mm-

> The problem is as follows:
> How many hand are there in bridge?
> Well, duh, C(52,13).
> How many such hands misses at least one color (void in one
> suite, the book says to be exact)?
> Well, duh, dunno... :)
Heres an approach from a different angle:
How many hands consisting only of hearts? One.
How many hands void in three arbitrary suits? Four.
How many hands void in hearts and spades (and no other
suits)? (26,13)
- 2 [thats 13 cards chosen from the 26 cards that arent
hearts or
spades, then subtract off the two cases where youre void in
three suits
and two of them are hearts and spades]
How many hands void in two suits? -- (4,2)((26,13) - 2) [pick
your two
suits that youre going to void in, multiply by the number of
hands void
in those two suits]
etc. Hopefully the above is correct.
-jwgh
--
They can track this putz through the usenet and just arrest
him.
===
Subject: Re: imposing ring structures on a abelian groups
> Does anyone know if there are any results that discuss
anything about
> the possible ways of turning an abelian group into a ring
(in the
> usual way - so that + in the ring is the original operation
of the
> abelian group)? Such as how many possible ways are there,
given a
> particular abelian group, to impose ring structures?
For abelian group G dene a mutiplication to be a bilinear map
m : G X G -> G.
That is, m(a + b, c) = m(a, c) + m(b, c) and
m(a, b + c) = m(a, b) + m(a, c).
Note that a multiplication gives G the structure of a
_non-associative_
ring.
With the obvious addition the set of all multiplications on
G, Mult(G),
is an abelian group. ( If you required associativity, Mult(G)
would not
be a group.)
L. Fuchs[1] shows Mult(G) is isomorphic to Hom( G tensor G,
G) and to
Hom(G, End(G)).
He goes on to observe It is not much known how the associative
multiplications are located in Mult(G).
[1] L. Fuchs, Innite Abelian Groups, vol 2, Academic Press,
1973,
--
Paul Sperry
Columbia, SC (USA)
===
Subject: Re: A question on a boundary value problem
Your argument works ne if r>0.
But I dont see how it works for r<0.
>Prove that for any real number r, the problem
>u = 2pi (1-2t)(1+u^2) + r, t in [0,2]
>u(0)=0
>has no solutions.
>i.e. every solution blows up before t=2. This is easy if
r=0, but I have
>trouble for nonzero r.
> Hint: Write it as
> v = 2 pi (1-2t) + f(t)
> where v(t) = arctan(u(t)), and f(t) = r/(1+u^2) always has
the same
> sign as r.
> The blow-up will actually occur by t=(1+sqrt(2))/2.
> Robert Israel israel@math.ubc.ca
> Department of Mathematics http://www.math.ubc.ca/~israel
> University of British Columbia
> Vancouver, BC, Canada V6T 1Z2
===
Subject: Re: A question on a boundary value problem
Never mind. I gured out.
> Your argument works ne if r>0.
> But I dont see how it works for r<0.
>Prove that for any real number r, the problem
>u = 2pi (1-2t)(1+u^2) + r, t in [0,2]
>u(0)=0
has no solutions.
>i.e. every solution blows up before t=2. This is easy if
r=0, but I
have
>trouble for nonzero r.
> Hint: Write it as
> v = 2 pi (1-2t) + f(t)
> where v(t) = arctan(u(t)), and f(t) = r/(1+u^2) always has
the same
> sign as r.
> The blow-up will actually occur by t=(1+sqrt(2))/2.
> Robert Israel israel@math.ubc.ca
> Department of Mathematics http://www.math.ubc.ca/~israel
> University of British Columbia
> Vancouver, BC, Canada V6T 1Z2
===
Subject: Re: Multitude of innities.
> Take the universal set. There is no set greater than that.
What about the power set of your universal set?
> Darren
> It would also contain the power set of the universal set.
> It may confuse you at rst, but I assigned this innity to
also have
> an innite cardinality.
> This means its power set would also be in itself. Meaning
it is more
> innite than any other innite set.
> If you consider this a contradiction, ne. I dont. Thats
because
> this is by denition not only a set which contains an
innite number
> of members but a set which contains everything.
Thats not a contradiction in and of itself. The contradictory
part comes from this:
Let U be the universal set.
Let S be in U such that S = {X in U: X is a set and is not in
itself}
Now try to gure out whether S is contained in itself. Youd
see that assuming one way necessarily implies its negation.
Dan.
===
Subject: Re: Multitude of innities.
>Well, it would be a type of innity of which no greater set
could be
>constructed. This would require that, if one were to try to
construct
>it in any way, one would simply get this same set. Meaning
for each
>set within this super innite set, any transformation that
can be
>done to it will also exist in this set.
> If you allow the transformation of taking the power set of
any set,
then
> Cantors diagonal proof shows that the size of the power
set is always
> strictly greater than the size of the original set. Are you
familiar
with
> Cantors proof? If so, then which step would you want to
invalidate in
> order to allow your super-innite set to be the same as its
power set?
What would make it strictly greater than the other set? What
if the
set size was so innite that adding to it would not make it
any
bigger?
http://mathworld.wolfram.com/CantorsParadox.html
So you are saying that we cannot have the set of all sets.
>(BTW, if modern set theory has a way of representing this,
then that
>would honestly be perfect.)
> Well, it has a way of showing that it cant be done. Is
that almost
perfect?
At, most, all its doing is showing it cant be done under the
premises of the given set theory. What if we allow different
premises?
>Now, conceptually, I see nothing wrong with explaining an
innitude
>set like that. But clearly its been told to me a number of
times that
>you simply cant do this. At least not with modern set
theory.
> Yes. But dont take our word for it. Cantors diagonal
proof is short
> and quite accessible, even from an intuitive
naive-set-theory point of
> view.
>I think maybe it has to do with the denition of the set. If
this is
>the case, then we need a more general type of set. Does
anybody know a
>type of set which isnt really a set but acts like a set in
most ways,
>except that it is open where the set is closed?
> I think a way for you to tighten up that question is to
study Cantors
> proof and see what aspect of sets youd like to change in
order to make
> the proof fail. Frankly, I very much doubt your chances of
nding a
> change that doesnt do fatal violence to some other aspect
of your
> intuitive notion of set.
What about the idea that making something bigger will always
change
it? Certainly if we take innity, and add one to it, it does
not
change, even though logically innity + 1 should be bigger
than
innity. Innity still remains the same.
Allowing an innite cardinality would mean that wed have
aleph
innity. Taking the power set would give us aleph innity +
1. But
innity + 1 and innity are the same number.
Nothings wrong with set theory unless we are operating under
the
standard, unwritten rules of common sense.
To me, common sense is a senseless notion.
(...Starblade Riven Darksquall...)
===
Subject: Re: Multitude of innities.
>>Well, it would be a type of innity of which no greater set
could be
>>constructed. This would require that, if one were to try to
construct
>>it in any way, one would simply get this same set. Meaning
for each
>>set within this super innite set, any transformation that
can be
>>done to it will also exist in this set.
>> If you allow the transformation of taking the power set of
any set,
then
>> Cantors diagonal proof shows that the size of the power
set is always
>> strictly greater than the size of the original set. Are
you familiar
with
>> Cantors proof? If so, then which step would you want to
invalidate in
>> order to allow your super-innite set to be the same as
its power set?
>What would make it strictly greater than the other set?
The meanings of power set and same size. Cantors proof
doesnt
rely
on much more than that.
>What if the
>set size was so innite that adding to it would not make it
any bigger?
What if circles had corners? The very meanings of the terms
used imply
(via Cantors diagonal argument) that the power set is
strictly larger.
It doesnt matter how large the initial set is, the power set
will always
be larger.
>http://mathworld.wolfram.com/CantorsParadox.html
>So you are saying that we cannot have the set of all sets.
Thats a consequence of the fact that a sets power set must
be larger
than the original set. Another consequence is that there is
no largest
innity.
>>(BTW, if modern set theory has a way of representing this,
then that
>>would honestly be perfect.)
>> Well, it has a way of showing that it cant be done. Is
that almost
perfect?
>At, most, all its doing is showing it cant be done under
the
>premises of the given set theory. What if we allow different
premises?
Go for it, as I suggested. You will almost certainly nd that
attempting
to jiggle some premise so as to frustrate Cantors proof does
fatal
violence to what you consider to be intuitively
unobjectionable properties
of sets.
Now would be a really good time to actually study Cantors
proof if you
have not done so already. Its short and accessible.
>>Now, conceptually, I see nothing wrong with explaining an
innitude
>>set like that. But clearly its been told to me a number of
times that
>>you simply cant do this. At least not with modern set
theory.
>> Yes. But dont take our word for it. Cantors diagonal
proof is short
>> and quite accessible, even from an intuitive
naive-set-theory point of
>> view.
>>I think maybe it has to do with the denition of the set.
If this is
>>the case, then we need a more general type of set. Does
anybody know a
>>type of set which isnt really a set but acts like a set in
most ways,
>>except that it is open where the set is closed?
>> I think a way for you to tighten up that question is to
study Cantors
>> proof and see what aspect of sets youd like to change in
order to make
>> the proof fail. Frankly, I very much doubt your chances of
nding a
>> change that doesnt do fatal violence to some other aspect
of your
>> intuitive notion of set.
>What about the idea that making something bigger will always
change
>it? Certainly if we take innity, and add one to it, it does
not
>change, even though logically innity + 1 should be bigger
than
>innity. Innity still remains the same.
Adding 1 element to a set doesnt necessarily make a larger
sized set
(indeed, one denition of innite set is that it is the same
size as
some proper subset of itself), but going to the power set
ALWAYS DOES make
a larger sized set.
>Allowing an innite cardinality would mean that wed have
aleph
>innity. Taking the power set would give us aleph innity +
1. But
>innity + 1 and innity are the same number.
This relates to the idea of large cardinals, i.e. innities
greater
than aleph-N for every N. But even a set whose size is a
large cardinal
must have a power set that is even larger. Cantors diagonal
proof
applies equally well *regardless* of how hugely
mind-bogglingly big the
set is.
>Nothings wrong with set theory unless we are operating
under the
>standard, unwritten rules of common sense.
>To me, common sense is a senseless notion.
Eh? But regardless, you are free to tweak the premises used
by Cantor, to
invalidate his proof. BUT, if you do that, you will almost
certainly
damage some aspect of the notion of set that you nd
intuitively
compelling.
Please, do take the time to study Cantors proof. You may get
an
impressive aha! experience. Why settle for the Classics
Illustrated or
Cliffs Notes version when you can have the real thing?
>(...Starblade Riven Darksquall...)
--
---------------------------
| BBB b barbara minus knox at iname stop com
| B B aa rrr b |
| BBB a a r bbb |
| B B a a r b b |
| BBB aa a r bbb |
-----------------------------
===
Subject: Re: Multitude of innities.
>>Well, it would be a type of innity of which no greater set
could be
>>constructed. This would require that, if one were to try to
construct
>>it in any way, one would simply get this same set. Meaning
for each
>>set within this super innite set, any transformation that
can be
>>done to it will also exist in this set.
>> If you allow the transformation of taking the power set of
any set,
then
>> Cantors diagonal proof shows that the size of the power
set is always
>> strictly greater than the size of the original set. Are
you familiar
with
>> Cantors proof? If so, then which step would you want to
invalidate
in
>> order to allow your super-innite set to be the same as
its power
set?
>What would make it strictly greater than the other set?
> The meanings of power set and same size. Cantors proof
doesnt
rely
> on much more than that.
>What if the
>set size was so innite that adding to it would not make it
any bigger?
> What if circles had corners? The very meanings of the terms
used imply
> (via Cantors diagonal argument) that the power set is
strictly larger.
> It doesnt matter how large the initial set is, the power
set will always
> be larger.
>http://mathworld.wolfram.com/CantorsParadox.html
>So you are saying that we cannot have the set of all sets.
> Thats a consequence of the fact that a sets power set
must be larger
> than the original set. Another consequence is that there is
no largest
> innity.

Thats not what Cantor proved. Since he didnt prove anything
about mathematicians, or their moronic real numbers.
===
Subject: Re: Multitude of innities.
> ...
>Allowing an innite cardinality would mean that wed have
aleph
>innity. Taking the power set would give us aleph innity +
1. But
>innity + 1 and innity are the same number.
> This relates to the idea of large cardinals, i.e. innities
greater
> than aleph-N for every N.
Small correction to your clear-headed post: if your N here is
ranging
over natural numbers, as it appears to be, then many cardinals
satisfying this condition (aleph-omega, for instance) are in
fact
small. Large cardinals are hugely mind-bogglingly bigger than
this
-- so large that they cannot be proved to exist in ZFC. The
simplest
condition (weak inaccessibility) for largeness is this: (i) k
>
aleph-0; (ii) k is a limit cardinal (i.e., has no immediate
cardinal
predecessor); and (iii) k is not the sum of fewer than k
cardinals each
of which is < k. (Note aleph-omega, for instance, is the sum
of
{aleph-0, aleph-1, ...}.) Weak inaccessibles are the smallest
of the
large cardinals; that is, there are many other types of large
cardinal,
the smallest of which (assuming there are any at all) is
larger than the
smallest weakly inaccessible cardinal.
Chris Menzel
===
Subject: Re: Multitude of innities.
>>Well, it would be a type of innity of which no greater set
could be
>>constructed. This would require that, if one were to try to
construct
>>it in any way, one would simply get this same set. Meaning
for each
>>set within this super innite set, any transformation that
can be
>>done to it will also exist in this set.
>> If you allow the transformation of taking the power set of
any set,
then
>> Cantors diagonal proof shows that the size of the power
set is always
>> strictly greater than the size of the original set. Are
you familiar
with
>> Cantors proof? If so, then which step would you want to
invalidate
in
>> order to allow your super-innite set to be the same as
its power
set?
>What would make it strictly greater than the other set?
> The meanings of power set and same size. Cantors proof
doesnt
rely
> on much more than that.
>What if the
>set size was so innite that adding to it would not make it
any bigger?
> What if circles had corners?
They do. However their angle approaches pi radians, so its
better to
describe the curvature at a point rather than the amount of
radians
subtended on an innitesimal region.
The very meanings of the terms used imply
> (via Cantors diagonal argument) that the power set is
strictly larger.
> It doesnt matter how large the initial set is, the power
set will always
> be larger.
What if the term larger loses its meaning? And anything
which is not
strictly zero size with respect to the set is automatically
neither
larger, smaller, or equal in size, yet are also all three
things?
Of course, thats an absurdity only because were using
denitive
mathematical structures. If we used indenitive mathematical
structures we would not have that problem.
So the set of all arbitrary sets is an undened. Mathematics
simply
cannot deal with it.
But if we had a mathematics that gave meaning to such things
it could
deal with it. Such a mathematics would give meaning to lim
x->0 f(x)
where f(x) = |x| and also lim x->0 f(x) where f(x) = 1/x.
Since ours doesnt, its no surprise we manage to nd such
beautiful
mathematical paradoxes as we tend to do.
>http://mathworld.wolfram.com/CantorsParadox.html
>So you are saying that we cannot have the set of all sets.
> Thats a consequence of the fact that a sets power set
must be larger
> than the original set. Another consequence is that there is
no largest
> innity.
Why is it necessarily larger? Why does larger have to have a
meaning?
Why cant we just say for innitely innite sets, larger,
smaller,
and same size are meaningless concepts?
>>(BTW, if modern set theory has a way of representing this,
then that
>>would honestly be perfect.)
>> Well, it has a way of showing that it cant be done. Is
that almost
perfect?
>At, most, all its doing is showing it cant be done under
the
>premises of the given set theory. What if we allow different
premises?
> Go for it, as I suggested. You will almost certainly nd
that
attempting
> to jiggle some premise so as to frustrate Cantors proof
does fatal
> violence to what you consider to be intuitively
unobjectionable
properties
> of sets.
Is one of these premises that any set can be uniquely
described in a
nite number of terms?
> Now would be a really good time to actually study Cantors
proof if you
> have not done so already. Its short and accessible.
I looked at it, but did not study it. I will go and do so.
And I
guarantee that I will object to it on the same fundamental
premises as
I did before.
>>Now, conceptually, I see nothing wrong with explaining an
innitude
>>set like that. But clearly its been told to me a number of
times
that
>>you simply cant do this. At least not with modern set
theory.
>> Yes. But dont take our word for it. Cantors diagonal
proof is
short
>> and quite accessible, even from an intuitive
naive-set-theory point of
>> view.
>>I think maybe it has to do with the denition of the set.
If this is
>>the case, then we need a more general type of set. Does
anybody know
a
>>type of set which isnt really a set but acts like a set in
most
ways,
>>except that it is open where the set is closed?
>> I think a way for you to tighten up that question is to
study Cantors
>> proof and see what aspect of sets youd like to change in
order to
make
>> the proof fail. Frankly, I very much doubt your chances of
nding a
>> change that doesnt do fatal violence to some other aspect
of your
>> intuitive notion of set.
>What about the idea that making something bigger will always
change
>it? Certainly if we take innity, and add one to it, it does
not
>change, even though logically innity + 1 should be bigger
than
>innity. Innity still remains the same.
> Adding 1 element to a set doesnt necessarily make a larger
sized set
> (indeed, one denition of innite set is that it is the
same size
as
> some proper subset of itself), but going to the power set
ALWAYS DOES
make
> a larger sized set.
I didnt ask about sets. I asked about numerical value.
Does innity+1 have a greater numerical value than innity?
Answer that question rst. Then answer this one: Isnt size
akin to a
numerical value? Taking the cardinality as a numerical
description of
a set, you are saying that the cardinality must always be
nite, and
can never be innite. Obviously if it was innite, then
adding 1
more size would not have any effect on it.
>Allowing an innite cardinality would mean that wed have
aleph
>innity. Taking the power set would give us aleph innity +
1. But
>innity + 1 and innity are the same number.
> This relates to the idea of large cardinals, i.e. innities
greater
> than aleph-N for every N. But even a set whose size is a
large cardinal
> must have a power set that is even larger. Cantors
diagonal proof
> applies equally well *regardless* of how hugely
mind-bogglingly big
the
> set is.
As long as the idea of big is that other things relative to
it must
have a denite nonzero though innitesimal size, then you are
correct. But if we allow all sizes relative to something that
is not
as big to be zero, then you are wrong.
>Nothings wrong with set theory unless we are operating
under the
>standard, unwritten rules of common sense.
>To me, common sense is a senseless notion.
> Eh? But regardless, you are free to tweak the premises used
by Cantor,
to
> invalidate his proof. BUT, if you do that, you will almost
certainly
> damage some aspect of the notion of set that you nd
intuitively
> compelling.
So in a set, we cannot have an innite ascenscion of
elements, just
like we cannot have an innite descencion of elements. So,
for every
set, we must be capable of describing an element that is not
in the
set.
I do not like this. While thats ne for a set, this means
that a set
is just a limiting case of something else. Something else we
might not
have the mathematical tools for at this point.
> Please, do take the time to study Cantors proof. You may
get an
> impressive aha! experience. Why settle for the Classics
Illustrated
or
> Cliffs Notes version when you can have the real thing?
I studied it. But it still relies on the idea that making
something
that SHOULD be bigger than it will actually have an effect on
it. If I
contest that notion, then while I do not necessarily
disproove Cantor,
I do disproove the idea that there is no innitely large
innity,
because for an innitely large innity, something large than
it
would be the same size. Also, if something was smaller than
it but not
nonzero, it would also be the same size, by logical extention.
Maybe you just dont like it because it gives meaning to the
notion of
1/0.
(...Starblade Riven Darksquall...)
===
Subject: Re: Multitude of innities.
>>Well, it would be a type of innity of which no greater set
could
be
>constructed. This would require that, if one were to try to
construct
>it in any way, one would simply get this same set. Meaning
for each
>set within this super innite set, any transformation that
can be
>done to it will also exist in this set.
>> If you allow the transformation of taking the power set of
any
set, then
> Cantors diagonal proof shows that the size of the power
set is
always
> strictly greater than the size of the original set. Are you
familiar
with
> Cantors proof? If so, then which step would you want to
invalidate
in
> order to allow your super-innite set to be the same as its
power
set?
>What would make it strictly greater than the other set?
>> The meanings of power set and same size. Cantors proof
doesnt
rely
>> on much more than that.
>>What if the
>>set size was so innite that adding to it would not make it
any
bigger?
>> What if circles had corners?
>They do. However their angle approaches pi radians, so its
better to
>describe the curvature at a point rather than the amount of
radians
>subtended on an innitesimal region.
> The very meanings of the terms used imply
>> (via Cantors diagonal argument) that the power set is
strictly larger.
>> It doesnt matter how large the initial set is, the power
set will
always
>> be larger.
>What if the term larger loses its meaning?
How would that happen? WRT sets, the notion seems very hard
to get rid
of. Two sets are the same size iff there is some bijection (a
1-1
correspondence) between them. If there isnt, then the set
with unpaired
elements left over is of larger size. So the only machinery
needed to
dene larger size are the basic notions of the elements of a
set and
of
relations between sets. Unless youre willing to discard one
or both of
these, youre stuck with also having a well-dened notion of
larger
size; it cant be avoided.
> And anything which is not
>strictly zero size with respect to the set is automatically
neither
>larger, smaller, or equal in size, yet are also all three
things?
Eh? I dont understand what you mean by size with respect to
the set.
>Of course, thats an absurdity only because were using
denitive
>mathematical structures. If we used indenitive mathematical
>structures we would not have that problem.
What do you mean by indenitive mathematical structures? Can
you give
an example or two?
>So the set of all arbitrary sets is an undened. Mathematics
simply
>cannot deal with it.
Oh, its a perfectly well-dened notion: a set whose elements
are all the
sets. The problem is that Cantors proof shows that a set
with that
property can not exist. So mathematics deals with it quite
nicely, by
proving that the notion is intrinsically incompatible with
other
more-basic notions (and intuitions) about sets.
>But if we had a mathematics that gave meaning to such things
it could
>deal with it. Such a mathematics would give meaning to lim
x->0 f(x)
>where f(x) = |x| and also lim x->0 f(x) where f(x) = 1/x.
One can extend the real numbers to include oo, which will
handle 1/0.
For special purposes one can also extend domains to include a
specic
undened element, which could take care of d|x|/dx at x=0.
What is
the
particular problem youre wanting to solve?
>Since ours doesnt, its no surprise we manage to nd such
beautiful
>mathematical paradoxes as we tend to do.
Which paradoxes do you mean? Modern axiomatic mathematics
adequately
deals with most traditional paradoxes.
>>http://mathworld.wolfram.com/CantorsParadox.html
>>So you are saying that we cannot have the set of all sets.
>> Thats a consequence of the fact that a sets power set
must be larger
>> than the original set. Another consequence is that there
is no
largest
>> innity.
>Why is it necessarily larger? Why does larger have to have a
meaning?
>Why cant we just say for innitely innite sets, larger,
smaller,
>and same size are meaningless concepts?
See above.
>(BTW, if modern set theory has a way of representing this,
then that
>would honestly be perfect.)
>> Well, it has a way of showing that it cant be done. Is
that almost
perfect?
>At, most, all its doing is showing it cant be done under
the
>>premises of the given set theory. What if we allow
different premises?
>> Go for it, as I suggested. You will almost certainly nd
that
attempting
>> to jiggle some premise so as to frustrate Cantors proof
does fatal
>> violence to what you consider to be intuitively
unobjectionable
properties
>> of sets.
>Is one of these premises that any set can be uniquely
described in a
>nite number of terms?
No. In fact, its provable that there are undescribable sets.
(Needless
to say, its a nonconstructive proof!)
>> Now would be a really good time to actually study Cantors
proof if you
>> have not done so already. Its short and accessible.
>I looked at it, but did not study it. I will go and do so.
Good!
> And I
>guarantee that I will object to it on the same fundamental
premises as
>I did before.
Yes, but then youll be able to cite some SPECIFIC STEP(s) of
Cantors
proof that you object to, rather than resort to generalities.
>Now, conceptually, I see nothing wrong with explaining an
innitude
>set like that. But clearly its been told to me a number of
times
that
>you simply cant do this. At least not with modern set
theory.
>> Yes. But dont take our word for it. Cantors diagonal
proof is
short
> and quite accessible, even from an intuitive
naive-set-theory point
of
> view.
>>I think maybe it has to do with the denition of the set.
If this
is
>the case, then we need a more general type of set. Does
anybody know
a
>type of set which isnt really a set but acts like a set in
most
ways,
>except that it is open where the set is closed?
>> I think a way for you to tighten up that question is to
study
Cantors
> proof and see what aspect of sets youd like to change in
order to
make
> the proof fail. Frankly, I very much doubt your chances of
nding a
> change that doesnt do fatal violence to some other aspect
of your
> intuitive notion of set.
>>
>>What about the idea that making something bigger will
always change
>>it? Certainly if we take innity, and add one to it, it
does not
>>change, even though logically innity + 1 should be bigger
than
>>innity. Innity still remains the same.
>> Adding 1 element to a set doesnt necessarily make a
larger sized set
>> (indeed, one denition of innite set is that it is the
same size
as
>> some proper subset of itself), but going to the power set
ALWAYS DOES
make
>> a larger sized set.
>I didnt ask about sets. I asked about numerical value.
>Does innity+1 have a greater numerical value than innity?
No. But 2^innity is always greater (for any innity).
>Answer that question rst.
Done.
> Then answer this one: Isnt size akin to a numerical value?
Yes, although the usual nite notion of numerical value needs
to be
extended with the aleph-Ns (and maybe also with large
cardinals).
> Taking the cardinality as a numerical description of
>a set, you are saying that the cardinality must always be
nite,
No, thats why you need the alephs.
> and
>can never be innite. Obviously if it was innite, then
adding 1
>more size would not have any effect on it.
>>Allowing an innite cardinality would mean that wed have
aleph
>>innity. Taking the power set would give us aleph innity +
1. But
>>innity + 1 and innity are the same number.
>> This relates to the idea of large cardinals, i.e.
innities greater
>> than aleph-N for every N. But even a set whose size is a
large cardinal
>> must have a power set that is even larger. Cantors
diagonal proof
>> applies equally well *regardless* of how hugely
mind-bogglingly big
the
>> set is.
>As long as the idea of big is that other things relative to
it must
>have a denite nonzero though innitesimal size, then you are
>correct. But if we allow all sizes relative to something
that is not
>as big to be zero, then you are wrong.
Eh? I think I covered this above; if not, please try
rephrasing it.
>>Nothings wrong with set theory unless we are operating
under the
>>standard, unwritten rules of common sense.
>>To me, common sense is a senseless notion.
>> Eh? But regardless, you are free to tweak the premises
used by Cantor,
to
>> invalidate his proof. BUT, if you do that, you will almost
certainly
>> damage some aspect of the notion of set that you nd
intuitively
>> compelling.
>So in a set, we cannot have an innite ascenscion of
elements, just
>like we cannot have an innite descencion of elements. So,
for every
>set, we must be capable of describing an element that is not
in the
>set.
Im not sure about this, but I think the set (or maybe class)
of all
describable sets can be dened, in which case it by denition
does NOT
have any non-element that is describable!
>I do not like this. While thats ne for a set, this means
that a set
>is just a limiting case of something else. Something else we
might not
>have the mathematical tools for at this point.
We have the tools. What we lack are the appropriate
intuitions,
especially about innities. One thing I like about rigourous
maths is
that is shows me places where I need to recalibrate my
intuitions.
>> Please, do take the time to study Cantors proof. You may
get an
>> impressive aha! experience. Why settle for the Classics
Illustrated
or
>> Cliffs Notes version when you can have the real thing?
>I studied it. But it still relies on the idea that making
something
>that SHOULD be bigger than it will actually have an effect
on it.
Eh? Note that not just any bigger will sufce; Cantors proof
only
deals with the power set, and shows that its size is bigger.
> If I
>contest that notion, then while I do not necessarily
disproove Cantor,
>I do disproove the idea that there is no innitely large
innity,
But how can you contest that notion while not doing violence
to basic
intuitions about sets?
>because for an innitely large innity,
Remember, for ANY cardinality (even a large innite one),
there is a
larger one. You might not like that, but its as unavoidable
as the fact
that for every nite number there is a larger one (which some
people also
dont like, believe it or not).
> something large than it
>would be the same size. Also, if something was smaller than
it but not
>nonzero,
Eh? If the size is not nonzero then its zero.
> it would also be the same size, by logical extention.
Eh?
>Maybe you just dont like it because it gives meaning to the
notion of
>1/0.
I have no problem with extending the reals to include oo,
which is then
the value of 1/0. Note that this doesnt relate to Cantors
diagonal
proof.
>(...Starblade Riven Darksquall...)
--
---------------------------
| BBB b barbara minus knox at iname stop com
| B B aa rrr b |
| BBB a a r bbb |
| B B a a r b b |
| BBB aa a r bbb |
-----------------------------
===
Subject: Re: Multitude of innities.
>>Well, it would be a type of innity of which no greater set
could
be
>constructed. This would require that, if one were to try to
construct
>it in any way, one would simply get this same set. Meaning
for
each
>set within this super innite set, any transformation that
can be
>done to it will also exist in this set.
>> If you allow the transformation of taking the power set of
any
set, then
> Cantors diagonal proof shows that the size of the power
set is
always
> strictly greater than the size of the original set. Are you
familiar with
> Cantors proof? If so, then which step would you want to
invalidate
in
> order to allow your super-innite set to be the same as its
power
set?
>What would make it strictly greater than the other set?
>> The meanings of power set and same size. Cantors proof
doesnt rely
>> on much more than that.
>>What if the
>>set size was so innite that adding to it would not make it
any
bigger?
>> What if circles had corners?
>They do. However their angle approaches pi radians, so its
better to
>describe the curvature at a point rather than the amount of
radians
>subtended on an innitesimal region.
> The very meanings of the terms used imply
>> (via Cantors diagonal argument) that the power set is
strictly larger.

>> It doesnt matter how large the initial set is, the power
set will
always
>> be larger.
>What if the term larger loses its meaning?
> How would that happen? WRT sets, the notion seems very hard
to get rid
> of. Two sets are the same size iff there is some bijection
(a 1-1
> correspondence) between them. If there isnt, then the set
with unpaired
> elements left over is of larger size. So the only machinery
needed to
> dene larger size are the basic notions of the elements of
a set and
of
> relations between sets. Unless youre willing to discard
one or both of
> these, youre stuck with also having a well-dened notion
of larger
> size; it cant be avoided.
What if we dene a eld of numbers for which for certain sets
of
which or other similar constructs, bijection cannot be
ascertained?
> And anything which is not
>strictly zero size with respect to the set is automatically
neither
>larger, smaller, or equal in size, yet are also all three
things?
> Eh? I dont understand what you mean by size with respect
to the
set.
Say you have two sets, the set of the digits from 0-9, and
the set of
natural numbers? The relative size of the second set to the
rst is
aleph 1, but the relative size of the rst set to the second
would be
aleph -1. This is basically like a directed distance. Even if
we
restrict ourselves to using positive, real numbers, the
directed
distance between two numbers would be positive in some cases
but
negative in others.
>Of course, thats an absurdity only because were using
denitive
>mathematical structures. If we used indenitive mathematical
>structures we would not have that problem.
> What do you mean by indenitive mathematical structures?
Can you
give
> an example or two?
The slope of the absolute value function at zero. One divided
by zero.
A number such that n = aleph-n or however one would write
that. A
number representing the derivative (non directional) of the
conjugate
of a complex number. Et cetera.
>So the set of all arbitrary sets is an undened. Mathematics
simply
>cannot deal with it.
> Oh, its a perfectly well-dened notion: a set whose
elements are all
the
> sets. The problem is that Cantors proof shows that a set
with that
> property can not exist. So mathematics deals with it quite
nicely, by
> proving that the notion is intrinsically incompatible with
other
> more-basic notions (and intuitions) about sets.
So we can have a set whose elements are all the sets but we
also cant
have it. We can conceptualize it, but in mathematics it does
not
exist. Then where do arbitrarily large sets come from?
>But if we had a mathematics that gave meaning to such things
it could
>deal with it. Such a mathematics would give meaning to lim
x->0 f(x)
>where f(x) = |x| and also lim x->0 f(x) where f(x) = 1/x.
> One can extend the real numbers to include oo, which will
handle 1/0.

I would not. I would restrict oo to something for which the
relative
size between an integer and oo would be innitesimal.
Innitesimal
would not be the same as zero.
> For special purposes one can also extend domains to include
a specic
> undened element, which could take care of d|x|/dx at x=0.
What is
the
> particular problem youre wanting to solve?
I want to see how one would use the undened element. For
example, if
there is the set of all sets which do not contain themselves,
would
the set be in itself? Id imagine we could give a value other
than
true or false which would represent this.
>Since ours doesnt, its no surprise we manage to nd such
beautiful
>mathematical paradoxes as we tend to do.
> Which paradoxes do you mean? Modern axiomatic mathematics
adequately
> deals with most traditional paradoxes.
I know. But I liked the paradoxes. While mathematically you
can get
rid of the paradoxes, its still something to consider.
>>http://mathworld.wolfram.com/CantorsParadox.html
>>So you are saying that we cannot have the set of all sets.
>> Thats a consequence of the fact that a sets power set
must be larger
>> than the original set. Another consequence is that there
is no
largest
>> innity.
>>
Why does the power set have to be larger? If we have the set
of all
the sets, then the power set would be in the set.
For which kinds of sets are power series worked on? Do not
say all
sets. If you cannot have a set with arbitrary elements in it
then you
cannot have a mathematics which works arbitrarily well on ALL
sets
since youre automatically discluding those sets which do have
arbitrary elements, contradicting the assumption that your
mathematics
applies to all sets.
You can say all existing sets and then dene existing
meaning that
which when logically inspected follows certain axioms of
logic, but by
doing so, youre discluding nonexisting sets.
So mathematics would apply to real mathematical formalisms
but would
have nothing to do with ideal, nonformalisms.
>Why is it necessarily larger? Why does larger have to have a
meaning?
>Why cant we just say for innitely innite sets, larger,
smaller,
>and same size are meaningless concepts?
> See above.
Bijection. What if you could not be sure of the relative
largeness
between two types of sets, perhaps two ideal sets, because
when
judging the effects of bijection, whether one is bigger or
smaller,
and which one, or whether they are equal, would depend on how
you
judged it? Are there such situations in mathematics, at least
for
ideal sets?
>(BTW, if modern set theory has a way of representing this,
then
that
>would honestly be perfect.)
>> Well, it has a way of showing that it cant be done. Is
that
almost
> perfect?
>At, most, all its doing is showing it cant be done under
the
>>premises of the given set theory. What if we allow different
premises?
>> Go for it, as I suggested. You will almost certainly nd
that
attempting
>> to jiggle some premise so as to frustrate Cantors proof
does fatal
>> violence to what you consider to be intuitively
unobjectionable
properties
>> of sets.
>Is one of these premises that any set can be uniquely
described in a
>nite number of terms?
> No. In fact, its provable that there are undescribable
sets. (Needless
> to say, its a nonconstructive proof!)
Alright, so if these sets are undescribable, do they have a
size?
And do larger and smaller have anything to do with them?
>> Now would be a really good time to actually study Cantors
proof if
you
>> have not done so already. Its short and accessible.
>I looked at it, but did not study it. I will go and do so.
> Good!
I have looked at some of the mathematics and I am catching on
real
quickly. But do you have any pages, like historical
discussions
between Cantor and others, where they raise their objections
to his
set theory and then he revises the set theory? Id also like
to see
that.
> And I
>guarantee that I will object to it on the same fundamental
premises as
>I did before.
> Yes, but then youll be able to cite some SPECIFIC STEP(s)
of Cantors
> proof that you object to, rather than resort to
generalities.
I am perfectly ne with that. At this point I basically know
what Im
looking for, even if I have not found it. If I can nd a pair
of sets
for which the attempting a bijection results in uncertainty
then I
will effectively show that a class of sets can be imagined
with
abnormal, and in fact contradictory properties can be
idealized, but
which do not exist according to the given mathematical
formalisms.
Furthermore, I could probably show that subsets of these sets
do
exist.
>Now, conceptually, I see nothing wrong with explaining an
innitude
>set like that. But clearly its been told to me a number of
times
that
>you simply cant do this. At least not with modern set
theory.
>> Yes. But dont take our word for it. Cantors diagonal
proof is
short
> and quite accessible, even from an intuitive
naive-set-theory point
of
> view.
>>I think maybe it has to do with the denition of the set.
If this
is
>the case, then we need a more general type of set. Does
anybody
know a
>type of set which isnt really a set but acts like a set in
most
ways,
>except that it is open where the set is closed?
>> I think a way for you to tighten up that question is to
study
Cantors
> proof and see what aspect of sets youd like to change in
order to
make
> the proof fail. Frankly, I very much doubt your chances of
nding
a
> change that doesnt do fatal violence to some other aspect
of your
> intuitive notion of set.
>>What about the idea that making something bigger will
always change
>>it? Certainly if we take innity, and add one to it, it
does not
>>change, even though logically innity + 1 should be bigger
than
>>innity. Innity still remains the same.
>> Adding 1 element to a set doesnt necessarily make a
larger sized set
>> (indeed, one denition of innite set is that it is the
same size
as
>> some proper subset of itself), but going to the power set
ALWAYS DOES
make
>> a larger sized set.
>I didnt ask about sets. I asked about numerical value.
>Does innity+1 have a greater numerical value than innity?
> No. But 2^innity is always greater (for any innity).
Alright... so you use n^innity. Now use n^(innity^innity),
and
then use n^(innity^(innity^innity)). If you do that an
innite
number of times, such as
n^(innity^(innity^...^(innity^infninity)...)), then does
doing
it once more have any effect on the size?
>Answer that question rst.
> Done.
Okay then.
> Then answer this one: Isnt size akin to a numerical value?
> Yes, although the usual nite notion of numerical value
needs to be
> extended with the aleph-Ns (and maybe also with large
cardinals).
Does aleph-n where n = innity have a meaning? If so, then
wouldnt
taking its power set be akin to taking aleph-m where m = n+1?
And if
that is true, then since n is innity, wouldnt saying that m
= n+1
be an essentially meaningless statement?
> Taking the cardinality as a numerical description of
>a set, you are saying that the cardinality must always be
nite,
> No, thats why you need the alephs.
Oh, so the cardinality isnt which aleph it is. I thought
aleph-0 had
a cardinality of 0, aleph-1 had a cardinality of 1, and so
on. That
clears things up. So, does aleph-innity have any meaning?
And would
aleph-(aleph-1) be different from aleph(aleph-2)? And could
you have a
bizzarre number called W, for weird, for which W = aleph-W?
> and
>can never be innite. Obviously if it was innite, then
adding 1
>more size would not have any effect on it.
>>Allowing an innite cardinality would mean that wed have
aleph
>>innity. Taking the power set would give us aleph innity +
1. But
>>innity + 1 and innity are the same number.
>> This relates to the idea of large cardinals, i.e. innities
greater
>> than aleph-N for every N. But even a set whose size is a
large
cardinal
>> must have a power set that is even larger. Cantors
diagonal proof
>> applies equally well *regardless* of how hugely
mind-bogglingly big
the
>> set is.
>As long as the idea of big is that other things relative to
it must
>have a denite nonzero though innitesimal size, then you are
>correct. But if we allow all sizes relative to something
that is not
>as big to be zero, then you are wrong.
> Eh? I think I covered this above; if not, please try
rephrasing it.
The relative size of a nite number to, say, something that is
same as zero. Then go on from here to show that aleph-n where
n is any
number cannot be so big that the size of a nite number
relative to
it is ACTUALLY 0, even though it is arbitrarily close.
If this is possible, then I can construct a new type of
number, that
has very strange properties. I mentioned such a number above.
Another
thing is that W could be said to be reciprocal of 0, and that
even the
innities of cardinality-n would not have this property.
>>Nothings wrong with set theory unless we are operating
under the
>>standard, unwritten rules of common sense.
>>To me, common sense is a senseless notion.
>> Eh? But regardless, you are free to tweak the premises
used by Cantor,
to
>> invalidate his proof. BUT, if you do that, you will almost
certainly
>> damage some aspect of the notion of set that you nd
intuitively
>> compelling.
>So in a set, we cannot have an innite ascenscion of
elements, just
>like we cannot have an innite descencion of elements. So,
for every
>set, we must be capable of describing an element that is not
in the
>set.
> Im not sure about this, but I think the set (or maybe
class) of all
> describable sets can be dened, in which case it by
denition does NOT
> have any non-element that is describable!
What? It does not have any non-element that is describable?
Did I
misread that? That seems like a rather odd statement.
>I do not like this. While thats ne for a set, this means
that a set
>is just a limiting case of something else. Something else we
might not
>have the mathematical tools for at this point.
> We have the tools. What we lack are the appropriate
intuitions,
> especially about innities. One thing I like about
rigourous maths is
> that is shows me places where I need to recalibrate my
intuitions.
I know. Ive had to do this too.
>> Please, do take the time to study Cantors proof. You may
get an
>> impressive aha! experience. Why settle for the Classics
Illustrated or
>> Cliffs Notes version when you can have the real thing?
>I studied it. But it still relies on the idea that making
something
>that SHOULD be bigger than it will actually have an effect
on it.
> Eh? Note that not just any bigger will sufce; Cantors
proof only
> deals with the power set, and shows that its size is bigger.
Well I didnt STUDY study it. But still, the denition of
bigger is
using the fact that n^x>x. What about a number W for which
n^W=W or
something outlandishly bizzare like that?
> If I
>contest that notion, then while I do not necessarily
disproove Cantor,
>I do disproove the idea that there is no innitely large
innity,
> But how can you contest that notion while not doing
violence to basic
> intuitions about sets?
Well, what if I say real, physical sets have some
properties, but
then dene a putitive class of sets that are nonphysical?
>because for an innitely large innity,
> Remember, for ANY cardinality (even a large innite one),
there is a
> larger one. You might not like that, but its as
unavoidable as the fact
> that for every nite number there is a larger one (which
some people
also
> dont like, believe it or not).
I made the mistake of assuming that cardinality had to do
with which
aleph a sets size was.
So what is the term for a given innity? That is, the
aleph-ness of a
given type of innity?
> something large than it
>would be the same size. Also, if something was smaller than
it but not
>nonzero,
> Eh? If the size is not nonzero then its zero.
If the size is not nonzero then its zero? What is it?
> it would also be the same size, by logical extention.
> Eh?
Maybe same Im using the wrong term, same size. That is one
of the
things Im trying to prove. That for some pairs of sets, we
cannot
judge the size because using different methods will give
different
results, and they are irreconcilable with eachother,
completely
unsolvable, not because we dont know how to, but because
there is no
solution.
>Maybe you just dont like it because it gives meaning to the
notion of
>1/0.
> I have no problem with extending the reals to include oo,
which is
then
> the value of 1/0. Note that this doesnt relate to Cantors
diagonal
> proof.
Well I will not allow the symbol oo to be used for the
value of 1/0
because thats not what Im trying to show. Or, when you say
oo, do
you mean an innity that transcends aleph notation and not
only
mathematical intuition but any mathematical formalisms, as a
TRUE
innity should?
(...Starblade Riven Darksquall...)
===
Subject: Re: Multitude of innities.
> Say you have two sets, the set of the digits from 0-9, and
the set of
> natural numbers? The relative size of the second set to the
rst is
> aleph 1, but the relative size of the rst set to the
second would be
> aleph -1
Nonsense. Where did you hear this stuff?
> The slope of the absolute value function at zero. One
divided by zero.
How do you gure that? If you approach zero from the positive
side,
the slope isnt dened is because the left and right limits
are
different. Theres no division by zero.
===
Subject: Re: Multitude of innities.
> What if the term larger loses its meaning? And anything
which is not
> strictly zero size with respect to the set is automatically
neither
> larger, smaller, or equal in size, yet are also all three
things?
Hello? Anyone home? Cardinal numbers can be compared. The
cardinal
number of set A is smaller than the cardinal number of set B
iff there
exists a one-one mapping from A onto a proper subset of B but
no such
mapping exists from B to a proper subset of A. See
Schroeder-Bernstein
Theorem.
> Of course, thats an absurdity only because were using
denitive
> mathematical structures. If we used indenitive mathematical
> structures we would not have that problem.
> So the set of all arbitrary sets is an undened.
Mathematics simply
> cannot deal with it.
That is because it leads to a contradiction. Mathematics does
not permit
contradictions. A mathematical system must be consistent or
it is nonsense.
> But if we had a mathematics that gave meaning to such
things it could
> deal with it. Such a mathematics would give meaning to lim
x->0 f(x)
> where f(x) = |x| and also lim x->0 f(x) where f(x) = 1/x.
One can compactify the real number system so that limits like
that would
exist. Of course the points added to the real numbers would
not behave
like nite real numbers algebraically, but they would have
the right
topological properties.
> Since ours doesnt, its no surprise we manage to nd such
beautiful
> mathematical paradoxes as we tend to do.
What paradoxes? As far as we know, mathemtics is consistent.
The know
paradoxes have been purged by modifying the math.
> Why is it necessarily larger? Why does larger have to have
a meaning?
> Why cant we just say for innitely innite sets, larger,
smaller,
> and same size are meaningless concepts?
See above for denition of less than or greater than for
cardinal numbers.
> I looked at it, but did not study it. I will go and do
so. And I
> guarantee that I will object to it on the same fundamental
premises as
> I did before.
All the objections to which you have alluded were raised
against
Cantors set theory. The theory has be worked on and modied
so the
known paradoxes are removed. If one wishes to restrict
mathematics to
nitary systems, one can, but certain results can no longer
be derived.
> Answer that question rst. Then answer this one: Isnt size
akin to a
> numerical value? Taking the cardinality as a numerical
description of
> a set, you are saying that the cardinality must always be
nite, and
> can never be innite. Obviously if it was innite, then
adding 1
> more size would not have any effect on it.
Nonesense. An unbounded hierarchy of innite cardinals exists.
> So in a set, we cannot have an innite ascenscion of
elements, just
> like we cannot have an innite descencion of elements. So,
for every
> set, we must be capable of describing an element that is
not in the
> set.
Innite ordinals exist. There are your ascensions. There are
non-archimedean number elds in which non-zero elements small
than any
given non zero real number exist. Look up non-standard
analysis on Google.
> I studied it. But it still relies on the idea that making
something
> that SHOULD be bigger than it will actually have an effect
on it. If I
> contest that notion, then while I do not necessarily
disproove Cantor,
> I do disproove the idea that there is no innitely large
innity,
> because for an innitely large innity, something large
than it
> would be the same size. Also, if something was smaller than
it but not
> nonzero, it would also be the same size, by logical
extention.
> Maybe you just dont like it because it gives meaning to
the notion of
> 1/0.
Are you familiar with projective geometry? We can construct
spaces with
a line at innitity added so that all pairs of lines
intersect. The
line at innity provides intersection points for pairs of
parallel lines.
Consider the following system of co-ordinates for a plane (x,
y, t)
where this is mapped into the normal cartesian co-ordate
(x/t, y/t). The
point (x,y,0) corresponds to a point at innity. These
so-called
homogeneous co-ordinates can be used to co-ordinatize
projective planes.
Bob Kolker
===
Subject: Re: Multitude of innities.
> What if the term larger loses its meaning? And anything
which is not
> strictly zero size with respect to the set is automatically
neither
> larger, smaller, or equal in size, yet are also all three
things?
> Hello? Anyone home? Cardinal numbers can be compared.
What if I use noncardinal types of mathematics? That is, a
mathematics
dealing with numbers that cannot be compared in the ordinary
way?
The cardinal
> number of set A is smaller than the cardinal number of set
B iff there
> exists a one-one mapping from A onto a proper subset of B
but no such
> mapping exists from B to a proper subset of A. See
Schroeder-Bernstein
> Theorem.
Is there a way to measure the mapping properties? That is, to
use
relative ordinals that measure how innite or innitesimal
one set
is to the other? So that if one set was innite to another,
say it
was aleph 2 to aleph 0, the relative size of the innite set
to the
noninnite set is aleph 2, and the reverse is aleph -2? Aleph
-2 not
actually being a negative set size but rather a relative size
of the
zeroth to the second aleph?
Then if we could prove that no matter how negative we made
the aleph
we would not get zero I will have shown how I was right. At
least,
right that if we used unphysical numbers wed get the result
I was
aiming for.
Of course, thats an absurdity only because were using
denitive
> mathematical structures. If we used indenitive mathematical
> structures we would not have that problem.
So the set of all arbitrary sets is an undened. Mathematics
simply
> cannot deal with it.
> That is because it leads to a contradiction. Mathematics
does not permit
> contradictions. A mathematical system must be consistent or
it is
nonsense.
What if we classify contradictions? Allow mathematics to work
in
specic circumstances with made up numbers with made up
properties,
that represent the solution to a contradiction, but cannot be
actualized in classical mathematics? Then what?
But if we had a mathematics that gave meaning to such things
it could
> deal with it. Such a mathematics would give meaning to lim
x->0 f(x)
> where f(x) = |x| and also lim x->0 f(x) where f(x) = 1/x.
> One can compactify the real number system so that limits
like that would
> exist. Of course the points added to the real numbers would
not behave
> like nite real numbers algebraically, but they would have
the right
> topological properties.
This intruiges me. Where did you learn of such numbers?
Since ours doesnt, its no surprise we manage to nd such
beautiful
> mathematical paradoxes as we tend to do.
> What paradoxes? As far as we know, mathemtics is
consistent. The know
> paradoxes have been purged by modifying the math.
Yes, but by eliminating the paradoxes all we have done is
limit
ourselves to certain type of numbers, that is, actualizable
numbers.
Why is it necessarily larger? Why does larger have to have a
meaning?
> Why cant we just say for innitely innite sets, larger,
smaller,
> and same size are meaningless concepts?
> See above for denition of less than or greater than for
cardinal
numbers.
Why do all numbers have to have a cardinality? What if there
was a
number such that the only comparison that could be made
between it and
other numbers were that all other numbers were size zero
relative to
it? Not simply innitesimal, but literally size zero?
I looked at it, but did not study it. I will go and do so.
And I
> guarantee that I will object to it on the same fundamental
premises as
> I did before.
> All the objections to which you have alluded were raised
against
> Cantors set theory. The theory has be worked on and
modied so the
> known paradoxes are removed. If one wishes to restrict
mathematics to
> nitary systems, one can, but certain results can no longer
be derived.
I know. But I liked the paradoxes. They were neat, and hinted
at
numbers that could not be resolved using actualizable
situations.
Meaning that we might stumble onto nonclassical mathematics.
> Answer that question rst. Then answer this one: Isnt size
akin to a
> numerical value? Taking the cardinality as a numerical
description of
> a set, you are saying that the cardinality must always be
nite, and
> can never be innite. Obviously if it was innite, then
adding 1
> more size would not have any effect on it.
> Nonesense. An unbounded hierarchy of innite cardinals
exists.
So what would the notation for this be? If we used limit
notation, we
would get something like lim n n->(inf) aleph-n, so we could
designate
the limiting case as an abstract quantity that mathematics
cannot be
used on.
So in a set, we cannot have an innite ascenscion of
elements, just
> like we cannot have an innite descencion of elements. So,
for every
> set, we must be capable of describing an element that is
not in the
> set.
> Innite ordinals exist. There are your ascensions. There are
> non-archimedean number elds in which non-zero elements
small than any
> given non zero real number exist. Look up non-standard
analysis on
Google.
Are these innitesimal numbers or zeroes? One property of
zero is
that if you multiply anything by it you would get itself, a
number
that is not distinct from zero. Would multiplying any number
by these
non-archimedian numbers produce distinct results or result in
the same
number?
> I studied it. But it still relies on the idea that making
something
> that SHOULD be bigger than it will actually have an effect
on it. If I
> contest that notion, then while I do not necessarily
disproove Cantor,
> I do disproove the idea that there is no innitely large
innity,
> because for an innitely large innity, something large
than it
> would be the same size. Also, if something was smaller than
it but not
> nonzero, it would also be the same size, by logical
extention.
Maybe you just dont like it because it gives meaning to the
notion of
> 1/0.
> Are you familiar with projective geometry? We can construct
spaces with
> a line at innitity added so that all pairs of lines
intersect. The
> line at innity provides intersection points for pairs of
parallel
lines.
> Consider the following system of co-ordinates for a plane
(x, y, t)
> where this is mapped into the normal cartesian co-ordate
(x/t, y/t). The
> point (x,y,0) corresponds to a point at innity. These
so-called
> homogeneous co-ordinates can be used to co-ordinatize
projective planes.
> Bob Kolker
So if we considered the bound to be an unreal, abstract
number, for
example, n such that aleph-n=n, could we then say that this is
equivilent to n*0=m where m a number from cardinal
innitesimal
numbers to regular numbers to cardinal innite numbers but
which is
all of these at once?
In the mathematics youre familiar with, this is a completely
meaningness scenario. What if I developed a mathematics for
which this
kind of notation has what I call an unreal meaning?
(...Starblade Riven Darksquall...)
===
Subject: Re: Multitude of innities.
> Thats a consequence of the fact that a sets power set
must be larger
> than the original set. Another consequence is that there is
no
largest
> innity.
Why is it necessarily larger? Why does larger have to have a
meaning?
> Why cant we just say for innitely innite sets, larger,
smaller,
> and same size are meaningless concepts?
Read Cantor. He gave answers to all those questions.
===
Subject: Re: Multitude of innities.
>>Well, it would be a type of innity of which no greater set
could be
>>constructed. This would require that, if one were to try to
construct
>>it in any way, one would simply get this same set. Meaning
for each
>>set within this super innite set, any transformation that
can be
>>done to it will also exist in this set.
>> If you allow the transformation of taking the power set of
any set,
then
>> Cantors diagonal proof shows that the size of the power
set is always
>> strictly greater than the size of the original set. Are
you familiar
with
>> Cantors proof? If so, then which step would you want to
invalidate in
>> order to allow your super-innite set to be the same as
its power set?
>Well there would be a single innty type, the size of the
set is innite
and
>the size of the power set of this innite set is also
innite.
You apparently dont know that Cantors proof shows that an
innite sets
innite power set is of a strictly GREATER innity. Thus
although both
sets are innite, the power set is still strictly larger.
Cantor introduced the modern notion of innity. Before that,
when people
used innitary language (for ever and ever, the everlasting
hills,
innite in mercy, etc.) all it meant was big -- really big --
you
just
wont believe how vastly, hugely mind-bogglingly big it is
[D. Adams]
--
---------------------------
| BBB b barbara minus knox at iname stop com
| B B aa rrr b |
| BBB a a r bbb |
| B B a a r b b |
| BBB aa a r bbb |
-----------------------------
===
Subject: Re: Multitude of innities.
> Here is your answer:
> Because mathmatics is the contingent application of a
function for which
an
> outcome must be dened, there is no contingency to create
integers if
that
> outcome is not dened. The outcome that is not dened, is
not
innity!
> The outcome that is not dened means not to look for an
outcome. And
so
> we leave the realm of mathematics alltogether. The set of
integers is a
set
> of numerals, or signs.
Perhaps Jones would benet from review of what a function is
http://mathworld.wolfram.com/Function.html
===
Subject: Multitude of innities. 2
Come on now.
Point out my error please, I beg, its just quicker and
easier, thats all.
JJ
> Here is your answer:
> Because mathmatics is the contingent application of a
function for
which
an
> outcome must be dened, there is no contingency to create
integers if
that
> outcome is not dened. The outcome that is not dened, is
not
innity!
> The outcome that is not dened means not to look for an
outcome. And
so
> we leave the realm of mathematics alltogether. The set of
integers is a
set
> of numerals, or signs.
> Perhaps Jones would benet from review of what a function is
> http://mathworld.wolfram.com/Function.html
===
Subject: Re: Multitude of innities.
> we leave the realm of mathematics alltogether. The set of
integers is a
set
> of numerals, or signs.
Numerals and signs can be used to represent integers, but the
integers
exist whether or not they are represented.
Bob Kolker
===
Subject: contingent application function
> Numerals and signs can be used to represent integers, but
the integers
> exist whether or not they are represented.
Quite right. And the form of an integer is contingent upon
the application
of a function.
JJ
> we leave the realm of mathematics alltogether. The set of
integers is a
set
> of numerals, or signs.
> Numerals and signs can be used to represent integers, but
the integers
> exist whether or not they are represented.
> Bob Kolker
===
Subject: Re: any shorter form of this formula?
>f(n)=sum from 1 to innity of (n div 2^i)
>offcourse innity can be changed to something like
downto(log2n)
>but i was kind of thinking of a form where there is no sum
>thanx in advance
If n div 2^i means the truncated quoteint FLOOR(n / 2^i),
then your sum is the power of 2 dividing n!.
This is n - (number of 1s in binary expansion of n).
--
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: Fiber bundle having Lorentz space as its ber
When gauge theory is discussed on the principal ber bundle
and its
connection, the base space is generally Lorentz space and
inner space
is regarded as the ber on it. In the following webpage:
http://139.134.5.123/tiddler2/gauge4/gauge.htm
gauge theory is treated with reverse way, i.e. the ber
bundle has
inner space as its base space and Lorentz space as ber on
it.
===
Subject: contingent application count innity
> Can you dene contingent application? Are you saying that
whenever
> innity is and has been used in physics all those theories
are negated
> because the dependence upon applicability for testing is
not possible?
When a mathematical manoevure is made: contingent
application. The
manoevure
to use maths is not mathematical.
You cant, also, reach innity through maths. Innity is not a
mathematical
operation.
You cant say the count is innite because innity cant say
what a full
count is.
> For an innite regress in nite space, so called, you dont
have to
> construe Koch curves or triangles, you can have a function
for say,
+1.
> Here, the space is always nite, while the steps become
proportionally
> smaller. But in all these examples, the construction can
only go so far
as
> contingent application can go. Thats as far as maths goes.
All the
> functions
> of maths are contingent applications. There is no
mathematical
application
> to innity.
> JJ
> Can you dene contingent application? Are you saying that
whenever
> innity is and has been used in physics all those theories
are negated
> because the dependence upon applicability for testing is
not possible?
> But we could construe any number of strange maths:
> How long is the coast-line of Great Britain? Given a map
one can sit down
> with a ruler and soon come up with a value for the length.
The problem is
> that repeating the operation with a larger scale map yields
a greater
> estimate of the length. If we actually went to the coast
and measured
them
> directly, then still greater estimates would result. It
turns out that as
> the scale of measurement decreases the estimated length
increases without
> limit. Thus, if the scale of the (hypothetical)
measurements were to be
> innitely small, then the estimated length would become
innitely
large!
> The ultimate map is the terrain itself. The particular
method we use to
> search through the possibility space determines how close
we look and
how
> accurately we model the terrain.
> -------------------
> Take the coastline of Britain for example. How long is it ?
> Nobody knows.
> Of course they do you say ! Ah, but they know roughly the
area of the
> country so they must, by Euclid, know the minimum boundary
surely, an
> equivalent circle ? Yes, but the actual boundary is innite
! To see
this,
> go in your mind to the seaside with a metre rule and
measure a section of
> rock. You will skip over a few crevices will you not ? Now
use a
kilometre
> ruler instead - this skips over a lot more resulting in a
different,
lower
> reading. Take now a 1 cm measurement, this will go around
most
> irregularities and give a much bigger total. So, the length
is variable
> isnt it ? But not innite surely ?
> Move up the coast and you nd a river. What do you measure
now ? Well,
just
> continue up the left bank until you re-emerge on the right,
the coastline
> must be continuous ! But now you are following all the
tributaries and
> streams and rivulets and....you will never appear on the
other side I
think
> ! If you do, take an micron sized measuring stick and try
again, by the
time
> innite time. Britain thus has an coastline approaching
innity but a
> nite area. A 2 dimensional paradox.
> More Paradoxes
> Now take a tree and measure its volume (easy, just dunk it
in a big bath
and
> see how much water overows, like Archimedes would). But
what is its
> surface area ? Yes, thats right its nearly innite again
(dont forget
to
> measure down every pore or stomata in every leaf...). A 3
dimensional
> paradox now !
> How many dimensions did I say ? Two ? Three ? Neither are
right, both
items
> have what is called Fractal Dimension - a fractional
number. Ignoring
> technicalities, this is a measure of how irregular an
object is. A Koch
> snowake (a triangle, with other third size triangles stuck
midway on
each
> side, and so ad innitum - illustrated) has for example a
fractal
dimension
> of 1.26. In the extreme cases we have seen the dimension
become one more
> than we would expect, a single dimensional line becomes two
dimensional,
a
> two dimensional surface becomes three. Ill leave to your
imagination a 3
> dimensional fractal in 4 dimensions !
> Self-Similarity
> Can we go the other way, reducing a three dimensional
object to two ? Yes
we
> can. Take a pyramid and drill a hole in it. We increase the
surface area
and
> reduce the volume. Keep drilling smaller holes in what is
left until you
run
> out of material, you now have a solid without any volume,
made up only of
> edges. The formal version of this is called the Sierpinski
gasket in
three
> dimensions, but has a fractal dimension of only two !
> While measuring the coast with the different ruler sizes
something may
have
> struck you. The shape of the coast at 1km scale is the same
as at 1m
scales,
> and again the same at 1cm scale. Think of Africa from
space, an island on
a
> map, a rock pool beneath your feet. The shape of the
coastline always
> appears the same, equally jagged. This is a feature of
fractal systems
which
> we call Self-Similarity. We cannot tell just by looking
at a system
what
> scale it really is. Yet every view is slightly different,
the objects are
> identical in form but also not identical in detail ! Why is
this ?
> Mathematical Roots
> To see let us look at something quite different. Many
people know that
you
> can nd the roots of an equation involving squares of x by
using a
simple
> formula, and it gives two values. Generalising, we can
expect n roots for
an
> equation of x to the power of n. How do we solve these
bigger equations ?
> Often Newtons method is used, this is an iterative
solution and gives a
> better approximation by putting the last answer back into
the equation,
> repeating until the answer doesnt change - that being a
solution. For
other
> solutions we try different starting values and, if lucky,
the formula
will
> converge to an alternative solution. Luck ? This is precise
mathematics
> isnt it ? Not quite, it is another fractal !
> The boundary between the range of values converging on one
root and those
> converging on another is a fractal curve. If we guess a
value close to
this
> boundary we cannot be sure which root it will converge
towards. It make
no
> difference if we magnify a graph of value versus root, the
irregularity
and
> unpredictability of the boundary is still there, big as
ever (as
illustrated
> in this plot of x6). It is this boundary between part of a
system moving
in
> one direction and part moving in another which is at the
heart of
fractals.


> While this is my view, this may not be your view. I am
eager to
hear
> your view on things. I posted this to four different
newsgroups
> because these are the four newsgroups which would be most
interest
> in
> this topic. I hope you enjoy debating this topic with
eachother.
I
> know I had fun starting this debate!
There is an innite hierarchy of innite (transnite)
cardinals.
> Also
> there is an innite hierarchy of innite ordinals.
The matter is dealt with in Cantors theory of transnite
sets,
about
> which there are scores of good treatises. You can easily
nd one
on
> amazon.com or abebooks.com.
Bob Kolker
When you run the algorithms for Koch curves and Cantor sets
it has
the
> apparent potential to create and innite amount of
information in a
> nite
> space.
Physical vs Design - Space
In physical space we observe objects which are stable
processes or
for
> particular durations stable enough to be building blocks in
larger
> constructions with other duration potentials.
In design space we observe the possible changing congurations
through
> durations.
A nite number of congurationally stable durations [atoms]
likely
> have
> an
> innite congurational possibility space.
Cantor Sets & Koch Curves:
The Von Koch curve. Its construction is almost as easy as the
Sierpinski
> Triangle. You start with a triangle (equilaterality is
really more
> important
> in this case than for the Sierpinski Triangle), and, on
each side,
tack
> on
> little equilateral triangles, so you end up with a shape
very much
like
> Star of David. Now tack on triangle on all straight
segments and
iterate
> smaller and smaller...
The length of the Koch curve is innite. Assuming that we
started
with
> unit line segment the length of the curve in the k^th stage
is equal
to
> the
> number of line segments times the length of each segment.
The number
of
> pieces in the kth stage is 4k and the length of a segment
is (1/3)k.
> Thus,
> the the total length of the curve is (4/3)k at the k^th
stage of
> construction. Since, the Koch curve, by denition, is the
limiting
set
> of
> the above geometric construction scheme, when k goes to
innity. The
> length
> of the Koch curve is innite.
http://chaos.phy.ohiou.edu/~thomas/fractal/frac1.html
To make an Cantor sets take a line of a certian length and
take out
the
> middle third, draw the two resulting lines missing middle
space below
it
> and
> then do this again and again and we seem to get an innite
space out
of
> and
> nite line.
http://www.math.sunysb.edu/~scott/Book331/Cantor_sets.html
Georg Cantor, a German mathematician, in 1883, introduced a
set, now
> called
> the Cantor set that had some exceptional properties.
Following is a
> simple
> geometric construction scheme to visualize the Cantor set.
The
> construction
> of the Cantor set (C) is done by starting with a unit line
[0,1].
Divide
> the
> unit line segment into three equal parts and then remove
the middle
> third
> (leaving the end points)to form new segments that exist at
[0, 1/3]
and
> [2/3, 1]. This is the rst stage in the construction of the
C set.
In
> the
> next stage repeat the above process of removing the middle
third of
the
> each
> of the two line segments that was obtained in the last step
to get
four
> smaller line segments. This process of removing the middle
third from
> the
> remaining segments from the previous stage, is continued
adinnitum.
> The
> gure below shows the rst six stages in the construction
of the C
> set.
> The points marked by the white vertical line (in stage 0
and stage 1)
> are
> the line segments that are removed.
http://chaos.phy.ohiou.edu/~thomas/fractal/frac1.html
The same phenomena happens with the Koch curve which is an
design
used
> by
> snow ake formation with freezing H2O molecules. Take a
triangle,
and
> along
> each side put another triangle precicely one third the size
and then
you
> get
> a star of david. Repeat the process over and over and u get
this
> seemingly
> innite (down to molecule size) line of triangles within
the same
space
> as
> the orgional nite triangle.
Some claim that in design space (space of possible
[irreducible]
> *circuit
> properties) their is innite variability within a nite
space with
> nite
> stabilities:
The snowake never escapes the dashed square you see in
gures 1-4,
so
> it
> encloses a nite amount of area no larger than a credit
card. On the
> other
> hand, at each step building the new little triangles adds
more than
one
> unit
> of length to the curve. To be precise, [4 3]n - 1
units are added at
> the
> nth step, so the length of the snowake is larger than 3 +
1 + 1 + 1
+
> +
> 1 + ....... = innity.
The snowake curve is innitely long, yet it would t in your
wallet!
http://scidiv.bcc.ctc.edu/Math/Snowake.html
Student: So I can think of innity as being larger than any
counting
> number? And iterating innitely many times is the idea of
repeating
the
> steps forever?
Mentor: For now these are good ways to think. Here is a more
standard
> way
> to
> say repeat innitely many times:
Let the number of iterations approach innity.
http://www.shodor.org/interactivate/discussions/innity.html
Now, the real trouble with innity is much the same: we cant
count
> that
> high! Cantors insight was that, even though we cant
enumerate an
> innite
> set, we can nonetheless apply the same procedure to any
well-dened
> innite set that we applied above to determining if our
hands have
the
> same
> number of ngers. In other words, we can determine if two
innite
sets
> are
> the same size (equinumerous) by seeking to nd a one-to-one
match-up
> between the elements of each set. Now, remember Galileos
Paradox?
> Galileo
> noticed that we can do the following:
http://www.mathacademy.com/pr/minitext/innity/

> Cantors Comb Example:
>
http://www.shodor.org/interactivate/activities/cantor/
index.html
Kochs Innite Snowake Example:
>
http://www.shodor.org/interactivate/activities/koch/index.html
> http://www.shodor.org/MASTER/fractal/software/Snowake.html
Sierpinskis Gasket with 100,000 iterations:
> http://www.cs.wisc.edu/~richm/cs302/applets/gasket.html
Innity - Denitions:
> http://scidiv.bcc.ctc.edu/Math/innity.html
> http://members.shaw.ca/quadibloc/math/innt.htm
Innite Perimeter, Finite Area:
> http://www.zeuscat.com/andrew/chaos/vonkoch.html


>
===
Subject: Re: contingent application count innity
>> Can you dene contingent application? Are you saying that
whenever
>> innity is and has been used in physics all those theories
are negated
>> because the dependence upon applicability for testing is
not possible?
>When a mathematical manoevure is made: contingent
application. The
manoevure
>to use maths is not mathematical.
>You cant, also, reach innity through maths. Innity is not a
mathematical
>operation.
>You cant say the count is innite because innity cant say
what a full
>count is.
>> For an innite regress in nite space, so called, you
dont have to
>> construe Koch curves or triangles, you can have a function
for say,
+1.
>> Here, the space is always nite, while the steps become
proportionally
>> smaller. But in all these examples, the construction can
only go so
far
>> a
>> contingent application can go. Thats as far as maths goes.
All the
>> functions
>> of maths are contingent applications. There is no
mathematical
>application
>> to innity.
>> JJ
>> Can you dene contingent application? Are you saying that
whenever
>> innity is and has been used in physics all those theories
are negated
>> because the dependence upon applicability for testing is
not possible?
>> But we could construe any number of strange maths:
>> How long is the coast-line of Great Britain? Given a map
one can sit
down
>> with a ruler and soon come up with a value for the length.
The problem
is
>> that repeating the operation with a larger scale map
yields a greater
>> estimate of the length. If we actually went to the coast
and measured
them
>> directly, then still greater estimates would result. It
turns out that
as
>> the scale of measurement decreases the estimated length
increases
without
>> limit. Thus, if the scale of the (hypothetical)
measurements were to be
>> innitely small, then the estimated length would become
innitely
large!
>> The ultimate map is the terrain itself. The particular
method we use to
>> search through the possibility space determines how close
we look
and
>how
>> accurately we model the terrain.
>> -------------------
>> Take the coastline of Britain for example. How long is it ?
>> Nobody knows.
>> Of course they do you say ! Ah, but they know roughly the
area of the
>> country so they must, by Euclid, know the minimum boundary
surely, an
>> equivalent circle ? Yes, but the actual boundary is
innite ! To see
>this,
>> go in your mind to the seaside with a metre rule and
measure a section
of
>> rock. You will skip over a few crevices will you not ? Now
use a
kilometre
>> ruler instead - this skips over a lot more resulting in a
different,
lower
>> reading. Take now a 1 cm measurement, this will go around
most
>> irregularities and give a much bigger total. So, the
length is variable
>> isnt it ? But not innite surely ?
>> Move up the coast and you nd a river. What do you measure
now ? Well,
>just
>> continue up the left bank until you re-emerge on the
right, the
coastline
>> must be continuous ! But now you are following all the
tributaries and
>> streams and rivulets and....you will never appear on the
other side I
>think
>> ! If you do, take an micron sized measuring stick and try
again, by the
>time
>> innite time. Britain thus has an coastline approaching
innity but a
>> nite area. A 2 dimensional paradox.
>> More Paradoxes
>> Now take a tree and measure its volume (easy, just dunk it
in a big bath
>and
>> see how much water overows, like Archimedes would). But
what is its
>> surface area ? Yes, thats right its nearly innite again
(dont
forget
>> measure down every pore or stomata in every leaf...). A 3
dimensional
>> paradox now !
>> How many dimensions did I say ? Two ? Three ? Neither are
right, both
>items
>> have what is called Fractal Dimension - a fractional
number. Ignoring
>> technicalities, this is a measure of how irregular an
object is. A Koch
>> snowake (a triangle, with other third size triangles
stuck midway on
>each
>> side, and so ad innitum - illustrated) has for example a
fractal
>dimension
>> of 1.26. In the extreme cases we have seen the dimension
become one more
>> than we would expect, a single dimensional line becomes
two dimensional,
a
>> two dimensional surface becomes three. Ill leave to your
imagination a
3
>> dimensional fractal in 4 dimensions !
>> Self-Similarity
>> Can we go the other way, reducing a three dimensional
object to two ?
Yes
>> can. Take a pyramid and drill a hole in it. We increase
the surface area
>and
>> reduce the volume. Keep drilling smaller holes in what is
left until you
>run
>> out of material, you now have a solid without any volume,
made up only
of
>> edges. The formal version of this is called the Sierpinski
gasket in
three
>> dimensions, but has a fractal dimension of only two !
>> While measuring the coast with the different ruler sizes
something may
>have
>> struck you. The shape of the coast at 1km scale is the
same as at 1m
>scales,
>> and again the same at 1cm scale. Think of Africa from
space, an island
on
>> map, a rock pool beneath your feet. The shape of the
coastline always
>> appears the same, equally jagged. This is a feature of
fractal systems
>which
>> we call Self-Similarity. We cannot tell just by looking
at a system
what
>> scale it really is. Yet every view is slightly different,
the objects
are
>> identical in form but also not identical in detail ! Why
is this ?
>> Mathematical Roots
>> To see let us look at something quite different. Many
people know that
you
>> can nd the roots of an equation involving squares of x by
using a
simple
>> formula, and it gives two values. Generalising, we can
expect n roots
for
>> equation of x to the power of n. How do we solve these
bigger equations
?
>> Often Newtons method is used, this is an iterative
solution and gives a
>> better approximation by putting the last answer back into
the equation,
>> repeating until the answer doesnt change - that being a
solution. For
>other
>> solutions we try different starting values and, if lucky,
the formula
will
>> converge to an alternative solution. Luck ? This is
precise mathematics
>> isnt it ? Not quite, it is another fractal !
>> The boundary between the range of values converging on one
root and
those
>> converging on another is a fractal curve. If we guess a
value close to
>this
>> boundary we cannot be sure which root it will converge
towards. It make
no
>> difference if we magnify a graph of value versus root, the
irregularity
>and
>> unpredictability of the boundary is still there, big as
ever (as
>illustrated
>> in this plot of x6). It is this boundary between part of a
system moving
>> one direction and part moving in another which is at the
heart of
>fractals.
>> While this is my view, this may not be your view. I am
eager to
>hear
>> your view on things. I posted this to four different
newsgroups
>> because these are the four newsgroups which would be most
interest
>> in
>> this topic. I hope you enjoy debating this topic with
eachother.
I
>> know I had fun starting this debate!
>> There is an innite hierarchy of innite (transnite)
cardinals.
>> Also
>> there is an innite hierarchy of innite ordinals.
>> The matter is dealt with in Cantors theory of transnite
sets,
>about
>> which there are scores of good treatises. You can easily
nd one
on
>> amazon.com or abebooks.com.
>> Bob Kolker
>> When you run the algorithms for Koch curves and Cantor
sets it has
the
>> apparent potential to create and innite amount of
information in a
>> nite
>> space.
>> Physical vs Design - Space
>> In physical space we observe objects which are stable
processes or
for
>> particular durations stable enough to be building blocks
in larger
>> constructions with other duration potentials.
>> In design space we observe the possible changing
congurations
>through
>> durations.
>> A nite number of congurationally stable durations
[atoms] likely
>> have
>> an
>> innite congurational possibility space.
>> Cantor Sets & Koch Curves:
>> The Von Koch curve. Its construction is almost as easy as
the
>Sierpinski
>> Triangle. You start with a triangle (equilaterality is
really more
>> important
>> in this case than for the Sierpinski Triangle), and, on
each side,
>tack
>> on
>> little equilateral triangles, so you end up with a shape
very much
>like
>> a
>> Star of David. Now tack on triangle on all straight
segments and
>iterate
>> smaller and smaller...
>> The length of the Koch curve is innite. Assuming that we
started
>with
>> a
>> unit line segment the length of the curve in the k^th
stage is equal
>> the
>> number of line segments times the length of each segment.
The number
>> pieces in the kth stage is 4k and the length of a segment
is (1/3)k.
>> Thus,
>> the the total length of the curve is (4/3)k at the k^th
stage of
>> construction. Since, the Koch curve, by denition, is the
limiting
>set
>> of
>> the above geometric construction scheme, when k goes to
innity.
The
>> length
>> of the Koch curve is innite.
>> http://chaos.phy.ohiou.edu/~thomas/fractal/frac1.html
>> To make an Cantor sets take a line of a certian length and
take out
>the
>> middle third, draw the two resulting lines missing middle
space
below
>> and
>> then do this again and again and we seem to get an innite
space
out
>> and
>> nite line.
>> http://www.math.sunysb.edu/~scott/Book331/Cantor_sets.html
>> Georg Cantor, a German mathematician, in 1883, introduced
a set, now
>> called
>> the Cantor set that had some exceptional properties.
Following is a
>> simple
>> geometric construction scheme to visualize the Cantor set.
The
>> construction
>> of the Cantor set (C) is done by starting with a unit line
[0,1].
>Divide
>> the
>> unit line segment into three equal parts and then remove
the middle
>> third
>> (leaving the end points)to form new segments that exist at
[0, 1/3]
>and
>> [2/3, 1]. This is the rst stage in the construction of
the C set.
In
>> the
>> next stage repeat the above process of removing the middle
third of
>the
>> each
>> of the two line segments that was obtained in the last
step to get
>four
>> smaller line segments. This process of removing the middle
third
from
>> the
>> remaining segments from the previous stage, is continued
adinnitum.
>> The
>> gure below shows the rst six stages in the construction
of the C
>> set.
>> The points marked by the white vertical line (in stage 0
and stage
1)
>> are
>> the line segments that are removed.
>> http://chaos.phy.ohiou.edu/~thomas/fractal/frac1.html
>> The same phenomena happens with the Koch curve which is an
design
used
>> by
>> snow ake formation with freezing H2O molecules. Take a
triangle,
and
>> along
>> each side put another triangle precicely one third the
size and then
>you
>> get
>> a star of david. Repeat the process over and over and u
get this
>> seemingly
>> innite (down to molecule size) line of triangles within
the same
>space
>> as
>> the orgional nite triangle.
>> Some claim that in design space (space of possible
[irreducible]
>> *circuit
>> properties) their is innite variability within a nite
space with
>> nite
>> stabilities:
>> The snowake never escapes the dashed square you see in
gures
1-4,
>> it
>> encloses a nite amount of area no larger than a credit
card. On
the
>> other
>> hand, at each step building the new little triangles adds
more than
>one
>> unit
>> of length to the curve. To be precise, [4 3]n -
1 units are added at
>> the
>> nth step, so the length of the snowake is larger than 3 +
1 + 1 +
1
>> 1
>> +
>> 1 + ....... = innity.
>> The snowake curve is innitely long, yet it would t in
your
>wallet!
>> http://scidiv.bcc.ctc.edu/Math/Snowake.html
>> Student: So I can think of innity as being larger than any
counting
>> number? And iterating innitely many times is the idea of
repeating
>the
>> steps forever?
>> Mentor: For now these are good ways to think. Here is a
more
standard
>> way
>> to
>> say repeat innitely many times:
>> Let the number of iterations approach innity.
>>
http://www.shodor.org/interactivate/discussions/innity.html
>> Now, the real trouble with innity is much the same: we
cant count
>> that
>> high! Cantors insight was that, even though we cant
enumerate an
>> innite
>> set, we can nonetheless apply the same procedure to any
well-dened
>> innite set that we applied above to determining if our
hands have
>the
>> same
>> number of ngers. In other words, we can determine if two
innite
>sets
>> are
>> the same size (equinumerous) by seeking to nd a one-to-one
>match-up
>> between the elements of each set. Now, remember Galileos
Paradox?
>> Galileo
>> noticed that we can do the following:
>> http://www.mathacademy.com/pr/minitext/innity/
>> Cantors Comb Example:
>>
http://www.shodor.org/interactivate/activities/cantor/
index.html
>> Kochs Innite Snowake Example:
>>
http://www.shodor.org/interactivate/activities/koch/index.html
>> http://www.shodor.org/MASTER/fractal/software/Snowake.html
>> Sierpinskis Gasket with 100,000 iterations:
>> http://www.cs.wisc.edu/~richm/cs302/applets/gasket.html
>> Innity - Denitions:
>> http://scidiv.bcc.ctc.edu/Math/innity.html
>> http://members.shaw.ca/quadibloc/math/innt.htm
>> Innite Perimeter, Finite Area:
>> http://www.zeuscat.com/andrew/chaos/vonkoch.html
>>
I dont necessarily need to be able to count to innity to
show that
one innity is larger than another.
===
Subject: Re: contingent application count innity
> When a mathematical manoevure is made: contingent
application. The
manoevure
> to use maths is not mathematical.
> You cant, also, reach innity through maths. Innity is not
a
mathematical
> operation.
Establishing mappings between two set IS a mathematical
operation.
> You cant say the count is innite because innity cant
say what a
full
> count is.
When you establish and one to one mapping of one set onto
another you
have established that the sets have the same cardinality.
Are you aware that you are a mathematical ignoramus?
Bob Kolker
===
Subject: Re: contingent application count innity
> When a mathematical manoevure is made: contingent
application. The
manoevure
> to use maths is not mathematical.
> You cant, also, reach innity through maths. Innity is not
a
mathematical
> operation.
> Establishing mappings between two set IS a mathematical
operation.
[mathematical operation]
n : calculation by mathematical methods; the problems at the
end of the
chapter demonstrated the mathematical processes involved in
the
derivation;
they were learning the basic operations of arithmetic [syn:
mathematical
process, operation]
http://dictionary.reference.com/search?q=mathematical%
20operation
--------------------------------
[mathematical process]
n : calculation by mathematical methods; the problems at the
end of the
chapter demonstrated the mathematical processes involved in
the
derivation;
they were learning the basic operations of arithmetic [syn:
mathematical
operation, operation]
http://dictionary.reference.com/search?q=mathematical%
20process
--------------------------------
[operation]
n. The act or process of operating or functioning.
The state of being operative or functional: a factory in
operation.
A process or series of acts involved in a particular form of
work: the
operation of building a house.
An instance or method of efcient, productive activity: That
restaurant is
quite an operation.
An unethical or illegal business: a fencing operation for
stolen goods.
Medicine. A surgical procedure for remedying an injury,
ailment, defect, or
dysfunction.
Mathematics. A process or action, such as addition,
substitution,
transposition, or differentiation, performed in a specied
sequence and in
accordance with specic rules.
A logical operation.
Computer Science. An action resulting from a single
instruction.
-A military or naval action, campaign, or mission.
-operations The headquarters or center from which a military
action,
ights
into and out of an aireld, or other activities are
controlled.
operations The division of an organization that carries out
the major
planning and operating functions.
http://dictionary.reference.com/search?q=operation
--------------------------------
[methods]
A means or manner of procedure, especially a regular and
systematic way of
accomplishing something: a simple method for making a pie
crust; mediation
as a method of solving disputes. See Usage Note at
methodology.
Orderly arrangement of parts or steps to accomplish an end:
random efforts
that lack method.
The procedures and techniques characteristic of a particular
discipline or
eld of knowledge: This eld course gives an overview of
archaeological
method.
Method A technique of acting in which the actor recalls
emotions and
reactions from past experience and uses them in identifying
with and
individualizing the character being portrayed.
http://dictionary.reference.com/search?r=2&q=methods
> You cant say the count is innite because innity cant
say what a
full
> count is.
> When you establish and one to one mapping of one set onto
another you
> have established that the sets have the same cardinality.
> Are you aware that you are a mathematical ignoramus?
> Bob Kolker
===
Subject: Re: Electrical Storm Sets Three Houses Ablaze
> Brian Quincy Hutchings top-posted:
incidentally, I decry the automatic insertion
> of entire postings in ones reply, but
> what is wrong with top-posting?
> http://www.4qd.org/philos/Netiquette.html
I understand the points made by this author justifying his
opinion.
But keep in mind, it is ONLY an OPINION. The RFCs say
nothing about
top-posting, and THEY are the only canonical description of
what
constitutes a properly formed reply. I sometimes think that
Netiquette
promulgators are the equivalent of the 18th-century English
grammarians who made up their rules out of whole cloth (e.g.
Never
end a sentence with a preposition, which was a pure invention
of its
author).
Your authors main objection to top-posting is that it
doesnt simulate
the ow of conversation. But one could easily take the
contrarian
view: conversation is a sequence of replies, which one has
usually
followed from the beginning (or nearly the beginning), and
typically
one is only replying to the IMMEDIATELY prior reply (which,
alas, very
frequently is the only relevant one). In such a situation,
top-posting
is an enormous time-saver. To mimic a comment of the author,
I cant
tell you how many posts Ive given up on because I couldnt
nd what
was different about THIS reply through all the excessive
verbiage at
the front.
Finally, if top-posting is such a Bad Thing, why do so many
newsreaders
default to it?
I do agree that MIXING top- and bottom-posting is ugly. But
thats
just one guys opinion...
--Ron Bruck
===
Subject: Re: Electrical Storm Sets Three Houses Ablaze
> Finally, if top-posting is such a Bad Thing, why do so many
newsreaders
> default to it?
Because theyre Micro$oft?
--
Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html
His mind has been corrupted by colours, sounds and shapes.
The League of Gentlemen
===
Subject: Re: Question about triangle puzzle (??)
> Hi all,
> I have a picture of two triangles here :
> http://home.mindspring.com/~cj-bubba/
> My question is: shouldnt the sum of the areas of the
polygons equal the
> area of the triangle. In the second triangle one unit has
to be added
for
> its area to equal the rst. Why is that so?
There are many puzzles of this type. But why do these puzzles
always seem
to have dimensions which involve Fibonacci numbers? In this
case, the
large
triange is 13 x 5, and the smaller one is 8 x 3.
--Mark
===
Subject: Re: Continuous fractions
>> (i) Its enough (why?) to show that A intersects the
interval (0,
>> eps).
>I got confused here. I couldn really realize why this is
not a loss
>of generality.
> Suppose x is in A intersect (0, eps). Think about what the
sequence
> x, 2x, 3x, ... looks like.
Now its clear. Its an arithmetic progression. For every r,
there is
always some m so that mx is in (r, r+eps). Seems evident, now
that you
gave the answer (well, whats evident is a matter of
experience).
(snip)

>Proof (by induction on n).
>First, its easy to see that, for every real a, frac(a) =
frac(2a) iff
>a is an integer.
> _Exactly_ how do you show that? Doesnt seem easy to me...
> no, doesnt seem at all easy.
It sufces to consider the case a>=0, right? We have a = I +
frac(a),
where I is an integer (the oor of a). There are 3 cases to
deal
with:
(i) frac(a)=0 -- Then, a is integer and so is 2a, so that
frac(a) =
frac(2a) =0.
(ii 0<frac(a)<1/2 -- Then, 2a = 2I + 2 frac(a) and 0<frac(a)<
2*frac(a) <1, so that 2I is the oor of 2a and frac(2a) = 2
frac(a)>
frac(a). In this case, a is not integer and frac(2a)<> frac(a)
(iii) 1/2<=frac(a)<1 -- Then, frac(a)= 1/2 +s for some
0<=s<1/2.
Therefore, 2a = 2I + 2frac(a) = 2I + 1 + 2s and 0<=2s<1, so
that 2I+1
is the oor of 2a and frac(2a)=2s. Since 0<=s<1/2, it follows
frac(a)>frac(2a). Like in (ii), a is not integer and
frac(2a)<>
frac(a)
We have covered all possibilities and concluded that frac(a)=
frac(2a)
if and only if frac(a)=0, that is iff a is integer. A bit
clumsy (I
agree this proof is not elegant at all) but not difcult,
right? All
we used was high school algebra, nothing to compare with
Lebesgue
integrals, sigma-algebras and the like... Probably theres a
better
proof based on congruences or on theorems of Number Theory.
> But since in our case a is irrational, this is
>impossible. Therefore, frac(a) <> frac(2a). Now, suposse
that, for
>some natural n, frac(i*a)<>frac(j*a) whenever i,j =1,...n
and i<>j.
>Admit, by way of contradiction, that frac((n+1)*a)= f, where
f =
>frac(i*a) for some natural i, 1<=i<=n. Then, (m+1)a =I_m+1 f
and i*a =
>I_i + f, where I_m+1 and I_i are integers. It follows that
(m+1-i)a =
>I_m+1 - I_i and a = (I_m+1 - I_i)(m+1-i), contrarily to the
basic
>assumption that a is irrational. This completes the
induction and
>shows that the terms of the sequence (frac(n)) are pairwise
distinct.
>Therefore, S = {frac(n*a) : n is an integer} is a countable
innite
>set, as stated.
> Theres really no need to get all inductive here. You only
need
> to show that frac(n*a) <> frac(m*a) if n <> m. Suppose that
> frac(n*a) = frac(m*a). Then n*a - m*a is an integer, and
this
> implies that a is rational.
Yes. I made things more complicated than they really are...
Amanda
===
Subject: Re: Continuous fractions
>[...]
>>First, its easy to see that, for every real a, frac(a) =
frac(2a) iff
>>a is an integer.
>> _Exactly_ how do you show that? Doesnt seem easy to me...
>> no, doesnt seem at all easy.
>It sufces to consider the case a>=0, right? We have a = I +
frac(a),
Sorry. I actually thought that what youd said was wrong, and
I
was being coy, saying I didnt know how to prove it instead of
saying it was wrong.
Not being coy to be nasty, thought it would be more
educational
for you to discover for yourself why it was wrong in your
attempt
to prove it carefully. But of course I was wrong, its _not_
wrong,
and yes its easy to see.
(If youre curious, I was sort of thinking, without thinking
too
hard about it, that x = 1/2 was a solution to 2x - 1 = x...)
Just goes to show you cant believe everything you read.
************************
David C. Ullrich
===
Subject: Re: Continuous fractions
>(snip)
>>Proof (by induction on n).
>>First, its easy to see that, for every real a, frac(a) =
frac(2a) iff
>>a is an integer.
>> _Exactly_ how do you show that? Doesnt seem easy to me...
>> no, doesnt seem at all easy.
>It sufces to consider the case a>=0, right? We have a = I +
frac(a),
>where I is an integer (the oor of a). There are 3 cases to
deal
>with:
Rather than three cases, we have
a = I1 + frac(a)
2a = I2 + frac(2a)
for some integers I1, I2. Assuming frac(a) = frac(2a),
subtract to get
a = 2a - a = I2 - I1
so a is an integer.
--
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: Continuous fractions
>(snip)
>Proof (by induction on n).
>>First, its easy to see that, for every real a, frac(a) =
frac(2a)
iff
>>a is an integer.
>> _Exactly_ how do you show that? Doesnt seem easy to me...
>> no, doesnt seem at all easy.
>It sufces to consider the case a>=0, right? We have a = I +
frac(a),
>where I is an integer (the oor of a). There are 3 cases to
deal
>with:
> Rather than three cases, we have
> a = I1 + frac(a)
> 2a = I2 + frac(2a)
> for some integers I1, I2. Assuming frac(a) = frac(2a),
subtract to get
> a = 2a - a = I2 - I1
> so a is an integer.
Yes! Much simpler!
Amanda
===
Subject: Number theory problem
How do you solve the problem:
What is the greatest integer n such that n divides p^4-1 for
all primes p >
5?
The choices are
12, 30, 48, 120, 240
I dont know what the answer is, nor how to solve it, but Im
most
interested in knowing the basic method to solve this problem.
It seems
like
Fermats Little Theorem might have something to say about
this problem but
I
cant gure out how to apply it in this case. (By the way,
this comes
from
an old GRE Subject Test, Im not trying to cheat on my
homework or
anything.)
Zach
===
Subject: Re: Number theory problem
>How do you solve the problem:
>What is the greatest integer n such that n divides p^4-1 for
all primes p
The choices are
>12, 30, 48, 120, 240
>I dont know what the answer is, nor how to solve it, but
Im most
>interested in knowing the basic method to solve this
problem. It seems
like
>Fermats Little Theorem might have something to say about
this problem but
I
>cant gure out how to apply it in this case. (By the way,
this comes
from
>an old GRE Subject Test, Im not trying to cheat on my
homework or
>anything.)
Start by noting that the rst primes exceeding 5 are 7, 11,
13.
7^4 - 1 = 2400 (you might note that
7^4 - 1 = (7^2 - 1)*(7^2 + 1) = 48*50 = 2400).
Can you eliminate any of the ve choices?
No, all ve divide 2400.
The next prime is 11. Perhaps use the binomial theorem to
expand (10 + 1)^4 - 1 = 14640. GCD(2400, 14640) = 240
by Euclidean algorithm since 14640 - 6*2400 = 240
and 240 divides 2400.
We suspect the answer is 240. Checking 13, we see
13^4 - 1 = 168 * 170. Since 168 is divisible by 24 and
170 by 10, 13^4 - 1 is also divisible by 24 * 10 = 240.
Since the GRE is a multiple choice test, you might
guess 240 now and go on to the next question.
Or you can prove this result, as done in other postings.
--
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: Number theory problem
> How do you solve the problem:
> What is the greatest integer n such that n divides p^4-1
for all primes p
5?
> The choices are
> 12, 30, 48, 120, 240
> I dont know what the answer is, nor how to solve it, but
Im most
> interested in knowing the basic method to solve this
problem. It seems
like
> Fermats Little Theorem might have something to say about
this problem but
I
> cant gure out how to apply it in this case. (By the way,
this comes
from
> an old GRE Subject Test, Im not trying to cheat on my
homework or
> anything.)
> Zach
p^4 - 1 = (p-1)(p+1)(p^2+1)
For primes other than 2, one of p-1 and p+1 is divisible by 2
but
not 4 and the other is divisible by 4 but not aways 8, and
p^2+1 is
divisible by 2, but not by 4, so p^4-1 is always divisible by
16 but
not always by 32. This eliminates 12, 30, and 120 as
posibilities.
Since x^4 ==1 mod 5 for x =/= 0 mod 5, that means p^4-1 is
divisible
by 5 and that eliminates 48, leaving only 240 possible.
Since x^4 == 1 mod 3 for x =/= 0 mod 3, that means p^4-1 is
divisible by 3 and that conrms 240.
===
Subject: Re: Zeta function, product of summations
Jim,
Diana
>>I am trying to test my formulas on Mathematica, and
understand how to
>>program regular summations. But, I am unsure how to code
the summation
of
>>divisors d of n with Mathematica.
>
AAAAAAAAAAARRRRRRRRRRRRRRGGGGGGGGGHHHHHHHHHHHHH!!!!!!!!!!!!!!
!!!!
> Forget Mathematica!!!
> You dont have to give large quantities of money to Stephen
Wolfram
> to obtain mathematical insight. Better spend a little on a
pencil
> and some paper.
> Given a function f[n,d], the sum of f[n,d] over all
(positive)
> divisors d of n can be expressed as
> Plus@@(f[n,#]& /@ Divisors[n])
> Keep using Mathematica! Throw away that pencil! Bwwaaahh
hah hah!
> --
> | Jim Ferry | Center for Simulation |
> +------------------------------------+ of Advanced Rockets |
> | http://www.uiuc.edu/ph/www/jferry/
+------------------------+
> | jferry@[delete_this]uiuc.edu | University of Illinois |
===
Subject: Re: Zeta function, product of summations
I will compute this for small n as you suggest.
Diana
> I am trying to show that the inverse of the Zeta(s)
function is the
> innite summation as n goes from 1 to innity of
MobiusMu(n)/(n^s).
> OK
> The Zeta(s) function is dened as the innite summation as
n goes
from
> 1 to innity of 1/(n^s).
> To do this, I am rst calculating the product of two generic
summations:
> {the innite summation as n goes from 1 to innity of
a_n/(n^s)}
times
> {the innite summation as n goes from 1 to innity of
b_n/(n^s)}.
> What I believe is the answer is:
> {the innite summation as n goes from 1 to innity of the
summation
of
> divisors d of n of [a_d * (b_(n/d))]/(n^s)}.
> Thats it.
> Using this formula, I am trying to show that:
> {the innite summation as n goes from 1 to innity of
1/(n^s) of
the
> summation of divisors d of n of MobiusMu(n/d)/(n^s)}
> is one.
> Thats right.
> I am trying to test my formulas on Mathematica, and
understand how to
> program regular summations. But, I am unsure how to code
the summation
of
> divisors d of n with Mathematica.
>
AAAAAAAAAAARRRRRRRRRRRRRRGGGGGGGGGHHHHHHHHHHHHH!!!!!!!!!!!!!!
!!!!
> Forget Mathematica!!!
> You dont have to give large quantities of money to Stephen
Wolfram
> to obtain mathematical insight. Better spend a little on a
pencil
> and some paper. As you point out in your product the
coefcient of
> 1/n^s is
> sum_{d divides n} mu(n/d).
> Why not just take a large piece of paper and see what
happens when you
> compute this sum for some small n. Say up to 20 or 30. What
patterns
> do you see emerging? Can you prove that these patterns
persist for
> all n?
> --
> Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html
> His mind has been corrupted by colours, sounds and shapes.
> The League of Gentlemen
===
Subject: Re: A simple combinatorics...
> I think this problem is a hard one, if you want to get to
the optimum(Im
> not sure, I think its NP-complete). Anyway, you can get a
simple 4/3
> aproximation, relatively fast. A 4/3 means that the number
of bags you
> got as solution from the algorithm is no more than 4/3 of
the optimum
> value. And its really easy: rst, order the elements you
want to get
> into the bags, smallest last. Now, you put as many elements
as you can in
> the rst bag. When the next one doesnt t, change to the
next bag, and
> so on. It takes about time n*log(n) (ordering) and m*n
(tting).
I have got the same algorithm but do not know how to prove it.
(i.e., I dont know whether it is a true algorithm or not..)
Can we prove that the problem is NP? Why do you say it is 3/4
optimum?
I would be very surprised if nobody has never studied this
problem.
(We need a combinatorics mathematician to answer this
question..)
I think the problem becomes more complicated if all the
maximum weights
that the bags can hold are different.
For example,
Objects : 4 kg, 3 kg, 3 kg, 1 kg.
bags(max) : 6 kg, 5 kg.
The decreasing algorithm wont work in this case.
-- Danny
===
Subject: Re: probability
$UBKtM1C6Tk-Kn1I|)j#wr;2I&}g~I^uI_<f(Y87E_3&[%uE$!>ytA2a.gS
zI0VL[?-(Be5zw<Qan&/EvR@rzhK*Q9~HU5j=2U!
> Howdy,
> I was curious on the probability on something. Let us say
that I have 2
> dice. They are normal 6 sided dice. How many times would I
have to
throw
> them to GUARANTEE that I will, at least once, come up with
2 and 3.
As others have said, to the question as stated above the
answer is that
the appearence of a 2 and 3 can never be guaranteed.
However, later on you contradict your statement that they are
normal 6
sided dice.
> The only catch is that throws are always unique. 1 and 2,
or 2 and 1,
will
> only occur once.
Real dice dont act that way, obviously.
> I believe that under these circumstances, the probability
after X throws
> would be P = X*P1*P2. Which for guarantee, 100%, the
equation would look
> like:
> 1 = X * 1/6 * 1/6
> X = 36
I wouldnt really approach it this way. Instead, I would ask:
1) When you throw two dice, how many possible outcomes are
there?
Since you are treating 1 and 2 and 2 and 1 as the same,
the
answer is not 36 but is 6 + 5 + 4 + 3 + 2 + 1 = 6(6+1)/2 = 21.
2) How many of these outcomes full the conditions that you
want to
full?
That condition is throwing a 2 and a 3 (in either order). This
matches only one of the outcomes in (1) above.
So, you throw the dice multiple times with the understanding
that each
new throw cannot duplicate any of the previous outcomes.
Whats the
maximum number of throws before you get a 2 and a 3 (in any
order)? 21
(at which point you will have gone through all the possible
outcomes).
Probability doesnt really enter into this.
I wonder what happens if you throw them a 22nd time.
-jwgh
--
They can track this putz through the usenet and just arrest
him.
===
Subject: Re: Factorial/Exponential Identity, Innity
Im writing software to factor numbers, factorization
software.
I read about how there are a variety of methods, many
inuenced by
the work of Fermat, these are efcient methods, but I thought
I would
try and start from scratch to see what foundation the
research into
the question area would give me in understanding the
contemporary
methods.
One thing I saw recently that I thought was really great was
the
quadratic sieve. Basically it plots x=y^2, then for it
connected with
a line each point with integer coordinates. Then, where those
lines
cross the integers on the x-axis there are only composites,
and and
all composites, and all primes are missed.
So anyways I go to implement a factorization method. Im
working with
numbers stored using the java.math.BigInteger arbitrary
precision
represent the integer in signed 2s complement form, binary.
I call
the input variable n and the result of the method is to be an
array
(multiset) of the prime factors of n, besides n (if prime)
and not
including one.
I gure rst I should factor any twos of the integer. I use
the
method that gets the index of the rightmost set bit of the
binary
representation of n, and it returns the result of how many
twos are in
the prime factorization of n. I then shift n right that many
places
and proceed, having added that many twos to the list of prime
factors.
Next, I think it would be efcient to try and remove small
prime
factors three, ve, etcetera. I think I can use the method
where the
digits of a number add to be a somewhat smaller number, and
that sum
is a multiple of the factor when n is a multiple of the
factor, and
trial division is faster on the smaller number. This is about
casting
nines in decimal, where decimal is base b=10 and nine is is
b-1. For
example, sum the digits of a decimal number, if the sum is a
multiple
of nine, then the number is a multiple of nine.
That method appears to work for b-1 and roots of b-1. Anyways
I can
group the bits of the binary representation into digits of
higher
bases that are powers of two, and then sum those digits to
see if the
sum is a multiple of b-1. For example, here is a short list
of powers
of two and one less than them:
2^2 = 4 4-1=3
2^3 = 8 8-1=7
2^4 = 16 16-1=15
The numbers 2^n-1 are called Mersenne numbers. They are binary
repunits. They are composite unless n is prime, in which case
they
might be prime.
2^5 = 32 32-1=31
2^6 = 64 64-1=63
2^7 = 128 128-1=127
2^8 = 256 256-1=255
2^9 = 512 512-1=511
2^10 = 1024 1024-1=1023
Anyways for 2^2 and 2^3, groups of 2 and 3 bits, it is
possible to sum
those digits into an integer result and then check the
remainder of
the sums division by 3 or 7 respectively for congruence of n
modulo 3
or 7. For small numbers, its usefulness is trivial, but for
3^10000000000 * 7^100000000000, it might save quite a bit of
time
compared to trial division dividing the entire number in base
two by 3
or 7 until the remainder is non-zero, from knowing that the
remainder
is zero. The method says that the Mersenne number is a factor
of the
integer, then division can be done to get the product of the
remaining
multiplicative partitions.
Of the numbers on that short list of Mersenne numbers, none
are
perfect squares or cubes which would allow checking for the
square or
cube root of that number, and some of them are prime and some
composite, for example 31 and 127 are primes, as is 8191, in
using the
composite values then their use would only work when the
composite is
a factor of the number, and it might be worthwhile to try to
factor
the composites from n rst, in that way leaving, for example,
3, in
the number, where 3 is a factor of those composites, eg, 63
and 255.
I dont know the least Mersenne number that is a square or
cube.
I was hoping that the digit summation congruence method would
have a
method for 5, then with the binary power of two then all
primes less
than ten would be factored by those methods. Instead, I still
have to
test divide 5, 11, and 13, etcetera, the digit summation
congruence
method only works for 3 and 7, small primes.
15 = 5 * 3
31 = 31
63 = 7 * 3 * 3
127 = 127
255 = 17 * 5 * 3
511 = 73 * 7
1023 = 31 * 11 * 3
2047 = 89 * 23
4095 = 13 * 7 * 5 * 3 * 3
The digit summation congruence method for 2^12, 4095 will
factor out a
13, a 7, a 5, and two 3s if theyre there. So in using the
digit
summation congruence method, if n is greater than (2^x)-1 for
x from
twelve to two then Ill try it.
Still, the computer is fast at that binary arithmetic, its
good to
get as much as can be accomplished with its most efcient
atomic
operations. I gure I can go to digits of about twelve bits,
base
b=4096, where the sum would t in a 32-bit signed integer a
minimum
of 524288 bytes worth of integer, 2^4194304. Its probably
fastest for
summing the base 2^8 or 2^16 digits.
I wonder if theres a good way to consider overow, without
actually
doing the work.
After that so far I only have the notion of using trial
division by
primes, testing each prime by division into the number until a
remainder is returned, and then going on to the next prime
until the
number left to factorize is equal to one and the multiset
contains a
copy of each of the prime factors.
To that end I should develop a list of prime numbers. One way
to do
this is to use the trial division sieve on integers from one
towards
innity, and as primes are discovered store them in the list
of
primes, and as composites are factored store their factors in
the list
of composites and their factors. What concerns me about that
is that
it would take some storage. The good thing about it would be
that in
factoring any number, if a composite in a range of numbers
that was
pre-factorized was the result then it would probably take less
computation to load the precomputed result than to
recalculate the
result.
Heres something about perfect squares, obviously it factors
into its
square roots, but one less than a perfect square x^2-1
factors into
x+1and x-1, I guess thats obvious enough, yet its enough of
a
problem to factor a number in the rst place without
determining it a
square. On the other hand, for populating a list of prime
factorizations, where the idea is to get the prime
factorizations for
a given range, that would allow ready factorizations of
perfect
squares.
Fermats method and other more modern methods use the square
root of n
as a jumping-off point for trial division. They get to better
than
%50 on something and bet on it.
The cryptography prime factorization problem is about
factoring a
large composite into exactly two primes that are each nearer
the
square root than one. Most numbers have more than two prime
factors,
a very few only have one.
I cobble away on the digit summation congruence
implementation.
[space:~/math/Stirling] space% java Factorize
17080198121677824
2* 2* 2* 2* 2* 2* 2* 2* 2* 2* 7*3*3* 7*3*3* 7*3*3* 7*3*3*
7*3*3* 7* 7*
7* 7* 7* 1*
[space:~/math/Stirling] space% java Pow 63 60
63^60 =
91292051633079798989750131910067116342455228306074831463666747
880705514289315
2629668193590354000850926342401
[space:~/math/Stirling] space% java Factorize
91292051633079798989750131910067116342455228306074831463666747
88070551428931
52629668193590354000850926342401
7*3*3* 7*3*3* 7*3*3* 7*3*3* 7*3*3* 7*3*3* 7*3*3* 7*3*3*
7*3*3* 7*3*3*
7*3*3*
7*3*3* 7*3*3* 7*3*3* 7*3*3* 7*3*3* 7*3*3* 7*3*3* 7*3*3*
7*3*3* 7*3*3*
7*3*3*
7*3*3* 7*3*3* 7*3*3* 7*3*3* 7*3*3* 7*3*3* 7*3*3* 7*3*3*
7*3*3* 7*3*3*
7*3*3*
7*3*3* 7*3*3* 7*3*3* 7*3*3* 7*3*3* 7*3*3* 7*3*3* 7*3*3*
7*3*3* 7*3*3*
7*3*3*
7*3*3* 7*3*3* 7*3*3* 7*3*3* 7*3*3* 7*3*3* 7*3*3* 7*3*3*
7*3*3* 7*3*3*
7*3*3*
7*3*3* 7*3*3* 7*3*3* 7*3*3* 7*3*3* 1*
Well, it seems to be a good computational method for
determining prime
factors of numbers that are powers of 3, 7, 15, 31, 63, 127,
255, 511,
1023, or products thereof. Its still necessary to divide the
determined factor out of the number to get the quotient to
recursively
factorize.
Im considering something along the lines of binary coded
decimal as I
want a method to determine if 5, 11, or 13 are factors
besides where
3*5, 3*11*31, and 3*3*5*7*13 are factors.
Heres one method to consider, say the number is not a
multiple of 3.
I want to check it for being a multiple of 5. I multiply the
number
by 3 and then check using digit summation congruence whether
it is a
multiple of 15. For eleven is is similar, I use the DSC
method to see
if the number is a multiple of 3 or 31. Where it isnt, it
might
still be a multiple of 11. I multiply the number by 3*31 and
can then
use DSC(31*11*3) on that to see if it is a multiple of 11. It
might
be faster to simply test divide the number by 5 and check the
remainder for zero, but multiplication might be much faster
than
division.
So, after attempting factorization of 15 and 3, I can
multiply the
number to factorize by 3 and then check that for a factor of
15 using
digit summation congruence to see if the original number was a
multiple of 5.
[space:~/math/Stirling] space% java Pow 5 20
5^20 = 95367431640625
[space:~/math/Stirling] space% java Factorize 95367431640625
5* 5* 5* 5* 5* 5* 5* 5* 5* 5* 5* 5* 5* 5* 5* 5* 5* 5* 5* 5* 1*
Then for each resulting factor of the number, its multiplied
by
three, determined that product had a factor of fteen,
divided by
ve, and then repeated. Using plain test division, it would
divide
the number by ve and check the remainder for equality with
zero.
[space:~/math/Stirling] space% java Factorize
100000000000000000000000000
2* 2* 2* 2* 2* 2* 2* 2* 2* 2* 2* 2* 2* 2* 2* 2* 2* 2* 2* 2*
2* 2* 2*
2* 2* 2*
5* 5* 5* 5* 5* 5* 5* 5* 5* 5* 5* 5* 5* 5* 5* 5* 5* 5* 5* 5*
5* 5* 5*
5* 5* 5*
1*
For 11 it is similar. The factors 31 and 3 would have already
been
factorized from the number by DSC(31) and DSC(3), before that
DSC(1023) would have determined a factor of 1023 = 31*11*3.
So where
it is known that the number has no factors of 31 or 3, it
might still
have factors of 11. Thus, the number is multiplied by 93, and
that
product has applied to it the DSC(1023) method, which if
returning
true on the product means that the number is a multiple of
eleven.
This is useful where the cost of performing a multiplication
(by 3 or
93) and the DSC test is less than performing division and
test of the
remainder. Multiplication by three can be accomplished with a
one-bit
leftshift and an add. I think a digit summation congruence is
an
effective method to factor small primes from random integers,
after
all, a third of the integers are multiples of three, and a
fth are
multiples of ve. After that then the contemporary methods
are very
good.
I wonder about tests for factors that could make use of the
relative
numbers of ones and zeros in the base two number, for example
a test
that would be more or less likely to get a factor given a
number with
3/4 ones, nearer in Hamming distance to a Mersenne number,
than 1/2
ones, or 1/4 ones, nearer in Hamming distance to a power of
two.
I havent yet implemented trial division by primes. Partially
thats
because I dont yet have a method to return the nth prime.
Returning to (sum n)^x - sum (n^x) and s(n+1, n-x+1), I begin
to
factorize some of those values. Now, for x=2 the difference
always
has a factor of 2, and the ratio in the limit appears to be
equl to 2.
For x=3, the difference always has factors 2 and 3, and the
ratio in
the limit appears to be equal to 2*3. For x=4, the difference
does
not always have factors of 2, 3, and 2 and 2, yet the ratio
in the
limit does appear to be 24. The rst value of n where the
difference
is a multiple of 24 is n=17.
n=17 %= (2*2*2*3*13*17*79*1307*)/(2*7*3*17*19463*)
n=17 %= (2*2*13*79*1307) / (7*19463)
My prime factorization implementation is very slow.
n=17
%=
39.40925272128067175079454789674180312828003317650340205958558
7304849494645
51786907
I look for patterns in the factorizations.
n=32 %= (2*2*2*2*31*7*5*11*19*21419*)/(2*2*31*11*23*29*2753*)
n=32 %= (2*2*7*5*19*21419)/(23*29*2753)
Here are the values of n that contain a 7 in the
factorization of the
numerator:
4
6, 7, 8
11
13, 14, 15
18
20, 21, 22
25
27, 28, 29
32
34, 35, 36
39
41, 42, 43
46(x2)
...
Here are values of n that contain an 11 in the factorization
of the
numerator:
10, 11, 12
21, 22, 23
32, 33, 34
43, 44, 45
...
For prime factor 13:
4
12, 13, 14
17
25, 26, 27
30
39, 40
43...
For prime factor 17:
13
16, 17, 18
30
33, 34, 35
...
Those patterns would seem to predict some of the factors of
the
difference, but it appears that the simple pattern of groups
of three
in a row does not hold.
For |s(n+1, n+2)|, here are the values of n where 7 is a
prime factor
of |s(n+1, n+2)|:
6, 7, 8, 9, 10
13, 14, 15, 16, 17
20, 21, 22, 23, 24
27, 28, 29, 30, 31
34, 35, 36, 37, 38
41, 42, 43, 44, 45
...
Initially it appears that 7 is a prime factor of |s(n+1,
n-3)| for n =
7z-1, 7z, 7z+1, 7x+2, and 7z+3 for positive integer z. For
prime
factor 11:
10, 11, 12, 13, 14
21, 22, 23, 24, 25
32, 33, 34, 35, 36
43, 44, 45, 46, 47
...
Initially it appears that 11 is a prime factor of |s(n+1,
n-3)| for n
= 11z-1, 11z, 11z+1, 11x+2, and 11z+3 for positive integer z.
For
prime factor 13:
12, 13, 14, 15, 16
25, 26, 27, 28, 29
38, 39, 40, 41, 42
...
Initially it appears that 13 is a prime factor of |s(n+1,
n-3)| for n
= 13z-1, 13z, 13z+1, 13x+2, and 13z+3 for positive integer z.
The same appears to hold for 17. The same appears to hold for
19.
For n=21=1*19+2 and n=40 = 2*19+2 the denominator has two
prime
factors of 19.
The number 7 is the fourth prime, 11 the fth, etcetera. A
short
list of primes from 7 onwards is 7, 11, 13, 17, 19, 23, 29,
31, 37.
The prime factor 29 is a factor of the Stirling cycle number
for n=28,
29, 30, 31, 32, and also for 35. The prime factor 31 is a
factor of
the Stirling cycle number for n=30, 31, 32, 33, 34, and also
for 28.
My factorization method is only thus far to n=47.
Five is not a prime factor of |s(n+1, n-3)| until n=24.
24, 25, 26, 27, 28
...
For prime factor 3:
4,
8, 9, 10, 11, 12, 13
17, 18, 19, 20, 21, 22x2
26x2, 27x2, 28x2, 29x2, 30x2, 31
35, 36, 37, 38, 39, 40
44, 45, 46, 47, ...
Well, I suppose I can conjecture that for prime p greater
than or
equal to 7 that for n=pz-1, pz, pz+1, pz+2, pz+3 that p
divides
|s(n+1, n-3)|, but that does not do much good in determining
the form
of the polynomials to directly calculate |(s(n+1, n-3)| from
(sum n)^4
- sum (n^4).
For x=5, the prime factor 11, the fth prime number, is
present as a
factor in n=11z-1, 11z, 11z+1, 11z+2, 11z+3, and 11z+4, 6
consecutive
Stirling numbers |s(n+1, n-4)|, with two occurrences in 11z-1
and 11z.
Those are the only occurrences of 11 in the prime
factorizations for
n < 45. The same holds true for p=17. For p=19, there are
those
prime factors and then also occurrences in n=9, and an extra
occurrence in each 19z+4.
The prime factorization for x=5 seems to be much faster than
that for
x=4. The largest prime factors for x=5 are smaller than those
for
x=4. The prime factorization for x=5 for a hundred values
completed
rapidly, that for x=4 didnt after hours.
Here are reduced factorizations in ratio for x=5, ((sum n)^5
- sum
(n^5)) / |s(n+1, n-4)|:
n=5 %= (7*5*719)/(2*2)
n=6 %= (5*5*277)/(3)
n=7 %= (2*2*3*3*5*487)/(67)
n=8 %= (2*2*17*5*5*47)/(89)
n=9 %= (2*2*11*887)/(3*19)
n=10 %= (2*5*3*9239)/(7*71)
n=11 %= (2*5*5*13*379)/(3*173)
n=12 %= (2*5*7*3697)/(3*3*3*23)
n=13 %= (2*5*5*5023)/(11*61)
n=14 %= (2*2*2*173*193)/(11*71)
n=15 %= (2*2*17*5*43*1013)/(3*11*13*109)
n=16 %= (2*2*5*3*5*3*1597)/(13*373)
n=17 %= (2*2*5*19*67)/(7*13)
n=18 %= (5*5*2521)/(3*79)
n=19 %= (7*3*108869)/(17*23*23)
n=20 %= (5*11*132929)/(17*3*587)
n=21 %= (5*5*5*23*59*109)/(3*3*3*3*3*17*19)
n=22 %= (2*5*38557)/(19*89)
n=23 %= (2*2*5*5*5*6553)/(19*19*41)
...
n=100 %= (17*5*3*10931807)/(2*7*97*14947)
n=101 %= (5*5*43*103*353*1049)/(2*7*7*3*11*97*953)
n=102 %= (13*47*50333)/(2*2*7*3*3*3*3*3*3*11)
n=103 %= (2*2*7*5*5*3443051)/(11*101*15859)
n=104 %= (2*2*53*89451179)/(17*5*19*23*37*101)
...
So, in assuming that those ratios represent two polynomials
g(n) and
h(n), that are the same for each value n, then I am looking
for
something, for example multiplying the ratio by any x/x to
get a
polynomial g(n) that matches that value for each n, and also
the
polynomial h(n).
I thought I might discover some orthogonal polynomials, but I
dont
know.
The evaluation of g(n) starts much larger than h(n) and as n
grows
g(n)/h(n) goes to approximate 120. This implies that the
univariate
polynomials g(n) and h(n) have the same degree and the
coefcient a_0
of g(n) is 120 and of h(n) is 1, or rather, coefcient g_0 =
120 h_0.
I assume a variety of things, among them are that the
coefcients are
integer, that a pair of polynomials with the required
properties
exists, etcetera. Given a few prospective values of g_0, then
here
are values of h_0.
30 1/4
60 1/2
120 1
180 3/2
240 2
360 3
480 4
600 5
720 6
n=5, % =
25165 / 4
50330 / 8
75495 / 12
100660 / 16
n^3 = 125
120 n^3 = 15000
600 n^3 = 75000
n=6,
n^3 = 216
600 n^3 = 129600
% = 6925/3
131575 / 57
1975
495 19...
1975 54.86
Here I am just trying to determine a polynomial g(n) with its
rst
coefcient g_0 being a multiple of 120, and, also, I dont
know the
order of the rst term. This is for x=5, for x=3 the order of
the
polynomials is 1, it might be the case that it is 2 for x=4,
but that
is not yet determined, and it might be 3 for x=5, but it
could be 4.
n=5
720 * n^3 = 90000
720 n^3 + 420 n^2 = 100500
720 n^3 + 420 n^2 + 30 n = 100660
n=6
720 n^3 + 420 n^2 + 30 n = 155520 + 15120 + 180 = 170820
170820/6925 = 24.66715, bzzt.
n=5
720 n^3 + 400 n^2 + 130 n = 100660
n=6
720 n^3 + 400 n^2 + 130 n = 155520 + 14400 + 780 = 170700
170700/6925 = 24.64982...
n=5
720 n^3 + 360 n^2 + 330 n = 100660
n=6
720 n^3 + 360 n^2 + 330 n = 155520 + 12960 + 1980 = 170460
24.61516...
I guess I can enumerate many possible values of the
coefcients,
assuming that the order of the polynomial is three, for a
rst term
of g_0 n^3, and determine how to get various values multiples
of the
numerator of the ratio, and then see if the same equation
works for
n=6 and even integer multiples of that numerator.
I write a program for x=4 to try and determine a polynomial
g(n) that
is valid for n=4, 5,6. I assume that it is of the form g_0
n^2 + g_1n
+ g_3 where g_0 is a multiple of 24. I get quite a few
polynomials
that work with n=4 and 5, now Im trying for n=6.
I run the program for a reasonable amount of time, it doesnt
get any
results. I try for multiple of 4 and 6 instead of 24, still
no luck,
luck in mathematics is arbitrary, luck in mathematical
pursuit is part
of inspiration.
Maybe I should divide the numerator by n^2, n^3, etcetera, to
see
which one is proportional to the value.
I try to nd a polynomial of rank 4, g_0 n^4 + g_1 n^3 + g_2
n^2 +
g_3 n + g_4. Say, how does one determine the rank and
coefcients of
a polynomial given values of the expression for variables?
I guess it is pretty simple:
(n(n+1)/2)^4 -n(n+1)(2n+1)(3n^2-3n+1)/30
(n^8 +4n^7 +6n^6 +4n^5 +n^4)/8
-(6n^5 + 15n^4 +10n^3 -n)/30
(15n^8 + 60n^7 + 90n^6 +60n^5 +15n^4)/120
-(24n^5 +60n^4 +40n^3 -4n)/120
(15n^8 +60n^7 +90n^6 +36n^5 -45n^4 -40n^3 +4n)/120
Assuming I correctly transformed the algebraic expression,
the above
is the polynomial that is (sum n)^4 - sum(n^4).
Then, I plan to divide that by a ratio of polynomials that
results in
the unsigned Stirling cycle number |s(n+1, n-3)|, the ratio
evaluates
to 4! = 24 in the limit. Im going to just write s(n+1,
n-x+1) for
unsigned numbers. These polynomials exist for s(n+1, n-1) and
s(n+1,
n-2), that is s(n+1, n-x+1) for x=2 and 3, but I dont know
if the
polynomials with real, integer coefcients g(n) and h(n)
exist for
higher values of x, which would allow direct computation of
the
Stirling cycle number as a ratio of polynomials without
summing the
products of x-subsets of an n-set, summing over indexes, or
via the
recurrence relation, except where sum (n^x) is expressed as a
Bernoulli polynomial, the coefcients of which are currently
generated by a recurrence relation.
Its unclear to me whether the polynomials g(n) and h(n)
exist, or
whether there might be polynomials satisfying the expression
with
non-integer coefcients, or not, one of the few reaons that I
would
think they do is as they exist for x=2, 3.
The polynomial ratio g(n)/h(n) for x=2 is 2. The polynomial
ratio
g(n)/h(n) for x=3 is 6(n+2)/(n-2), but the polynomials might
be 3(n+2)
and 1/2 (n-2).
Perhaps it is so then that the rst polynomials for g(n) and
h(n) are
as so:
g_2(n) = 1
g_3(n) = 3 (n+2)
h_2(n) = 1/2
h_3(n) = 1/2 (n-2)
Then Im trying to determine if there are polynomials g_4(n)
and
h_4(n), in the limit g_x(n)/h_x(n) = x!.
I guess I would think something along the lines of
g_4(n) = 6 (n+2)(n+...)
h_4(n) = 1/4 (n-2)(n-...)
or
g_4(n) = 3 (n+2)(2n+1)...
h_4(n) = 1/2 (n-2) 1/2 (n-2)...
Yet, the polynomial ratio g_4(4)/h_4(4) would have to be
equal to
9646/24 = 4823/12. So, the rank of each g_4 and h_4 would
have to be
higher, as they are equal, to get to 9646 as a function of 4.
Oh well, at least s(n+1, n-3) ~ n^8/192, or s(n+1, n-x+1) ~
n^(2x) /
(2^x x!).
[space:~/math/Stirling] space% java Stirling1 200 195
s(200, 195) = -24311576926544407500
[space:~/math/Stirling] space% java Pow 199 10
199^10 = 97393677359695041798001
[space:~/math/Stirling] space% java Multiply 32 120
32 * 120 = 3840
[space:~/math/Stirling] space% java Divide
97393677359695041798001
3840
97393677359695041798001 / 3840 = 25362936812420583801
They have the same number of digits, its off by about 5%.
[space:~/math/Stirling] space% java ApproxStirling1 199 5
|s(199+1, 199-5+1)| ~ 25362936812420583801
|s(199+1, 199-5+1)| = 24311576926544407500
[space:~/math/Stirling] space% java ApproxStirling1 1000 10
|s(1000+1, 1000-10+1)| ~
269114445546737213403880070546737213403880070546737
|s(1000+1, 1000-10+1)| =
255944587934420402547409683097079125714353332944500
Not very accurate, but closer than n^(2x+1) / (2^x x!). Its
worse
for higher values of x. There are easy modications to make
it more
accurate.
Back to g_x(n) and h_x(n), Im still trying to gure out
g_4(n) and
h_4(n), if they exist, and of course generally g_x(n) and
h_x(n).
Do you know any polynomials in these forms?
g_2(n) = 1
g_3(n) = 3 (n+2)
h_2(n) = 1/2
h_3(n) = 1/2 (n-2)
If so, what are g_x(n) and h_x(n)?
How do I t a polynomial to known data values? Please dont
suggest
polynomial regression.
Have a nice day,
Ross
===
Subject: Mechanics question
Hi group, Im a high school student attempting to
realistically model the
motion of a ball sliding down a helical ramp with a side wall.
First Ill present my current understanding of the concept,
please correct
any mistakes here as Im still learning. In the absence of
friction, the
tangential and normal components of acceleration are given
respectively by
(1) a_T = g sintheta T
(2) a_N = frac{v^2 cos^2theta}{rho} N
= frac{g^2sin^2theta t^2cos^2theta}{rho} N
where (T, N, B) is the righthanded orthonormal trihedral
associated with
the
helix and rho the radius of curvature.
If however, the ball rolls without slipping, then the ball is
subject to
two
frictional forces directed antiparallel to T, which arise due
to the static
friction between the ball and the planes spanned by TN and
BT. The
frictional forces in these planes would seem to give rise to
two components
of angular acceleration in the directions of N and B
respectively, with the
direction of rotation determined by the right-hand rule (a
diagram may help
here).
According to one reference [1], a ball rolling down a simple
inclined plane
will feel a frictional force directed antiparallel to its
linear velocity
given by
(3) f = frac{Ialpha}{r}
where I is the moment of inertia of a uniform shperical
object of radius r.
Since the ball in the helix rotates simultaneously about N
and B with the
same angular acceleration, the resultant angular acceleration
is in the
plane spanned by BN and of magnitude
(4) |alpha| = sqrt{2}|alpha_N| = sqrt{2}frac{|a_T|}{r}
Am I thus justied in substituting the above expression into
Eq. (3) to
obtain the resultant frictional force?
James
[1] http://theory.uwinnipeg.ca/physics/rot/node9.html
===
Subject: Re: Collatz (3x+1) as diophantine problem
> Hi -
> Im just reading these 3x+1-threads and after ddling a bit
> on older ideas, I stuck with a - possibly not difcult -
idea
> for a partial proof.
> It depends on a certain class of equations, and it is the
> question, whether there is another x (besides x=1), which
> solves the actual equation in nonnegative integers
> (x,a,b,c,...)
> It is simple to show that there is always the solution
> x = 1 with a=b=c=...=2, and never a solution for instance
> with a=b=c=...=0
> 1
> 1) x = -------
> 2^a -3^1
> 2^b + 3^1
> 2) x = ------------
> 2^(b+a) -3^2
> 2^(c+b) + 2^c*3^1 + 3^2
> 3) x = -----------------------
> 2^(c+b+a) -3^3
> 2^(d+c+b) + 2^(d+c)*3^1 + 2^d*3^2 + 3^3
> 4) x = ----------------------------------------
> 2^(d+c+b+a) - 3^4
> If I can prove, that - except the above indicated solutions
-
> there exist no others, then there is no loop except the
root-
> loop 1-2-4-1 caused by the collatz (3x+1)-transformation.
> It looks simple in a rst view, since the different powers
of 3
> make seemingly the nominator and denominator incompatible
for
> generating integers in x, I remember weakly something like
the
> Eisenstein-criterion (but which I cannot apply rmly just
now).
> Examples:
> 1) solves simply:
> the only natural number x, which can be result of the
equation
> is 1, since
> 1 1
> x = --------- = --------------
> 2^a -3^1 (2^a -4) + 1
> and for any a<>2 result a negative or positiv fraction.
> Conclusion: there is no one-step-loop except (1)-(4-2-1)-...
> (a loop of x1 = (3 x0 + 1) / 2^a )
> 2) is already more complicated, when tried by simply
reformulating:
> 2^b + 3^1 2^b-4 + 7
> 2) x = ------------ = --------------
> 2^(b+a) -3^2 2^(b+a)-16 + 7
> indicates at least the standard solution for a=b=2 giving
x=1 again,
> but to disprove other solutions I currently have to check
min/max-
> conditions for a and b (not too difcult) and to check each
discrete
> combination.
> Conclusion: there is no two-step-loop except
(1)-(4-2-1)-(4-2-1)-...
> (a loop of x1 = (3 x0 + 1) / 2^a )
>
> 3) similar, but not done.
> Instead Im ddling with a method to make an induction from
> equation type n to equation type n+1.
> But - after some hours of sketching - it looks too much
complicated,
> if there could be a more simple proven criterion, which is
applicable
> here ...
> Gottfried
Cant offer constructive feedback; Im sorry. But I nd it
interesting to study.
The Collatz problem is like an Elepahnt and we are innite
blind
men.
We can never be sure how many of the same Elephants we have
explored
if we were alert I allways think. What that has to to do with
anything well? I dono.
Still a reply was in order.
Ernst
===
Subject: Re: Collatz (3x+1) as diophantine problem
> Cant offer constructive feedback; Im sorry. But I nd it
> interesting to study.
Hi Ernst -
short explanation.
One transformation of Collatz, starting at an odd integer x0
and going to the next odd integer x1 can be written as

3 x0 + 1
x1 = --------
2^a
where a is the highest exponent, keeping x1 integer.
If you repeat the operation, you always need a new exponent
for the denominator.

If you go 2 transformations you get
2 1 a
3 (x1) + 1 3 x0 + 3 + 2
x2 = -------------- = -----------------------
2^b 2^(a+b)
and so on with more transformations and corresponding many
variables
a,b,c,...
(That made me use the notation of C for an arbitrary
x0;a,b,c,d,... z
Collatz-number)
If a series of transformations, let say k, generates a loop,
that means,
that
xk = x0,
or
C = C
x0;a,b,c,0 x0,a,b,c,a,b,c,0
and for the simple case of the above equation with two
transformations
we can ask:
assuming x2=x0 (being the same positive integer)
is there any possible solution for integer a,b ?
(Among some restrictions there is one special useful, that
all a,b,c must
be higher than zero, otherwise we wont get proper
Collatz-numbers)
Testing that for the question of existence of a 2-loop:
-------------------------------------------------------
So, if we say x2 = x0 = x , we can rearrange the above
equation for x
2 1 a
3 x + 3 + 2
x = -----------------------
a+b
2
a+b 2 1 a
x * 2 = 3 x + 3 + 2
a 1
2 + 3
x = ---------------
a+b 2
2 - 3

The same can be generated for a 3-length or a general
k-length loop,
we only have then 3 variables (a,b,c) or n variables
(a,b,c,...,z) (n
variables)
involved.
For n variables ( a,b,c,... z) this equation looks like
n-1 n-2 z n-3 z+y 1 z+y+...+c z+...c+b
3 + 3 2 + 3 2 + ... + 3 2 + 2
x =
-------------------------------------------------------------
--
z+...+c+b+a n
2 - 3
and it must give an integer solution as x for integers
a,b,c...z if
there were any loop of length n in the natural numbers under
collatz-
transformation
Example, how to apply this using a known solution.

For instance, if all variables a..z are equal (a=b=c=d
...=z), then
this equation simplies remarkably to

n-1 n-2 z1 n-3 2z 1 z(n-2) z(n-1)
3 + 3 2 + 3 2 + ... + 3 2 + 2
x =
--------------------------------------------------------------
z*n n
2 - 3
and the denominator is nothing else than (2^z -3)* nominator
so we have
nominator 1
x = --------------------- = ----------
(2^z - 3) * nominator 2^z - 3
where we can conclude both remaining parameters:
x cannot be greater than 1, but must be a positive integer,
so it must
be 1
thus 2^z-3 must be 1, so z must be 2.
Conclusion:
-----------
If we select all variables equal, then we can produce loops
of arbitray length, if all variables = 2, and x is always 1.
That reects the recursion of (1-2-4)-1... which is a 1-step
loop in this transformation, or - in my
Collatz-number-notation
C = x --> C = C = C = C
x ;a,b,c,...,0 1;0 1;2,0 1;2,2,0 1;2,2,2...,2,0
> The Collatz problem is like an Elepahnt and we are innite
blind
> men.
Yes, for the moment I found, that it is equivalent to the
question,
whether an arbitrary digit-string in a numbersystem to a
certain
base (in collatz to the base of some derivate of 2/3) can be
an integer
factor to a repunit of the same length (in collatz we have
some
constraints on the digits, additionally),
For instance, for a given base, can the number, described by
the
digit-string
A B C D E F = x* 1 1 1 1 1 1 where x is an integer and
A,B,C are not bounded to be in [0.. base[
But this seems to be too high for me, being blind in countable
innity...
> We can never be sure how many of the same Elephants we have
explored
> if we were alert I allways think. What that has to to do
with
> anything well? I dono.
> Still a reply was in order.
Not bad ;-)
Gottfried
--
Gottfried Helms
Univ Kassel
===
Subject: Re: without mirror
Uytkownik Robert Israel <israel@math.ubc.ca>
napisa w wiadomo.a6ci
>There is a circuit and two points (A and B lying outside the
circuit)
given.
>How to nd the point (X) on the circuit so that the
distance: |AX|+|XB|
is
>minimal.
> Make the circuit into a mirror. Put a light source at A,
stand at B and
> look for the reection of the light...
but how to construct that point without using a mirror.You
can also presume
that A is (x1,y1) B(x2,y2) and the circle x^2+y^2=1
> Robert Israel israel@math.ubc.ca
> Department of Mathematics http://www.math.ubc.ca/~israel
> University of British Columbia
> Vancouver, BC, Canada V6T 1Z2
===
Subject: Re: without mirror
>There is a circuit and two points (A and B lying outside the
circuit)
>given.
>>How to nd the point (X) on the circuit so that the
distance:
|AX|+|XB|
>
>>minimal.
>
> Make the circuit into a mirror. Put a light source at A,
stand at B
and
> look for the reection of the light...
>>
>but how to construct that point without using a mirror.You
can also
presume
>that A is (x1,y1) B(x2,y2) and the circle x^2+y^2=1
This is a straightforward constrained minimization problem in
two
variables. Use a Lagrange multiplier.
--
Stephen J. Herschkorn herschko@rutcor.rutgers.edu
===
Subject: Re: A Trigonometric Challenge?
> So as {cos(n)| n in N} is dense it isnt well-ordered WITH
RESPECT TO
<,
> but N is, so you cannot match the two. So Y doesnt exist.
Changing to
> the set {cos(n+1-Pi) | n in N} doesnt change this
Absolutely. So the core of proving that Y does not exist
resides in
proving
that the cosines of the naturals constitutes a dense set. As
a challenge,
I
wanted to come up with my own original proof of this and have
done so over
these past couple of days. (Ive never seen any proofs of
this, nor did I
want to, but others may have thought along these lines.) My
proof is a
monster and I will be happy to post it if anybody wants to
see it. Its
still in the process of being nalized over the next day or
two.
Randy
http://www.rlgerl.com
===
Subject: The Unied Field
When y(x,t) is the transverse displacement of a vibrating
string,
y(x,t) can be determined by the wave equation
[@^2y/@t^2] = v^2[ @^2 y/@x^2] , where @ denotes the partial
derivative symbol.
Standing waves can be set up in an 1-dimensional string,
analogous to
that in a violin string. The form of the standing wave
becomes y(x,t)
= psi(x) sin (wt)
Two sinusoidal travelling waves with the same amplitude and
wavelength
moving in opposite directions on a string, become resonating
standing
waves:
y(x,t) = y1(x,t) + y2(x,t) = Asin(kx - wt) + Asin(kx + wt) =
[2Asin(kx)]*cos(wt).
As the entropy continues to increase in the universe, and if
the
universe is a closed system, the entropy may be considered to
be the
result of a damping force. This damping force may also be one
possible solution to the dark matter enigma.
Solve the Schwarzschild solution for the entire universe,
since the
universe can be postulated to be a closed system with nothing
outside
itself. The condition of nothingness leads one to ask What
are the
properties of nothingness? Of course there are no measurable
properties, but nothingness in itself must be a type of
massless
solid. A condition that has no distance - metric scales. In
other
words, there is no outside to the universe, no measurable
border
between something and nothing.
Nothing then becomes analogous to a perfectly symmetrical
pressure
force on the surface of existence.
-(F)^2 ---->|U|<---- +(F)^2
Simple harmonic oscillation given by the equation (F)^2 =
-(K*X)^2
What is K ? What is X ?
U stands for universe. So it becomes reasonable to assume
that the
entire universe is analogous to that which is inside the
event horizon
of a black hole. The cosmos becomes a quantum superposition
of states,
collapsing under the crushing force of nothingness.
Analytically continue the Schwarzschild solution to the
imaginary
values of the time variable. The Schwarzschild solution
becomes
periodic in the imaginary time direction.
All waves would then be standing waves in the closed
universe. A
Schrodinger wave equation in one dimension is of the form:
d^2 psi/dx^2 + (2m/hbar^2) [E - U(x)] psi(x)
U(x) is the potential energy and E is the total energy.
psi(x) is the wave function for a state in which the energy E
is
constant in time. Such states are called stationary states.
Certain
denite vibration frequencies are allowed multiples of
fundamental
wavelengths
lambda = h/p
certain state between the region x and x+dx
psi^2 = psi psi* . When psi is complex, psi* is the complex
conjugate
of psi. psi^2 (x) is the probability density.
An equation for the damped oscillator in one dimension:
X = A[exp[-(b/2m)t]]*cos[wt + theta]
Why not describe Einsteins equation as a rule that tells the
geometry
of space how to evolve as function of time? Lorentzian
manifolds M,
diffeomorphic to R x S, where the manifold S represents
space, and t,
an element of R, represents time. So spacetime is sliced into
instants
of time as an arbitrary choice, or possibly boundary limits,
imposed
by Plancks constant.
F: M---> R x S
Spacetime becomes quantized or sliced up but that could be
what
nature really does. According to relativity, an objects
position and
momentum can only be dened with respect to a frame of
reference,
i.e. another object. Yet the universe as a whole has no frame
of
reference outside of itself, so how can its momentum be
dened? It
can only be dened with reference to itself. Worldlines ll up
spacetime and the criss crossing of world lines mark events
beyond the
need for coordinate systems or coordinates. Points in
spacetime are
given the name events so there is a coordinate independence.
The geometric view of physics means that the laws of physics
are the
same in every Lorentz reference system. Local Lorentz
invariance. But
since the universe has no exterior reference frame, and it
must refer
to itself, its world line intersects with itself. This
quantized-evolution of spacetime dictated by GR and QM, means
that the
world line of the past intersects with the world lines of the
present,
for the universe. A geometric stacking of space like slices,
parameterized by t, The universe is a function of itself.
Spacetime
becomes compressed. As the time evolution proceeds in the
thermodynamic direction of t, the space like sheets
continually
increase in density. The information storage of space time.
(->(->(->(U)<-)<-)<-)
This increasing refractive spacetime density must be
background
independent. The increasing density functions are, in a sense,
equivalent to the non-Euclidean geometry of Riemann and
Einstein.
Russell E. Rierson
analog57@yahoo.com
===
Subject: Re: The Unied Field
> When y(x,t) is the transverse displacement of a vibrating
string,
> y(x,t) can be determined by the wave equation
> [@^2y/@t^2] = v^2[ @^2 y/@x^2] , where @ denotes the partial
> derivative symbol.
> Standing waves can be set up in an 1-dimensional string,
analogous to
> that in a violin string. The form of the standing wave
becomes y(x,t)
> = psi(x) sin (wt)
> Two sinusoidal travelling waves with the same amplitude and
wavelength
> moving in opposite directions on a string, become
resonating standing
> waves:
> y(x,t) = y1(x,t) + y2(x,t) = Asin(kx - wt) + Asin(kx + wt) =
> [2Asin(kx)]*cos(wt).
> As the entropy continues to increase in the universe, and
if the
> universe is a closed system, the entropy may be considered
to be the
> result of a damping force. This damping force may also be
one
> possible solution to the dark matter enigma.
It cant, since the assumption of the idiotic dark matter
itself violates all the assumptions of entropy.
The Dark Matter enigma has already been solved.
Its simply a Relativists excuse for having a weak mind,
a weak force, a weak computer, an unemployed Chemist as a
lab assistant, a Democratic National Convention,
and an inquiring mind that even the National Enquirer
and Oprah Winfrey would be proud of.
===
Subject: CANT Order Reals
another for my numerology list,
Cantor Cant Order Reals
Peano unit 1 pen*s
Godel the unprovable
name one billionaire?
what did Lady Di do?
Tiger :: golf :: ?
star wars program was introduced by which president ?
Nic Cage stars in what kind of movies ?
Who is the smartest man, Haw.... ?
proof of numerology at www.adamskingdom.com
Herc
sigh nearly 2 years of proof and not one believer
===
Subject: Re: CANT Order Reals
> proof of numerology at www.adamskingdom.com
> Herc
> sigh nearly 2 years of proof and not one believer
--
The second greatest error in reasoning is mistaking evidence
for proof.
The greatest error is mistaking testimony for evidence.
--
Correlation alone does not prove causality. Nor does
coincidence prove
correlation.
--
http://www.crbond.com
===
Subject: Re: CANT Order Reals
>sigh nearly 2 years of proof and not one believer
Probably because your post is incoherent, as usual.
===
Subject: Re: CANT Order Reals
> proof of numerology at www.adamskingdom.com
> Herc
> sigh nearly 2 years of proof and not one believer
Im hurt.
I already told you that the palindromic symmetries associated
with
sphere packings made me receptive to your claims. On the
other hand,
the fact that I do not have your awareness with regard to the
matter
reects the complexity of encodings associated with the Leech
lattice.
Unconditional belief is not my style.
:-)
mitch
===
Subject: Re: CANT Order Reals
> proof of numerology at www.adamskingdom.com
> Herc
> sigh nearly 2 years of proof and not one believer
> Im hurt.
> I already told you that the palindromic symmetries
associated with
> sphere packings made me receptive to your claims. On the
other hand,
> the fact that I do not have your awareness with regard to
the matter
> reects the complexity of encodings associated with the
Leech lattice.
Youre into 4s and 6s,
Im into 7s and 10s
Itd never work :)
> Unconditional belief is not my style.
if thats what you believe
Herc
===
Subject: Re: CANT Order Reals
: I already told you that the palindromic symmetries
associated with
: sphere packings made me receptive to your claims. On the
other hand,
: the fact that I do not have your awareness with regard to
the matter
: reects the complexity of encodings associated with the
Leech lattice.
May I steal this for a .sig?
--
---
Its difcult ... you need to be united to have any
strength, but internal issues have to be addressed.
--- E. Ray Lewis, on liberalism in America
===
Subject: Re: CANT Order Reals
> : I already told you that the palindromic symmetries
associated with
> : sphere packings made me receptive to your claims. On the
other hand,
> : the fact that I do not have your awareness with regard to
the matter
> : reects the complexity of encodings associated with the
Leech
lattice.
> May I steal this for a .sig?
lol
:-)
mitch
===
Subject: Re: CANT Order Reals
> another for my numerology list,
> Cantor Cant Order Reals
You can order the reals; you just cant count them.
Hope this helps.
> Peano unit 1 pen*s
> Godel the unprovable
> name one billionaire?
> what did Lady Di do?
> Tiger :: golf :: ?
> star wars program was introduced by which president ?
> Nic Cage stars in what kind of movies ?
> Who is the smartest man, Haw.... ?
> proof of numerology at www.adamskingdom.com
> Herc
> sigh nearly 2 years of proof and not one believer
Then again, maybe it wont.
===
Subject: Re: CANT Order Reals
> another for my numerology list,
> Cantor Cant Order Reals
> You can order the reals; you just cant count them.
how can you order something you cant count?
> Hope this helps.
> Peano unit 1 pen*s
> Godel the unprovable
> name one billionaire?
> what did Lady Di do?
> Tiger :: golf :: ?
> star wars program was introduced by which president ?
> Nic Cage stars in what kind of movies ?
> Who is the smartest man, Haw.... ?
> proof of numerology at www.adamskingdom.com
> Herc
> sigh nearly 2 years of proof and not one believer
how can you order something you cant count?
Herc
===
Subject: Re: CANT Order Reals
>>You can order the reals; you just cant count them.
> how can you order something you cant count?
Given any two distinct reals, you can determine which is the
greater,
and in a way that is consistent with arithmetic operations.
This isnt
a trivial property - you cant do it with complex numbers.
You dont need to be able to enumerate the set of reals as a
sequence
to do this.
===
Subject: Re: CANT Order Reals
In sci.math, spakka
<usenet_spam@m16.demon.co.uk>
<9nC8b.9093$cw2.79259778@news-text.cableinet.net>:
>You can order the reals; you just cant count them.
>> how can you order something you cant count?
> Given any two distinct reals, you can determine which is
the greater,
> and in a way that is consistent with arithmetic operations.
This isnt
> a trivial property - you cant do it with complex numbers.
> You dont need to be able to enumerate the set of reals as
a sequence
> to do this.
Also, the ordering satises the usual properties:
[1] Irreexibility: For any a, a is not < a and a is not > a.
[2] Anti-commutativity: For any a and b, a < b implies b > a,
and
a > b implies b < a.
[3] Transitivity: For any a, b, and c, a > b and b > c
implies a > c.
For reals, there are additional issues:
[4] For any a and b, a > b, there are an uncountable number
of cs
such that a > c > b. (Namely, c = a + (b-a)r, 0 < r < 1.
Im not sure what replaced Cantors diagonalization proof for
the uncountability of the interval (0,1), but Cantors proof
worked to a certain extent.
[5] For any a and b, a > b, there exist a and b, a > a >
b > b,
such that a and b have nite decimal representations.
(This follows trivially from [4] and the fact that
lim(N->+oo) 10^(-N) = 0.) The converse is *not* true but
its usually not a problem. :-) (I suppose one could extend
this to two innite approximation series or something. The
main issue is at the limit; the difference of corresponding
terms must be greater than 0, and further the limit of the
difference must be greater than 0, otherwise ones only found
two series for one number.)
--
#191, ewill3@earthlink.net
Its still legal to go .sigless.
===
Subject: Re: CANT Order Reals
> In sci.math, spakka
> <usenet_spam@m16.demon.co.uk
<9nC8b.9093$cw2.79259778@news-text.cableinet.net>:
>You can order the reals; you just cant count them.
>> how can you order something you cant count?
> Given any two distinct reals, you can determine which is
the greater,
> and in a way that is consistent with arithmetic operations.
This isnt
> a trivial property - you cant do it with complex numbers.
> You dont need to be able to enumerate the set of reals as
a sequence
> to do this.
in mathematics the terms order and countable are distinct but
that
is a specialisation of the term order, both synonyms for list.
You can order reals (nite subset), you cant order the reals.
> Also, the ordering satises the usual properties:
> [1] Irreexibility: For any a, a is not < a and a is not >
a.
> [2] Anti-commutativity: For any a and b, a < b implies b >
a, and
> a > b implies b < a.
> [3] Transitivity: For any a, b, and c, a > b and b > c
implies a > c.
> For reals, there are additional issues:
> [4] For any a and b, a > b, there are an uncountable number
of cs
> such that a > c > b. (Namely, c = a + (b-a)r, 0 < r < 1.
> Im not sure what replaced Cantors diagonalization proof
for
> the uncountability of the interval (0,1), but Cantors proof
> worked to a certain extent.
> [5] For any a and b, a > b, there exist a and b, a > a >
b > b,
> such that a and b have nite decimal representations.
> (This follows trivially from [4] and the fact that
> lim(N->+oo) 10^(-N) = 0.) The converse is *not* true but
> its usually not a problem. :-) (I suppose one could extend
> this to two innite approximation series or something. The
> main issue is at the limit; the difference of corresponding
> terms must be greater than 0, and further the limit of the
> difference must be greater than 0, otherwise ones only
found
> two series for one number.)
> --
> #191, ewill3@earthlink.net
> Its still legal to go .sigless.
thats a partial order, so is my family tree
partial order C total order C countable order
<!--StartFragment-->-----------------------------------------
---------------
-----------------------
-
Randi will test you when you properly apply to be tested.
Sign up here:
http://www.randi.org/research/challenge.html
-----------------
Rich Shewmaker
CNote
Wanda
Rust
-------------------------------------------------------------
---------------
----
It really all depends on the situation.
-----------------
Shanx
See You In Hell My Friend.
Someone
Greg Neill
-------------------------------------------------------------
---------------
----
If ever I actually found myself in that situation, Id hold
it upright,
with the intent of attacking my assailants knife hand.
-----------------
cliff86
Rust
Shanx
NormDePloom
-------------------------------------------------------------
---------------
----
These 3 posts are just ordinary replies to me all on the same
day!
But they all have a phenomenal charateristic, you can tell
who the poster
is, have a go!
This is just an indicator that I have a supernatural power,
one of the 4
names
Randi will test you when you properly apply to be tested.
Sign up
here:
http://www.randi.org/research/challenge.html
-----------------
Rich Shewmaker
CNote
Wanda
Rust
check the answer for yourself
waii.net&lr=&hl=en
Herc
===
Subject: Re: CANT Order Reals
> in mathematics the terms order and countable are distinct
but that
> is a specialisation of the term order, both synonyms for
list.
No, thats not true. You can have uncountable ordered sets.
===
Subject: Re: CANT Order Reals
> In sci.math, spakka
> <usenet_spam@m16.demon.co.uk
<9nC8b.9093$cw2.79259778@news-text.cableinet.net>:
>>You can order the reals; you just cant count them.
>how can you order something you cant count?
>>Given any two distinct reals, you can determine which is
the greater,
>>and in a way that is consistent with arithmetic operations.
This isnt
>>a trivial property - you cant do it with complex numbers.
>>You dont need to be able to enumerate the set of reals as
a sequence
>>to do this.
> Also, the ordering satises the usual properties:
> [1] Irreexibility: For any a, a is not < a and a is not >
a.
> [2] Anti-commutativity: For any a and b, a < b implies b >
a, and
> a > b implies b < a.
> [3] Transitivity: For any a, b, and c, a > b and b > c
implies a > c.
Hmm. Id express this differently, so as not to introduce
both symbols
< and >. Also, for the reals we get the useful property
that for
any a, b one of a < b, a = b , b < a is true - i.e. the order
is total.
> For reals, there are additional issues:
> [4] For any a and b, a > b, there are an uncountable number
of cs
> such that a > c > b. (Namely, c = a + (b-a)r, 0 < r < 1.
> Im not sure what replaced Cantors diagonalization proof
for
> the uncountability of the interval (0,1), but Cantors proof
> worked to a certain extent.
What do you think was the problem with it? Proofs dont tend
to work
to a certain extent.
> [5] For any a and b, a > b, there exist a and b, a > a >
b > b,
> such that a and b have nite decimal representations.
> (This follows trivially from [4] and the fact that
> lim(N->+oo) 10^(-N) = 0.) The converse is *not* true but
> its usually not a problem. :-)
What do you mean by the converse here?
> (I suppose one could extend
> this to two innite approximation series or something. The
> main issue is at the limit; the difference of corresponding
> terms must be greater than 0, and further the limit of the
> difference must be greater than 0, otherwise ones only
found
> two series for one number.)
Im not sure any of this helps the OP. Schizophrenic
Numerology was
never my strong subject.
===
Subject: Re: Question - Special properties of 4-D
> Im trying to compile a list of mathematical properties,
objects and
> relationships which are peculiar to a 4-dimensional space
(of course,
> the same would be of interest for a general dimension, n).
For e.g.,
> are there certain theorems that are easy to prove for all
dimensions
> but n=4? Or a certain type of nontrivial mathematical
object whose
> denition constrains its existence to 4-D only?
One thinks of the regular 4-dimensional object: the 24-cell.
It has no
analog in dimensions above 4. The closest analog in 3
dimensions is the
rhombic dodecahedron, but that is not regular.
--
Clive Tooth
http://www.clivetooth.dk
===
Subject: Re: Question - Special properties of 4-D
NNTP-Posting-User: [AReYkpoZ/SAw59BdZIp5eby2uHeOp+gW]
> Im trying to compile a list of mathematical properties,
objects and
> relationships which are peculiar to a 4-dimensional space
(of course,
> the same would be of interest for a general dimension, n).
For e.g.,
> are there certain theorems that are easy to prove for all
dimensions
> but n=4? Or a certain type of nontrivial mathematical
object whose
> denition constrains its existence to 4-D only?
Fake R^4 springs immediately to mind. It is known that for n
<> 4 (of
course n is a nonnegative integer), all smooth structures on
R^n are
diffeomorphic to one another. On the other hand, there exist
*uncountably
many* non-diffeomorphic smooth structures on R^4. (These are
called fake
R^4s.) Im sorry but I am unable to provide a citation just
now; all my
books are at the ofce.
> -riskbert
-david
===
Subject: Re: Question - Special properties of 4-D
> Im trying to compile a list of mathematical properties,
objects and
> relationships which are peculiar to a 4-dimensional space
(of course,
> the same would be of interest for a general dimension, n).
For e.g.,
> are there certain theorems that are easy to prove for all
dimensions
> but n=4? Or a certain type of nontrivial mathematical
object whose
> denition constrains its existence to 4-D only?
> Fake R^4 springs immediately to mind. It is known that for
n <> 4 (of
> course n is a nonnegative integer), all smooth structures
on R^n are
> diffeomorphic to one another. On the other hand, there
exist *uncountably
> many* non-diffeomorphic smooth structures on R^4. (These
are called
fake
> R^4s.) Im sorry but I am unable to provide a citation
just now; all
my
> books are at the ofce.
Oh, I thought the margin was too small.
--
Giuseppe Oblomov Bilotta
Cant you see
It all makes perfect sense
Expressed in dollar and cents
Pounds shillings and pence
(Roger Waters)
===
Subject: Re: Question - Special properties of 4-D
Ive seen it argued that a universe that supports life has to
be 4-D (3+1).
One reason being that you do not get stable orbits in more
than 3 space
dimensions. But thats physics I guess.
> Im trying to compile a list of mathematical properties,
objects and
> relationships which are peculiar to a 4-dimensional space
(of course,
> the same would be of interest for a general dimension, n).
For e.g.,
> are there certain theorems that are easy to prove for all
dimensions
> but n=4? Or a certain type of nontrivial mathematical
object whose
> denition constrains its existence to 4-D only?
> -riskbert
===
Subject: Re: Basic Calculus Question
===
>Subject: Re: Basic calculus questions
>> (1) (dy/dx)= 3x
>> (2) dy = 3xdx (the same as dy = 3x*dx ?)
>> What steps are needed to justify going from line (1) to
line (2)?
>> What does line (1) mean, and what does line (2) mean?
>Line 1 means that if y = f(x), then f(x) = 3x. Line 2 is
pretty much
>meaningless.
Come on now, line 2 isnt meaningless. Its what you
integrate both
sides of to get y(x). It means that if I change x a little
bit the
change in y is approximately proportional to the change in
x... with
the proportionality being (dy/dx).
(Small change in y) = (dy/dx)*(small change in x)
+ O(change in x squared)
adam
===
Subject: Re: Basic Calculus Question
===
>Subject: Re: Basic calculus questions
>> (1) (dy/dx)= 3x
>> (2) dy = 3xdx (the same as dy = 3x*dx ?)
>> What steps are needed to justify going from line (1) to
line (2)?
>> What does line (1) mean, and what does line (2) mean?
>Line 1 means that if y = f(x), then f(x) = 3x. Line 2 is
pretty much
>meaningless.
> Come on now, line 2 isnt meaningless. Its what you
integrate both
> sides of to get y(x). It means that if I change x a little
bit the
> change in y is approximately proportional to the change in
x... with
> the proportionality being (dy/dx).
> (Small change in y) = (dy/dx)*(small change in x)
> + O(change in x squared)
> adam
Of course, its meaningful.
It seemed to be the original posters intent to ask why its
Consistent
with
the rest of calculus.
Several posters have answered this with varying degrees of
rigor,but thats
OK
Bob Pease
===
Subject: Re: comparing two lists
> M is from N, so ignore N. I would load M into a map that
keeps track of
> the frequency, then use an iterator over the map to count
the number of
> distinct elements.
You can just insert each element of M into a map if and
only if it
isnt already there. Then just keep count of how many
insertions you made.
The map can be implemented as a letter tree, hash, btree, or
any of various
other ways.
DS
===
Subject: Re: A simple combinatorics problem?
I found a good tutorial on the bin packing problem:
http://www.york.cuny.edu/~malk/tidbits/tidbit-bin-packing.html
<I re-discovered the bin packing problem. :-) >
-- Danny
> Suppose I have n objects of different weights ( each < = W
kilograms)
> and m (> = n) bags such that each bag can hold at most W
kilograms.
Question: Is there an efcient algorithm to nd a way to put
> the n objects into these m bags that would minimize the
number
> of bags used? (well, ... besides exhaust search)
> This is original and good:
> (1) Put weights exceeding W/2 in separate bags.
> (2) Repeatedly scan through weights. Place weight w in bag
containing
> weight W - w if one or more of the following three
conditions holds.
> (a) there is no remaining weight <= w/2
> (b) the total weight of all the remaining objects that are
lighter
> than w does not exceed w
> (c) w = 2.
> Repeat (2) until stagnation. To save computation, completed
bags can
> be binned.
> (3) If any weights remain, place heaviest remaining weight
into
> fullest bag possible (starting a new bag if necessary). Go
to step
> (2).
> (4) Solution certainly optimal if T > W*(B-1) or B = L
where T is the
> total of all weights, B is the number of bags used and L is
the number
> of heavy weights (weights strictly exceeding W/2). If
neither is
> satised, further analysis required.
===
Subject: Death Rattle Of Neoclassical Theory Of Value (was
Re: Is supply and
demand fundamentally awed?)
Supply and demand do not seem capable of explaining
equilibrium
prices.
-- Robert Vienneau
<http://csf.colorado.edu/pkt/pktauthors/Vienneau.Robert/
Sraffa3.pdf>
p 21.
I do not nd any elementary errors...
Supply and demand is a fundamentally awed theory. (Mark
Witte has
agreed on another thread - Death Rattle Of Neoclassical
Theory Of
Value.) But it takes lots of study to nd out how supply and
demand
is a awed theory.
I was wondering what Ari Fleischer was up to these days!
--
Try
http://csf.colorado.edu/pkt/pktauthors/Vienneau.Robert/
Bukharin.html
To solve Linear Programs: .../LPSolver.html
r c A game: .../Keynes.html
v s a Whether strength of body or of mind, or wisdom, or
i m p virtue, are found in proportion to the power or
wealth
e a e of a man is a question t perhaps to be discussed
by
n e . slaves in the hearing of their masters, but highly
@ r c m unbecoming to reasonable and free men in search of
d o the truth. -- Rousseau
===
Subject: Re: Death Rattle Of Neoclassical Theory Of Value
> The neoclassical theory of value and distribution is
mistaken. I have
> updated my demonstration of this proposition at:
<http://csf.colorado.edu/pkt/pktauthors/Vienneau.Robert/
Sraffa3.pdf>
Heres the sort of comments I had for the previous version:
I got Sraffa3 to print beautifully! Now to nd the time to
work
> on it.
> -- Mark Patrick Witte, 4 December 1998
> [ Long winded nothing - deleted. ]
> I do not nd any elementary errors that
> should stop the reader in his or her tracks.
> [ Long winded nothing - deleted. ]
> This paper visits the currently sleep area of value theory
> [ Long winded nothing - deleted. ]
Perhaps Mr. Witte is correct. Mainstream economists do not
currently
have any theory of how many and for how much commodities are
sold.
And they do not care that they do not have any such theory.
> reposting the same URL does not make it any more correct.
> -- John J. Weatherby, 15 May 2002
Well have another hearty laugh at Sraffa3.pdf another time.

> But nothing substantial. I dont expect substantial
comments on
> this version either. Most of the economists that post to
sci.econ
> dont seem to be the sort that are willing or capable of
giving
> substantial comments. (One could take umbrage at this
remark, or
> one could prove me wrong.)
When one person says a paper is correct and another says it is
wrong, a puzzle is raised.
Anyways, I have shown the following remark is mistaken:
Reswitching poses a serious problem for capital aggregation
and measurement, and thats about it.
-- Poor Chris Auld, 9 April 1999
Perhaps Mr. Witte can tell us if he agrees that Chris Aulds
statement
is incorrect.
--
Try
http://csf.colorado.edu/pkt/pktauthors/Vienneau.Robert/
Bukharin.html
To solve Linear Programs: .../LPSolver.html
r c A game: .../Keynes.html
v s a Whether strength of body or of mind, or wisdom, or
i m p virtue, are found in proportion to the power or
wealth
e a e of a man is a question t perhaps to be discussed
by
n e . slaves in the hearing of their masters, but highly
@ r c m unbecoming to reasonable and free men in search of
d o the truth. -- Rousseau
===
Subject: Daniel Kane, the most talented mathematicians to
come along in
the last 20 years
Daniel Kanes home page:
Does anybody know what important conjecture Daniel has proved?
-------------------------------------------------------------
--------------
Madison grad described as genius mathematician
Associated Press
MADISON, Wis. - A math prodigy whos been dubbed one of the
brightest
young minds in the country was awarded a $50,000 scholarship
from a
nonprot group that nurtures the profoundly gifted.
of 15 students named this year as a fellow by the Davidson
Institute
in Reno, Nev.
A profoundly gifted person has an intellectual precocity at
least
three standard deviations above the norm, or about one in
every 10,000
people, the institute says.
Ofcials said Kane may be one of the most talented
mathematicians to
come along in the last 20 years.
The fellows have the potential to become the next Einsteins,
Marie
Curies and Mozarts, said Marie Capurro, the institutes
director of
programs and services.
Kane, a National Merit Scholar nalist who has twice
represented the
United States at the International Mathematical Olympiad,
will attend
the Massachusetts Institute of Technology this fall, majoring
in math.
He applied to be a fellow by submitting a work called Two
Papers on
the Theory of Partitions, a branch of additive number theory.
What I see in Daniel is real, raw talent of the type Ive
never seen
before, said Ken Ono, a University of Wisconsin-Madison math
professor who mentors Kane. He wakes up every day wanting to
prove a
new theorem.
Kane already has proved a conjecture posed by national
experts in the
eld, including George Andrews of Penn State, arguably the
worlds
leading authority on the theory of partitions, Ono said.
===
Subject: paper on ellipsoid method
Can anyone help me nd a (digital) copy of :
M. Grotschel, L. Lovasz and A. Schrijver, The ellipsoid
method and its
consequences in combinatorial optimization, Combinatorica 1,
169-197,
1981.
or give me a link to a site/paper where the ellipsoid method
and the
equivalence between optimality and separation is explained?
Thanx,
Dion Bongaerts
===
Subject: a little help about metric spaces
will you help me in proving that
every Cauchy s succession in a metric space is a bounded set?
tern_@libero.it
===
Subject: Re: a little help about metric spaces
> will you help me in proving that
> every Cauchy s succession in a metric space is a bounded
set?
A hint: assume that it is not bounded and prove that it cant
be
Cauchy.
--
Giuseppe Oblomov Bilotta
Cant you see
It all makes perfect sense
Expressed in dollar and cents
Pounds shillings and pence
(Roger Waters)
===
Subject: Re: a little help about metric spaces
wNC8b.315560$lK4.9906120@twister1.libero.it...
> will you help me in proving that
> every Cauchy s succession in a metric space is a bounded
set?
e>0.
There exists a positive integer N, for any n>=N,
d(x_n,x_N)<=e, which means
(x_n) is bounded (any element of (x_n) is contained in the
ball B(x_N, r)
where the radius r = max(d(x_0,x_N), ..., d(x_(N-1),x_N), e).
--
Julien Santini
===
Subject: Re: a little help about metric spaces
> wNC8b.315560$lK4.9906120@twister1.libero.it...
> will you help me in proving that
> every Cauchy s succession in a metric space is a bounded
set?
> e>0.
> There exists a positive integer N, for any n>=N,
d(x_n,x_N)<=e, which
means
> (x_n) is bounded (any element of (x_n) is contained in the
ball B(x_N, r)
> where the radius r = max(d(x_0,x_N), ..., d(x_(N-1),x_N),
e).
Its interesting to remark that, actually, (x_n) is totally
bounded,
because the ball B(x_N, e) contains all but nitely many
terms of
(x_n).
Artur
===
Subject: Legal logic ? puzzle.
When reading certain high court judgement reasons, it is
clear to me
that often some quite profound thinking in being exercised.
But when I dialog with legal people: real live ones or via
*.legal.*
newsgroups; it seems that they are just phrase matching
clerks.
Im convinced of the left-brain/right-brain theory which
divides
humanity into spacial/verbal thinkers.
The matter which I am desperately struggling with, concerns a
time sequence of event; with the assumption that cause does
NOT
follow effect.
As a science graduate, I nd it best and natural to
demonstrate my
argument by a minimalist model. None of the legal people
know
what Im talking about and worse still they dont volunteer
that
my model is (to them) not understandable.
Is the following understandable ? How can I impove it ?
Please answer some of the questions.
The following conditions apply:-
1. The suppliers billing system is wrong: over charging by
10%.
2. The corrrect consumption/billing is $10 per period/month.
3. The client with-holds payment (after repeated written
complaints
and written acknowledgement of errors in the suppliers
billing system)

in order to get a court hearing to expose the suppliers
billing
system.
4. A debtor is said to NOT have a bona de defense if he
admits owing
at least as much as the claim (amount sued for).
Month True Billed Event
1 10 11
2 20 22
3 30 33 A
4 40 44 B
5 50 55 C
6 60 66
7 70 77 D
8 80 88
9 90 99 E
-----
The events labeled in the table are as follows:
A = demand letter from supplier ( for $33 obviously)
B = demand letter from suppliers lawer - for $33
C = summons served on absent client for $33 - not received by
debtor.
D = Default judgment - because of undefended; because summons
not received.
E = ignorant of the served summons, the client delivers a
letter (to
the supplier - not the lawer) with this calculation-table,
writing:
it is NOT about non payment of due debt, in fact I now admit
owing $90, which is more than the $33 claim.
Please accept my calculations or show were they are wrong, so
that I may settle my account.
-----
Q1. Has the client got a bona de defence for with-holding
payment
at period 3 ?
Q2. If the client had a bona de defence for with-holding
payment
at time 3, has this been removed at time 9, because he admits
owing $90, whereas the claim is for $33 ?
Q3. When is the date of accrual of the cause of action ?
(which is dened as: when the material facts on which it is
based
have been discovered or ought to have been discovered by the
plaintiff,
by the exercise of reasonable diligence. )
Q4. If at the appeal level the justice department
acknowledges the
absurdity of Q2 (ie. claiming that events AFTER the default
judgment
[month 9] can justify the default judgment [month 7]), is the
bona de defence of month 3 lost, since at summons commencing
action [month 5] the client admits owing $50;
which is more than the claim of $33 ?
Q5. If so, does this mean that the summons has been
decoupled from
the demand letters ?
Q6. What are the implications of the fact that the threshold
date
(when the calculated/admitted amount owing exceeds the claim)
could happen arbitrarly, at any time between A and E.
to: eas-lab@absamail.co.za)
== Chris Glur.
===
Subject: Re: Geodetic spheres
That message is not avaible to me. Maybe its removed from
the server;
maybe
its only my connection which is removed from the server?
Surng on the net looking for these things I found two online
book called
Synergetics I and II by inventor R. Buckminster Fuller (at
www.b.org).
Has
someone read them? Are they valueable for a mathematician or
are they
hopelessly out of date? They seem a bit strange but maybe
they contain
something cool?
===
Subject: Re: Fast prime counting implementation
Stan Gula <sgula@bellatlantic.net.remove.invalid> a .8ecrit
dans le
message
[...
> The biggest problem is a difference in the packaging of the
math
functions.
> In g++ pow and sqrt arent in the std namespace so I had to
remove the
std::
> and add main.h to the include list.
Including cmath is easier...
BTW, the program core dumps on my machine (Linux, gcc 3.2.1)
with
and without optimization.
Laurent
===
Subject: Re: Fast prime counting implementation
> Stan Gula <sgula@bellatlantic.net.remove.invalid> a .8ecrit
dans le
message
> [...
> The biggest problem is a difference in the packaging of the
math
> functions.
> In g++ pow and sqrt arent in the std namespace so I had to
remove the
> std::
> and add main.h to the include list.
> Including cmath is easier...
> BTW, the program core dumps on my machine (Linux, gcc
3.2.1) with
> and without optimization.
Any idea why? So far it works ne for me on CodeWarrior for
OS X,
CodeWarrior for Windows, Project Builder with gcc.
===
Subject: Re: Fast prime counting implementation
> BTW, the program core dumps on my machine (Linux, gcc
3.2.1) with
> and without optimization.
> Any idea why? So far it works ne for me on CodeWarrior for
OS X,
> CodeWarrior for Windows, Project Builder with gcc.
I sent you an e-mail. If you dont receive it, contact me
directly
(my e-mail does not contain spam counter measures :).
Laurent
===
Subject: Re: Afne and Riemann
>[...]
>I also revel in ludicrous complication, and I rather resent
the
>suggestion that any putative misquotation was necessary to
inspire
>my ludicrously complicated attempt at clearing up some of the
>confusion, or vice versa.
Can we quote you on that? (Assuming we do so precisely, of
course...)
>Lee Rudolph
************************
David C. Ullrich
===
Subject: Re: Afne and Riemann
|
|...
|>> perhaps lee was making fun of your habit of misquoting
people. the
|>> construction is ludicrously complicated only because of
your
|>> misquotation.
|>I didnt misquote anyone, nor do I make a habit of
misquoting people,
|>and I rather resent the suggestion that I do.
|
|I do have a habit of making fun of people and those of their
|habits which are not my habits, but neither was I making fun
of
|Timothy Murphy (or any of his habits) nor do I hold the
opinion
|that he either has a habit of misquoting people or was
misquoting
|anyone in this thread.
perhaps misquotation _is_ a slightly too strong way of
putting it.
what he keeps doing is echoing things that i (or other
people) say,
except with some peculiar distortions introduced, with a
pretty clear
implication that he thinks that hes faithfully repeating the
substance of the original statement. i mention the orbit
space of a
certain operator, he asks what exactly is the orbit of an
operator?.
i mention that the operator x->x*2 on the open half-line
has a
circle as orbit space, he asks how did the orbit space of the
operator x->x^2 on the open half-line turn out to be a
circle?. i
dont think hes going to get very useful answers to his
questions if
he continues this habit of distorting things that people say.
--
[e-mail address jdolan@math.ucr.edu]
===
Subject: Re: Afne and Riemann
>dont think hes going to get very useful answers to his
questions if
>he continues this habit of distorting things that people say.
One persons distortion is another persons change of
coordinates.
Lee Rudolph
===
Subject: Re: Afne and Riemann
>i
>dont think hes going to get very useful answers to his
questions if
>he continues this habit of distorting things that people say.
> One persons distortion is another persons change of
coordinates.
Projective afnity ...
--
Giuseppe Oblomov Bilotta
Cant you see
It all makes perfect sense
Expressed in dollar and cents
Pounds shillings and pence
(Roger Waters)
===
Subject: Hard problem...
Hi
Is there any prime number p and positive integers with 0
(0,1,2,3...) a,
b, c which satisfy equation:
(12a + 5)(12b + 7) = p^c
any hint?
If there isnt a,b,c and p which satisfy this equation how
can I prove
this?
cheers
Nat
===
Subject: Re: Hard problem...
> Hi
> Is there any prime number p and positive integers with 0
(0,1,2,3...)
a,
> b, c which satisfy equation:
> (12a + 5)(12b + 7) = p^c
> any hint?
> If there isnt a,b,c and p which satisfy this equation how
can I prove
this?
> cheers
> Nat
SPOILER:
No, because u = 12a + 5 == 4.Z + 1 and v = 12b + 7 == 4.Z +
3, which
requires
that your prime p == 4.Z + 3 and u, v are (resp) even and odd
powers of p.
But, likewise u == 3.Z + 2 and v == 3.Z + 1 requires that p
== 3.Z + 2 and
u, v are (resp) odd and even powers of p.
-------------------------------------------------------------
--------------
John R Ramsden (jr@adslate.com)
-------------------------------------------------------------
--------------
A stockbroker is someone who invests your money until its
all gone.
Woody Allen
electron-dot-cloud are galaxies
===
Subject: When laying block, better than string for
straightness
I am building my own concrete block garage. And when it came
time to
lay the rst course I did not like the string method for it
depends too
much
on eye judgement. So what I did was haul out two very long
and stout
plumbing pipes. Very long stiff and rm and layed the block
loosely for
the rst row
and then instead of string and eyeball I simple rolled the
two pipes,
one on the inside of the row and one on the outside of the
row to make a
perfect line.
That method is great for the rst course because the pipes
are on the
concrete slab.
Now, I am trying to gure a way to use the pipe method going
up the
wall
so that I never have to use the time wasting and imprecise
string
method.
Thought I might share that with you so that we do replace the
old string
method with something easier, faster and better. A method
where the test
uses the material instead of a eyeball judgement call
Archimedes Plutonium, a_plutonium@hotmail.com
whole entire Universe is just one big atom where dots
of the electron-dot-cloud are galaxies
===
Subject: Re: When laying block, better than string for
straightness
Seems like alot of work would it not be easier to buy two
cheap laser
pointers mount two to form a 90 deg angle, one pointing down
one pointing
in
the direction of the row of bricks. Mark the ground so the
pointers are in
the same location on the rst brick and line the bricks up
against the
lightr pointing down the row you could even mount a bubble
level on them
to
make sure its level
> I am building my own concrete block garage. And when it
came time to
> lay the rst course I did not like the string method for it
depends too
> much
> on eye judgement. So what I did was haul out two very long
and stout
> plumbing pipes. Very long stiff and rm and layed the block
loosely for
> the rst row
> and then instead of string and eyeball I simple rolled the
two pipes,
> one on the inside of the row and one on the outside of the
row to make a
> perfect line.
> That method is great for the rst course because the pipes
are on the
> concrete slab.
> Now, I am trying to gure a way to use the pipe method
going up the
> wall
> so that I never have to use the time wasting and imprecise
string
> method.
> Thought I might share that with you so that we do replace
the old string
> method with something easier, faster and better. A method
where the test
> uses the material instead of a eyeball judgement call
> Archimedes Plutonium, a_plutonium@hotmail.com
> whole entire Universe is just one big atom where dots
> of the electron-dot-cloud are galaxies
===
Subject: Re: Need help with solution: SL Dixons Text
as 1/[pi*Po1*ao1^3] where ao1 = sqrt(gamma*R*To1).
David
> Did you miss the reference to Eqn 7.11? The LHS of the eqn
is
> mdot*Omega^2/[pi*k*gamma*Po1*sqrt(gamma*R*To1)]. If you
take out
> mdot*omega^2/k, whats left,
> 1/[pi*gamma*Po1*sqrt(gamma*R*To1)], is equal to 0.6598e-8.
Did you
actually
> try to plug in numbers?
> Pete K.
> A solution to problem 7.4 in SL Dixons Fluid Mechanics and
> Thermodynamics of Turbomachinery 4th Ed. has me stumped and
I would
> appreciate any help understanding the steps that were left
out.
> The compressible ow relation between mass ow rate, speed
of
> rotation and the ow parameters at the eye tip is given by
eqn.
> (7.11),
> 0.6598 x 10^-8 x (mdot x rads ^2)/k = ...
> Ive not gured out yet how were to get 0.6598 x 10^-8
> David
===
Subject: Re: elementary identity
> My theorem:
> Let x,y be points in real line and let f be increasing
function
> then f(max(x,y))= max(f(x),f(y)).
> I wonder if the converse is also true i.e.
> If f(max(x,y))=max(f(x),f(y)) then f is increasing.
> Is there someone who d like to prove it?
> Is the condition f is increasing the only characterization
or most
general
> one such that equation f(max(x,y))=max(f(x),f(y)) is still
valid?
> most general is nondecreasing...
Here is a proof; Im using increasing as a synonym for
nondecreasing (i.e., meaning f.x stays the same or becomes
larger
if x becomes larger).
Sikari wonders whether
(0) f is increasing == <Ax,y:: f.(x max y) = f.x max f.y>
(Some notations borrowed from E.W.Dijkstra here: == is logical
equivalence; <Ax:P:Q> means for every x such that P, Q holds;
and
f.x is function application, the same as the usual f(x).)
We let x and y range over some totally ordered domain, and f
is a
function from that domain to some (possibly different)
totally ordered
domain. We will use <= to indicate the total order in both
cases.
We use the following denition, which seems to be the
simplest one:
(1) f is increasing == <Ax,y: x<=y: f.x <= f.y>
Obviously we have to relate max to <=. The simplest relation
between these two is
(2) x<=y == y = x max y
We use this to prove (0), by transforming f.x <= f.y to f.(x
max y)
= f.x max f.y, as follows:
f is increasing
== denition (1)
<Ax,y: x<=y: f.x <= f.y>
== using (2) on the RHS
-- this is the simplest thing we can do which leads us
towards our goal
<Ax,y: x<=y: f.y = f.x max f.y>
== by (2) the LHS implies y = x max y; substitute this in
the leftmost f.y -- this gives the RHS the right shape
<Ax,y: x<=y: f.(x max y) = f.x max f.y>
== add symmetrical statement; see subproof
-- we need to get rid of the LHS condition, and we
observe that the RHS is symmetric in x and y

<Ax,y: x<=y: f.(x max y) = f.x max f.y>
== rename x,y:=y,x
<Ay,x: y<=x: f.(y max x) = f.y max f.x>
== simplify RHS using symmetry of max (twice)
<Ax,y: y<=x: f.(x max y) = f.x max f.y>

<Ax,y: x<=y: f.(x max y) = f.x max f.y>
/ <Ax,y: y<=x: f.(x max y) = f.x max f.y>
== merge quantications
<Ax,y: x<=y / y<=x: f.(x max y) = f.x max f.y>
== the LHS is true, because <= is a total order
<Ax,y:: f.(x max y) = f.x max f.y>
Groetjes,
<><
Marnix
--
Marnix Klooster
self emailAddress: ln.naab@retsoolkm reverse
===
Subject: Re: Set theory is seriously awed
> All of modern matthematics is based on sets. You need sets
to dene
> what a function is.
No you dont.
===
Subject: Re: Set theory is seriously awed 3
> A function is dened by the limits of its operation. Thats
not a set.
> We cannot speak of all in mathematics and so we cannot
attempt to group
> all in a set.
> A set of all numerals has an error that must be
questioned. The
members
> or marks of a set, do not assume the properties of its
members or
marks.
> The counting is not done by the set, but by ourselves, when
we use
> mathematics.
What is a set? What is a limit? What is an operation? What is
a
member? What is a mark? What is ourselves? What is
mathematics?
Why am I aksing all these questions? (Because Im curious to
know if
you will reply to each of them.)
--
Giuseppe Oblomov Bilotta
Cant you see
It all makes perfect sense
Expressed in dollar and cents
Pounds shillings and pence
(Roger Waters)
===
Subject: Re: Set theory is seriously awed 3
> A function is dened by the limits of its operation. Thats
not a set.
> We cannot speak of all in mathematics and so we cannot
attempt to
group
> all in a set.
> A set of all numerals has an error that must be
questioned. The
members
> or marks of a set, do not assume the properties of its
members or
marks.
> The counting is not done by the set, but by ourselves, when
we use
> mathematics.
> What is a set?
A set is anything thats not zero.
What is a limit?
A limit is a question with no answer. Hence its a physicist
on drugs.
What is an operation?
An operation is a map from nowhere to nowhere. Hence
its everything thats not Geometry.
What is a member?
A member is sorta like a nger. Except that its attached.
What is a mark?
Nobody knows. Since algebra tarts still havent told
us what a slash is.
What is ourselves?
Ourselves is those things which is looking for extraterrestial
intelligence, since we know theres none in math class.
What is mathematics?
Mathematics is second-hand ction. Hence its a rumor of
humor.
> Why am I aksing all these questions? (Because Im curious
to know if
> you will reply to each of them.)

Youre not curious, youre inquisitive. Curious people
dont frequent math class.
===
Subject: Re: Set theory is seriously awed 3
> A function is dened by the limits of its operation. Thats
not a
set.
> We cannot speak of all in mathematics and so we cannot
attempt to
group
> all in a set.
> A set of all numerals has an error that must be
questioned. The
members
> or marks of a set, do not assume the properties of its
members or
marks.
> The counting is not done by the set, but by ourselves, when
we use
> mathematics.
What is a set?
> A set is anything thats not zero.
> What is a limit?
> A limit is a question with no answer. Hence its a
physicist on drugs.
> What is an operation?
> An operation is a map from nowhere to nowhere. Hence
> its everything thats not Geometry.
> What is a member?
> A member is sorta like a nger. Except that its attached.
> What is a mark?
> Nobody knows. Since algebra tarts still havent told
> us what a slash is.
> What is ourselves?
> Ourselves is those things which is looking for
extraterrestial
> intelligence, since we know theres none in math class.
> What is mathematics?
> Mathematics is second-hand ction. Hence its a rumor of
humor.
I still have to decide if youre just a troll, an idiot, or
someone
who tries to be witty without success.
> Why am I aksing all these questions? (Because Im curious
to know if
> you will reply to each of them.)
> Youre not curious, youre inquisitive. Curious people
> dont frequent math class.
I know a lot of curious people that frequent math classes.
Curious,
isnt it?
--
Giuseppe Oblomov Bilotta
Cant you see
It all makes perfect sense
Expressed in dollar and cents
Pounds shillings and pence
(Roger Waters)
===
Subject: Re: Set theory is seriously awed 3
> A function is dened by the limits of its operation.
A function is a set of pairs. The rst element from the
domain, the
second from the co-domain.
You are an idiot.
Bob Kolker
===
Subject: Set theory is seriously awed 3b
A function is still dened by the limits of its operation, or
no function
is applied!
Idiot!
JJ
> A function is dened by the limits of its operation.
> A function is a set of pairs. The rst element from the
domain, the
> second from the co-domain.
> You are an idiot.
> Bob Kolker
===
Subject: Re: Set theory is seriously awed 3b
> A function is still dened by the limits of its operation,
or no
function
> is applied!
> Idiot!
It doesnt matter how mathematicians dene functions,
since the idiots only ever dene binary relations,
concerning their retarted binary logic.
A function is an application of power to power.
And since mathemastooges are as clueless about power today
as they were Cantor and Hilbert invented sets for morons,
it is irrelevent.
===
Subject: Re: Set theory is seriously awed 3b
> A function is still dened by the limits of its operation,
or no
function
> is applied!
> Idiot!
A function is a set of pairs. Given sets A and B and function
from A to
B is a subset R of the cartesian product A x B (set of all
pairs, the
rst element from A, the second from B). R has the property
that if
(x,y1) and (x,y2) in R then y1 = y2.
This is the standard mathematical denition of a function
from A to B.
Why dont you learn some mathematics?
Bob Kolker
===
Subject: Re: Set theory is seriously awed 3b
<bjpkdj$lfmin$1@ID-76471.news.uni-berlin.de>
<bjpttp$kr64a$1@ID-76471.news.uni-berlin.de>
<bjqcs3$lkiol$1@ID-76471.news.uni-berlin.de>
<1g38b.779$%G3.56@newsread2.news.atl.earthlink.net>
<bjqmr1$rnb$1@hercules.btinternet.com>
<bjqrd5$mcssp$1@ID-76471.news.uni-berlin.de>
<bjtneg$ggn$1@hercules.btinternet.com>
<bjtnqe$n6c3h$1@ID-76471.news.uni-berlin.de>
<bjtpbn$e2m$1@sparta.btinternet.com>
<bjtr2g$mrhja$1@ID-76471.news.uni-berlin.de>
<bju5lr$7du$1@hercules.btinternet.com>
<bjuvar$nevr2$2@ID-76471.news.uni-berlin.de A function is a
set of pairs.
Hows a pair dened? I have always used to dene a pair as a
function with the set {1,2} (or {0,1} or something similar) as
its domain. But this obviously leads to a circular denition
if
used to dene a function.
===
Subject: Re: Set theory is seriously awed 3b
> A function is a set of pairs.
> Hows a pair dened? I have always used to dene a pair as a
> function with the set {1,2} (or {0,1} or something similar)
as
> its domain. But this obviously leads to a circular
denition if
> used to dene a function.
Formally, in set-theory, a pair is not a function but a set
p = {a, {a, b}}
that is the union of two singletons ({a} and {{a, b}}) such
that
1. the only element in one of them (a in {a}) is an element
of the
only element of the other singleton (a is an element of {a,b}
which
is the only element of {{a,b}}).
This is called the rst element of the pair.
2. the only element of the other singleton ({a,b} in {{a,b}})
is the
union of two singletons ({a}, {b}) such that.
2.1. the only element of one of them is the rst element
(see above)
2.2. the only element of the other one is what is called
the second element of the pair.
Its easier to understand it if you go the other way around:
take to
elements a, b; to form the pair, you do:
p = {a} union { {a} union {b} }
Also, formally, a function is not a pair but a relation
between two
sets, that is a particular subset of the cartesian product of
the two
sets.
--
Giuseppe Oblomov Bilotta
Cant you see
It all makes perfect sense
Expressed in dollar and cents
Pounds shillings and pence
(Roger Waters)
===
Subject: Re: Set theory is seriously awed 3b
>>A function is a set of pairs.
> Hows a pair dened? I have always used to dene a pair as a
> function with the set {1,2} (or {0,1} or something similar)
as
> its domain. But this obviously leads to a circular
denition if
> used to dene a function.
The ordered pair (a,b) = {{a}, {a,b}} where curly brackets {,
} mean an
unordered collection.
Bob Kolker
===
Subject: Skewed mean value, skew-induced partial order?
Given natural numbers i and j such that i <= j
let k be maximal integer satisfying i*2^k <= j
Is there a term for i*2^k value? If there isnt, then, any
suggestions? (Ariphmetic-geometric mean is already taken:-(
Next, Im looking for a term for the following partial order
relation
between natural numbers.
Again, given natural numbers i and j such that i <= j (in the
standard total ordering;-), let k be maximal integer
satisfying i*2^k
<= j. Consider binary relation le satisfying
i le j <=> j-i*2^k < k or i = j.
In other words we require ariphmetic distance between j and
i*2^k to
be smaller than geometric distance. It is easy to see that le
is a
partial order.
Any suggested term for le? Appreciate your help.
===
Subject: innity dimensional vector space
Can you prove that the vector space of real-valued function
over a
differentiable manifold M is innity-dimensional
Tern_
===
Subject: Re: innity dimensional vector space
>Can you prove that the vector space of real-valued function
over a
>differentiable manifold M is innity-dimensional
No, and nor can anyone else, unless you add some hypothesis
that
rules out the possibility that M is a compact 0-dimensional
manifold (that is, a nite set with the discrete topology
and the only possible differentiable structure).
Thinking about why that case *doesnt* work may give you a
big clue as to why all other cases *do* work. But it isnt
going to be completely trivial; youre going to have to know
*something* (some sort of extension theorem, or some sort of
partition-of-unity theorem, or...).
Lee Rudolph
===
Subject: Re: innity dimensional vector space
> Can you prove that the vector space of real-valued function
over a
> differentiable manifold M is innity-dimensional
Suggestion: if n is the dimension of M, then start by proving
that
the vector space of real-valued function over an open subset
of R^n
is innity-dimensional. Since there is an obvious injective
linear
function from such a space into your space...
Jose Carlos Santos
===
Subject: Re: innity dimensional vector space
>> Can you prove that the vector space of real-valued
function over a
>> differentiable manifold M is innity-dimensional
>Suggestion: if n is the dimension of M, then start by
proving that
>the vector space of real-valued function over an open subset
of R^n
>is innity-dimensional. Since there is an obvious injective
linear
>function from such a space into your space...
And that obvious injective linear function is *what*,
precisely?
...Oh, ah. I see that I missed the fact (also in my direct
reply to tern) that tern didnt specify *differentiable*
function.
Without any such restriction, I agree that there *is* an
obvious
injection.
I bet he was supposed to specify that, though. (Otherwise the
question is bizarrely phrased.)
Lee Rudolph
===
Subject: Re: Math - Education Path
>>The *teachers* consider these examinations to be demeaning.
They
>>have taught their subjects successfully
>FSVO successfully. Maybe your expectations are set too low.
>>apparently without signicant complaints about their
performance,
>Apparent to whom? Determined how?
As I have pointed out, complaints about the quality of
curriculum, including requiring children to sit through
material they know well, are NOT respected. Furthermore,
the ones in charge do not understand mathematics, or in
fact any other honest subject.
>>and are then asked to
>>subject themselves to an authority that is newly formed and
holds
>>power of authority over them.
Not only now, but in the past, mathematicians have
volunteered to help improve the curriculum and raise
standards. They have consistently been rebuffed.
The teachers of mathematics in the public schools do not
know mathematics. They only know formal words and how
to carry out routine manipulations.
>Im more concerned about the kids that they have power of
authority
>over.
As everyone should be.
>>And it is extremely probable
>>that some top-notch third-semester calculus students could
teach
>>rst- and second-semester calculus without having degrees or
>>credentials.
>Its also extremely probable that some of the teachers
complaining
>about the tests took Mathematics for dummies[1] instead of
real
>Mathematics classes. Those top notch 3rd semester student
would
>probably do a better job than the certied teachers.
Few of the entering college students now, including
those intending to major in the sciences or engineering,
understand induction or proofs. A half century ago,
those planning to major in English at a good college
were expected to have had a good course in Euclid.
>[1] Im suspicious of any class that is not accepted for
departmental
>credit. Thats usually a red ag indicating that the
contents have
>been watered down.
It is worse than that. The pressure has caused even
the courses for departmental credit to be watered
down. Administrators are more concerned with quantity
and student ratings than quality.
--
This address is for information only. I do not claim that
these views
are those of the Statistics Department or of Purdue
University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
===
Subject: Re: Mathematical induction and the use of n and k (?)
>Hi all,
>After reading some of the mathematical induction problems, I
tried to
>remember why it is necessary to replace the n with k in the
induction
step.
>Can somebody help me? Is it that k in N is supposed to be a
particular
>element; whereas, n in N is meant as a general element. How
is this
>described in logic?
As you are using it, N is a constant, the set of integers.
The idea that one should even use mnemonic letters for
variables is greatly to be decried. It makes no difference
if one uses n or k or q or z or T or alpha for a variable
symbol. Sometimes, more than one needs to be used.
--
This address is for information only. I do not claim that
these views
are those of the Statistics Department or of Purdue
University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
===
Subject: Re: Mathematical induction and the use of n and k (?)
>>Hi all,
>>After reading some of the mathematical induction problems,
I tried to
>>remember why it is necessary to replace the n with k in the
induction
step.
>>Can somebody help me? Is it that k in N is supposed to be a
particular
>>element; whereas, n in N is meant as a general element. How
is this
>>described in logic?
>As you are using it, N is a constant, the set of integers.
>The idea that one should even use mnemonic letters for
>variables is greatly to be decried.
Id be interested in your reasons why. ISTM that mnemonic
variable names
are common precisely because they *are* helpful.
> It makes no difference
>if one uses n or k or q or z or T or alpha for a variable
>symbol. Sometimes, more than one needs to be used.
There are well-established coventions for variable names,
presumably as
memory aids: <delta> and <epsilon> in limit proofs, <theta>
for an angle,
x and y for reals, z for complex, etc.
Although formally the variable names make no difference,
well-chosen ones
certainly do seem to aid readability.
--
---------------------------
| BBB b barbara minus knox at iname stop com
| B B aa rrr b |
| BBB a a r bbb |
| B B a a r b b |
| BBB aa a r bbb |
-----------------------------
===
Subject: Re: Mathematical induction and the use of n and k (?)
>> Hi all,
>> After reading some of the mathematical induction problems,
I tried to
>> remember why it is necessary to replace the n with k in
the induction
>step.
>> Can somebody help me? Is it that k in N is supposed to be a
particular
>> element; whereas, n in N is meant as a general element.
How is this
>> described in logic?
>I think the short answer is, to avoid the appearance of
circular
argument.
>It really doesnt matter what variable, n or k or some other
variable is
>used,
>so long as one is careful about the (often implied)
quantication of
these
>variables.
>Lets review the outline of mathematical induction and point
out why it
>might appear to an unfamiliar eye that the principle
espouses a form of
>circular proof that could be used to prove anything at all.
>We refer to a basis step and an induction step as the
ingredients
>that make up a proof by mathematical induction. Although
some may
>disagree, I hold with zero as being the least natural number
(it makes N
>both an additive as well as multiplicative monoid). Thus to
prove by
>(weak) induction for some property P of natural numbers that:
>for all n in N, P(n)
>we should show both ingredients:
>(basis step) P(0)
>(induction step) for all k in N, P(k) implies P(k+1)
>The second ingredient, the induction step, is often
explained in
>high school algebra by a somewhat dubious notion that we are
>assuming the truth of P(k) in order to show that P(k+1) is
true.
>While this does amount to the procedural reasoning involved,
>the bright student will begin to suspect that by assuming the
>truth of P(k), we are indulging in a circular argument.
>And of course, we _almost_ are. Its a delicate point. The
change
>of variable from n to k is really a distraction, designed to
suppress
>what is really an intelligent but difcult to answer
question.
>Your note hits pretty close to the mark in raising the issue
of
>whether n may be a general element, while k is to be a
particular
>element. This is precisely what quantication is about in the
>theory of symbolic logic. When asserting that:
>for all n, P(n)
>the meaning of n is intuitively a general element.
>The meaning of k in:
>for all k, P(k) implies P(k+1)
>is also that of a general element. However, what happens is
>that once we begin to assume the induction hypothesis, P(k),
>a logical procedure whose correctness is justied by
something
>in the theory of symbolic logic called the Deduction Theorem,
>we no longer have complete generality for k. At that point
the
>meaning of k has been limited to a particular element for
>which P(k) happens to be true.
> Hmm. Of course I agree about the particular element bit -
> where I come from we do two things: (i) prove P(1), (ii)
> assume P(n) and prove P(n+1). We say three times that
> we do _not_ assume P(n) for all n, one or two students
> do anyway, and we say it again...
> But I dont follow your distaste for using the Deduction
> Theorem. I wouldnt call that something in the theory
> of symbolic logic, although of course it is that. Seems
> to me that the simplest _denition_ of a correct proof
> of if A then B is A proof that looks like the following:
> Assume A.
> Hence B.
> Leaving aside induction for a moment. How
> would you have them prove, say, if n is
> even then n^2 is even.? Seems to me a
> correct proof is this:
> [1] Pf: Assume n is even. Then n = 2k, so
> n^2 = 2(2k^2), hence n^2 is even,
> while Id call
> [2] Pf Since n = 2k, n^2 = 2(2K^2), so n^2
> is even.
> _wrong_, because we dont know that n = 2k,
> because we never assumed that n was even.
> (While [2] _is_ a correct proof if what is to be
> proved was phrased Suppose n is even. Then
> n^2 is even.
> Seems to me to give a correct proof _without_
> explicitly assuming the hypotheses are true
> one has to begin every sentence with
> if n is even then..., which is awkward in
> a longer proof.
> (Explictly stating what were assuming seems
> more important if, say, were proving
> If n is even then n^2 is even, while if n is odd
> then n^2 is odd.)
>Hope this somewhat wordy discussion helps to clarify the
>matter for you...
> ************************
> David C. Ullrich
No pejorative connotation intended. I was noting that the
logical procedure, which were all familiar with, of switching
into Assume P(k), then proceeding to argue for P(k+1),
as a means to proving:
|- P(k) implies P(k+1)
is indeed rigorously justied by something (presumably not
well known, at least by name) called the Deduction Theorem.
Really, Im fond of it. Taken together with our other stock
in trade, Proof by Contradiction, these form the basis for
the proof systems of natural deduction.
Of course since youve already invoked the Compactness
Theorem at least once this week (for rst-order logic), I
had no doubts youd know immediately whereof I speak.
The subtle point here is that from a proof theoretic POV,
the variable in P(k), when we make this assumption, is
not universally quantied. It is restricted by that very
assumption.
The same variable will be universally quantied once we
get to:
|- for all k, P(k) implies P(k+1)
but thats a different place, a different time.
If P(k) were an axiom rather than merely an assumption
for the sake of applying the Deduction Theorem, then
|- for all k, P(k)
would be a proper application of the schema of Axiom
Generalization.
More pointedly the fact of being able to prove:
|- P(k) implies P(k+1)
without assumptions (as the Deduction Theorem will
guarantee us) allows the inference that:
|- for all k, P(k) implies P(k+1)
an instance of the rule of generalization. Its validity
requires that k is not restricted
===
Subject: Re: Mathematical induction and the use of n and k (?)
>[...]
>>The second ingredient, the induction step, is often
explained in
>>high school algebra by a somewhat dubious notion that we are
>>assuming the truth of P(k) in order to show that P(k+1) is
true.
>>[long complaint about the idea that this is dubious
snipped...]
>No pejorative connotation intended.
Ok, I guess I read the word dubious as meaning more than
you intended.
>I was noting that the
>logical procedure, which were all familiar with, of
switching
>into Assume P(k), then proceeding to argue for P(k+1),
>as a means to proving:
>|- P(k) implies P(k+1)
>is indeed rigorously justied by something (presumably not
>well known, at least by name) called the Deduction Theorem.
>Really, Im fond of it. Taken together with our other stock
>in trade, Proof by Contradiction, these form the basis for
>the proof systems of natural deduction.
>Of course since youve already invoked the Compactness
>Theorem at least once this week (for rst-order logic), I
>had no doubts youd know immediately whereof I speak.
>The subtle point here is that from a proof theoretic POV,
>the variable in P(k), when we make this assumption, is
>not universally quantied. It is restricted by that very
>assumption.
>The same variable will be universally quantied once we
>get to:
>|- for all k, P(k) implies P(k+1)
>but thats a different place, a different time.
>If P(k) were an axiom rather than merely an assumption
>for the sake of applying the Deduction Theorem, then
>|- for all k, P(k)
>would be a proper application of the schema of Axiom
>Generalization.
>More pointedly the fact of being able to prove:
>|- P(k) implies P(k+1)
>without assumptions (as the Deduction Theorem will
>guarantee us) allows the inference that:
>|- for all k, P(k) implies P(k+1)
>an instance of the rule of generalization. Its validity
>requires that k is not restricted
************************
David C. Ullrich
===
Subject: Re: This just in: elementary proof of FLT [not JSH]
>Christopher J. Henrich
>> I tried the Safari browser for the rst time today. It
seems to have
>> an area for calling ones attention to interesting Web
sites. One of
>> them announces an elementary proof of Fermats Last
Theorem.
>> http://www.multiwire.net/fermat/welcome.html
>J. Edgar Harris wont like this guy:
>http://www.coolissues.com/mathematics/Fermat/fermat.htm
Looking at this page, I note several invalid steps. His
so-called
New Theorem is contradicted by the simple counterexample
sum_{n=1}^innity 1/2^n, which satsises what sense can be
made
of his premises, but contradicts his conclusion. Where his
proof
falls down (for those reading the page) is that his variables
p
and q are dependent on n, but he assumes that they are
independent of n.
>http://www.coolissues.com/mathematics/Riemann/riemann.htm
Again, if you are looking at the page, he gives incorrect
expressions for u, v, u, v, and he also assumes that if u
and u
are distinct functions, then they cannot simultaneously take
the
value 0 (and similarly for v and v), So he made two
unsupportable assertions. This is no proof of Riemanns
Hypothesis.
>http://www.coolissues.com/mathematics/Goldbach/goldbach.htm
First, he claims to have a fast factorization method:
apparently one
factorizes a product of primes p = ab by nding a suitable
even number s
such that a and b are the solutions of x^2-sx+p=0. This is
well and good,
except for the fact that it is a hard problem computationally
to nd an
appropriate value for s. He hasnt saved any work since he
has merely
replaced one hard problem with another which is equally hard.
I actually
found a method of the same type that he suggests in a book,
and I think
that the book pointed out the technique only works if the
difference
between a and b is of the order of the square root of a.
He then goes on to prove Goldbachs conjecture by nding an
appropriate value for p such that the equation x^2-sx+p=0 has
solutions a and b, which are integers. Of course, this proof
requires the assumption that such a value for p exists. He
does not bother to prove that such a value for p exists, and
so
he has failed to supply a proof for Goldbachs Conjecture. In
Shor, in order to prove Goldbachs Conjecture, he made an
unproven assumption which is equivalent to Goldbachs
Conjecture.
>http://www.coolissues.com/mathematics/Tprimes/tprimes.htm
He states that for odd primes a and b (with a > b), there
exist
n and k such that a+b = 2n and a-b = 2k, so that a = n+k and
b = n-k. He then claims that the Prime Number Theorem
determines
that the density of primes a = n+k and b = n-k is about
2k/log n,
and so primes whose difference is 2k (any specic k> 0) must
go
on forever. He then just takes the special case of k = 1,
So his conclusion supposed to come from the Prime Number
Theorem.
Since the Prime Number Theorem has no bearing on the
distribution
of primes which are separated by a specied amount, then his
proof collapses.
In short, all the proofs are consequences of unsupported
assumptions.
He has not proved anything.
David McAnally
--------------
===
Subject: Re: Sets. - Complementary Associations Theory - A
formal solution
to Hilberts 1st and 6th problems.
Here you can nd an example of a next step mathematical
theory:
http://www.geocities.com/complementarytheory/CATpage.html
===
Subject: Re: Sets. - Complementary Associations Theory - A
formal solution
to Hilberts 1st and 6th problems.
> Here you can nd an example of a next step mathematical
theory:
> http://www.geocities.com/complementarytheory/CATpage.html
I dont see how this is next step. Its just old style
senseless
babbling.
Where is the new style senseless babbling?
===
Subject: Re: Sets. - Complementary Associations Theory - A
formal solution
to Hilberts 1st and 6th problems.
> Here you can nd an example of a next step mathematical
theory:
> http://www.geocities.com/complementarytheory/CATpage.html
>I dont see how this is next step. Its just old style
senseless
babbling.
>Where is the new style senseless babbling?
Well, its in HTML. Thats something.
Lee Rudolph
===
Subject: Re: Sets.
> To me, it seem that we conceive of sets as a box that we
can put
> things in and which automatically contains whatever
elements its
> supposed to have. IE, the set of all natural numbers is a
set that you
> put all the natural numbers into. However, there are two
things you
> cannot do with a set.
> Firstly, you cannot have a set inclusive of all things.
This is
> because in set theory each set has a related power set,
which is
> always bigger. We have no set theory which can simply
contain the
> power reformulation to the innite recusion, because set
theory just
> cannot deal with a situation like that.
> Secondly, you cannot exactly have the set of all sets which
do not
> contain themselves as a member of a set, because then you
couldnt
> answer whether or not it would be a member of itself. It is
if and
> only if it is not.
> More generally, you cannot have a set of all sets
satisfying P, where P
> is some arbitrary property. In fact, this restriction
covers both of
> yours. What you can have instead is the principle of bounded
> comprehension, which says that if you have a set A and a
property P, then
> you can form the subset B = { x in A : P(x) }. The set A
serves as an
> upper bound for the size of the new set being constructed.
The other
> kind is called unbounded comprehension and is what leads to
paradoxes
> such as those of Russell, Cantor, and Burali-Forti.
What if we allow unbounded comprehensive sets? Or sets that
do not
follow strict logic. That is, we allow something to exist in
states
other than either existing or not existing.
What would happen if we injected fuzzy logic into Zermelo
Fraenkel Set
theory?
http://mathworld.wolfram.com/Zermelo-FraenkelAxioms.html
This only has exists and not exists.
http://mathworld.wolfram.com/FuzzyLogic.html
This does.
Has anybody ever tried formulating fuzzy mathematics, or
other
abstract forms of mathematics? More specically, have they
ever tried
to do something like expand Zermelo-Fraenkel Set theory such
that
states beyond existing and not existing can have a meaning?
(...Starblade Riven Darksquall...)
===
Subject: Re: Sets.
>> More generally, you cannot have a set of all sets
satisfying P, where P
>> is some arbitrary property. In fact, this restriction
covers both of
>> yours. What you can have instead is the principle of
bounded
>> comprehension, which says that if you have a set A and a
property P,
then
>> you can form the subset B = { x in A : P(x) }. The set A
serves as an
>> upper bound for the size of the new set being constructed.
The other
>> kind is called unbounded comprehension and is what leads to
paradoxes
>> such as those of Russell, Cantor, and Burali-Forti.
> What if we allow unbounded comprehensive sets?
Your question doesnt make sense. I was using comprehension
as a noun,
not an adjective.
>Or sets that do not
> follow strict logic. That is, we allow something to exist
in states
> other than either existing or not existing.
> What would happen if we injected fuzzy logic into Zermelo
Fraenkel Set
> theory?
We would get functions with codomain [0,1]. Not a big deal.
--
Dave Seaman
Judge Yohns mistakes revealed in Mumia Abu-Jamal ruling.
<http://www.commoncouragepress.com/index.cfm?action=book&
bookid=228>
===
Subject: Re: Sets.
> To me, it seem that we conceive of sets as a box that we
can put
> things in and which automatically contains whatever
elements its
> supposed to have.
> No. A set of cows does not form a herd. A set of recipes
does does not
need
> to be cooked. A set of boiling liquids is not hot. So you
are wrong: A
set
> is not a collection.
A set of cows is not a herd. But a set of cows on the same
eld with
perhaps another modier or two is equivalent to a description
of a
herd. But an arbitrary set of cows, say one from Indiana, one
from
California, and one from England, in fact, even taking a few
cows here
and there such that there are a million cows, does not form a
herd.
But a thousand cows on a single eld is a herd. Or, at least,
the
herd is the set and the bunch of cows that were on the same
eld that
I mentioned earlier are the members.
Also, a set of boiling liquids cannot be hot or cold unless
compared
to something else, and even then it only makes sense to speak
of it if
the liquids are in a dening area, such that you can place it
next to
something else that is hotter or colder. Hot doesnt really
exist. Hot
is a property, and it only has a relative meaning. However,
here if we
consider temperature a propery, then what we get into is
something
that is conceptually abstract.
> JJ
(...Starblade Riven Darksquall...)
===
Subject: Sets. 4
> But a thousand cows on a single eld is a herd. Or, at
least, the
> herd is the set and the bunch of cows that were on the same
eld that
> I mentioned earlier are the members.
The set of a herd of cows is not derived from a set of cows,
it is simply
not derived. It is chosen.
JJ
> To me, it seem that we conceive of sets as a box that we
can put
> things in and which automatically contains whatever
elements its
> supposed to have.
> No. A set of cows does not form a herd. A set of recipes
does does not
need
> to be cooked. A set of boiling liquids is not hot. So you
are wrong: A
set
> is not a collection.
> A set of cows is not a herd. But a set of cows on the same
eld with
> perhaps another modier or two is equivalent to a
description of a
> herd. But an arbitrary set of cows, say one from Indiana,
one from
> California, and one from England, in fact, even taking a
few cows here
> and there such that there are a million cows, does not form
a herd.
> But a thousand cows on a single eld is a herd. Or, at
least, the
> herd is the set and the bunch of cows that were on the same
eld that
> I mentioned earlier are the members.
> Also, a set of boiling liquids cannot be hot or cold unless
compared
> to something else, and even then it only makes sense to
speak of it if
> the liquids are in a dening area, such that you can place
it next to
> something else that is hotter or colder. Hot doesnt really
exist. Hot
> is a property, and it only has a relative meaning. However,
here if we
> consider temperature a propery, then what we get into is
something
> that is conceptually abstract.
> JJ
> (...Starblade Riven Darksquall...)
===
Subject: Re: Sets.
> To me, it seem that we conceive of sets as a box that we
can put
> things in and which automatically contains whatever
elements its
> supposed to have.
No. A set of cows does not form a herd. A set of recipes does
does not
need
> to be cooked. A set of boiling liquids is not hot. So you
are wrong: A
set
> is not a collection.
A set of cows is not a herd.
Can the cows be seen? Then they are seen and not herd.
===
Subject: Re: Sets.
> To me, it seem that we conceive of sets as a box that we
can put
> things in and which automatically contains whatever
elements its
> supposed to have.
No. A set of cows does not form a herd. A set of recipes does
does not
need
> to be cooked. A set of boiling liquids is not hot. So you
are wrong: A
set
> is not a collection.

> A set of cows is not a herd.
> Can the cows be seen? Then they are seen and not herd.
ROTFL.
An un-set-tling comment for sure.
--
Giuseppe Oblomov Bilotta
Cant you see
It all makes perfect sense
Expressed in dollar and cents
Pounds shillings and pence
(Roger Waters)
===
Subject: Re: Fundamental Reason for High Achievements of Jews
Should we think that american white people are cowards
because they ran
away instead of ghting for their religious beliefs?
Or should we think that Chinese, Irish and Italian imigrants
are more
honoured because they went to US searching for a job? While
the white ones
were just running away?)
Or maybe we should conclude that US blacks are more patriots
because they
didnt want to leave their home continent, they were *really*
forced to.
So, uncle al, stop your hatred prejudice.
And get your head out of your ass.
Uncle Al <UncleAl0@hate.spam.net> escreveu na mensagem
> question
> that people in colleges, academia, and sciences will
encounter. Why
> are there
> so many disproportionate number of jews in sciences? Is it
because
> they are
> a superior race and conversely lending truth to racism and
thus
> Shockleys
> fallacy that blacks are inferior? Do the jews have greater
stamina
> than
> others?
> They have Jewish mothers. Read Portnoys Complaint by
Philip Roth.
> Chinese mothers are equally lethal and man-hating. That
Jews from the
> Pale of Settlement average 115 IQs doesnt hurt. That
Hitler killed
> off the bottom 90% of the bell curve for European Jews
doesnt hurt
> either - evolution is a hoot if you are one of the
survivors. The
> best of the best emigrated to the New World and were
scythed again by
> anti-Semitism. The survivors are intellectually superior in
every way
> (on the average).
> Selective breeding works the other way, too. The absolutely
most
> stupid Blacks in West Africa were captured by their
marginally more
> intelligent and vicious brethren, then sold to Arabs who in
turn sold
> them to the Triangular Trade and New World slavery. Uppity
Darkies
> were weeded out. The result is US Inner Cities as
celebrations of
> ethnic diversity.
> --
> Uncle Al
> http://www.mazepath.com/uncleal/eotvos.htm
> (Do something naughty to physics)
---
Outgoing mail is certied Virus Free.
Checked by AVG anti-virus system (http://www.grisoft.com).
===
Subject: a question about Relative Numbers
In Z, I can multiply (a, b) and (a, b) in this way:
(a, b) * (a, b) := (a*a + b*b, a*b + a*b)
I want to prove that this operation has not affected by the
specic
choice of the couples.
So I can take:
(c, d) R (a, b) so that c+b = a+d
(c, d ) R (a, b) so that c+b = a+d
and then: (c,d) * (c,d) = (c*c + d*d, c*d + c*d)
How can I prove that (c*c + d*d, c*d + c*d) R (a*a +
b*b, a*b +
a*b) ?
Luigi
===
Subject: Re: a question about Relative Numbers
Visiting Assistant Professor at the University of Montana.
>In Z, I can multiply (a, b) and (a, b) in this way:
>(a, b) * (a, b) := (a*a + b*b, a*b + a*b)
You mean, in ZxZ. (In fact, you probably mean N x N, pairs of
nonnegative integers).
>I want to prove that this operation has not affected by the
specic
>choice of the couples.
This makes no sense as written. What you mean is that you
will dene
an equivalence relation on ZxZ, and you want to show that the
operation above is well dened on equivalence classes. Thats
what
you do below:
>So I can take:
>(c, d) R (a, b) so that c+b = a+d
>(c, d ) R (a, b) so that c+b = a+d
You have an equivalence relation, (a,b) R (x,y) if and only if
a+y=b+x. You want to show that if (a,b)R(c,d) and
(a,b)R(c,d),
then (a,b)*(a,b) R (c,d)*(c,d), according to the
denition of *
given above.
>and then: (c,d) * (c,d) = (c*c + d*d, c*d + c*d)
>How can I prove that (c*c + d*d, c*d + c*d) R (a*a +
b*b, a*b +
>a*b) ?
Well, you need to show that
cc+dd + ab + ab = cd + cd + aa + bb,
obviously. What you know is that c+b=a+d and c+b=a+d.
Try to understand what your pairs mean and what the operation
is. You think of the pair (a,b) as representing the integer
a-b. Multiplying a-b by c-d gives you ac+bd-(ad+bc), that is,
the
integer (ac+bd,ad+bc).
Multiply the two together to get
(c-d)*(c-d) = (b-a)*(b-a); or
cc + dd - (cd + cd) = bb + aa - (ab + ab).
Therefore,
cc + dd + ab + ab = bb + aa + cd + cd.
Its not denial. Im just very selective about
what I accept as reality.
--- Calvin (Calvin and Hobbes)
Arturo Magidin
magidin@math.berkeley.edu
===
Subject: Re: Second Call For Primecounter Contest Entries
In sci.math, Stan Gula
<sgula@bellatlantic.net.remove.invalid>
<3Q68b.7432$ej1.3051@nwrdny01.gnilink.net>:
> of
>> fun.
>> Dont be so sure. :-) An anonymous submitter memoized my
legendrephi
>> algorithm and so far that ones in third place. :-)
> Oh, sure, if youre going to cheat ;-)
*grin* Cheat? Ive already cheated. None of you have managed
to budge my binarysearch algorithm. :-) ;-) :-)
Then again, I cant say binarysearch of a table of
20 million primes is horribly elegant, mathematically.
So Im not going to count it as a real contestant although
Ill note it in the nal results, as it is theoretically
one of the fastest methods to answer the question, albeit
in a rather inelegant fashion. Ill also simply build an
array piarray[n] = pi(n) and use that as another entry, but
both are cheating to a very large degree, and horribly
inelegant mathematically. :-) A third entry is a stupid
bitcounter using another prefab table.
Im curious which one is faster and which one bigger, as the
binary search has O(log(N)) time, whereas the piarray has
O(1) time. The bitcounter will have O(N) time.
And I thought my 5MB prime table was a memory hog. Ill have
to
feed the generated assembly directly to the assembler using
a pipe as the source les way too big to store. Fortunately,
Gnus assembler is perfectly happy with this. Unfortunately,
it seems to be building the image in its own VM space, which
So I think that answers the bigger question. Ill have to
wait a bit for the answer to the faster. :-) Im still
not sure how to answer the question regarding recursion
but Ill probably work on that Tuesday and doctor everyones
source code with macros or something.
Ive now waited and it turns out binarysearch is faster,
although both are reading 0 on my user times anyway so
Id need a higher-precision timer metric than the ticks
provided by Unix. Yipes! (Bitty didnt perform as well,
which isnt surprising. Of course bitty is another cheat;
its using a precomputed primetable which means one should
conceptually add bitty and sievebool together. How to
generalize that idea, I have no clue at this time.)
In any event, Im an engineer, not a mathematician (despite
my getting a BA from UCSB in math -- and that was 20
years ago!). The solutions Ive thrown into the pot dont
require much mathematical skullwork; the most challenging
and elegant one Ive worked on is the legendrephi, and I
stumbled through that as you may well recall, needing help
from somebody (I think it was Christian Bau but would have
to look); the submissions Ive received are more elegant,
although I cant explain some of em. :-)
Its been a very interesting contest but Ill have to make a
note on memoization, and of course my entries that are using
prebuilt tables arent exactly the most honest; at best,
theyre milestone markers of some sort, if the milestone
concept makes any sense at all here. At worst, theyre
an illustration of my programming and mathematical abilities.
:-)
(Or lack thereof.)
--
#191, ewill3@earthlink.net
Its still legal to go .sigless.
===
Subject: double factorials replacing factorials
I have encountered a linear combination of the two functions
1 + x^2/2!! + x^4/4!! + x^6/6!! + ...
and
x + x^3/3!! + x^5/5!! + x^7/7!! + ...
with double factorials
5!!=5*3*1, 9!!=9*7*5*3*1,
6!!=6*4*2, 10!!=10*8*6*4*2
etc.
Are they related to classical functions,
hypergeometric,etc?
George Marsaglia
===
Subject: Re: double factorials replacing factorials
> I have encountered a linear combination of the two functions
> 1 + x^2/2!! + x^4/4!! + x^6/6!! + ...
> and
> x + x^3/3!! + x^5/5!! + x^7/7!! + ...
> with double factorials
> 5!!=5*3*1, 9!!=9*7*5*3*1,
> 6!!=6*4*2, 10!!=10*8*6*4*2
> etc.
> Are they related to classical functions,
> hypergeometric,etc?
> George Marsaglia
Of course they are hypergeometric.
Now if k is even, k=2*n, then k!! = 2^n*n!, and
1 + x^2/2!! + x^4/4!! + x^6/6!! + ...
= 1 + x^2/2/1! + x^4/2^2/2! + ...
= exp(x^2/2);
And if k is odd, k=2*n+1, then k!! = (2*n+1)!/(2^n*n!) and
x + x^3/3!! + x^5/5!! + x^7/7!! + ...
Maple does this one as:
> sum(x^(2*n+1)/(2*n+1)!*2^n*n!,n=0..innity) assuming x>0;
2
- 1/2 sqrt(2) exp(1/2 x ) sqrt(Pi) (-1 + erfc(1/2 sqrt(2) x))
===
Subject: Re: double factorials replacing factorials
> Of course they are hypergeometric.
...
I should have been more specic. They are of course
hypergeometric
since the ratio of successive coefcients is a rational
function of the
index.
But that is a large class of functions with many delightful
and unexpected
relations.
I hoped that a few groupies might provide relations that
might lead to
better ways to evaluate the two functions for large arguments
or ties to
other functions that might be of interest.
They arose in relation to the standard normal integral;
indeed a linear
combination of the
two functions, multiplied by the density of a standard normal
X,
provides Prob(X>x) to 15 decimal places for x<5., with far
simpler
methods
that standard ones based on the error function with its
annoying
x/sqrt(2).
I would like to nd easy evaluations for 5<x<10, beyond which
asymptotics
provide the desired 15-or-more-digit accuracy.
Besides, with n! replaced by n!!, they come from sinh(x) and
cosh(x),
with sum exp(x), and there might be surprises that hold for
the n!! forms.
George Marsaglia
===
Subject: Symmetric polynomials (I need help)
In his Basic Algebra, vol. 1, Jacobson proves the following
theorem:
THEOREM: Every symmetric (homogeneous) polynomial is
expressible as a
polynomial in the elementary symmetric polynomials p_i.
The idea of the demonstration is to use that the highest
monomial in
every
p_r is x_1*x_2*x_3*...*x_r, so the highest monomial (using
the total degree
to
choose which monomial is the highest and lexicographic order
to decide
draws)
of p_1^{d_1}*p_2^{d_2}*...*p_r*{d_r} will be
x_1^{d_1+...+d_r}*x_2^{d_2+...+d_r}*...*p_r*{d_r}.
Given a symmetric polynomial f(x), let
x_1^{k_1}*x_2^{k_2}*...*x_r*{k_r}
be
the highest monomial in f(x). Then
p_1^{k_1-k_2}*p_2^{k_2-k_3}*...*p_{r-1}^{k_{r-1}-k_r}*p_r^{k_
r} =
x_1^{k_1}*x_2^{k_2}*...*x_r*{k_r}
Hence:
f_1:=f-p_1^{k_1-k_2}*p_2^{k_2-k_3}*...*p_{r-1}^{k_{r-1}-k_r}*
p_r^{k_r}

is such that his highest monomial is less than the highest
monomial ocurring
in
f.
I agree with all that. The problem is that, at this point, he
simply says
We
can repeat the process with f1. Since there are only a nite
number of
monomials of degree m, a nite number of aplications of the
process yields
a
representation of f as a polynomial in p_1,p_2,...,p_r.
Thats ok, as long as you know that f_1 is symmetric, which
is not obvious
to
me. Can someone explain me that? Could I be missing something
trivial in
here?
language]
Anderson Brasil
andersbrasil@aol.com
Its not that Im afraid to die, I just dont want to be there
when it happens. (Woody Allen)
===
Subject: Re: Symmetric polynomials (I need help)
> Hence:
>
f_1:=f-p_1^{k_1-k_2}*p_2^{k_2-k_3}*...*p_{r-1}^{k_{r-1}-k_r}*
p_r^{k_r}
> is such that his highest monomial is less than the highest
monomial
> ocurring in f.
> I agree with all that. The problem is that, at this point,
he simply
> says We
> can repeat the process with f1. Since there are only a
nite number of
> monomials of degree m, a nite number of aplications of the
process
> yields a representation of f as a polynomial in
p_1,p_2,...,p_r.
> Thats ok, as long as you know that f_1 is symmetric, which
is not
> obvious to
> me. Can someone explain me that? Could I be missing
something trivial in
> here?
Any product of symmetric polynomials is symmetric
(from the denition of symmetric --
invariant under any permutation of x_1,...,x_n),
and so is any difference of symmetric polynomials.
--
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: Math Question: roots of a parabola (or any equation)
Take the equation: x^2 - 2x -35
This has two real solutions (where the parabola crosses the x
axis)
What is the reason that solutions/roots are on the X axis.
Why are they not on the y axis?
Why are they not on the line of Y=1?
Math Learner
===
Subject: Re: Math Question: roots of a parabola (or any
equation)
> Take the equation: x^2 - 2x -35
> This has two real solutions (where the parabola crosses the
x axis)
> What is the reason that solutions/roots are on the X axis.
> Why are they not on the y axis?
> Why are they not on the line of Y=1?
You omitted something important from your equation. The
equation is
y = x^2 - 2x -35
If you set y=0 and solve, you nd out the x values for which
y=0. Any
point with y=0 must be on the x-axis.
===
Subject: Math Question: roots of an equation and square roots
Is there any relation between the roots/solutions of an
equation and
the square root of a number?
Math Learner
===
Subject: Re: Math Question: roots of an equation and square
roots
> Is there any relation between the roots/solutions of an
equation and
> the square root of a number?
> Math Learner
It depends. If I have the equation
x^2-4=0 and I solve it for x, I will get
(x+2)(x-2)=0
x+2=0 or x-2=0 by zero product property
x=-2 or x=2 which are also the x intercepts of the graph of
this function.
Higher degree polynomials, I dont know of any relation
between the roots
and the square root of a number. I think that only works if
you have an
equation of the form x^2-a^2=0.
David Moran
===
Subject: Re: The Grand Facade
charset=iso-8859-1
> Frankly does the tech sector need revitalizing? Part of the
reason
> for the high unemployment is the abnormally large number of
people who
> ocked to the sector during the boom.
Yup. And creating a huge glut of labor in the workforce for
those jobs. Damn ockers.
===
Subject: Re: The Grand Facade
spicedham2@dualboot.net says...
> Frankly does the tech sector need revitalizing? Part of the
reason
> for the high unemployment is the abnormally large number of
people who
> ocked to the sector during the boom.
> Yup. And creating a huge glut of labor in the workforce for
> those jobs. Damn ockers.
Beware the revolution...
--
Do not meddle in the affairs of dragons, for | Doug Van Dorn
thou art crunchy and taste good with ketchup |
dvandorn@mn.rr.com
===
Subject: Re: The Grand Facade
charset=Windows-1252
> There are many who dispute that Dubya was either duly
elected or has
> been acting properly as U. S. President. The economy has
certainly
> gone to hell since he was sworn in.
Yeah! Exactly! I mean, its not like he got more votes,
a larger percentage of the vote, or a larger number of
votes as a percentage of the population than Clinton did.
So theres NO reason to call his election legitimate ....
Oh wait, he did get all that. Erm, well, the economy
still sucks, theres still that! I mean, unemployment is
almost as high as it was during, errr, the mid-1990s! And
we all remember how hellish it was living then, right?
Grueling. Dickensian almost.
Yup, that stupid hick cowboy Dubya sure has farked up this
country but good.
===
Subject: Re: The Grand Facade
>> There are many who dispute that Dubya was either duly
elected or has
>> been acting properly as U. S. President. The economy has
certainly
>> gone to hell since he was sworn in.
>Yeah! Exactly! I mean, its not like he got more votes,
>a larger percentage of the vote, or a larger number of
>votes as a percentage of the population than Clinton did.
>So theres NO reason to call his election legitimate ....
Gore, but otherwise your statement is accurate. But off-topic.
- Randy
===
Subject: Re: The Grand Facade
>> There are many who dispute that Dubya was either duly
elected or has
>> been acting properly as U. S. President. The economy has
certainly
>> gone to hell since he was sworn in.
>Yeah! Exactly! I mean, its not like he got more votes,
>a larger percentage of the vote, or a larger number of
>votes as a percentage of the population than Clinton did.
>So theres NO reason to call his election legitimate ....
> Gore, but otherwise your statement is accurate. But
off-topic.
And how many of Gores voters were the dead and illegal
immigrants*,
thus most loyal of the Democrat constituants?
*A situation now to get much worse since Gray Davis signed
the bill that
gives illegals drivers licences and thus voting priveleges.
--
Scott Lowther, Engineer
Any statement by Edward Wright that starts with You seem to
think
that... is wrong. Always. Its a law of Usenet, like
Godwins.
- Jorge R. Frank, 11 Nov 2002
===
Subject: Re: The Grand Facade
> There are many who dispute that Dubya was either duly
elected or has
> been acting properly as U. S. President. The economy has
certainly
> gone to hell since he was sworn in.
> Yeah! Exactly! I mean, its not like he got more votes,
> a larger percentage of the vote, or a larger number of
> votes as a percentage of the population than Clinton did.
> So theres NO reason to call his election legitimate ....
I assume you mean Gore?
And actually I thought when all was said and done, he DID get
a larger
popular vote, but it was the electroral college that matters.
My personal take is, regardless of how one feels about Bush,
we settled in
a
civil manner (rapid left/right wing talkshow hosts aside).
There were no
troops in the street, no military coops, etc. People
disagreeing with the
outcome arent being shot in the streets or taken for
parachute training at
10,000 over the Atlantic sans parachute.
Was it a situation the founding fathers planned for the in
Constitution.
Not really. Was it an improvisation, a hack even. Sure. But
it generated
an out come we could all live through, if not with.
> Oh wait, he did get all that. Erm, well, the economy
> still sucks, theres still that! I mean, unemployment is
> almost as high as it was during, errr, the mid-1990s! And
> we all remember how hellish it was living then, right?
> Grueling. Dickensian almost.
> Yup, that stupid hick cowboy Dubya sure has farked up this
> country but good.
===
Subject: Math Question: graphs of equivalent equations
I have two equations:
a) (x-1)^2 + 1
b) x^2 - 2x + 3
assumption:
these two equations are algebraically equivalent
(a expands into b)
Why do these two equivalent equations result in different
graphs?
Math Learner
===
Subject: Re: Math Question: graphs of equivalent equations
> I have two equations:
> a) (x-1)^2 + 1
> b) x^2 - 2x + 3
Those arent equations.
===
Subject: Re: Math Question: graphs of equivalent equations
> I have two equations:
> a) (x-1)^2 + 1
> b) x^2 - 2x + 3
> assumption:
> these two equations are algebraically equivalent
> (a expands into b)
Check your algebra.
===
Subject: Re: Math Question: graphs of equivalent equations
> I have two equations:
> a) (x-1)^2 + 1
> b) x^2 - 2x + 3
assumption:
> these two equations are algebraically equivalent
> (a expands into b)
Check your algebra.
Well, its still true in Z_2 :)
--
Giuseppe Oblomov Bilotta
Cant you see
It all makes perfect sense
Expressed in dollar and cents
Pounds shillings and pence
(Roger Waters)
===
Subject: Re: Math Question: graphs of equivalent equations
> I have two equations:
> a) (x-1)^2 + 1
> b) x^2 - 2x + 3
> assumption:
> these two equations are algebraically equivalent
> (a expands into b)
> Why do these two equivalent equations result in different
graphs?
> Math Learner
If I look at equation b, I can rewrite it in the form
(x-h)^2+k. To get it
in this form, I need to complete the square. Ignore the 3 for
a moment and
look at x^2-2x. To get it into the form described, I need to
make x^2-2x a
perfect square. To do this, I take half of the x coefcient
and square it.
Half of 2 is 1 and 1^2 is 1 so I add that and subtract it
from the
constant,
giving (x-1)^2+2. Therefore, the two equations are not
equivalent, but they
have different vertical shifts.
David Moran
===
Subject: Last Call For Primecounters
Its been an interesting and slightly weird contest so far,
and Ill denitely have to write a blurb on memoization
somewhere in the results. (To put it crudely, to memoize
a function, usually recursive, means to add a cache
thereto, so that a result isnt computed more than once.
The speedup can be amazing.)
As always, the rules and preliminary results are available at
http://home.earthlink.net/~ewill3/math/primecounters/
index.html
and, boiled down, the rules are fairly simple: correctly
implement the function
unsigned long long pi(unsigned long long x)
in C or C++, as fast, as clearly, and as elegantly as
possible. (I should mention that Ive got a lock on the
fast with my binarysearch implementation, but its far
from elegant!)
Since my equipment is limited pi(7,900,000) is currently
the highest value, though I might rebuild and compute
pi(10,000,000) if time permits.
See
http://mathworld.wolfram.com/PrimeCountingFunction.html
if you need details on what pi() is supposed to be. Note
that Im not looking for approximations.
(basically, very late Monday night), as measured by my
newsspool, give or take 4 hours or so (as I poll my uplink
every 4 hours and am not sure I can see when it received
your message).
If your compiler cant accept unsigned long long for some
reason, dont worry too much about it; the rst rule
allows me to make reasonable edits, and Ill preserve your
post for posterity (or at least as long as the web subnode
stays up), if you submit it by posting your algorithm
on Usenet; that should help resolve any issues.
No, theres no prize as such beyond the happy glow of
recognition from your fellow denizens on Usenet. :-)
Happy hacking.
--
#191, ewill3@earthlink.net
Its still legal to go .sigless.
===
Subject: Re: Way off topic
>Vuos cryoer que clea macrhe asusi en frnaias ? ;-)
>J-L.
My French is not that great, but I can read it.
It is as I have claimed, the brain does process
the information after it is read.
We can read even more distorted words.
>The Last Danish Pastry <TheLastDanishPastry@yahoo.com> a
crit dans le
>> Aoccdrnig to rscheearch at an Elingsh uinervtisy, it
deosnt mttaer in
>what
>> oredr the ltteers in a wrod are, the olny iprmoetnt tihng
is taht the
>frist
>> and lsat ltteers are in the rghit pclae. The rset can be a
toatl mses
and
>> you can sitll raed it wouthit a porbelm. Tihs is bcuseae
we do not raed
>> ervey lteter by itslef but the wrod as a wlohe.
--
This address is for information only. I do not claim that
these views
are those of the Statistics Department or of Purdue
University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
===
Subject: Re: Way off topic
>Aoccdrnig to rscheearch at an Elingsh uinervtisy, it deosnt
mttaer in
what
>oredr the ltteers in a wrod are, the olny iprmoetnt tihng is
taht the
frist
>and lsat ltteers are in the rghit pclae. The rset can be a
toatl mses and
>you can sitll raed it wouthit a porbelm. Tihs is bcuseae we
do not raed
>ervey lteter by itslef but the wrod as a wlohe.
I do not believe this. We may take in the word as a whole,
but the internal processor still does it letter by letter,
unless the word is recognized quickly. Those with larger
vocabularies will nd more words to separate out.
--
This address is for information only. I do not claim that
these views
are those of the Statistics Department or of Purdue
University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
===
Subject: Re: Way off topic
In sci.math, The Last Danish Pastry
<TheLastDanishPastry@yahoo.com>
<bjsgsu$mt6lr$1@ID-11651.news.uni-berlin.de>:
> Aoccdrnig to rscheearch at an Elingsh uinervtisy, it
deosnt mttaer in
what
> oredr the ltteers in a wrod are, the olny iprmoetnt tihng
is taht the
frist
> and lsat ltteers are in the rghit pclae. The rset can be a
toatl mses and
> you can sitll raed it wouthit a porbelm. Tihs is bcuseae we
do not raed
> ervey lteter by itslef but the wrod as a wlohe.
Fascinating, and quite readable. :-) Ill admit this is
probably
more appropriate to sci.psychology or some such but the idea
has occurred to me personally.
I might have to whip up a Perl script to test this at some
point.
Of course spelling checkers will blow chunks over such
submissions. :-)
--
#191, ewill3@earthlink.net
Its still legal to go .sigless.
===
Subject: Re: Way off topic
(I am reposting this reply, since it is not appearing on Math
Forum,
which sci.math posts usually do immediately when posted
through
Google.)
> Aoccdrnig to rscheearch at an Elingsh uinervtisy, it
deosnt mttaer in
what
> oredr the ltteers in a wrod are, the olny iprmoetnt tihng
is taht the
frist
> and lsat ltteers are in the rghit pclae. The rset can be a
toatl mses and
> you can sitll raed it wouthit a porbelm. Tihs is bcuseae we
do not raed
> ervey lteter by itslef but the wrod as a wlohe.

Since we are being off-topic here, I might as well copy/paste
part of
a reply to a puzzle/joke I posted to rec.puzzles. (They like
this
kind of stuff at rec.puzzles. Perhaps TLDP, or someone else,
should
cross-post this thread there. {I myself will not personally.})
> --> (syk) -- ^ |
> | V
> ^ |
> | V
> <---<---<---<-
> (View with xed-font.)
> Leroy Quet
Okay, okay, I will give a clue. (Yes, this is not suppose to
be
obvious as to what I mean here.)
First, the ascii-image may be messed up. So I will simplify
it a bit:
--> (syk) -->
<------------
...Or in BASIC-programming-mode:
1) (syk)
2) goto 1
>...
(BTW, no one has really gotten yet my joke. Can you??...)
(Clue: This puzzle/joke is not really that off-topic to THIS
thread
anyway...)
Leroy
Quet
===
Subject: Please help with Induction.
Hello
Im trying to prove the following using Mathematical
Induction. This is as
far as I have got:
Show that 1^3 + 2^3 + ... + n^3 = [n(n+1)/2] ^2 whenever n is
a positive
integer.
Basis Step. let n = 1
n^3 = 1 and [n(n+1)/2] ^2 = (1(2))^2 / 2 = 1 This step is
true.
Induction Hypothesis. let n = k
1^3 + 2^3 + ... + k^3 = [k(k+1)/2] ^2
Induction Proof. n = (k+1)
1^3 + 2^3 + ... + (k+1)^3 = [((k+1)((k+1)+1))/2] ^2
===
Subject: Re: Please help with Induction.
> Hello
> Im trying to prove the following using Mathematical
Induction. This is as
> far as I have got:
> Show that 1^3 + 2^3 + ... + n^3 = [n(n+1)/2] ^2 whenever n
is a positive
> integer.
> Basis Step. let n = 1
> n^3 = 1 and [n(n+1)/2] ^2 = (1(2))^2 / 2 = 1 This step is
true.
> Induction Hypothesis. let n = k
> 1^3 + 2^3 + ... + k^3 = [k(k+1)/2] ^2
> Induction Proof. n = (k+1)
> 1^3 + 2^3 + ... + (k+1)^3 = [((k+1)((k+1)+1))/2] ^2
You want to show Sum[j^3,{i,1,n+1}] = ((n+1)(n+2)/2))^2 (Call
this equ. 1)
Now we add the (n+1)th term, so we get:
((n)(n+1)/2))^2 + (n+1)^3
Simplifying, we get (((n+1)(n+2))/2)^2
Look like equ 1? Sure does
See if you can follow the steps as it is a bit tricky, but
hopefully this
will give enough guidance.
The trick is to take out the IH, and then add the n+1 term to
it and
simplify.
Since we have k^3 evaluated at (n+ 1), that is where the
(n+1)^3 term comes
from.
Make sense? HTH, Flip
===
Subject: Re: Please help with Induction.
> Hello
> Im trying to prove the following using Mathematical
Induction. This
> is as far as I have got:
:
You might start by searching on induction in this newsgroup,
since
there
have been numerous threads on induction.
===
Subject: Re: David Ullrich on Identity
> You should check out http://megafoundation.org/
> Harris has nally found people who understand him there.
What part of Ax[Qu(x) -> ~(x = x)] do YOU fail to understand?
>David Ullrich says:
>And yes, identity is in _fact_ reexive. To
>refute that statement you need to give an
>example of something which is not identical
>to itself. The idea that there is something
>which is _not_ identical to itself is simply
>ludicrous: Thats what identity _means_: A
>thing is identical to itself and to nothing
>For a contrasting standpoint, see
>************************************************************
**
>David Ullrich asks:
>Whats an example of something thats not identical
>************************************************************
**
>David Ullrich dares:
>Exhibit of proof of Ex~(x=x) from
>C1-C4 and someone will point out the error.
>C1 AxAy[x=y -> Az(z in x <-> z in y)] LL1
>C2 AxAy[Az(z in x <-> z in y) -> Az(x in z <-> y in z)] LL2
>C3 EyAx[x in y <-> Et(x in t) & A] (with y not free in A)
>Classication
>C4 AxAy[Az(z in x <-> z in y) -> {Et(x in t & y in t) <->
x=y}]
> Weak
>Would someone be kind enough help David out with a proof?
>*************************************************************
>David Ullrich remonstrates:
>I blunder by saying that equality is reexive by denition?
>Huh. Do you have any idea what the word denition means?
>Homework for David Ullrich:
>1) What philosopher said:
> ...denitions are available only for transforming
> truths, not for founding them.
>2) In your own words, explain why (or why not) you think
> this is true.
>--John
> ************************
> David C. Ullrich
===
Subject: Re: Legal logic ? puzzle.
> When reading certain high court judgement reasons, it is
clear to me
> that often some quite profound thinking in being exercised.
> But when I dialog with legal people: real live ones or
via *.legal.*
> newsgroups; it seems that they are just phrase matching
clerks.
> Im convinced of the left-brain/right-brain theory which
divides
> humanity into spacial/verbal thinkers.
> The matter which I am desperately struggling with, concerns
a
> time sequence of event; with the assumption that cause
does NOT
> follow effect.
> As a science graduate, I nd it best and natural to
demonstrate my
> argument by a minimalist model. None of the legal people
know
> what Im talking about and worse still they dont volunteer
that
> my model is (to them) not understandable.
> Is the following understandable ? How can I impove it ?
> Please answer some of the questions.
> The following conditions apply:-
> -- [snip details]-
> This is too complicated for me to want to read. Also you
seem to be using
> differnent terms for the same thing - client, debtor which
is not
appealing
> to a math person.
Well part of applied maths is realising that client and
debtor
are the same entity.
And its not a maths problem, like mechanically manipulating
some symbols. Rather one needs to be able to visualise the
sequence of events and analyse their interrelationships in the
context of the questions.
If I cant explain this to scientically trained persons, how
will
I explain the legal-types ?!
== Chris Glur.
===
Subject: karnaugh maps
Theyre trying to foist Karnaugh maps off on me with only a
handwaving
explanation of how youre supposed to use them; along these
lines:
1) each loops must comprise 2^n squares of the map, for some n
2) every 1 on the map must belong to at least one loop
3) try to use as few loops as possible
4) try to make loops as big as possible
script in Maple that is supposed to verify whether a given
solution uses
the
minimal number of loops (ie: whether the proposed solution is
optimal when
condition (4) is left out of consideration), but it (the
script) is too
complex
to run. (And, like I said, once condition (4) enters the
picture, I dont
even
know what optimal means anymore.) Can anybody point me to an
exact
account of
this technique?
===
Subject: Re: karnaugh maps
===
>Subject: karnaugh maps
>Message-id:
<49M8b.954202$3C2.21581237@news3.calgary.shaw.caTheyre
trying to foist Karnaugh maps off on me with only a handwaving
>explanation of how youre supposed to use them; along these
lines:
>1) each loops must comprise 2^n squares of the map, for some
n
>2) every 1 on the map must belong to at least one loop
>3) try to use as few loops as possible
>4) try to make loops as big as possible
>script in Maple that is supposed to verify whether a given
solution uses
the
>minimal number of loops (ie: whether the proposed solution
is optimal when
>condition (4) is left out of consideration), but it (the
script) is too
complex
>to run. (And, like I said, once condition (4) enters the
picture, I dont
even
>know what optimal means anymore.) Can anybody point me to an
exact
account
>of this technique?
The goal is to reduce the number of terms in the boolean
equation. Each
term
becomes a piece of hardware, see
http://members.aol.com/owagiveaway/add_opt.gif
Fewer terms means less hardware, less complexity in the
circuit wiring and
better propagation delay. The whole idea behind K-maps is to
nd common
terms
that can be factored out to make the circuit simpler. If your
boolean
equation
is
Y = ABC + ABE
you can implement this with two three-input AND gates plus a
two-input OR
gate.
Now you may not have any three-input AND gates. So it would
take 4
two-input
AND gates plus an Or gate and use three propagation delays.
But the
optimized
boolean expression is
Y = AB(C + E)
which can be implemented with three two-input gates and
brings us back to
two
propagation delays. Making the loops as big as possible means
that
Y = AB(C + E)
is a better solution than
Y = A(BC + BE)
--
Mensanator
2 of Clubs
http://members.aol.com/mensanator666/2ofclubs/2ofclubs.htm
===
Subject: Re: karnaugh maps
Originator: jgamble@ripco.com (John M. Gamble)
>Theyre trying to foist Karnaugh maps off on me with only a
handwaving
>explanation of how youre supposed to use them; along these
lines:
>1) each loops must comprise 2^n squares of the map, for some
n
>2) every 1 on the map must belong to at least one loop
>3) try to use as few loops as possible
>4) try to make loops as big as possible
>script in Maple that is supposed to verify whether a given
solution uses
the
>minimal number of loops (ie: whether the proposed solution
is optimal when
>condition (4) is left out of consideration), but it (the
script) is too
complex
>to run. (And, like I said, once condition (4) enters the
picture, I dont
even
>know what optimal means anymore.) Can anybody point me to an
exact
account of
>this technique?
Yeah, if your description is complete, they did a lousy job
of explaining
it.
A rst-year electrical engineering text on digital logic
should have a
chapter on the subject. This should satisfy some of your
questions.
Did they just drop the K-map on you with no background? The
purpose
of the thing is to make handling truth tables easier. Boolean
equations have 1 and 0 outputs depending upon their inputs.
Sometimes
you want to go in the other direction, and create a boolean
equation
from the outputs. (Note to other readers with trigger-happy
reply
ngers: yes, i know that im simplifying a lot).
For example: lets say that you want to nd the equation for
the
very simple truth table (monospace font to read this is
advised):
A B| Result
-----|-------
0 0| 1
0 1| 1
1 0| 1
1 1| 0
The K-map could look like:
_ |
B | B
-------------
_ | | |
A | 1 | 1 |
-----------------
| | |
A | 1 | 0 |
-------------
Im going to use ~ for Not, * for And, and + for Or.
Sorry, it will look a little messy.
Each square with a 1 represents an output that you want,
gotten by looking at the row and column it is in. The
upper left hand corner is in the ~A and ~B column, so
its an output that you want, and its term will be
~A*~B.
Now obviously, you could circle each individual 1, and come
up with the solution Result = ~A*~B + ~A*B + A*~B.
But this could be simplied. With a small equation like this
its pretty obvious how, but its even easier if one circles
the pairs of ones. If you do that, youll see that, in this
example, one circle cuts across the ~B and B columns, while
the other circle cuts across the ~A and A rows. This is
one way of showing that the not-X and X variables cancel
each other out, and can be dropped. This leaves you
with ~A (the B columns cancelled out) and ~B (the A rows
cancelled out).
The solution is what you have left in the circles:
Result = ~A + ~B.
This is a simpler equation.
K-maps can be drawn for equations with more inputs, of course.
A four-variable K-map would look like:
_ |
B | B
-------------------------
| | | | | _
_ | | | | | D
A -----------------------------
| | | | |
| | | | |
----------------------------- D
| | | | |
| | | | |
A -----------------------------
| | | | | _
| | | | | D
-------------------------
_ | | _
C | C | C
And i personally found six variables to be the limit of what
i could do by hand. The more ones that you can circle in a
group of 2, 4, 8 (it has to be a power of 2), the more outputs
can be grouped together in a single, small term.
--
-john
February 28 1997: Last day libraries could order catalogue
cards
from the Library of Congress.
===
Subject: Re: Question about i
> When working with complex numbers, is there any reason that
we cannot
take
> even roots of negative integers, other than the square root
of -1? Or is
it
> just some sort of dened convention? Or does no such
restriction exist,
and
> in some space (-1)^(.25) has meaning?
> --riverman
Thats the neat thing about i. We dont have to invent a new
numeral
called j to represent the square root of i. The reason why we
have to
take the square root of -1 to be i is because we take 1 to be
the
multaplicative identity, and since the additive opposite of
-1 is not
the multaplicative identity, but also produces a number on the
opposite side, 1*1 and -1*-1 are both 1. So 1 is even and -1
is odd.
However, since i is neither even nor odd, it in fact rotates
the
system, or mixxes it up differently. sqrt(i) can be any
number which,
when squared, gives the result i. Take a+bi... then square
it. It has
the form a^2 + 2*a*bi + (bi)^2. However, this is just
(a^2-b^2) +
2abi. If |a|=|b| then the (a^2-b^2) term disappears.
Furthermore, if
2abi=i, then 2ab=1, and ab=1/2. Since |a|=|b|, and 1/2 is
even, since
it is positive, then a and b are the same, or are equal to c.
So
c^2=1/2, or c = +-1/sqrt(2), which can be written equally
well as
+-sqrt(2)/2. Taking c to be the positive case, the square
root of i
would then be equal to (1+i)*sqrt(2)/2. And thus, you have
your square
root of i.
Of course, now, with some matrix type stuff, things have four
square
roots, and we do have 1, i, j, and k. But thats a different
matter,
which involves octonians... I think. Well, youll learn it
eventually... maybe. :P
(...Starblade Riven Darksquall...)
===
Subject: Re: Question about i
> When working with complex numbers, is there any reason that
we cannot
take
> even roots of negative integers, other than the square root
of -1? Or is
it
> just some sort of dened convention? Or does no such
restriction exist,
and
> in some space (-1)^(.25) has meaning?
Yes, (-1)^(.25) has meaning to computers.
Where the confusion usually comes is that Quantum Mechanics
wackos
believe that complex numbers have something to do with
physics.
As in they believe that there are negative probabilites.
So, whats usually done is to take the square root of Pascal,
and add two to get all answers to complex number questions.

> --riverman
===
Subject: Re: Question about i
> Theres no convention for sqrt for non-real arguments,
> though I suppose it wouldnt be too hard to make one up
> when one needed it.
Taking the principal branch is a very common convention.
> But you cant do it in such a way that
> sqrt(zw) = sqrt(z) sqrt(w)
> is always true :-)
Right, since you and Randy are thinking of sqrt as a function
returning
a single complex value.
But if we think of sqrt as a set-valued function, returning a
set of
two
complex values for any nonzero argument, then we do always
have
sqrt(zw) = sqrt(z) sqrt(w).
David
===
Subject: Re: Question about i
> at 06:59 PM, riverman <nospam@sorry.com> said:
>to me how to manage negative bases to reducible rational
powers?
> The same as any exponential. You must at all times
distinguish
> bewtween single valued expressions and expressions that
represent
> multiple values.
Good advice.
> For example, Ive
>been teaching the advanced algebra kids that (-16)^(2/4)
does not
>exist,
> Which is wrong.
Agreed.
> It can refer to two distinct values.
True. Or perhaps its actually nicer to think of it as
possibly referring
to a _unique_ set of values, that set having two members.
Of course, another possibility is that it refers to a single
principal
value: 4i.
So back to what I called Good advice above. We would need to
be clear
as
to how x^y is dened in such a case; whether x^y is to be
taken as
principal-valued or set-valued.
>as it is not well-dened
> It is up to factors of cos Pi/2 + i sin Pi/2.
??? Maybe you meant up to factors of -1.
In any event, its important to realize that it _can_ be
well-dened.
Whether riverman and his text have done so is a different
matter.
> (-16)^(2/4) should equal (-16)^(1/2) = 4i,
And so it does if were dealing with principal values.
> Thats only one of the four possible values.
I hope you meant to say Thats only one of the two possible
values., in
agreement with your previous statement that It can refer to
two distinct
values.
David
===
Subject: Re: Question about i
Robin Chapman <rjc@ivorynospamtower.freeserve.co.uk>
scribbled the
following:
>> Theres no convention for sqrt for non-real arguments,
>> though I suppose it wouldnt be too hard to make one up
>> when one needed it.
> But you cant do it in such a way that
> sqrt(zw) = sqrt(z) sqrt(w)
> is always true :-)
Because otherwise:
1 == sqrt(1) == sqrt(-1 * -1) == sqrt(-1) * sqrt(-1) == i * i
== -1,
right?
--
/-- Joona Palaste (palaste@cc.helsinki.)
---------------------------
| Kingpriest of The Flying Lemon Tree G++ FR FW+ M- #108 D+
ADA N+++|
| http://www.helsinki./~palaste W++ B OP+ |
----------------------------------------- Finland rules!
------------/
This isnt right. This isnt even wrong.
- Wolfgang Pauli
===
Subject: Re: Question about i
> Robin Chapman <rjc@ivorynospamtower.freeserve.co.uk>
scribbled the
> following:
>> Theres no convention for sqrt for non-real arguments,
>> though I suppose it wouldnt be too hard to make one up
>> when one needed it.
> But you cant do it in such a way that
> sqrt(zw) = sqrt(z) sqrt(w)
> is always true :-)
> Because otherwise:
> 1 == sqrt(1) == sqrt(-1 * -1) == sqrt(-1) * sqrt(-1) == i *
i == -1,
> right?
True, if we want sqrt to yield a single complex value. But,
as I just
noted in my response to Robin, if we take sqrt to be
set-valued, there
is no problem:
{-1,1} = sqrt(1) = sqrt(-1 * -1) = sqrt(-1) * sqrt(-1) =
{-i,i} * {-i,i} =
{-1,1}
David
===
Subject: Re: Question about i
> When working with complex numbers, is there any reason that
we cannot
take
> even roots of negative integers, other than the square root
of -1? Or is
it
> just some sort of dened convention? Or does no such
restriction exist,
and
> in some space (-1)^(.25) has meaning?
> --riverman
Complex roots of complex numbers are dened:
[(a+bi)^(c+di)]=e^[(c+di)ln(a+bi)], for any real a,b,c,d.
(where e is the base of the natural logarithims (ln))
i.e. [(sqrt(-2))^(sqrt(-2)]= a+bi, for some real a and b.
Witt
===
Subject: Re: programming matlab problem
> Maybe not the best newsgroup, but I dont know a better one.
> I cant gure out how to program the following in matlab:
> (iow, to ll hyper-matrices of arbitrary size)
> function x = ll(N = positive integer)
> H(1,1, ..., 1).matrix = 1 % (H has N arguments)
> % end function
> The problem is that you need N for-loops.
> Wilbert
Heres a trick to ll a matrix of given dimensions with
entries equal to
scalar x:
zeros(m1,m2,...,mN) + x
===
Subject: i * (3,4)=( - 4, 3)= - 4 +3 * i and the Manifesto di
Bombelli
Introducing the dot-multiplication in the vector-space (
R2,+,r.s.m )
results in an Euklidian vectorspace, inside you can nd a
commutative
eld (R2,+,* ).
Here * is the Bombelli-multiplication: i * i = - 1 , or in
Hmiltons
notation:
(a,b)*(c,d)=(ac-bd,ad+bc), and i rotates a vector 90 degrees
to the
left by
i*(a,b)=(-b,a).
In modern-talking math:A vector, an ordered tupel by
denition, can
be expressed in the terms of a vectorspace-basis. In R(=R1)
the vector
(a) with the standard-basis ( 1 ) gives: (a)=a*1 and thats
-of
course - equal to a. Adjungate a foreign element to R, you go
2D.(a,b)
can be notated as a linear combination of the basis-elements:
c*1+d*element. With the standard-basis
( (1,0) , (0,1) )=( 1 , (0,1) ) or ( 1 , i ) you get a + b *
i,
looking different from (a,b), but identical.
You can write and calculate vectors in mixed mode now.
I propose the name Bombelli vector-space.

Real arrows showing winds on a weather-map.
Attach (add) to a number of points the difference in
coordinates from
one central point,multiplied by (0,1)=i.Improve this rst
model by
multiplying
(cos @,sin @)=cos @+i*sin@ - allowing for friction alpha from
10
(over sea) to 35 (over land)(-alpha on the southern
hemisphere), and
scale everything by a factor 1/10 (rsp. -1/10) and add a
common
velocity to the east. It models
the winds of the inner part of a depression
- done solely with Bombellis operations - and ...........
we stayed plain real.

Two questions arise, sensitive questions for some - and the
answer is
left to them :
Is there any difference to (C,+,*)? -
can you tell this difference to your computer?
If you call the y-axis an imaginary axis, do you add any
mathematical properties, or do you just change the name? Is
the
Gauss-plane, complex
plane or the Aragand-diagramm more than just a
Fata-Motgana-Reection
of the real plane?
Why - twohundred years after Wessel - this question
.9fberhaupt ?
In a math-lesson you get sometimes the impression, an
extraterrestical
element has been adjungated to R. My opinion:
i - is - no - longer - imaginary
Copy this Hero
As in sci.math you can not display applets,you might take a
look at
www.i-z.eu.tt or
www.i-is-no-longer-imaginary.gmxhome.de
There You nd the Manifesto, i quoted above, in a printable
version
for the
pinboard and a (free) mixed-mode calculator.
As the computer makes no difference - do You have to ?
Hero.
===
Subject: f(n^3 + m^3) = f(n)^3 + f(m)^3
It is known that if a map f from N to N satises f(1) > 0 and
f(n^2 + m^2) = f(n)^2 + f(m)^2
for all n, m in N, then f is the identity map.
(Here N is the set of natural numbers including 0.)
What happens if we consider cubes instead of squares...?
And in case the conditions become too weak, let us extend the
domain and
range
to all the integers to make the conditions a little bit
stronger.
So, to be exact, if a map f from Z to Z satises f(1) > 0 and
f(n^3 + m^3) = f(n)^3 + f(m)^3
for all n, m in Z, then can we conclude that f is the
identity map?
TIA.
Tad
===
Subject: Re: How Many Blinkin Lights On The Tree?
===
>Subject: Re: How Many Blinkin Lights On The Tree?
>Message-id: <3f63aff6$0$28119$afc38c87@news.optusnet.com.au>
God isnt responsible for the bad things that happen, Satan
is. Read
>> your damn Bible, specically, the Book of Job.
>Lol and god created Satan knowing *FULL* well what would
happen
>(unless hes not omniscient).
Did God create Satan?
Does the Bible say God is omniscient?
--
Mensanator
2 of Clubs
http://members.aol.com/mensanator666/2ofclubs/2ofclubs.htm
===
Subject: Re: How Many Blinkin Lights On The Tree?
>Neither of these messages belong on a board about
MATHEMATICS.
>This board is about MATHEMATICS.
>You know, the science thing with numbers, you may have heard
of it.
>Sometimes called maths or math.
>Im sure there are many boards dealing with God or the
absence of God
>or the Book of Job or Numerology, please utilise them
At least I dont top-post.
--
Mensanator
2 of Clubs
http://members.aol.com/mensanator666/2ofclubs/2ofclubs.htm
===
Subject: Re: How Many Blinkin Lights On The Tree?
===
>Subject: Re: How Many Blinkin Lights On The Tree?
>Message-id: <iqj6mvcq81o2i7pe4pjtp92shgnvbbj5@4ax.com If
God starts 800 plus forest res with lightning strikes in
British
> Columbia and devastates several hundreds of homes and
business with
> re, how many blinkin bulbs should we put on our decorated
trees and
> houses this winter?
>>God isnt responsible for the bad things that happen, Satan
is. Read
>>your damn Bible, specically, the Book of Job.
>To be fair, God gives his permission to Satan to send Job
all those
>calamities (the only limit is that Job cant be killed) , so
in a
>sense God is responsible too. In fact, chap.1 v.11 implies
that it is
>God who really sends the calamities: But put forth Your hand
now and
>touch all that he has; he will surely curse You to Your face.
Ok, Satan asks God to smite Job and watch what happens. But
does God take
Satan
up on that request? He does not. In the very next verse you
have
And the Lord said unto Satan Behold, all that he hath is in
_thy_ power;

(emphasis added)
so yes, God _allows_ bad things to happen, but God does _not_
actually do
them.
Obviously, people have been asking how can God do so many bad
things since
the
beginning of religion. The writers of the Bible gured out a
clever
explanation that lets God off the hook. Thus, the Book of
Job came to be
written.
And later in v.16 one of the calamities is described as the
re of God
but
thats because Job was unaware of the deal between God and
Satan. Job just
assumed all natural disasters are the work of God and like
your typical
naive
person, blames God for the disaster (it may have been Gods
SUV, but Satan
was
driving it).
--
Mensanator
2 of Clubs
http://members.aol.com/mensanator666/2ofclubs/2ofclubs.htm
===
Subject: oh....my problem...
nd all orbit of cyclic subgroup H=<(1 3 5)> in symmetric
group S5
-------------------------
H=<(1 3 5)>={ e , (1 3 5), (1 5 3)}
1. {2}, {4}, {1,3,5}
2. {1},{2},{3},{4},{5},{1,3,5}
which is correct answer 1 or 2 ??
===
Subject: Re: Four Color Theorem Simplied
> ALL YOU NEED TO KNOW ABOUT THE FOUR COLOR THEOREM!!!
> DEFINITIONS:
> CHI(G) is the chromatic number of graph G.
> If CHI(G) <= k, then graph G is k-colorable
> If CHI(G) = k, then graph G is k-chroma.
> A graph is planar if it contains no subgraph homeomorphic to
> K5 or K3,3.
> Every maximal planar graph [MPG] has exactly 3n-6 edges.
> If every MPG is 4-colorable; then every planar graph is
4-colorable.
> All MPGs are generated by the intersection of two
triangulations of an
> n-sided polygon.
> Every triangulation of every convex polygon is 3-colorable.
> Every 5-colorable graph has a subgraph homeomorphic to the
complete graph
K5.
> All 4-partite graphs, both planar and non-planar, are
4-colorable.
How about; Every 5-chroma non planar graph has a subgraph
homeomorphic (or is it isomorphic) to the complete graph
K5.? Is
this a valid statement?
===
Subject: Re: Four Color Theorem Simplied
> ALL YOU NEED TO KNOW ABOUT THE FOUR COLOR THEOREM!!!
> DEFINITIONS:
> CHI(G) is the chromatic number of graph G.
> If CHI(G) <= k, then graph G is k-colorable
> If CHI(G) = k, then graph G is k-chroma.
> A graph is planar if it contains no subgraph homeomorphic to
> K5 or K3,3.
> Every maximal planar graph [MPG] has exactly 3n-6 edges.
> If every MPG is 4-colorable; then every planar graph is
4-colorable.
> All MPGs are generated by the intersection of two
triangulations of an
> n-sided polygon.
> Every triangulation of every convex polygon is 3-colorable.
> Every 5-colorable graph has a subgraph homeomorphic to the
complete graph
K5.
> All 4-partite graphs, both planar and non-planar, are
4-colorable.
Your pentultimate statement is clearly wrong.
===
Subject: Re: Four Color Theorem Simplied
> ALL YOU NEED TO KNOW ABOUT THE FOUR COLOR THEOREM!!!
> Every 5-colorable graph has a subgraph homeomorphic to the
complete
graph
> K5.

> Your pentultimate statement is clearly wrong.
Thars penultimate! Anyway, what is the smallest n for which
it is
not true? An example, please?
===
Subject: Re: Four Color Theorem Simplied
> ALL YOU NEED TO KNOW ABOUT THE FOUR COLOR THEOREM!!!
Every 5-colorable graph has a subgraph homeomorphic to the
complete
graph
> K5.
> Your pentultimate statement is clearly wrong.
> Thars penultimate! Anyway, what is the smallest n for
which it is
> not true? An example, please?
You give no special denitions of terms, so presumably your
usage is
standard for treatments of the four-color problem.
In standard parlance, four-colorable implies ve-colorable,
so a trivial
example (e.g. of a planar graph) may be adduced.
Even granting that you intended ve-colorable in a stronger
sense of
ve- but not four-colorable, the statement would still be
incorrect.
Here is an example which may be drawn on the surface of a
torus.
Imagine three circles running parallel around the hole in a
doughnut.
Subdivide one of the resulting three regions so obtained with
ve segments
cutting perpendicularly across that region.
The map now has 7 regions, lacks any embedded subgraph K5,
and is
ve-colorable but not four-colorable.
Your question about the smallest n is interesting. Obviously
if there were
a map with 5 regions that was ve- but not four-colorable, as
a graph it
would be (isomorphic to) K5 (the complete graph on 5
vertices). Is there a
counterexample with n=6? I suspect not, but apart from an
exhaustive
examination of cases, I cannot think off-hand of an approach
to showing
this.
Perhaps you will have better insight.