must be all of G.
> On second thought, a^k =1 for some k>1 is enough to prove that
> a has an inverse
And then prove that any inverse of a and inverse of b is the same
element.
--
Jon Haugsand
Dept. of Informatics, Univ. of Oslo, Norway, mailto:jonhaug@ifi.uio.no
http://www.ifi.uio.no/~jonhaug/, Phone: +47 22 85 24 92
===
Subject: Re: Conway's comment on large numbers
>> I was reading a paper by Conway and Doyle that proves that division by
>> three is possible (without the axiom of choice).
>> http://math.dartmouth.edu/~doyle/docs/three/three.pdf
>> In the last paragraph the authors explain their doubts about the ZF
> axioms, saying,
>> Indeed, we're somewhat doubtful whether large natural numbers (like
>> 80^5000, or even 2^200) exist in any very real sense...
So Conway has finally flipped ...
> There was a thread about computability where the question
> of theoretical vs actual computers came up.
> A theoretical computer can perform an unbounded, but finite,
> number of operations. I pointed out that finite was pretty
> small for an actual computer (where actual means conforms
> to the laws of physics as we know them).
> Assume a computer can perform an operation in 3 x 10^-44 sec.
> (Planck's time - theoretically, the shortest time an action can occur
in).
> Let this computer count for a billion years (10^9).
> (The universe is estimated to be about 13 billion years old.
> In 3 billion years, the Milky Way will crash into Andromeda.)
> Google says 1 year = 31 556 926 seconds (3 x 10^7).
> So, in a billion years, the computer will count to
> 3 x 10^44 x 10^9 x 3 x 10^7 = approximately 10^61.
You might as well say that Venus doesn't exist because you can't walk
there.
Surely one doesn't normally count up to 2^200
when working out if 2^200 + 1 is prime or not?
It seems to me to have even less to do with
The belief that 2^200 doesn't exist
makes the intuitionists seem like airy-fairy idealists.
--
Timothy Murphy
e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie
tel: +353-86-2336090, +353-1-2842366
s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland
===
Subject: Re: Conway's comment on large numbers
>> I was reading a paper by Conway and Doyle that proves that division
by
>> three is possible (without the axiom of choice).
>> http://math.dartmouth.edu/~doyle/docs/three/three.pdf
>> In the last paragraph the authors explain their doubts about the ZF
> axioms, saying,
>> Indeed, we're somewhat doubtful whether large natural numbers
(like
>> 80^5000, or even 2^200) exist in any very real sense...
> So Conway has finally flipped ...
> There was a thread about computability where the question
> of theoretical vs actual computers came up.
> A theoretical computer can perform an unbounded, but finite,
> number of operations. I pointed out that finite was pretty
> small for an actual computer (where actual means conforms
> to the laws of physics as we know them).
Assume a computer can perform an operation in 3 x 10^-44 sec.
> (Planck's time - theoretically, the shortest time an action can occur
in).
Let this computer count for a billion years (10^9).
> (The universe is estimated to be about 13 billion years old.
> In 3 billion years, the Milky Way will crash into Andromeda.)
Google says 1 year = 31 556 926 seconds (3 x 10^7).
So, in a billion years, the computer will count to
> 3 x 10^44 x 10^9 x 3 x 10^7 = approximately 10^61.
> You might as well say that Venus doesn't exist because you can't walk
there.
I don't see where Russell made any claim about the existence of large
numbers. In any case, someone who worked with what later turned out
to be modular arithmetic might confuse the number 2^200 with 0. Or 10.
Or any number less than 2^200.
> Surely one doesn't normally count up to 2^200
> when working out if 2^200 + 1 is prime or not?
> It seems to me to have even less to do with
> The belief that 2^200 doesn't exist
> makes the intuitionists seem like airy-fairy idealists.
2^200 obviously exists. The trouble is determining what exactly one
_means_ by the word exist. This is not a question of existence, but
a question of status and access.
And do you exactly know what a philosophical idealist is? (hint:
intuitionists usually commit themselves to some form of idealism in
formulating their position)
'cid 'ooh
===
Subject: [help!] Polynomial Equations
Let m_i, n_i be rational numbers and f(x) a polynomial equation of order
greater than 4 where each root has the form m_i + n_i^0.5.
Is there a polynomial-time algorithm to find the exact roots?
Perhaps that problem can be reformulated in this way:
let R be a floating point number with infinite mantissa (i.e. big as you
like) that approximates a value of the form a + b^0.5 where a, b are
rational numbers, is it possible to estimate the exact a and b?
If you had not guessed it already, I need a polynomial equation with roots
of a known form, but impossible to find if not approximate.
===
Subject: Re: [help!] Polynomial Equations
>Let m_i, n_i be rational numbers and f(x) a polynomial equation of order
>greater than 4 where each root has the form m_i + n_i^0.5.
>Is there a polynomial-time algorithm to find the exact roots?
>Perhaps that problem can be reformulated in this way:
>let R be a floating point number with infinite mantissa (i.e. big as you
>like) that approximates a value of the form a + b^0.5 where a, b are
>rational numbers, is it possible to estimate the exact a and b?
>If you had not guessed it already, I need a polynomial equation with roots
>of a known form, but impossible to find if not approximate.
Your R is a root of the quadratic (X - a)^2 - b = 0
where a, b are rational.
Assume R is irrational (equivalently, b is not a rational square),
so this monic quadratic is unique.
After removing denominators, you have a quadratic
with integer coefficients and root R.
If you can discover this quadratic,
you can use the quadratic formula to find a, b.
Using lattice basis reduction or other means, one can
recover this quadratic given a high-precision estimate of R.
If you have maple, look up
?polytools,minpoly
(or ?PolynomialTools[MinimalPolynomial] with Version 9].
When R is complex, you know R and its conjugate
satisfy the same monic quadratic. You can
get high-precision values of a and b quickly, and
figure out the the rational values for a, b.
You say degree(f(x)) > 4 but don't subsequently reference f(x).
Many different f's will be divisible by the quadratic for R.
If f is known, you should simply factor it over Z,
and then find which factor corresponds to your R.
--
John Adams served two terms as Vice President and one as President, but
lost
reelection. Later his son became President despite losing the popular
vote.
That son lost his reelection attempt badly. Now history is repeating
itself.
pmontgom@cwi.nl Microsoft Research and CWI Home: San Rafael,
California
===
Subject: Re: [help!] Polynomial Equations
Kiuhnm
===
Subject: Re: [help!] Polynomial Equations
In the previous post I used the term impossible, but I meant
computationally impractical (e.g. NP-hard).
Kiuhnm
===
Subject: Re: ring isomorphism in relation to tensor products question
> Alternatively, once you've defined a map from Z[i](x) R to C by
> inducing it from a bilinear map from Z[i] x R to C, you could try to
> define its inverse from C to Z[i] x R in order to show bijectivity,
> but that seems hard off the top of my head.
That is really quite easy: map a+bi to 1 (x) a + i (x) b.
--
Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html
Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9
Francis Wheen, _How Mumbo-Jumbo Conquered the World_
===
Subject: Re: ring isomorphism in relation to tensor products question
Adjunct Assistant Professor at the University of Montana.
>> Alternatively, once you've defined a map from Z[i](x) R to C by
>> inducing it from a bilinear map from Z[i] x R to C, you could try to
>> define its inverse from C to Z[i] x R in order to show bijectivity,
>> but that seems hard off the top of my head.
>That is really quite easy: map a+bi to 1 (x) a + i (x) b.
Sigh. Not my week.
--
It's not denial. I'm just very selective about
what I accept as reality.
--- Calvin (Calvin and Hobbes)
Arturo Magidin
magidin@math.berkeley.edu
===
Subject: Re: Replacement to MMA and Maple
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision:
1.9 primary) id i3LCvlN14149;
>Then, I tried Mathematica 5.0.
>At my very first try, I found
>bugs. I cannot use it for my
>research.
Yes. I switched back to 4.x; stable, bug free, and
fewer annoying features.
>Maple and MMA are to CAS what
>M$ Word is to word processing.
>Do you know a CAS that is more
>into the LATEX philosophy?
>How robust, advanced, etc.,
>is it?
Depends what you are looking for. If you are looking for
a commutative algebra package, then there are some free cutting egde
tools (cocoa, macauly2, singular). In number theory the best tool
around is the free pari/gp. However, if you need a general tool,
the only options are mathematica and maple.
HTH
Daivd
===
Subject: Re: Replacement to MMA and Maple
>Then, I tried Mathematica 5.0.
>At my very first try, I found
>bugs. I cannot use it for my
>research.
> Yes. I switched back to 4.x; stable, bug free, and
> fewer annoying features.
>Maple and MMA are to CAS what
>M$ Word is to word processing.
>Do you know a CAS that is more
>into the LATEX philosophy?
>How robust, advanced, etc.,
>is it?
> Depends what you are looking for. If you are looking for
> a commutative algebra package, then there are some free cutting egde
> tools (cocoa, macauly2, singular). In number theory the best tool
> around is the free pari/gp. However, if you need a general tool,
> the only options are mathematica and maple.
> HTH
> Daivd
Well there is maxima, at maxima.sf.net.
===
Subject: are the following Noetherian rings
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision:
1.9 primary) id i3LCvlk14153;
Could you please help me detemine if the following rings are Notherian?
1-The ring of rational functions of z having no pole on the circle |z|=1.
2-The ring of polynomials in z,w all of whose partial derivatives with
respect to w vanish for z=0.
In all cases the coefficients are complex.
===
Subject: Re: are the following Noetherian rings
> Could you please help me detemine if the following rings are Notherian?
> 1-The ring of rational functions of z having no pole on the circle |z|=1.
> 2-The ring of polynomials in z,w all of whose partial derivatives with
respect to w vanish for z=0.
> In all cases the coefficients are complex.
1. If R is a commutative ring and T is a subset closed under products,
containing 1 and not containing 0, let R_T denote the ring of all
fractions r/t, r in R and t in T. It is called a ring of fractions
over R. Addition and multiplication are defined as usual for
fractions. If R has no zero divisors and T consists of all non-zero
elements, R_T is the field of all fractions. An example is Z_T where
Z is the ring of integers and T is all powers of 2. You get all
fractions a/b where b is a power of 2. The answer to the first is
settled by observing that any ring of fractions over a Noetherian ring
is Noetherian. If I is an ideal, any element has the form a/t where t
in T and then a in I as well. Conversely, if a is an element of R
cap I, then a/t is in I for any t in T. Therefore any set of
generators of the R-ideal I cap R is also a set of generators for I.
2. I don't have any idea on this one.
===
Subject: difference equations
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision:
1.9 primary) id i3LCvnn14208;
===
Subject: Re: difference equations
>i'm looking for online references to difference equations, especially
nonlinear ones. recommendations for books also welcome.
Luenberger, Introduction to Dyanmic Systems, provides a nice
introduction to linear difference equations.
--
Stephen J. Herschkorn herschko@rutcor.rutgers.edu
===
Subject: Re: Gender issues on 60 Minutes, Women especially respond!
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision:
1.9 primary) id i3LCvlh14145;
Single sex education is all the things that you were written about here and
alot more. Single sex education is, and should be, an OPTION for any one
who
desires it. The benefits are there and deserve to be offered. Putting
young
men and women in situations where they do not have the pressures and
distrations that go with coeducation can,in many cases, provide the best
opportunities for success in academics for both sexes.
As an educator and a man who went to an all boys middle school, I would
highly recommend it to any parent of a Middle School child. I think that
the
social requirements of growing up demand coeducation...but that those
years...6th -9th..are prime years where intensive instruction with little
distraction can have dramatic results.
Again...Single Sex classes should not be for everyone and not mandatory, but
should be available. After all, it can not be any worse than something like
German Immersion schools.
>Shouldn't they be allowed to set up single - sex classes for girls?
>> I've heard that *some* girls do better in single-sex schools, and
>> that *some* boys (e.g., those from inner city, high crime areas,
>> perhaps with no father in the home) may do better in a highly
>> disciplined, single-sex situation almost like boot camp.
>> I'm not at all sure that *public* schools should be doing this,
>> though. It seems awfully divisive to me. It gives very negative
>> messages, like girls and boys are a lot more different than they
>> actually are, that girls and boys have to be separated because the
>> girls just can't compete with the boys.
>>Women here in sci.math, would a single-sex class have been better
>>for you in math and science?
>> No. For me it would have been a lot worse. I excelled in math and
>> science, and knew that I was smarter than all the boys *and* girls,
>> which no doubt contributed to the self-confidence that I have today.
>> I had about equal numbers of male and female friends, and would
>> probably have been less comfortable socially in an all girl setting
>> (a good friend went off to a private, all girls high school, which I
>> remember visiting, and finding weird--it was not a place where I
>> would have been happy).
>> Thinking back on college, most of my math classes (and computer
>> science etc.) were overwhelmingly male (though I hardly noticed this
>> at the time). I was completely comfortable and confident in such
>> classes, and continued to excel. This might not have been the case
>> had I been isolated in all-girls classes up until then, and been
>> given the message that girls can't compete with the boys.
>> --
>> Helen Read
>> The issues concerning gender and nature-vs.-nurture are so clouded by
>> strong opinions and politics and subconscious agendas( someone might be
>> implying something in what (s)he says, and not even realize it, or
>> it could be VERY subtle( somethings that Rush Limbaugh says are like
>> this ), etc. ) and biases. Sometimes the accusations are leveled that
>> cause is confused with effect and there exists a rebuttal to that
>> thatsays that what those people call the cause( social stereotypes ) had
>> to have originated somehow, and that what those people call the effect
>> is actually the cause of that, itself being the effect corresponding to
>> a cause which is said to be innate(e.g., closer to the level of genetic
>> or hormone-influenced, etc. ).
>> I too would disagree with the sexual segregation of education because it
>> would reinforce the notion, in the minds of those in it( of both sexes
>> ), that the sexes live in vastly different worlds that bare little if
>> any resemblance to each other. Instead of eliminating it, it would be
>> likely to engender( I am aware of the pun; react to it as you wish )
>> sexism in the minds of all who are in this system as opposed to a
>> single-sex, traditional, eductaion system. School is supposed to reflect
>> reality; if we institute sexually segregated education systems at lower
>> levels of education, students in these systems will find themselves
>> somewhat unprepared for the upper levels( however far up you have to go:
>> if universities and journals are segregated then you would have to go
>> all the way to the ultimate level, which is the collective present state
>> of the mathemtical knowledge of humanity, and that would be very hard if
>> not impossible to segregate sexually, given the state of today's
>> society ), and the difficulties will occur there, especially, if, in the
>> single-gender system, they were given the message that one sex was
>> inherently better than the other at math, computers, science, and
>> related fields.
>> Environment can have a profoundly powerful effect on a person's
>> perceptions, especially where these issues with respect to gender are
>> concerned. Parents, in finding it so cute that their little boy is
so
>> into the usual male gender-role stereotypes and/or that their little
>> girl is so into the usual female gender-role stereotypes at such a young
>> age, may be inadvertently sowing the seeds of sexism into the next
>> generation.
>> I will say more later, but now there is a thunderstorm outside.
>> ---- David Harden
>Continuing, as I said I would:
>Another influence that, as I have read, promotes gender-role stereotypes
>is television. Studies show that girls who watch more television at a
>young age are more likely to view women as better suited to home life,
>happier with children, etc. than girls who watched less television at a
>young age. There is no reason not to believe that analogous things hold
>for boys and thus, from an early age, often without their awareness,
>boys and girls are channeled by society into having different sets of
>interests and tendencies. As they grow older, the process continues and
>boys are more harangued about not making a big show of emotions than
>girls are( among other things ), and in adolescence, acting within
>traditional gender-role stereotypes can be compunded by social
>peer pressure to affirm oneself as a member of one's own sex and also as
>a heterosexual( if you behave too much 'like a member of the opposite
>sex', you might be a homosexual, so the reasoning goes ). By the time
>adulthood is reached, one is usually well set in thinking patterns and
>the differences remain, and on to the next generation...
>The mathematical/scientific fields of study are distinguished from
>others in that they transcend culture, society, politics, and gender.
>They have an objective existence all over the world in all time periods,
>historical events, political, revoltuions, fads, and for both men and
>women.
>Anyone who has what it takes can investigate mathematics, and that
>ability should be evaluated in a gender-blind way, purely
>meritocratically, without previous reservations or inclinations. If this
>is done, then let's see what happens, and see what difference it makes.
>---- David Harden
>
===
Subject: Re: the MVT -- - -- applications
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision:
1.9 primary) id i3LCvmQ14184;
>Hey! I'm learning about the MVT, and am stuggling with being able to apply
it to solve these problems.
>Question 1:
>Prove that tan x > x for 0 < x < pi/2.
>I think there must be an f'(c) = (tan(pi/2) - tan 0)/(pi/2)
>however, since tan(pi/2) is undefined i am having difficulty trying to
answer the question.
But you want to prove that tan x> x for ALL x in the interval so a value
of the derivative of tan x at some specific c doesn't help. Since tan x>
x is the same as tan x- x is NOT 0, it might be better to turn it
around: Certainly tan x- x= 0 for x= 0. IF tan x- x=0 for some x< pi/2,
them, by MVT, f'(c)= 0 (here f(x)= tan x- x). Can you show that that is NOT
true?
>Question 2:
>By using the MVT with f(x)=x^.5 on [25,26], show that
>26^.5 - 25^.5 < 1
>f'(c) = 26^.5 - 25^.5
>and f'(x) = 1/(2(x^.5)), so f'(c) = 1/(2(x^.5)).
>but how do I prove the answer?
You know that 26^(.5)- 25^(.5)= 1/(2(c^.5)) for some c between 5 and 6.
Okay, how large can that right hand side be? (Hint: since x to a negative
power is a DECREASING function, the maximum is at the left end point!)
===
Subject: Re: Existing of a triangle
>Let ABCD is a rectangle and X is an inner point for ABCD.
>Is there a triangle, the faces of which are amongst XA, XB, XC and XD?
Any three non-collinear points form a triangle.
If X is in the interoir, it is clearly not collinear with any two adjacent
vertices of the rectangle. So you have triangles XAB, XBC, XCD, and XDA.
--Keith Lewis klewis {at} mitre.org
The above may not (yet) represent the opinions of my employer.
===
Subject: Re: Existing of a triangle
I believe that the OP was referring to the trigonal inequality,
not the tiling of a rectangle!
> If X is in the interoir, it is clearly not collinear with any two
adjacent
> vertices of the rectangle. So you have triangles XAB, XBC, XCD, and XDA.
--ils duces d'Enron!
http://tarpley.net
===
Subject: Re: Existing of a triangle
>>Let ABCD is a rectangle and X is an inner point for ABCD.
>>Is there a triangle, the faces of which are amongst XA, XB, XC and XD?
>Any three non-collinear points form a triangle.
>If X is in the interoir, it is clearly not collinear with any two adjacent
>vertices of the rectangle. So you have triangles XAB, XBC, XCD, and XDA.
But the edges of e.g. XAB include AB, which is not in the set of four
candidates, so this doesn't answer the question.
Three edges form a triangle if the longest is shorter than the sum of
the other two. Well, we can draw coordinates around the rectangle
ABCD so that the vertices have x-coordinates 0 or a and y-coordinates
0 or b, and so that X=(x,y) with 0 < x < a/2, 0 < y < b/2. In that
case the longest edge has length sqrt( (a-x)^2 + (b-y)^2 ) and the
next two have lengths sqrt( x^2 + (b-y)^2 ) and sqrt( (a-x)^2 + y^2 );
it's easy to check that the sum of these is less than the longest length.
So, yes, the three longest lengths always form a triangle.
dave
===
Subject: Re: Existing of a triangle
>Let ABCD is a rectangle and X is an inner point for ABCD.
>Is there a triangle, the faces of which are amongst XA, XB, XC and XD?
>>Any three non-collinear points form a triangle.
>>If X is in the interoir, it is clearly not collinear with any two
adjacent
>>vertices of the rectangle. So you have triangles XAB, XBC, XCD, and XDA.
> But the edges of e.g. XAB include AB, which is not in the set of four
> candidates, so this doesn't answer the question.
> Three edges form a triangle if the longest is shorter than the sum of
> the other two. Well, we can draw coordinates around the rectangle
> ABCD so that the vertices have x-coordinates 0 or a and y-coordinates
> 0 or b, and so that X=(x,y) with 0 < x < a/2, 0 < y < b/2. In that
> case the longest edge has length sqrt( (a-x)^2 + (b-y)^2 ) and the
> next two have lengths sqrt( x^2 + (b-y)^2 ) and sqrt( (a-x)^2 + y^2 );
> it's easy to check that the sum of these is less than the longest length.
> So, yes, the three longest lengths always form a triangle.
> dave
Hmm. I discarded that as being a too-trivial interpretation.
Perhaps the OP intended that the triangle be made by translating
the edges, keeping their orientations. In that case, my intuition
is that the answer is no, but I don't have a proof.
Rick
EC-FUNDED RESEARCH VISIT OPPORTUNITIES IN HIGH PERFORMANCE COMPUTING
www.hpc-europa.org
HPC-Europa, a consortium of 11 leading European centres, is calling
for applications from researchers working in Europe to visit any of
the 6 centres in its Transnational Access programme. Visitors will
use HPC-Europa's High Performance Computing (HPC) facilities to
advance their research, while working in collaboration with a
scientific researcher from an appropriate local research institute.
The programme is fully funded by the EC's Structuring the European
Research Area programme, and offers:
- access to some of the most powerful High Performance Computing
(HPC) facilities in Europe;
- HPC consultancy from experienced staff;
- opportunities to collaborate with scientists working in related
fields at a local research institute;
- travel costs, subsistence expenses and accommodation (may
be in a shared flat);
The 6 HPC-Europa centres offering Transnational Access are:
CEPBA-CESCA-IBM (Spain), CINECA (Italy), EPCC (UK), HLRS (Germany),
IDRIS (France) and SARA (the Netherlands).
APPLICATION AND ELIGIBILITY
Applications for the programme are welcomed from academic or
industrial researchers at any level, from senior postgraduate students
to the most senior researchers, who are working on non-proprietary
research in any discipline to which access to HPC facilities is
beneficial.
Applicants must be working in a research group based in an eligible
country, i.e. any EU member state or Associated State (see full list
at http://www.hpc-europa.org/presentation.html). Note that researchers
are NOT eligible for visits to centres in the country where they are
currently carrying out their research. Researchers who do not have
access to similar HPC facilities are particularly encouraged to apply.
The selection panel approves applications on the basis of scientific
merit. For this reason, applicants are strongly encouraged, where
possible, to identify a suitable researcher with whom to collaborate
before applying, as this greatly increases the chances of the
application being approved.
Applications should be made using the on-line application form at
selection meeting in May. Successful applicants will visit for between
meetings will be held 4 times per year.
Further information and the on-line application form are available at:
http://www.hpc-europa.org/ If you have any questions, please contact
the HPC-EUROPA team at access@hpc-europa.org
===
Subject: Re: Commutative rings aren't physical spaces (This Week's Finds
205)
> Take for example supposedly curved
> space as in general relativity. Whitehead showed you could do
> relativity in flat space.
Did he? Not in what is usually known as Whitehead's theory of gravity,
because spacetime is not flat in it, just differently curved than in
Einstein's theory.
--
Esa Peuha
student of mathematics at the University of Helsinki
http://www.helsinki.fi/~peuha/
===
Subject: Re: the MVT -- - -- applications
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision:
1.9 primary) id i3LF4tt03787;
Hey All,
Here's kind of a fun MVT application:
Prove that for integers n >1 that there is always a square between n^3 and
(n+1)^3.
Mark
>Q1.
>Pick x so that 0 < x < pi/2. (This eliminates the need for tan(pi/2).)
>The MVT in this instance states
> (tan x - tan 0)/(x - 0) = (sec^2)(c)
>where c satisfies 0 < c < x. Now (sec^2)(c) >= 1 . . . .
>Hope this helps.
>__________________________________________________________
>> Hey! I'm learning about the MVT, and am stuggling with being able to
>apply it to solve these problems.
>> Question 1:
>> Prove that tan x > x for 0 < x < pi/2.
>> I think there must be an f'(c) = (tan(pi/2) - tan 0)/(pi/2)
>> however, since tan(pi/2) is undefined i am having difficulty trying to
>answer the question.
>> Question 2:
>> By using the MVT with f(x)=x^.5 on [25,26], show that
>> 26^.5 - 25^.5 < 1
>> f'(c) = 26^.5 - 25^.5
>> and f'(x) = 1/(2(x^.5)), so f'(c) = 1/(2(x^.5)).
>> but how do I prove the answer?
===
Subject: Re: Mathematical Language Equations
> Here is a language equation puzzle where the solutions are
mathematical.
> [So I will cross-post this to sci.math. ]
> ...
1) 160 T of 187.
> this one is a bit hard to fathom. is there an equation here?
2) S of D of 70 is 144.
> Sum Divisors
3) The sixth H N is 49/20.
> Harmonic Number
4) 12th F N is 12th positive S.
> Fibonacci Number Square
5) F of N L of 10000 is 9.
> Floor Natural Logarithm
6) 8 C 3 is 56.
> Choose
7) 10th T N is 10th F N.
> Triangular Number Fibonacci Number
8) S of 1st 5 positive S is 10th F N too.
> Sum Squares Fibonacci Number
9) (m+2) C 3 P (m+2) C 2 is S of 1st m positive S.
> Choose Plus Choose Sum Squares
> S's aren't always equal in an equation: S=Sum or Square.
10) For m >= 2, S of T of m is m T P m D B 2.
> ???
>
> 11) The N of P L T 50 is 15.
> Number Primes Less Than
12) The P of 1st 4 H N is 275/48.
> Product Harmonic Numbers
13) D I B 0 and 1 of -ln(1-x)/x is P S D B 6.
> Definite Integral Between Pi Squared Divided By
14) L, as m A I, of S of 1st m positive C D B m R B 4 is 1/4.
> Limit Approaches Infinity Sum Cubes Divided By Raised By
15) (2m) F D B m F S is A to 4 R B m D B S R of P.
> Factorial Divided By Factorial Squared Asymptotic Raised By
> Divided By Square Root Pi
You got all of your answers correct!...
including (15), which I stated wrong.
(15) should have been:
(2m) F D B m F S is A to 4 R B m D B S R of (m P).
(because I forgot about dividing by sqrt(m) too.)
As for (1) and (10), the T in (1) is the same the first T in (10).
In (1), perhaps it would have been less confusing if I had written:
1) There are 160 T of 187.
Leroy Quet
===
Subject: Re: Mathematical Language Equations
Cc: hiraga@slis.tsukuba.ac.jp
> As for (1) and (10), the T in (1) is the same the first T in (10).
> In (1), perhaps it would have been less confusing if I had written:
> 1) There are 160 T of 187.
Totatives
> 10) For m >= 2, S of T of m is m T P m D B 2.
Sum of Totatives of m is m Times Phi(m) Divided By 2.
# Perhaps P m should have been T of m
Confession 1:
Though I got the answer quick (observing 187 = 11*17 and 160 =
(11-1)*(17-1)),
I didn't know the word totative and had to look it up.
Confession 2:
Couldn't get as much answers as Mitch Harris did.
- Yuzuru Hiraga
===
Subject: Finite set containment question
S is a set. {S_n} is a sequence of subsets of S such that S_n's are
increasing and U (n=1-->oo)S_n = S. If A is a proper subset of S, is
any help in proving or disproving.
===
Subject: Re: Finite set containment question
Adjunct Assistant Professor at the University of Montana.
>S is a set. {S_n} is a sequence of subsets of S such that S_n's are
>increasing and U (n=1-->oo)S_n = S. If A is a proper subset of S, is
>any help in proving or disproving.
If either S or A are finite, then yes. (In either case, A is finite;
for each x in A there exists n(x) such that x is in S_n(x); let M =
max{n(x): x in A}).
But if A is infinite, then the answer is there need not be. Let
{x_1,...,x_n,...} be an denumerable subset of A, and let
S_n = S - {x_(n+1),x_{(n+2)},...}
--
It's not denial. I'm just very selective about
what I accept as reality.
--- Calvin (Calvin and Hobbes)
Arturo Magidin
magidin@math.berkeley.edu
===
Subject: Re: Finite set containment question
> S is a set. {S_n} is a sequence of subsets of S such that S_n's are
> increasing and U (n=1-->oo)S_n = S. If A is a proper subset of S, is
> any help in proving or disproving.
Consider S=N (the natural numbers), S_n={0..n} and A={2k | k in N}
Marc
===
Subject: Calculating tide times and heights
Node Factors and Equilibrium Arguments, together with Harmonic
Constituents are used in the calculation of tides. The standard source
for such data (XTide) tabulates the node factors and EA's for each
constituent for a range of years 1970 - 2038. Can anyone advise how to
CALCULATE node factors and EA's - then in my tide prediction s/w I
won't have to look up the values, and won't be restricted to the
tabulated years.
Chris Allonby.
===
Subject: 2 Part Recursion Puzzle
Here is a 2 part (implicit) recursion puzzle. Solve the first puzzle,
then use the answer to solve the second puzzle.
(Partial credit for getting only the first sequence.)
:)
{a(m,k)} is an infinite array defined as follows:
Each a(m,m) = m.
For every positive integer m and n,
sum{k=1 to m} floor[a(m,k) *n /m]
=
sum{k=1 to n} a(k,m).
(Note that order of m and k in the a()'s has been exchanged and
depends on which side of the equation the a() is on.)
What is the closed-form for {a(k,m)}?
--
b(1) = 1;
b(m) = sum{k=1 to m} a(k,m) *b(a(k,m))
for all positive integers m.
What is {b(m)}?
My opinion is the answers to this puzzle are surprising and not too
complicated.
Leroy
Quet
===
Subject: Re: Mass versus Weight
> The ratio of the gravitational weight [w], or heavieness of any
> object;
Hey Dumb Donny Head, you ing STILL don't know the difference
among gravitational, inertial, active, and passive mass. You are
dysfunctional intellectual dog , Dumb Donny Head.
--
Uncle Al
http://www.mazepath.com/uncleal/
(Toxic URL! Unsafe for children and most mammals)
Quis custodiet ipsos custodes? The Net!
===
Subject: Re: Mass versus Weight
> The ratio of the gravitational weight [w], or heavieness of any
> object; body, or mass of matter, divided by the acceleration [g] at
> which it will free fall is its mass: For any given body: Its mass m =
> w/g = f/a; anywhere; anytime!
Please notice that since Newton made up the quantity of matter as
consisting of its bulk and density, and said that it was what he meant
when he referred to bodies, and masses of matter; that the two
definitions are compatable:
That is for any given volume, the weight-density of matter varies in
proportion to the weight-density of water at its maximum density here
in a laboratory on Earth; where the weight-density of water has been
arbitrarily established as the standard.
===
Subject: Checking a purported Cailey table
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision:
1.9 primary) id i3LFrxn13260;
Is there an O(n^2) algorithm to verify that an n by n table
is the multiplication table of a group ?
In the general case, verifying associativity would require
O(n^3) time.
However, in a group, we are in a particular case. Identity
and inverses can be verified in O(n^2), and we can do a few
more things in O(n^2). For example, we can check that the
table is a Latin square (this corresponds to the
simplification rule, whose proof depends on associativity),
or we can check things like a*(a^(-1)*b) = b, whose validity
also requires associativity. Maybe those tests give us
partial information about associativity, and we could make
use of that information ?
I wonder if it would be possible to combine a set of
necessary conditions, each verifiable in O(n^2), to get a
sufficient condition.
J. Willekens
===
Subject: Re: Checking a purported Cailey table
> Is there an O(n^2) algorithm to verify that an n by n table
> is the multiplication table of a group ?
Snip
There is no general method. I have a procedure in [1] that can
identify any Moufang Loop (groups plus octonions), with up to 73
elements, presented as a Cayley table. It checks for Frobenius
conservative multiplication in O(n^2) (I think - I have not checked
this), and uses a generalised signature to identify most tables. A
battery of procedures is needed for difficult groups. In the worst
cases the presence of a specific subgroup or quotientgroup must be
tested. I do not know if GAP includes an identification procedure for
Cayley tables.
Roger Beresford.
[1] http://wolfram.com.Infocenter/MathSource/4894
===
Subject: Re: schools for topology
> Can anyone tell me which schools in southern california is good in
> topology/algebraic topology?
I cannot attest to how good the schools are in these topics, but you can
look at (I assume by Southern Cal you mean the LA area as opposed to San
Diego). These are in no order of precedance.
1.
http://math.usc.edu/programs/currentGradDegrees.html#Ph.D.%20in%20Mathematic
s
2. http://www.math.ucla.edu/grad_programs/faculty/research_areas/top.html
3. http://www.math.caltech.edu/people/profs.html
I am sure there are others at other UC and CS schools.
It might also help to peruse this listing as well for many of the great
schools from san fran down to san diego are represented.
http://www.gradschools.com/listings/cal/mathematics_cal.html
HTH, Flip
===
Subject: Re: Antidiagonal, Infinity
> There are no cases to consider here. The hypothesis says we are
given
> a mapping f: N -> R. That means the elements of the list are real
> numbers, not digit strings. We may choose to represent the numbers in
> that list in any way we find convenient.
Ok, and if someone chooses a list in such a way that the diagonal
number is the same as the anti-diagonal number then we have a seperate
case, that only comes up when using the binary representation of the
reals.
> There seems to be a widespread misconception that the diagonal argument
> is not complete until we have explained how to apply it in every
> conceivable base. That's tantamount to suggesting that the argument is
> not complete until we have translated it into every known language,
> including Sanskrit and Klingon. Just one correct proof is quite
> sufficient. In fact, if your 12-year-olds have trouble grasping this
> point, I submit that it is more important that the diagonal proof itself,
> since it gets to the heart of the matter of what constitutes a valid
> proof.
I'd go furthers, and say there is a widespread misconception that
Cantor's Arguement is false! And what if Klingon's only count in
binary? How do you propose to convince them of the validity of the
proof? Again I'm simply looking for more tools to explain the proof in
any base given any example. I realize mathamaticians go for the once
it's solved, it's solved sort of reasoning but that isn't my goal, I
simply want to understand more about the nature of the proof so I can
explain it more convincingly to doubters. If they hand me a special
case like the binary dual representation example, I want to be able to
contruct a counter example on my own that they're more likely to agree
to.
> If you really think your five-paragraph explanation is more
> understandable to 12-year-olds than any of my three one-sentence
> descriptions, then have at it.
Well my 5 paragraphs of arugmentation isn't ment to convince th middle
school student, it's ment to explain it to you. If I was explaining it
to a 12 year old, I would have drawn the list, showing from there how
to select a number from P to fill each hole in the number Q. I can
use the pair wise swapping to convince most 12 year olds, but at this
point my main concern is, did I manage to construct a number not on
the list? Is my reasoning valid here? Is the contruction I gave
sufficent to demonstrate the uncountability of the reals?
===
Subject: Re: Antidiagonal, Infinity
>> There are no cases to consider here. The hypothesis says we are
given
>> a mapping f: N -> R. That means the elements of the list are real
>> numbers, not digit strings. We may choose to represent the numbers in
>> that list in any way we find convenient.
Choosing a list means specifying a mapping f: N -> R. Notice that
choosing a list does not include specifying a base. It is possible to
specify real numbers without using a base at all, for example as e-1-1 or
as sup{x-1: x > 0 and x*x < 1+1}. More importantly, it is not part of
the hypothesis that the value of f(n) is specified at all for any n;
all that is required is that f(n) exist and be a real number for each n.
It's not necessary that I be able to write it down and hand it to you.
The point is, I am not thinking in terms of a list of digit strings in
some implied base when I begin the diagonal argument. That's why there
are no cases to consider. All I need to know is that for each n, f(n)
is a real number and therefore has a decimal representation.
Construction of the table of digit strings is a part of the diagonal
process, not a given. I choose to use decimal and not binary because the
argument works better that way, and it's my argument, so I get to choose.
For each n, f(n) has a corresponding decimal representation. All I need
to know is that this representation exists. Again, I may not be able to
write it down and hand it to you.
> Ok, and if someone chooses a list in such a way that the diagonal
> number is the same as the anti-diagonal number then we have a seperate
> case, that only comes up when using the binary representation of the
> reals.
I have been avoiding using terms such as diagonal number and
anti-diagonal number in my discussion throughout, since I consider them
misleading and confusing.
Perhaps you mean that the diagonal number is the number obtained directly
from the diagonal, without changing any digits, and the anti-diagonal
number is the number obtained from the diagonal number in some systematic
way, such as changing each each 1 to a 2 and each non-1 to a 1. Please
correct me if this is not what you mean by those terms.
Thus the diagonal depends on the original function f: N -> R and the base
we choose to represent the numbers. The anti-diagonal is derived from
the diagonal by changing each digit according to a specific rule of our
choosing.
If I have correctly interpreted your terminology, then I have three
objections to your premise stated here. First, if I am using a properly
constructed diagonal rule, it is impossible for anyone to choose a list
in such a way that the anti-diagonal number is a dual-representation
number. That's because every dual-representation number in base 10 ends
in all 0's or all 9's, and we have specifically avoided those digits in
constructing the anti-diagonal.
Second, the validity of the argument does not depend on anyone choosing
anything. In order for the argument to be valid, it must be the case
that the anti-diagonal is not a member of the range of f for any f: N ->
R. To be sure of that, we need to know that the anti-diagonal is not a
dual-representation number for any f.
Third, the failure is not caused by the diagonal being the same as the
anti-diagonal. Failure is caused by the anti-diagonal being in the range
of f, and the diagonal need not be in the range of f.
>> There seems to be a widespread misconception that the diagonal argument
>> is not complete until we have explained how to apply it in every
>> conceivable base. That's tantamount to suggesting that the argument is
>> not complete until we have translated it into every known language,
>> including Sanskrit and Klingon. Just one correct proof is quite
>> sufficient. In fact, if your 12-year-olds have trouble grasping this
>> point, I submit that it is more important that the diagonal proof
itself,
>> since it gets to the heart of the matter of what constitutes a valid
>> proof.
> I'd go furthers, and say there is a widespread misconception that
> Cantor's Arguement is false! And what if Klingon's only count in
> binary? How do you propose to convince them of the validity of the
> proof? Again I'm simply looking for more tools to explain the proof in
> any base given any example. I realize mathamaticians go for the once
> it's solved, it's solved sort of reasoning but that isn't my goal, I
> simply want to understand more about the nature of the proof so I can
> explain it more convincingly to doubters. If they hand me a special
> case like the binary dual representation example, I want to be able to
> contruct a counter example on my own that they're more likely to agree
> to.
There are plenty of proofs of the uncountability of the reals that don't
depend on decimal representation or any other base. All that's needed is
the definition of the reals, and in particular the completeness property.
I suspect Klingon mathematicians would understand such arguments.
Even if Klingons count in binary, it shouldn't be hard to explain the
concept of decimal representation to a Klingon mathematician.
Fortunately, that's not a problem you will have in explaining the proof
to 12-year-old earthlings.
>> If you really think your five-paragraph explanation is more
>> understandable to 12-year-olds than any of my three one-sentence
>> descriptions, then have at it.
> Well my 5 paragraphs of arugmentation isn't ment to convince th middle
> school student, it's ment to explain it to you. If I was explaining it
> to a 12 year old, I would have drawn the list, showing from there how
> to select a number from P to fill each hole in the number Q. I can
> use the pair wise swapping to convince most 12 year olds, but at this
> point my main concern is, did I manage to construct a number not on
> the list? Is my reasoning valid here? Is the contruction I gave
> sufficent to demonstrate the uncountability of the reals?
I have not gone through all the details of your construction, but I
notice that someone else in this thread has already posted a
counterexample to your claim. The basic idea is that you don't know what
P and Q might be when you are constructing your rule.
--
Dave Seaman
Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling.
===
Subject: Re: Antidiagonal, Infinity
>> There are no cases to consider here. The hypothesis says we are
given
>> a mapping f: N -> R. That means the elements of the list are real
>> numbers, not digit strings. We may choose to represent the numbers in
>> that list in any way we find convenient.
> Choosing a list means specifying a mapping f: N -> R. Notice that
> choosing a list does not include specifying a base. It is possible to
> specify real numbers without using a base at all, for example as e-1-1 or
> as sup{x-1: x > 0 and x*x < 1+1}. More importantly, it is not part of
> the hypothesis that the value of f(n) is specified at all for any n;
> all that is required is that f(n) exist and be a real number for each n.
> It's not necessary that I be able to write it down and hand it to you.
> The point is, I am not thinking in terms of a list of digit strings in
> some implied base when I begin the diagonal argument. That's why there
> are no cases to consider. All I need to know is that for each n,
f(n)
> is a real number and therefore has a decimal representation.
> Construction of the table of digit strings is a part of the diagonal
> process, not a given. I choose to use decimal and not binary because the
> argument works better that way, and it's my argument, so I get to choose.
> For each n, f(n) has a corresponding decimal representation. All I need
> to know is that this representation exists. Again, I may not be able to
> write it down and hand it to you.
> Ok, and if someone chooses a list in such a way that the diagonal
> number is the same as the anti-diagonal number then we have a seperate
> case, that only comes up when using the binary representation of the
> reals.
> I have been avoiding using terms such as diagonal number and
> anti-diagonal number in my discussion throughout, since I consider
them
> misleading and confusing.
> Perhaps you mean that the diagonal number is the number obtained directly
> from the diagonal, without changing any digits, and the anti-diagonal
> number is the number obtained from the diagonal number in some systematic
> way, such as changing each each 1 to a 2 and each non-1 to a 1. Please
> correct me if this is not what you mean by those terms.
> Thus the diagonal depends on the original function f: N -> R and the base
> we choose to represent the numbers. The anti-diagonal is derived from
> the diagonal by changing each digit according to a specific rule of our
> choosing.
> If I have correctly interpreted your terminology, then I have three
> objections to your premise stated here. First, if I am using a properly
> constructed diagonal rule, it is impossible for anyone to choose a list
> in such a way that the anti-diagonal number is a dual-representation
> number. That's because every dual-representation number in base 10 ends
> in all 0's or all 9's, and we have specifically avoided those digits in
> constructing the anti-diagonal.
> Second, the validity of the argument does not depend on anyone choosing
> anything. In order for the argument to be valid, it must be the case
> that the anti-diagonal is not a member of the range of f for any f: N - R.
To be sure of that, we need to know that the anti-diagonal is not a
> dual-representation number for any f.
> Third, the failure is not caused by the diagonal being the same as the
> anti-diagonal. Failure is caused by the anti-diagonal being in the range
> of f, and the diagonal need not be in the range of f.
>> There seems to be a widespread misconception that the diagonal
argument
>> is not complete until we have explained how to apply it in every
>> conceivable base. That's tantamount to suggesting that the argument
is
>> not complete until we have translated it into every known language,
>> including Sanskrit and Klingon. Just one correct proof is quite
>> sufficient. In fact, if your 12-year-olds have trouble grasping this
>> point, I submit that it is more important that the diagonal proof
itself,
>> since it gets to the heart of the matter of what constitutes a valid
>> proof.
> I'd go furthers, and say there is a widespread misconception that
> Cantor's Arguement is false! And what if Klingon's only count in
> binary? How do you propose to convince them of the validity of the
> proof? Again I'm simply looking for more tools to explain the proof in
> any base given any example. I realize mathamaticians go for the once
> it's solved, it's solved sort of reasoning but that isn't my goal, I
> simply want to understand more about the nature of the proof so I can
> explain it more convincingly to doubters. If they hand me a special
> case like the binary dual representation example, I want to be able to
> contruct a counter example on my own that they're more likely to agree
> to.
> There are plenty of proofs of the uncountability of the reals that don't
> depend on decimal representation or any other base. All that's needed is
> the definition of the reals, and in particular the completeness property.
> I suspect Klingon mathematicians would understand such arguments.
> Even if Klingons count in binary, it shouldn't be hard to explain the
> concept of decimal representation to a Klingon mathematician.
> Fortunately, that's not a problem you will have in explaining the proof
> to 12-year-old earthlings.
>> If you really think your five-paragraph explanation is more
>> understandable to 12-year-olds than any of my three one-sentence
>> descriptions, then have at it.
> Well my 5 paragraphs of arugmentation isn't ment to convince th middle
> school student, it's ment to explain it to you. If I was explaining it
> to a 12 year old, I would have drawn the list, showing from there how
> to select a number from P to fill each hole in the number Q. I can
> use the pair wise swapping to convince most 12 year olds, but at this
> point my main concern is, did I manage to construct a number not on
> the list? Is my reasoning valid here? Is the contruction I gave
> sufficent to demonstrate the uncountability of the reals?
> I have not gone through all the details of your construction, but I
> notice that someone else in this thread has already posted a
> counterexample to your claim. The basic idea is that you don't know what
> P and Q might be when you are constructing your rule.
Even if someone insists on using binary, use pairs of digits, instead of
single digits, to make it effectively base 4, or use triples of digits
for base 8, and so on.
In any base greater than 3, it works.
===
Subject: Re: Antidiagonal, Infinity
> I'd go furthers, and say there is a widespread misconception that
> Cantor's Arguement is false!
It's only widespread on this newsgroup.
===
Subject: Re: Geometric interpretation of the trace of a matrix?
>> If you start with unitary eigenvectors of the transformation, as sides
>> of a paralelepiped, the trace is certainly what I said in the last
>> post, i.e., the sum of the edges of the image paralelepiped divided by
>> 2^{n-1}.
>Almost correct.
Your willingness to state that MJimenez's quoted statement is
almost correct entails that you have an idea of what it is
supposed to mean. I would be grateful for any effort you'd
be willing to put into explaining it to me (and explaining
why it's almost correct); apparently I'm missing something
very badly, because I cannot make any sense of it at all.
>You have to be careful with the signs of the eigenvalues.
>A modification like: if the transformation reverses top and bottom
>(or left and right etc), then the length of the corresponding edge
>has to be counted with a minus sign.
>When your transformation has complex eigenvalues, you are more or less
>out of luck (ok, you may be able to do something with rotations)
>If you make the further assumption that the transformation is symmetric,
>then you can choose the parallelpiped to be rectangular. All the
eigenvalues
>of a symmetric transformation are real, so in that case a geometric
interpretation
>is possible.
I am totally lost.
Lee Rudolph
===
Subject: Re: Periodic function
>A function like cos x^2 produces a cosine wave that compresses itself in
>both directions away from x=0. What function would produce regularly
>periodic compressions and rarefactions?
Since no one else appears to be pointing out what you might think is
obviously, I'll do it: any periodic function is a sum of simple
sines and cosines, that is, it may be expressed as
sum ( a_n sin( n x ) + b_n cos( n x ) )
where, unlike other posters, I am not including anything more complicated
than multiples of x as arguments to sine and cosine. You choose the
coefficients a_n and b_n to make the curve have the desired shape
on one period, and after that, periodic behaviour is guaranteed.
You won't get the infinite wiggling that some people seem to like but
you can match something like cos(x^2) pretty well. For example, an
11-term
sum gives a good approximation which clearly exhibits the periodic
bunching:
f(x) = 0.1874450324 + 0.5095889344 cos(x) + 0.5262043454 cos(2 x)
+ 0.08496928957 cos(3 x) - 0.5839140525 cos(4 x) + 0.3871804531 cos(5
x)
- 0.1330855634 cos(6 x) + 0.01110860988 cos(7 x) + 0.03019173362 cos(8
x)
- 0.04150460650 cos(9 x) + 0.02181582400 cos(10 x)
dave
===
Subject: Re: Periodic function
Just a clarification. Referring to Ignacio's previous relation on
2002-12-15,is
f(x+Pi/2)=(1+f(x))/(1-f(x)) right for all 2 Pi period functions?
Functions of sin,cos do not fit in here ..
===
Subject: Re: Periodic function
> Just a clarification. Referring to Ignacio's previous relation on
> 2002-12-15,
Are you referring to a post by ILC in this newsgroup on 2002-12-15?
I cannot find any such post. It would have been nice if you had provided
a link.
> is f(x+Pi/2)=(1+f(x))/(1-f(x)) right for all 2 Pi period
> functions? Functions of sin,cos do not fit in here ..
Of course it's not right. Why did you bother to ask?
DWC
===
Subject: Re: Periodic function
>> Just a clarification. Referring to Ignacio's previous relation on
>> 2002-12-15,
> Are you referring to a post by ILC in this newsgroup on 2002-12-15?
> I cannot find any such post. It would have been nice if you had provided
> a link.
I think that it is
or some other post from the same thread.
Jose Carlos Santos
===
Subject: Re: Periodic function
>> Just a clarification. Referring to Ignacio's previous relation on
>> 2002-12-15,
> Are you referring to a post by ILC in this newsgroup on 2002-12-15?
> I cannot find any such post. It would have been nice if you had
> provided a link.
> I think that it is
> @ID-137122.news.dfncis.de
> or some other post from the same thread.
> Jose Carlos Santos
Doh! I've learned something today. Google has put this new thread onto
the end of that old thread, presumably because they have the same title,
making it all seem to be just one big thread! That's useful information
(and it's probably why GLN didn't think any link would be needed).
Anyway, as to the clarification requested by GLN:
> if for all x
> f(x+a) = (1+f(x))/(1-f(x))
> then function f is periodic.
but of course, the reverse implication does not hold.
David
===
Subject: Re: Periodic function
>> Just a clarification. Referring to Ignacio's previous relation on
>> 2002-12-15,
Are you referring to a post by ILC in this newsgroup on 2002-12-15?
> I cannot find any such post. It would have been nice if you had
> provided a link.
> I think that it is
> @ID-137122.news.dfncis.de
> or some other post from the same thread.
> Jose Carlos Santos
> Doh! I've learned something today. Google has put this new thread onto
> the end of that old thread, presumably because they have the same title,
> making it all seem to be just one big thread! That's useful information
> (and it's probably why GLN didn't think any link would be needed).
Yes,sorry about that confusion.
> Anyway, as to the clarification requested by GLN:
> if for all x
> f(x+a) = (1+f(x))/(1-f(x))
> then function f is periodic.
> but of course, the reverse implication does not hold.
> David
But when tan(Pi/4+x)=(1+tan(x))/(1-tan(x)) by componendo dividendo we
get
tan(x)=tan(Pi/4 + x)-1)/(tan(Pi/4 + x)+1).
You mean to generalize this per ILC original posting, we can't now
===
Subject: transform PDF from plane to sphere
Hi all,
I have a question regarding the tranformation of a probability density
distribution (pdf) taken at a plane to the surface of a sphere.
An exampel : A distribution is uniform (isotropically) distributed on a
x,y-plane
with marginal pdf's
p(x) = 1; p(y) = 1;
On a sphere, the marginal pdf for the isotropic distribiution is
p(theta) = 1/2 sin(theta) according to
mathworld.wolfram.com/SpherePointPicking.html
My question: For a pdf on the plane that has a (truncated) Gaussian as
marginal pdf in x
direction and a uniform distribution in y direction, how do I map such
a distributon onto the sphere, or better on a hemisphere (which should
make it easier) ?
The strips parallel to the equator should be uniform again (y).
I know that the Fisher distribution on the sphere is the analogue to
the Gaussian in the plane, but I have no idea if I can perform a
transformation and how.
My first idea was to do the same as with the isotropical distribution :
perfoming a weighting of the pdf on the plane by sin(theta) to map it
to the sphere. But this is just an idea, I do not have a glue if this is
the right way.
Any hints are very welcome,
Dirk
===
Subject: OT: The Kabatoff Posts
It was very informative. I have killfile'd Mr. Kabatoff to avoid his
endless
off-topic posts on numerology.
Now my only problem is that every Kabatoff post gets a refutation post, so
that even after ponking Mr. Kabatoff, the newsgroups are still cluttered
with Kabatoff-related posts. One such post is certainly helpful, but can't
we now assume that everyone here knows who Daryl Kabatoff is, and let us
get
back to the topic?
===
Subject: Re: The Kabatoff Posts
> It was very informative. I have killfile'd Mr. Kabatoff to avoid his
endless
> off-topic posts on numerology.
> Now my only problem is that every Kabatoff post gets a refutation post,
so
> that even after ponking Mr. Kabatoff, the newsgroups are still cluttered
> with Kabatoff-related posts. One such post is certainly helpful, but
can't
> we now assume that everyone here knows who Daryl Kabatoff is, and let us
get
> back to the topic?
frequently quotes this libel using the name Thomas... and you
appreciate this? Following is some mathematics that you similarly
don't care to see:
243 <- 1 Samuel 7
274 <- 2 Samuel 7
---
517
298 <- 1 Kings 7
320 <- 2 Kings 7
---
618 <-verses in Bible Book 7
345 <- 1 Chronicles 7
374 <- 2 Chronicles 7
---
719
See above that First Kings 7 and Second Kings 7 are chapters 298 and
320,
together for the 618 verses of Bible Book 7. And First Samuel 7 and Second
Samuel 7 averages with First Chronicles 7 and Second Chronicles 7 for the
618 verses of Bible Book 7 (243+274+345+374=618+618). And First Chronicles
7
and Second Chronicles 7 are chapters 345 and 374, together for the 719
verses of Bible Book 12, pretty as 12 is the 7th non-prime while 719 is the
128th or the 2 to the 7th prime. See Books by Bonnie Gaunt for a Gematric
discussion of the number 618.
The first 4 primes plus the first
4 non-primes add together for the
36 chapters of Bible Book 4 Numbers:
Primes Non-Primes
2 1
3 4
5 6
7 <-4th-> 8
-- --
17 19
The first 4 primes in prime
positions add together for the
36 chapters of Bible Book 4:
Primes
In Prime
Primes Positions
1 2
2 3 <- 3
3 5 <- 5
4 7
5 11 <- 11
6 13
7 17 <- 17
--
36
Seven plus the 7th prime plus the
7th non-prime adds together for
the 36 chapters of Bible Book 4,
pretty as 7 is the 4th prime:
Primes Non-Primes
2 1
3 4
5 6
7 8
11 9
13 10
17 <-7th-> 12
-- --
58 50
===
Subject: Re: OT: The Kabatoff Posts
> It was very informative. I have killfile'd Mr. Kabatoff to avoid his
endless
> off-topic posts on numerology.
> Now my only problem is that every Kabatoff post gets a refutation post,
so
> that even after ponking Mr. Kabatoff, the newsgroups are still cluttered
> with Kabatoff-related posts. One such post is certainly helpful, but
can't
> we now assume that everyone here knows who Daryl Kabatoff is, and let us
get
> back to the topic?
Depending on your newsreader, you may be able to tell it to ignore a
thread started by Kabatoff, rather than simply filtering out his post.
--
Will Twentyman
email: wtwentyman at copper dot net
===
Subject: Simple question
A question about sets: Is there any topological difference between Q (the
rationals) and A (or Q bar) (the algebraics)?
Barnaby
===
Subject: Re: Simple question
>A question about sets: Is there any topological difference between Q (the
>rationals) and A (or Q bar) (the algebraics)?
No (as others have said) if you mean the real algebraics. But
algebraics by itself means algebraic complex numbers. In that case
I think it's not so obvious.
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia
Vancouver, BC, Canada V6T 1Z2
===
Subject: Re: Simple question
>>A question about sets: Is there any topological difference between Q (the
>>rationals) and A (or Q bar) (the algebraics)?
>>
>No (as others have said) if you mean the real algebraics. But
>algebraics by itself means algebraic complex numbers. In that case
>I think it's not so obvious.
Correct, it is not as obvious. But we have discussed here before that
Q^2 is homeomorphic to Q. See the thread
--
Stephen J. Herschkorn herschko@rutcor.rutgers.edu
===
Subject: Re: Simple question
> A question about sets: Is there any topological difference between Q (the
> rationals) and A (or Q bar) (the algebraics)?
> Barnaby
If I remember correctly:
In terms of order, if a set is
countable,
linearly ordered (every two members are comparable),
densely ordered (between every two different members there
is at least one, different from the two),
without a maximal and without a minimal member,
then it is order isomorphic to the rationals.
Since the presumed topologies (inherited from the reals)
are generated by open intervals, the sets Q and A must
be homeomorphic.
===
Subject: Re: Simple question
>> A question about sets: Is there any topological difference between Q
(the
>> rationals) and A (or Q bar) (the algebraics)?
>> Barnaby
>If I remember correctly:
>In terms of order, if a set is
> countable,
> linearly ordered (every two members are comparable),
> densely ordered (between every two different members there
> is at least one, different from the two),
> without a maximal and without a minimal member,
>then it is order isomorphic to the rationals.
>Since the presumed topologies (inherited from the reals)
>are generated by open intervals, the sets Q and A must
>be homeomorphic.
QUESTION: Can we find an explicit bijection
f : Q -> A such that f(x) > f(y) precisely when x > y?
--
John Adams served two terms as Vice President and one as President, but
lost
reelection. Later his son became President despite losing the popular
vote.
That son lost his reelection attempt badly. Now history is repeating
itself.
pmontgom@cwi.nl Microsoft Research and CWI Home: San Rafael,
California
===
Subject: Re: Simple question
>A question about sets: Is there any topological difference between Q (the
>rationals) and A (or Q bar) (the algebraics)?
I don't remember the details right now (I'm tired today), but I think
any countable, totally ordered, dense set without a mimimum or maximum
element is order-isomorphic to Q. (Dense means there is an element
between any two distinct elements.) Since the order topology is used,
this fact implies that the answer to your question is negative. I might
be wrong, though.
--
Stephen J. Herschkorn herschko@rutcor.rutgers.edu
===
Subject: Re: Simple question
>> A question about sets: Is there any topological difference between Q
>> (the
>> rationals) and A (or Q bar) (the algebraics)?
> I don't remember the details right now (I'm tired today), but I think
> any countable, totally ordered, dense set without a mimimum or maximum
> element is order-isomorphic to Q. (Dense means there is an element
> between any two distinct elements.) Since the order topology is used,
> this fact implies that the answer to your question is negative. I
> might be wrong, though.
I have napped by now, so I'm thinking about how to prove the first fact.
I know how to show that the rationals can be order-embedded in any
totally ordered dense set with at least two elements. The proof I have
in mind uses choice; can we show this fact without choice?
By the same method, any countable, dense, totally ordered set with at
least two elements can be embedded in another. Does the existence of
embeddings in both directions imply that two such sets are
order-isomorphic? Again, can the first part be done without choice?
If each of two topological spaces can be embedded in the other, are the
two spaces isomorphic?
I'm asking about order and topological versions of Schroeder-Bernstein
here. Is there a general categorical Schroeder-Bernstein theorem?
--
Stephen J. Herschkorn herschko@rutcor.rutgers.edu
===
Subject: Re: Simple question
> I know how to show that the rationals can be order-embedded in any
> totally ordered dense set with at least two elements. The proof I
> have in mind uses choice; can we show this fact without choice?
> By the same method, any countable, dense, totally ordered set with at
> least two elements can be embedded in another. Does the existence of
> embeddings in both directions imply that two such sets are
> order-isomorphic? Again, can the first part be done without choice?
Actually, I have realized a direct proof that any two c.d.t.-o. sets
each with at least two elements are isomorphic. It does not use choice.
I still don't know whether the first fact above requires choice. For
example, although such a set is infinite, do we know without choice that
it is Dedekind infinite?
--
Stephen J. Herschkorn herschko@rutcor.rutgers.edu
===
Subject: Re: Sorry
>If xyz = 1, prove that (xy)^2 + (yz)^2 + (zx)^2 - 2x - 2y - 2z + 3 >= 0.
>>Writing z = 1/(xy), we want to show that
>> 0
>> 2 2 1 1 2
>> <= x y + --- + --- - 2x - 2y - --- + 3
>> x^2 y^2 x y
>> 2 2 1 1 2
>> = x y - 2x - 2y + 3 + ( - - - )
>> x y
>> 1 1 2
>> = f(x,y) + ( - - - )
>> x y
>>Setting the partials of f(x,y) to 0, we get that its minimum occurs at
>>(x,y) = (1,1). Since f(1,1) = 0, the inequality holds.
>>Rob Johnson take out the trash before replying
> The partials of f do vanish at (1, 1), but this is a saddle
>point, not a minimum. For example, x = 0.9 and y = 1.1 give
> f = 0.9801 - 1.8 - 2.2 + 3 = -.0199 < 0
Right you are. My mistake; I only looked at part of the bounding circle
near infinity and missed the small parts near the x and y axes where
f(x,y) is negative. Looking at the rest of the circle, f(x,y) appears
to grow like r^4, which would make (1,1) a minimum, but for those pesky
>Easier for hand computation is x = 0 and y = 2, giving -1.
However, x = 0 makes xyz = 1 kind of hard. x = 1/2 and y = 2 is almost
as easy, also giving -1.
Here is a proper attempt. Let u = xy and v = x+y. Then
0
2 2 1 1 2
<= x y + --- + --- - 2x - 2y - --- + 3
x^2 y^2 x y
2 2 1 1 2
= x y - 2(x+y) + 3 + ( - - - )
x y
2 v^2-4u
= u - 2v + 3 + ------
u^2
1 2 2 4
= --- v - 2v + u + 3 - - [1]
u^2 u
Let us fix u. [1] is a positive quadratic in v whose minimum is at
v = u^2. Plugging this back into [1] yields the minimum for a given u
to be 3 - 4/u. So unless 0 < u < 4/3, the minimum of the parabola is
positive.
Note that v^2-4u = (x-y)^2 >= 0. Thus, for any u, we must have that
v^2 >= 4u. Since the minimum of the parabola is when v = u^2, when
0 < u < 4/3, v^2 = u^4 < 64/27 u < 4u. Therefore, when 0 < u < 4/3,
the minimum of the parabola cannot be reached. Thus, the minimum must
be reached when v^2 = 4u. Plugging this back into [1] yields a minimum
of u^2 - 4sqrt(u) + 3, which has a minumum of 0.
Rob Johnson
take out the trash before replying
===
Subject: Functional system
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision:
1.9 primary) id i3LInLg09157;
Function f(x) defined in the following system:
|f(x + 1) = (sqrt(2))^f(x)
|f(-f(x)) = -x
|f(0) = 1
Does it exist a direct deterministic way (algorithm) which find the lowest
value of f(x) for x in (-2; inf)?
Does it exist a continuous non-recursive function g such that g(x) = the
lowest value of f(x).
Leonel
===
Subject: Re: EINSTEIN'S RELATIVITY THEORY PROBED
einstein did not eat meat for most part of his life.
> There is no present. Learn from the past
> and plan to apply the lessons in the future.
> The future turns into the past continuously.
> Some call this process time.
> - Jai Maharaj
> http://www.mantra.com/jyotish
> Einstein's relativity theory probed
> By Anna Salleh
> ABC Science Online
> Einstein's theory of general relativity, fundamental to
> our understanding of the cosmos, is about to be put to a
> rigorous test, using a satellite due to be launched early
> this week.
===
Subject: Re: Existing of a triangle
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision:
1.9 primary) id i3LIt2810154;
===
Subject: Iteration Theory by Julia/Fatou
Does anyone know of any online refs that contain a good outline of the
results of the Theory of Iteration by Fatou and Julia?
transcendants entieres?
PS: I am particularly interested in theorems that analytically expand the
iterates f^(n)(z) around fixed points.
--
Ioannis Galidakis
http://users.forthnet.gr/ath/jgal/
------------------------------------------
Eventually, _everything_ is understandable
===
Subject: Re: Why is it ln on the calculator?
Originator: jeyadev@kaveri
>> This is one of the examples which can remind you that mathematics
>>is international. We must expect to find abbreviations for words in
>>languages other than English. Another example is the set of all integers
>> {.... -3, -2, -1, 0, 1, 2, 3, 4, ....}
>>whose standard modern name is Z, short for the German word Zahlen
>>(numbers). Then there's the well-known use of the Greek letter pi,
>>which is short for the Greek word for periphery, i.e. the distance
>>around a circle of unit diameter.
>As coined by such well-known Greeks as Oughtred, Jones & Euler... :-)
But many of these chaps did things in Latin and they were not even
Mexican or Argentine or .....
--
Surendar Jeyadev jeyadev@wrc.xerox.bounceback.com
Remove 'bounceback' for email address
===
Subject: Re: Groebner basis calculation
> I have an ideal generated by 75 binomials in 110 variables for which
> I'd like to find a Groebner basis. I have no idea whether or not this
> is a tractable problem. I've left it running in Singular for 2 days on
> a 2.4GHz Pentium with no result so far (though it hasn't run out of
> memory yet). However my problem is a special case with only binomial
> generators so I'm wondering if there is some special purpose code out
> there that can solve this problem. I tried just implementing
> Buchberger's algorithm with some code specialised for binomials but
> that seems far too slow too. Or is this problem just too difficult on
> todays PCs? Maybe it would take years or centuries? Is there something
> faster than Singular out there that can handle 110 variables? What is
> the state of the art in Groebner basis computations?
> I is the ideal, defined over the field Q, is given by all binomials of
> the form
> c_i * A_j - B_i,j
> Where j is in {1,2,...,31}, i is in {0,1,2,3,4}, and 2^i appears in
> the binary expansion of j.
> For example c_0A_3-B_0,3 appears in I because 3=2^0+2^1 but
> c_4A_13-B_4,13 doesn't as 13=2^0+2^2+2^3 which doesn't contain 2^4.
> I'd like to eliminate the c_i so I'd like to use something like the
> 5th elimination order though just plain lexicographic would do as long
> as the c_i are greater than the other monomials.
> I hope I haven't made too many errors in the above. This isn't really
> my field.
I'll make a few remarks. First, you might get a better response if you
post this sort of question to sci.math.symbolic (since Groebner basis
computation methods are among the interests of people who lurk on that
group). I'm cross-posting thereto.
Second, your ideal has a couple of nice qualities (by the way, I get
80 generators, not 75, assuming I interpreted the criterion
correctly). For one, it is toric. For another, the generators already
comprise a Groebner basis with respect to term orders wherein the 'b'
variables are weighted more heavily than all others (that is, term
orders that are elimination orders for those variables). This gives
rise to two possible approaches.
(1) Use a method that is dedicated to handling toric ideals.
(2) Use a Groebner basis conversion method such as the Groebner walk.
I do not know offhand whether either or both of these are available in
Singular but I suspect this is not hard to learn from the
documentation.
To get some idea of difficulty I tried this out in Mathematica. I set
the range of 'j' to be a parameter so as to test smaller related
problems.
1
toricProblem[n_, a_:a, b_:b, c_:c] := Module[
{avars,bvars,cvars, allACproducts, usedACproducts, polys},
cvars = Array[c, n, 0];
avars = Array[a, 2^n-1];
allACproducts = Flatten[Outer[Times, cvars, avars]];
usedACproducts = Cases[allACproducts, a[j_]*c[i_] /;
isInBinaryExpansion[i,j]];
polys = Map[#-(#/.a[j_]*c[i_]->b[i,j])&, usedACproducts];
bvars = Cases[Variables[polys], b[__]];
GroebnerBasis[polys, Join[avars,bvars], cvars,
MonomialOrder->EliminationOrder, Sort->True]
]
Not surprisingly it is easy for n = 3.
In[5]:= Timing[gb3 = toricProblem[3];]
Out[5]= {0.03 Second, Null}
In[7]:= gb3 // InputForm
Out[7]//InputForm=
{a[7]*b[1, 3] - a[3]*b[1, 7], a[7]*b[0, 3] - a[3]*b[0, 7],
a[7]*b[2, 5] - a[5]*b[2, 7], a[7]*b[0, 5] - a[5]*b[0, 7],
-(a[5]*b[0, 3]) + a[3]*b[0, 5], a[7]*b[2, 6] - a[6]*b[2, 7],
-(a[6]*b[2, 5]) + a[5]*b[2, 6], a[7]*b[1, 6] - a[6]*b[1, 7],
-(a[6]*b[1, 3]) + a[3]*b[1, 6], a[7]*b[0, 1] - a[1]*b[0, 7],
a[5]*b[0, 1] - a[1]*b[0, 5], a[3]*b[0, 1] - a[1]*b[0, 3],
a[7]*b[1, 2] - a[2]*b[1, 7], a[6]*b[1, 2] - a[2]*b[1, 6],
a[3]*b[1, 2] - a[2]*b[1, 3], a[7]*b[2, 4] - a[4]*b[2, 7],
a[6]*b[2, 4] - a[4]*b[2, 6], a[5]*b[2, 4] - a[4]*b[2, 5],
-(b[1, 7]*b[2, 6]) + b[1, 6]*b[2, 7], -(b[0, 7]*b[2, 5]) + b[0,
5]*b[2, 7],
-(b[0, 7]*b[1, 3]) + b[0, 3]*b[1, 7], -(a[7]*b[1, 3]*b[2, 6]) +
a[3]*b[1, 6]*b[2, 7], -(a[7]*b[0, 3]*b[2, 5]) + a[3]*b[0, 5]*b[2,
7],
-(a[3]*b[1, 6]*b[2, 5]) + a[5]*b[1, 3]*b[2, 6],
-(b[0, 3]*b[1, 7]*b[2, 5]) + b[0, 5]*b[1, 3]*b[2, 7],
-(b[0, 3]*b[1, 6]*b[2, 5]) + b[0, 5]*b[1, 3]*b[2, 6]}
For n = 4 the computation is far more substantial, but still not too
hard.
In[9]:= Timing[gb4 = toricProblem[4];]
Out[9]= {18.57 Second, Null}
The result is fairly large but again, not out of bounds by any means.
In[10]:= Length[gb4]
Out[10]= 321
In[11]:= LeafCount[gb4]
Out[11]= 5983
Now n = 5 is another matter entirely. I ran out of memory on one
attempt (after several minutes). I am now trying it again on a bigaram
machine. No telling what will happen. Groebner basis computations can
be funny that way.
Daniel Lichtblau
Wolfram Research
===
Subject: Re: Groebner basis calculation
> I have an ideal generated by 75 binomials in 110 variables for which
> I'd like to find a Groebner basis. I have no idea whether or not this
> is a tractable problem.
Simplifying such a system is indeed intractable -in gereral-.
The associated problem of deciding if a polynomial is a member of an
likely has a lower case GB computation that is exponential time (or worse,
and the best algorithms so far take doubly exponential times in the number
of variables.
> I've left it running in Singular for 2 days on
> a 2.4GHz Pentium with no result so far (though it hasn't run out of
> memory yet). However my problem is a special case with only binomial
> generators so I'm wondering if there is some special purpose code out
> there that can solve this problem.
Special cases is a different matter altogether. If your system is
conveniently structured, you may be able to do a lot of elimination
theoretically (use symmetry, systematic elimination, induction)
> I tried just implementing
> Buchberger's algorithm with some code specialised for binomials but
> that seems far too slow too. Or is this problem just too difficult on
> todays PCs? Maybe it would take years or centuries? Is there something
> faster than Singular out there that can handle 110 variables? What is
> the state of the art in Groebner basis computations?
GB computation in general reduces to GB computation with binomials (that
is, binomial GB computation is just as hard as the unrestricted case).
> I is the ideal, defined over the field Q, is given by all binomials of
> the form
> c_i * A_j - B_i,j
> Where j is in {1,2,...,31}, i is in {0,1,2,3,4}, and 2^i appears in
> the binary expansion of j.
> For example c_0A_3-B_0,3 appears in I because 3=2^0+2^1 but
> c_4A_13-B_4,13 doesn't as 13=2^0+2^2+2^3 which doesn't contain 2^4.
> I'd like to eliminate the c_i so I'd like to use something like the
> 5th elimination order though just plain lexicographic would do as long
> as the c_i are greater than the other monomials.
Because of the digit thing, there is a lot of structure here,
so there might be hope.
Can you do a much much smaller system by program and by hand
(to check the program)? like j in {1,2,3}, i in {0,1}?
--
Mitch Harris
(remove q to reply)
===
Subject: Re: Frequentist probability confusion
Right; the problem would be solved if our
number system had genuine
>>infinitesimals. Then each integer could be assigned a probability
>>1/aleph_0 and everything would work out ok.
>Wrong. In non-standard analysis, there exist
>infinitesimals, but all of them are much smaller than
>anything looking like that. All non-standard positive
>integers have at least as many smaller integers as there
>are ordinary real numbers. But they behave like finite
>integers within the model.
I was thinking of the surreals rather than the hyperreals;
I believe they have well-defined multiplicative inverses for
every non-zero number (including aleph_0) along with the same
rules of distributivity as real numbers. I'm don't know
very much about this, but wouldn't that be sufficient?
> We could
>>even imagine a formal definition of probability distributions as
>>equivalence classes of algorithms for producing numbers from random
>>seeds. Then we could honestly claim to be talking about processes
>>generating numbers when we do probability.
>This is already the case in probability as we have it
>now, but with random seeds being real numbers uniform
>between 0 and 1.
If that were true there would be a distribution for my process which
generates random integers, but there isn't.
>> If one allows finitely
>> additive probability measures, then one can defined a probability
measure
>> over the natural numbers that some might want to label uniform.
However,
>> the construction of this measure uses the the axiom of choice (or at
least
>> some large portion of the axiom of choice).
>>Do you have a reference for this?
>One does not need much of the axiom of choice, but some
>is needed. If one only wants some of the sets to be
>measurable, nothing is needed; consider the field of
>sets which are periodic from some point on, and give
>it the limiting frequency. But what are you going to
>do with it?
Just some of the sets being measurable isn't satisfying, but
R.
===
Subject: Help, please: an equation on complex exponentional numbers
I really don't know how to proceed with this:
Let p>2 be an arbitrary prime number, prove that for all k=1,2,...,p
there exist convenient integers a_k and b_k such that for k:=1 to
k:=p:
{SUM of a_k*[e^(i 2k*PI/p)]}^2 + {SUM of b_k*[e^(i 2k*PI/p)]}^2 + 1 =
0
Danny.
===
Subject: Re: Help, please: an equation on complex exponentional numbers
>I really don't know how to proceed with this:
>Let p>2 be an arbitrary prime number, prove that for all k=1,2,...,p
>there exist convenient integers a_k and b_k such that for k:=1 to
>k:=p:
>{SUM of a_k*[e^(i 2k*PI/p)]}^2 + {SUM of b_k*[e^(i 2k*PI/p)]}^2 + 1 =
>Danny.
What is e^(i * Pi) ?
--
Jeremy Boden
===
Subject: Re: Help, please: an equation on complex exponentional numbers
>I really don't know how to proceed with this:
>Let p>2 be an arbitrary prime number, prove that for all k=1,2,...,p
>there exist convenient integers a_k and b_k such that for k:=1 to
>k:=p:
>{SUM of a_k*[e^(i 2k*PI/p)]}^2 + {SUM of b_k*[e^(i 2k*PI/p)]}^2 + 1 =
>0
> What is e^(i * Pi) ?
Are you asking because you don't know, or because you think
the OP doesn't know?
e^(i*Pi) is minus one, but the problem involves numbers of the form
e^(i 2k*PI/p), which is a different kettle of fish.
--
Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email)
===
Subject: Re: Theory behind algebra
conversation with Dave Rusin.]
I went to public school in Nassau County, New York. I *think* we were
taught about induction in eleventh-grade math in the mid 70's. I am
fairly certain it was never covered at college. Perhaps it was taught
in twelfth-grade math, which was not a required coursed. Now that I
think about it, that is more likely the case. Have entering college
freshmen generally taken high-school-senior-level math?
True, I did pick up induction on my own before it was ever covered in
any class, so maybe I'm just imagining that my classmates learned it as
well. I have no high-school math text at my disposal to see if it is in
the curriculum.
--
Stephen J. Herschkorn herschko@rutcor.rutgers.edu
===
Subject: Re: Theory behind algebra
>Where can I find books that rigorously proves all of the facts,
>theorems, and properties one is taught in High School algebra?
You might enjoy thinking about the following issue, treated carefully
in a recent preprint by Wilkies, http://www.maths.ox.ac.uk/~wilkie/ .
This is apparently a question of Tarski: are there true statements
you can make about natural numbers using the symbols + * ^ 1 which do
not follow from the axioms of high school algebra? That is, we ask
students to know the 2 commutative, 2 associative, and 1 distributive laws,
the properties of 1 (1 for * and 2 for ^), and the 3 laws of exponents
(add exponents, multiply exponents, multiply bases). We sort of promise
them that this is all they need to use to complete their calculations.
Well guess what: the following statement is true for all natural numbers
x,y:
[(x + 1)^x + (x^2 + x + 1)^x]^y * [(x^3 + 1)^y + (x^4 + x^2 + 1)^y]^x =
[(x + 1)^y + (x^2 + x + 1)^y]^x * [(x^3 + 1)^x + (x^4 + x^2 + 1)^x]^y
However, it does _not_ follow from the 11 first-order axioms cited!
So rigorously prov[ing] all of the facts, theorems, and properties one
is taught in High School algebra is perhaps a harder task than one might
think!
dave
===
Subject: Re: Theory behind algebra
>You might enjoy thinking about the following issue, treated carefully
>in a recent preprint by Wilkies, http://www.maths.ox.ac.uk/~wilkie/ .
>This is apparently a question of Tarski: are there true statements
>you can make about natural numbers using the symbols + * ^ 1 which do
>not follow from the axioms of high school algebra? That is, we ask
>students to know the 2 commutative, 2 associative, and 1 distributive
laws,
>the properties of 1 (1 for * and 2 for ^), and the 3 laws of exponents
>(add exponents, multiply exponents, multiply bases). We sort of promise
>them that this is all they need to use to complete their calculations.
>Well guess what: the following statement is true for all natural numbers
x,y:
> [(x + 1)^x + (x^2 + x + 1)^x]^y * [(x^3 + 1)^y + (x^4 + x^2 + 1)^y]^x =
> [(x + 1)^y + (x^2 + x + 1)^y]^x * [(x^3 + 1)^x + (x^4 + x^2 + 1)^x]^y
>However, it does _not_ follow from the 11 first-order axioms cited!
>So rigorously prov[ing] all of the facts, theorems, and properties one
>is taught in High School algebra is perhaps a harder task than one might
>think!
Sorry about being too lazy to think through this. What other axiom(s)
do you need? Is induction derivable from the stated axioms?
--
Stephen J. Herschkorn herschko@rutcor.rutgers.edu
===
Subject: Re: Theory behind algebra
>>Where can I find books that rigorously proves all of the facts,
>>theorems, and properties one is taught in High School algebra?
> You might enjoy thinking about the following issue, treated carefully
> in a recent preprint by Wilkies, http://www.maths.ox.ac.uk/~wilkie/ .
> This is apparently a question of Tarski: are there true statements
> you can make about natural numbers using the symbols + * ^ 1 which do
> not follow from the axioms of high school algebra? That is, we ask
> students to know the 2 commutative, 2 associative, and 1 distributive
> laws,
> the properties of 1 (1 for * and 2 for ^), and the 3 laws of exponents
> (add exponents, multiply exponents, multiply bases). We sort of promise
> them that this is all they need to use to complete their calculations.
> Well guess what: the following statement is true for all natural numbers
> x,y:
> [(x + 1)^x + (x^2 + x + 1)^x]^y * [(x^3 + 1)^y + (x^4 + x^2 + 1)^y]^x
=
> [(x + 1)^y + (x^2 + x + 1)^y]^x * [(x^3 + 1)^x + (x^4 + x^2 + 1)^x]^y
> However, it does _not_ follow from the 11 first-order axioms cited!
> So rigorously prov[ing] all of the facts, theorems, and properties one
> is taught in High School algebra is perhaps a harder task than one
might
> think!
Hmmm. Well that's because Wilkie's theory avoids subtraction
(natural in this first-order set-up as using negatives rapidly takes
you out of Z) so I shouldn't think a bright schookid would be unable
to do this :-)
--
Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html
Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9
Francis Wheen, _How Mumbo-Jumbo Conquered the World_
===
Subject: Erdos number for sale
Seems to be for real!
http://cgi.ebay.com/ws/eBayISAPI.dll?ViewItem&item=3189039958
===
Subject: Re: calculators
|>
|> How does a calculator do complicated math computaitons so quickly?
|> For example, finding the sine of 23.71 degrees.
|>
|>
|> I read that many calculators for trig and log functions simply use
|> lookup tables so finding the sine of any number is just a matter of
|> pulling a value from a permanent memory, which is why it can be
computed
|> so quickly.
|> I'm pretty sure that it uses a taylor series, because if you try to
|> find the sine of very large angles, you will see they become
|> increasingly inaccurate. I use TI-83+ Silver Edition, and TI-30X
There is a trivial reason why the calculation of sines of large
arguments
gets increasingly more inaccurate: since the argument is kept in a finite
representation (typically 32bit or 64 bit on modern computers),
it looses about half a decimal digit (2*pi is about 10/2 here..)
in accuracy for each additional period added to the argument. These bits
that store the most significant bits of the argument are lost and not
available to represent the phase, but the least significant bits would be
the
ones actually needed (and the ones left after reduction to [0,2*pi]).
Example: supposed you use a 32bit floating point representation with
about 8 decimal digits of accuracy. If you are to compute the sine of the
number 12345678.9, the binary representation allows to store this
as something close to 12345678, but does not allow to store the
fraction 0.9. Without the knowledge on the trailing fractions, you have
no way to figure out what the sine of the full representation
would be.
===
Subject: Re: calculators
Keywords: sine function, series expansion, pocket calculator
|> How does a calculator do complicated math computaitons so quickly?
|> For example, finding the sine of 23.71 degrees.
There exist Chebyshev series expansions for the sine,
in Math Comp 34 (149) (1980) pp 237-244 or the one by Clenshaw
in Math Tabl Aids Comput 8 (47) (1954) pp 143-147. To obtain
an IEEE single precision accuracy for sin(Pi*x/2)/x
one needs 8 terms, for double precision 16 terms.
===
Subject: Re: calculators
>|> How does a calculator do complicated math computaitons so quickly?
>|> For example, finding the sine of 23.71 degrees.
>|There exist Chebyshev series expansions for the sine,
> in Math Comp 34 (149) (1980) pp 237-244 or the one by Clenshaw
> in Math Tabl Aids Comput 8 (47) (1954) pp 143-147. To obtain
> an IEEE single precision accuracy for sin(Pi*x/2)/x
> one needs 8 terms, for double precision 16 terms.
The CORDIC algorithm is commonly used for calculating trig functions. See
for a
brief description in connection with the original electronic slide rule
calculator. A Google search for CORDIC turns up many hits.
--
Dave Seaman
Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling.
===
Subject: Re: calculators
Originator: jeyadev@kaveri
>> I read that many calculators for trig and log functions simply use
>> lookup tables so finding the sine of any number is just a matter of
>> pulling a value from a permanent memory, which is why it can be computed
>> so quickly.
>I'm pretty sure that it uses a taylor series, because if you try to
>find the sine of very large angles, you will see they become
>increasingly inaccurate. I use TI-83+ Silver Edition, and TI-30X
It may be a TI thing. I remember seeing this way back in 1976. But,
even back then, the HP calculators gave the right answer. Why?
Because they were smart enough to reduce all angles larger than
2 pi to the range 0 to 2 pi. Not hard to do, at all. Not doing it
is not very smart.
--
Surendar Jeyadev jeyadev@wrc.xerox.bounceback.com
Remove 'bounceback' for email address
===
Subject: Re: Help Needed Understanding Article
> Of course? If you're familiar with the system of logic used in
> the paper then why do you keep asking people to explain it
> to you?
Now David, be nice. You know I'm not really asking them to explain
the system to me. I'm asking where this elusive formal proof of the
Halting Problem is. If I asked, Where's the beef? would that make
my question more clear? And it's not about explaining unsolvability
either. It's about finding out whether or not Boyer and Moore did
what they claim in the title of the paper.
> When did I say that the paper convinced me that the halting
> problem was unsolvable?
in the summary that reads ëThe unsolvability theorem is proved
in a
constructive logic like those of Skolem[7] and Goodstein[4]'. I
assumed that you wouldn't quote from a paper without believing in it
as well. Earlier, to expose those BS-ers who claim that it does what
its title says, I pointed out that a proof, by definition, must be
convincing, and asked for what was convincing about it. This is to
keep them from getting into an infinite regress of quoting other BS
papers to justify previous ones. To be convincing, you have to be for
real. But if you're now distancing yourself from that paper (seeing
that public opinion is turning against it, perhaps), then never
mind.
> I do otoh have an advantage over some people: I understand
> that there are things that I don't understand
How do you know you don't understand it if you don't understand it?
(I think this is related to Godel's 2nd Incompleteness Theorem,
actually.)
> and I don't
> go around saying that a paper I don't understand simple
> cannot contain what it claims to contain. Or that I'm the only
> one who's ever done something, when there exist papers I
> don't understand that claim to do the same thing. If I'd
> done something, thought I was the only one who'd done
> it, and was told about a paper where it had been done
> previously I'd make certain to understand that paper
> _before_ informing the world that the authors were lying.
I agree (that I do that, not that I necessarily believe that you do.)
> (No, you _don't_ understand the contents of the paper.
> This is evident from the way you continually ask people
> to explain it to you. You're reminding people of James
> Harris, explaining why Wiles' proof of Fermat's last
> theorem is wrong although he doesn't even know what
> a ring is.)
Don't know the guy myself. In any case, saying You don't understand
does not substantiate a technical assertion! But now you realize I am
playing innocent so people will be more willing to cooperate, right?
But you make a good point, actually. If I say it's BS and then act
like I don't understand it in order to get the nurturing types to join
in, I am implying that I condemned something that I don't understand.
That's a good one. I'll have to rememeber that in the future (and
fine-tune my subterfuges.)
Ok, I'll take you off my list of BS-ers who're defending the
Boyer-Moore paper. (What's that - I made your day?)
Charlie Voilkstorf
Cambridge, MA
> ************************
> David C. Ullrich
===
Subject: Re: Help Needed Understanding Article
> The authors inverted the LISP complier and applied the relation
> proving it is solvable to appear the solution using symbols. A key
> concept is the letter or byte as the symbol or the machine state. The
> authors have done nothing but confuse everybody.
I am afraid that some of your terminology is foreign to me. Could you
define these terms in more detail?
I agree that they have done nothing - except perhaps saying some
confusing things.
> Now adays the existence of the LISP self compilation proves the
> halting problem solved.
By the halting problem solved, do you mean the unsolvability of the
halting problem? And what exactly do you mean by self compilation?
I don't see offhand how the existence of one program can prove the
nonexistence of another program. Could you give me a simple example
of how that could be .9a a simple use of that principle?
The existence of a program proves that the function that it computes
is recursive. The existence of a program with a particular input and
output proves that the function that it computes, when applied to that
input, produces that output. But how does the existence of a program
per se prove that a particular function is NOT recursive?
(Note that programs that prove theorems are an example of the latter:
the existence of a program with a particular input and output proves
that the function that it computes, typically rules of inference, when
applied to its input, typically axioms, produces the output, typically
a theorem. But, again, you have to explain what those axioms, rules
and theorem mean for it to mean anything.)
BTW These happen to be fundamental rules that are formalized by my
axiomatization and used over and over in formal proofs. This is part
2 of my system. Part 1, which proves the unsolvability of the
halting problem and tons of variations (which I actually produce,
rather than offering excuses), has to do with the existence or non
existence of certain programs. Part 2, which is really a
generalization, has to do with properties of the syntax of the program
itself. This proves such theorems as the Recursion Theorem, the Fixed
Point Theorem, and various extensions of these, especially those
developed by Raymond Smullyan.
I # yes(I) Every program computes the set of values on which it
halts yes.
wr(I) # I Function wr returns a program that computes the input to
wr.
wr(I) Function wr is recursive.
N # f(I) => s11(N,M) # f(M) Substitution function
s11(I,J) Substitution is recursive.
(eN) N # N Theorem: There is a program that outputs itself.
f(I) => (eN) N # f(N) Recursion Theorem
f(I) => (eN) N ~ f(N) Fixed Point Theorem
It's really a great system. Simple, very powerful, easy to use and a
lot of fun. It's lots of fun to write formal proofs for various
theorems, especially the seemingly unlimited supply coming from
Raymond Smullyan. (You also uncover mistakes in published works along
the way.)
It's a refreshing break from all the BS that gets published - e.g.
theorems without proofs (and the idiots who defend that.)
If anyone is interested in more details, please email me directly. I
can always use helpers. The main task is to develop proofs of
published theorems, as well as to show variations that don't seem to
have been published. Occasionally that involves adding an additional
axiom, which is always Oh, of course! What a perfectly reasonable
axiom!
Charlie Volkstorf
Cambridge, MA
===
Subject: Re: Help Needed Understanding Article
> I am convinced that there is no computable procedure for determining
> if any arbitrary program written in some general purpose language can
> halt.
I agree completely.
> I see the title and abstract of this paper. I know Boyer and Moore
> have an absolutely stellar reputation in the area of systems for
> formal proofs.
Ok. And what specifically makes you feel that way?
> There's lots of technical details in the paper,
> which I have not read. So I am more than willing to accept that
> their system has verified it.
Would you then agree that your belief in the authenticity of their
paper is more of a personal opinion rather than being based on a
mathematical or logical proof?
> I don't maintain that the paper must present a proof of the
> unsolvability of the halting problem.
Then would you agree that they do not present a proof of the
unsolvability of the halting problem?
Charlie Volkstorf
Cambridge, MA
===
Subject: Re: Help Needed Understanding Article
> The theorem is proven using a Theorem prover. They show the proof found
by
> the theorem prover for each of the theorems. The theorem prover is
described
> in the references, but just from this paper you can make some good
guesses
> about how it operates. If you are serious about understanding the paper
you
> would go read the references. Have you done so?
> Stephen
Of course, but it is just more of the same, with no proof of the
Halting Problem. Showing the EVAL function that evaluates an
expression does not prove that the Halting Problem is unsolvable. Nor
does a description of a variation of the LISP programming language.
Any proof of the unsolvability of the Halting Problem would contain
some sort of logic and reasoning which reaches that conclusion.
Does this paper (including its references) convince you that the
Halting Problem is unsolvable? If so, could you explain the logic and
reasoning that convinces you? If not, do you believe that they proved
that the Halting Problem is unsolvable?
Charlie Volkstorf
Cambridge, MA
===
Subject: Re: Help Needed Understanding Article
:> The theorem is proven using a Theorem prover. They show the proof found
by
:> the theorem prover for each of the theorems. The theorem prover is
described
:> in the references, but just from this paper you can make some good
guesses
:> about how it operates. If you are serious about understanding the paper
you
:> would go read the references. Have you done so?
:> Stephen
: Of course, but it is just more of the same, with no proof of the
: Halting Problem. Showing the EVAL function that evaluates an
: expression does not prove that the Halting Problem is unsolvable.
Your sentence does not make sense to me.
: does a description of a variation of the LISP programming language.
: Any proof of the unsolvability of the Halting Problem would contain
: some sort of logic and reasoning which reaches that conclusion.
The final theorem proved by their theorem prover is that if
H equals the result of HALT(X,X), then if H is true then X does
not halt on X, and if H is false then X does halt on X. That
sounds like the Halting Problem to me.
Are you saying that because they use LISP it is not the Halting
Problem? LISP is Turing Complete. What language do you use?
: Does this paper (including its references) convince you that the
: Halting Problem is unsolvable? If so, could you explain the logic and
: reasoning that convinces you? If not, do you believe that they proved
: that the Halting Problem is unsolvable?
I believe that their theorem prover works as advertised and
that what is presented in the paper is the actual output of
their theorem prover. They work through all the lemmas they
proved along the way, so you can backtrack from the final theorem
and figure out how the theorem prover proved it.
If you really want to understand the paper, you can go read
the references, download the theorem prover, and try it yourself.
Stephen
===
Subject: Re: Help Needed Understanding Article
>>The theorem is proven using a Theorem prover. They show the proof found
by
>>the theorem prover for each of the theorems. The theorem prover is
described
>>in the references, but just from this paper you can make some good
guesses
>>about how it operates. If you are serious about understanding the paper
you
>>would go read the references. Have you done so?
>>Stephen
> Of course, but it is just more of the same, with no proof of the
> Halting Problem. Showing the EVAL function that evaluates an
> expression does not prove that the Halting Problem is unsolvable. Nor
> does a description of a variation of the LISP programming language.
Why isn't the standard proof satisfactory to you?
I think, most people who are exposed to the standard
proof accept the result and don't feel the need to
find a formal version.
In that sense, the halting problem proof (at least for me)
is no different than other simple proofs, such as:
(1) The non-existence of a greatest prime number.
(2) The existence of the square root of 2.
(3) The non-existence of a bijection between N and R
,and so on...
I've never seen a formal version of any of the proofs above,
but I don't doubt that they can be formalized, and I wouldn't
consider their formalization to be revolutionary or anything
like that.
If I had doubts that a (simple) proof could be formalized,
it would have to be because I was in disagreement with the
informal version. In other words, I don't see any reason to
require a formal proof for the sake of being convinced, unless
you think you've detected a subtle inconsistency which might
be exposed by a formal version.
===
Subject: Re: Help Needed Understanding Article
> Why isn't the standard proof satisfactory to you?
> I think, most people who are exposed to the standard
> proof accept the result and don't feel the need to
> find a formal version.
> In that sense, the halting problem proof (at least for me)
> is no different than other simple proofs, such as:
> (1) The non-existence of a greatest prime number.
> (2) The existence of the square root of 2.
> (3) The non-existence of a bijection between N and R
> ,and so on...
> I've never seen a formal version of any of the proofs above,
> but I don't doubt that they can be formalized, and I wouldn't
> consider their formalization to be revolutionary or anything
> like that.
> If I had doubts that a (simple) proof could be formalized,
> it would have to be because I was in disagreement with the
> informal version. In other words, I don't see any reason to
> require a formal proof for the sake of being convinced, unless
> you think you've detected a subtle inconsistency which might
> be exposed by a formal version.
Consider a statement T and its proof. If the proof can be formalized, it
means that a machine can prove T. This shows that as far as T goes,
machines are not any inferior than human minds. So, the issue now is -
Are there any interesting statements whose proofs cannot be formalized?
--
-----------------------------------------
Remove obviously faked punctuation from
burningspear79 a.t ly-cos peri-od c.om
and ignore the rest of this sig
bspear79@angelfire.com
-----------------------------------------
===
Subject: Re: Help Needed Understanding Article
fuminggreen@yahoo.com schreef:
>> Hello all,
>> I am interested in the Theory of Computation and also the use of
>> formal methods of proof. So naturally I became very excited when I
>> acquired a copy of the paper, A Mechanical Proof Of The
>> Unsolvability of The Halting Problem at
>> http://citeseer.ist.psu.edu/boyer82mechanical.html (click on PDF in
>> the upper right hand corner.)
>> However, I am having a hard time understanding how they actually
>> reach
>> the conclusion that the Halting Problem is unsolvable. I know about
>> Turing's original proof in 1937 and don't have any problem with that.
>> I just don't see that the above paper proves it using their method.
>> Is there anyone here who believes that this paper does prove that the
>> Halting Problem is undecidable, and if so, could you explain the
>> logic
>> and reasoning that is used? Are there axioms and rules of inference?
>> What do they state, in plain English?
>> Charlie Volkstorf
>> Cambridge, MA
Hello Charlie,
Technically you're right - this paper doesn't prove it. BUT...
Even though I didn't check refs. 1 and 2, I'm pretty positive that
they're referring to the famous Boyer-Moore theorem prover. So what the
and to-be-proven lemmas/conjectures/theorems) they gave to this theorem
prover, and (implicitly) assert that the prover was able to prove the
lemmas with these definitions. One of these theorems is the halting
problem's unsolvability.
There are no rules in the paper because they are embedded in the
Boyer-Moore theorem prover. The system has predefined logical rules
that needn't be stated or (re)defined. If you study the prover you'll
know what they are (and hence what kind of logic they use - it appears
to be classical)
The idea behind many theorem provers is the so-called
propositions-as-types (or Curry-Howard) correspondence. That is, to a
mathematical statement (lemma, conjecture, theorem) there will
correspond a type of a type system---you know, the same kind of thing
they use in programming languages, although maths needs a more general
type system than any of the programming languages you're likely to
know.
Now in the propositions-as-types correspondence, a term (that is, a
function, in programmer speak) of a certain type is just a proof of the
theorem corresponding to that type. What the theorem prover thus does
is finding you a term of the required type. And in fact, if you do
things in the right way, a proof of a statement there exists x with
property P will actually encode a +function+ that provides you with
such an x.
So we could make the following scheme:
term : type
proof : theorem
function : output type
Now since type checking is standard technology, at least checking
whether something is a valid proof of some statement is easy: it's just
typechecking the corresponding term and seeing whether it has the
correct type!
The problem of finding a term is more difficult, and there are automatic
theorem provers (of which Boyer-Moore's seems to be a variant) and
interactive ones (these are also called proof assistants - although in
my opinion they don't actually +assist+ ;-) ); also the underlying
logical system may differ (classical logic, constructive, ...) and
sometimes even type checking is undecidable.
Of course one may wonder why we should trust the programmer's skills in
writing such type checkers. This is a valid question - but we may apply
the so-called de Bruijn criterion: the actual type checking code must
be a very small program, so that we can verify it by hand. And in fact
such systems do exist; e.g. the HOL-Light system has a kernel of only
some 500 lines of code (including comments etc.). I don't know the
facts on the Boyer-Moore prover though.
Some other systems you may want to investigate are called Coq, Oleg
(formerly called Lego, until the toy makers complained :-S), Mizar,
PVS, Isabelle, Twelf; and there are many more.
> The authors inverted the LISP complier and applied the relation
> proving it is solvable to appear the solution using symbols. A key
> concept is the letter or byte as the symbol or the machine state. The
> authors have done nothing but confuse everybody.
Quite the contrary - I think they describe very well what they did and
how they did it.
> Now adays the existence of the LISP self compilation proves the
> halting problem solved.
Eh...? Care to expand on that?
Jasper
--
The problem with having an open mind is that people toss in garbage
===
Subject: Re: Help Needed Understanding Article
> Hello Charlie,
> Technically you're right - this paper doesn't prove it.
And do you know what that makes me think? Well, # 1, the fact is they
did not present a formal proof of the unsolvability of the Halting
opposed to so many others.) So the title, A Mechanical Proof of the
Unsolvability of the Halting Problem, is a misnomer?
And for # 2, well, should I say it? Maybe I should just leave it at #
1. But, just between you and me, when I write a paper, I always put
my formal results right up front. Why? Because that is what I look
for myself. I'd rather see formal wffs, axioms, rules and proofs than
pages of English (with an occasional formal (?) expression mixed in.)
And besides, if they presented it, then they could honestly say that
they did present a formal proof of this theorem. I mean, if you're
going to write a paper and have it published, why leave out the best
part? Sounds pretty fishy. So I can only assume that they probably
just don't have it (gasp!) Just makes sense to me.
> So what the
> and to-be-proven lemmas/conjectures/theorems) they gave to this theorem
> prover, and (implicitly) assert that the prover was able to prove the
> lemmas with these definitions. One of these theorems is the halting
> problem's unsolvability.
> There are no rules in the paper because they are embedded in the
> Boyer-Moore theorem prover.
But what rules were used to derive the proof of the unsolvability of
the Halting Problem, then? I can't tell that from reading the theorem
prover's user's manual.
And what did those lemmas/conjectures/theorems state? Why not list
them and the rules used? Why not list the proof itself?
> Some other systems you may want to investigate are called Coq, Oleg
> (formerly called Lego, until the toy makers complained :-S), Mizar,
> PVS, Isabelle, Twelf; and there are many more.
As you can see from the links that I posted earlier, there are
literally hundreds! But who has formally proved the unsolvability of
the Halting Problem (besides http://www.arxiv.org/html/cs.lo/0003071)?
Charlie Volkstorf
Cambridge, MA
> Jasper
===
Subject: Re: Help Needed Understanding Article
: But what rules were used to derive the proof of the unsolvability of
: the Halting Problem, then? I can't tell that from reading the theorem
: prover's user's manual.
If you would just go type the theorem into the theorem prover
it would show you what rules were used.
Stephen
===
Subject: Re: Anti-compact
William Elliot a .8ecrit dans le message de
> A space is anti-compact when every compact subset is finite.
> Discrete spaces are anticompact, Hausdorff.
> Uncountable cocountable spaces are anticompact, T1 and not Hausdorff.
> Are all anticompact, Hausdorff spaces discrete?
Intersting definition. Anticompact Hausdorf fspace must be discrete but we
can find a connected anticompact space.
a) Anticompact + Hausdorff => discrete
As you point out, in an anticompact space an injective sequence cannot have
any limit point.
So infinite set cannot have accumulation point.
In an Hausdorff space, that mean they must be closed.
But in Hausdorff space finite space are closed too.
So the space is discrete.
b) An anticompact connected space
Take N with the topology with open sets [0,n] with all n in N.
It is connected because every open contain 0 so they are no disjoint open
set (hyperconneted)
It is anticompact because
([0,n]) is an open covering with no finite subcovering and every infinite
subset is homeomorph to him.
===
Subject: Re: Anti-compact
>William Elliot a .8ecrit dans le message de
>> A space is anti-compact when every compact subset is finite.
>> Discrete spaces are anticompact, Hausdorff.
>> Uncountable cocountable spaces are anticompact, T1 and not Hausdorff.
>> Are all anticompact, Hausdorff spaces discrete?
>Intersting definition. Anticompact Hausdorf fspace must be discrete but we
>can find a connected anticompact space.
>a) Anticompact + Hausdorff => discrete
>As you point out, in an anticompact space an injective sequence cannot
have
>any limit point.
>So infinite set cannot have accumulation point.
Careful there. An accumulation point need not be the limit of a sequence.
E.g. let e0 be the first uncountable ordinal. Take the set X of
ordinals <= e0, with the order topology. Then e0 is an accumulation
point of X {e0}, i.e. every neighbourhood of e0 contains a member of
X {e0}, but there is no sequence in X {e0} whose limit is e0.
Of course this doesn't answer the question, because X is not anticompact.
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia
Vancouver, BC, Canada V6T 1Z2
===
Subject: Monkey typing ABRACADABRA
I have a rather simple question (so it seems):
A monkey types only capital letters and is on every occasion equally
like to type any of the 26 letters in the alphabet. Each second the
monkey types one letter. How long on average will it take him to
produce the sequence 'ABRACADABRA' ?
I have heard that the solution should be:
26^11+26^4+26 seconds.
However this is not very intuitive to me, i would rather say 26^11
seconds.
Unfortunately I also cannot derive the above result.
Any help from you would be appreciated very much!
Best wishes, Peter.
===
Subject: Re: Monkey typing ABRACADABRA
>I have a rather simple question (so it seems):
>A monkey types only capital letters and is on every occasion equally
>like to type any of the 26 letters in the alphabet. Each second the
>monkey types one letter. How long on average will it take him to
>produce the sequence 'ABRACADABRA' ?
>I have heard that the solution should be:
>26^11+26^4+26 seconds.
You have heard right.
>However this is not very intuitive to me, i would rather say 26^11
>seconds.
>Unfortunately I also cannot derive the above result.
No, it's good to not be able to derive wrong results.
Hint:
In fancier language, we're talking about the expected time to
reach a certain state in a Markov chain. The state is the longest
suffix of the monkey's current output that is a prefix
of ABRACADABRA. For example, if the monkey has typed ABRACAXABR
the state is ABR. Initially it is `` (the empty string). The situation
is complicated by the fact that from a given state, as well as adding
another letter or going back to ``, you might be able to go to various
previous states, e.g. from ABRA to A and AB as well as ABRAC and ``.
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia
Vancouver, BC, Canada V6T 1Z2
===
Subject: Re: Monkey typing ABRACADABRA
>I have a rather simple question (so it seems):
Not really that simple - I didn't learn this until graduate school.
>A monkey types only capital letters and is on every occasion equally
>like to type any of the 26 letters in the alphabet. Each second the
>monkey types one letter. How long on average will it take him to
>produce the sequence 'ABRACADABRA' ?
>I have heard that the solution should be:
>26^11+26^4+26 seconds.
>However this is not very intuitive to me, i would rather say 26^11
>seconds.
>Unfortunately I also cannot derive the above result.
The first answer you cited is correct. One can show this with renewal
theory, Markov chains, or martingales. Here's the basic
renewal-theoretic argument:
Since the probablity that ABRACADABRA appears in consecutive letters is
26^-11, the expected time to get from one instance thereof to the next
is 26^11. (That's Blackwell's theorem from renewal theory.) However,
when you start at one instance, you already have ABRA towards the next
instance (so, for example, it is possible to reach the next instance in
just seven more letters). Starting from scratch, you don't have these.
So you have to get to ABRA first, whence the 26^4. Similarly, ABRA both
ends and begins with A. So what you're computing is the time to get to
A plus the time to get from (ABR)A to ABRA plus the time to get from
(ABRACAD)ABRA to ABRACADABRA.
The expected time to get to the pattern ABRACADABRR *is* 26^11.
--
Stephen J. Herschkorn herschko@rutcor.rutgers.edu
===
Subject: Re: Monkey typing ABRACADABRA
>>I have a rather simple question (so it seems):
>Not really that simple - I didn't learn this until graduate school.
Hmm, then perhaps I should fear to tread here instead of rushing in...
But still...
>>A monkey types only capital letters and is on every occasion equally
>>like to type any of the 26 letters in the alphabet. Each second the
>>monkey types one letter. How long on average will it take him to
>>produce the sequence 'ABRACADABRA' ?
>>I have heard that the solution should be:
>>26^11+26^4+26 seconds.
>>However this is not very intuitive to me, i would rather say 26^11
>>seconds.
>>Unfortunately I also cannot derive the above result.
>The first answer you cited is correct. One can show this with renewal
>theory, Markov chains, or martingales. Here's the basic
>renewal-theoretic argument:
>Since the probablity that ABRACADABRA appears in consecutive letters is
>26^-11, the expected time to get from one instance thereof to the next
>is 26^11. (That's Blackwell's theorem from renewal theory.) However,
>when you start at one instance, you already have ABRA towards the next
>instance (so, for example, it is possible to reach the next instance in
>just seven more letters).
But then, that wouldn't *be* the next instance. It would overlap the
first instance.
If you want to allow overlaps, then it seems to me you should take the
2^11 figure and *reduce* it accordingly here, by putting minus signs
in the o.p.'s expression. The availability of a solution with seven
letters means that the monkey will get to ABRACADABRA *faster* on
average than to a different 11-character string.
Starting from scratch, you don't have these.
>So you have to get to ABRA first, whence the 26^4. Similarly, ABRA both
>ends and begins with A. So what you're computing is the time to get to
>A plus the time to get from (ABR)A to ABRA plus the time to get from
>(ABRACAD)ABRA to ABRACADABRA.
>The expected time to get to the pattern ABRACADABRR *is* 26^11.
But that would imply that it's easier to type an R than an A, contrary
to hypothesis.
===
Subject: Re: Monkey typing ABRACADABRA
>>
>I have a rather simple question (so it seems):
>
>>Not really that simple - I didn't learn this until graduate school.
>>
>Hmm, then perhaps I should fear to tread here instead of rushing in...
>But still...
>A monkey types only capital letters and is on every occasion equally
>like to type any of the 26 letters in the alphabet. Each second the
>monkey types one letter. How long on average will it take him to
>produce the sequence 'ABRACADABRA' ?
>I have heard that the solution should be:
>26^11+26^4+26 seconds.
>However this is not very intuitive to me, i would rather say 26^11
>seconds.
>Unfortunately I also cannot derive the above result.
>
>>The first answer you cited is correct. One can show this with renewal
>>theory, Markov chains, or martingales. Here's the basic
>>renewal-theoretic argument:
>>Since the probablity that ABRACADABRA appears in consecutive letters is
>>26^-11, the expected time to get from one instance thereof to the next
>>is 26^11. (That's Blackwell's theorem from renewal theory.) However,
>>when you start at one instance, you already have ABRA towards the next
>>instance (so, for example, it is possible to reach the next instance in
>>just seven more letters).
>>
>But then, that wouldn't *be* the next instance. It would overlap the
>first instance.
By instance, I mean a keystroke where the pattern ends. There are three
instances in
FDKABRACADABRAERNVABRACADABRACADABRABRRRVDKJABRCFJHKSF.
The important thing for the theory is that the times from one instance
to next are i.i.d.
>If you want to allow overlaps, then it seems to me you should take the
>2^11 figure and *reduce* it accordingly here, by putting minus signs
>in the o.p.'s expression. The availability of a solution with seven
>letters means that the monkey will get to ABRACADABRA *faster* on
>average than to a different 11-character string.
Your last sentence is exactly right. On average, it takes less time to
get from ABRACADABRA to ABRACADABRA than to get from nothing to
ABRACADABRA.
> Starting from scratch, you don't have these.
>>So you have to get to ABRA first, whence the 26^4. Similarly, ABRA both
>>ends and begins with A. So what you're computing is the time to get to
>>A plus the time to get from (ABR)A to ABRA plus the time to get from
>>(ABRACAD)ABRA to ABRACADABRA.
>>The expected time to get to the pattern ABRACADABRR *is* 26^11.
>>
>But that would imply that it's easier to type an R than an A, contrary
>to hypothesis.
Not at all.
Simplify the discussion by switcihing the setting to flips of a fair
coin. The expected time to reach the pattern TH is 4. The expected
time to reach the pattern TT is 6. Why? In the first case (TH) ,
suppose your first flip is a tail. Great - your almost there. Then
your next flip is another tail - d'oh! You missed it. But at least
now, with the most recent tail, you are started on your way to TH again.
Now consider the second case (TT). First you flip a tail - good. Then
you flip a head, lousing things up. Now you have to start all over from
scratch again. So here you gain nothing from the second flip.
In fact, we see that the time until TH is the time until the first T
plus the time until the first H thereafter. It is not so simple in the
case of TT. With more than two outcomes (e.g., twenty-six letters),
this analysis is not as simple, but the general effect is the same.
In renewal-theoretic terms, the process with a pattern whose end does
not overlap is an ordinary reneewal process. With the overlap, it is a
delayed renewal process. (In the latter, the time until the first
instance differs in distribution from the time between instances.)
--
Stephen J. Herschkorn herschko@rutcor.rutgers.edu
===
Subject: Unofficial announcement of MathML on Mathforge
Well, we're proud to announce our message boards and wikis are
mathml-enabled. Type a Tex-like string, e.g. $e^(ipi)+1=0$ in your
message and you will see it marked up like actual math!
Go to http://www.mathforge.net right now and try it out! Additionally,
you'll find the latest in math news.
Soon, we'll have the capability to create marked up text online, which
should be useful for homework, marking up class notes, and
collaborating on documents.
Joe Kauzlarich
http://www.mathforge.net
Tasteful Mathematics for Sensitive People
===
Subject: Re: Unofficial announcement of MathML on Mathforge
> Well, we're proud to announce our message boards and wikis are
> mathml-enabled. Type a Tex-like string, e.g. $e^(ipi)+1=0$ in your
> message and you will see it marked up like actual math!
I find this a somewhat baffling remark.
Surely there is a vast difference between TeX and MathML?
> Go to http://www.mathforge.net right now and try it out! Additionally,
> you'll find the latest in math news.
--
Timothy Murphy
e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie
tel: +353-86-2336090, +353-1-2842366
s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland
===
Subject: Re: Unofficial announcement of MathML on Mathforge
>> Well, we're proud to announce our message boards and wikis are
>> mathml-enabled. Type a Tex-like string, e.g. $e^(ipi)+1=0$ in your
>> message and you will see it marked up like actual math!
> I find this a somewhat baffling remark.
> Surely there is a vast difference between TeX and MathML?
>> Go to http://www.mathforge.net right now and try it out! Additionally,
>> you'll find the latest in math news.
Naturally there is a distinct difference between TeX and MathML. What this
equation editor does is convert the TeX-like input--with which many people
are familiar--to MathML for convenient viewing in Web browsers. This is
also much more accessible to the TeX-nically (hah! Get it?) unskilled
without erasing the productivity inherent in a GUI-less equation editor.
===
Subject: Re: Yes, these are homework questions!
> because it's so obvious now... can anyone recommend a book on algebra -
for
Notice how you are dealing mostly with linear things. Even your
> polynomial problem, could be split into linear factors over an
> appropriate splitting field. So first of all you need to get a book
> on linear algebra. Since you are dealing with polynomials, it is
> essential that you also get a book on galois theory, which is the
> study of polynomials. Once you finish those, you are ready to go a
> little deeper, and you should seek a book on abstract algebra.
> If these books seem a little bit over your head (I'm not sure how good
> you are), you might need to step back and start with I.N. Herstein's
> Topics in Algebra, which is written for folks like you getting
> started in math.
> Good luck in all your math endeavors!
> Your dear friend,
> Bruce B.
Ack, I can't believe nobody replied yet. What a failed troll I am. I
was hoping to create a flame war as everyone jumped in to exclaim that
the books above are NOT appropriate for you. Which they aren't-- I
was counting on people to bite and correct me, but it seems to have
failed. So don't follow the advice above, as it is quite inaccurate.
===
Subject: Re: Yes, these are homework questions!
Converter schreef in bericht
> MRe schreef in bericht
I was going to ask my teacher, but ran out of time... and as for
working
> these out myself, well there's only so may times I can shake my fist
at
> them, I'm lost.
> Anyone care to help me out, please?
Express a in terms of b and c
> a=(b-a)/c
> -- Don't have a clue where to begin.
Variable a represents a tablevalue.
> First determine the weight of each variable.
> b>c and b for b=(a*c)+a
> c for c=1 b has to be as big as twice to a:
> (c*a)+a=b
> or coming back to a=(b-a)/c:
> (c*a)=b-a
> In any table c will be the multiplier to tablevalue a where b will
adept
by
> +a upgoing the table (from c=1 b=a+a, c=2 b=a+a+a etc.).
Given a factor, find k
> x^3-x^2+kx+n=0
k = (x^2-x)
> I don't see the point in n.
> Geere
> I don't understand how you came about k = (x^2-x), I got k = (-x^2+x).
> (removing n from the equation because you saw no point in it)
> 1. x^3-x^2+kx = 0
> 2. x(x^2-x+k) = 0
> 3. x(x^2-x+k)/x = 0/x
> 4. x^2-x+k = 0
> -x^2 + x^2
> 5. k = (-x^2+x)
Well done! I don't have to explain anymore, as if you know:
x(x^2-x)=x(-x^2+x)
Geere
===
Subject: Re: Four Color Theorem Proof
>sounds like a novel approach; is it?
>> Every vertex in every maximal planar graph is completely enclosed
>> by a cycle graph; ie, a ring of edges. Essentially, every vertex is
>> the hub of
>> its own wheel graph.
I thought I read recently that someone had found a map requireing more
than four colours, but I don't recall the source.
--
John W Hall
Cochrane, Alberta, Canada.
Helping People Prosper in the Information Age
===
Subject: Re: Four Color Theorem Proof
> sounds like a novel approach; is it?
I thought of it all by my self! I do not know if I am the first.
> Every vertex in every maximal planar graph is completely enclosed
> by a cycle graph; ie, a ring of edges. Essentially, every vertex is
> the hub of
> its own wheel graph.
===
Subject: big-Oh notation...
Say that {X_n} a sequence of random variables satisfies X_n = O(1) if, given
e
> 0, there exists M_e and an integer N_e such that
P { |X_n| > M_e } < e
for all n >= N_e.
Also, suppose f is strictly increasing on the positive reals, with f(0) =
0,
lim(t --> oo) f(t) = 1.
Show X_n = O(1) if and only if limsup E f(|X_n|) < 1.
I can easily show that, if the big-Oh condition is satisfied, then limsup E
f(|X_n|) <=1, but I don't know how to eliminate the possibility of equality
holding. Any ideas? Hopefully this'll illuminate how to show the converse,
which I'm entirely in the dark on....
===
Subject: Re: central limit theorem, equivalent statements
===
>Subject: central limit theorem, equivalent statements
>CLT and related topics):
>Let {X_n,i} be a triangular array of random variables (independent
>within each row). Suppose
> (*) given epsilon > 0, sum(over i) E( X_n,i I(|X_n,i| <= epsilon) )
>---> m
>(**) given epsilon > 0, sum(over i) Var(X_n,i I(|X_n,i| <= epsilon) )
>---> s^2 finite
>(***) max(over i) |X_n,i| ---> 0 in probability,
>where I(A) is the indicator function of the set A.
>Then sum(over i) X_n,i ---> N(m,s^2) in distribution. I'm going to
>call this result R1.
>The problem is that I haven't been able to convince myself that this
>follows from (or really is an equivalent formulation of) any of the
>version of the CLT for triangular independent arrays (probably
>Lindeberg's) that I'm familiar with.
>I'm guessing there's no loss of generality if we set m=0, s^2=1. Then
>I'm acquainted with the following:
>Let {X_n,i} be a triangular array of random variables (independent
>within each row). Suppose
>(1) E(X_n,i) = 0
>(2) Sum(i=1,...,k_n) E((X_n,i)^2) = 1
>(3) Given epsilon > 0, Sum (i=1,...,k_n) E( (X_n,i)^2 I(|X_n,i| epsilon)
) ---> 0.
>Then Z_n = Sum(i=1,...,k_n) X_n,i ---> N(0,1) in distribution. I'm
>going to call this result R2.
>Can someone help show me how R1 is equivalent to one of the more
>popular formulations of the CLT for triangular arrays (say R2)? I've
>tried doing various manipulations of the hypotheses of R1 (i.e.
>flipping around the indicator functions to I(|X_n,i| > epsilon) but
>I still can't put it in just the right form. please help.....
This looks pretty close to the version of Lindeberg's theorem stated in
Billingsley's book, Probability and Measure (3rd ed), theorem 27.4.
Can someone confirm these two statements are the same??
===
Subject: How to calculate the dirivative of the following vector valued
function
I have a vector valued function (cos(s*t),sin(s*t)) , s and t are 2
parameters,
How to calculate its derivative? Is there any connection between
GG
===
Subject: Re: How to calculate the dirivative of the following vector valued
function
> I have a vector valued function (cos(s*t),sin(s*t)) , s and t are 2
> parameters,
> How to calculate its derivative? Is there any connection between
> derivative and Jacobian Matrix?
It's (total) derivative at any (s_0, t_0) is the linear transformation
whose matrix is the Jacobian Matrix.
===
Subject: Re: Garry Donald Rabbitskin - June 20th 1972
> 12 13 14 15 16 17 18 19 20 21 <-positions
> R A B B I T S K I N
> 18 1 2 2 9 20 19 11 9 14 = 105 <-value
> Today God brought Garry and I together, he is from some reserve near
Big
> River and grew up in North Battleford. He was able to sit, watch and
listen
> for great length while I worked on his stats, which he wouldn't have if he
> had known he was listening to a convicted stalker/pedophile scumbagink.
<<
The following (courtesy of Waxy.org) is sort of an unofficial FAQ
explaining the psychotic nonsense posted to Usenet by Shawn Daryl
Kabatoff AKA Dar, AKA Probababbilities. And now AKA marcia and
me.
WARNING: Read below before even thinking about responding to this
twit.
http://www.waxy.org/archive/2002/05/21/dar_kaba.shtml#000643
Usenet has the tendency to provide a public forum for those who would
normally be scribbling in a closet. For example, take Daryl Shawn
Kabatoff. For the last few years, he's methodically gathered
statistics from various sources, ranging from local newspaper
obituary pages to the food court of the Saskatoon Midtown Plaza mall.
With all the raw data he's collected, he's attempting to prove daily
that our full names are in mathematical harmony with our birthdays.
His rants normally focus on a single individual he's met or read
about, starting with calculations related to their birthdate and full
names, blending in whatever other personal information about their
family members, spouses, birthplace, and career he's been able to
zealotry, and personal torment. I've never seen anything like it.
With all the prime numbers, Fibonacci sequences and biblical
references, it's like reading the notebooks of Maximillian Cohen and
John Nash combined. Unsurprisingly, several posts unfold to reveal a
history of painful mental illness. If you have some time, take a look.
I've detailed his posting history and a several sample posts below.
Usenet Posting History:
January 27, 1999 to July 5, 2000 as Catsco@home.com
December 9, 2000 to May 4, 2001 as s.kabatoff@sk.sympatico.ca
Oct 30, 2001 to Oct 31, 2001 as kabatoff@the.link.ca
January 20, 2002 to April 17, 2002 as s_kabatoff@hotmail.com (original
posts have been
removed from Google Groups archive)
April 26, 2002 to Present as dar_kabatoff@hotmail.com
Selected Posts:
Tessa Lynne Smith
Dastageer Sakhizai and Helen Smith
Brett David Maki
Andrew Meredith Cotton
Kathryn Lee Hipperson
Amanda Dawn Newton
Mona Marie Etcheverry
Tony Peter Nuspl
Lisa Charlene McMillan
Grant Allyn Wood
Comments
scarier still is that saskatoon is my hometown, though not my current
residence. and every single place he's mentioned in his posts (most
notably nervous harold's and the roastary) were either places i've
been (as it's a small city of 200K) or hangouts, ie. the two places
mentioned. chances are i could email some friends back home and find
out if they know of him, they (my friends that is) being of the
broadway-centred slacker ilk. myself, too, until i got out of there.
eh, anyways. thought it odd to see all this. midtown mall. i ate my
meals there, whilst waiting several days in line for star wars episode
one, at the theatre across the street.
posted by andy raad on May 22, 2002 06:20 PM
Fascinating. It's like he's trying to take chaos and bind it into
whatever rules he can find, religious, logical and otherwise. Numbers
and math have a reliable pattern, something that can always be proven
to true or false. People and religion do not. It reminds me of Darren
Aronofsky's movie Pi. It's the story of an paraniod genius who is
trying to find a pattern in Pi. A group that takes interest in his
work is convinced that the existence of Pi, a number whose existence
can be proven but no quantified, is proof of the existence of God.
Kabatoff's hunt for patterns in something as random as name selection
is a way to reconcile his deeply logical thought process with his
conflicting religious views.
Exactly. I probably shouldn't have, but I e-mailed Daryl yesterday,
asking him if he'd be willing to create a numerological analysis for
me. I also asked him if he had seen either Pi or A Beautiful Mind, and
what he thought of them. If he replies, I'll be sure to post it.
I baked many pumpkin pies for Shawn (he likes pumpkin pies). I rubbed
pumpkin pie all over my breasts for him, and my breasts turned orange.
I am a pumpkin for Shawn.
posted by Trisha Blondie on July 24, 2002 10:41 PM
Um, that's swell. So, you're in love with him?
Shawn once went to a funeral for a Jehovah Witness that shot himself
and the lemon tarts were very bad, they were not only sour but were
rubbery as well. Shawn said that the guy was some kind of Jehovah
Witness prophet, he saw in advance that the lemon tarts at his funeral
were to be very very bad, and so he shot himself. Shawn said that he
never ate pumpkin pie at a funeral but would like to some day. Shawn
likes pumpkin pie and so I have been practicing to make very good
pumpkin pies.
posted by Trisha Blondie on July 25, 2002 02:49 PM
Shawn said that the lemon tarts were sour, bitter and rubbery.
I don't think this guy takes notes. I think he has Total Recall, and
it has driven him insane...
Oh... I almost forgot... I didnt spend thousands of dollars a day
tormenting Daryl... We got a deal on tormenting that fiscal year, it
only came to about 37cents a day....
Mr. Kabatoff attempts to portray himself as a victim, but in fact he
is a violent predatory pedophile who is well known to his local law
enforcement. In his post to multiple newsgroups with the subject
Collecting Mail For The Coming Anti-Christ, he encourages mothers to
send him photos of their naked daughters. Mr Kabatoff explains, I
personally did not want photographs being mailed to (the coming
Ant-Christ) that were of underage children unless the parent was
signing consent. He is banned from virtually all the shopping malls
in his community because he stalks young people and sexually harasses
them. He has an extensive arrest record which includes sexual
molestation charges. He's been hospitalized in mental institutions
about his contact with young girls in many posts. Search newsgroup
archives for posts by him containing the word nubile. As part of his
harrassment, he provides personal details in a public forum, such as
the real names of real children, in these and other posts. About one
wanted her and her sister dead.
He not only curses children and prays for their death in his posts, he
also enjoys attending the funerals of young people: And so, since
nubile sweeties are found in greatest abundance at the funerals of
high school students, then it is the funerals of high school students
that make the very very best funerals, especially if there is food...
I stuff my face (and my pockets) with all the good food and look at
all the pretty nubile sweeties and have the time of my life..
r=&ie=UTF-8&scoring=d&selm=LfXN8.63042%24R53.25142039%40twister.socal.rr.
com&rnum=1
Many of his posts are sent to alt.teens.advice. However, he liberally
missives to countless newsgroups. Some people HAVE problems and some
folks ARE problems. Don't dismiss Mr. Kabatoff as a harmless nut. When
he sends these posts to any newgroup, please help by reporting him to
I knew of him when I was attending the University of Saskatchewan.
He'd hang out in the Arts computer lab and all you'd see is screens of
numbers racing by on his laptop. I have an original copy of his
Collecting Mail for the Coming Anti-Christ pamphlet, and have seen
him be hauled away by campus security on more than one occasion. My
friends and I refer to him as Crazy Number Man.
I've been posting to (and about) Shawn for over two years with big
gaps in between. He has seen Pi and didn't like it and didn't think it
resembled him at all. (Wrong, it fits him to a tee) He doesn't have
total recall and has stated that he travels with a lap top to notate
items. Also, he uses cut n' paste a lot if you read all the way
through his ramblings. He is anti-social as shown by his angry
statements towards those who, by his own admission, have been kind
(but not kind enough) to him. Still, he's intelligent and seems to be
able to take a joke on occassion. That's where I came in.
ALOHA
Reply to group
(Unsolicited e-mail is deleted from the server unread
if it comes from anyone not already in my addressbook.
I'll never even see it)
===
Subject: Appropriateness of Principles of Real Analysis by Aliprantis
I wanted to study graduate level Real Analysis on my own this Summer. I
found
this book and noticed that it had a comprehensive solutions manual to go
along
with it. One thing caught my attention when I was reading the synopsis on
amazon.com: focus on measure theory via the semiring approach. Is
this
different than the normal approach to measure theory? I don't really want
to
learn a non-standard method and then start all over again next year.
===
Subject: Re: Cantor: ignorant, harmless fool or intentional liar?
> Wittgenstein - now there's a level headed guy for you.
> I really do not understand why anyone takes Wittgenstein seriously. Is
> puzzlement.
> Bob Kolker
Perhaps it's because he taught Turing logic, or came up with the truth
incredibly original thoughts, or taught a course called Philosophy
for Mathematicians, or was just a straight up bad ass?
'cid 'ooh
===
Subject: Re: Cantor: ignorant, harmless fool or intentional liar?
> On the other hand, Sagan in this chapter talks about the fallacy of
> argument from authority, which in view of your slavish quotes from
> Wittgenstein, you might want to brush up on (I'd include your quotes from
> Hilbert and Sagan, but you don't seem to understand what they are
saying.)
You sound like you might know a thing or two. Why does anyone take what
Wittgenstein says seriously?
Bob Kolker
===
Subject: Re: Cantor: ignorant, harmless fool or intentional liar?
> On the other hand, Sagan in this chapter talks about the fallacy of
> argument from authority, which in view of your slavish quotes from
> Wittgenstein, you might want to brush up on (I'd include your quotes
from
> Hilbert and Sagan, but you don't seem to understand what they are
saying.)
> You sound like you might know a thing or two. Why does anyone take what
> Wittgenstein says seriously?
Actually I take what he says seriously, the little I can understand. I
commented on the slavishness of the quote, not its content. The OP also
commented derisively on the mental states of Cantor and Goedel but
conveniently left out Wittgenstein's own leanings in that direction.
===
Subject: Re: Cantor: ignorant, harmless fool or intentional liar?
> Aus dem Paradies das Cantor uns geschaffen hat soll niemand uns
> vertreiben k.9annen.
> I would rater read this as follows:
> Hopefully nobody will be able to expel us from the paradise ...
In context, however, it is clear that he is expressing a
determination to preserve Cantor's paradise. The comment is made, it
should be noted, in the very same paper (On the infinite in English
translation) where Hilbert presents his finitism, according to which
the infinite is nowhere to be found in either physical or mathematical
reality. This view did not prevent him from describing Cantor's set
theory as the most admirable flower of the mathematical intellect and
in general one of the highest achievements of purely rational
human activity.
===
Subject: Re: Cantor: ignorant, harmless fool or intentional liar?
>>Hopefully nobody will be able to expel us from the paradise ...
> In context, however, it is clear that he is expressing a
> determination to preserve Cantor's paradise. The comment is made, it
> should be noted, in the very same paper (On the infinite in English
> translation) where Hilbert presents his finitism, according to which
> the infinite is nowhere to be found in either physical or mathematical
> reality. This view did not prevent him from describing Cantor's set
> theory as the most admirable flower of the mathematical intellect and
> in general one of the highest achievements of purely rational
> human activity.
Yes, and he also denied the human possibility to grasp the essence of
the infinitely large as well as the infinitely small.
Those who are interested in the original Cantor papers (in Latin, German
and French) might look into a link currently provided at
de.sci.mathematik by Hermann Kremer:
http://134.76.163.65/agora_docs/49439TABLE_OF_CONTENTS.html
In brief (p.116-18): Cantor's paradise is a construct out of numbers
that essentially deviates from continuum.
Admittedly, I am not a mathematician. While my physical reasoning
collided with common tenets dating back at least to Cantor, I am not
aware of any case where stuff like cardinality proved immediately useful
in practice. I wonder why there is so much fruitless criticism against
what I consider a mathematical caprice.
I would appreciate those who could tell me whether or not Robinson with
his *IR took issue against the holy caprice and offered an alternative
which saves Buridan's donkey.
Eckard Blumschein
===
Subject: Re: Cantor: ignorant, harmless fool or intentional liar?
> Admittedly, I am not a mathematician.
As you keep insisting on demonstrating.
But why do you do so in sci.math?
--
Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html
Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9
Francis Wheen, _How Mumbo-Jumbo Conquered the World_
===
Subject: Re: Cantor: ignorant, harmless fool or intentional liar?
> While my physical reasoning
> collided with common tenets dating back at least to Cantor, I am not
> aware of any case where stuff like cardinality proved immediately useful
> in practice.
It proved immediately useful in mathematical practice, for example
by establishing that almost all real numbers are transcendental.
===
Subject: Re: Is English Abelian?
Originator: dmoews@ccrwest.org (David Moews)
|When I was still taking mathematics seriously, there was a conjecture
|being worked on by some of the graduate students and research fellows
|in their spare time.
|
|Consider the free group W generated by all the letters a to z (only a
|to z, no accents, etc.) Now define the equivalence E using u ~ v [E]
|iff u and v are English words (Standard American English spelling;
|proper nouns excluded!) AND are permutations of each other.
|(If you'd rather, replace Standard American English with British. But
|don't mix the spellings.)
|
|Examples
| - tea = ate = eat
| - boer <> bore since Boer is in fact a proper noun
| - theater <> theatre since theatre is the British spelling.
|
|Conjecture: The quotient group W/E is abelian
|As a start: ore = roe => or = ro
|
|This is of course not the kind of problem that brings fame or fortune.
|
|Does anybody know whether or not it's been solved?
As far as I know, this conjecture has not been resolved. For an earlier
discussion of this problem, you can look at a thread on rec.puzzles
with subject `Group Theory' around 2000-VII-5
--
David Moews dmoews@xraysgi.ims.uconn.edu
===
Subject: Re: infinite series overflow problem
> fn=summation(an*x^n)
> but I want to use more number of terms in my apllication (may be
> around 500 terms) and coefficients (an) are of alternative sign,
a(2n+1)=0
I assume you mean alternating signs? That is numerically unstable --
problems. One of the reasons why power series are not used to
evaluate functions for large arguments.
> is it possible to overcome this problem by normalizing x??
If the an are sufficiently small, sure. Just set bn = an*100^n and
calculate fn = summation(bn*(x/100)^n). If they are not, you will
need to rewrite your series in some other way (depends on the an) or
solve the underlying problem with other methods.
--
+--+
+--+|
|+-|+ Christopher Creutzig (ccr@mupad.de)
+--+ Tel.: 05251-60-5525
===
Subject: Re: MuPAD 2.5.3 and Scilab errors, etc.
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision:
1.9 primary) id i3LF0gH02906;
Hello ,
I have recently downloaded the Mupad light v2.5.3 for windows with
scilab. However I similarly get the error message when trying to
Execute scilab::start(), returns Error: Unknown slot
scilab::start [slot], as does scilab::avail().
I have checked scilab is there and it is in the packages
sub-directory, uner sciface in the program files in the root
directory. Pardon my stupidity but any suggestions on how I can sort
this one out.
Will.
>>I'm having trouble using MuPAD's (2.5.3) Scilab integration. The
package
>>seems to be loaded (*maybe*), but it's decidedly not working.
>There exist a problem with MuPAD and the Scilab link if there is no
>ld installed on the machine. Then Scilab cannot load it
communication
>modules which are necessary for the link. We recognized this problem
>too late. So be sure that ld is installed on your system and it is
>accessable via your program search path. So try 'which ld' to see if
>it is there. BTW for the next version of the link it is not
>necessary to have the linker ld installed.
>>Executing scilab::start(), however, returns Error: Unknown slot
>>scilab::start [slot], as does scilab::avail().
>That's really strange. Are you sure the Scilab package is installed?
>>On a completely different note, I have Scilab 2.7 installed on the
same
>>machine, and would love to be able to use it with MuPAD. Is there
any way
>>I can edit the included Scilab .mu files and/or UDX scripts to use
it
>>instead?
>You can use Scilab 2.7 by setting the environment variable MUP_SCI
>to the path where Scilab is installed. This is the Scilab main
directory
>where e.g. the Scilab bin directory can be found. MuPAD then uses
this
>Scilab distribuiton and it works together with Scilab 2.7
>If your problem is not solved by installing ld please contact me
directly
>so that we can try to solve your problem.
>ToM
===
Subject: how jump to in derive 5
Does anyone know how to jump to an expresion number/name in Derive 5
for
Wiondows as is done in Derive for Dos by simply typing a j and
get
to?.
Else
2) does anyone know a source for Derive for Dos 3 with 373 page User
Manual, for my students?
3) Is any other program quite like that Derive 3?
4) When does Dos copyright expire; is it a perpetual monoply?
===
Subject: Re: how jump to in derive 5
Sorry to be so vague in 3}, meant quite like its succint handling of 3 and
4
row matrices and vector list of symbolic elements.
> Does anyone know how to jump to an expresion number/name in Derive
5
for
> Wiondows as is done in Derive for Dos by simply typing a j and
get
> to?.
> Else
> 2) does anyone know a source for Derive for Dos 3 with 373 page User
> Manual, for my students?
> 3) Is any other program quite like that Derive 3?
> 4) When does Dos copyright expire; is it a perpetual monoply?
===
Subject: Mupad 2.5.3 and Scilab ????
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision:
1.9 primary) id i3LFrwj13231;
Hello ,
I have recently downloaded the Mupad light v2.5.3 for windows with
scilab. However I get the error message when trying to
Execute scilab::start(), returns Error: Unknown slot
scilab::start [slot], as does scilab::avail().
I've tried call scilab, load scilab etc..... but frustratingly to no
avail.
I have checked scilab is there and it is in the packages
sub-directory, under sciface in the program files in the root
directory. Pardon my stupidity but any suggestions on how I can sort
this one out.
Will.
===
Subject: Appropriate contour for an improper integral containing a Bessel
function Jo
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision:
1.9 primary) id i3LGvTs25348;
The following integral (Ewing and Jardetzky page 44 eq 2.86b) is
causing us a lot of problems. Any help is very much appreciated.
The integral is from 0 to inf over the variable k.
The integrand is
k*b^2*(k^2-b^2)*exp(-sqrt(k^2-a^2)*h)*Jo(k*r)/((2*k^2-b^2)^2-4*k^2*sqrt(k^2-
a^2)*sqrt(k^2-b^2))
a and b are complex numbers. r and h > 0 and real.
This is an odd function. It has two poles, four branch points, four
hyperbolic branch cuts (fig 2.6 from the above reference). The biggest
problem we have is the presence of the Bessel function. I don't know
what kind of a contour would be appropriate for an improper integral
containing an integrand that has a bessel function. Bessel J function
blows up for imaginary arguments. I think that would be a serious
problem if I use a circular contour and make the radius go to inf to
make the integral vanish on this circular contour. It also causes
problems when finding the conribution on the branch cuts which go to
+/- inf.
Yigit Yazicioglu
University of Illinois at Chicago
Acoustics and Vibrations Laboratory
===
Subject: Re: Replacement to MMA and Maple
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision:
1.9 primary) id i3LIDkB04514;
You are right 100%.
A person who responded is the author of both MACSYMA...
and MockMMA ;)
Best wishes,
Vladimir Bondarenko
http://www.cybertester.com/
http://maple.bug-list.org/
http://www.CAS-testing.org/
......................................................
===
Subject: Re: Replacement to MMA and Maple
superb new graphics, hundreds of graphical attributes allowing to
> manipulate all details of a plot interactively, export to MS Word,
> etc. If I want nice graphics, for example, I will use a software which
> is specialized in graphics.
Export to Word is a very isolated interface feature that doesn't
affect the rest of the system at all. If you don't want graphics,
well, don't use them. In that case, MuPAD 2.5.3 will be almost the
same for you as 3.0 -- just much cheaper.
> I do not understand this need for multipurpose softwares (this does
I do understand the requirement of having a mathematical graphics
tool that knows a whole lot about mathematics -- especially after
having (been part of the group that) implemented such a tool. For
starters, what's a CAS there for? Manipulating mathematical objects,
usually formulae, right? And if you want to visualize what you have
just done inside the CAS, it is pretty helpful if the system you use
for display knows about the structures you have just been using. I
certainly wouldn't start using MuPAD for non-mathematical graphics or
technical drawings, I've got Metapost, Photoshop etc. for that. But
for graphing mathematical topics, I haven't found a better (in my
personal opinion, of course) tool yet.
> Who will bring his car to a garage where the mechanician use a
> Swiss-army knife as only tool?
To be honest: I wouldn't care. As long as the result is good, he
need not use any tools at all, as far as I am concerned.
> Why is it different with sofwares?
Software is different from just about anything else. I fully agree
that there is really no need for do-it-all software such as an editor
trying to do typesetting, animation, maybe e-Mail, web layout etc. at
the same time. OTOH, there are tasks which require a common basis.
(How should a system that has never seen something called besselJ
plot besselJ(v, x) for v in [0, 10] and x in [-10,10]?) These need
not be inside the same binary, but it is certainly reasonable to
share a common code basis in these cases. MuPAD is mostly written in
the MuPAD language, so in this case it makes sense to combine the two
into a single product. Note that only those parts actually used
will ever be loaded into memory.
On the other hand, me and probably you are not standard users. It
seems that most users want to have everything in one interface, which
leads to CASs that are praised for their ability to produce
more-or-less printer-ready papers or online presentations and I can
understand a commercial company that feels this sort of praise (of
competitors' software) and the resulting explicit customer demands as
important and spend time and money implementing things the two of us
feel to be not really part of a CAS. (Who needs yet another editor
inside every programmable program?)
>> I'm sorry, but the more exotic special functions have probably
>> never
>> been asked for in MuPAD, at least I haven't seen requests, so you
>> won't find e.g. StruveH. (Although MeijerG and hypergeom are there,
>> so you might be able to get some of your functions.)
> I agree, the market is not big enough for such a request. That's why,
There is only about a dozen people involved in developing MuPAD.
Sorry, but we simply can't do everything and therefore need input
from our users. Note that I am specifically not talking about
customers. It is very rare that I care whether a user spent money
on MuPAD or uses one of the freely available versions. (Then again,
I'm not doing interface programming, where the differences lie.)
> in my opinion, a high-quality CAS has to be free in order to avoid, or
> at least limit, the market size effects.
Write one. Or take part in improving one of the existing free
systems. Or ask SciFace about a free license for the MuPAD library.
--
+--+
+--+|
|+-|+ Christopher Creutzig (ccr@mupad.de)
+--+ Tel.: 05251-60-5525
===
Subject: Re: product of Bessel functions, mean value of exp(i*periodic
function)
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision:
1.9 primary) id i3LIDkr04518;
look on p. 148 (section 5.41) of Watson's book on Bessel Functions.
Or look in the index under products. I'm interested in the same
integral. If you have figured out how to do it, let me know...
Ariel
>Hi everybody,
>Is there anyone knowing how to express the product of two Bessel
>functions (ie Jm(a*x)*Jp(b*x)) with a series of Bessel functions ?
>In fact the main problem to solve is to calculate the integral :
> / Pi
> |
> |
> 1 | i* Sum (over q) of Aq cos(qx)
> --- | dx e
> 2Pi |
> |
> |
> _/ (-Pi)
>where Aq are known coefficients
> i is the complex number such as iî=-1
> cos is the function cosinus
>I ve tried to search a solution but I hasn't found any
>Thierry Ferrus.
===
Subject: Re: A proof for Goldbach's conjecture
>I' ve got the proof of the Goldbach's conjecture.
What a coincidence! I've got a counterexample.
You can't write 389965026819938 as the sum of two primes.
Take a look at your proof and ask yourself how _your proof_
proves that I'm mistaken.
dave
[Followups to sci.math]
===
Subject: MathML on Mathforge.net
Well, we're proud to announce our message boards and wikis are
mathml-enabled. Type a Tex-like string, e.g. $e^(ipi)+1=0$ in your
message and you will see it marked up like actual math!
Go to http://www.mathforge.net right now and try it out! Additionally,
you'll find the latest in math news.
Soon, we'll have the capability to create marked up text online, which
should be useful for homework, marking up class notes, and
collaborating on documents.
Joe Kauzlarich
http://www.mathforge.net
Tasteful Mathematics for Sensitive People
===
Subject: Re: Trigonometric Equations
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision:
1.9 primary) id i3LCvmh14157;
>can someone help me with the equation: cos2(x+55)=0.4673, and please
>include some workings!
First- be clear! Do you mean cos of 2 times (x+ 55) (which you
ought to write as cos(2(x+55))) or cos squared (x+55) (which you ought
to write as cos^2(x+55)?
If it is the former, it's easy: Take the inverse cosine (also called
arccosine): 2(x+55)= arccos(0.4673)= 62.14 (according to my
calculator) so x+55= 62.14/2= 31.07 and x= 31.07- 55= -23.93 degrees.
If it is the latter, it's still easy: First take the square root of
each side: cos(x+55)= sqrt(0.4673)= 0.6836. NOW find the inverse
cosine: x+ 55= arccos(0.6836)= 46.87 so x= 46.86- 55= -8.13 degrees.
By the way: since there is no mention of angles and you are
treating cosine as a function, normally I would assume that I should
have my calculator in radian mode. However, that 55 made me think
it should be in degrees. You might want to check on that.
===
Subject: Re: Heating efficiency
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision:
1.9 primary) id i3LD8CF16648;
>Our textbook (Washington, /Basic Technical Mathematics/, 7th ed)
>sets the following problem (sec 21.2, prob 47), and I 've assigned
>it to my students. But I don't think there's actually enough
>information to solve it:
>One heating unit uses x gallons of fuel with 72% efficiency, and
>another uses y gallons at 90% efficiency. If 135,000 Btu of heat is
>delivered by these units together, express y as a function of x.
>The problem wants students to begin with
> .72x + .90y = 135,000
>and then solve for y. But I question whether that's correct.
>Shouldn't there be some sort of conversion factor from gallons to
>Btu? Surely at 100% efficiency, 1 gallon of fuel delivers rather
>more than one Btu!
>--
>Stan Brown, Oak Road Systems, Cortland County, New York, USA
> http://OakRoadSystems.comnecessary. -- IRS Form 1040 line 23 instructions
Unless efficiency is taking that into account. It is quite
possible that saying that a fuel is 90% efficient is intended to
MEAN that one-gallon 90% of that fuel produces 90% of the BTUs
one-gallon of 100% efficient fuel would. You don't say at what level
this course is, but I would think that unless this is specifically a
course for heating engineers, that kind of simplification would be
reasonable. Of course, it would a good idea to SAY that in the
problem!
===
Subject: Re: Heating efficiency
> Unless efficiency is taking that into account. It is quite
>possible that saying that a fuel is 90% efficient is intended to
>MEAN that one-gallon 90% of that fuel produces 90% of the BTUs
>one-gallon of 100% efficient fuel would.
I think your explanation makes sense. (The 135,000 applies only to
kerosene, not other type of heating fuel.)
>Of course, it would a good idea to SAY that in the problem!
Indeed yes!
> You don't say at what level
>this course is, but I would think that unless this is specifically a
>course for heating engineers, that kind of simplification would be
>reasonable.
It's second-semester technical math at a two-year college. Students
are aiming at careers in electrical technology or construction
technology.
Instead of making it homework, I'll use it as a classroom example,
where I can make the additional explanations. With the extra info, I
think it's a decent example.
--
Stan Brown, Oak Road Systems, Cortland County, New York, USA
http://OakRoadSystems.com
An expense does not have to be required to be considered
necessary. -- IRS Form 1040 line 23 instructions
===
Subject: Re: Probability issue; please help!
>probability
>Suppose that 52% of all register voters prefer George Bush to Al Gore.
>a. In a random sample of 100 voters, what is the probability that the
>sample will indicate that Bush will win the elections (that is, there
>will be more votes in the sample for Bush?)
.6184
>b. In a random sample of 100 voters, what is the probability that the
>sample will indicate that Gore will win the election?
.3082
>c. In a random sample of 100 voters, what is the probability that the
>sample will indicate a dead heat? (50-50)
.0735
>d. In a random sample of 100 voters, what is the probability that
>between 40 and 60 (inclusive) voters will prefer Bush?
.9500
There -- now your homework's done without any need for thought or
work on your part.
--
Stan Brown, Oak Road Systems, Cortland County, New York, USA
http://OakRoadSystems.com
An expense does not have to be required to be considered
necessary. -- IRS Form 1040 line 23 instructions
===
Subject: Re: DeMoivre's Theorem
>tan t = -2(sqr 3)/2 = -(sqr 3); arc tan t = -pi/3 already is in Quad IV.
>Why the shift to another branch of arctan?
So that the principal root will have a positive angle.
--
Stan Brown, Oak Road Systems, Cortland County, New York, USA
http://OakRoadSystems.com
An expense does not have to be required to be considered
necessary. -- IRS Form 1040 line 23 instructions
===
Subject: Formula please
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision:
1.9 primary) id i3LGEnH18900;
opening ammount = O
number of years = X
closing amount = C
average annual % return on investment is ?
Perry
===
Subject: Re: Formula please
> opening ammount = O
> number of years = X
> closing amount = C
> average annual % return on investment is ?
Try to work on it. Assume the annual return
on investment is r. Then after one year
the amount you have is
O + rO = O(1 + r)
After two years the amount you have is
(amount after one year) + r(amount after one year)
= (amount after one year)(1 + r)
= [O(1 + r)](1 + r)
= O(1 + r)(1 + r)
= O(1 + r)^2
You should be able to carry that forward to establish
the relationship beteen O, X, C, and r, and then
solve the resulting equation for r.
--
Rich Carreiro rlcarr@animato.arlington.ma.us
===
Subject: Re: Formula please
>>opening ammount = O
>>number of years = X
>>closing amount = C
>>average annual % return on investment is ?
> Try to work on it. Assume the annual return
> on investment is r. Then after one year
> the amount you have is
> O + rO = O(1 + r)
> After two years the amount you have is
> (amount after one year) + r(amount after one year)
> = (amount after one year)(1 + r)
> = [O(1 + r)](1 + r)
> = O(1 + r)(1 + r)
> = O(1 + r)^2
> You should be able to carry that forward to establish
> the relationship beteen O, X, C, and r, and then
> solve the resulting equation for r.
That depends, is the annual percentage supposed to be compound or simple
interest?
--
Will Twentyman
email: wtwentyman at copper dot net
===
Subject: Re: Formula please
>>opening ammount = O
>>number of years = X
>>closing amount = C
>>average annual % return on investment is ?
> Try to work on it. Assume the annual return
> on investment is r. Then after one year
> the amount you have is
> O + rO = O(1 + r)
After two years the amount you have is
> (amount after one year) + r(amount after one year)
> = (amount after one year)(1 + r)
> = [O(1 + r)](1 + r)
> = O(1 + r)(1 + r)
> = O(1 + r)^2
You should be able to carry that forward to establish
> the relationship beteen O, X, C, and r, and then
> solve the resulting equation for r.
That depends, is the annual percentage supposed to be compound or simple
> interest?
The original question, asking for the annualized rate of return, is
usually answered in terms of the compound interest rate, though for
small enough rates and short enough periods, simple and compound rates
do not differ enough to quibble over.
===
Subject: Re: Formula please
>>opening ammount = O
>>number of years = X
>>closing amount = C
>>average annual % return on investment is ?
>Try to work on it. Assume the annual return
>on investment is r. Then after one year
>the amount you have is
> O + rO = O(1 + r)
>After two years the amount you have is
> (amount after one year) + r(amount after one year)
> = (amount after one year)(1 + r)
> = [O(1 + r)](1 + r)
> = O(1 + r)(1 + r)
> = O(1 + r)^2
>You should be able to carry that forward to establish
>the relationship beteen O, X, C, and r, and then
>solve the resulting equation for r.
>>That depends, is the annual percentage supposed to be compound or simple
>>interest?
> The original question, asking for the annualized rate of return, is
> usually answered in terms of the compound interest rate, though for
> small enough rates and short enough periods, simple and compound rates
> do not differ enough to quibble over.
True. Knowing the context of the problem would make the approach more
obvious.
--
Will Twentyman
email: wtwentyman at copper dot net
===
Subject: probability
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision:
1.9 primary) id i3LGvTb25352;
The grades on the midterm examination given in a large managerial
statistics class are normally distributed with mean 75 and standard
deviation 9. The instructor of this class wants to assign an A grade
to the top 10% of the scores, a B grade to the next 10% of the scores,
a C grade to the next 10% of the scores, a D grade to the next 10% of
the scores, and an F grade to all the scores below the 60th percentile
of this distribution. For each possible letter grade, find the lowest
acceptable score within the established range. For example, the
lowest acceptable score for an A is the score at the 90th percentile
of this normal distribution
===
Subject: Re: probability
>The grades on the midterm examination given in a large managerial
>statistics class are normally distributed with mean 75 and standard
>deviation 9. The instructor of this class wants to assign an A grade
>to the top 10% of the scores, a B grade to the next 10% of the scores,
>a C grade to the next 10% of the scores, a D grade to the next 10% of
>the scores, and an F grade to all the scores below the 60th percentile
>of this distribution. For each possible letter grade, find the lowest
>acceptable score within the established range. For example, the
>lowest acceptable score for an A is the score at the 90th percentile
>of this normal distribution
Please post what you have done to solve this problem on your own.
--
Stan Brown, Oak Road Systems, Cortland County, New York, USA
http://OakRoadSystems.com
An expense does not have to be required to be considered
necessary. -- IRS Form 1040 line 23 instructions
===
Subject: Re: Finding the vertex of a parabola based on the foci and a point
>> The mounting point is a point on the hyperbolic mirror itself, so its
>> coordinates (20,20) can be used in the formula for the hyperbola
itself.
>> The problem I had was that, after putting everything into the general
for
>> of a hyperbola, I had a 4th power equation that just seemed to not be
>> right:
>>
>> 400/a^2 - 400/(400-a^2)^2 = 1 ; basic setup, with (400-a^2) being b
> I think this may be your problem. If the hyperbola has equation
x^2/a^2 - y^2/b^2 = 1, and the focus is at (c,0) then a^2 + b^2 = c^2,
> so you should have derived
400/a^2 - 400/(400-a^2)= 1
> That would actually be 400/a^2 - 400/(400-a^2)^2 = 1.
So that b^2 = (c^2 - a^2)^2?
===
Subject: Fourier Graphing
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision:
1.9 primary) id i3LL2mk28228;
Try this site. http://home.messiah.edu/~barrett/c3_fourier.html
===
Subject: probability/statistics ouch
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision:
1.9 primary) id i3LMY1203379;
need yesterday. This is a data analysis class utilizing excel
spreadsheet and statpro. I'd be satisfied with an answer in excel
spreadsheet format, tables, explanation, the whole nine yards
Suppose that a popular hotel for vacationers, in Orlando, Florida has
a total of 300 identical rooms. Like many major airline companies,
this hotel has adopted an overbooking policy in an effort to maximize
the usage of its available lodging capacity. Assume that each
potential hotel customer holding a room reservation, independently of
the hotel on a given night with the probability of 0.15.
a. Find the largest number of room reservations that this hotel can
book and still be at least 95% sure that everyone who shows up at the
hotel will have a room on any given night
b. Given that the hotel books the number of reservations found in
answering part a, find the probability that at least 90% of the
available rooms will be occupied on a given night.
c. Given that the hotel books the number of reservations found in
answering part a, find the probability that at most 90% of the
available rooms will be occupied on a given night.
d. How does your answer to part a change as the required assurance
rate increases from 95% to 97%? How does you answer to part a change
as the required assurance rate increases from 95% to 99%.
varies between 5% and 25% (in increments of 5%) Assume now that the
required assurance rate is held fixed at 95%.
===
Subject: Re: A proof for Goldbach's conjecture
>I' ve got the proof of the Goldbach's conjecture.
> What a coincidence! I've got a counterexample.
> You can't write 389965026819938 as the sum of two primes.
> Take a look at your proof and ask yourself how _your proof_
> proves that I'm mistaken.
> dave
> [Followups to sci.math]
5569 + 389965026814369. Brute Force.
Do I win the Donald Trump price?
===
Subject: CombinatoricaGraphics packages for MMA. and other q's
I posted this in steve's forum, but i wanted to post in this group
also since many of you are mathematicians and experienced with mma.
package.
below is a replica of email sent to mathforum. i hoe you guys don't
ind me cross posting with same email.
thank you all for helpful comments.
sean
-----------
Hello group and david.
I love david's Package for improving Combinatorica
graphics.
and obviously by the looks of this email, and the fact
that I'm writing, yes.. I have a few problems.
please consider the following digraph.
In[12]:=
lg = Graph[{
{{1, 2}}, {{2, 3}},
{{3, 2}}, {{3, 1}},
{{1, 4}}, {{1, 4}}, {{1, 5}}, {{1, 5}},
{{4, 5}}, {{4, 2}}, {{4, 2}},
{{5, 2}}, {{5, 2}}, {{5, 4}}, {{5, 4}}},
{{{0.0, 1.5}, VertexLabel -> a1},
{{0.0, 0.0}, VertexLabel -> b2},
{{0.5, 0.75}, VertexLabel -> c3},
{{-0.5, 0.75}, VertexLabel -> d4},
{{-1.5, 0.75}, VertexLabel -> e5}}, EdgeDirection-> On]
ShowGraph[lg]
CombinatoricaPlot[
{DrawGraphEdges[lg][All],
DrawGraphVertices[lg][All]},
Background -> White];
above will produce two graphics. onedefault and one
produced by david's new package.
1. How come the CombinatoricaPlot isn't drawing the
directed edges using DrawGraphBowArrow as a default?
2. As far as I understand it, a tour is a path in a
digraph where you only cross every vertex once,
before returning to the starting pt( in this case, a1)
How do I show all tours that are possible withthe
digraph above?
3. how do I show that all the tours are isomorphic?
or is that an obvious question?
sean
===
Subject: Re: how jump to in derive 5
You might try asking on the yahoo e-Derive Users Group:
http://groups.yahoo.com/group/eDUG/
> Sorry to be so vague in 3}, meant quite like its succint handling of 3
and
4
> row matrices and vector list of symbolic elements.
> Does anyone know how to jump to an expresion number/name in Derive
5
> for
> Wiondows as is done in Derive for Dos by simply typing a j and
get
> to?.
> Else
> 2) does anyone know a source for Derive for Dos 3 with 373 page
User
> Manual, for my students?
> 3) Is any other program quite like that Derive 3?
> 4) When does Dos copyright expire; is it a perpetual monoply?
===
Subject: Re: Replacement to MMA and Maple
First, a precision: I have nothing against
MuPaD in particular, and I didn't want to
start a war.
Now, let me precise my point.
Yes, it is very nice to plot easily
within a mathematical software. It is
very convenient and helps a lot for
debugging. I think that CAS should be
able to do graphics. But not pretty
graphics.
I am using MATLAB and plotting every
days. I am very happy with its basic
features.
On the other, each time I try to use
MATLAB for doing pretty graphics, i.e.
high quality for publication, it is
a huge struggle. I am spending a LOT
of time trying to do clean graphics,
the way I want them to be.
Moreover, what I see is NEVER what I get.
MATLAB's basic graphic features are nice,
advanced ones suck. It's the same with Maple,
and probably with MuPaD too.
I repeat: I do not understand this effort
for advanced graphic capabilities.
For other features, I think it is important
that programs can exchange datas, via some
normalized protocol. I don't think it is
necessary for a program to do everything.
Using the right tool for the right job
is important. That's what professionals
in any field do, except in informatics,
it seems.
D.
===
Subject: Re: Laplace transformation software?
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision:
1.9 primary) id i3MDxE722906;
It seems that Matlab herits the symbolic mathematics modules from
Maple. I wonder whether there exists any big difference (for example,
the intrinsic function library for Laplace transformation) between
Matlab and Maple especially in symbolic mathematics.
Why do I ask the question? Because the problem I will solve is
difficult and also involves special functions. If you want to feel the
problem, I think the paper in AICHE JOURNAL,1981,27(6):1032 could be
an axample. Therefore, the modules of the 'laplace/inverse laplace'
transformation are what I most want to know. After all, the inverse
laplace transformation in which the special functions are involved are
curbsome. However, I found that the Maxima is far inadequate in
laplace transformation.
Looking forward to your comments or suggestions.
===
Subject: Re: Laplace transformation software?
> It seems that Matlab herits the symbolic mathematics modules from
> Maple. I wonder whether there exists any big difference (for example,
> the intrinsic function library for Laplace transformation) between
> Matlab and Maple especially in symbolic mathematics.
I'm not 100% sure, but according to
http://www.mathworks.com/access/helpdesk/help/toolbox/symbolic/symbolic.html
Maple's inttrans package (for integral transforms such as laplace) isn't
available for MATLAB. Maybe someone else can confirm.
> Why do I ask the question? Because the problem I will solve is
> difficult and also involves special functions. If you want to feel the
> problem, I think the paper in AICHE JOURNAL,1981,27(6):1032 could be
> an axample. Therefore, the modules of the 'laplace/inverse laplace'
> transformation are what I most want to know. After all, the inverse
> laplace transformation in which the special functions are involved are
> curbsome. However, I found that the Maxima is far inadequate in
> laplace transformation.
Maple's help on Laplace transform says (partial quoting):
Expressions involving a wide variety of functions including
exponentials, trigonometrics, Bessel functions, error functions, and
many others can be transformed.
Both laplace and invlaplace recognize the Dirac-delta (or unit-impulse)
function as Dirac(t) and Heaviside's unit step function as Heaviside(t).
Users can add their own functions to laplace's internal lookup table by
using the addtable function.
So for more exotic functions you're probably out of luck...
--
Thomas Richard
Maple Support
Scientific Computers GmbH
http://www.scientific.de
===
Subject: Re: Laplace transformation software?
> I'm not 100% sure, but according to
>
http://www.mathworks.com/access/helpdesk/help/toolbox/symbolic/symbolic.html
> Maple's inttrans package (for integral transforms such as laplace) isn't
> available for MATLAB. Maybe someone else can confirm.
Sorry, I was wrong here.
The Extended Symbolic Math Toolbox is not necessary for this task.
The Symbolic Math Toolbox contains fourier and ifourier.
You can find examples, starting from the URL above.
--
Thomas Richard
Maple Support
Scientific Computers GmbH
http://www.scientific.de
===
Subject: Re: Computer Algebra System
...
> I can only speak for MuPAD, where a power is represented like this:
> prog::exprtree(a^2)
> _power
> |
> +-- a
> |
> `-- 2
>Polynomials use a denser representation, where a reference to some
>arbitrary object (the coefficient) is followed by the exponent vector
>of the current term, stored in C integers. But general expressions
>are not stored as polynomials.
The denser vector of polynomials is perhaps minor.
> What do you mean by the exponent of a number? Integers are not
>stored as floats.
An exponent can put into all nodes. It would be an integer. It seems
like a good idea since most of the algorithms can't cope with an
exponent and shifting it into a node createss simple special cases.
...
>> * How is information about the identity of the CPU stored?.
> Why should it be stored at all?
There would be at least a stack for each CPU.
...
>> * Are there any pointers in expressions except to names and
>> subtrees?.
> Certainly in an OO system such as MuPAD each object has a pointer to
>its domain type. (Kernel types do not have a pointer but rather a
>small number, but that's an implementation detail.) Oh, and you did
>not include numbers, functions etc. in the list, which are of course
>pointed to as well.
In Ada 95 a record can be extended without changing its type and by
adding adjacent fields (with no pointers).
...
>> * What is the interpretation of out-of-bounds lengths of the array?.
> I'm sorry? Which array are you talking about?
Have you got the source code. A URL would be nice.
...
>> I request that you make availanle the source code that data structure
> You *request*? I certainly hope you did not really mean to use this
>word. If you did, please explain on which basis you think to have
>the right to request such a thing.
At this moment I do not have the file/document that shows exactly the
design of the data structure. The request has failed.
I request the reasoning for not getting source file released.
You said you were an employee so the request is one that could fail.
...
> It's not really hard to guess, no. Just invoke ?iszero and read the
>documentation.
The topic was made by me to be on the 3GL internal design of Mupad.
I.e. the exact design and not summaries or clarifications of source
code that is not in the public domain. You kept writing to
sci.math.symolic and advocating use of Mupad.
>> The webpage of the 2nd provided Mupad URL does not say that nodes
>> of expressions actually even contain a field saying what the mass
>> is. Is 3x implemented with 3 records or one, in Mupad ?.
> I don't understand what you mean by mass. 3*x is stored as an
>expression having the operands _mult (the operator symbol, available
>as op(3*x, 0), x, and 3.
An example of a problem with storing the 3 and the x in 3x, in different
nodes/records:
simplify (3*x - 5*y <= -y),
versus: simplify (x*3 - 5*y <= -y)
Having more atoms/nodes increases the number of cases. There is no need
to have a separate number but instead there is a dimensionless x that
has no name.
>> The data has been online in the past ?.
> Again, I do not understand what you mean. Please clarify.
A yes or no would be a preferred form for the response. I want to
get to the information speedily. Perhaps Mupad is not selling well
enough.
===
Subject: This week in the mathematics arXiv (29 Mar - 2 Apr)
Originator: bergv@math.uiuc.edu (Maarten Bergvelt)
Here are this week's titles in the mathematics arXiv, available at:
http://front.math.ucdavis.edu/
http://front.math.ucdavis.edu/submissions
This week in the mathematics arXiv may be freely redistributed
with attribution and without modification.
Titles in the mathematics arXiv (29 Mar - 2 Apr)
------------------------------------------------
AC: Commutative Algebra
-----------------------
math.AC/0403535
Xinxian Zheng: Monomial ideals arising from distributive lattices
math.AC/0403534
Juergen Herzog, Takayuki Hibi: Level rings arising from meet-distributive
meet-semilattices
math.AC/0403531
Susumu Oda: On unramified finitely generaed extensions of polynomial
rings
over a field
AG: Algebraic Geometry
----------------------
math.AG/0404012
Edoardo Ballico, Elizabeth Gasparim: Bundles near a negative curve
hep-th/0403272
Ron Donagi, Alon E. Faraggi: On the Number of Chiral Generations in Z2 X
Z2
Orbifolds
math.AG/0403553
Arsen Elkin: Hyperelliptic Jacobians with real multiplication
math.AG/0403550
Chad Schoen: Specialization of the torsion subgroup of the Chow group
math.AG/0403543
E. Artal, J. Carmona, J.I. Cogolludo, M.A. Marco: Invariants of
Combinatorial Line Arrangements and Rybnikov's Example
math.AG/0403530
Mike Develin, Jeremy L. Martin, Victor Reiner: Classification of Ding's
Schubert varieties: finer rook equivalence
math.AG/0403526
Henning Krause: The stable derived category of a noetherian scheme
math.AG/0403525
Razvan Veliche: Genus 2 mapping class groups are not Kahler
math.AG/0403519
Kieran G. O'Grady: Involutions and linear systems on holomorphic
symplectic
manifolds
math.AG/0403516
Christopher Deninger, Annette Werner: Vector bundles on p-adic curves and
parallel transport
math.AG/0403478
Igor Dolgachev, JongHae Keum: Finite groups of symplectic automorphisms
of
K3 surfaces in positive characteristic
math.AG/0403471
Ivan Dimitrov, Ivan Penkov: Ind--varieties of generalized flags as
homogeneous spaces for classical ind--groups
math.AG/0403464
Ivan Petrakiev: Specializing Multiple Points to Elliptic Curves
math.AG/0403460
H. Hakopian: The Multivariate Fundamental Theorem of Algebra and
Algebraic
Geometry
math.AG/0403459
H. Hakopian, M. Tonoyan: On an Algebraic System of Equations
math.AG/0403449
D. Arcara: A lower bound for the dimension of the base locus of the
generalized theta divisor
AP: Analysis of PDEs
--------------------
math.AP/0404011
Damiano Foschi: Maximizers for the Strichartz inequality
math.AP/0404006
Hiroyuki Chihara: The initial value problem for a third order dispersive
equation on the two-dimensional torus
math.AP/0404005
Hiroyuki Chihara: Third order semilinear dispersive equations related to
deep water waves
math.AP/0403549
Benjin Xuan: The solvability of Brezis-Nirenberg type problems of
singular
quasilinear elliptic equation
math.AP/0403507
Gregory Eskin, James Ralston: Inverse boundary value problems for systems
of
partial differential equations
math.AP/0403469
C M Doria: Boundary Value Problems for the $2^{nd}$-order Seiberg-Witten
Equations
math.AP/0403468
Alexandru Tamasan: On the scattering for the $bar{partial}$- equation
and
reconstruction of convection terms
AT: Algebraic Topology
----------------------
math.AT/0404003
Ezra Getzler: Lie theory for nilpotent L-infinity algebras
math.AT/0403540
Martin Jakob: A note on the Thom isomorphism in geometric (co)homology
math.AT/0403486
Ralph L. Cohen: Multiplicative properties of Atiyah duality
math.AT/0403451
J. Daniel Christensen, Daniel C. Isaksen: Duality and Pro-spectra
CA: Classical Analysis and ODEs
-------------------------------
math.CA/0404013
Allan Pinkus: Strictly Hermitian Positive Definite Functions
math.CA/0403533
Jonathan Coussement, Walter Van Assche: Gauusian quadrature for multiple
orthogonal polynomials
math.CA/0403532
Kelly Postelmans, Walter Van Assche: Multiple little q-Jacobi polynomials
math.CA/0403520
Jean Ludwig, Detlef Muller, Sofiane Souaifi: Holomorphic L^p-type for
sub-Laplacians on connected Lie groups
math.CA/0403510
Raimundas Vidunas: Expressions for values of the gamma function
math.CA/0403482
Alejandro S. Jakubi: The branching problem in generalized power solutions
to
differential equations
math.CA/0403481
George Gasper, Michael Schlosser: Some curious q-series expansions and
beta
integral evaluations
math.CA/0403476
Detlef Mueller, Christoph Thiele: Wave equation and multiplier estimates
on
ax+b groups
math.CA/0403472
S. Denisov, S. Kupin: The Szego class with a polynomial weight
CO: Combinatorics
-----------------
math.CO/0403551
R.M. Green, J. Losonczy: Schubert varieties and free braidedness
math.CO/0403546
Herbert E. Scarf, Kevin M. Woods: Neighborhood complexes and generating
functions for affine semigroups
math.CO/0403541
Francois Bergeron, Riccardo Biagioli, Mercedes H. Rosas: Inequalities
between Littlewood-Richardson Coefficients
math.CO/0403513
Frederic Bosio: Suppressing nonrevisiting paths
math.CO/0403508
Elchanan Mossel: Distorted metrics on trees and phylogenetic forests
math.CO/0403505
Bilal Khan, Kiran R. Bhutani, Delaram Kahrobaei: A graphic generalization
of
arithmetic
math.CO/0403502
Miklos Bona: The limit of a Stanley-Wilf sequence is not always an
integer,
and layered patterns beat monotone patterns
math.CO/0403494
Michael Joswig, Frank H. Lutz: One-Point Suspensions and Wreath Products
of
Polytopes and Spheres
math.CO/0403450
Vladimir Nikiforov: The asymptotics of strongly regular graphs
CV: Complex Variables
---------------------
math.CV/0403539
Joel Merker: On envelopes of holomorphy of domains covered by Levi-flat
hats
and the reflection principle
DG: Differential Geometry
-------------------------
math.DG/0404004
A. Rod Gover: Conformal de Rham Hodge theory and operators generalising
the
Q-curvature
math.DG/0403555
Andre Diatta: Left Invariant Contact Structures on Lie Groups
math.DG/0403547
Chongying Dong, Kefeng Liu, Xiaonan Ma, Jian Zhou: $K$-theory associated
to
vertex operator algebras
math.DG/0403545
Colin Guillarmou: Resonances and scattering poles on asymptotically
hyperbolic manifolds
math.DG/0403544
Ronaldo Garcia, Romildo Pina: Ricci Tensors with Rotational Symmetry on
R^n
math.DG/0403528
Nobuhiro Honda: Self-dual metrics and twenty-eight bitangents
math.DG/0403500
Ronaldo Garcia, Jorge Sotomayor: Lines of Mean Curvature on Surfaces
Immersed in R3
math.DG/0403485
Claus Gerhardt: The inverse mean curvature flow in ARW spaces--transition
from big crunch to big bang
math.DG/0403483
David L. Johnson, Penelope Smith: Partial regularity of mass-minimizing
Cartesian currents
math.DG/0403479
Bogdan Alexandrov: On weak holonomy
math.DG/0403473
Maxim Braverman, Valentin Silantyev: Kirwan-Novikov inequalities on a
manifold with boundary
math.DG/0403465
Daniel Azagra, Juan Ferrera: Proximal calculus on Riemannian manifolds,
with
applications to fixed point theory
math.DG/0403452
S.P. Novikov: On the Metric Independent Exotic Homology
DS: Dynamical Systems
---------------------
math.DS/0403552
Alberto Abbondandolo, Pietro Majer: When the Morse index is infinite
math.DS/0403537
Jose F. Alves, Armando Castro, Vilton Pinheiro: Backward volume
contraction
for endomorphisms with eventual volume expansion
math.DS/0403536
Jose F. Alves, Krerley Oliveira, Ali Tahzibi: On the continuity of the
SRB
entropy for endomorphisms
math.DS/0403523
the
Topology of Solenoidal Attractors of the Cylinder
math.DS/0403518
Stefano Marmi, Pierre Moussa, Jean-Christophe Yoccoz: The cohomological
equation for Roth type interval exchange maps
math.DS/0403501
Tien-Cuong Dinh, Christophe Dupont: Dimension de la mesure d'equilibre
d'applications meromorphes
math.DS/0403454
Nikos Frantzikinakis, Bryna Kra: Polynomial Averages Converge to the
Product
of Integrals
math.DS/0403453
Nikos Frantzikinakis: The Structure of Strongly Stationary Systems
FA: Functional Analysis
-----------------------
math.FA/0404014
Vassiliki Farmaki: Ramsey and Nash-Williams combinatorics via Schreier
families
math.FA/0403490
Lev Sakhnovich: Integrable Operators and Canonical Differential Systems
math.FA/0403474
Cleon S. Barroso, Eduardo V. Teixeira: A Fixed Point Theorem for Sum of
Operators and Applications
GM: General Mathematics
-----------------------
math.GM/0403521
Sergei Fedotov, Stephanos Panayides: Stochastic arbitrage return and its
implications for option pricing
GN: General Topology
--------------------
math.GN/0403495
Mathieu Baillif: The homotopy classes of continuous maps between some
non-metrizable manifolds
GR: Group Theory
----------------
math.GR/0403514
Robert Shwartz: Ordered Generating Systems of Mathieu Groups
math.GR/0403480
Robert L. Griess Jr: Pieces of 2^d: Existence and uniqueness for
Barnes-Wall
and Ypsilanti lattices
math.GR/0403456
Sarah J. Campbell, Graham A. Niblo: Hilbert space compression and
exactness
for discrete groups
GT: Geometric Topology
----------------------
math.GT/0404007
Mary Rees: The geometric model and coarse Lipschitz equivalence direct
from
Teichmuller geodesics
math.GT/0403554
Frank Quinn: Cores of s-cobordisms of 4-manifolds
math.GT/0403489
Joan S. Birman: Braids, knots and contact structures
math.GT/0403475
Jozef H.Przytycki, Witold Rosicki: The topological interpretation of the
core group of a surface in S^4
math.GT/0403470
J'er^ome Dubois: Non abelian Reidemeister torsion and volume form on
the
SU-representation space of knot groups
math.GT/0403455
Kevin P. Knudson: On the kernel of the Gassner representation
KT: K-Theory and Homology
-------------------------
math.KT/0403511
Uuye Otgonbayar: The Baum-Connes Conjecture for KK-theory
MG: Metric Geometry
-------------------
math.MG/0403503
Dragutin Svrtan, Darko Veljan, Vladimir Volenec: Geometry of pentagons:
from
Gauss to Robbins
MP: Mathematical Physics
------------------------
math-ph/0404005
A.Zabrodin: Whitham hierarchy in growth problems
math-ph/0404004
R. Cartas-Fuentevilla: The Euler characteristic and the first Chern
number
in the covariant phase space formulation of string theory
math-ph/0404003
S. Dobrokhotov, K. Pankrashkin, E. Semenov: On Maslov Conjecture about
Square Root Type Singular Solutions of the Shallow Water Equations
math-ph/0404002
Alessandra Bianchi, Pierluigi Contucci, Andreas Knauf: Stochastically
Stable
Quenched Measures
math-ph/0404001
R. Cartas-Fuentevilla: Topological terms and the global symplectic
geometry
of the phase space in string theory
quant-ph/0403210
Randall Espinoza, Tom Imbo, Paul Lopata: Translation of Quantum Texts
quant-ph/0403208
Randall Espinoza, Tom Imbo, Paul Lopata: Enscription of Quantum Texts
math-ph/0403063
Jeffrey H Schenker: Holder equicontinuity of the integrated density
of
states at weak disorder
math-ph/0403062
Nicolae Cotfas: On the self-similarities of the Penrose tiling
hep-th/0403118
Kurusch Ebrahimi-Fard, Li Guo, Dirk Kreimer: Integrable Renormalization
II:
the general case
cond-mat/0403644
Ernesto P. Borges, Constantino Tsallis, Jose G. V. Miranda, Roberto F. S.
Andrade: Mother wavelet functions generalized through q-exponentials
math-ph/0403061
Oliver Maspfuhl: Gauge fields and Sternberg-Weinstein Approximation of
Poisson Manifolds
math-ph/0403060
Alexandre Fedotov, Frederic Klopp: Operateurs de Schrodinger
quasi-periodiques adiabatiques : Interactions entre les bandes spectrales
d'un operateur periodique
math-ph/0403059
R.Parthasarathy, R.Sridhar: A q-Generalization of Product Densities and
Janossy Functions in Stochastic Point Processes
gr-qc/0403068
Sergei Kopeikin, Igor Vlasov: Parameterized Post-Newtonian Theory of
Reference Frames, Multipolar Expansions and Equations of Motion in the
N-body Problem
quant-ph/0403179
Gavriel Segre: Noncommutative Bayesian Statistical Inference from a wedge
of
a Bifurcate Killing Horizon
math-ph/0403058
Miloslav Znojil: An asymptotic intertwining of the undelayed and delayed
Fibonacci numbers
math-ph/0403057
Michel Planat, Metod Saniga, Haret Rosu: Mutually Unbiased Bases and
Finite
Projective Planes
math-ph/0403056
J. Golenia, A.K. Prykarpatsky, Y.A. Prykarpatsky: The structure of
Gelfand-Levitan-Marchenko type equations for Delsarte transmutation
operators of linear multi-dimensional differential operators and operator
pencils. Part 1
math-ph/0403055
Y.A. Prykarpatsky, A.M. Samoilenko, A.K. Prykarpatsky, V.Hr. Samoylenko:
The
Delsarte-Darboux type binary transformations and their
differential-geometric and operator structure. Part 1
math-ph/0403054
Yarema Prykarpatsky, Anatoliy Samoilenko, Anatoliy K. Prykarpatsky:
Differential-geometric and topological structure of multidimensional
Delsarte transmutation operators
math-ph/0403053
Doug Pickrell: The radial part of the zero-mode Hamiltonian for sigma
models
with group target space
math-ph/0403052
Dmitri V. Vassilevich: Spectral problems from quantum field theory
math-ph/0403051
Avinash Khare, Arul Lakshminarayan, Uday Sukhatme: Cyclic Identities
Involving Ratios of Jacobi Theta Functions
math-ph/0403050
Klaus Kirsten, Alan J. McKane: Functional determinants for general
Sturm-Liouville problems
hep-th/0309068
Roger E. Behrend, David E. Evans: Integrable Lattice Models for Conjugate
$A^{(1)}_n$
gr-qc/0402121
Gavriel Segre: The topological undecidabilities hidden in the Principle
of
Equivalence of Classical General Relativity
math-ph/0403049
Guido Carlet: The Hamiltonian structures of the two-dimensional Toda
lattice
and R-matrices
math-ph/0403048
Christian Gerard Christian Jaekel: Thermal Quantum Fields without
Cut-offs
in 1+1 Space-time Dimensions
math-ph/0403047
C. Quesne, V.M. Tkachuk: Deformed algebras, position-dependent effective
masses and curved spaces: An exactly solvable Coulomb problem
hep-th/0403207
Christoph Bergbauer, Dirk Kreimer: The Hopf algebra of rooted trees in
Epstein-Glaser renormalization
cond-mat/0403625
Satoshi Morita, Hidetoshi Nishimori, Pierluigi Contucci: Griffiths
Inequalities for the Gaussian Spin Glass
astro-ph/0403586
A. Bernui, G.I. Gomero, B. Mota, M.J. Reboucas: A Note on the Robustness
of
Pair Separations Methods in Cosmic Topology
NA: Numerical Analysis
----------------------
math.NA/0403517
Folkmar Bornemann, Christian Rasch: Finite-Element Discretization of
Static
Hamilton-Jacobi Equations Based on a Local Variational Principle
math.NA/0403492
H. Hakopian, M. Tonoyan: Polynomial Interpolation and a Multivariate
Analog
of Fundamental Theorem of Algebra
math.NA/0403484
H. Hakopian: Normal Systems of Algebraic and Partial Differential
Equations
NT: Number Theory
-----------------
math.NT/0404002
Jay Jorgenson, Cormac O'Sullivan: Convolution Dirichlet series and a
Kronecker limit formula for second-order Eisenstein series
math.NT/0403548
David Joyner, Salahoddin Shokranian: Remarks on codes from modular
curves:
MAGMA applications
math.NT/0403542
Victor Abrashkin: Galois modules arising from Faltings's strict modules
math.NT/0403522
Pietro Corvaja, Umberto Zannier: On the rational approximations to the
powers of an algebraic number
math.NT/0403515
Luis Dieulefait: Computing the level of a modular rigid Calabi-Yau
threefold
math.NT/0403498
Peter Schneider, Jeremy Teitelbaum: Duality for admissible locally
analytic
representations
math.NT/0403458
Sinya Kitani, Eiki Sawada, Kimio Ueno: Finite automata and relations of
multiple zeta values
math.NT/0403457
Michitomo Nishizawa, Kimio Ueno: Connection formulas for the confluent
hypergeometric functions and the functional relation for the Hurwitz zeta
function
OA: Operator Algebras
---------------------
math.OA/0404001
David Kruml, Pedro Resende: On quantales that classify C*-algebras
OC: Optimization and Control
----------------------------
math.OC/0403467
Christopher King, Michael Nathanson: On the existence of a common
quadratic
Lyapunov function for a rank one difference
math.OC/0403466
Joao Jose de Farias Neto: Some Results on Algebraic and Geometric
Characterization of Linear Systems Models for Time Series Analysis
PR: Probability Theory
----------------------
math.PR/0404015
James Allen Fill, Robin Pemantle: Percolation, first-passage percolation,
and covering times for Richardson's model on the n-cube
math.PR/0403538
Carl Graham: Functional central limit theorems for a large network in
which
customers join the shortest of several queues
math.PR/0403504
Mark Adler & Pierre van Moerbeke: PDE's for the joint distributions of
the
Dyson, Airy and Sine processes
math.PR/0403499
Jacques Franchi, Yves Le Jan: Relativistic Diffusions
math.PR/0403497
D. Feyel, A.S. Ustunel: Solution of the Monge-Ampere Equation on Wiener
Space for log-concave measures
math.PR/0403461
Jean M'{e}min, Franc{c}ois Coquet, Adam Jakubowski, Leszek Slominski:
Natural decomposition of processes and weak Dirichlet processes
QA: Quantum Algebra
-------------------
math.QA/0404010
Gwenael Massuyeau, Florin F. Nichita: Yang-Baxter operators arising from
algebra structures and the Alexander polynomial of knots
math.QA/0404008
I. Heckenberger: Finite dimensional rank 2 Nichols algebras of diagonal
type
II: Classification
hep-th/0403277
Anastasia Doikou: On reflection algebras and boundary Yangians
math.QA/0403527
Marta M. Asaeda, Jozef H. Przytycki, Adam S. Sikora: Khovanov homology of
links in I-bundles over surfaces
math.QA/0403488
Alexander V. Karabegov: On the inverse mapping of the formal symplectic
groupoid of a deformation quantization
math.QA/0403477
Tomoyuki Arakawa: Quantized Reductions and Irreducible Representations of
W-Algebras
math.QA/0403463
Gaetano Fiore: On the hermiticity of q-differential operators and forms
on
the quantum Euclidean spaces R_q^N
RA: Rings and Algebras
----------------------
math.RA/0404009
Nikolai L. Gordeev, Vladimir L. Popov: Automorphism groups of finite
dimensional simple algebras
math.RA/0403509
Michael K. Kinyon: Leibniz algebras, Lie racks, and digroups
math.RA/0403493
Friedrich Knop: Graded cofinite rings of differential operators
RT: Representation Theory
-------------------------
math.RT/0403529
Julia Gordon: Motivic nature of character values of depth-zero
representations
math.RT/0403524
Gregory D. Landweber: Twisted representation rings and Dirac induction
math.RT/0403496
Wolfgang Soergel: Kazhdan-Lusztig-Polynome und unzerlegbare Bimoduln
uber
Polynomringen
math.RT/0403487
Alexander Stasinski: Representations of reductive groups over finite
rings
and extended Deligne-Lusztig varieties
SG: Symplectic Geometry
-----------------------
math.SG/0403512
Fernando Etayo, Rafael Santamar'{i}a, Uju'e R. Tr'{i}as: The
geometry
of a bi-Lagrangian manifold
math.SG/0403506
Matvei Libine: On equivariant cohomology for non-compact group actions
SP: Spectral Theory
-------------------
math.SP/0403491
Radu Cascaval, Fritz Gesztesy: J-Self-Adjointness of a Class of
Dirac-Type
Operators
math.SP/0403462
Rakib Feyruz Efendiev: Inverse problem for one class of nonselfadjoint
operator's bunches with nonperiodic coefficients
--
/ Greg Kuperberg (UC Davis)
/
/ Visit the Math ArXiv Front at http://front.math.ucdavis.edu/
/ * All the math that's fit to e-print *
===
Subject: Re: Lower bound on probability
Originator: bergv@math.uiuc.edu (Maarten Bergvelt)
> sort of material, what would it be?
Try
Whittle, Probability via expectations.
Arnold Neumaier
===
Subject: Re: Functional equation
Epigone-thread: bolgrangphoi
Originator: bergv@math.uiuc.edu (Maarten Bergvelt)
>|> f(r1,r2,r3) = f(-r1/(1+r2),-r2/(1+r2),-r3/(1+r2)) (1)
>I presume you want f defined on R^3 except for the plane r2=-1.
>|> f = g(|ln(1+r2)|,r1/r2,r3/r2) (2)
>|> I was wondering if (2) is the most general solution and if not
>|> how to obtain it.
>Note that the linear fractional transformation T: x -> -x/(1+x) is
its
>own inverse, and takes
>(-infinity, -2] 1-1 onto [-2, -1),
>(-1,0] 1-1 onto [0,infinity).
>Just define f(r1,r2,r3) arbitrarily for -infinity < r2 <= -2 and
>-1 < r2 <= 0, and
> f(r1,r2,r3) = f(-r1/(1+r2),-r2/(1+r2),-r3/(1+r2)) for all others.
>Robert Israel israel@math.ubc.ca
>Department of Mathematics (604) 822-3629
>University of British Columbia fax 822-6074
>Vancouver, BC, Canada V6T 1Y4
I've worked a bit on this functional equation .
The fact that -r2/(1+r2) is a cyclic mapping - involutive - yields no
useful invariants on R .Let me,please explain what I think.
Changing,for instance all minus signs ,that is solving :
f(r1,r2,r3)=f(r1/(1+r2),r2/(1+r2),r3/(1+r2)), gives an easier solution
f(r1,r2,r3)=g(m(1/r2 modulo1),r1/r2,r2/r3) ; a particular case might
be :g(sin(2Pi/r2),r1/r2,r2/r3).Anyway your solution seems a very good
one.
Amiti.8es Alain.
===
Subject: Metrizable groups and differentiability
Originator: bergv@math.uiuc.edu (Maarten Bergvelt)
hi,
anybody knows good books for metrizable groups and differentaboility
properties concerning them.
===
Subject: Paper published by Geometry and Topology
Originator: bergv@math.uiuc.edu (Maarten Bergvelt)
The following paper has been published:
URL:
http://www.maths.warwick.ac.uk/gt/GTVol8/paper16.abs.html
Title:
Units of ring spectra and their traces in algebraic K-theory
Author(s):
Christian Schlichtkrull
Abstract:
Let GL_1(R) be the units of a commutative ring spectrum R. In this
paper we identify the composition
BGL_1(R)->K(R)->THH(R)->Omega^{infty }(R),
where K(R) is the algebraic K-theory and THH(R) the topological
Hochschild homology of R. As a corollary we show that classes in
pi_{i-1}(R) not annihilated by the stable Hopf map give rise to
non-trivial classes in K_i(R) for igeq 3.
Secondary: 19D10, 55P48
Keywords:
Ring spectra, algebraic K-theory, topological Hochschild homology
Proposed: Thomas Goodwillie
Seconded: Ralph Cohen, Haynes Miller
Author(s) address(es):
Department of Mathematics, Oslo University
PO Box 1053, Blindern
NO-0316 Oslo, Norway
Email: krull@math.uio.no
===
Subject: 1-foliations
Originator: bergv@math.uiuc.edu (Maarten Bergvelt)
Is it possible for a compact Riemann 3-manifold (M^3,g) to be foliated by
non-
trivial (i.e. neither points nor circles) recurrent geodesic orbits of a
nowere zero vector field xi, none of which realizes the distance between a
point p in M^3 and its cut locus C_{m}(p) and none of which has conjugate
points?
===
Subject: Re: matrix inequality
> I'm having a little difficulty showing that (if it is indeed true)
> Tr_A[ Tr_B(PU^*) Tr_B(UP) ] <= Tr_A[ (Tr_B P)^2 ]
Actually, it's not even true!
===
Subject: March your way into the weekend!
March your way into the weekend!
Join Nader supporters in DC, April 24-25,
as they rally for justice and equal rights
Washington, DC:
Saturday, April 24th, IMF/World Bank Protest - meet at 11am, 15th & H St,
NW.
Sunday, April 25th, March for Women's Lives - meet at 9:30am, Smithsonian
Metro Stop.
If you arrive late, ask a volunteer to point you towards the Ralph Nader
group.
See you there!
Nader: Oil the Source of Two of Earth.89s Major Problems:
Wars for Oil and Climate Change
Protect the Environment and the People of Our Planet by Ending Our Addiction
to Oil
Ralph Nader highlighted as two priorities: ending the Iraqi occupation
and facing up to the immediate crisis of global climate change. Future
geopolitical crises involving oil resources and environmental problems
will be diminished by finding alternatives to fossil fuels, said Nader.
Nader linked the two issues on Earth Day because he sees them as
symptomatic of the corporate oil-based, planet destructive behavior of
the Bush Administration. Nader has called for the United States to
withdraw from Iraq - including military, private military contractors,
oil industry and other corporations. Nader noted: The war in Iraq is
unleashing widening cycles of violence. The potential for escalation of
violence increases every day the US military remains in Iraq. The way to
reverse the spiral of violence is for the United States to go back home.
The US presence serves as a magnet for the insurrection, kidnapping,
terrorism and destruction. Nader released a three-step strategy for
withdrawal earlier this week. See www.votenader.org for details.
--
For further information, contact:
Kevin Zeese
1-202-265-4000
Matt Ahearn
ahearn@votenader.org
1-201-314-9747
--
to perfection!
1 high quality limb, rack, or roast
Potatoes, carrot
Oil
celery
onions
green onions
parsley
garlic
salt, pepper, etc
2 cups beef stock
Marinate meat (optional, not necessary with better cuts).
Season liberally and lace with garlic cloves by making incisions,
and placing whole cloves deep into the meat.
Grease a baking pan, and fill with a thick bed of onions,
celery, green onions, and parsley.
Place roast on top with fat side up.
Place uncovered in 500Á oven for 20 minutes, reduce oven to
325Á.
Bake till medium rare (150Á) and let roast rest.
Pour stock over onions and drippings, carve the meat and
place the slices in the au jus.
Bisque .88 l?Enfant
Honor the memory of Grandma with this dish by utilizing her good
silver soup tureen and her great grandchildren (crawfish, crab or
lobster will work just as well, however this dish is classically
made with crawfish).
Stuffed infant heads, stuffed crawfish heads, stuffed crab or lobster
shells;
make patties if shell or head is not available
(such as with packaged crawfish, crab, or headless baby).
Flour
oil
onions
bell peppers
garlic salt, pepper, etc.
3 cups chicken stock
2 sticks butter
3 tablespoons oil
First stuff the heads, or make the patties (see index)
then fry or bake.
Set aside to drain on paper towels.
Make a roux with butter,
===
Subject: Re: probability
> The grades on the midterm examination given in a large managerial
> statistics class are normally distributed with mean 75 and standard
> deviation 9. The instructor of this class wants to assign an A grade
> to the top 10% of the scores, a B grade to the next 10% of the scores,
> a C grade to the next 10% of the scores, a D grade to the next 10% of
> the scores, and an F grade to all the scores below the 60th percentile
> of this distribution. For each possible letter grade, find the lowest
> acceptable score within the established range. For example, the
> lowest acceptable score for an A is the score at the 90th percentile
> of this normal distribution
Why? Why not let the instructor do his own work? Or, better yet, let the
students in the class do it as part of their grade.
Why fail 60% of the students? Isn't it a bit disingenuous to accept their
tuition and then fail them? Either they're really not capable of the work,
in which case they shouldn't be accepted into the program, or they are
capable of the work, in which case by flunking them out you aren't really
communicating anything valuable to the students or potential employers
about
the students. Except maybe about ruthlessness. Is that really a valuable
managerial skill? I know Chainsaw Al Dunlap thought so, but I think
he's
still unemployed.
Jon Miller
===
Subject: Re: Combinations and Binary Strings
> It's possible to count the solutions to
> A + B + C + D + E + F = n
>
> in non-negative integers with B, C, D, and E strictly
> positive without needing to know about generating functions,
> though, and Bernd might find the approach useful.
> Think of A, B, C, D, E, and F as boxes into which you are to
> distribute n identical marbles, with the additional
> requirement that boxes B, C, D, and E must receive at least
> one marble each. To meet this requirement, put one marble
> into each of those four boxes; you now have n - 4 marbles to
> be distributed arbitrarily amongst the six boxes with no
> restrictions at all. Each distribution -- so many marbles
> in box A, so many in box B, etc. -- corresponds to a
> solution to the equation.
> Let's look at the general problem of distributing m marbles
> amongst the six boxes. Suppose, for instance, that you have
> 3 in box A, 2 in box B, 0 in box C, 1 in box D, 2 in box E,
> and 0 in box F; you can represent this schematically as
> o o o | o o | | o | o o |.
>
> With a little thought you can see that each possible
> distribution of m marbles corresponds uniquely to a string
> of m o's, representing the marbles, and 5 |'s, representing
> the spaces between adjacent boxes. There are C(m+5, 5) such
> strings, since there are C(m+5, 5) ways of picking 5 slots
> for the |'s.
> In your problem m = n - 4, so the desired number of
> solutions is C(n+1, 5).
> Brian
Wow. You sir are very good. I like this solution a lot. Very intuitive.
I wish I had the intuition to come up with stuff like that.
So much for induction.
Bernd
===
Subject: Re: i need help with 3 riddels
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision:
1.9 primary) id i3MBnpv02935;
the first on could be either illusion, silence, will, riddle,
darkness, consitration, embrace, stride, fall, cycle, routine,
promise, tension, spirit, spell or a hundred other things.
the answer to the second one is infinity.
and the third is sign.
===
Subject: can anyone help
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision:
1.9 primary) id i3MBnjZ02823;
investor wants to invest 30,000 in bonds that are rated x,y,z. the
lower ones pay higher interest but are higher risk as well the average
yeild is 5%on x,6% on y, and 10% on z investor wants to invest 2 as
much in x bonds than z. To get a return of 2000 how much should he
invest in each?
===
Subject: Re: can anyone help
> investor wants to invest 30,000 in bonds that are rated x,y,z. the
> lower ones pay higher interest but are higher risk as well the average
> yeild is 5%on x,6% on y, and 10% on z investor wants to invest 2 as
> much in x bonds than z. To get a return of 2000 how much should he
> invest in each?
Set up a system of equations.
x+y+z=30000
.05x+.06y+.1z=2000
z=2x
3x+y=30000
.25x+.06y=2000
-.18x-.06y=-1800
.25x+.06y=2000
.07x=200
x=2857.14
z=2(2857.14)=5714.29
y=30000-3(2857.14)=21428.57
David Moran
===
Subject: Re: can anyone help
> investor wants to invest 30,000 in bonds that are rated x,y,z. the
> lower ones pay higher interest but are higher risk as well the average
> yeild is 5%on x,6% on y, and 10% on z investor wants to invest 2 as
> much in x bonds than z. To get a return of 2000 how much should he
> invest in each?
> Set up a system of equations.
> x+y+z=30000
> .05x+.06y+.1z=2000
> z=2x
<...>
Should be x=2z.
y+3z=30000
.06y+.2z=2000 ==> x=20000, y=0, z=10000
--
Darrell
===
Subject: Pi and ACT
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision:
1.9 primary) id i3MDlEt21467;
First, the speed of light (@186,000 mps) squared gives you pi. Not
quite, not yet. A large decimal point problem. The only way to move
that decimal point is by exerting great force of Big Bang magnitude.
But how to circumvent my ACT score? When ACT and PELL both moved to
Iowa City, was that not in fact Big Bang II?
Someone check my math.
===
Subject: Re: Pi and ACT
> First, the speed of light (@186,000 mps) squared gives you pi. Not
> quite, not yet. A large decimal point problem. The only way to move
> that decimal point is by exerting great force of Big Bang magnitude.
> But how to circumvent my ACT score? When ACT and PELL both moved to
> Iowa City, was that not in fact Big Bang II?
> Someone check my math.
Somebody check his head!
===
Subject: Re: Pi and ACT
> First, the speed of light (@186,000 mps) squared gives you pi. Not
> quite, not yet. A large decimal point problem.
No, it doesn't. It doesn't even come close. 186000^2 is 34,596,000,000. Pi
would be 3.14159
> The only way to move
> that decimal point is by exerting great force of Big Bang magnitude.
...or an overactive imagination...
> But how to circumvent my ACT score? When ACT and PELL both moved to
> Iowa City, was that not in fact Big Bang II?
> Someone check my math.
Did you take any math classes?
--
Darryl L. Pierce
Visit the Infobahn Offramp -
What do you care what other people think, Mr. Feynman?
===
Subject: Distance: Related Rates
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision:
1.9 primary) id i3ME6Uf24268;
Let x and y be differentiable functions of t and let s=
squarert.(x^2+y^2) be the distance between the points (x,0) and (0,y)
in the xy-plane.
(a) How is ds/dt related to dx/dt if y is a constant?
(b)How is ds/dt related to dx/dt if neither x nor y is a constant?
(c) How is dx/dt related to dy/dy if s is a constant?
Can someone please assisst me with this problem by today April 22,
===
Subject: Re: Distance: Related Rates
> Let x and y be differentiable functions of t and let s=
> squarert.(x^2+y^2) be the distance between the points (x,0) and (0,y)
> in the xy-plane.
> (a) How is ds/dt related to dx/dt if y is a constant?
> (b)How is ds/dt related to dx/dt if neither x nor y is a constant?
> (c) How is dx/dt related to dy/dy if s is a constant?
> Can someone please assisst me with this problem by today April 22,
Use the change rule.
===
Subject: Re: Distance: Related Rates
> Let x and y be differentiable functions of t and let s=
> squarert.(x^2+y^2) be the distance between the points (x,0) and (0,y)
> in the xy-plane.
> (a) How is ds/dt related to dx/dt if y is a constant?
> (b)How is ds/dt related to dx/dt if neither x nor y is a constant?
> (c) How is dx/dt related to dy/dy if s is a constant?
> Can someone please assisst me with this problem by today April 22,
> Use the change rule.
And if that fails try the chain rule ;-)
Hint: it is easier to work with s^2. If you're not too confident, write
s(t), x(t) and y(t) to remind yourself that t is the variable. (BTW,
take another look at (c)).
--
Paul Sperry
Columbia, SC (USA)
===
Subject: Re: Second Derivative
> ....
> The question reads: Given: 4x^2 - 5y^2 = -1 . Simplify Fully.
> The answer is: 4/25 y^3 <----How do i get this?....
First, I think you meant the question to include Find the second
derivative y'' of y with respect to x. Second, the answer is
actually 4/(25 y^3) i.e. with the y^3 in the denominator.
You need to think of the equation 4x^2 - 5y^2 = -1 as defining a
function intrinsically, and differentiate the whole equation:
8x - 10 y y' = 0.
The term 10 y y' comes from the chain rule: (d(5 y^2)/dy)(dy/dx).
Now you can find y' in terms of x and y, differentiate again to
find y'', and simplify. This problem is artificially rigged so that
the answer includes (4 x^2 - 5 y^2) which you can simplify from the
original equation.
Can you make progress now?
Ken Pledger.
===
Subject: Re: Math Help: Second Derivative
> ....
> And could you show me the work for this question
> I'm sure he could, as could I. But this is a problem that _you_ are
> supposed to do. JVN has already provided you with a nice outline of
> the necessary steps; you should be able to do the rest yourself if
> you are to deserve the extra credit....
Right. I saw the question under a different title, and probably
gave too much help before finding this thread. It may be too late to
matter anyway.
Ken Pledger.
===
Subject: [MMA/Combinatorica] enumerations and graphs
I posted this in MMA forum, but like i said in my last post, i wanted
to post in here too. I hope someone can shed soem light.
sorry for cross posting.
------
hello group,
this is kinda like a follow up on my last post.
i have a way to enumerate the problem i posted earlier.
I wanted to find all tours that are possible with the edges given. Now
I have to plot them.
consider the following. it will generate 64 lists of edges within a
list.
la= {
{1, 4}, a,
{1, 5}, a,
{4, 2}, a,
{4, 5}, a,
{5, 2}, a,
{5, 4}, a
};
li ={
{1, 4}, i,
{1, 5}, i,
{4, 2}, i,
{4, 5}, i,
{5, 2}, i,
{5, 4}, i
};
lai=Outer[List,Sequence@@Transpose[{la,li}]]//Flatten[#,Length[la]-1]&//ToSt
ring;
StringReplace[lai,{a,[Rule] EdgeColor ->
Blue},{,i,->EdgeColor-> Red},{, a-> EdgeColor [Rule]
Blue,
i->EdgeColor-> Red}]//ToExpression;
Partition[%, 6]
%//Length
now what I woudl like to do is use combinatorica to show all 64
enumerations as 64 separate graphs using the same vertices as follows.
{
{{0.0, 1.5}, VertexLabel -> a1},
{{0.0, 0.0}, VertexLabel -> b2},
{{0.5, 0.75}, VertexLabel -> c3},
{{-0.5, 0.75}, VertexLabel -> d4},
{{-1.5, 0.75}, VertexLabel -> e5}
}
how do I do that? at least how do map those vertices to those edge
sets generated by the enueration above so that i canuse in ShowGraph
command?
sean
===
Subject: Re: Replacement to MMA and Maple
> On the other, each time I try to use
> MATLAB for doing pretty graphics, i.e.
> high quality for publication, it is
> a huge struggle. I am spending a LOT
> of time trying to do clean graphics,
> the way I want them to be.
> Moreover, what I see is NEVER what I get.
I don't know what the standards are in your community. But exporting
Matlab graphics to eps and using them within LaTeX works quite OK for
me.
> MATLAB's basic graphic features are nice,
> advanced ones suck. It's the same with Maple,
> and probably with MuPaD too.
If you need advanced graphics in Maple, Mathematica and (probably)
Mupad, you sit down and use the Maple/Mathematica/Mupad language to
program these graphics on your own. These systems are programmable.
As is their graphics subsystem. It's up to you what they do for you.
> I repeat: I do not understand this effort
> for advanced graphic capabilities.
> For other features, I think it is important
> that programs can exchange datas, via some
> normalized protocol. I don't think it is
> necessary for a program to do everything.
If you are visualizing a formula (as opposed to some numbers in an
array) your data may be a very complex program. To visualize it
properly within finite time you may have to adapt your sampling rate
(evaluate more points where the function is complicated and less where
it's simple and smooth, depending on the size and type of the graphics
you generate, etc.).
That's why it's good if the tool which knows how to evaluate
expressions in the Mupad, Maple, Mathematica language also knows how
to plot more or less decent graphics.
Besides, some users prefer to purchase one program instead of two.
And they prefer to learn one language instead of two. And they may
prefer to use their CAS of choice to create completely new types of
graphics.
===
Subject: Re: Replacement to MMA and Maple
...
> If you need advanced graphics in Maple, Mathematica and (probably)
> Mupad, you sit down and use the Maple/Mathematica/Mupad language to
> program these graphics on your own. These systems are programmable.
> As is their graphics subsystem. It's up to you what they do for you.
I think I know Mma quite well, and I know perfectly well how I might
get the system to produce decent publication-quality graphics. In my
experience, however, it just isn't worth the effort if I want, or
need, to control the graphics and its labels with a lot of detail.
What I usually do is just export the numerical data and use it with
metapost, which has the added advantage of perfect integration with
TeX / LaTeX.
As for the value of visualization: in my work, this is of only minor
importance - in fact, I usually run Mma in an emacs buffer on the
console, and the ASCII graphics there is more than sufficient for me.
Also, there is some danger that shiny graphics lead some people to
forget about the need to understand their functions - I have seen more
than one case where people produced a lot of beautiful pictures but
had a hard time interpreting them. (Which is also why I am sceptical
about too much emphasis on visualization in teaching, but that is a
different topic.)
...
...
> For other features, I think it is important
> that programs can exchange datas, via some
> normalized protocol. I don't think it is
> necessary for a program to do everything.
Doesn't more or less everything allow you to write a plain text file
with the data? That should be sufficient for just about anything, and
if it isn't, some simple sed or Perl or whatever scripting goes a long
way.
Albert.
===
Subject: Re: Replacement to MMA and Maple
Originator: bellenot@haar
>What I usually do is just export the numerical data and use it with
>metapost, which has the added advantage of perfect integration with
>TeX / LaTeX.
This sounds interesting, is there a url with some examples?
--
Steven Bellenot http://www.math.fsu.edu/~bellenot
Department of Mathematics real.email@line.below
Florida State University bellenot at math.fsu.edu
Tallahassee, FL 32306-4510 USA +1 (850) 644-7189 (FAX: 4053)
===
Subject: Re: Replacement to MMA and Maple
MuPaD in particular, and I didn't want to
> start a war.
I didn't think you wanted to.
> Yes, it is very nice to plot easily
> within a mathematical software. It is
> very convenient and helps a lot for
> debugging. I think that CAS should be
> able to do graphics. But not pretty
> graphics.
graphics are a valuable tool for exploration, which is a large part of
(my) research. And the step from good and useful graphics to
pretty graphics is actually a very small one -- even features of
pretty graphics yet missing in MuPAD such as texture mapping could
come in handy every now and then for actual research. And don't
forget teaching -- images are a very important aspect of teaching
mathematics.
> MATLAB's basic graphic features are nice,
> advanced ones suck. It's the same with Maple,
> and probably with MuPaD too.
It depends on the precise results you want to achieve. I have no
experience with MATLAB graphics, but I can assure you that MuPAD's
graphics are a very long way ahead of Maple's. Have a look at the
small galleries at www.mupad.de/gallery.html (actual commands
included) and http://www.mupad.com/?&ID=66737&menu=2 to get an
impression.
> I repeat: I do not understand this effort
> for advanced graphic capabilities.
Most of the intelligence for advanced mathematical plots is there
to make exploration easy. Getting shiny models in 3D is nowadays not
much more difficult than drawing anything at all in a way you can do
interactive manipulations such as rotating and zooming -- again
things you really want to do when looking at some yet unknown
structure.
> For other features, I think it is important
> that programs can exchange datas, via some
> normalized protocol. I don't think it is
> necessary for a program to do everything.
Remember: Standards are great -- everyone should have one.
Sure, I'd love to be able to export MuPAD graphics to, say, POVray or
something like that, but I haven't seen a tool yet outside a
mathematical program that I'd wish to use for interactive manipulation
of such objects.
--
+--+
+--+|
|+-|+ Christopher Creutzig (ccr@mupad.de)
+--+ Tel.: 05251-60-5525
===
Subject: Re: Replacement to MMA and Maple
> I repeat: I do not understand this effort
> for advanced graphic capabilities.
May be becuase many see scientific visulization as
a very important tool to help understand the data they
analyze?
A CAS system without good support to plotting and
powerfull graphics (2D, 3D, animation, zooming, view angles,
exporting to different image formats, etc...) would not be
too successfull these days.
> Using the right tool for the right job
> is important. That's what professionals
> in any field do, except in informatics,
> it seems.
So, you'd rather export your data to disk file, then try to
import it to another application specialized just for
plotting, which hopefully will be able to read the format
from the first application, rather than have everything
integrated into one application?? good luck selling
such a CAS system.
Check matlab 7 (to be released soon) new grphics features,
they have added more support to better manage these
things. Which is a good thing (tm). Matlab has always been
strong on graphics, and this is one of its strong selling
points.
this is a first look screen shots:
http://homepage.mac.com/samchops/B733177502/C1647066619/E649021504/
http://www.mathworks.com/products/new_products/R14_prerelease.html
===
Subject: Re: Laplace transformation software?
> It seems that Matlab herits the symbolic mathematics modules from
> Maple. I wonder whether there exists any big difference (for example,
> the intrinsic function library for Laplace transformation) between
> Matlab and Maple especially in symbolic mathematics.
My information may be outdated, but AFAIK Matlab uses (parts of)
Maple for symbolic calculations, so the only difference should be
that you might not be able to access everything Maple offers from
within Matlab and that the Maple version used by Matlab might be an
older one than you can buy separately.
--
+--+
+--+|
|+-|+ Christopher Creutzig (ccr@mupad.de)
+--+ Tel.: 05251-60-5525
===
Subject: Re: Laplace transformation software?
>> It seems that Matlab herits the symbolic mathematics modules from
>> Maple. I wonder whether there exists any big difference (for example,
>> the intrinsic function library for Laplace transformation) between
>> Matlab and Maple especially in symbolic mathematics.
> My information may be outdated, but AFAIK Matlab uses (parts of)
> Maple for symbolic calculations, so the only difference should be
> that you might not be able to access everything Maple offers from
> within Matlab and that the Maple version used by Matlab might be an
> older one than you can buy separately.
The Symbolic Math Toolbox does use the Maple kernel. As of version 3.0 of
the toolbox, Maple 8's kernel is being used for the symbolic calculations.
--
Steve Lord
slord@mathworks.com
===
Subject: Re: Computer Algebra System
<92rf80hfki7lqrqlcd4rdqq33d3rs1nld2@4ax.com>Polynomials use a denser
representation, where a reference to some
> The denser vector of polynomials is perhaps minor.
It depends on what you want to achieve. For understanding a general
system, true. For understanding polnomial arithmetic algorithms, I
would recommend such a structure. And for faster calculations, it is
almost required.
> An exponent can put into all nodes. It would be an integer. It seems
> like a good idea since most of the algorithms can't cope with an
> exponent and shifting it into a node createss simple special cases.
It would be difficult to support this transparently to the user and I
would not expect significant benefits -- to the contrary.
> * How is information about the identity of the CPU stored?.
>> Why should it be stored at all?
> There would be at least a stack for each CPU.
Oh, sorry -- MuPAD is not multithreaded, so the question does not arise.
>> Certainly in an OO system such as MuPAD each object has a pointer to
>>its domain type. (Kernel types do not have a pointer but rather a
>>small number, but that's an implementation detail.) Oh, and you did
>>not include numbers, functions etc. in the list, which are of course
>>pointed to as well.
> In Ada 95 a record can be extended without changing its type and by
> adding adjacent fields (with no pointers).
There must be information stored somewhere about the type. In an
interpreted language (with lazy evaluation), this means having some
sort of pointer (actual pointer, index into list, ...) to the domain
stored inside the data structure.
> * What is the interpretation of out-of-bounds lengths of the array?.
>> I'm sorry? Which array are you talking about?
> Have you got the source code. A URL would be nice.
I understand this question even less than the one before that.
> I request the reasoning for not getting source file released.
You question the very idea of commercial, closed-source software --
fair enough, but this is certainly not the right forum for that.
> The topic was made by me to be on the 3GL internal design of Mupad.
> I.e. the exact design and not summaries or clarifications of source
> code that is not in the public domain. You kept writing to
> sci.math.symolic and advocating use of Mupad.
I still don't understand your question. The design is that each
domain type may have its own idea of what is or is not zero and that
it can implement this idea by overloading the function iszero. If
you want more detail, see ?iszero for what the kernel domains regard
as zero and read the source code for the library domains not covered
by the documentation, if there are any. Your question is completely
unrelated to closed source vs. open source and I certainly will
continue to advocate the use of MuPAD, since I regard it as a
valuable tool -- I do not think narrow-minded criteria like but it's
not open-source should limit my choice of tools.
> An example of a problem with storing the 3 and the x in 3x, in different
> nodes/records:
> simplify (3*x - 5*y <= -y),
> versus: simplify (x*3 - 5*y <= -y)
As you can easily verify (and is documented in ?*), products of
basic types are reordered to a canonical form. Quoting from the
documentation:
,----
| In particular, a numerical factor is internally stored as the last
| operand. On the screen, a numerical factor is displayed in front of the
| remaining terms:
`----
> The data has been online in the past ?.
>> Again, I do not understand what you mean. Please clarify.
> A yes or no would be a preferred form for the response. I want to
> get to the information speedily. Perhaps Mupad is not selling well
> enough.
I do not understand the question, so yes or no would be
inappropriate. Which data are you talking about?
--
+--+
+--+|
|+-|+ Christopher Creutzig (ccr@mupad.de)
+--+ Tel.: 05251-60-5525
===
Subject: Particular solution to the Navier's equation
Hallo!
Is there anybody who could help me? For I'm on a desperate search for
months, without success so far and also with dwindling hope that it
ever will turn secceed.
I'm seeking for a particular solution (in closed form) to the Navier's
equation in 3D space with non-constant body forces being described by
some function with decay behavior, i.e. some function which influence
decreases as the distance from its focus increases. Example of such a
function f is e.g. f = exp(-r), f = exp(-(r*r)), f = 1/(1+r), f =
1/(1+r*r) or alike (here r is the Euclidian distance, or radius from
the source point to the observation point where the three displacement
components ux, uy, uz are sought).
On this place I should point out the difference between the Navier's
equation (which describes the static equilibrium in a unbounded
elastic body) and the Navier-Stokes equation (which describes the
dynamic equilibrium or fluid flow). I'm interested only in the
Navier's equation which is less complicated being time-independent.
Is there any solution for some f function with decay behavior already
known? Or does somebody have any idea for which function f could it be
possible to find a closed-form solution?
As a second alterantive to the suggested f functions, would come into
question also the so-called compactly supported radial basis functions
(CS RBFs). They have form of a cut-off function and thus the integral
over such a function is bounded, what's the necessary property for the
functions I need.
I've already tried to solve the Navier's equation for the
aforementioned functions f directly using Mathematica and also for
many other similar functions using the known fundamental solution as a
volume integral over the product of the fundamental solution and the f
function. Everything without success.
I would be thankful for any answer, whether positive or negative,
though positive would be preferable.
Andrej
===
Subject: got stuck with an integration
Hi !
I am working on a problem in acoustics, and I need to
express a pressure function from the time domain to
the frequency domain, i.e. through a Fourier transform.
/
the integration is: |exp(iwt)/sqrt(t*(t+a))/(t+b) dt
/
from 0 to infinity, with a > b > 0.
I have tried all possible ways, with no success so far.
The question is if it is possible at all to make get
an explicit form to this integral.
Any hint is warmly welcome !
D. Ouis
===
Subject: Re: got stuck with an integration
> Hi !
> I am working on a problem in acoustics, and I need to
> express a pressure function from the time domain to
> the frequency domain, i.e. through a Fourier transform.
> /
> the integration is: |exp(iwt)/sqrt(t*(t+a))/(t+b) dt
> /
> from 0 to infinity, with a > b > 0.
> I have tried all possible ways, with no success so far.
> The question is if it is possible at all to make get
> an explicit form to this integral.
> Any hint is warmly welcome !
> D. Ouis
Here is a special case... b=1, a=2
> lprint(%);
Int(exp(I*w*t)/((t+1)*t^(1/2)*(t+2)^(1/2)), t = 0 .. infinity) =
(1/2)*exp(-I*w)*(Pi+I*w*( 2*BesselK(0, -I*w)
-Pi*StruveH(1, w)*BesselK(0, -I*w)
-I*Pi*StruveH(0, w)*BesselK(1, -I*w) ))
in the notation of Maple.
--
G. A. Edgar
http://www.math.ohio-state.edu/~edgar/
===
Subject: Re: testing programs in lapack does not run with ifc 7.1
> LAPACK/TESTING directory. In this directory make single complex
> complex16 works but when I do
make double
the computer hangs at
> ./xlintstd < dtest.in > dtest.out 2>&1
I am using Intel(R) Fortran Compiler for 32-bit applications, Version
> I built the blas and lapack libraires using -O2 -Vaxlib options when
> optimization is needed. I used -O2 -Vaxlib when optimization is not
needed.
You need to recompile a couple of subroutines without optimization for
> this too work. Try:
> ifc -O ?lamch.f xerbla.f
> ar -r liblapack.a ?lamch.o xerbla.o
> ranlib liblapack.a
> otherwise the testing executable goes into an infinite look. Also I
> would add a:
> -mp -u
> to your make file (and perhaps a -tpp[4,5,6] and/or -x[i,M,K] for better
> performance).
> Jason
The correct way to achieve that is to explicitly specify
NOOPT = -O0
(That's -O zero) in the Make.inc file in LAPACK main directory.
Whenever a hook exists, use it.
And yes, I've been hit by this too; apparently you have to force zero
optimization onto the Intel compiler, not telling it anything is not
enough.
Hope this helps
Salvatore Filippone
===
Subject: Re: testing programs in lapack does not run with ifc 7.1
> The correct way to achieve that is to explicitly specify
> NOOPT = -O0
> (That's -O zero) in the Make.inc file in LAPACK main directory.
> Whenever a hook exists, use it.
> And yes, I've been hit by this too; apparently you have to force zero
> optimization onto the Intel compiler, not telling it anything is not
> enough.
> Hope this helps
> Salvatore Filippone
Oops, typo.. should be -O0, however I did have NOOPT= -O0 -FI in my
makefile and I still needed to do the little trick in the last post to get
the test suite to run. I don't think all of the routines that need it are
compiled without optimization.
Jason
===
Subject: controlling relative errors when solution = 0
In many general numerical codes for solving ordinary or partial
differential equations, one is given an option of controlling
relative errors of the solution obtained. However, in many cases
of practical interest, the solution may approach zero values,
so that the relative error control surely encounters a problem
with divide by zero (or equivalently the problem with never satisfying the
convergence condition). My question is: are there any known tricks
or methods to avoid this problem while designing a general code?
I address this question to those who have experience with writing
such programs. I'd be happy to find relevant references or program
descriptions explaining the workarounds.
L.Bieniasz
*-------------------------------------------------------------------*
| Dr. Leslaw Bieniasz, |
| Institute of Physical Chemistry of the Polish Academy of Sciences,|
| Department of Electrochemical Oxidation of Gaseous Fuels, |
| ul. Zagrody 13, 30-318 Cracow, Poland. |
| tel./fax: +48 (12) 266-03-41 |
| E-mail: nbbienia@cyf-kr.edu.pl |
*-------------------------------------------------------------------*
| Interested in Computational Electrochemistry? |
| Visit my web site: http://www.cyf-kr.edu.pl/~nbbienia |
*-------------------------------------------------------------------*
===
Subject: Re: Solving Linear Systems in loops
Sorry for posting myself. I noticed a profoundly mistake in my Question:
I'm looking for numeric computation methods of linear TIME-INVARIANT
systems, such that can be written in the form
x'(t) = A*x(t) + B*u(t)
y(t) = C^T*x(t) + d*u(t)
(where A is for example a state space matrix of 3rd or even 4th order)
Once again, I would be grateful for any kind of entrance point to that
topic.
Thanx!
_uli
--
> Hi all!
> I'm a newbie to computational numeric algebra. I want to simulate linear
> (and later on nonlinear) systems with C++, MTL and CLAPACK. I know that
for
> solving linear systems there are some good algorithm I could use quite
> comfortable together with MTL or ITL to solve my systems.
> Actually my systems (like filters) are mostly of the 3rd order and are
> within loops. I would be grateful for some information, hints and
literatur
> citations about the ways to solve those problems within a bigger system
most
> efficiently (but with some comfort, since I'm just an engineer ;)
> I am of the opinion that for fixed Systems of 3rd order within loops
(i.e.
a
> loopfilter) it is still the best way to solve them by hand just computing
> the result with my state space coefficients and eigenvalues instead of
using
> matrix computations. (Together with Matlab it's definitly the way)
> I'm primary interested in learning more about Numeric Computation, so
I'am
> thankful for any information!
> _uli
===
Subject: Contraction Mappings
I am trying to solve the following question
Question: Let X = {x elementOf R: x>1} and let the mapping T:X->X be
defined by T(x)=x/2 + 1/x, Show that T is a contraction mapping.
How do I go about showing T is a contraction mapping? What are the
steps involved here?
I have searched the net for examples of showing a mapping is a
contraction mapping but to no avail. Does anyone have references to
such examples.
any help appreciated,
pat
===
Subject: Re: Contraction Mappings
> I am trying to solve the following question
> Question: Let X = {x elementOf R: x>1} and let the mapping T:X->X be
> defined by T(x)=x/2 + 1/x, Show that T is a contraction mapping.
> How do I go about showing T is a contraction mapping? What are the
> steps involved here?
> I have searched the net for examples of showing a mapping is a
> contraction mapping but to no avail.
Calculate a Lipschitz constant (upper bound of the norm of the
derivative) and show it is smaller than 1.
Arnold Neumaier
===
Subject: Re: System of ODEs
> My aim is the solution of a huge, sparse system of
ordinary
> first
> order differential equations of the general form
> > dy/dt = A y = (B + c D) y
> > Here, A and B are Hermitian matrices and D is a diagonal
> matrix.
> The
> solution is to be obtained for a series of equally spaced
> values of
> the real parameter c. How can I make use of the special
> structure
> of
> the problem? Assuming a solution via the matrix exponent
is
> attempted,
> is it possible to use the eigenvectors obtained for one
> value
> of c
> to
> aid the calculation of y for subsequent values of c? By
the
> way, c
> spans several orders of magnitude and, hence, c D may not
> always be
> regarded a small perturbation. However, adjacent values
of
> c
> differ
> only slightly.
> > Any ideas?
> > Johan
I assume D is a positive matrix (not really necessary, but
> helpful),
> and
> that B and D are time-independent. Then
Step 1: transform the dependent variable, x = sqrt(D)
y;
Step 2: now the equation is dx/dt = Cx + cIx where I =
> unit
> matrix
> and C = [sqrt(D)]^{-1} B [sqrt(D)]^{-1};
> and since [C,cI]=0, x(t)=exp(ct)exp(Ct)x(0). If [B,D]=0 then
> y(t)=exp(cDt)exp(Bt). So c is not the problem but rather
having
> to
> calculate exp(Ct) or exp(Bt) when A is huge and sparse
> (selfadjointness
> is
> of no advantage except that the eigenvalues of A are real) is
a
> problem
> and
> is usually done via Kyrov subspace techniques.
Indeed, B and D to not commute. I forgot to mention that
before.
> I
> consider using expokit in order to calculate exp(A t)*y_0
before.
> Still I hoped I could somehow make use of previous results when
> stepping c.
> calculate
> the vector exp(Ct)D^1/2y(0) (computationally expensive) and have
> D^-1/2exp(ct) (computationally inexpensive) operate on it over
the
> range of
> c. Doesn't come much better than that, :-).
Yes, you are absolutely right. Unfortunately Julian's
transformation
> does
> not apply.
> Johan
Johan,
Julian's rotating frame transformation is indeed applicable. D is
> diagonal.
> You need D^1/2 = (exp(D_i,j) d_i,j) where D = (D_i,j) and d_i,j is
the
> Kronocker delta. You also need D^-1/2 = 1/D^1/2 = (exp(-D_i,j)
d_i,j).
> You
> *don't* need D^-1 (there was a typo in Julian's first post but the
> physics
> was bang on).
Good luck Johan,
Gerry T.
No, I messed up here (hey, the Toronto Maple Leafs won): exp should
have
> been sqrt.
> But all is not lost, having gone from the y-representation to the
> x-picture
> and subsequently to the z-picture there is nothing to compel you to go
> back
> to the y-picture (other than that D^1/2 is noninvertible but physics
> doesn't care so long as you know y(0), a given). This is all you need:
> x(t) = exp(ct)exp(Ct)D^1/2y(0)
> and you're done.
> Oops, did it again: C needs D^-1/2.
> How about using the Trotter formula
> exp(B + cD)t = Lim exp(Bt/n)exp(cDt/n) as n --> oo? n doubles as a
scaling
> factor in computing the exponential matrices.
Indeed, that is an excellent idea! Initially I'm going to try
Trotter-Suzuki's formula, which is correct to fourth order. Ideally, I
imagine to calculate exp(B+cm D)t at an intermediate value of c, cm,
and to obtain an approximation to the matrix exponent at adjacent
values of c using the above approximation. I'm going to do some
numerical testing and return in order to report on problems or
success.
Johan
===
Subject: Re: choice of program/language
Matlab is very good, but very expensive.
> There is 'octave' of course the open source, free as in free speach.
IMHO, I would avoid Object Oriented languages (e.g., C++) unless you
can
> identify clear benefits from using them.
The clear benefits tend to come as the program gets bigger.
> Since most programs start small and manageable, but grow big,
> complicated and unmanageable, when do you decide to go OO,
> when it is too late?
> In the end it is mostly a matter of person choice. It is
> possible to use non-OO languages in a OO sort of way, but
> why make it hard for yourself, unless the client wants
> to avoid OO languages.
> Jack
supports OOP.
info. Fortran 95 has derived types (analogous to C structures, but
allowing private data and functions), and OOP in Fortran 95 has been
discussed in a book Object-Oriented Programming via Fortran 90/95 by
John Akin.
Fortran, like and C and C++ but unlike Java, does not force you to use
structures/classes. Often, you can do the job using scalars and arrays
(possibly multidimensional) of the basic types.
In numerical work, the structures/classes you do want to use will
often be pretty simple (as opposed to those found in a graphical user
interface), and you will often use multidimensional arrays of them.
Fortran 95 makes it easy to access the components (including array
sections) of arrays of derived types, as shown in the following code.
I do not think C++ or Python has this functionality.
module ij_mod
implicit none
type, public :: ij
integer :: i=0,j=0
end type ij
end module ij_mod
program xij
! demonstrate arrays of derived types in Fortran 95
use ij_mod, only: ij
implicit none
integer :: i
integer, parameter :: n = 3
type(ij) :: aa(n)
character (len=*), parameter :: fmt_i=(100i4)
real :: xi
do i=1,n
aa(i) = ij(i,i**2)
end do ! OUTPUT:
print fmt_i,aa ! 1 1 2 4 3 9
print fmt_i,aa%i ! 1 2 3
print fmt_i,aa(2:)%j ! 4 9
print fmt_i,aa((/1,3/))%j ! 1 9
end program xij
===
Subject: Re: choice of program/language
John,
Obviously if people misuse any tool or method the result
is disaster.
I find many programmers do not use OO because they are
ignorant of OO and think they can normal block structuring is
enough. When the application is mostly implementing
formulae (as in stats and so on) then you are probably
correct, normal block structuring is adequate.
How many applications stop at just implementing formulae?
At the very least data must be stored and made available at
the appropriate time.
I could name a big open source C library project that is very
difficult to use because it is written in very old fashion
block structuring. If parameters are not set up just so,
followed by initialisation functions in the correct order,
then nothing works. The library has been extended to add
wrappers around the base functions, but this hardly helps
at all, just another level of complication.
As I said at the beginning it comes down to personal choice.
If you have vast experince in block structured programs then
moving to OO will be a big investment, and probably not worth
it for you. But for new entrants to programming I think they
should seriously consider OO as a necessary skill.
Jack
===
Subject: Re: choice of program/language
>I would only question whether the application requires the use of OO
>techniques. Just because a program is big doesn't mean OO techniques
are
>required. If one cannot identify a need for encapsulation, inheritance
and
>polymorphism in an application, why bother using OO? I just think its a
>little too parochial to believe that every program in the world will
benefit
>from OO. Lapack is a perfect example of F77 code (non-OO) that is
>representative of good structured programming practices. It is not
unusual
>for OO programs (especially in C++) to become bloated and buggy because of
>the improper application of OO techniques. OO requires a large investment
>of time up-front before a single line of code is written. Most people
skimp
>on that investment and proceed directly to developing a plethora of
classes
>and creating obscure inheritance heirarchies that quickly lead to a
>quagmire. Just ask Bill Gates.
Yes. I'm sure there are plenty of hard core OO programmers out there who
could
probably make their codes cleaner and more maintainable simply by dropping
their OO in favor of GOTO's all over the place.
On a more serious note, I use GOTO on rare occasions. There are certainly
times where a GOTO can make code a LOT cleaner and better than jumping
through
hoops to avoid the GOTO.
Dan :-)
===
Subject: linear function
can someone please help me with this problem
Let F:V-->R be a linear function. Let W=kerF and {x1,...,xs} a basic
of W. Let y be element of V and assume that F(y) is not equal to 0. We
wish to show that {x1,...xs,y} span V. Let v be an arbitrary element
in V. If v is in W then v is the sum aixi, ai is eleent of R since
{x1...xs} is a basic of W. If v is not in W, then we try to find beta
so that v-betay element of W. If we succeed at this, then we can write
v-betay = sum aixi, or v = sum aixi + betay . This is what we wish to
establish. So we have to show that such beta exists. A vector is in W
if F evaluated on that vector is zero. So we need to show there exist
a betat so that F(v-betay)=0.
===
Subject: Taylor expansions at different expansion points!
I've a question regarding the taylor expansion around two different
expansion points.
I can say for example :
f(x0+h)=sum_n=0^Inf (d^nf/dx^n)|x=x0 h^n/n!.
If I expand this function around another point x1 such that x0+h=x1+h'
I get:
f(x1+h')=sum_n=0^Inf (d^nf/dx^n)|x=x1 h'^n/n!
Both these functions are evaluted at tha same position so they must be
equal. Therefore there is a relation ship between the different
expansion coefficients. It is this relationship that I'm after.
I aspecially want to understand the relation ship for a finite series
instead of an infinite.
If someone can give me a clue in which field of math I've to look I'm
happy.
Gr,
Maurice
===
Subject: Re: Taylor expansions at different expansion points!
>I've a question regarding the taylor expansion around two different
>expansion points.
>I can say for example :
>f(x0+h)=sum_n=0^Inf (d^nf/dx^n)|x=x0 h^n/n!.
>If I expand this function around another point x1 such that x0+h=x1+h'
and hence x0=x1
>I get:
>f(x1+h')=sum_n=0^Inf (d^nf/dx^n)|x=x1 h'^n/n!
>Both these functions are evaluted at tha same position so they must be
>equal. Therefore there is a relation ship between the different
>expansion coefficients. It is this relationship that I'm after.
>I aspecially want to understand the relation ship for a finite series
>instead of an infinite.
>If someone can give me a clue in which field of math I've to look I'm
happy.
>Gr,
>Maurice
maybe we change this to x0+h=x1-h=x and evaluate both series at x?
and change h to (-h) in the second series
if both series converge then you have of course the possibility
to express all derivatives at x0 and x1 also by a series at x
and this gives you the relation you are searching?
hth
peter
===
Subject: Re: MPI SPOOLES question
> If the data has to be partitioned
> before SPOOLES can use it, is there a particular free(open source)
> partitioner that you recomend?
Parmetis and Chaco are popular choices.
V.
--
email: lastname at cs utk edu
homepage: cs utk edu tilde lastname
===
Subject: Re: MathML runtime
I would suggest to write an XSLT Stylesheet
to transform MathML to C# code (or other) that
uses your specific math function library.
As MathML input control you could use
MathML .NET Control.
Bernhard
www.soft4science.com
> Do you have a solution ?
> AC
> from
>C++ to a more open standard - MathML to be more specific. Is anyone
>aware of a piece of software that can read in MathML and create code
> (C,
>C++, C# etc.. ) from it?
>This would mean we could remove any gaps between the algorithm we say
>we're using, and the actual implementation.
>4Space
===
Subject: Covergence...
If i have a sequence generated by a mapping by which the (k+1)-th term
of the sequence is generated by T( k-th term ) e.g. x1 = T(x0)...etc
How do i show that it converges to a fixed point on the real line in
the interval [a,b] if |T'(x)| < 1 forall x elementOf [a,b]
Where do i start? What steps to take?
Any help appreciated!
pat
===
Subject: Re: Covergence...
>If i have a sequence generated by a mapping by which the (k+1)-th term
>of the sequence is generated by T( k-th term ) e.g. x1 = T(x0)...etc
>How do i show that it converges to a fixed point on the real line in
>the interval [a,b] if |T'(x)| < 1 forall x elementOf [a,b]
>Where do i start? What steps to take?
>Any help appreciated!
>pat
never try the impossible.
what about
T(x)=log(1+exp(x))
for arbitrary real x?
lookup some introductory text e.g. on numerical analysis or functional
analysis and search for the contraction mapping theorem. you will find
another condition , named selfmapping. this one may be harder to show than
|T'(x)|<1, which is alo not sufficient, you need |T'(x)|<=L<1, L independent
of x
hth
peter
===
Subject: Re: Square Number Problem
<6U9jc.28$li2.126@news.oracle.com> I am distantly acquainted with a
participant in this year's
>> Mathematics Olympiad. When I heard about this I looked up some of the
>> qualification problems here http://www.bmoc.maths.org/ to see what the
>> standard was. This was the first problem, from 1993:
>> Find, showing your method, a six-digit integer n with the following
>> properties:
>> (i) n is a perfect square,
>> (ii) the number formed by the last three digits of n is exactly one
>> greater than the number formed by the first three digits of n. (Thus n
>> might look like 123124, although this is not a square.)
>> I didn't have any flashes of inspiration so I started churning out
>> algebra eg letting sqrt(n) = 100a+10b+c and then squaring, and got
>> nowhere. Generally when I resort to pages of algebra I figure I've
>> lost and the answer lies elsewhere. Can anyone here solve this problem
>> and provide *hints* about how to do it.
>> cheers
>> dd
>I can't solve it, but I can do some algebra.
>n is square, so n=y*y
>n is also of the form x*1000+x+1
>so: x*1001-y*y+1=0
1001 * x = y*y - 1
1001 * x = (y+1)*(y-1)
7 * 11 * 13 * x = (y+1)*(y-1)
Since y*y is a 6-digit number, 316 < y < 1000.
Neither y-1 nor y+1 can be 1001, so the 3 prime factors must be distributed
between the 2 terms. That cuts the number of y's to check down to a few
dozen:
multiples of 77 +- 1
multiples of 91 +- 1
multiples of 143 +- 1
I found some. (Spoilers below)
183184 is one solution (y=11*13*3-1).
328329 is another (y=11*13*4+1).
There are probably a more, since 2 out of every [7,11,13] will work.
Exactly 6 of them, at a guess.
--Keith Lewis klewis {at} mitre.org
The above may not (yet) represent the opinions of my employer.