>So, I think that integral of v dt =
dr/dt dt = dr over the same>interval. That's what I mean by
in'nitessimal or instanteous>integral.I can't 
make sense of v
dt = dr/dt dt = dr. Do you mean v * dt = (dr/dt) *dt =
dr?Even if so, it still doesn't give meaning to 
in'nitesimal
integral. Anintegral is a function, not a value; but dr is a
value. You can evaluate theintegral for speci'ed values of r
and t, of course.-- Best Wishes,Wolf Kirchmeir, Blind River
ONNot that brains are everything --you'll also need a skull
to put them in. (Nancy Franklin, 1997)
Ah, c'mon, Wolf.
A time derivative is just the ratio dr/dt as t --> 0.>>No, 
it's
as dt --> 0, which is not quite the same thing.I think 
you're
correct here. But I don't see that it changes the case.>So,
I think that integral of v dt = dr/dt dt = dr over the
same>>interval. That's what I mean by 
in'nitessimal or
instanteous>>integral.>>I can't make sense of v dt = dr/dt 
dt
= dr. Do you mean v * dt = (dr/dt) *>dt = dr?I mean that
the integral of a over the period of time dt as dt -->
0produces v and that the integral of v over the period of time
dt as dt--> 0 produces dr and that all are centripetal in
direction.I would like to stay away from the question as to
whether the resultsare vectors or scalars or scalar multiples
of vectors because I'm justtrying at this point to establish
the values involved. I think theresults are scalar multiples
of vectors but am just trying toconcentrate on the general
idea of instantaneous integral values,which are just the
integrals over the same period of time dt used toform
derivatives as dt --> 0.>>Even if so, it still doesn't give
meaning to in'nitesimal integral. An>integral is a
function, not a value; but dr is a value. You can evaluate
the>integral for speci'ed values of r and t, of course.Unless
we are talking about in'nitesimal values of r and t in
whichcase we are not talking about 'nite magnitudes of v, r,
or t. Theintegral is a function. It's just that taking the
integral over thesame dt as is used in forming the derivative
produces an instanteousvalue for the result rather than the
more conventional 'niteintegration over a range.>>-- >Best
Wishes,>Wolf Kirchmeir, Blind River ON>Not that brains are
everything -->you'll also need a skull to put them in. 
(Nancy
Franklin, 1997)>>
dr/dt dt = dr. Do you mean v * dt = (dr/dt) *>dt = dr? I mean that the integral of a over the period of time dt as
dt --> 0> produces v and that the integral of v over the
period of time dt as dt> --> 0 produces dr and that all are
centripetal in direction.Hmm. When I was taught about
integration and stuff, we were warnedVery Sternly that if we
thought we could treat dr and dt as thoughthey were ordinary
quantities, we would soon produce nonsense. I'mglad I got 
that
warning.> I would like to stay away from the question as to
whether the results> are vectors or scalars or scalar
multiples of vectors because I'm just<niggle>Can you point 
to
a vector that is not a scalar multiple of avector?</niggle>>
Unless we are talking about in'nitesimal values of r and t in
which> case we are not talking about 'nite magnitudes of v, 
r,
or t. The> integral is a function. It's just that taking the
integral over the> same dt as is used in forming the
derivative produces an instanteous> value for the result
rather than the more conventional 'nite> integration over a
range.It does? Hmm, an interesting idea. OK, so consider this
body movinground this circle, a snail gliding around the
periphery of a clock, ifyou like.[nf] stands for normal
'nite - this is the one the rest of us talkabout.At 12
o'clock, r, the [nf] position vector, points up, v the
[nf]velocity vector points left, a, the [nf] acceleration
vector iscentripetal, and points down.However, you say,
integrating dv/dt [down] over dt gives theinstantaneous
velocity, which must of course be in the same direction- i.e.
down. By exactly the same argument, integrating dr/dt
[left]over dt gives the instantaneous position, which must of
course be inthe same direction - i.e. left. So although the
[nf] snail is munchinga [nf] lettuce at the top of the clock,
its instantaneous positionis somewhere along the hour hand.
(I omitted to mention that ithappens to be nine o'clock.)I
think your argument could also easily show that a braking car
isactually going backwards. Somewhere I fancy there should be
a usefullegal application of all this.Brian
Chandler----------------geo://Sano.Japan.Planet_3Jigsaw
puzzles from Japan
at:http://imaginatorium.org/shop/
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small screen area of the device. Most of the screen area is
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other calculators where most of the screen is populated by
buttons. For a complete list of features (with lots of screen
shots) and some more information go to:
http://www.poliplus.com/handheldproducts.htm Poliplus Software
is excited about this release as it demonstrates the
versatility and power of its F1 application. Carlos
Bazzarella, CEO of Poliplus Software commented Years ago,
our vision was aimed at the convergence of mobile computing.
The P800 is one of the devices leading the way in that
direction, and we are proud to offer our CAS for it as it
further validates our strength and innovative software written
in Java. Poliplus Software is dedicated to offering powerful
educational handheld applications. About Poliplus Software
Poliplus Software is an innovative developer of non-trivial,
feature rich applications for handheld devices. Poliplus'
comprehensive suite of applications in the 'elds of
Mathematics, Science and Engineering add tremendous value to
handheld devices. All of our products are developed in 100%
pure Java using various standard pro'les including MIDP,
PersonalJava, JavaSE and JavaEE. For more information please
visit www.poliplus.com. -30-Contact: Rafael Bazzarella
Poliplus Software rbazzaTAKE@OUTpoliplus.com
 .... > Find
the general solution of the non-linear differential equation: 2y'y''' - 3 
y''^2 - y'^4 = 0> .... I 
shan't spoil other
people's fun by listing the succession of trickswhich led to
the solutiony = a +/- ln|(x + b)/(x + c)|where a, b, c are
constants of integration. Has it any singular solutions as
well? Ken Pledger.
=hi,Could you guys recommend some good
texts on 3-manifolds which are not toohard to start with? I
'nd Hample's classic text is a little bit hard 
tostart
with.
hi,>>Could you guys recommend some good texts on
3-manifolds which are not too>hard to start with? I 'nd
Hample's classic text is a little bit hard to>start
with.What's your background? What's your 
reason for wanting
theread such a text?Depending on the answers, you might 'nd
the material on thesubject in Rolfsen's _Knots and Links_
good, or not. (Itcouldn't hurt you, but you might 
'nd it
off-target, or youmight know it all already, or something.)I
agree that Hampel is somewhat daunting. Jaco's _Lectures
on3-Manifold Topology_ might be less so, but as I remember
themboth, both are rather dry. (It's a subject which can 
*be*
verydry.) Jaco is less condensed, though. Lee Rudolph
>>hi,>>Could you guys recommend some good texts on
3-manifolds which are not too>>hard to start with? I 'nd
Hample's classic text is a little bit hard to>>start with.>
What's your background? What's your reason for 
wanting the>
read such a text?> Depending on the answers, you might 'nd
the material on the> subject in Rolfsen's _Knots and Links_
good, or not. (It> couldn't hurt you, but you might 
'nd it
off-target, or you> might know it all already, or something.) I agree that Hampel is somewhat daunting. Is this contagious?
I mean, spelling Hempel's name wrong. 
;-)Jaco's _Lectures on>
3-Manifold Topology_ might be less so, but as I remember them>
both, both are rather dry. (It's a subject which can *be* 
very>
dry.) Jaco is less condensed, though. > Yes, I agree, Jaco's
book is not as condensed, but I think if Hempel istoo hard for
the OP then Jaco will be even harder. My recollection isthat
the material that overlaps with Hempel is far more involved.
Ofcourse what may make Hempel intimidating is that it touches
on quite anumber of topics, whereas Jaco appears friendlier
since it focuses oncertain topics.A problem with both these
books is that unless you already know some3-manifolds, it can
be hard to see what the point is, to see the forest,instead of
only the trees. I have some suggestions for this, which I
willdefer to a post made directly to the OP.
=[Sarah says
Hample, I say Hampel, Chan-Ho calls the whole thing
off]>Is this contagious? I mean, spelling Hempel's name
wrong. ;-)Hey, my typo-recognition algorithm worked perfectly.
Thetypo-*correction* algorithm is still a bit buggy.>A problem
with both these books is that unless you already know
some>3-manifolds, it can be hard to see what the point is, to
see the forest,>instead of only the trees. I have some
suggestions for this, which I will>defer to a post made
directly to the OP.Oh, go ahead, spill the beans in public.
More 3-manifold topologyin sci.math can only raise the
tone.Lee Rudolph
 hi,> Could you guys recommend some
good texts on 3-manifolds which are not too> hard to start
with? I 'nd Hample's classic text is a little 
bit hard to>
start with.> You might try Allen Hatcher's notes on
3-manifolds. They are incomplete,stopping short before
explaining Haken manifolds, but they cover thebasics of
incompressible surfaces, Seifert manifolds, prime
decomposition,JSJ decomposition (simple geometric version),
torus bundles, andthe loop and sphere theorems. Hatcher's
writing style, I think, makesthese notes very
accessible.http://www.math.cornell.edu/~hatcher/3M/
3Mdownloads.htmlFor some reason, the Cornell math site 
hasn't
been up lately, buthopefully it should work soon.Also, try
Marc Lackenby's 3-manifold
notes:http://www.maths.ox.ac.uk/~lackenbyproof of 
Waldhausen's
theorems. A noteworthy feature is the 'rst fewchapters are
devoted to nice, informal explanations of the basic PL and
bundle technology that you need to get started studying
3-manifolds. If you haven't studied PL topology or the
analogous smooth theory, thisshould be handy.I found reading
parts of Hatcher's and Lackenby's notes made 
readingHempel far
easier, especially since I think it gave me a sense of
whatparts of Hempel to read 'rst. I wouldn't 
recommend reading
Hempel fromcover to cover, although reading the 'rst few
chapters (on Heegardsplittings and prime decomposition) is
recommended. Oh, and I de'nitelywouldn't 
recommend spending
more than a few minutes on Chapter 1. Afterall, aren't the
transversality and regular neighborhood stuff
fairlybelievable? ;-) Not to mention, speaking from
experience, learning that stuff really requires a whole summer
by itself, and after you've done that, you feel it was 
fairly
laborious and obvious after all. You shouldde'nitely learn 
the
material, but it's best to pick it up in pieces asyou need 
it.
[I recommend a book by Leslie Glaser for this stuff] What you
want to do is get as quickly as you can to Haken manifolds
andWaldhausen's theorems (I think that's in 
chapters 11 and
12?). They areat the very core of 3-manifold theory,
motivating many times what peopleare doing today. Hempel's
writing style (since he references the mainresults he uses)
should make it clear what sections of previous chaptersyou
should know. I would say the material on Haken hierarchies and
Waldhausen theorems are written very well in Hempel. In
general, Hempelis a very nicely written book, and I am at a
loss to explain why initiallyI found it so repugnant. It might
be Chapter 1. Or it might be becausethere are only three
pictures in the entire book (unless you countcommutative
diagrams). Oh, and that's another thing I 
don't think
isemphasized often enough. For this kind of thing,
low-dimensionaltopology, 3-manifolds, or whatever, drawing
pictures is really important. Much of Hempel's prose becomes
much clearer when you draw the appropriatepicture, so I
suggest you do so.Unfortunately, there aren't many books on
3-manifolds in general, or otherthan Hempel (or possibly Jaco)
books that are meant to ground beginners inthe subject. And of
course there are all these books (like Hempel) thatwere
written in the 70s, before Bill Thurston's work had sifted
down to bemainstream 3-manifold stuff. At some point you'll
need to learn what Thurston's work has to do with thestuff 
in
Hempel, and then the scarcity of good books becomes even
worse. 
i;@/WO(?;[KC9sW;wG/4@H[_VFFH4?QHJ#O(?m}7fQMrJ,]0THA'|e-EPG
_>56Mi}_RRhBS'a2}u_7jm)0_+'=$V#E2r4#IQE/d)yMv3
_4@hl<)mA&*tDN/
=In sci.math, Will
Self<wself@msubillings.edu><vfjjfoescte7e6@corp.supernews.com>
:>> Does 3*pi/4 = 7*arctan(1/3)+arctan(29/278)?>>
Just curious,>> Rich Burge>> Yes.>> arctan(29/278) is
the angle of 278+29i.>> Now consider the factorization of
these Gaussian integers into primes.>> -- >> Chris Henrich No need to do any factoring. Just multiply (278+29i)(3+i)^7>
PARI/GP (from the PARI group), a GNU algebraic calculator,
gives29*i^8 + 887*i^7 + 11319*i^6 + 79947*i^5 + 344925*i^4+
936117*i^3 + 1566621*i^2 + 1482057*i + 607986Since i^4 is 1
this can be reexpressed as29 + 887*i^3 + 11319*i^2 + 79947*i +
344925+ 936117*i^3 + 1566621*i^2 + 1482057*i + 607986(I do this
to reduce the chance I inadvertantly §ip the wrong
sign).Collecting terms we get937004*i^3 + 1577940*i^2 +
out, this is a vector at angle 135 degrees (3 pi / 4)of length
625000 * sqrt(2), and PARI/GP has a complex mode anywayusing
ïI' instead of ïi', 
giving the same result.GP could spoil me
rotten. :-)-- #191, ewill3@earthlink.netIt's still legal to 
go
news@aol.com
<57ngt-onk.ln1@lexi2.athghost7038suus.net>
In sci.math, Will
Self><wself@msubillings.edu>><vfjjfoescte7e6@
corp.supernews.com>:> Does 3*pi/4 =
7*arctan(1/3)+arctan(29/278)?>> Just curious,> Rich Burge> Yes.> arctan(29/278) is the angle of
278+29i.>> Now consider the factorization of these
Gaussian integers into primes.>> -- > Chris Henrich>
No need to do any factoring. Just multiply (278+29i)(3+i)^7>>PARI/GP (from the PARI group), a GNU algebraic calculator,
gives>>29*i^8 + 887*i^7 + 11319*i^6 + 79947*i^5 + 344925*i^4>+
936117*i^3 + 1566621*i^2 + 1482057*i + 607986>>Since i^4 is 1
this can be reexpressed as>>29 + 887*i^3 + 11319*i^2 + 79947*i
+ 344925>+ 936117*i^3 + 1566621*i^2 + 1482057*i + 607986>>(I do
this to reduce the chance I inadvertantly §ip the wrong
sign).>>Collecting terms we get>>937004*i^3 + 1577940*i^2 +
turns out, this is a vector at angle 135 degrees (3 pi / 4)>of
length 625000 * sqrt(2), and PARI/GP has a complex mode
anyway>using ïI' instead of 
ïi', giving the same result.>>GP
could spoil me rotten. :-)>Another choice would be
lindep([arctan(1),arctan(1/3),arctan(29/278)]).The way I
discovered the identity was using the following:Given
x[0],y[0], and z[0] are real numbers, compute x[i],y[i] and
z[i], i>0 asfollows: x[i+1]=y[i]-int(y[i]/x[i])*x[i]
y[i+1]=z[i]-int(z[i]/x[i])*x[i] z[i+1]=x[i].There would seem
to exist integers a,b,c s.t. a*x[0]+b*y[0]+c*z[0]=0 iff
x[i]=0for some i>0. (The appropriate generalization, if true,
answers a recentquestion from sci.math.research about algebraic
numbers.) It does work for the case above, anyway. Finding
a,b,c is easy. The methodalso generalizes quite nicely. I
don't really know how fast it is. A simpleUBASIC
implementation takes about twice as long as PARI/gp lindep for
7variables with the same precision but doesn't always get 
the
same answer.Curiously, if the method doesn't agree with
PARI/gp's lindep, replacing x[i+1]with 2*x[i+1], ect
sometimes helps. I suspect that a good implementation wouldbe
both fast and accurate.R. Burge
=Let G be a 'nite abelian
group generated by x and y (i.e. G = <x, y>) and H asubgroup,
such that G = <x - y> + H. Suppose |G| = m n, |H| = m. Then
it's truethatG/H = {H, x - y + H, 2 (x - y) + H, ..., (n - 
1)
(x - y) + H}.I'm trying to come up with an automorphism f_h: 
G
---> G such that, for g in G,h in H, f_h (g) and g are in the
same coset of H in G. Also, you should be ableto adjust the
value of h such that, given an element k in the coset, there
is avalue of h such that f_h (g) = k. Can anyone give such an
automorphism? Ihaven't been able to write one down yet, and 
it
would be a big help for somework I'm doing.Thx.
Let G be
a 'nite abelian group generated by x and y (i.e. G = <x, y>)
and H a>subgroup, such that G = <x - y> + H. Suppose |G| = m
n, |H| = m. Then it's true>that>>G/H = {H, x - y + H, 2 (x -
y) + H, ..., (n - 1) (x - y) + H}.>>I'm trying to come up 
with
an automorphism f_h: G ---> G such that, for g in G,>h in H,
f_h (g) and g are in the same coset of H in G.I cannot make
any sense of this question at all.1. You introduce h as a
subscript in f_h, and then later write for h in H.2. For g
in G is in any case ambiguous. Do you mean for all g in G
or for some g in G ?3. It is unclear in what way f_h is
supposed to depend on h.>Also, you should be able>to adjust
the value of h such that, given an element k in the coset,
there is a>value of h such that f_h (g) = k. Can anyone give
such an automorphism? IBy 3. above, it is unclear how f_h
depends on h, and so it is unclearwhat the effect of adjusting
h will be.But you cannot expect there to be a group
automorphism of G with thisproperty for arbitrary g and k in
the same coset. g and k won't necessarilyhave the same
order.Derek Holt.>haven't been able to write one down yet, 
and
it would be a big help for some>work I'm 
news@aol.com
=I think the original poster meant the
following (I'm omitting the descriptionof G and H this time,
but they're in previous posts):What is an automorphism f: G
----> G such that, given a coset C, f(C) = C, andsuch that
given c' in C, there exists c in C such that f(c) = 
c'.
=I
think this question can be easily answered, provided we assume
H iscyclic, i.e. H = <h>.....>Let G be a 'nite abelian group
generated by x and y (i.e. G = <x, y>) and H a>subgroup,
such that G = <x - y> + H. Suppose |G| = m n, |H| = m. Then
it's true>that>>G/H = {H, x - y + H, 2 (x - y) + H,
..., (n - 1) (x - y) + H}.>>I'm trying to come up with an
automorphism f_h: G ---> G such that, for g in G,>h in H,
f_h (g) and g are in the same coset of H in G.> I cannot
make any sense of this question at all.> 1. You introduce h
as a subscript in f_h, and then later write for h in H.> 2.
For g in G is in any case ambiguous. Do you mean for all g
in G or> for some g in G ?> 3. It is unclear in what way
f_h is supposed to depend on h.>Also, you should be ableto adjust the value of h such that, given an element k in the
coset, there is a>value of h such that f_h (g) = k. Can
anyone give such an automorphism? I> By 3. above, it is
unclear how f_h depends on h, and so it is unclear> what the
effect of adjusting h will be.> But you cannot expect there
to be a group automorphism of G with this> property for
arbitrary g and k in the same coset. g and k won't
necessarily> have the same order.> Derek Holt.>haven't
been able to write one down yet, and it would be a big help
news@aol.com
=Hehe, yes, by original poster I did indeed
mean me.
=This is completely the kind of thing I was
posting a while ago. Butthe closest thing I can easily 'nd is
at:http://mathforum.org/discuss/sci.math/a/t/423825But if I
have already posted this particular result, I apologize.Let
{b(k)} be any sequence de'ned for all positive integer 
indexes
k,where the limit exists.For r = positive real, let a(r,k)
=limit{m->oo} (1/m) sum{j=1 to m} (sum{1<=h<=r/k, GCD(h,j)=1}
b(hk)).(The inner sum is over those positive integers h, h <=
r/k, which arerelatively prime to j.)Then, for n = any
positive integer:sum{k|n} mu(k) a(r,k) = sum{k=1 to §oor(r)}
(sum{1<=h<=r/k, GCD(h,n)=1} b(hk)) mu(k)/k,where the top sum
is over the positive divisors, k, of n; and wheremu() is the
Mobius function.The result rewritten to perhaps be easier to
read:Let c(k,n) = sum{1<=h<=r/k, GCD(h,n)=1} b(hk).For r =
positive real, let a(r,k) =limit{m->oo} (1/m) sum{j=1 to m}
c(k,j).Then, for n = any positive integer:sum{k|n} mu(k)
a(r,k) = sum{k=1 to §oor(r)} c(k,n) mu(k)/k,(I am now starting
to have slight doubts about this. It should be easyto prove, if
true.)Leroy Quet
=As we all here know, mathematics can have
some aspects which revealthemselves in visually beautiful ways
(such as with many fractals).are already pretty famous.But I am
suggesting here that math-promoters draw from manyless-famous
visual math sources to inspire humanity as to math'sbeauty. 
(I
know that many math-promoters already do this.)This is not my
idea as all, to promote math like this. But I might aswell
post this.(And related: With budget-cuts, arts-education in
the US is beingcarried out in many non-art classes {as
art-classes themselves arecut}, including in math-class:Leroy
Quet
=Leroy Quet> As we all here know, mathematics can
have some aspects which reveal> themselves in visually
beautiful ways (such as with many fractals).>> aesthetically
pleasing to even many non-math people.>> Now, the Mandelbrot
set, the golden-spiral, and some function-plots> are already
pretty famous.>> But I am suggesting here that math-promoters
draw from many> less-famous visual math sources to inspire
humanity as to math's> beauty. (I know that many
math-promoters already do this.)I wonder to what extent the
use of graphics has been inhibited bymathematicians
themselves. We are all taught that a proof must not rely on
adiagram; more precisely, it must not make illicit use of
intuition. Butrigorous proofs are not the whole meaning of
mathematics. For the purposesof teaching, rigour can be
overdone, especially toward novices. No onestarts to learn
mathematics by reading diagramless Bourbaki or the
like,clearly. Every one of us learned quite a bit, and took an
interest, beforewe ever saw a real proof. And some diagrams
certainly are enticing, likePascal's hexagrammum mysticum,
or the diagram in Morley's theorem.Personally I am a great
believer in the use of diagrams, although not somuch in the
use of computers for what is intended to be
enlighteninganimation. If I were dictator...Larry
=just
testing> As we all here know, mathematics can have some
aspects which reveal> themselves in visually beautiful ways
(such as with many fractals).>> aesthetically pleasing to even
many non-math people.>> Now, the Mandelbrot set, the
golden-spiral, and some function-plots> are already pretty
famous.>> But I am suggesting here that math-promoters draw
from many> less-famous visual math sources to inspire humanity
as to math's> beauty. (I know that many math-promoters 
already
do this.)>> This is not my idea as all, to promote math like
this. But I might as> well post this.>> (And related: With
budget-cuts, arts-education in the US is being> carried out in
many non-art classes {as art-classes themselves are> cut},
including in math-class:>> Leroy Quet electron-dot-cloud are
galaxies not mutation; Stonethrowing behaviour led to
bipedalism
=It is interesting to note what little
understanding biologists have ofEvolutionary Processes. I say
this in connection with Stonethrowingtheory having led the
wayfor ancient apes to eventually become bipedal and
eventually become thehumanspecies. That a ** behavioural trait
** was the accelerating factor ofchange.Some 8-12 million years
ago in southern Europe there lived an ancientapespecies who had
members that began throwing stones and rocks for morefoodand
more mates. They were quadrapeds but with this
stonethrowingbehaviour would cause to accelerate the changes
in the body anatomy toincrease the abilityto stonethrow. On
such gradual change would be to become biped fromquadraped.I
'nd it utterly amusing to how dumb the biology community can
stoop tothink that the 'rst land walking creatures were due 
to
mutationalchanges of sea dwelling 'sh whereas the real truth 
of
the matter isthat behaviour propelled some sea dwellers to
continually venture ontothe land and this behaviour
accelerated the changes in the body anatomyof sea dwellers to
become land dwellers. Most of us realize thatamphibians or
reptiles come from ancient 'shwho had their 'ns 
converted to
legs. But how silly the biologycommunitythinks that mutations
and not behavioural pressure turned sea dwellersintoland
dwellers. The Behaviour was the prime mover of turning sea
dwellersinto land dwellers. The Behaviour enforced and
reinforced the selectionofmutations of the body to increased
behaviour of venturing on the land.Same scenario for §ying.
How silly and stupid is the biologycommunityto think that
Behaviour was not a factor in turning many creatures
into§yingcreatures. The behaviour of §ying came 'rst 
and not a
complex set ofmutationsthat may or may not be good for the act
of §ying. When Earth had no§yingcreatures, then, what 
created
the 'rst §ying creatures was aBehavioural trait insome 
animals
to venture off their tree limb for whatever reason
(a)accident(b) pursued by predator (c) time saving. A creature
began with abehaviourof pretending to §y, much like our gliding
squirrels. And glidingsquirrels ofmodern day are perhaps the
equivalent of the 'rst animals to graduallyhavebecome the 
'rst
true §yers, analogously, modern chimps are very muchsimilarof a
creature to the 'rst prehumans that began stonethrowing but
werestillquadrapeds. So for §ying, the behaviour came 'rst 
and
many creaturesventuredon §ying and some were indeed successful
and they had more offspringduein large part to their §ying
events. And as time rolled on mutations ofbody change that
made §ying even better -- the rise of birds -- wouldoccurr.So,
there is a pattern in biology which the believers of
DarwinEvolution seem to have a blind spot about. They fail to
realize that themajor number one primemover of change in
biology is not mutations and genetic recombinationwithnatural
selection. But rather instead, the major prime factor in
changein biology is a Behavioural Change by a species. For 
'sh
that ventureonto land is a Behavioural entity which would
select for mutations thatenhance that Behavioural trait.Now I
picked the number 66% not out of the air but picked it from
themathprobability theory of 1/e. That the chances of getting
the maximum bestitemout of 100 items is a 66% chance.
Interesting: special note: I thinkhere I havea key idea to
expand the VonNeumann Game theory in that the number 1/eis
fundamental to that theory.New Math conjecture: in Gametheory
of optimal strategies, the number 1/eand the number 1-(1/e)
are a measure of the diverse range of that game.Exampleto make
clear. Take chess. It is an OS game with an optimal
strategy.Now,modify chess to change one of its rules ever so
slightly, such as notcastling.This modi'ed chess is also an 
OS
game. And the number 1/e connectsthosetwo games. (perhaps more
on this later)So, what I am conjectureing is that of the
forces of biological changeinDarwin Evolution theory that 66%
change is due primarily from Behaviourof individuals in a
species and that 33% change is due to
mutation,geneticrecombination change in the body anatomy
itself to enhance thatbehaviouraltrait.Some may say that
Behaviour trait falls under Darwin Evolution inselectingbrains
of individuals who do the behaviour. That behaviour is
genetics.I wouldsay the answer is no. I would say that a
behaviour trait hasindependent existence apart from the
members of a species and theirentire genetics. Thebehaviour of
venturing out of water and onto land, and the behaviour
ofattempting to §y when not a §yer and the behaviour of
stonethrowingwhennot well equiped to do that. I would say that
behaviour is like anAttractorof Physics and that once such a
Attractor arises then the mutation andgenetic recombination of
physical body anatomy accelerates and enhancesthat behavioural
trait.Biologists call behaviour in animals mostly as
instinct. I do not callit instinctbut call it
Superdeterminism where the brain and mind of every animal
islikea radio antennae receiver. But here I am talking of the
failings ofDarwinEvolution Theory.whole entire Universe is
just one big atom where dotsof the electron-dot-cloud are
galaxies
=I have a B.S. in Computer science and started my
P.hd. program inMathematics last January. Before starting
gradaute school, I worked as asoftware developer for 4 years.
Programming to me is extremely interestingand I really loved
my job, but I really wanted to go back to school to get
aP.hd. and I chose Mathematics because I believe it very
beautiful and I knowI have the brains for it. However, I
suffer from ADD and because of mycondition, I am always
looking for something exciting, otherwise myattention span is
usually short. For example I always found programming tobe
very exciting and as a result I read tons of books on many
computerlanuages! Most applicable sciences to me are exciting,
since when readingbooks in them my mind is usually focusing on
using what I'm reading for someapplicable result. Ofcourse
the nature of pure mathematics is different. I'mtrying to
'gure out how to read/study mathematics so that it becomes
veryexciting to me! One thing that I 'nd exciting in
Mathematics is thinkingabout research problems, I could think
about a research problem for weeks oreven months, but reading
mathematics books isn't as exciting as I would likeit to be. 
I
have tried thinking about what I read more and that has
helpedme a bit, but I still seem to lack that excitement I
need (My ADD isstrong). I would like to hear from any
mathematicians regarding this issue:How in general should one
study Mathematics? What makes reading Mathematicsbooks
exciting? What goes on in your mind as you are reading? When
was itthat you truly found Mathematics very exciting?Maybe
there is a more enjoyable way to study Mathematics, than just
reading.Any sort of feedback would be helpful.Bill
 I
have a B.S. in Computer science and started my P.hd. program
in> Mathematics last January. Before starting gradaute school,
I worked as a> software developer for 4 years. Programming to
me is extremely interesting> and I really loved my job, but I
really wanted to go back to school to get a> P.hd. and I chose
Mathematics because I believe it very beautiful and I know> I
have the brains for it. However, I suffer from ADD and because
of my> condition, I am always looking for something exciting,
otherwise my> attention span is usually short. For example I
always found programming to> be very exciting and as a result
I read tons of books on many computer> lanuages! Most
applicable sciences to me are exciting, since when reading>
books in them my mind is usually focusing on using what I'm
reading for some> applicable result. Ofcourse the nature of
pure mathematics is different. I'm> trying to 
'gure out how
to read/study mathematics so that it becomes very> exciting
to me! One thing that I 'nd exciting in Mathematics is
thinking> about research problems, I could think about a
research problem for weeks or> even months, but reading
mathematics books isn't as exciting as I would like> it to
be. I have tried thinking about what I read more and that has
helped> me a bit, but I still seem to lack that excitement I
need (My ADD is> strong). I would like to hear from any
mathematicians regarding this issue:> How in general should
one study Mathematics? What makes reading Mathematics> books
exciting? What goes on in your mind as you are reading? When
was it> that you truly found Mathematics very exciting?>
Maybe there is a more enjoyable way to study Mathematics, than
just reading.> Any sort of feedback would be helpful.In grad
school you spend the 'rst couple of years power-slogging
through the basic curriculum until you either (a) rise above
through genius, (b) get by through persistence and study
habits, or (c) end up forgetting why you thought you liked
math in the 'rst place. If you're ADD, then the 
only options
are (1) and (3). If you are on meds for ADD they may help, but
ultimately if you are not a genius, then you need to develop
killer discipline.The only way to transcend this system is to
be really talented. If you're the next Galois, or Newton, or
Beautiful Mind guy John Nash, then you will show up at grad
school and immediately start working on research projects. If
you're normal -- an undergrad hotshot hitting his limit in
grad school -- then either you develop REALLY good disciplined
study habits, or you go back to programming.The boredom of the
endless de'nition-theorem-proof-homework system is grad
school's way of weeding out the geniuses from the rest of 
us.
Like Barbie once said, Math class is tough!
 I have a
B.S. in Computer science and started my P.hd. program in>
Mathematics last January. Before starting gradate school, I
worked as a> software developer for 4 years. Programming to me
is extremely interesting> and I really loved my job, but I
really wanted to go back to school to get a> P.hd. and I
chose Mathematics because I believe it very beautiful and I
know> I have the brains for it. However, I suffer from ADD
and because of my> condition, I am always looking for
something exciting, otherwise my> attention span is usually
short.You're not ADD as you show below. You've 
great
concentration abilities.What you've got is early training of
entertain me, excite me to motivateme. This comes of the media
and thence to the schools who have gradeschoolers who are
raised upon I've got to be entertained.> One thing that I 
'nd
exciting in Mathematics is thinking> about research problems,
I could think about a research problem for weeks or> even
months, but reading mathematics books isn't as exciting as I
would like> it to be.Research problems are exciting, just
think of the glory and fame you'dhave were you to solve 
them.
To bad math doesn't interest you. Neitherwould a climb to 
the
top of Mt. Everest interest you. Too much onelaborous foot
ahead of another laborous foot. If the rush of excitementis
all that motivates you, then reexamine the media training that
hasleft you so de'cient of motivation and so manipulable by
excitableprattle. Most of life is more than just a §ash in the
pants.> I have tried thinking about what I read more and that
has helped> me a bit, but I still seem to lack that excitement
I need (My ADD is> strong).No, you're craving for excitement 
is
addictive.> I would like to hear from any mathematicians
regarding this issue:> How in general should one study
Mathematics? What makes reading Mathematics> books exciting?
What goes on in your mind as you are reading? When was it>
that you truly found Mathematics very exciting?>> Maybe there
is a more enjoyable way to study Mathematics, than just
reading.Yes, it's call philosophy, philos-sophia, love of
wisdom.Perhaps this will appeal to you: Sophia is a beautiful
Greek goddess,the goddess of wisdom. Philosophy is loving
Sophia, have an orgy.> Any sort of feedback would be
helpful.Once upon a time there were students who enjoyed
learning for the sake o§earning. Perhaps you should study
history: intrigue, murder, mystery,war, famine, pestilence,
conquest, pillage, rape, discovery, invention,exploration,
protests, revolts, rebellions and much much more than
they'dever dared to tell you in school such as the history 
of
the labormovement.
 I have a B.S. in Computer science and
started my P.hd. program in> Mathematics last January. Before
starting gradaute school, I worked as a> software developer
for 4 years. Programming to me is extremely interesting> and
I really loved my job, but I really wanted to go back to
school to get a> P.hd. and I chose Mathematics because I
believe it very beautiful and I know> I have the brains for
it. However, I suffer from ADD and because of my> condition, I
am always looking for something exciting, otherwise my>
attention span is usually short. For example I always found
programming to> be very exciting and as a result I read tons
of books on many computer> lanuages! Most applicable sciences
to me are exciting, since when reading> books in them my mind
is usually focusing on using what I'm reading for some>
applicable result. Ofcourse the nature of pure mathematics is
different. I'm> trying to 'gure out how to 
read/study
mathematics so that it becomes very> exciting to me! One
thing that I 'nd exciting in Mathematics is thinking> about
research problems, I could think about a research problem for
weeks or> even months, but reading mathematics books isn't 
as
exciting as I would like> it to be. I have tried thinking
about what I read more and that has helped> me a bit, but I
still seem to lack that excitement I need (My ADD is> strong).
I would like to hear from any mathematicians regarding this
issue:> How in general should one study Mathematics? What
makes reading Mathematics> books exciting? What goes on in
your mind as you are reading? When was it> that you truly
found Mathematics very exciting?> Maybe there is a more
enjoyable way to study Mathematics, than just reading.> Any
sort of feedback would be helpful.Try to use your programming
background. There are many areas in bothpure and applied
mathematics where you can use computers by applyingthe
advanced math that you learn. Would you like to program
thingslike Mathematica can do?Some math departments might
consider work in arti'cial integelligenceto be a computer
topic (although this might be more appropriate fora Ph.D.
degree in computer science). I am especially interestedin
temporal difference learning (but have not work with it
forsome time).You might also try to work with computer
veri'cation of proofs orcomputer generation of proofs.An idea
that I would wish to implement is to have a computer
programuse arti'cial intelligence techniques to verify proofs
of students,say in group theory, where the students could
leave certain obvioussteps out in a proof of a theorem, but
the number of steps omittedbetween two successive statements
must be a small number like 1 or2 or 3 (preference set by
prover).Look in the math section of the university library or
in themath library itself for books that deal with computer
applicationto such areas as algebra, analysis, etc. See if you
'nd some area
=