> > I apologize if this request is too basic for this
forum, but
here goes...
>> > I am looking for more info on pythagorean triples.
I have been
playing
>> > around and noticed some patterns, and become
curious.
>> >
>> > for example:
>> > a b c
>> > 3 4 5
>> > 5 12 13
>> > 7 24 25
>> > 9 40 41
>> > 11 60 61
>> >
>> > a is the odd numbers
>> > b has a pattern
>> > c is one greater than b
>> >
>> > for your help
>If you look, you'll find that all of those triples can be
found with: c =
>sqrt((x^4 + 2x^2 + 1) / 4).Example:
> take any odd number for 'a', (we'll use 3 for now). [3^4 =
81] [2*3^2
=
>18] [81+18+1= 100] [100 / 4 = 25]
> so we have a = 3 and c = 5. b is just c - 1 for 4. 3^2 + 4^2
= 5^2.
>The above equation comes from this:
> b + c = a^2 c-b = 1
> a^2 = k (k-1)/2 = b and ((k-1)/2) + 1 = c
> so, a^2 + ((a^2 - 1) / 2)^2 = (((a^2 - 1) / 2) + 1)^2
> then, k^2 + (k^4 - 2k^2 + 1) / 4 = (k^2 - 1) / 2 + 1
> next, (4k^2) / 4 + (k^4 - 2k^2 + 1) / 4 = ((k^2 + 1)/2)^2
> finally, (k^4 + 2k^2 + 1) / 4 = (k^4 + 2k^2 + 1) / 4
> I think everything is correct up there. Of course, this only
works to
find
>primitive triples with twin b and c.
>Let's check with your little table.
> a b c
> 5 12 13
> 7 24 25
> 9 40 41
> 11 60 61
>[5^4 = 625] [2*5^2 = 50] [625 + 50 + 1 = 676] [676 / 4 = 169]
[sqrt(169) =
13]
>[5^2 + 12^2 = 13^2]
>[7^4 = 2401] [2*7^2 = 98] [2401 + 98 + 1 = 2500] [2500 / 4 =
625]
[sqrt(625) =
>25] [7^2 + 24^2 = 25^2]
>[9^4 = 6561] [2*9^2 = 162] [6561 + 162 + 1 = 6724] [6724 / 4
= 1681]
>[sqrt(1681) = 41] [9^2 + 40^2 = 41^2]
>[11^4 = 14641] [2*11^2 = 242] [11461 + 242 + 1 = 14884]
[14884 / 4 = 3721]
>[sqrt(3721) = 61] [11^2 + 60^2 = 61^2]
>There is a similar formula for Pythagorean Quadruples.
Whether or not any
of
>this is helpful, I don't know. But it looks like what you may
have
wanted.
>
===
Subject: Re: God=G_uv (God is also an eigenvector)
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id h8RLSLH10555;
What is the physical significance of Eigen vectors. What its
magnitude
represents.
sai gavirneni
>Whaddaru here to do, copy a whole elementary Linear
>>Algebra text onto Usenet?
>> You appear to need the background.
>> Actually, the above (snipped) paragraph was intended
>> for Scott McDermid, who asked about the physical
>> significance of eigenvectors when he introduced this
>> thread into sci.math.
>> My original response was more geometry-oriented,
>> though I mentioned covariance matrices in passing.
>> After realizing that covariance matrices were the context
>> under discussion, I decided more amplification in
>> that context was needed.
>> I'm not going to claim to be a statistician, but it
>> frequently comes up on the job and I've had to pick
>> up enough to learn the language and speak to the
>> real (Ph.D.) statisticians when questions come up.
>> There are a half-dozen mathematical statistics and
>> probability books on my shelf, and I've taken, oh,
>> perhaps 18 graduate credits of related course material
>> in statistics and probability theory.
>> I *will* claim to be fairly proficient at linear
>> algebra and matrix analysis, as I use it on a nearly
>> daily basis.
>> I don't think our esteemed SPOGhead is one of those
>> Ph.D. statisticians whose counsel I seek periodically. I
>> class him more with the hapless education doctoral
>> candidate I heard about at my school, who had based
>> his entire thesis on SPSS analyses. The statistician
>> on his committee asked what is a regression?
>> He failed.
>> - Randy
>100 years of Psychometry is also called into question,
>not to mention the constant claim that factor analysis
>can be used to prove anything.
>http://courses.albion.edu/Archived_Fall2001/eng337diedrick/
gould.htm
===
Subject: Re: ref to solutions of homogenous quadratic equations
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id h8S2Luh27522;
>> Is the complete solution to (1) known?
>> (1) ax^2 + bxy + cy^2 = f^2
>> where integers a,b,c are known, with b^2 > 4ac, but f is
not known and
x,y are
>> to be integer also?
>> Dario Alpern's javascript program at http://
www.alpertron.com.ar/QUA
D.HTM is
>> almost there, but his program requires that f be known in
advance.
>> Is it possible to solve (1) so that all possible solutions
of f^2 are
located?
>> for references to this.
>Actually John Cremona and David Rusin published a nice paper
on this in
1997,
>which appeared in Mathematics of Computation* as Efficient
Solution of
>Rational Conics in 2001 basically parametrizing all
solutions, once one
>solution was known.
>Cremona's program is very efficient at locating a rational
solution and
>parametrizing the others, see his tconic C++ program under the
mwrank3.gz
>package (http:
//www.maths
.nott.ac.uk/personal/jec/ftp/progs/)
>See http://
eprints.not
tingham.ac.uk/archive/00000060/ for the paper.
>I can see from the lack of response to my inquiry that the
internet
newsgroup is
>quite shallow, this is sad.
>*MathComp Efficient solution of rational conics
> J. E. Cremona; D. Rusin
For whatever reason, I think your original post did not appear
on
some newsreaders. I see the above on MathForum, but not on
AOL's
sci.math. And I do not recall seeing your original post on
MathForum.
I think interest among this newsgroup on these kinds of
equations
is quite high.
See also
http://hometown.aol.com/jpr2718/
Solving the equation ax^2 + bxy + cy^2 + dx + ey + f = 0 - PDF
File
which gives a method, and references to the method in writings
of Weil and Serre, for solving the above equation in rational
numbers. Your equation is that above with d = e = 0, f = -1,
and x, y rational.
The method given above is suitable for smaller a, b, c, while
the
method of Cremona and Rusin is suitable for larger
coefficients.
And the method determines when the equation has a solution,
and finds all of them when it does. As you note, this is in
contrast to the method of Cremona and Rusin, which assumes a
solution is known.
Actually, I think if you search sci.math, you will find that
this question has been answered before.
John Robertson
===
Subject: Re: A structural question
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id h8S2LuZ27518;
>Take for example these 5 equations:
1 1 1
> + 2 = + 2 = +
> 1 1 1
>4 = + 4 = + 4 = +
> 1 1 1
> + + 2 = +
> 1 1 1
> 1 1
> + 2 = +
> 3 = 1 3 = + 1
>4 = + + 4 = +
> 1 1
> 1 1
Set A = {{x1},{x2},{x3},{x4}}, where each x# is some number.
>Now, let us say that the above equations represent some
cardinal's
>equation-trees of set A.
What cardinal? What is a cardinal's equation tree?
>Let us say that any cardinal which is > 1 is the continuous
side
>of the cardinal's equation-tree.
May I ask what this definition is supposed to represent?
>Let us say that any cardinal which is = 1 is the discrete side
>of the cardinal's equation-tree.
1 is the only cardinal that is equal to 1. What do you mean by
the
cardinal's equation tree?
>Let x#' be a dummy variable of
xor(|{x1}|,|{x2}|,|{x3}|,|{x4}|) .
What is a dummy variable? What does xor mean when applied to
cardinals? Why do you write |{x1}| when it is obvious that this
expression is equal to 1 (as {x1} is a set with one element)?
> 1 is xor(x1',x2',x3',x4')
> +
> 1 is xor(x1',x2',x3',x4')
> 4 = +
> 1 is xor(x1',x2',x3',x4')
> +
> 1 is xor(x1',x2',x3',x4')
> 1 is xor(x1',x2')
> 2 = +
> 1 is xor(x1',x2')
> 4 = +
> 1 is xor(x1',x2',x3',x4')
> +
> 1 is xor(x1',x2',x3',x4')
> 1 is xor(x1',x2')
> 2 = +
> 1 is xor(x1',x2')
> 4 = +
> 1 is xor(x1',x2')
> 2 = +
> 1 is xor(x1',x2')
> 1 is xor(x1',x2',x3')
> +
> 3 = 1 is xor(x1',x2',x3')
> 4 = + +
> 1 is xor(x1',x2',x3')
> 1 is |{x4}|
> 1 is |{x1}|
> 2 = +
> 3 = + 1 is |{x2}|
>4 = +
> 1 is |{x3}|
> 1 is |{x4}|
>As you can see above, the quantity in each cardinal's
>equation-tree is being kept, while the structural
symmety-degree
>and the information's clarity-degree of each tree are changed.
What is a sturctural symmetry-degree? What is an information
clarity-degree? What were they to begin with, and how did they
change?
>My question is:
>What mathematical's branch deals with this kind of
>information's structures ?
No branch of mathematics deals with such undefined concepts as
you
have described in this post. Perhaps if you clarify the
definitions,
this could qualify as a mathematical theory (either existing
already
or new). Right you do not meet the standards for precise
mathematical
definitions. In my opinion you don't even meet the standards
for
comprehensible text.
>Doron Shadmi
Cron
===
Subject: Re: about zero divider, help~!
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id h8S2Lvw27526;
The part where you ask to add a-2ab to a=ab,
you're obviously doubling the value right there.
The result:
2(a-ab)=a-ab
It's another obvious thing, you didn't really had to make the
last part and
ask me to divide it, because if I wanted to get somewhere with
this equation
the most obvious thing to do would be to cut out the a-abs,
finally to get 2=1.
when the equation says to add a-2ab to a=ab
(which, in case, is a2-ab) you're doubling the equation and
getting 2=1
===
Subject: Math. Proof of Existance of God??? Minus some babble,
&HRs
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id h8SLVOx30008;
>>Is there anything you can think of that exists that does not
have
>>some context associated with it? I cannot think of any such
item.
>>Now take the context of the Universe. Is there a better
known way to
>> model the context of the Universe than God?
Don't know of a better way to say it. Maybe it could be stated
more Mathematically. But that is all.
>>Context as far as the Universe goes could mean:
>>Where did it come from. Common properties. Common
Informational
>>Properties. Common non temporal informational properties.
Common
>>origin, cause. Etc.
>>The existance of God could be demonstrated by the usefull
knowledge
>> and applications derived by actually possessing the
knowledge found
>>in Math. , Science, Bible. As for the knowledge of God as
well as in
>>Mathematics and Sciences, you must actually possess the
knowledge in
>> order for God to actually exist as demonstrated thru usefull
>> applications of this knowledge.
Babbled a little bit here. Just trying to say that as in
Mathematics,
you must possess the knowledge of Mathematics for it to be
true. Same
with God. And the existance of God is proven by the usefull
applications derived by knowledge of God.
>>For example:
>>In the Bible it says: Man is made in the image of God.....
>>This makes some intuitive sense to me by experience also.
>>So according to modern Mathematics: A Transformation must
exist....
The Transformation would be like Mathematics. Which would give
truth of God, Science, Math., only if you understood the
Transformation.
>>And according to this, I think: Truth is Possible.
>>But the Truth exists and is usefull only if you possess the
knowledge of
it.
>>Zim Olson
http://www.zimmathematics.comYou are babbling.
I don't think I did too bad. Do you know a better Math. Proof
of God?
Look at this post on PROOFOFGOD
http://www.mathforum.org/discuss/sci.math/t/537735
Here is some real cyber babbling to me.
http://www.zimmathematics.com
===
Subject: Re: Minimum sequence for keypad code
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id h8SLVVQ30079;
>> I've searched the net for an answer but have come up with
nothing...
>> All around us there are keypads where you enter a 4 digit
number to
>> open a door (for example). For a keypad requiring 4 digits
there are
>> 10,000 different combinations - 0000-9999. But in reality
because the
>> numbers only have to be entered in the correct sequence;
when you
>> enter 123456 you're actually testing 1234, 2345 and 3456 at
the same
>> time.
>> My question is simple - Is there an algorith for
calculating the
>> optimum sequence to solve this problem with as few
keystrokes as
>> possible? And how many numbers would you have to enter?
>> These keypad locks are all around us and it would be nice
to know how
>> safe they really are.
>> Per
>On the keypads that I use, after you enter 4 digits the lock
checks
>them. If they are right, it opens, if wrong, all 4 are
discarded and
>you MUST start over. The trick you list above does not work
for this
>type of lock.
>--
>Will Twentyman
>email: wtwentyman at copper dot net
===
Subject: This problem is probably old as old as chocholate...
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id h8SLVS530056;
It is well known that a ping pong ball can be made to levitate
by a gust of wind blowing vertically upward. What other objects
besides a sphere exist with this property? I know of at
least one other because I constructed it myself out of paper
and
levitated it for about 5 seconds at about a steady 7cm distance
from my own mouth (don`t know what its called geometrically...
but it
doesn`t have any wholes, topologically speaking).
.
===
Subject: Re: polysigned numbers
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id h8SLVY930099;
>I suppose a definition is in order.
>polysigned means a number system having n signs. It includes
the reals
>as they have two signs. It also includes a set of numbers
that best
>represent time: one signed, and a set of numbers that
represent the
>plane: three-signed, and a set of four-signed numbers that
represent
>3D space.
>The crux of the polysigned approach lies in accepting an
increase in
>dimensionality. This comes as a result of summation. In
general a zero
>sum always has identical component magnitudes in every sign
yielding
>For example, in two-signed math (the reals) for any magnitude
x:
> - x + x = 0.
>In three-signed math for any magnitude x:
> - x + x * x = 0. (where * is a new sign)
>In four-signed math for any magnitude x:
> - x + x * x # x = 0. (where # is a new sign)
>General sums in two signs are one-dimensional.
>General sums in three signs are two-dimensional.
>General sums in four signs are three-dimensional.
>General sums in n signs are (n-1)-dimensional.
>General sums in one sign are zero-dimensional?
>Product rules exist and work much like the real numbers.
>In effect each sign has a number that represents how many
extremities
>away from the identity sign to travel. The identity sign is
always the
>maximum sign. The smallest sign is - (one), then + (two),
then *
>(three), then # (four), and I don't know what character to
use next.
>This system is logical and coexists with the traditional real
numbers.
>It also works for the complex plane.
>I have created a three-signed arithmetic that produces
exactly the
>same results for product and summation as would traditional
complex
>arithmetic. If you search for three-signed within sci.math
you will
>find that thread. Is anyone interested in this?
>I will try to publish this in a more formal way but first
have to get
>a linux box up and running some latex tools. My math
terminology may
>not be very precise but the math is so simple that all of the
math
>thus far is easy.
>I would like to find some help with this.
I did some experimenting with something similar years ago.
When I later learned of hypercomplex numbers, I realized that
what I was doing coincided with certain types of hypercomplex
numbers.
Hypercomplex numbers are represented in more that two
dimensions,
and you always end up with (atleast) one of the following
situations:
(1) non-zero numbers multiply to give zero,
or
(2) multiplication is not commutative.
===
Subject: Re: Minimum sequence for keypad code
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id h8SLVWB30084;
i need a list from 0000-9999 of all possible combinations
ASAP!!!
!
===
Subject: Re: Quaternion Extensions
> I have recently extended the Quaternions to larger sets by
requiring some
> (new) group elements to commute. In doing so, I found this
process and
its
> results to be very asthetic. For one, the law of association
is regained.
> However, the algebra involved is no longer a division
algebra, i.e. we
may
> not always follow x = 0 or y = 0 from xy=0 (when x and y are
certain
> elements taken from a linear combination of group vectors).
...
> Has this type of thing been done before and are its
conclusions of
> interest?
It's obvious that there exist extensions of the quaternions H,
eg H + H (direct sum),
pr algebras of matrices with quaternion elements.
You'd have to say what properties your extension has
before anyone could say if it is of interest.
--
Timothy Murphy
e-mail: tim@birdsnest.maths.tcd.ie
tel: +353-86-233 6090
s-mail: School of Mathematics, Trinity College, Dublin 2,
Ireland
===
Subject: Re: study algebraic number theory
> I want to study algebraic number theory,However most of the
book is
> difficult for me .For example Lang, Serge Algebraic number
theory.So, if
> there are more easy book for studying?
I think the little book by Ian Stewart is pretty
straightforward.
--
Timothy Murphy
e-mail: tim@birdsnest.maths.tcd.ie
tel: +353-86-233 6090
s-mail: School of Mathematics, Trinity College, Dublin 2,
Ireland
===
Subject: Re: The Avatar (was): Learning [was Basic Aspects of
AI]
>GS: But Longley's views are mostly consistent with a
scientific and
>philosophic tradition that is embraced by a minority of
practicing
>scientists and philosophers. Longley's thesis (and I am not
endorsing
every
>thing that David says, mind you) is not of the hollow Earth
variety -
it
>is still a minority view, granted, but I can certainly find
hundreds of
>people (most of whom teach in universities and do government
funded
Rickert: Name a bunch of them.
GS: No need to - just check out the editorial board of JEAB.
Now, not every
person there named will agree with everything Longley says, or
with
everything I say, or everything that each other says. But they
are mostly
behaviorists, and most of what Longley says is behavioristic.
In fact, most
of them are likely to be a bit more critical of mainstream
psychology than
Longley.
Rickert: If Longley's views are compatible with credible
scientists in
psychology related fields, why is he not posting in the
sci.psychology newsgroups?
GS: Perhaps you need remedial reading lessons, Rickert.
Behaviorists are in
the minority, here and elsewhere. The difference is that, in
AI, people
must
eventually put their money where their mouths are - at least
those claiming
some ties to what real organisms do. Mainstream psychologists
don't have to
really develop any real control over their subject matter, and
are held to
the lowest standard of proof imaginable - the rejection of the
null
hypothesis. Mainstream psychology will never give up mentalism
because it
doesn't really have to produce good experimental control or a
technology.
AI
is important because it will, I think, eventually show that
reinforcement
of
spontaneously occuring behavior and the establishment of
stimulus control
over such behavior can explain complex behavior.
Rickert: It is my impression that he posts in
c.a.p because he was was considered a crank by many of those
(including behaviorists) who do post in the psychology groups.
GS: I have seen little evidence that your powers of
self-observation are
much more advanced than a 3 year old. Perhaps the strain of
pretending to
be
a real scientist when you are merely a glorified programmer
has gotten to
you?
Rickert: Longley regularly takes *interpretations* of
experimental results,
and declares them to be evidence. He seems to not even be
aware that
the interpretations were made on the assumption of the folk
psychological views that Longley decries.
GS: He reinterprets the results of experiments conducted in
the mentalist
tradition. The interaction between mentalistic concepts and the
experimental
procedures are not as straightforward as you are making them
out to be. I
have already suggested that you read Facts, Theories, and
Concepts: The
Shape of Psychology's Epistemic Triangle, where this is,
perhaps, made a
little clearer. But I would say that Longley's interpretations
are likely
to
stick closer to the observations than the interpretations
given by the
authors themselves. After all, it is the mentalist authors
that manipulate
one thing (and that is clearly stated in the procedure
section) but claim
they are actually manipulating something else - something
unobserved or
unobservable.
Rickert: For example, he cites work of Kahneman and Tversky. I
don't have
any
problem with their experimental work. It is sound research. But
what Longley asserts, is that this provides evidence of
irrationality. But he can only claim this based on
folk-psychological accounts of rationality. If he wants to
reject
folk psychology, he should equally reject those accounts of
rationality.
GS: As I have told you many times (and you have apparently
ignored) one
must
mention folk-psychological terms sometimes in order to point
to the
BEHAVIORAL PHENOMENA said to require them. The phenomena are
real enough,
but the mentalistic accounts are rubbish.
Rickert: He regularly cites Quine. But Quine's accounts of
science (as in
folk-psychological assumptions. Likewise, much of Quine's
writing on
language is derivative of folk-psychological assumptions.
Strictly speaking, radical behaviorism also is heavily
dependent on
folk psychological assumptions. If you removed all such
assumptions
from radical behaviorism, little of use would remain.
GS: On the contrary, radical behaviorism is useful precisely
because it
treats the phenomena said to require mentalistic notions
without pointing
to
such entities as causes. For example, it points to
contingencies of
reinforcement to explain behavior said to show possession of a
concept
etc. etc. etc. etc. etc. etc. You don't understand this,
because you don't
understand radical behaviorism. I know you claimed to have
read Verbal
Behavior, but I am asserting that you lied. I can tell from
the way you
talk
that you could not, and would not, have read the book.
>Now, we are all familiar with minority positions within
established
natural
>sciences; as you say, the individuals are frequently regarded
as a little
>quirky, but we have learned to be tolerant of the minority
view, as long
as
>it is not too far-fetched.
Rickert: And there we see an important point. Namely, Longley
is intolerant
of any view that does not agree with his. This should count as
evidence that Longley is a crank.
GS: We are all intolerant, to some extent, with views that
disagree with
ours. I give no credence to those who cannot observe their own
behavior
(like you) and see that this is true.
> In the case of Longley's endeavor's - and mine as well - the
issues
>are conceptual; they revolve around the very definition of
the subject
>matter and how it is to be studied.
Rickert: I see little discussion of actual definitions by
either Longley or
you.
GS: To claim to be a behaviorist is to define the subject
matter of
psychology.
> The rift that I am talking about is
>nothing less than the ongoing debate between behaviorism and
mainstream
>psychology and philosophy. The mainstream side of things
(cognitivism,
>mentalism) seeks to convince its neophytes that the debate is
over and the
>facts decided.
Rickert: .. Whereas the behaviorist side of thinkgs seeks to
convince its
neophytes that the debate is over and the facts decided in
favor of
behaviorism.
GS: True, but behaviorism does not perpetuate
misrepresentations of
cognitive science and occasionally encourages its students to
critically
read the literature. Can you imaging any cognitive people
telling their
students to read Verbal Behavior or Science and Human
Behavior? I can't.
Rickert: If you are making a distinction here, it would appear
to be a
distinction without a difference.
GS: Of course it appears that way to you, but you only appear
to have any
commerce with those responsible for the de facto censorship of
behaviorism.
>facts decided. You will see this position clearly expressed
in some of
the
>posts that are responses to mine. The upshot of this approach
is the de
>facto censorship and continued misrepresentation of
behaviorism, much like
>that of the creationists with respect to evolutionary theory.
The only
>difference is that the evolutionary view predominates in
mainstream
biology,
>but the psychological form of creationism called cognitive
psychology
>predominates in mainstream psychology and associated
philosophies.
Rickert: By contrast, the behaviorist form of creationism
predominates in
radical behaviriorist psychology. Again you appear to make a
distinction without a difference.
GS: We all know what biological creationism is. Psychological
creationism
simply holds that behavior is caused by people themselves, or
is a product
of a host of indwelling entities. For behaviorists behavior is
a product of
the selection that operates on species, cultures, and
individual behavioral
repertoires. If you can't see a difference, it is because you
doggedly
refuse to look.
>GS: But Longley's views are mostly consistent with a
scientific and
>philosophic tradition that is embraced by a minority of
practicing
>scientists and philosophers. Longley's thesis (and I am not
endorsing
every
>thing that David says, mind you) is not of the hollow Earth
variety
-
it
>is still a minority view, granted, but I can certainly find
hundreds of
>people (most of whom teach in universities and do government
funded
> Name a bunch of them.
> If Longley's views are compatible with credible scientists in
> psychology related fields, why is he not posting in the
> sci.psychology newsgroups? It is my impression that he posts
in
> c.a.p because he was was considered a crank by many of those
> (including behaviorists) who do post in the psychology
groups.
> Longley regularly takes *interpretations* of experimental
results,
> and declares them to be evidence. He seems to not even be
aware that
> the interpretations were made on the assumption of the folk
> psychological views that Longley decries.
> For example, he cites work of Kahneman and Tversky. I don't
have any
> problem with their experimental work. It is sound research.
But
> what Longley asserts, is that this provides evidence of
> irrationality. But he can only claim this based on
> folk-psychological accounts of rationality. If he wants to
reject
> folk psychology, he should equally reject those accounts of
> rationality.
> He regularly cites Quine. But Quine's accounts of science
(as in
> folk-psychological assumptions. Likewise, much of Quine's
writing on
> language is derivative of folk-psychological assumptions.
> Strictly speaking, radical behaviorism also is heavily
dependent on
> folk psychological assumptions. If you removed all such
assumptions
> from radical behaviorism, little of use would remain.
>Now, we are all familiar with minority positions within
established
natural
>sciences; as you say, the individuals are frequently regarded
as a
little
>quirky, but we have learned to be tolerant of the minority
view, as long
as
>it is not too far-fetched.
> And there we see an important point. Namely, Longley is
intolerant
> of any view that does not agree with his. This should count
as
> evidence that Longley is a crank.
> In the case of Longley's endeavor's - and mine as well - the
issues
>are conceptual; they revolve around the very definition of
the subject
>matter and how it is to be studied.
> I see little discussion of actual definitions by either
Longley or
> you.
> The rift that I am talking about is
>nothing less than the ongoing debate between behaviorism and
mainstream
>psychology and philosophy. The mainstream side of things
(cognitivism,
>mentalism) seeks to convince its neophytes that the debate is
over and
the
>facts decided.
> .. Whereas the behaviorist side of thinkgs seeks to convince
its
> neophytes that the debate is over and the facts decided in
favor of
> behaviorism.
> If you are making a distinction here, it would appear to be a
> distinction without a difference.
>facts decided. You will see this position clearly expressed
in some
of
the
>posts that are responses to mine. The upshot of this approach
is the de
>facto censorship and continued misrepresentation of
behaviorism, much
like
>that of the creationists with respect to evolutionary theory.
The only
>difference is that the evolutionary view predominates in
mainstream
biology,
>but the psychological form of creationism called cognitive
psychology
>predominates in mainstream psychology and associated
philosophies.
> By contrast, the behaviorist form of creationism
predominates in
> radical behaviriorist psychology. Again you appear to make a
> distinction without a difference.
===
Subject: Re: [Newbie] Ker(f) visual representation
> What is Im and Ker in linear algebra concerning
> the 3D geometry ?
> Is it linked to a transformation matrix ?
> Which one ?
The image and the kernel. Given any linear transformation T
between two
vector spaces A and B apply T to each point of A and look at
the resultant
point in B, put all those points in B together and you have
the image. The
image is a vector space too, in fact a subspace of B.
The kernel is those points of A which get mapped to zero in B.
It is a
vector space too, in fact a subspace of A.
Example: Let A be an ordinary 3D space with coordinates
(x,y,z) and B a 2D
space with coordinates (u,v) Let T take (x,y,z) and map it to
(x,0) in B
The image of T is the 'u' axis in B, the kernel of T is the yz
plane.
===
Subject: Re: [no subject]
> hello:
i want study algebraic number theory. i have chosen the book
(Lang,S
> algebraic number theory),but it is too difficult for me.So,
if there are
> more easier books for studying?
You could try the text by Stewart and Tall.
--
Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html
Needless to say, I had the last laugh.
Alan Partridge, _Bouncing Back_ (14 times)
===
Subject: Re: Radical center hypothesis
> Given a general triangle ABC. Erect three similar
> isosceles triangles ABc,BCa,CAb (in the same direction)
> on AB,BC,CA. The lines Aa,Bb,Cc concur in R (known -
> cf. Kiepert hyperbola).
> Draw a circle C_a around a through B and C. Likewise
> define C_b and C_c.
> Hpyothesis: The radical center of the C_i is R.
Your hypothesis is generally wrong.
Suppose that theta is the common basis angle of your three
isoceles
triangles (theta > 0 if the triangles are erected externally,
theta <0
otherwise)
Your lines Aa, Bb, Cc intersect at R(theta) and your radical
center is
R(Pi/2-theta), ie the second intersection of the line OR with
the Kiepert
hyperbola
where O = circumcenter(ABC)
===
Subject: Re: Radical center hypothesis
Oops! One shouldn't do statistics on a n=1 base,
accidental coincidencies abound :-)
But THX anyway, the correct version is as good
for my purposes as the case I thought of. Do you have a
literatur reference?
--
Hauke Reddmann <:-EX8
Private email:fc3a501@math.uni-hamburg.de
For our chemistry workgroup,remove math from the address
===
Subject: Re: Radical center hypothesis
> Oops! One shouldn't do statistics on a n=1 base,
> accidental coincidencies abound :-)
> But THX anyway, the correct version is as good
> for my purposes as the case I thought of. Do you have a
> literatur reference?
at
http://forumgeom.fau.edu/FG2001volume1/FG200118index.html
===
Subject: Radical center hypothesis
Given a general triangle ABC. Erect three similar
isosceles triangles ABc,BCa,CAb (in the same direction)
on AB,BC,CA. The lines Aa,Bb,Cc concur in R (known -
cf. Kiepert hyperbola).
Draw a circle C_a around a through B and C. Likewise
define C_b and C_c.
Hpyothesis: The radical center of the C_i is R.
--
Hauke Reddmann <:-EX8
Private email:fc3a501@math.uni-hamburg.de
For our chemistry workgroup,remove math from the address
===
Subject: Re: Fixed points and topology
> you describe are in fact open (which I don't see why). Why
do you say
that
> these two sets are in fact open? Can they be
> written as the intersection of X and an open set in R? I
guess the
question
> is which open set in R does this?
Consider g:[0,1]->R given by g(x)=f(x)-x. It's continous, and
{x|f(x)>x}={x|g(x)>0}=g^{-1}(0,infty) is open. The other one's
just
the same with inequalities reversed.
Philipp
===
Subject: Re: Fixed points and topology
hello,
assume f(0)>0 and f(1)<1 (otherwise we are done). consider
C := {x : f(x)> x }
then C is not empty by our assumption and C is bounded, so it
must
have an supremum A = sup(C). by continuity A cannot lie in C (
f(x)-x
is continous too !).
choose now a series (a_n) in C, that converges to A. then we
have f(A)
<= A (because A is not in C) and we have lim f(a_n) = f(A) >=
A since
f(a_n) >= a_n. together this gives f(A) = A, q.e.d.
T.Salesch
===
Subject: Re: Fixed points and topology
> Is it possible to prove this problem without using the
intermediate
> value
> theorem?
Let X = [0,1] which is a subset of R (the reals) and let X
have the
> subspace
> topology. f : X --> X is continuous. Prove that f has a fixed
point.
It's easy of course with the intermediate value theorem. Any
thoughts?
Let g(x) = f(x) - x (as the usual proof starts). Then if f(0)
= 0,
> we're
> done. Otherwise, g(0) > 0 and g(1) < 0 (unless g(1) = 0
where we are
> done
> again). Let g(0) = a and g(1) = b.
> Maybe I should start taking inverse images of open sets in X
under f?
Look at the two open sets {x|f(x)>x} and {x|f(x) that [0,1] is connected.
>
! Got the answer, but this is contingent on knowing that the
two
sets
> you describe are in fact open (which I don't see why). Why
do you say
that
> these two sets are in fact open? Can they be
> written as the intersection of X and an open set in R? I
guess the
question
> is which open set in R does this?
> Philipp
We might just consider the set S={x:f(x)>x}. The other is an
analogy.
suppose some x_0 belong to S.
We now offer some b>0 such that
a=f(x_0)>x_0+2b.
By uniform continuity of f on the interval X, we are free to
find some
d>0,
such that
|f(x_1)-f(x_0)|--
>
--
--
===
Subject: Re: Extension fields & Galois Theory Questions
Visiting Assistant Professor at the University of Montana.
>> I have a 4 questions about a couple of issues that are
confusing me. Any
>> answers to them would be very helpful:
>> 1- Let the field extension F(alpha) be constructed by
adjoining a root
alpha
>> of the irreducible polynomial f(x) of degree n in F[x].
Suppose that
>> F(alpha) contains more than 1 root of f(x) (possible
splitting field).
Is
>> there any way to tell whether the other roots other than
alpha will be
>> powers of alpha, c*alpha, or an F-linear combination of
>> alpha^(n-1),.....,alpha, 1?
>Not sure I understand the question.
>Every element of F(alpha) is an F-linear combination
>of alpha^(n-1), ..., alpha, 1 (and of course every such
F-linear
>combination is an element of F(alpha)) so the other roots of
f(x)
>are of that form if and only if F(alpha) is a splitting field
for
>f over F. It's unusual, but not impossible, for the other
roots
>to be powers of alpha, but I'm not sure how to tell (or even
what
>the question means - it depends on what information you are
given).
>The same for c alpha (where I take it you mean for c to be in
F).
But you can figure out a few things more or less easily. For
example,
if both alpha and alpha^k are roots of the irreducible f(x),
then let
N:F(alpha)->F be the norm function; then
N(alpha)=N(alpha^k)=(-1)^n*f(0), where n = deg(f).
Since N is multiplicative, we also know that
N(alpha^k)=N(alpha)^k, so
we must have N(alpha)=N(alpha)^k, or N(alpha)(N(alpha)^{k-1} -
1) =
0. So either alpha=0, or else N(alpha) is a (k-1)-st root of
unity in F.
Likewise, you would have N(c*alpha)=N(c)*N(alpha) = N(alpha)
if both
alpha and c*alpha are roots of f(x); if c in F, then N(c)=c^n,
so we
would have N(alpha) = c^n*N(alpha), hence alpha=0 or c^n=1,
making c
an n-th root of unity in F.
===
Subject: Re: Extension fields & Galois Theory Questions
> I have a 4 questions about a couple of issues that are
confusing me. Any
> answers to them would be very helpful:
1- Let the field extension F(alpha) be constructed by
adjoining a root
alpha
> of the irreducible polynomial f(x) of degree n in F[x].
Suppose that
> F(alpha) contains more than 1 root of f(x) (possible
splitting field). Is
> there any way to tell whether the other roots other than
alpha will be
> powers of alpha, c*alpha, or an F-linear combination of
> alpha^(n-1),.....,alpha, 1?
Not sure I understand the question.
Every element of F(alpha) is an F-linear combination
of alpha^(n-1), ..., alpha, 1 (and of course every such
F-linear
combination is an element of F(alpha)) so the other roots of
f(x)
are of that form if and only if F(alpha) is a splitting field
for
f over F. It's unusual, but not impossible, for the other roots
to be powers of alpha, but I'm not sure how to tell (or even
what
the question means - it depends on what information you are
given).
The same for c alpha (where I take it you mean for c to be in
F).
2- Suppose I have a cubic irreducible inseperable polynomial
f(x). Let
alpha
> be a root of multiplicity 2
That can't happen, can it? If f is cubic, irreducible, and
inseparable,
and alpha is a root in some extension field, doesn't alpha
have to be
of multiplicity 3?
> 3- Let f(x) = x^3 - 2 and let cbr(2) denote the cubic root
of 2, z(3) the
> 3rd primitive root of unity and let Q denote the rationals.
Now, Q(z(3),
> cbr(3)) is the splitting field of f(x) and the roots of f(x)
are cbr(2),
> z(3)*cbr(2) z(3)^2*cbr(2). I have 2 questions about this:
a) In Q(cbr(3)) : f(x) = (x - cbr(2))(x^2 + x + 1).
Nonsense.
--
Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email)
===
Subject: more generel kronecker symbol?
hello to all,
as you know the kronecker symbol (n/2) is defined as 1 if n =
1 mod 8,
0 if n = 0,4 mod 8 and -1 if n = 5 mod 8.
if you put n = D, where D = t^2 - 4n is the discriminant of
x^2 - tx +
n then
(D/2) tells you if the polynomial has 2,1 or 0 different roots.
my question: if i take a 2-adic number field K over the 2-adic
numbers
Q_2 of dimension n = ef, the ring A of its 2-whole numbers of
K and
its unique primideal P, is there a kronecker symbol (D/P),
that gives
me the same information as in the
above special case? for any answer
T.Salesch
===
Subject: Re: Smallest S_n in which a group G embeds.
>[...]
>> this; it is equivalent to finding the maximal order of a
>> subgroup of G that does not contain any proper normal
>> subgroup...
>I meant the maximum among the orders of subgroups of G that...
>(or indeed, the index of a largest subgroup not containing a
proper
>normal subgroup, as suggested by J.McKay via email.)
But it isn't right is it!! There are lots of examples of
groups G for which
the smallest S_n in which G embeds is smaller than the index
of the largest
subgroup not containing a proper normal subgroup. Think about
it.
Derek Holt.
===
Subject: Re: Smallest S_n in which a group G embeds.
> But it isn't right is it!! There are lots of examples of
groups G for
which
> the smallest S_n in which G embeds is smaller than the index
of the
largest
> subgroup not containing a proper normal subgroup. Think
about it.
ok, ok, I meant transitive embeddings of course.
Intransitive ones are even harder to find IMHO.
E.g. one has to factor integers to do this
just for cyclic groups.
> Derek Holt.
===
Subject: Re: Smallest S_n in which a group G embeds.
>> Given a finite group G, let s(G) be the smallest number n
such that G
>> embeds in the symmetric group on n letters, or
equivalently, the
>> cardinality of the smallest set on which G acts faithfully.
>For an arbitatry G there is nothing like a uniform answer to
>this; it is equivalent to finding the maximal order of a
>subgroup of G that does not contain any proper normal
>subgroup...
Yes, it is a difficult problem in general, but your final
statement:
>it is equivalent to finding the maximal order of a
>subgroup of G that does not contain any proper normal
>subgroup...
is not always correct.
Exercise: find an example in which this is not correct.
Derek Holt.
>> Question: I have been looking for some references, even
very basic,
>> on this topic. Any ideas?
>A basic reference would be P.Cameron's Permutation groups.
>In particular, the chapter on the O'Nan-Scott theorem.
>The latter would give one an idea of what is going on
>at least in the case of G a simple group.
>HTH,
>Dmitrii
===
Subject: Re: Smallest S_n in which a group G embeds.
[...]
> this; it is equivalent to finding the maximal order of a
> subgroup of G that does not contain any proper normal
> subgroup...
I meant the maximum among the orders of subgroups of G that...
(or indeed, the index of a largest subgroup not containing a
proper
normal subgroup, as suggested by J.McKay via email.)
===
Subject: Re: Smallest S_n in which a group G embeds.
> Given a finite group G, let s(G) be the smallest number n
such that G
> embeds in the symmetric group on n letters, or equivalently,
the
> cardinality of the smallest set on which G acts faithfully.
For an arbitatry G there is nothing like a uniform answer to
this; it is equivalent to finding the maximal order of a
subgroup of G that does not contain any proper normal
subgroup...
> Question: I have been looking for some references, even very
basic,
> on this topic. Any ideas?
A basic reference would be P.Cameron's Permutation groups.
In particular, the chapter on the O'Nan-Scott theorem.
The latter would give one an idea of what is going on
at least in the case of G a simple group.
HTH,
Dmitrii
===
Subject: Re: Expressing a polynomial in terms of y
> Is it possible to express a polynomial such as y = Ax^3 +
Bx^2 + Cx + D
in
> terms of y (i.e. x = f(y)). If yes, what is the process; if
no then why?
> in advance for any help.
> Andrew Milne
Sometimes it is possible. For example one can solve y=x^3+4*x
for x and get:
x=(1/6)*((108*y+12*(768+81*y^2)^(1/2))^(2/3)-48)/(108*y+12*(
768+81*y^2)^(1/2
))^(
1/3)
This sort of thing was first done in the 1500's by Italian
mathematicians.
See, for example: http://www.m-a.org.uk/eb/mg/mg077ch.pdf
If the degree of the polynomial is larger than 4, it is only
possible to
express
the solution(s) in terms of elementary functions in special
cases.
Jim Buddenhagen
------------
To reply copy jbuddenh@REMOVEtexas.net to address bar and edit
out REMOVE
===
Subject: Re: Expressing a polynomial in terms of y
> See, for example: http://www.m-a.org.uk/eb/mg/mg077ch.pdf
That link is no longer valid. The correct link is:
http://www.m-a.org.uk/docs/library/2059.pdf
--Jim Buddenhagen
===
Subject: Re: Expressing a polynomial in terms of y
> If you want to stay in the reals and (as you noted in
another post)
> want to limit yourself to intervals for functions y |-> x,
you can
> break the x-range up into three parts by the two points
where dy/dx =
> 0. You can always find two intervals (-infty, y0], [y1,
infty)
> such that the desired function f (still not a polynomial) is
> well-defined on the union of these two intervals, but,
except in
> special cases, you will not be able to find a well-defined
function
> f with domain all of R. (Consider, for example, y = x^3 - x.)
For that parenthetical example, Mathematica(R) yields
In[1]:= Solve[y==x^3-x,x]
1/3
2
Out[1]= {{x -> -(--------------------------------) -
2 1/3
(-27 y + Sqrt[-108 + 729 y ])
2 1/3
(-27 y + Sqrt[-108 + 729 y ])
> --------------------------------},
1/3
3 2
which is the unique real value of x if |y| > 2 sqrt(3)/9.
--
Stephen J. Herschkorn herschko@rutcor.rutgers.edu
===
Subject: Re: Math. Proof of Existance of God??? Minus some
babble
>[purported mathematical proof of the existence of God snipped]
>Zim Olson
>http://www.zimmathematics.com>You are babbling.
>I don't think I did too bad.
Of course you don't.
> Do you know a better Math. Proof of God?
Of course not. So what? The point was not that your proof
is worse than others, the point is that it's babbling nonsense.
>Look at this post on PROOFOFGOD
>http://www.mathforum.org/discuss/sci.math/t/537735
>Here is some real cyber babbling to me.
So? The fact that someone else is in your opinion babbling
even worse shows somehow that what you're saying is
not babbling nonsense? Doesn't follow.
>Zim Olson
>http://www.zimmathematics.com
************************
===
Subject: Re: Math. Proof of Existance of God??? Minus some
babble
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id h8SD2tH00903;
>>Is there anything you can think of that exists that does not
have
>>some context associated with it? I cannot think of any such
item.
>>Now take the context of the Universe. Is there a better
known way to
>> model the context of the Universe than God?
Don't know of a better way to say it. Maybe it could be stated
more
Mathematically. But that is all.
>>Context as far as the Universe goes could mean:
>>Where did it come from. Common properties. Common
Informational
>>Properties. Common non temporal informational properties.
Common
>>origin, cause. Etc.
>>The existance of God could be demonstrated by the usefull
knowledge
>> and applications derived by actually possessing the
knowledge found
>>in Math. , Science, Bible. As for the knowledge of God as
well as in
>>Mathematics and Sciences, you must actually possess the
knowledge in
>> order for God to actually exist as demonstrated thru usefull
>> applications of this knowledge.
Babbled a little bit here. Just trying to say that as in
Mathematics,
you must possess the knowledge of Mathematics for it to be
true. Same with
God. And the existance of God is proven by the usefull
applications derived
by knowledge of God.
>>For example:
>>In the Bible it says: Man is made in the image of God.....
>>This makes some intuitive sense to me by experience also.
>>So according to modern Mathematics: A Transformation must
exist....
The Transformation would be like Mathematics. Which would give
truth of
God, Science, Math., only if you understood the Transformation.
>>And according to this, I think: Truth is Possible.
>>But the Truth exists and is usefull only if you possess the
knowledge of
it.
>>Zim Olson
http://www.zimmathematics.comYou are babbling.
I don't think I did too bad. Do you know a better Math. Proof
of God?
Look at this post on PROOFOFGOD
http://www.mathforum.org/discuss/sci.math/t/537735
Here is some real cyber babbling to me.
http://www.zimmathematics.com
===
Subject: Re: Oh, God. Please help me!
>Yes Chris. Your solution is correct.
> Argument: If the square would be complete the number would
be 2n(n+1).
> If I cut by a diagonal the left region have n(n+1), but in
this case
> it lacks the right staircase, which have 2n. Total : n(n+1)+
2n = n^2+
> 3n
> Pick this one!
> Show 2n(n+1) works for a full square first.
Herc
Are you blind?
There are n+1 rows with n horiz. And n rows with n+1 verticals.
Luis
===
Subject: Re: Oh, God. Please help me!
>Yes Chris. Your solution is correct.
> Argument: If the square would be complete the number would be
2n(n+1).
> If I cut by a diagonal the left region have n(n+1), but in
this case
> it lacks the right staircase, which have 2n. Total : n(n+1)+
2n =
n^2+
> 3n
> Pick this one!
> Show 2n(n+1) works for a full square first.
Herc
> Are you blind?
no
> There are n+1 rows with n horiz. And n rows with n+1
verticals.
> Luis
that's better,
though I thought of it as n^2 * 2, the bottom and left parts
of the small
squares,
plus n along the top, plus n along the right. thats 2n * n +
2n = 2n (n +
1).
Uni level maths you'd get away with it, anywhere else pulling
2n(n+1) out of
nowhere
is a lost mark.
Herc
===
Subject: Re: Subfield of algebraics?
> Let F be the smallest subfield of R which contains Q and is
> closed under power operation, x, y in F ==> x^y in F when x^y
> takes on a real value. (Define it to be the positive value
when there
> are two possible values for the power.) Since not all
algebraics can
> be expressed via radicals, F is strictly contained in the
> algebraics, no? Is there a name for F?
>>As ANN and WDH has pointed out, this is not a subfield of the
>>algebraics. Mea culpa. Still,
>>-Does it still have a name? It is countable, no?
>I think it must be countable, by downward Lowenhein-Sk.9alem.
Of course it's countable - you don't need anything like L-S
for that!
If you start with a countable set and take the closure under
finitely
many finitary operations what you get is countable, just
because
a countable union of countable sets is countable (and the
product of finitely many countable sets is countable.)
>It embeds in the the prime model of the theory of R, in the
language
>L:=(+,-,.,^,0,1,>) which is countable.
>>-Different definition: Let G be the smallest subfield of R
which
>>contains 1 and has the property that x in G, n in N, x and n
>>positive ==> x^(1/n) in G. That is certainly (?) contained
in the
>>real algebraics. Name?
>>--
>>Stephen J. Herschkorn herschko@rutcor.rutgers.edu
************************
===
Subject: Re: Subfield of algebraics?
Let F be the smallest subfield of R which contains Q and is
> closed under power operation, x, y in F ==> x^y in F when x^y
> takes on a real value. (Define it to be the positive value
when there
> are two possible values for the power.) Since not all
algebraics can
> be expressed via radicals, F is strictly contained in the
> algebraics, no? Is there a name for F?
>>As ANN and WDH has pointed out, this is not a subfield of the
>>algebraics. Mea culpa. Still,
>>-Does it still have a name? It is countable, no?
I think it must be countable, by downward Lowenhein-Sk.9alem.
Of course it's countable - you don't need anything like L-S
for that!
> If you start with a countable set and take the closure under
finitely
> many finitary operations what you get is countable, just
because
> a countable union of countable sets is countable (and the
> product of finitely many countable sets is countable.)
It embeds in the the prime model of the theory of R, in the
language
>L:=(+,-,.,^,0,1,>) which is countable.
>-Different definition: Let G be the smallest subfield of R
which
>>contains 1 and has the property that x in G, n in N, x and n
>>positive ==> x^(1/n) in G. That is certainly (?) contained
in the
>>real algebraics. Name?
>>--
>>Stephen J. Herschkorn
herschko@rutcor.rutgers.edu
************************
Dave,
What is that supposed to mean? You're exact, you're implying
that the
product of infinitely many countable sets may be uncountable.
Is it
not always? For example, consider NxNxNx.... There's a map
from N
over NxN like there's a map from R over RxR, express the
number as a
binary sequence a1 a2 a3 a4 and map that to the element (a1 a3
... ,
a2 a4...). Similarly there's a map from non-negative Q (the
set of
ordered pairs of elements of N, or NxN) over NxNxNxN, for each
rational p/q express the coordinate (p1 p3 ..., p2 p4 ..., q1
q3 ...,
q2 q4 ...). Similarly there's a map from N to NxNxNxNx...xN for
finitely many dimensions. If I can map N over the product of
finitely
many instances (copies) of N, why can't I map Q over the
product of
infinitely many copies of N?
Are you trying to tell me that R doesn't map to the product of
infinitely many copies of R? What about RxR? Does |RxRx...xR|
thus
equal aleph_2? Would the product of that product with itself
countably infinitely many times thus have a cardinality to
equal
aleph_3?
How does R biject with NxNxNx...xN for countably infinitely
many
dimensions? Does it?
Can I call NxNxNx...xN something like N^N? Does N equal omega
plus
one?
Today I saw ducks, pheasants, grouse, a bluejay with black
wings, a
hawk, and a flock of turkeys. I didn't see the owl or the
ravens
today, although I heard some woodpeckers. I saw this owl that
was
three feet tall. They do fly silently.
How about the first definition where they're elements of the
algebraics.
Ross
===
Subject: Re: Subfield of algebraics?
>> Let F be the smallest subfield of R which contains Q and is
>> closed under power operation, x, y in F ==> x^y in F when
x^y
>> takes on a real value. (Define it to be the positive value
when there
>> are two possible values for the power.) Since not all
algebraics can
>> be expressed via radicals, F is strictly contained in the
>> algebraics, no? Is there a name for F?
>As ANN and WDH has pointed out, this is not a subfield of the
>algebraics. Mea culpa. Still,
-Does it still have a name? It is countable, no?
>>I think it must be countable, by downward Lowenhein-Sk.9alem.
>> Of course it's countable - you don't need anything like L-S
for that!
>> If you start with a countable set and take the closure
under finitely
>> many finitary operations what you get is countable, just
because
>> a countable union of countable sets is countable (and the
>> product of finitely many countable sets is countable.)
>>It embeds in the the prime model of the theory of R, in the
language
>>L:=(+,-,.,^,0,1,>) which is countable.
>-Different definition: Let G be the smallest subfield of R
which
>contains 1 and has the property that x in G, n in N, x and n
>positive ==> x^(1/n) in G. That is certainly (?) contained in
the
>real algebraics. Name?
--
>Stephen J. Herschkorn
herschko@rutcor.rutgers.edu
>> ************************
>>
>Dave,
>What is that supposed to mean? You're exact,
Yes.
>you're implying that the
>product of infinitely many countable sets may be uncountable.
No, I didn't imply that. It's true, of course, but doesn't
follow from
anything I said.
>Is it
>not always? For example, consider NxNxNx.... There's a map
from N
>over NxN like there's a map from R over RxR, express the
number as a
>binary sequence a1 a2 a3 a4 and map that to the element (a1
a3 ... ,
>a2 a4...). Similarly there's a map from non-negative Q (the
set of
>ordered pairs of elements of N, or NxN) over NxNxNxN, for each
>rational p/q express the coordinate (p1 p3 ..., p2 p4 ..., q1
q3 ...,
>q2 q4 ...). Similarly there's a map from N to NxNxNxNx...xN
for
>finitely many dimensions. If I can map N over the product of
finitely
>many instances (copies) of N, why can't I map Q over the
product of
>infinitely many copies of N?
Curiously, I'm a little confused by what you write here - are
you
claiming that NxNx... is countable or uncountable? In fact it's
uncountable, and that's why you can't map Q _onto_ it.
>Are you trying to tell me that R doesn't map to the product of
>infinitely many copies of R?
No, why would you think I was trying to tell you that?
> What about RxR? Does |RxRx...xR| thus
>equal aleph_2? Would the product of that product with itself
>countably infinitely many times thus have a cardinality to
equal
>aleph_3?
>How does R biject with NxNxNx...xN for countably infinitely
many
>dimensions? Does it?
>Can I call NxNxNx...xN something like N^N? Does N equal omega
plus
>one?
>Today I saw ducks, pheasants, grouse, a bluejay with black
wings, a
>hawk, and a flock of turkeys. I didn't see the owl or the
ravens
>today, although I heard some woodpeckers. I saw this owl that
was
>three feet tall. They do fly silently.
_Now_ you're making sense.
>How about the first definition where they're elements of the
>algebraics.
>Ross
************************
===
Subject: Re: Subfield of algebraics?
> Today I saw ducks, pheasants, grouse, a bluejay with black
wings, a
> hawk, and a flock of turkeys. I didn't see the owl or the
ravens
> today, although I heard some woodpeckers. I saw this owl
that was
> three feet tall. They do fly silently.
> Ross
At least in his birdwatching, Ross has included no obviously
bad
mathematics.
===
Subject: Re: Subfield of algebraics?
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id h8SLVPE30020;
>> Let F be the smallest subfield of R which contains Q and is
>> closed under power operation, x, y in F ==> x^y in F when
x^y
>> takes on a real value. (Define it to be the positive value
when there
>> are two possible values for the power.) Since not all
algebraics can
>> be expressed via radicals, F is strictly contained in the
>> algebraics, no? Is there a name for F?
>As ANN and WDH has pointed out, this is not a subfield of the
>algebraics. Mea culpa. Still,
>-Does it still have a name? It is countable, no?
I think it must be countable, by downward Lowenhein-Sk.9alem.
It embeds in
the the prime model of the theory of R, in the language
L:=(+,-,.,^,0,1,>) which is countable.
>-Different definition: Let G be the smallest subfield of R
which
>contains 1 and has the property that x in G, n in N, x and n
>positive ==> x^(1/n) in G. That is certainly (?) contained in
the
>real algebraics. Name?
>--
>Stephen J. Herschkorn herschko@rutcor.rutgers.edu
===
Subject: Re: Why Riemann's Hypothesis may be indecidable.
> how can one base a proof upon pseudorandomness?
> the only randomness that exists is,
> That whose period is too long for you to compute.
>
I never say I have a proof. I say : IT MAY BE IF the Moebius
function
forms a pseudorandom sequence.That was an heuristic argument.
But if you accept the criterium of length of period for
randomness, it
is easy to show that the sequence of gaps between primes have
have
periods indefinitely long.
Luis
===
Subject: Re: total variation norm
>Define the total variation norm || || of a function on the
line to be
>||f|| = sup( sum over all j |f(x_{j+1})-f(x_j)| ),
_where_ x_1 < x_2 < ... < x_n.
>How do you show that, if u(x,t) is the solution to the heat
equation
>u_t=u_{xx}, u(x,0)=f(x), then u satisfies
>||u(.,t)|| <= ||f|| ?
>I know that u(x,t) = 1/sqrt(4 pi t) int(-oo,oo) exp(
-(x-y)^2/(4t) )
>f(y) dy, but with the x in the exponent I haven't been able
to make
>||u(.,t)|| look like anything I can start writing inequalities
>withany suggestions?
Make a change of variables, to show that
u(x,t) =
1/sqrt(4 pi t) int(-oo,oo) exp( -y^2/(4t) ) f(x-y) dy .
Use the fact that
1/sqrt(4 pi t) int(-oo,oo) exp( -y^2/(4t) ) dy = 1.
************************
===
Subject: Re: total variation norm
Hmmm...I see why making the change of variables makes sense,
but now I
can't proceed from
||u|| = sup (sum) |1/sqrt(4 pi t) INT(-oo,oo)
e^(-y^2/(4t))[f(x_{i+1}-y)-f(x_i-y)] dy|.
If f was in L^1, it seems like I could use some sort of
Cauchy-Schwarz
inequality in the integral, but it's still a mystery how to
end up
comparing ||u|| to ||f|| rather than ||INT(-oo,oo) f||...
Define the total variation norm || || of a function on the
line to be
||f|| = sup( sum over all j |f(x_{j+1})-f(x_j)| ),
_where_ x_1 < x_2 < ... < x_n.
>How do you show that, if u(x,t) is the solution to the heat
equation
>u_t=u_{xx}, u(x,0)=f(x), then u satisfies
||u(.,t)|| <= ||f|| ?
I know that u(x,t) = 1/sqrt(4 pi t) int(-oo,oo) exp(
-(x-y)^2/(4t) )
>f(y) dy, but with the x in the exponent I haven't been able
to make
>||u(.,t)|| look like anything I can start writing inequalities
>with?any suggestions?
Make a change of variables, to show that
u(x,t) =
> 1/sqrt(4 pi t) int(-oo,oo) exp( -y^2/(4t) ) f(x-y) dy .
> Use the fact that
1/sqrt(4 pi t) int(-oo,oo) exp( -y^2/(4t) ) dy = 1.
> ************************
===
Subject: Re: total variation norm
>Hmmm...I see why making the change of variables makes sense,
but now I
>can't proceed from
>||u|| = sup (sum) |1/sqrt(4 pi t) INT(-oo,oo)
>e^(-y^2/(4t))[f(x_{i+1}-y)-f(x_i-y)] dy|.
>If f was in L^1, it seems like I could use some sort of
Cauchy-Schwarz
>inequality in the integral, but it's still a mystery how to
end up
>comparing ||u|| to ||f|| rather than ||INT(-oo,oo) f||...
Well I'll just do it then - make certain to mention my name
when
you hand it in:
|u(x_1, t) - u(x_2, t)| + ...
<= 1/sqrt(4 pi t) INT(-oo,oo) e^(-y^2/(4t))
(|f(x_1-y)-f(x_2-y)| + ...) dy
<= 1/sqrt(4 pi t) INT(-oo,oo) e^(-y^2/(4t)) ||f|| dy
= ||f|| .
>>Define the total variation norm || || of a function on the
line to be
>>||f|| = sup( sum over all j |f(x_{j+1})-f(x_j)| ),
>> _where_ x_1 < x_2 < ... < x_n.
>>How do you show that, if u(x,t) is the solution to the heat
equation
>>u_t=u_{xx}, u(x,0)=f(x), then u satisfies
>>||u(.,t)|| <= ||f|| ?
>>I know that u(x,t) = 1/sqrt(4 pi t) int(-oo,oo) exp(
-(x-y)^2/(4t) )
>>f(y) dy, but with the x in the exponent I haven't been able
to make
>>||u(.,t)|| look like anything I can start writing
inequalities
>>with?any suggestions?
>> Make a change of variables, to show that
>> u(x,t) =
>> 1/sqrt(4 pi t) int(-oo,oo) exp( -y^2/(4t) ) f(x-y) dy .
>> Use the fact that
>> 1/sqrt(4 pi t) int(-oo,oo) exp( -y^2/(4t) ) dy = 1.
>> ************************
>>
************************
===
Subject: total variation norm
Define the total variation norm || || of a function on the
line to be
||f|| = sup( sum over all j |f(x_{j+1})-f(x_j)| ).
How do you show that, if u(x,t) is the solution to the heat
equation
u_t=u_{xx}, u(x,0)=f(x), then u satisfies
||u(.,t)|| <= ||f|| ?
I know that u(x,t) = 1/sqrt(4 pi t) int(-oo,oo) exp(
-(x-y)^2/(4t) )
f(y) dy, but with the x in the exponent I haven't been able to
make
||u(.,t)|| look like anything I can start writing inequalities
withany suggestions?
===
Subject: Irony
Looking at a book on fractal image compression.
Towards the start he says something about closed and bounded
subsets of the plane - he's trying to be friendly to people
who know
no math, so in a footnote on that page he says that closed and
bounded are just technicalities needed to make the proofs work.
He states
(i) the reason for the words closed and bounded is to minimize
complaints from mathematicians.
Then a little later he gives the definition (not making this
up):
(ii) a metric space is compact if it's closed and bounded.
Looks like those words are not doing what they're intended to
do...
************************
===
Subject: Re: Irony
|(i) the reason for the words closed and bounded is to minimize
| complaints from mathematicians.
Those pesky mathematicians! Always complaining. What we
really need is a no-complain list for mathematicians. That way
you can sign up and no mathematicians will spoil your fun.
|Then a little later he gives the definition (not making this
up):
|
|(ii) a metric space is compact if it's closed and bounded.
Sometimes it works better when they give up entirely on
precision. It reminds me of the way some people appearing
on judge shows talk, as they try to talk more formally.
Keith Ramsay
===
Subject: Re: Irony
> Sometimes it works better when they give up entirely on
> precision.
Absolutely. There's nothing wrong with being imprecise
(except, of course,
when precision is required). But (see below) you only get into
trouble
when
you try to snow the judge.
> It reminds me of the way some people appearing
> on judge shows talk, as they try to talk more formally.
Jon Miller
===
Subject: Re: Irony
In sci.math, KRamsay
|(i) the reason for the words closed and bounded is to minimize
> | complaints from mathematicians.
Those pesky mathematicians! Always complaining. What we
> really need is a no-complain list for mathematicians. That
way
> you can sign up and no mathematicians will spoil your fun.
Except for those mathematicians who work for charity or for
political campaigns. :-)
|Then a little later he gives the definition (not making this
up):
> |
> |(ii) a metric space is compact if it's closed and bounded.
Sometimes it works better when they give up entirely on
> precision. It reminds me of the way some people appearing
> on judge shows talk, as they try to talk more formally.
Keith Ramsay
>
--
#191, ewill3@earthlink.net
It's still legal to go .sigless.
===
Subject: Re: Computational Evolution without velocities
> The underlying assumptions allowing to neglect the velocity
> term is that it always remains negligible with respect to
the force
> term. This situation occurs effectively in very viscous
fluids,
> to random fluctuations.
The formula is still dimensionally incorrect, even in the
viscous
limit.
To make it even approximately correct, one needs to introduce
some sort of
relaxation time tau, and assume that the basic timestep is
very large
compared to tau, but that the force does not change
significantly over
the
distance travel during the timestep. Then, to a very crude
approximation,
x_final = x_initial + delta_T * tau * F / m
Hoever, even if one assumes that delta T has been scaled in
units of
tau,
this still does not reproduce the OP's formula,
OP> x_new = x_old - Force DeltaTime / 2 Mass
Which (in addition to the mysterious factor of `2') for some
bizarre reason
-- Gordon D. Pusch
===
Subject: Re: Computational Evolution without velocities
>>The underlying assumptions allowing to neglect the velocity
>>term is that it always remains negligible with respect to
the force
>>term. This situation occurs effectively in very viscous
fluids,
>>to random fluctuations.
> The formula is still dimensionally incorrect, even in the
viscous
limit.
Yes, it seems necessary to repeat a previous post remark:
the correct formula needs a (Delta t)^2 instead of (Delta t).
Dan
===
Subject: Re: Computational Evolution without velocities
>The underlying assumptions allowing to neglect the velocity
> term is that it always remains negligible with respect to
the force
> term. This situation occurs effectively in very viscous
fluids,
> to random fluctuations.
>> The formula is still dimensionally incorrect, even in the
viscous
limit.
Yes, it seems necessary to repeat a previous post remark:
> the correct formula needs a (Delta t)^2 instead of (Delta t).
That would make it dimensionally correct, but still would not
represent the
direction of the applied force !!!
-- Gordon D. Pusch
===
Subject: Re: Computational Evolution without velocities
The underlying assumptions allowing to neglect the velocity
term is that it always remains negligible with respect to the
force
term. This situation occurs effectively in very viscous fluids,
to random fluctuations.
Dan
> Sorry, there was a typo! I should have squared DeltaTime.
This is by no means cartoon physics. People are using this in
research
> simulations and I'm wondering if there is a sound basis or
not.
!!
> Free your mind. There is no spoon.
> ************************************************
> Dr. Patrick Bangert
> http://www.knot-theory.org
> Research Instructor for Mathematics
> International University Bremen
>>I have heard the folklore that when one wants to simulate
some structure
>>Newton's laws (i.e. non-relativistic) one can neglect the
fact that the
>>update formula
>>x_new = x_old - Force DeltaTime / 2 Mass
>>holds for any simulation and we do not need to take care of
velocities.
What
>I would like to know is does anyone know whether this holds
only under
>>certain conditions (evolution at equilibrium, etc.) and/or
knows of
research
>papers that deal with this issue and perhaps give a proof of
this claim
as
a
>theorem. The strange thing is that apparently this is very
well known and
>>widely practised but no one I've spoken to knew where to
find the
details.
>>Pat
>>Free your mind. There is no spoon.
>>************************************************
>>Dr. Patrick Bangert
>>http://www.knot-theory.org
>>Research Instructor for Mathematics
>>International University Bremen
>
===
Subject: Re: PigeonHole Principle Problem
> The (a) part can be fairly easily done with PHP: You have 9
pigeons and
> 2 holes so at least one hole contains at least ciel(9/2) = 5
pigeons.
hah! thats made my week. i always thought it was an old wives
tale that
americans dont know what a pigeonhole is.
===
Subject: Re: PigeonHole Principle Problem
>> The (a) part can be fairly easily done with PHP: You have 9
pigeons and
>> 2 holes so at least one hole contains at least ciel(9/2) =
5 pigeons.
>hah! thats made my week. i always thought it was an old wives
tale that
>americans dont know what a pigeonhole is.
Observe the answer is ceil(9/2) rather than floor(9/2) pigeons.
Pigeons can escape through a hole in the ceiling but not the
floor.
--
Wanted: Experts at choosing the best of 100+ applicants for a
position.
Register as a California voter by September 22, and vote on
October 7.
Peter-Lawrence.Montgomery@cwi.nl Home: San Rafael, California
Microsoft Research and CWI
===
Subject: Re: PigeonHole Principle Problem
> Hello
> We are studying the PigeonHole principle and though I have
answered the
> first part of the question, the second is a little more
tricky. It says:
> Suppose there are 9 students in a class
> a)
> Show the class must have at least 5 male or 5 female
students.
> I simply stated that as there are only 2 options, male or
female, if the
> above is false then there would be > 5 of the second option
making the
> statement true.
> b)
> Show the class must have at least 3 male or at least 7
female students.
> Im a little sketchy on this. How do i attack this? .
Look at problem (a). Set up two cubbies - one with 5 holes for
boys and
one
with 5 holes for girls. Place the maximum number of students
in the first
cubby without filling it (4) then do the same in the second
cubby (again
4).
There is one student left over. What happens if you place them
in the boys
cubby? What happens if you place them in the girls cubby?
Problem (b) is solved the exact same way.
===
Subject: Re: metric space equivalence:B is open <=> ?
> In a metric space X, a subset S is closed iff S^c is open.
> Proof
> (=>):
> 1 S is closed
> 2 x in S^c
> --3 S^c is not open (assumption that gives contradiction)
> | 4 Ae>0,Ey in X:y in (U(x,e) n S)
> | 5 x in Cl(S) (by def and line 4)
> | 6 x in S (by line 1,5)
> | 7 contradiction (by line 2,6)
> --------------------------------------
> 8 S^c is open
Do you agree?Is everything correct?
I am puzzled if by S^c you mean the complement of S in X then
the statement
is trivially true. A Metric space is a special kind of
topological space
and
from topology, closed sets are defined as the compliment of
open sets.
---------
Yes;S^c is a complement of S in X.Yes you are right but I
don't want to use
that definition of closed sets;instead I define it as a set of
all limit
points.
I think it would help if it was a little less cryptic. I have
no problem
until line 4.
I meant that line 4 equals line 3 i.e. negation of S^c is open
gives
line 4.
i.e. S^c is not open <=> Ae>0,Ey in X:y in (U(x,e) n S)
===
Subject: Re: metric space equivalence:B is open <=> ?
> In a metric space X, a subset S is closed iff S^c is open.
> Proof
> (=>):
> 1 S is closed
> 2 x in S^c
> --3 S^c is not open (assumption that gives contradiction)
> | 4 Ae>0,Ey in X:y in (U(x,e) n S)
> | 5 x in Cl(S) (by def and line 4)
> | 6 x in S (by line 1,5)
> | 7 contradiction (by line 2,6)
> --------------------------------------
> 8 S^c is open
> Do you agree?Is everything correct?
> I am puzzled if by S^c you mean the complement of S in X
then the
statement
> is trivially true. A Metric space is a special kind of
topological space
and
> from topology, closed sets are defined as the compliment of
open sets.
> ---------
> Yes;S^c is a complement of S in X.Yes you are right but I
don't want to
use
> that definition of closed sets;instead I define it as a set
of all limit
points.
> I think it would help if it was a little less cryptic. I
have no problem
> until line 4.
> I meant that line 4 equals line 3 i.e. negation of S^c is
open gives
line 4.
> i.e. S^c is not open <=> Ae>0,Ey in X:y in (U(x,e) n S)
This doesn't follow; in the metric space R^2 with the usual
distance
metric,
imagine S was the closed unit ball radius <=1centred at (0,0)
and take x to
be the point at (100,100) in S^c. In line 4 you're trying to
telling me for
all e>0, there exists a y in U(x,e) that also happend to be in
S? Any e
less
than 99 will show that thats not true.
I think you're being led astray by formalism, draw pictures,
try and
visualise what you are saying. 'Mathematics is a visual
science'.
===
Subject: Re: metric space equivalence:B is open <=> ?
In a metric space X, a subset S is closed iff S^c is open.
Proof
(=>):
1 S is closed
2 x in S^c
--3 S^c is not open (assumption that gives contradiction)
| 4 Ae>0,Ey in X:y in (U(x,e) n S)
| 5 x in Cl(S) (by def and line 4)
| 6 x in S (by line 1,5)
| 7 contradiction (by line 2,6)
--------------------------------------
8 S^c is open
This doesn't follow; in the metric space R^2 with the usual
distance
metric,
imagine S was the closed unit ball radius 1 centred at (0,0)
and take
x to
be the point at (100,100) in S^c. In line 4 you're trying to
telling
me for
all e>0, there exists a y in U(x,e) that also happend to be in
S? Any
e less
than 99 will show that thats not true.
Yes but isn't that just another way to get a contradiction?I'd
like to
know if
the following is true?
S^c is not open <=> Ae>0,Ey in X:y in (U(x,e) n S)
you say that it's not true but then it means that the
following is
also not true
S^c is open <=> Ee>0,Ay in X:y in U(x,e) => y in S^c
But this is a definition of open set or is it? This is the
crucial
point as I see it.
===
Subject: Re: metric space equivalence:B is open <=> ?
> In a metric space X, a subset S is closed iff S^c is open.
> Proof
> (=>):
> 1 S is closed
> 2 x in S^c
> --3 S^c is not open (assumption that gives contradiction)
> | 4 Ae>0,Ey in X:y in (U(x,e) n S)
> | 5 x in Cl(S) (by def and line 4)
> | 6 x in S (by line 1,5)
> | 7 contradiction (by line 2,6)
> --------------------------------------
> 8 S^c is open
> This doesn't follow; in the metric space R^2 with the usual
distance
> metric,
> imagine S was the closed unit ball radius 1 centred at (0,0)
and take
> x to
> be the point at (100,100) in S^c. In line 4 you're trying to
telling
> me for
> all e>0, there exists a y in U(x,e) that also happend to be
in S? Any
> e less
> than 99 will show that thats not true.
> Yes but isn't that just another way to get a
contradiction?I'd like to
> know if
> the following is true?
It isn't true.
> S^c is not open <=> Ae>0,Ey in X:y in (U(x,e) n S)
> you say that it's not true but then it means that the
following is
> also not true
> S^c is open <=> Ee>0,Ay in X:y in U(x,e) => y in S^c
> But this is a definition of open set or is it?
No it isn't, but if you insist on writing it like that it
would be:
S^c is open <=> Ax in S^c, Ee>0: U(x,e) contained in S^c
===
Subject: Re: Aren't Free Logicians Crass Formalists? (Was Re:
David Ullrich
on Identity)
> No one other than Karel Lambert (and his followers)
> would ever confuse an assertion about Vulcan with one
> about Vulcan. If Lambert weren't a Very Important
> Philosopher, it would have escaped nobody's attention
> that Robin Chapman's criticism of Charlie (for being
> unable to distinguish between a thing and its
> applies in spades--to Karel Lambert and to every
> mother and son of a 'free'logician, whose collective
> merit it has been to have parlayed one cock-eyed idea
> of Carnap's--that Honesty is a virtue REALLY means
> 'Honesty' is a virtue-word, into an even more cock-eyed
> idea, which has since become the calling-card of traffickers
> in 'Free' Logic: the idea that Pegasus doesn't exist
> REALLY means: 'Pegasus' doesn't denote
> (or something like that).
GET NON-SELF-IDENTICAL, JOHN!
PH
===
Subject: Re: Aren't Free Logicians Crass Formalists? (Was Re:
David Ullrich
on Identity)
If Lambert weren't a Very Important
>> Philosopher,
Never heard of him. Can't be a Very Important Philosopher
in the same sense that Socrates, Plato, Aristotle, Anselm,
Aquinas,
Descartes, Hume, Leibniz, Locke, Russell, Wittgenstein, Popper,
Quine and Kripke are.
Of course, any ludicrous conclusion can follow from a false
premise :-)
--
Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html
Needless to say, I had the last laugh.
Alan Partridge, _Bouncing Back_ (14 times)
===
Subject: Re: Aren't Free Logicians Crass Formalists? (Was Re:
David Ullrich
on Identity)
>
> If Lambert weren't a Very Important
>> Philosopher,
Never heard of him. Can't be a Very Important Philosopher
> in the same sense that Socrates, Plato, Aristotle, Anselm,
Aquinas,
> Descartes, Hume, Leibniz, Locke, Russell, Wittgenstein,
Popper,
> Quine and Kripke are.
Of course, any ludicrous conclusion can follow from a false
premise :-)
You're even more clueless than I thought. Is Exeter where they
stash
Brits who are otherwise unemployable? Or is that just the case
for the math department?
--John
===
Subject: Aren't Free Logicians Crass Formalists? (Was Re:
David Ullrich on
Identity)
> If identity is not reflexive, what does this say about the
possibility
> of contradiction? If something can be what it is not, there
is a
> problem.
Is there anything contradictory about there being no such thing
> as Vulcan, (~Ex(Vulcan = x))? If not, how can any
*consequence* of
> such an assertion give rise to a contradiction?
> Indeed, in Negative Free Logic (NFL) we have
~(Vulcan = Vulcan),
> from
> ~Ex(Vulcan = x),
but NFL is provable consistent (i.e. there are no
contradictions).
> F.
No one other than Karel Lambert (and his followers)
would ever confuse an assertion about Vulcan with one
about Vulcan. If Lambert weren't a Very Important
Philosopher, it would have escaped nobody's attention
that Robin Chapman's criticism of Charlie (for being
unable to distinguish between a thing and its
applies in spades--to Karel Lambert and to every
mother and son of a 'free'logician, whose collective
merit it has been to have parlayed one cock-eyed idea
of Carnap's--that Honesty is a virtue REALLY means
'Honesty' is a virtue-word, into an even more cock-eyed
idea, which has since become the calling-card of traffickers
in 'Free' Logic: the idea that Pegasus doesn't exist
REALLY means: 'Pegasus' doesn't denote
(or something like that).
--John
===
Subject: A Tensor problem
Given a 2nd order tensor a_ij with matrix:
(2 0 3)
(5 1 2)
(4 5 7)
We know, of course, that:
a_ii = tr (a_ij)
What then would be the associated matrix for a_ll?
(double letter l indices)
in advance.
===
Subject: Re: how does one prove that all even numbers can be
represented in
binary?
> how does one prove that all even numbers can be represented
in binary?
You prove first that all numbers are representable in binary.
To then show that an even number 2n is representable just
note that you can stick on a zero on the binary representation
of n to get one for 2n :-)
--
Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html
Needless to say, I had the last laugh.
Alan Partridge, _Bouncing Back_ (14 times)
===
Subject: Re: how does one prove that all even numbers can be
represented in
binary?
In sci.math, Robin Chapman
:
> how does one prove that all even numbers can be represented
in binary?
You prove first that all numbers are representable in binary.
> To then show that an even number 2n is representable just
> note that you can stick on a zero on the binary
representation
> of n to get one for 2n :-)
>
Pedant Point:
There is the problem that all numbers are *not* representable
in (finite) binary: for example, 1/11(2) = .010101010101...(2)
However, all integers are easily representable in binary
through progressive digitation (and, in the case of negative
numbers, by writing '-' followed by its negative, which is
positive).
In short, let D_0 be the bit just to the left of the binary
point,
D_1 be the digit to its left, D_2 be to the left, etc.
D_0 = n mod 2.
Let n_0 = n.
Then iterate:
n_{k+1} = floor(n_k / 2)
D_k = n_k mod 2.
k = k + 1
until n_k = 0.
There are other methods for determining D_k, of course.
--
#191, ewill3@earthlink.net
It's still legal to go .sigless.
===
Subject: how does one prove that all even numbers can be
represented in
binary?
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id h8T1wpG13265;
how does one prove that all even numbers can be represented in
binary?
===
Subject: Re: heteroscedastic t-distribution
>All,
>Is there an extension of the heteroscedastic t-distribution
over a
>multi-variate normal distribution? I want to compare the
means of
>different n-variate normal distributions, each of which might
come
>with a different covariance. Any reference would be great.
Bjoern
I think you are better off posting this question to the stat
newsgroups. I have done so with this reply.
--
Stephen J. Herschkorn herschko@rutcor.rutgers.edu
===
Subject: heteroscedastic t-distribution
All,
Is there an extension of the heteroscedastic t-distribution
over a
multi-variate normal distribution? I want to compare the means
of
different n-variate normal distributions, each of which might
come
with a different covariance. Any reference would be great.
Bjoern
===
Subject: Re: how to approve this equation?
>> my error. Determinant =0 represents a plane through
(x1,y1,z1),(x2,y2,z2),(x3,y3,z3).
===
Subject: Re: how to approve this equation?
> Is there some geometric interpretation?
Determinant =0 represents a plane through
(x1,y1),(x2,y2),(x3,y3).
===
Subject: Re: Uncle Al is My Bitch .
<23i4u4abj7go$.dlg@__.Jeff.Relf>
<112o7niv16j7q.dlg@__.Jeff.Relf>
<6ojybtqoky31.dlg@__.Jeff.Relf>
<6188pxkbotuk.dlg@__.Jeff.Relf>
<7c3oa9j9mmle$.dlg@__.Jeff.Relf>
Hi Hanson ,
> You cite some Bitch :
> http://www.mazepath.com/uncleal/bung.jpg
Al's site isn't working right now ,
> so I can't see the picture .
Liar. The mazepath host has logged better than 99.9% up time.
The
referenced picture is an apt Relfie-boy depiction - as Relfie's
hundreds of trolled bull posts screamingly attest. Look at the
shovelful he added to his manure pile today.
For all that, Relfie-boy STILL cannot format an ASCII line. He
puts
on airs like a patholgical C-tabbing programmer. It's bathetic
(look
it up).
--
Uncle Al
http://www.mazepath.com/uncleal/
===
Subject: C Code for Solving Cubic Equation
I'm looking for a C function for solving cubic equations of
the form
a*x^3 + b*x^2 + c*x + d = 0
I'm only interested in finding numeric values for the real
roots; I
don't need an analytical solution and I'm not interested in the
complex roots (if any). I'd like to know if such a C function
is
available before writing it myself.
!
-Nick
===
Subject: Re: C Code for Solving Cubic Equation
Open matlab, type solve('a*x^3 + b*x^2 + c*x + d ')
ans =
[
1/6/a*(36*c*b*a-108*d*a^2-8*b^3+12*3^(1/2)*(4*c^3*a-c^2*b^2-18
*c*b*a*d+27*d^
2*a^2+4*d*b^3)^(1/2)*a)^(1/3)-2/3*(3*c*a-b^2)/a/(36*c*b*a-108*
d*a^2-8*b^3+12
*3^(1/2)*(4*c^3*a-c^2*b^2-18*c*b*a*d+27*d^2*a^2+4*d*b^3)^(1/2)
*a)^(1/3)-1/3*
b/a]
[
-1/12/a*(36*c*b*a-108*d*a^2-8*b^3+12*3^(1/2)*(4*c^3*a-c^2*b^2-
18*c*b*a*d+2
7*d^2*a^2+4*d*b^3)^(1/2)*a)^(1/3)+1/3*(3*c*a-b^2)/a/(36*c*b*a-
108*d*a^2-8*b^
3+12*3^(1/2)*(4*c^3*a-c^2*b^2-18*c*b*a*d+27*d^2*a^2+4*d*b^3)^(
1/2)*a)^(1/3)-
1/3*b/a+1/2*i*3^(1/2)*(1/6/a*(36*c*b*a-108*d*a^2-8*b^3+12*3^(1
/2)*(4*c^3*a-c
^2*b^2-18*c*b*a*d+27*d^2*a^2+4*d*b^3)^(1/2)*a)^(1/3)+2/3*(3*c*
a-b^2)/a/(36*c
*b*a-108*d*a^2-8*b^3+12*3^(1/2)*(4*c^3*a-c^2*b^2-18*c*b*a*d+27
*d^2*a^2+4*d*b
^3)^(1/2)*a)^(1/3))]
[
-1/12/a*(36*c*b*a-108*d*a^2-8*b^3+12*3^(1/2)*(4*c^3*a-c^2*b^2-
18*c*b*a*d+2
7*d^2*a^2+4*d*b^3)^(1/2)*a)^(1/3)+1/3*(3*c*a-b^2)/a/(36*c*b*a-
108*d*a^2-8*b^
3+12*3^(1/2)*(4*c^3*a-c^2*b^2-18*c*b*a*d+27*d^2*a^2+4*d*b^3)^(
1/2)*a)^(1/3)-
1/3*b/a-1/2*i*3^(1/2)*(1/6/a*(36*c*b*a-108*d*a^2-8*b^3+12*3^(1
/2)*(4*c^3*a-c
^2*b^2-18*c*b*a*d+27*d^2*a^2+4*d*b^3)^(1/2)*a)^(1/3)+2/3*(3*c*
a-b^2)/a/(36*c
*b*a-108*d*a^2-8*b^3+12*3^(1/2)*(4*c^3*a-c^2*b^2-18*c*b*a*d+27
*d^2*a^2+4*d*b
^3)^(1/2)*a)^(1/3))]
===
Subject: Re: C Code for Solving Cubic Equation
> I'm looking for a C function for solving cubic equations of
the form
> a*x^3 + b*x^2 + c*x + d = 0
> I'm only interested in finding numeric values for the real
roots; I
> don't need an analytical solution and I'm not interested in
the
> complex roots (if any). I'd like to know if such a C
function is
> available before writing it myself.
> !
> -Nick
Yup in fact I found one in Java....
http://jas.freehep.org/servlet/lcdcvs/log/lcd/Jama/
QRDecomposition.java/0
In fact I was just about to ask this group to help optimze the
QR
method in this class. Anyone?
===
Subject: Re: C Code for Solving Cubic Equation
> I'm looking for a C function for solving cubic equations of
the form
> a*x^3 + b*x^2 + c*x + d = 0
> I'm only interested in finding numeric values for the real
roots; I
> don't need an analytical solution and I'm not interested in
the
> complex roots (if any). I'd like to know if such a C
function is
> available before writing it myself.
> !
> -Nick
This may be extreme overkill, but you can find C software for
extracting
all roots of polynomials at http://www.crbond.com/roots.htm
including an
implementation of Bairstow's method and a full C port of the
Jenkins-Traub FORTRAN code.
--
There are two things you must never attempt to prove: the
unprovable --
and the obvious.
--
Democracy: The triumph of popularity over principle.
--
http://www.crbond.com
===
Subject: Re: C Code for Solving Cubic Equation
>I'm looking for a C function for solving cubic equations of
the form
>a*x^3 + b*x^2 + c*x + d = 0
>I'm only interested in finding numeric values for the real
roots; I
>don't need an analytical solution and I'm not interested in
the
>complex roots (if any). I'd like to know if such a C function
is
>available before writing it myself.
>!
>-Nick
Numerical Recipes in C (http://serv.ul.cs.cmu.edu/html/) has
algorithms for finding roots of polynomials. See in particular
the
section on Laguerre's method. You're not supposed to use their
code in
commercial packages, though.
You can also try
http://netlib.bell-labs.com/netlib/toms/index.html
(search for cubic equation roots). These are from the ACM
collected
algorithms.
John Mitchell
===
Subject: Re: C Code for Solving Cubic Equation
Nick:
You need to learn/understand Newton's Method of root finding.
Look it up in any text on numerical methods. Takes less than
10 lines of C
for your problem.
OTOH... Why not use a pocket calculator? It only takes a few
keystrokes.
--
Peter
> I'm looking for a C function for solving cubic equations of
the form
> a*x^3 + b*x^2 + c*x + d = 0
> I'm only interested in finding numeric values for the real
roots; I
> don't need an analytical solution and I'm not interested in
the
> complex roots (if any). I'd like to know if such a C
function is
> available before writing it myself.
> !
> -Nick
===
Subject: Need Advice on Math Careers
I'm an experienced software developer who is considering
making a career
change in the hopes of finding better
employment opportunities. One of the options I'm considering
is getting an
M.S./M.A. Degree in Mathematics or
Statistics. What is the present/future job/career outlook for
these degree
holders in the USA?
Peter
===
Subject: Re: Need Advice on Math Careers
I can speak from experience.
Prospects suck. I was once told by
the CEO of a company that mathematicians have a reputation of
not
being 'results oriented'. My experience
in this field (25 years now) suggests that
this attitude is prevalent.
You can lead a horse's ass to knowledge, but you can't make
him think.
===
Subject: Re: Need Advice on Math Careers
pubkeybreaker@aol.com.comstuff said:
>Prospects suck. I was once told by
>the CEO of a company that mathematicians have a reputation of
not
>being 'results oriented'. My experience
>in this field (25 years now) suggests that
>this attitude is prevalent.
>You can lead a horse's ass to knowledge, but you can't make
him think.
A trend in judging Mathematicians is one thing; a trend in
judging
Mathematics
skill is another. Some sharp employment interviewers and
personnel
coordinators assess mathematical skills of their job
candidates. Good
performance on these assessments usually is highly impressive
to the
employer
trying to choose a candidate. Realize that these assessments
are not
necessarily for Mathematicians, but for engineers,
technologists, and
scientists who are usually expected to apply significant
mathematical
knowledge
in the work.
G C
===
Subject: Matrix multiplication order
I have a question regarding the order of multiplication
for two systems.
The first one is given as y=ABx+n (1)
while the second one is y=BAx+n (2).
n is a noise vector with gaussian distributed entries, with
a mean of zero and unit variance. The elements of A also
follow the same identical gaussian distribution while B
is a correlation matrix.
To find x I can invert (AB) or (BA), and obtain a noisy
estimate from y.
The problem is with the correlation matrix B. If it's
ill-condtioned then with (1) _some_ elements of the estimated x
are distorted quite heavily.
In contrast, with (2) the noise-enhancement is typically
spread across _all_ elements.
However, on average, it looks to me that the noise enhancement
due to B is the same for both systems.
Is it possible to show that in some way or am I quite wrong
here ?
Should I expect that, on average, with (2) my estimates of x
will
be much worse than under (1) ? Ie upper bounded by the sum
of noise distortions inv(B)n ?
Any help will be appreciated.
===
Subject: Re: Matrix multiplication order
There is no reason, one should favor either approach (1) or
(2), in either
cases part of the information content in x is destroyed after
being
transformed by A and B.
You may use regularization to find a robust solution for x.
Google on Keyword Per Christian Hansen for his matlab toolbox
that
offer
techniques to obtain regularized solutions for ordinary least
squares
problems.
Alien+
> I have a question regarding the order of multiplication
> for two systems.
> The first one is given as y=ABx+n (1)
> while the second one is y=BAx+n (2).
> n is a noise vector with gaussian distributed entries, with
> a mean of zero and unit variance. The elements of A also
> follow the same identical gaussian distribution while B
> is a correlation matrix.
> To find x I can invert (AB) or (BA), and obtain a noisy
> estimate from y.
> The problem is with the correlation matrix B. If it's
> ill-condtioned then with (1) _some_ elements of the
estimated x
> are distorted quite heavily.
> In contrast, with (2) the noise-enhancement is typically
> spread across _all_ elements.
> However, on average, it looks to me that the noise
enhancement
> due to B is the same for both systems.
> Is it possible to show that in some way or am I quite wrong
here ?
> Should I expect that, on average, with (2) my estimates of x
will
> be much worse than under (1) ? Ie upper bounded by the sum
> of noise distortions inv(B)n ?
> Any help will be appreciated.
===
Subject: Re: Matrix multiplication order
I know, both systems are bad; but will both of them
give _equal_ performance on average ?
> There is no reason, one should favor either approach (1) or
(2), in
either
> cases part of the information content in x is destroyed
after being
> transformed by A and B.
You may use regularization to find a robust solution for x.
> Google on Keyword Per Christian Hansen for his matlab
toolbox that
offer
> techniques to obtain regularized solutions for ordinary
least squares
> problems.
Alien+
I have a question regarding the order of multiplication
> for two systems.
The first one is given as y=ABx+n (1)
> while the second one is y=BAx+n (2).
n is a noise vector with gaussian distributed entries, with
> a mean of zero and unit variance. The elements of A also
> follow the same identical gaussian distribution while B
> is a correlation matrix.
To find x I can invert (AB) or (BA), and obtain a noisy
> estimate from y.
The problem is with the correlation matrix B. If it's
> ill-condtioned then with (1) _some_ elements of the
estimated x
> are distorted quite heavily.
> In contrast, with (2) the noise-enhancement is typically
> spread across _all_ elements.
However, on average, it looks to me that the noise enhancement
> due to B is the same for both systems.
> Is it possible to show that in some way or am I quite wrong
here ?
Should I expect that, on average, with (2) my estimates of x
will
> be much worse than under (1) ? Ie upper bounded by the sum
> of noise distortions inv(B)n ?
Any help will be appreciated.
===
Subject: Re: Conservation of angular momentum not apparant
If you are in a canoe floating on perfectly idle water, are
you going
> to tell me that you can accelerate the canoe and its
contents without
> a) interacting with an external mass (other than water canoe
is
> floating on)
> b) reducing mass of canoe
This is simply not possible as you will violate Newtonian
mechanics.
> The net momentum is always inertially conserved (it may
oscillate
with
> centre of mass remaining constant) and it may not accelerate
> whatsoever.
I have myself moved a canoe in just such a manner. One sits
near one
> end of the canoe and bounces up and down, carefully. The
varying
> forces of the water on the canoe as it moves up and down
push the
> canoe towards the lighter end. You can't go very fast this
way, but
> you can go slowly.
The water becomes non-idle at that point.
Only if the canoeist becomes non-idle!
===
Subject: Re: Conservation of angular momentum not apparant
If you are in a canoe floating on perfectly idle water, are you
going
> to tell me that you can accelerate the canoe and its contents
without
> a) interacting with an external mass (other than water canoe
is
> floating on)
> b) reducing mass of canoe
This is simply not possible as you will violate Newtonian
mechanics.
> The net momentum is always inertially conserved (it may
oscillate
with
> centre of mass remaining constant) and it may not accelerate
> whatsoever.
I have myself moved a canoe in just such a manner. One sits
near one
> end of the canoe and bounces up and down, carefully. The
varying
> forces of the water on the canoe as it moves up and down
push the
> canoe towards the lighter end. You can't go very fast this
way, but
> you can go slowly.
The water becomes non-idle at that point.
> Only if the canoeist becomes non-idle!
If the mass of the canoeist is allowed to change, interesting
scatological
solutions are suggested!
)>:
RJ P