mm-225
===
Subject: Re: Homeomorpsms and compactness Adjunct Assistant
Professor at the University of Montana.>I have a small
question wch I have been unable to solve on my own. I>know
that if f: M->N is a homeomorpsm, thenK compact in M <=> f(K)
compact in Nand I'd like to know if the opposite way holds,
that is, if f is>bijective andK compact in M <=> f(K) compact
in Nis f then a homeomorpsm? I have proven ts to be true for
locally>compact spaces, but I would like to know whether the
result holds>generally.Any help?I feel dumb. Here's a
counterexample if your definition ofcompactness includes the
separation axiom.Let M and N both have as underlying sets the
2 elements set {x,y}. LetM be given the indiscrete topology,
and let N be given the Sierpinskytopology: the open sets are
the empty set, the total set, and {x}.Every subset satisfies
the finite subcover property. But the onlysubsets that satisfy
the separation axioms are the empty set and theone element
sets, in both M and N. Therefore, any bijection between Mand N
will satisfy that K compact in M if and only if f(K) is
===
H#SM:U1U-/6#NN83s6?Die557~]Dfifz~-|V:wSKGL6T-|!qk{U4/M7+k5Py!-{

q=2Q/%0@ E29yc_kQC&^> Actually, 10000!! is easy, using
Mathematica. The !! notation> represents the double factorial
function, such that n!! => n*(n-2)*(n-4)*....I wasn't aware 
of
that usage.> I tnk you meant (10000!)!,I did indeed.> wch is a
bit larger.Several bits.-- J.97n Fairbairn
===
initials Adjunct Assistant Professor at the University of
Montana.> The concept of free speech encompasses the right of
people to say stupid >> and hateful tngs. The Supreme Court
said you can burn the American >> ßag or claim that Falwell
had sex with s mother in an outhouse, >> and that those tngs
are protected speech. I for one defend your right >> to say
whatever the hell you want.You can burn the ßag, but you have
no right to say that Falwell>had sex with s mother in an
outhouse (unless it's true). There are>libel 
laws.->Context<-.
Fishfry's presentation is indeed way too 
superficial. The case
in question, wch ->did<- involve a claim that Falwell had had
sex with s mother in an outhouse, and a suit forlibel against
Hustler magazine, Hustler Magazine v. Falwell, 485U.S. 46
(1988)http://caselaw.lp.findlaw.com/scripts/getcase.pl?court=us

&vol=485&invol=46decided 8-0, with Cef Justice Rehnquist
writing the opinion for theCourt, involved Hustler 
magazine's
parody of a Campari liquor adabout first times. The opinion
states in part: We conclude that public figures and public
officials may not recover for the tort of intentional 
inßiction
of emotional distress by reason of publications such as the one
here at issue without showing in addition that the publication
contains a false statement of fact wch was made with actual
malice, i. e., with knowledge that the statement was false or
with reckless disregard as to whether or not it was true.
[...] Here it is clear that respondent Falwell is a public
figure for purposes of First Amendment law. 5 The jury found
against respondent on s libel claim when it decided that the
Hustler ad parody could not reasonably be understood as
describing actual facts about [respondent] or actual events in
wch [he] participated. App. to Pet. for Cert. C1. The Court of
Appeals interpreted the jury's finding to be 
that the ad parody
was not reasonably believable, 797 F.2d, at 1278, and in
accordance with our custom we accept ts finding. Respondent is
thus relegated to s claim for damages awarded by the jury for
the intentional inßiction of emotional distress by outrageous
conduct. But for reasons heretofore stated ts claim cannot,
consistently with the First Amendment, form a basis for the
award of damages when the conduct in question is the
publication of a caricature such as the ad parody involved
===
function >I have a second order transfer function that looks
like ts: >P(s) = k/((tau1*s+1)*(tau2*s+1)) * exp(-td*s) >In
order to reverse laplace transform ts I will have to multiply
P(s) with >1/s >The resulting Laplace transform will be: >
>p(t)=k*(1+tau1*exp(-(t-td)/tau1)/(tau2-tau1)-tau2*exp(-(t-td)
/tau2)/(tau2-tau1) >) >When tau1, tau2 and td are known it
is possible to construct the curve p(t). >Ts is not difficult.
>However, imagine that you have ts curve, but do not know
the values of tau1, >tau2 and td that constructed the curve. >
> In my field ts is the problem that I have. I have a second
order system that >generates a curve p(t) as output when a
step function is input into the system. >The problem is that I
do not know the time constants (tau1 and tau2) nor the >dead
time (td). I want to analyse the curve and based on ts find 
the
values >of tau1, tau2 and td. But, I have no idea how to do ts.
The general solution >for p(t), presented above, doesn't 
allow
me to solve for t. >For a first order system it is all much
easier as it will be possible to >linearize the function for t
and then it is just a question of solving a system >of linear
equations. Ts does not seem to be possible for a second order
>system. >How do I find tau1, tau2 and td given a curve with
values for t and p(t)? > in advance, >-- >Someone >given
a dataset (t(i),p(t(i)) you want to get td, tau1, tau2 such
that the above equations holds on the dataset as good as
possible:a typical job for nonlinear least squares.look
athttp://plato.la.asu.edu/topics/problems/nlolsq.html for
codes for doing ts (for example ELSUNC would be a good
candidate for ts).you write a function program for computing
the residual left hand side minus right hand side for a
givenparameter set x in the code, for example x(1)=td,
x(2)=-tau1, x(3)=tau2 and the optimizer will give you the good
parameters.but you should provide reasonable initial guesses
for them, otherwsie the outcomemight be a fault.
===
there?http://www.giganews.com/info/dmca.html>How many
continuous functions are there from R to R?It seems it must be
same as the number of reals, since a continuous>function is
determined by its value at the rationals, and c ^ a_0 =>(2 ^
a_0) ^ a_0 = 2 ^ (a_0 * a_0) = 2 ^ a_0 = c.Is that
right?Yes.>-- Richard-- >Spam filter: to mail me from a
.com/.net site, put my surname in the headers.FreeBSD
===
> My main point is to show that the writer of that book is not
a fraud. > s methods can be very useful, as opposed to current
methods taught> in schools. If current methods are indeed
useless, how did people build the Golden > Gate Bridge or the
Queen Mary?I did not expect ts sort of response from you,
Ranjit. Ts sort I> expect from the type of Mr Singh & Co.I
never mentioned in any post that the current methods are
useless; I> said that ts Vedic method is better. That did
*not* imply that the> current methods are *useless*. You are
twisting my words, the same way> Mr Singh does.If Mr. Singh
were to use your wording and say, methods taught inschools can
be very useful, as opposed to Vedic methods, would
youunderstand that he never mentioned that Vedic methods are
useless butthat he merely said methods taught in schools were
better? The arebetter implication would have been clear only
if you had saidsometng along the lines of s methods are more
useful than currentmethods taught in schools.> As for your
question, people built the Golden Gate Bridge and the> Queen
Mary the same way the Egyptian built pyramids. It is
generally> understood that the ancient Egyptians did not know
trigonometry, or> any modern mathematics.FYI, people designed
the bridge and sp in question WITH theapplication of
trigonometry and mathematics, not without it. So, theydid not
===
Re: JSH: My use of my initials Adjunct Assistant Professor at
the University of Montana.> I see, thx. I wasn't clear on 
who
was firing at who :) But JSH does>> sometimes threaten people
with litigation, and he had somebody in court one>> time, for
calling m a crank. The case was quickly tossed.I don't tnk
there was any such incident.He has threatened on occasion; but
I tnk Mr. Hammick may be gettingconfused with the case where
Underwood Dudley ->was<- sued for callingsomeone a crank.
Dilworth v.
Dudley:http://www.law.emory.edu/7circuit/jan96/95-2282.htmlThe
===
functions are there?> How many continuous functions are there
from R to R?It seems it must be same as the number of reals,
since a continuous> function is determined by its value at the
rationals, and c ^ a_0 => (2 ^ a_0) ^ a_0 = 2 ^ (a_0 * a_0) = 2
^ a_0 = c.Is that right?-- Richard> It's essentially right. 
The
only error in it is that at the moment you've only shown 
that
the cardinality is at most c - not every possible function
from Q to R extends to a continuous function. However it's
easy to check that the cardinality is at least c, so you have
the cardinality is exactly c. (I would prefer to tnk of it in
terms of injections of functions and apply
schroeder-bernstein, but it doesn't really matter - wchever
===
want to do your new students a favor you can show them ts
simple new proof of the fundamental theorem of the calculus
and then they will know what they are learning. Its at:
===
letters of recommendation> I provided each professor with a
labelled envelope, tYeah, that's about the extent of it. You
want to make their job as easyas possible, but there is tto
much variation. But I write my letters inWord (heresy!)
anyway, so with cut and paste that part is not too
muchtrouble. You might offer to send them the addresses over
email so thatthey can cut/paste that if they write the letters
electronically.V.-- email: lastname at cs utk eduhomepage: cs
===
of recommendation Adjunct Assistant Professor at the
University of Montana.>I am a senior and I will be needing to
fill out applications for graduate>school. My problem is that 
I
am on the quarter system and the quarter is about>to end. We
will not resume until the beginning of January. I'm pretty 
shy
and>have not asked any of my professors for letters. I'm 
just
not sure how to ask>them and now that I basically have no time
left, I was just wondering how I>should go up to them and ask.
I have had the same professors for a lot of>classes in some
cases but I'm not sure whether or not lower level classes
would>give a professor enough to say anytng about me. Is there
any standard way to>go about ts? Most schools seem to want
three and for me that is a lot.Asking for letters is about the
most uncomfortable tng I've had todo, both when applying for
grad school, and now for jobs. The goodnews is that most
people are very nice about it.First: do it, and do it SOON.
The letters have to be in, and youshould give the professors
at least a couple of weeks to do it; as thesemester winds
down, a professor's time supply is short. Ask for theletters
ts week or next, and don't delay.Pick upper division classes
in wch you did well. They don't all haveto be math classes,
but at least one (preferably two) should be.Go to the
professor, remind m who you are and how he knows you, andsay
simply that you are applying for grad school, and since you
didwell in s/her class, you would very much appreciate it if
he/shecould could write you a letter of recommendation for
yourapplication. Take the forms with you, and be ready with
informationabout deadlines and so on. Probably the best is to
catch them duringtheir regularly scheduled office hours. If 
the
professor only knows you from lower division courses, it
isprobably not a good bet for a letter, but if you had m for
bothupper and lower divisions, then feel free to go for
it.And, keep ts in mind: most professors understand that ts is
partof their job. If you did well in the class, they will be
===
two complex numbersWhy (a,b) x (c,d) = (ac - bd, ad + bc) ?
(a,b) x (c,d) = (2ac - bd, ad + bc) also will serve to solve
===
instantaneous help for you:> It only shows how depesrate you
are to attack me in such a cheap way> that you failed to
realize how you are revealing how low you are.>
>>http://www.virtual-vibrator.com/index-ns.shtml#You're 
trying
to fit your desparation shoes onto my feet?LOLClearly sometng
ails you. Feel free to judge with your experience.>The other
possibilities> are magintudes worse. Sounds like you are
speaking about yourself.> Feel free to tell us. Would you
rather not make assumptions .. like you have been doing? Why?I
do apologize for name calling; I forgot that you could be
mucholder.But..everytng else I said .. stands. You started
picking on meafter you assumed that I was a Muslim. Of course
you didn't say itexplicitly but your prejudice was obvious;
the timing of the offensivepost just after not siding with
Hanson's (Hanson cannot tnk with sbrain) plea to be on s
side.I remember a professor (he must be almost 60 at the time)
who reallylike ts student who doesn't look her typical 
ethnic
group (I meanliking as a person) till he found out the ethnic
background.Some people just enjoy being prejudiced. Poor
===
and so forth only exist insofar as they can demonstrate> their
authority. They say they're here to preserve order, but in
fact> they'd go absolutely mad if all the criminals of the
world went on> strike for only a month. They'd be on their
knees waiting for a> crime. That's the only existence they
have.William S. Burroughs (American writer)> Guardian,
1966huffyI'm beginning to tnk you're a 
one-trick pony.--Ron
===
ts is a curve fitting equation. So what I needto know is given
a known value for x I need to predict for y.Jo>Can someone
help me solve the following equation for x. Could you>please
put all the steps down so I can follow it (idiot's guide!)>y
= A1*exp(-x/t1) + A2*exp(-x/t2) + y0Ts looks like a function,
not an equation, unless y is supposed to> be a constant. I
assume you want to solve y = 0?Huh? An equation is sometng
with an equals sign in the middle, > so I'd say ts 
qualifies as
an equation. I assume when the poster > asks for help solving
for x, the poster wants the equation solved > for x. Of
course, the steps you gave are useful for my interpretation as
> well (except that you won't want to set y to zero). But 
when
you > write, A1 z^{t2} + A2 z^{t1} + y0 = 0.If t1, t2 are
integers, then ts is a polynomial in z. Solve it by> any of
the usual methods, the usual methods are going to be tricky
unless t1 & t2 have been > chosen (by whoever set the problem)
===
two complex numbers Adjunct Assistant Professor at the
University of Montana. >Why (a,b) x (c,d) = (ac - bd, ad + bc)
?Because you are tnking of the pair (a,b) as representing a+bi,
wherei^2 = 1.> (a,b) x (c,d) = (2ac - bd, ad + bc) also will
serve to solve x^2 + 1 = 0.Mostly because it is not
associative:[(a,b)x(c,d)] x (e,f) = = (2ac - bd, ad+bc) x
(e,f) = (2(2ac-bd)e - (ad+bc)f, (2ac-bd)f + (ad+bc)e ) = (4ace
- 2bde - adf - bcf, 2acf - bdf + ade + bce)(a,b) x [(c,d) x
(e,f)] = (a,b) x (2ce - df, cf+de) = ( 2a(2ce-df) - b(cf+de),
a(cf+de) + b(2ce-df) ) = (4ace - 2adf - bcf - bde, acf + ade +
2bce - bdf)On the other hand, (a,b) x (c,d) = (ac - 2bd,
ad+bc)would work, since then you are tnking of (a,b) as
===
factorization, from basic to advancedIn ts post I'll go from 
a
rather basic factorization to a far more> complicated one,
outlining how to use one polynomial to tell you about> the
roots of another.You demonstrably cannot balance your
checkbook much less performsometng complicated. You are like
Sarfatti or Saddam Husseinaimlessly firing into the sky hoping
to t sometng. Both of themmelted down their gunbarrels and t
notng. All you t is yourselfin the
face.http://www.crank.net/harris.html It's not every braying
jackass that gets a whole page at crank.net
http://b5.sdvc.uwyo.edu/bab5/snds/argcstpd.wavhttp://
w0rli.home.att.net/youare.swfhttp://www.mazepath.com/uncleal/
===
for ts series?It needed several try because at first i used 
the
wrong closed form for thegeometric serie, i used (1-A^N)/(1-A)
instead of the right one(1-A^(N+1))/(1-A) but at last i found
the right expression bydifferentiating and multiplying by A as
you suggested.It should be
(A-N*A^(N+1)+N*A(N+2)-A^(N+1))/((1-A)^2)It works with several
===
<87smkswqrm.fsf@pwumbda.org> <bovsmn$ujusk$1@athena.ex.ac.uk>
sha1:/WVc+qrQYi2l0MeVzwEPlv3zd6U=> In another post by huffy,
you signed (and clearly spoke as>> Correy). Presumably it
wants to avoid killfiling. I have no idea why it> tnks that 
the
opinion of ts late novelist, heroin addict> and wife-murderer
is at all relevant though. These huffy posts> are considerably
more tedious than the rantings of D S Kabatoff.Posting the
exact same quote 47 times is an odd way to avoidkillfiling.--
Jesse HughesBasically there are two angry groups. I am a harsh
force ofone. Against me is a society of mathematicians. So far
it's been adraw. -- JSH gives another display of keen
===
recommendation> Is there any standard way to go about ts? 
I've
had students walk into my office with the letter written for 
me
and asking me to sign it. Let's callthat not standard (and 
not
acceptable.)As others have said, professors do ts all the time,
andoften for students they barely know, except as a list
ofnumbers in a grade book. If you can help make the
connectiona bit, that is a good tng. Some students walk in
andask for recommendations and then walk out. Later I getsome
addresses. OK, I can deal with it, and they get thebest letter
I can put together.But other students hand me a little resume
of their careers,wch helps me immensely. It reminds me that
they were inthat production of Ot I enjoyed so much and that
theyare always lying next to me when I'm giving blood at the
blood drive and it helps me find their records amongst allmy
spreadsheets when they've listed the classes they took.Oh,
yeah, she DID win that freshman math contest.It helps keep the
letter from looking generic.Plus the fact that the student had
it together enough tohand me the resume says sometng in
itself. Grad school also takes perserverence, and the letter
ought to addressthe student's character if it can. So if I 
can
say sometngpositive that indicates that the student not only
has somemathematical talent, but also exbits the academic
maturityfor sticking tngs out, then I will. The brief inteview
you get when asking for the letter is a quick, but
importantchance to demonstrate those characteristics.So, my
advice is that wle you're asking for the recommendation,you
connect personally with the professor a bit. It's
anopportunity to ask m for advice on the choice of grad
schoolsor field of study or tips for getting through and it
willhelp m write the letter if he knows you a bit
===
but what about the OP's 10000! ;-)>That's 
large itself, but it
doesn't take long to>compute. computing 10000!! 
isn't feasible
whatever algorithm>you use. True [at least for (10000!)!, as
opposed to the Mathematicameaning]. OTOH, its last 10000!/4 or
so digits are easy to calculate,at least to base 10, wch gets
you part of the way; and so are thefirst few digits. 
It's just
those in the middle that are hard ....-- Andy Walker, School
of MathSci., Univ. of Nott'm,
===
theory> Let q1,q2 algebraic integers such that
q1^k=n(integer),q2^k=m(integer) let >>
K1=Q(q1),K2=Q(q2),Q:rational field.>> Also suppose that m,n
free from k powers and K1=K2. >> Is there any relation of
m,n?>>Do you have any examples where m and n are not
equal?k=3, m=2, n=4. Then Q(q2) is clearly contained in Q(q1),
and > 2^{1/3} = (1/2)(4^{1/3})^2 so you get the reverse
inclusion.If k=2, then m must equal n. If k=3, I believe that
you must have that all primes that divide m> divide n as well,
and vice-versa, and that will be both necessary and>
sufficient.>>Is Q(cube root 6) the same field as 
Q(cube root
12)?Accoreding to Magma, x^3 - 12 is irreducible over Q(cube
root of 6),and x^3 - 6 is irreducible over Q(cude root of 12),
so the answerwould appear to be no.Derek Holt.>Don't know. I
thought about it more, and must take it back (good tng>I said
I believe...)The best I could come up with was a sufficient
condition, wch I>->tnk<- may be necessary as well:Let P =
{positive rational primes}.Define a (set-theoretic) map from
the nonzero integers to Z^P as>follows: given a nonzero
integer n, let f(n) be the vector in Z^P>wch has ord_p(n) in
the p-th coordinate (i.e., the coordinate>corresponding to the
prime p).Let k>0. There is a natural map from Z^P to (Z/kZ)^P,
and let g(n) be>the image of f(n) under ts map. Then Q(q1) is
equal to Q(q2) if the following occurs: (1) If k is even, then
either n and m are both positive or else they> are both
negative. (2) The cyclic subgroup of (Z/kZ)^P generated by
g(n) is equal to> the cyclic subgroup of (Z/kZ)^P generated by
what I accept as reality.> --- Calvin (Calvin and
===
H#SM:U1U-/6#NN83s6?Die557~]Dfifz~-|V:wSKGL6T-|!qk{U4/M7+k5Py!-{

q=2Q/%0@ E29yc_kQC&^>> Yes, but what about the OP's 10000!
;-)>That's large itself, but it doesn't take 
long to>compute.
computing 10000!! isn't feasible whatever algorithm>you use.
True [at least for (10000!)!, as opposed to the Mathematica>
meaning]. OTOH, its last 10000!/4 or so digits are easy to
calculate,> at least to base 10, wch gets you part of the way;
and so are the> first few digits. It's just 
those in the middle
that are hard ....I wasn't tnking so much of the 
difficulty, as
where to storethem when you're done.-- J.97n Fairbairn
===
it yesterday.>In German.>Uhm, did the synchro botch everytng up
or is it>a lot of senseless mathbabble already in the original?
:-)>-- >Hauke Reddmann <:-EX8 >Private
email:fc3a501@math.uni-hamburg.de>For our chemistry
anytng elseSaw it too.> The math is not completely senseless:
the idea to interpret the> numbers as coordinates is quite
obvious, isn't it?> Later on they used the same numbers to
describe the path a subcube follows over time. It's not so
clear how to do that, although it's> possible.> Then 
suddenly
the property of being a prime power played a role in>
identifying those rooms, that are traps. At that point I guess
it> was just nonsense. It seems difficult to me to encode two
types of> information into the number triplets shown in the
movie.HAfter reading ts thread, I bought a copy. When I
brought it to the counter:salesgirl: Oh, Cube. It's actually 
a
tesseract, you know. me: What?salesgirl: A tesseract. It's 
like
a square, only three dimensional. me: A cube _is_ threee
dimensional.salesgirl: Oh, I meant four dimensional, where
each face of the cube is connected to another cube. me: I know
what a tesseract is, I'm wondering if you know what it is.At 
ts
point the transaction was complete so I walked away. Didn't
want tobe beaten with sticks by those in the queue bend me.If
JSH wants to make money, he should become a script writer. s
bullstsounds more plausible than that produced by Hollywood. I
can see it now:Aliens invade and because of the core error, the
mathematical establishment is rendered helpless and only JSH,
with s Advanced Polynomial Factorization,can save the
===
<87znf0wyzo.fsf@pwumbda.org>
<bp0b1v$1skq$1@agate.berkeley.edu> Discussion,
concept of free speech encompasses the right of people to say
stupid > and hateful tngs. The Supreme Court said you can burn
the American > ßag or claim that Falwell had sex with s mother
in an outhouse, > and that those tngs are protected speech. I
for one defend your right > to say whatever the hell you
want.>>You can burn the ßag, but you have no right to say that
Falwell>>had sex with s mother in an outhouse (unless it's
true). There are>>libel laws. ->Context<-. Fishfry's
presentation is indeed way too superficial. I 
didn't realize he
was referring to an actual case. for thedetails. When you add
the fact that the claim was not presented as afactual claim
but only as part of a parody, that rather changes tngs.-- Run
mathematicians, RUN!!! I'm coming for you. It may take a
fewmonths, but I'll get [computer verification 
of my proof] and
then yourlives will be ended as you previously knew it. -- JSH
===
I.e., it is possible to prove directly, using only the
definition of> derivative, that (e^x)' = e^x? 
No, you also need
the definition of e^x.No comment ...> Calculus texts often do 
it
ts way: First define ln(x) by integrating> 1/x. Then the
derivative of ln(x) is easy. Then define exp(x) as the> 
inverse
function. So you get its derivative.But the order usual in
secondary school (at least in Spain) is :1) Definition an
properties of the functions. Logarithms as inverses
ofexponentiations, a^x using rational aproximations of x. The
number e isintroduced as Lim(1 + 1/n)^n.2) Derivatives3)
IntegralsIn that environment, the usual way to get the
derivative of f(x) = e^x isthrought the logarithmic derivative
or the derivative of the inversefunction. The derivative of the
logarithmic function is easily obtained fromits properties.I
was looking for a alternative to get the derivative of the
exponentialfunction directly from the properties of that
function or the loarithmicfunctuion, but without make use of
the chain's rule. But I have it yet, as Isaid in a previous
message.-- Saludos,Ignacio Larrosa Ca.96estroA Coru.96a
===
NY Times Science 11/11/03 >> I tnk we are so confused that we
should keep an open mind to>> tinkering with gravity, said Dr.
Michael Turner, a cosmologist at>> the University of Ccago.>>
previously>> Q to Ed Witten: How can the cosmological constant
be so close to>> zero but not zero? DL> Close to zero? WTF does
close mean? Is there a maximum value> DL> we can compare it to
to see how close it is? The current estimates are that the
energy density contributed by the> cosmological constant (or
dark energy) is 0.7, in units of the closure> physics that
make predictions about the cosmological constant. The> typical
value predicted is 10^50 or so.Excellent! . (I'm going to
pretend to understand that.) ;-)--Denis
===
determing the inside of a set of pointsI've got a list of
points (in x,y form) that are the corners of> a polygon, in
order. It's not necessarily a convex polygon. Is> there a
standard algorithm for determining whether a given point> is
inside or outside the polygon? All ts takes place in> the
plane... For practical purposes, an algorithm is ok if it> can
handle up to 7 points decently.Covered adequately in Sedgewicks
Algorithms (in C).-- Chuck F (cbfalconer@yahoo.com)
(cbfalconer@worldnet.att.net) Available for
consulting/temporary embedded and systems.
<http://cbfalconer.home.att.net> USE worldnet
===
thoughts on creating a theory of sets prior to a theory of> :
> propositions and quantifiers:> : > : > Let's 
start with the
empty set, 0, and logical identity, =, then we can> : > define
T, for true, by> : > : > T =def 0 = 0> : > : > Let's 
define
ordered pair a la Kuratowski, then we can define> : >
conjunction by : > Kuratowksy defines <a,b> as {a,{a,b}}. But
how do you make sense of> : > that latter notation at ts stage
of the presentation? Note that in> : > ZF, you can only make
sense of it because of the Axiom of Pairing; you> : > can only
verify that it satisfies the ordered pair axiom because of> : 
>
the Axiom of Extensionality. Those axioms both seem to require
the> : > apparatus of first order logic to be formulated.I
already said that. Why don't *I* rate a reply? : Well, we
cannot define un-ordered pair in context by> :> : u in {x,y)
iff u = x or u = y> :> : since we don't have or. So 
let's take
{x,y} as primitive and define> : {x} =def {x,x}> : <x,y> =def
{{x},{x,y}}Nobody is impressed. You'd be much better off 
just
taking> <x,y> as primitive. Your mission, now, wch will be
much harder,> is, given that you've taken {x,y} as 
primitive,
how on earth is> anybody supposed to parse {x,y,z} ?I need <x,
y> to define &. _After_ I've got the logic I can 
use settheory
to define {x, y, ..., z} in the usual way.More to the point,
you said before that we don't have 
Ôor',> but you have a much
bigger problem: you don't have *in*, EITHER.in is in the set
theory, just as in any other set theory.> What good does it do
you to take {x,y} as primitive, and to claim> that you've
presumed some set theory, if you STILL have NO wayI presume
_all_ of set theory: it's axioms and their 
first
orderconsequences.> of deciding whether z is or isn't *in*
===
police and so forth only exist insofar as they can
demonstrate>their authority. They say they're here to 
preserve
order, but in fact>they'd go absolutely mad if all the
criminals of the world went on>strike for only a month. 
They'd
be on their knees waiting for a>crime. That's the only
existence they have.>William S. Burroughs (American writer)>
Guardian, 1966>huffyYou got it right that time. But that
other post where you> signed your name sort of gave it away.
Giggle.> For about a second ts morning it looked like you'd>
found someone who tends to agree with you. But you> had to
make m up.How sad, a grown man depending on imaginary
playmates that way.> Your Fuffy, my Pzuffy. Deal with
===
question)The whole problem is just that you're misquoting 
the
axiom. You say: Everynonempty B contains a y with (B intersect
y) = empty. I say: Every nonempty Bcontains a y for wch there
is no z with z in y and z in B. My axiom sayssubstantially the
same tng, but it allows for the possibility that there
existcitrus fruits that are not sets. Alternatively, you could
banish such citrusfruits from your theory. Citrus fruits that
have no elements, but aren't theempty set, are technically
called individuals. Whether you want them to existor not is
basically a matter of taste, wch I suppose would have to be
fairlysour. HTH| folks,|| Okay, ts is going to be some very
stupid question to most of you.|| The axiom of foundation says
that:|| ForAll B ~= emptyset, ThereExists y in B such that (y
intersect B)=| emptyset.|| Now suppose B = {lime, orange,
lemon}. What is such y among ts set| of citruses? Suppose y =
lime. How can lime be a set? (my| understanding is that only
sets can intersect).|| I told you, it is a stupid question.
===
personality assessmentIn another post by huffy, you signed
(and clearly spoke as> Correy).>> Presumably it wants to avoid
killfiling. I have no idea why it>> tnks that the opinion of 
ts
late novelist, heroin addict>> and wife-murderer is at all
relevant though. These huffy posts>> are considerably more
tedious than the rantings of D S Kabatoff.Posting the exact
same quote 47 times is an odd way to avoid> killfiling.Well I
have now killfiled Correy under its new alias. Choosinga new
username is a way of forcing one's postings on peoplewho 
have
chosen to killfile one. Kabatoff does ts too.Perhaps I should
jus killfile all posts from yahoo.comas ts ISP seems to attact
===
about upper bound of magnitude of a complex root of
polynomialCan anyone help me to solve the question? I have no
idea on it...Let f(t) be a polynomial with complex 
coefficients
of degree n and with leadingcoefficient = 1.Let a be a 
root.Show
that | a | <= n b, where b is the maximum of the magnitude of
===
Recommendations on Complex Analysis books?> THE classic text
on complex functions is Ahlfors, _Complex Analysis_ .> BArry
Mazur has recently published a popular book on complex>
numbers _Imagining Numbers_. I would also recommend _Complex
Numbers> and Geometry_ by Hahn. Finally, a great old book:
Theory of> Functions_ by Caratheodory.I second the Ahlfors
book. We used it as our Complex Analysistextbook when I was a
===
My use of my initials>Some of you may have realized that I
have remarkable power to draw>attention on several newsgroups
to the extent that I even have my own>dedicated repliers, like
or Ullrich, who like to>obsessively insult me! Anyone have a
good term for people who just>follow around a popular poster
insulting m? Well I call them critic>trolls.Around here we
===
Re: Need advice on letters of recommendation permission for an
emailed response.X-Zippy-Says: I selected E5... but I didn't
hear ``Sam the Sham and the Pharaohs''!> 
I've had students
walk into my office with the letter > written for me and 
asking
me to sign it. Let's call> that not standard (and not
acceptable.)Eek! The tng to do of course is to agree to write
a letter anyway,insist that it be confidential, and then
mention the details of thetransaction in the
===
open interval (0,2pi) define a function f byf(x) = (1-cos(x))
sum_{n=0}^{infty} 1/(2pi n + x)^2.The problem is to prove,
preferably without using any WMD's, thatf(pi + x) + f(pi - 
x)
= 1/2for all x in (0,pi]. (The function f is not
===
are there? permission for an emailed response. The GNU project
will probably not be Posix conformant, Tom said noncommittally>
How many continuous functions are there from R to R?It seems it
must be same as the number of reals, since a continuous>
function is determined by its value at the rationals, and c ^
a_0 => (2 ^ a_0) ^ a_0 = 2 ^ (a_0 * a_0) = 2 ^ a_0 = c.Is that
right?Exactly right.I have been tnking for sometime that many
of the weirdnesses thatfollow from CH or from AC of course
involve creepily non-topologicalproperties of the reals. (For
example, thethrowing-darts-at-the-real-line argument, or the
existence ofnonlinear functions where f(a+b)=f(a)+f(b)
everywhere.Cantor noted that the closed set analog of CH does
hold. (Everyclosed subset of a closed set is either countable
or cardinality c.)The AC-related weirdnesses that I am
familiar with all depend on thespecification of well-orderings
of the reals, and of course there isno construction of such a
well ordering. Indeed, if we had aconstruction of
well-ordering, that would generally amount to also afixing of
the cardinality of c.Indeed, the more one works with the
subject (though I am strictlyamateur here), the more it seems
to me that a central issue is thefollowing:We simply have no
intuitions for topologically crazy sets of reals.Our
intuitions for the reals connect very very heavily with
theirtopological properties, and we don't have a good sense
for them apartfrom those properties.I'm still trying to put 
ts
idea into a more careful form, but thegeneral thrust is that
most of the weirdnesses attributed to ACreally ought to be
attributed to the indeterminancy of CH; that theindeterminancy
of CH ought to be attributed to a lack of intuitionabout
===
How many continuous functions are there?> How many continuous
functions are there from R to R?It seems it must be same as
the number of reals, since a continuous> function is
determined by its value at the rationals, and c ^ a_0 => (2 ^
a_0) ^ a_0 = 2 ^ (a_0 * a_0) = 2 ^ a_0 = c.Is that right?I
don't tnk so. I may be wrong, but I believe two functions 
can
beidentical on the rationals but not on the irrationals and
both stillbe continuous. However, at best only one can be
differentiable. Sothe cardinality of differentiable functions
is c, but I tnk that thecardinality of continuous functions is
still 2^c.It's been a wle, so someone with knowledge of ts 
may
===
sft, core issues>Don't you agree, , that from C1-C4--indeed,
from C3,C4 alone--any>*competent* mathematician could have
deduced Ex~(x=x)?Two comments, the second for anyone who
missed your first few hundred> repetitions of ts question:(i)
your question has no relevance here - you seem to be
confusing> a statement and its converse.(ii) the original
claim was that Ex~(x=x) followed _in_ standard> set theory.
Since Ax(x=x) is a theorem of standard set theory, if> ts
follows then it also follows that C1-C4 are simply
inconsistent> with standard set theory.>C1 AxAy[x=y -> Az(z
in x <-> z in y)]>C2 AxAy[Az(z in x <-> z in y) -> Az(x in z
<-> y in z)] >C3 EyAx[x in y <-> Et(x in t) & A] (with y not
free in>A)Classification>C4 AxAy[Az(z in x <-> z in y) -> 
{Et(x
in t & y in t) <-> x=y}] (Weak>Extensionality)>someone will
point out the error.?>Did your Homies correct you? Or did
they *defend* your mistake-->and you for having made it? A few
people recently have pointed out that to conclude that what> I
said was erroneous you need to take it out of context. Wch is>
of course true.Don't *all* of the following indicate that 
you
had *no idea* (and*still* have no idea) wh Ex~(x=x) follows
from C1-C4? =When you said set theory I assumed you meantZF.
That's what set theory with no 
qualificationmeans these days.If
you meant NGB set theory then no, C1-C4 arenot inconsistent
with set theory. It does _not_follow that C1-C4 give an
example of sometngwch is not equal to itself, or an example
ofsometng wch does not exist. [DON'T SNIP--J.C.]It is 
correct
that I have no idea why Ex~(x=x)follows from C1-C4. Ts is
because (assumingthat NBG is consistent) NBG has a model
inFOL=. In that model everytng is equal toitself.[DON'T
SNIP--JC]C1 AxAy[x=y -> Az(z in x <-> z in y)]C2 AxAy[Az(z in
x <-> z in y) -> Az(x in z <-> y in z)] C3 EyAx[x in y <->
Et(x in t) & A] (with y not free inA)ClassificationC4 
AxAy[Az(z
in x <-> z in y) -> {Et(x in t & y in t) <-> x=y}]
(WeakExtensionality)>You have no idea why Ex~(x=x) follows
from C1-C4>because you are brain-dead analysis teacher who>can
only work with the routines he has memorized.Could be. Now show
us why Ex~(x=x) _does_ followfrom C1-C4.Exbit of proof of
Ex~(x=x) from C1-C4 and someone will pointout the error.IT'S
ULLRICH BAIL-OUT TIME! SNAP TO IT, MAGIDIN! ON THE
DOUBLE,HUGHES!> Giggle: When you set up a new yahoo account
with a silly name> like pzuffy the idea is to avoid using your
real name in posts> you make under the silly name.(I'm 
assuming
the point to pzuffy was to make it appear that> there's 
someone
other than you out there who agress with the> tngs you say. If
the point was sometng other than that> then never mind.
Although it's hard to see what other point> there could have
been - if the point was to set up a new> account because the
reason other than ding your> identity, it's hard to see why
you'd choose a name like> 
Ôpzuffy'.)> You've got your 
fuffy.
===
bound of magnitude of a complex root of polynomialdeath &
rebirth> Can anyone help me to solve the question? I have no
idea on it... Let f(t) be a polynomial with complex
coefficients of degree n and withleading> 
coefficient = 1.> Let
a be a root.> Show that | a | <= n b, where b is the maximum
of the magnitude of the> coefficients of f(t).Sketch: If we 
had
|a|>nb, the polynomial could not be zero at $a$, by thetriangle
===
stupid question)>Citrus fruits that have no elements, but
aren't the>empty set, are technically called individuals.
>Whether you want them to exist>or not is basically a matter
of taste, >wch I suppose would have to be fairly>sour. HTHIs
===
Sets before logicRandom thoughts on creating a theory of sets
prior to a theory of> propositions and 
quantifiers:[Definitions
of logical constants snipped.]Now, what theory of sets will
yield what logic of propositions and> quantifiers?What are the
rules of logical inference, and how are they expresed?
===
Collatz conjecture> What special numbers other than the easy
ones like 1,[case 0] any number whose binary form is1wch
becomes 1 after 0 iterations>2^k, [case 1] any number whose
binary form is (where {0} means krepetitions of 0)1{0}wch
reverts to [case 0] after k iterations>(2^(2k)-1)/3, [case 2]
any number whose binary form is{01}wch reverts to [case 1]
after 1 iteration>2^k(2^(2n)-1)/3, [case 3] any number whose
binary form is{01}{0}wch reverts to [case 2] after k
iterations>have been proved to be valid for the Collatz
conjecture?You can extend ts list forever. Once you have
proven it for 3 (wchreverts to [case 3] after one iteration),
you have[case 4] any number whose binary form is11{0}wch
reverts to 3[case 5] any number whose binary form is11{01}wch
reverts to [case 3] after one iteration[case 6] any number
whose binary form is11{01}{0}wch reverts to [case 5] wch
reverts to [case 3]So once you have shown that a binary
pattern w is true, then youautomatically have proved it true
for[case x] any number whose binary form isw{0}[case y] any
number whose binary form isw{01}[case z] any number whose
===
of recommendation Adjunct Assistant Professor at the
University of Montana.> I've had students walk into my 
office
with the letter >> written for me and asking me to sign it.
Let's call>> that not standard (and not acceptable.)Eek! The
tng to do of course is to agree to write a letter
anyway,>insist that it be confidential, and then mention the
details of the>transaction in the letter.I disagree. There's 
a
chance, at least, that the individual issincerely trying to be
helpful. I had, on occasion, had professors Iasked for letters
of recommendation tell me to just write it and I'llsign it.
Needless to say, I decided to go ask other people forletters
when that was the case, and I frown thoroughly on
thatpractice. But perhaps the student had encountered a
similarlyill-disposed professor, and thought he would save
time by bringing aletter ready? Absent other experience, he
may well tnk that it->is<- standard.I would certainly tell the
student that the practice is not standard,and should not be
acceptable. I would also suggest that any professorof s who
has suggested ts method to m is NOT a good referencewriter,
and he should seek to replace m. And I will usually try
torefuse to write letters of recommendation for admission to
graduateschool for people that I will not recommend (i.e.,
people I would notsay good tngs in balance). That does not
mean that my letters do notmention any problems, just that if
I have a generally bad opinion ofsomeone, then I recommend to
the student to find someone else to writeit. 
(It's happened
once or twice; once, I asked a student who had notdone very
well in my class why he was asking me. He explained that hewas
seeking to be admitted to a specific, applied math program, 
and
hewas hoping that by getting a letter from an algebraist, he
would shows breadth. I suggested he just get letters from
applied math and uses transcripts to show breadth, and he
acquiesced. I usually try tosuggest other courses of actions
rather than ßatly refuse, and oncethe student insisted and I
said that I would not be able to writeanytng glowing, and
would point out that on the balance s work
===
you may have realized that I have remarkable power to draw>
attention on several newsgroups to the extent that I even have
my own> dedicated repliers, like or Ullrich, who like to>
obsessively insult me! Anyone have a good term for people who
just> follow around a popular poster insulting m? Well I call
them critic> trolls.I tnk critic of trolls is more accurate,
as you of course are thetroll. That real mathematicians are
still willing to even look atyour posts is sometng you ought
to feel honored, since frankly theirtime is far more valuable
than yours.As for calling yourself a popular poster, I'm not
sure thatfrequency of posting equates to popularity. Although
your infamy inpoor proof-writing has made you well known, it
does not mean you areconsidered popular by any conceivable
===
Recommendations on Complex Analysis books?>>The subject says
it -- however, I'd like to clarify>>one detail: 
I'm looking
for a book on *analysis*, as>>opposed to Calculus (i.e., that
covers rigorously>>the concepts and proofs on Complex numbers
and>>Complex variables functions).I was amazed by the way
you've put it. Ts means that for you it is> obvious that
Calculus is *not* rigorous and it is *not* analysis. No,
that's not what I meant.When I say Calculus in that context,
I'm referring totypical College courses in math: a Calculus
course wouldtypically emphasizes on the applications, and
understandingtngs well enough to apply them in different
contexts(i.e., it is engineering oriented).Another way to put
it is that I want sometng that ismore like analysis for
mathematicians, instead ofcalculus for engineers> My favorite
one: Theory of Complex Functions by Reinhold
===
are there? permission for an emailed response.X-Zippy-Says: I
had pancake makeup for brunch!> How many continuous functions
are there from R to R?It seems it must be same as the number
of reals, since a continuous> function is determined by its
value at the rationals, and c ^ a_0 => (2 ^ a_0) ^ a_0 = 2 ^
(a_0 * a_0) = 2 ^ a_0 = c.Is that right?I don't tnk so. I 
may
be wrong, but I believe two functions can be> identical on the
rationals but not on the irrationals and both still> be
continuous.Nope, identity on the rationals and continuity
uniquely determines thefunction at all points. Consider some
irrational x; it is the limitof some increasing sequence of
rational x0, x1, x2, ... Bycontinuity, f(x) must be the limit
===
conjectureHow about 2^k -1? Has anyone proved for that yet?>
What special numbers other than the easy ones like1,2^k,
(2^(2k)-1)/3, 2^k> (2^(2n)-1)/3, have been proved to be valid
===
letters of recommendation>> I've had students walk into my
office with the letter >> written for me and asking me to sign
it. Let's call>> that not standard (and not acceptable.)Eek!
The tng to do of course is to agree to write a letter anyway,>
insist that it be confidential, and then mention the details 
of
the> transaction in the letter.Well, I should have said
student, not students. Andin ts case, I treated it as a
special situation. First,it was a (very) foreign student.
Second, the letter wasonly for a summer job, not an internsp
or grad school.Trd, the situation came up suddenly, where few
opportunitiesare available for (very) foreign students and it
was a rush job (due that day) and the student thought she was
doing me a favor AND was quite polite about it. Fourth, the
letterwas accurate and well-written, to a collaborationof some
friends. So I had no trouble signing my name toit. I would have
written a more glowing letter for her,since she was a stellar
student.Then we had a discussion about proper procedure. No
harm done. (She _was_ jerking her boyfriend around at the
time,so if that character trait manifests itself again in a
moreserious situation, then she _will_ get nuked. But it
won'tbe because someone wasn't frank with her 
at least
===
equation, such as f2(x,y,z)=c2, can describe in a 3d space>>a
surface, possibly a plane, but not a line.>> I could *swear*
that {(x,y,z) in R^3 : x^2+y^2=0} was a line,>> last time I
looked.>> Lee RudolphTs actually happened.>Many years ago a
problem similar to ts came up in a CalcIII class I
was>teacng.I mentioned that situations like ts are called
Degenerate caseA girl raised her hand and saidBut that's 
what
my Father says *I* am!!Bob Pease>There is a certain beauty to
ts. If you define a line as theintersections of two planes, 
and
these are defined by P1 = 0 and P2 = 0 then you can 
define the
line as a cylinder, P1^2 + P2^2 = R^2, where R = 0. Now take
the cylinder of radius 0 andexpand the radius. sooner or later
it itersects the surface f1 = 0.So it is possible, even without
using gradients, to solve for theintersection and then minimize
R. (Though to say the two gradientsmust point in the same or
===
Convergence on a space with no topology?> Probably been asked
before...Let X be a metric space (or just a topological one).
Let>> B* be any subset of P(X), the set of all subsets of X.
Choose an>> element A in B*. Then, without attempting to 
define
a topology on B,>> we just say A_n -> A in B* if for every 
a'
in A there exists a >> sequence (a'_n) in X wch converges to
a' in the metric >> topology of X with a'_n in 
A_n for all
n.Ts looks complicated. Somehow you want to say the maximal A
wchsatisfies ts condition, and wch sets of limit points are
containedin B* depends entirely on the definition of B*, so
there is noguarantee that maximal is meaningful. I tnk that to
simulateCauchy sequences would have to wait until some other
tngs gotstraightened out.Ts is related to the Hausdorff metric
on subsets of a metric space.>And for your limit you probably
want the set of all a' such that>there is a sequence as
written. In case your sets are all compact,>ts will coincide
with convergence in the Hausdorff metric,>as I recall.>>
Then we have (at least) a notion of convergence on a space >>
that is not directly equipped with a topology itself. Thus,>>
we may also speak of the continuity of functions from >> B* to
B* or from B* to any other (metric) spaces. >>> To further
demonstrate ts, let C be any finite subset of X and>> f_C: B*
-> N (the natural numbers) the counting function, i.e.f_C(A)
gives how many elements of A coincide with C. Is f_C>>
continuous? (not mathematition)>>I tnk he is saying that you
don't need the Hausdorff metric or anyother metric to 
define
convergence in B*. But then he's saying youneed the 
definition
of convergence from the first topology. Without using a
topology, you can define A_n as convergent-type-A ifthe
intersection is nonempty, and taking n to infinity [where
A_(n+1)adds one set to A_n ], its limit-point-type-A would be
the set, emptyor nonempty, of the intersection limit.What he
is looking for is convergence-type-B of some kind. I tnk
itmight result from selecting only closed sets from the first
topology,ie, Let B* be any subset of Q(X), the set of all
CLOSED sets in X.Then you do not have to refer afterwards to
the first topology orestablish a second. Maybe Compact set
should be substituted forclosed - some of you may know ts - >
In case your sets are all compact,>ts will coincide with
convergence in the Hausdorff metric,>as I recall. Cauchy-like
sequences could be deifined afterwards - I don't 
know howmuch
===
Cube (the movie)
wants to make money, he should become a script writer. s
bullst> sounds more plausible than that produced by Hollywood.
I can see it now: Aliens invade and because of the core error,
the mathematical> establishment is rendered helpless and only
JSH, with s Advanced> Polynomial Factorization, can save the
Earth.>The entire cast and crew were sealed on set for the
duration ofthe shoot.After post production was completed,
everyone involved with the projectwas 
Ôdisappeared'.All
materials connected with the production were destroyed.In
short, everytng was done to prevent knowledge of ts
featurebecoming public before it went on general
release.Explain, then, if you would, how *you* managed to post
a spoiler for tsforthcoming sci-fi[1] epic in ts froup.[1]: 
I'm
unsure whether to desribe it as Ôscience 
fiction' 
or'fictional
science'.-- P.A.C. SmithThe vast majority of Iraqis want to
live in a peaceful, free world.And we will find these people
===
and we will bring them to justice.Subject: Re: JSH: My use of
remarkable power to draw : >attention on several newsgroups to
the extent that I even have my own : >dedicated repliers, like
or Ullrich, who like to : >obsessively insult me! Anyone have
a good term for people who just : >follow around a popular
poster insulting m? Well I call them critic : >trolls. : :
Around here we call them mathematicians.Who we? People of good
will call THOSE people assholes.That is not inconsistent with
their also being mathematicians,but the overlap is a BAD tng.
The people who merit mathematicianin the honorific sense in 
wch
you are trying to use it are peoplelike Dik Winter, who, even
in the face of potent distractions, are willingto stay focused
on mathematics.-- --- It's difficult ... you 
need to be united
to have any strength, but internal issues have to be
addressed. --- E. Ray Lewis, on liberalism in
===
magnitude of a complex root of polynomialsorry but what is
$a$? Hammick <hammick@telus.net> b.a6l.97> death & rebirth>
Can anyone help me to solve the question? I have no idea on
it...> Let f(t) be a polynomial with complex coefficients of
degree n and with> leading> coefficient = 1.> Let a be a 
root.>
Show that | a | <= n b, where b is the maximum of the magnitude
of the> coefficients of f(t).> Sketch: If we had |a|>nb, the
polynomial could not be zero at $a$, by the> triangle
===
f(a+b)=f(a)+f(b) everywhere. Thomas,that sounds interesting.
Can you point me to an example or other resources on such
===
complex numbers>Why (a,b) x (c,d) = (ac - bd, ad + bc) ? (a,b)
x (c,d) = (2ac - bd, ad + bc) also will serve to solve x^2 + 1
= 0. Because witth ts definition and the addition, the complex
numbersbecome an akgebraic field. Thus one can have division
===
numbers...Some physicist will be daft enough to produce a
theory that ts is the> number of parallel universes!!!!!!Wait
===
about upper bound of magnitude of a complex root of
polynomial> Let f(t) be a polynomial with complex coefficients
of degree n and with > leading> coefficient = 1.> Let a be a
root.> Show that | a | <= n b, where b is the maximum of the
magnitude of the> coefficients of f(t).If |a| < 1, 
you're done.
So assume |a| >= 1. You have f(t) = t^n + r(t). So a^n = -r(a).
===
scenarios to get the final score of a hockey gameI once got
interested in the number of ways to get a particularbowling
score (e.g., 1 way to get a 0 and 1 way to get a 300).Wrote a
program to do ts (try it, it was kind of challenging).paper on
th topic that had already been written (in J Rec Math or Math
Mag orsomewhere similar). As I recall, the score that can be
attainedthe most number of ways is in the 70's.Cp
Klostermeyer> I use to know how to do ts with my eyes closed -
but it's been some> time and I forget the formulas. Can 
someone
help me figure out the> number of different combinations for 
the
following.> The Final score of the hockey game was Team A 13
Team B 4> I went to sleep with the score tied 2-2.> What I
am trying to figure out is, how many different ways could the>
two teams have scored to arrive at the 13-4 final score.There
were 13 goals scored after you went to sleep, and 2 of them
were> scored by Team B. The number of combinations is the
===
Re: uniform convergence & differentiation>In many analysis-type
texts I've found statements like since the series>converges
uniformly, we can apply term-by-term differentiation and
uniform>convergence allows us to differentiate term-by-term,
but I've never>actually seen a proof or explanation of WHY 
you
need uniform convergence to>legitimize term-by-term
differentiation. Can someone explain ts? As others have noted,
in general uniform convergence of a seriessum_n u_n(x) is not
sufficient to be able to differentiate term by term.Perhaps 
the
analysis was complex analysis? If sum_n u_n(z) isa sequence of
analytic functions converging uniformly on compact subsetsof
some domain in C, then you can differentiate
===
Reality> There was an Indian mathematican called Shakuntala
who toured> the west beating the computers at multiplying two
large numbers - must> have used vedic methods,Must have? Must
have?Cite, or retract.I'll give you a clue - if you go to 
her
website you won't see _one_ _single_ mention of vedic 
methods.
IIRC.She's had my respect for about 25 years, wch was when I
firstread her The Joy of Numbers, (I tnk that's 
what it's
called - it was 25 years ago, rememeber). However, almost all
of her computational feats have been bettered in recent
decades. And none of the guys who've beaten her have claimed
any usage of vedic methods either. Pl-- Unpatched IE
vulnerability: DNSError folder disclosureDescription: Gaining
access to local security zonesReference:
===
Subject: Re: [Set Theory] A set has a disjoint copy of itself
?>>I am looking for a proof that any set has a disjoint copy
of itself,>>i.e. that given a set A, there exists a set B with
the same>>cardinality as A, and such that A/B=0. I am hoping ts
is true with>>ZF or ZFC.>B = {0} x ATs need not be disjoint
from A. In fact, if we take a_0 = {0}, a_{n+1} = <0, a_n},
andlet A be the set of all a_n, then only a_0 is not anelement
===
Myth and Reality>Tell us about the grade-school method.O(N^2)
pairs for two N-digit numbers.>What it is, and how it is the
same as the method given.O(N^2) pairs for two N-digit numbers.
Everytng else is irrelevant.Better than that, it's
Theta(N^2).Or do I mean Ôworse than that'?Pl-- 
Unpatched IE
vulnerability: dragDrop invocationDescription: Arbitrary local
file reading through native Windows dragDrop
invocation.Reference:
http://msgs.securepoint.com/cgi-bin/get/bugtraq0302/12.
htmlExploit:
===
Multiplication definition of two complex numbers> Why (a,b) x
(c,d) = (ac - bd, ad + bc) ? (a,b) x (c,d) = (2ac - bd, ad +
bc) also will serve to solve x^2 + 1 = 0.Another
reason...numbers (a,0) are supposed to act like the real
===
en el> I.e., it is possible to prove directly, using only the
definition of> derivative, that (e^x)' = e^x?> 
Of course it
depends on how you define e^x> (and that I suppose depends on
whether you tnk of it as e^x, that is> a number e raised to
the power x or as an exponential function>> exp(x).)>> The
definition of the function f(x) = a^x, a real, a > 0, that I
always>> use, since secondary school to now, is the function
whose value at x is>> the number a raised to the power x. And
the definition of the number>> a^x is>> a^x = Lim(a^(x(n)), n,
inf)>> where Lim(x(n), n, inf) = x, and the x(n) are
rationals.>That requires proving that ts limit is independent
of the sequence>x(n) (under the condition that it converges to
x).>So then you have to decide what e is.There are various
direct ways to prove that a^x is differentiable; start with
a^(x+y) = a^x * a^y, andget bounds on difference quotients,
wch have to converge just using that a^z -> 1 as z -> 0.
Oneway of showing ts is that, if a > 1, and 0 < z < 1,a^z < z
* (a-1).But just assuming a^x is differentiable, one getsits
derivative is a^x*L(a), and L(ab) = L(a)+L(b).It is not too
hard to get that L is a logarithm tosome base, and we may as
===
(sorry, maths not psych)Expires: 28 days>> If fields acted
instantaneously, we would be>> able to use them for
instantaneous communication.>>Please explain how you can use
the fact that a force is>a static electric field, for
instantaneous communication.>>Paul, puzzled> I'm sure some
bright QMian would find a way.>>Backing out again, Henry?>>You
cannot defend your assertion,>>but you will repeat it, won't
you?>>And you will back out once more when>>asked to defend
your repeated assertion, won't you?>>That's 
Henry Wilson's
eternal circle of ßeeing>>restatements of ßed
assertions.>>Paul, not surpriseda static electric field in an
accelerator, the force doesn't act>the same instant as it
enters the field, ts could be>used in instant communication.>>
Paul, it should be obvious that if the effect of a field is
instantaneous>> throughout the universe, then switcng it on an
off in an intelligent manner>> can be used for instantaneous
communication.Funny, eh? :-)Paul, enjoying the acrobatic
show>Of course you are funny.If electric and magnetic fields
acted instantaneously, light would travel atinfinite
speed.Henri Wilson. See the Stupidity of
===
How many continuous functions are there?>>the existence
of>>nonlinear functions where f(a+b)=f(a)+f(b) everywhere.>
Thomas,that sounds interesting. Can you point me to an example
or other > resources on such functions?> ,> There's not 
really
that much to tell about them. You can't explicitly write 
them
down, you need to use the axiom of choice.Here's a rough
sketch of the existence of such a function:Use zorn's lemma 
to
prove every vector space has a basis.Consider the vector space
R over the field Q. Take a basis for it, B. w.l.o.g. let 1 be
an element of ts basis. Let x be any other element.Define a 
map
on the basis sending 1 to x, x to 1 and fixing every other
element of the basis. Extend ts to a Q-linear map, f.Now, f is
Q-linear so additive, but if f was R-linear then we would have
for all t, f(t) = t f(1) = t x.But f(x) = 1. So x^2 = 1, x =
+/- 1, wch isn't the case.That's about all you 
can do to
stretch the discussion of such functions out past 5 minutes.
:) A more interesting generalisation of ts is that linear
operators on infinite dimensional vector spaces need not be
===
complex numbers> Why (a,b) x (c,d) = (ac - bd, ad + bc) ?
(a,b) x (c,d) = (2ac - bd, ad + bc) also will serve to solve
x^2 + 1 =0.There is much more to complex numbers than simply
solving x^2 + 1 = 0.There are an infinite number of
Ômultiplications' that solve it, 
includingyours.But, that is
not an interesting find, with respect to complex
numbers.(0,1)^2 + 1 = 0 iff (0,1)^2 = -1.(a,b)x(c,d) =
(n(ac)-bd, ad+bc), works for any n.(0,1)x(0,1) =
(n(0.0)-(1.1), (0.1)+(1.0)) = (-1, 0) = -1Addition and
multiplication that satisfies the field postulates 
is
required.Your multiplication function does not
===
work.WittSubject: Re: Ex(~x=x), counterpart theory, and
self-different, it possesses some : > properties not possessed
by it and, hence, must be a self-inconsistent : > object: : > :
: you will find a description of a universe that is a proper
part of itself. : : He calls it counterintuitive. I view it
with about the same sense that you : suggest above.What
language is ts description in?Is the tng's being a proper 
part
of itself consistent, or inconsistent,with its being equal to
itself, or identical with itself, or non-distinctfrom itself?
: I believe it to be a failure to understand the relationsps
between mereology : (part), set theory (element), and definite
description (equals).How can you say that there is ANY failure
involed here AT ALL, if thedescription of the universe is both
syntactically grammatical AND logicallyconsistent? : It is not
a matter of seeing whose is right and who is wrong.As a general
rule, a consistent set of first-order axioms CANNOT POSSIBLYbe
wrong. It can at worst be inadequate to its intended task. :
It is not a : matter of ts alternative foundation is better
than that alternative : foundation. It is a matter of seeing
how the identity puzzles are resolving : themselves.You still
haven't stated the identity puzzles.If you are appealing to
sophomoric natural language stuff thenthe resolutions are
pretty trivial. : Of course, everyone is so busy comparing
deductive calculi and formal systems, : they seem to have
forgotten that these tngs arose from plosopcal :
considerations that were being debated prior to the
introduction of such : calculi.And good riddance.You,
conversely, seem to have forgotten that people couldn't even
HAVEcoherent discourse about most of the questions until AFTER
acevingformal language. The neologism was HELPFUL. : : Perhaps
Mr. Greene is right. Perhaps not.Never say that out of
context. Right ABOUT WHAT?WHAT DID I
SAY???????????????????????????????????I repeat, the ONE tng I
am CERTAINLY right about is thatyou HAVE to QUOTE the people
you are disagreeing with!IN context!-- --- It's 
difficult ...
you need to be united to have any strength, but internal
issues have to be addressed. --- E. Ray Lewis, on liberalism
===
there? permission for an emailed response.X-Tom-Swiftie: VI is
much better than EMACS, Tom said with joythe existence of>
nonlinear functions where f(a+b)=f(a)+f(b) everywhere.> that
sounds interesting. Can you point me to an example or other >
resources on such functions?It's a consequence of the
existence of a basis for every vector space,wch is a
consequence of the axiom of choice, with the property
thatevery element of the space is a linear combination of some
finitesubset of the basis.You can characterize the reals as a
vector space over the rationals(where the scalars are Q). The
basis is then some (infinite) set of reals, such that every
real rcan be expressed as q1*b1 + q2*b2 + q3+b3 + ... + qn*bn,
where q1...qnare nonzero and rational, and b1...bn are in the
basis and piecewiseunique. (N and the particular q's and 
b's
of course depend on r.)Ok, now let b be some element of the
basis. Then every real is:(1) q*b + q1*b1 + q2*b2 + ... +
qn*bn, where b, b1, b2, ..., bn arepiecewise unique, q1...qn
are nonzero and rational; q is rational butmaybe zero. Since
ts is a basis for R, ts decomposition isunique.Define f(x) =
the projection of x on the coordinate b. (That is, qin (1)
above.)Clearly f(a+b) = f(a) + f(b). (It's a vector space, 
and
b is a basiselement.) But since f(x) is always rational, we
===
Re: Need advice on letters of recommendation permission for an
emailed response.X-Zippy-Says: Yow! Am I in Milwaukee?> Well, I
should have said student, not students. And> in ts case, I
treated it as a special situation. First,> it was a (very)
foreign student. Second, the letter was> only for a summer
job, not an internsp or grad school.> Trd, the situation came
up suddenly, where few opportunities> are available for (very)
foreign students and it was a > rush job (due that day) and the
student thought she was doing > me a favor AND was quite polite
about it. Fourth, the letter> was accurate and well-written, to
a collaboration> of some friends. So I had no trouble signing
my name to> it. I would have written a more glowing letter for
her,> since she was a stellar student.Ah yes, that is a
===
Book>I am looking for supplemental text to aid in my learning
of ODE and>PDE next semester. Currently my undergraduate text
is Elememntary>Differential Equations with Boundry Value
Problems by Derrick and>Grossman. Ideally, I would appreciate
an text wch emphsizes>theoretical foundation over applied
examples. Any suggestions are>greatly appreciated.Wow, 
that's
a refresng change: most students these days are lookingfor
lots of examples and no theory.You might look at M. Braun,
Differential Equations and Their Applications, Applied
Mathematical Sciences 15, Springer-Verlag 1975.Or a somewhat
more challenging one: G. Birkhoff and G.-C. Rota, Ordinary
===
about upper bound of magnitude of a complex root of
polynomial> sorry but what is $a$?Like many a guilty party
===
on Complex Analysis books?The subject says it -- however, 
I'd
like to clarify> one detail: I'm looking for a book on
*analysis*, as> opposed to Calculus (i.e., that covers
rigorously> the concepts and proofs on Complex numbers and>
Complex variables functions).I was amazed by the way you've
put it. Ts means that for you it is> obvious that Calculus is
*not* rigorous and it is *not* analysis. Have> you ever read,
for instance, Michael Spivak's Calculus or Introduction> to
Calculus and Analysis by Richard Courant and Fritz ?[...]I tnk
he was referring to a fairly common convention; if fact,
IIRC,in the preface to the second or the trd edition to s book
Spivaksaid he should have called it Analysis or sometng like
that but itwere to late to change the
===
personality assessment> ...>Where huffy == Correy (and so is
self-identical)?>In another post by huffy, you signed (and
clearly spoke as> Correy). >Why the (additional)
pseudonym?> Most of the posts from huffy are not signed .
Seems clear to> me that he's tired of having so many people
disagree with m> so he's invented a imaginary playmate - he
slipped signing one> of huffy's posts .Perhaps an idea for
JSH? Setting up tens of imaginary playmates, all> agreeing
with m?Well, well, well! If it ain't (Suk My) Dik! 
What's up,
===
books?>>THE classic text on complex functions is Ahlfors,
_Complex Analysis_ .>> BArry Mazur has recently published a
popular book on complex>>numbers _Imagining Numbers_. I would
also recommend _Complex Numbers>>and Geometry_ by Hahn.
Finally, a great old book: Theory of>>Functions_ by
Caratheodory.>> I second the Ahlfors book. We used it as our
Complex Analysis>textbook when I was a grad student at the
University of Micgan.>I admit to never having looked at
Ahlfors, but I have heard that it is a rather intense book -
terse, with very difficult exercises. My impression is that it
would be at too gh a level for a mathematical hobbyist (as the
OP identified mself).-- Stephen J. Herschkorn
===
continuityim using the following definition of uniform modulus
of continuity off:X->Y, where d_x and d_y are the metrics in X
and Y respectively:omega_f (delta) = sup { d_y(f(x), f(a)) :
a,x in X and d_x(a,x)<=delta}problem: if limit of
omega_f(delta)/delta = 0 as delta goes to zero,then f is
constant. i have already proved that f is constant
iffomega_f(delta)=0 for all delta>=0. i also proved
thatomega_f(t/n) >= 1/n omega_f(t) for all n in N and t in
Rwch seemed useful at first, but now i can't get 
anywhere.can
===
of Gravity?Jack,Your recollection is absolutely correct. A
Levi-Civita connection maybe represented as the sum of Affine
connection (noncovariant entity) anda covariant tensor of
nonmetricity (wch, by the way, in my view,describes
gravitational field preserving energy-momentum, as describedin
my papers, wch I e-mailed you after our chat in SF).Best
metricity in Hagen Kleinert's senseguv^;v = 0i.e. Diff(4)
divergence vanishes.I rewrite Einstein's zero torsion 1915
geometrodynamic classical local field equatonGuv =
-(8piG/c^4)TuvasGuv = -alpha'Tuvalpha' = 
(string tension)^-1 =
Witten parameterInfinite string tension means no gravity 
because
space-time geometry is too stiff to bend.The local
stress-energy density tensor of pure geometry is then
triviallyTuv(Geometry) = 
(alpha')^-1GuvEinstein's field
equation is then simply the balanceTuv(Geometry) +
Tuv(Ordinary Mass-Energy) = 0Adding random micro-quantum zero
point energy density from all quantum fields of spin 1/2
lepto-quarks and spin 1 gauge force bosons gives additional
termtuv(zpf) = (alpha')-1/zpfguv/zpf > 0 is exotic vacuum 
dark
energy with w = -1 negative pressure./zpf < 0 is exotic vacuum
dark matter with w = -1 positive pressureDark matter detectors
will never click except by false positives in my
theory.Einstein's equation is thenTuv(Geometry) + 
Tuv(Ordinary
Mass-Energy) + tuv(zpf) = 0In the 1915 theory with /zpf =
0Tuv(Geometry)^;v = 0from the Bianc identities.However these
identities FAIL IMHO when /zpf =/= 0 and is variableand if
there are torsion fields.In my theory (with 
Witten's h = c =1
convention)/zpf = 
(alpha')^-1[(alpha')^3/2[|MACRO-QUANTUM
VACUUM COHERENCE|^2 - 1]guv = Minkowski metric + Kleinert
World Crystal Lattice Strain TensorMake the Levi-Civita
connection from guv in the usual way.World Crystal Lattice
Distortion Field = du(x) = alpha'(Goldstone Phase of
MACRO-QUANTUM VACUUM COHERENCE),uStrain Tensor = du(x),v +
dv(x),uDiff(4) Landau-Ginzburg eq for VACUUM COHERENCE in a
two-way feedback loop between IT World Crystal Lattice
Distortion Field andBIT VACUUM COHERENCE in sense of Bohm's
interpretation of IT(dden variable) + BIT(Pilot Wave of Active
Information)Torsion fields meandu(x),v - dv(x),u =/=
===
>I've had students walk into my office with the 
letter >written
for me and asking me to sign it. Let's call>that not 
standard
(and not acceptable.)>Eek! The tng to do of course is to
agree to write a letter anyway,>>insist that it be
confidential, and then mention the details of the>>transaction
in the letter.>> I disagree. There's a chance, at least, 
that
the individual is>sincerely trying to be helpful. I had, on
occasion, had professors I>asked for letters of recommendation
tell me to just write it and I'll>sign it. Needless to say, 
I
decided to go ask other people for>letters when that was the
case, and I frown thoroughly on that>practice. But perhaps the
student had encountered a similarly>ill-disposed professor, and
thought he would save time by bringing a>letter ready? Absent
other experience, he may well tnk that it>->is<- standard.I
would certainly tell the student that the practice is not
standard,>and should not be acceptable. I would also suggest
that any professor>of s who has suggested ts method to m is
NOT a good reference>writer, and he should seek to replace
m.>Perhaps (well, more than perhaps) I am just lazy, but I
disagree. It sometimes happens that a student has very few
choices amongst referees. Ts happened to me (as the referee)
once. It was te middle of fall semester, a first-year grad
student needed a reference for some scholarsp fast, and I had
only had m that semester. I looked at the form and what it
asked and I knew that I could not knowledgeably answer the
questions. I suggested he write a recommendation mself and
give it to me. Then I could modify it to what I thought was
appropriate. I would certainly not just sign it carte
blanche.However, in ts case, I also suggested as a better
procedure that the student contact professors at s
undergraduate department; ts is what he did.-- Stephen J.
===
for NTS on proving a limit[...]>> I don't tnk there is any
other way to obtain the> derivative of ln(x), is there? I
mean, there is no> closed form expansion for ln(x+dx). And
using the> Taylor series expansion of ln(x) (or ln(1+x)) is
most> definitely cheating, since that series is 
justified> by
what the derivatives of ln(x) are. (please do let> me know if
there is another way that I've always been> missing! The way 
I
remember learning it, is that the> derivative of ln is 1/x as a
direct consequence of ln> being the inverse of e^x, and the
derivative of e^x> being e^x). >> Another approach is to
*define* ln x as int(y=1..x, 1/y) > What could be the
justification for that? (other than> knowing that the
derivative of the inverse of e^x is 1/x)What I mean is that
how would you convince me that the> inverse of such function
is e^x? (without using any> properties of e^x, and in
particular, without using> the derivative of e^x or the fact
that (e^x - 1)/x> has limit 1 when x -> 0 ),Carlos> -- > I
like the functional equation approach.For log: we want
log(x)+log(y) = log(x*y).log'(x) = lim (log(x+d)-log(x))/d 
(as
d -> 0) = lim(log(1+d/x))/d = lim {(log(1+d/x)/(d/x)}/x =
log'(1)/xThe natural log is the one with 
log'(1) =
1.Similarly, for exp, we want exp(x)*exp(y) = exp(x+y), so
thatexp'(x) = lim (exp(x+d)-exp(x))/d as d -> 0 = lim
(exp(x)*exp(d) - exp(x))/d = exp(x)*lim(exp(d)-1)/d =
exp(x)*exp'(0) Cohen(I previously sent ts out, but I have 
not
===
factorization, from basic to advanced>In ts post I'll go 
from
a rather basic factorization to a far more