T.R | Title | User | Personal Name | Date | Lines |
---|
1286.1 | "Name" that theorem | RDVAX::NG | | Tue Aug 14 1990 21:28 | 4 |
| Does each of these theorems has somebody's name attach to it?
I am ashamed that I only know about half of them. The ones in
questions are 10, 13-19, 21-24.
|
1286.2 | | HPSTEK::XIA | In my beginning is my end. | Wed Aug 15 1990 00:20 | 64 |
| > 1. e^(i*PI) = 1
Uh Franklin, I don' think this is right.
I think this is known as Euler's identity.
> 2. Euler's formula for a pohyedron : V + F = E + 2
> 3. The number of primes is infinite.
No name for this, but was proved by Euclid.
> 4. There are 5 regular polyhedra.
I think this is a direct result of 2. Hence, by Euler.
> 5. 1 1 1 PI^2
> 1 + --- + --- + --- + ... = ------
> 2^2 3^2 4^2 6
Also proved by Euler.
> 6. A continuous mapping of the closed disk into
> itself has a fixed point.
Brower's (sp?) fixed point theorem.
> 7. There is no rational number whose square is 2.
Proved by some Greek guy of the Pathegorean school.
> 8. PI is transcendental.
Lindmann's theorem
> 9. Every plane map can be coloured with 4 colours.
The four color conjecture claimed to have been proven by the guys in
Illinois with a massive computer run.
> 10. Every prime of the form 4n + 1 is the sum of
> two integral squares in exactly one way.
Don't know about this one.
> 11. The order of subgroup divides the order of the group.
Lagrange's theorem. I am surprised that Sylow's theorem is not in
the list.
Don't know the rest.
...
I am thinking about coming up with my list of favorate theorems, but
can't do it...
Modern mathematics do not have beautiful easily understood, but somewhat
isolated theorems any more. Rather carving small individual sculptures,
modern mathematicians are more interested in laying fundations and
building esoteric monuments. In a sense, it was all David Hilbert's
fault.
Eugene
|
1286.3 | Sorry, 1. should be e^(i*PI) = -1 | NOEDGE::HERMAN | Franklin B. Herman DTN 291-0170 PDM1-1/J9 | Wed Aug 15 1990 12:09 | 16 |
| Re: .2:
>> 1. e^(i*PI) = 1
>>
>>>> Uh Franklin, I don' think this is right.
>>>>
>>>> I think this is known as Euler's identity.
Ooops!! Forgot the -1, i.e,
i*PI
e = -1
Thanks Eugene :^)
-Franklin
|
1286.4 | more candidates | HERON::BUCHANAN | combinatorial bomb squad | Wed Aug 15 1990 16:09 | 34 |
1286.5 | Couple more | VMSDEV::HALLYB | The Smart Money was on Goliath | Thu Aug 16 1990 17:03 | 27 |
1286.6 | a few more (it probably never stops) | ALLVAX::JROTH | It's a bush recording... | Thu Aug 16 1990 18:28 | 38 |
1286.7 | | GUESS::DERAMO | Dan D'Eramo | Fri Aug 17 1990 12:08 | 15 |
| re .5,
>> V(constant) = 0; V(x) = |a| + |b|; V(|x|) = |a| + |b| also; etc.
Are you sure it's not V(x) = V(|x|) = b - a? The sum looks
like a sum of | delta x_n | which would be b - a.
>> Also let N(y) = "the number of x for which f(x) = y"
>>
>> Clearly N(y) is only integer-valued. Here a and b are implicit parameters.
He must be ruling out functions like f(x) = 0 over [a,b] with a < b,
for which N(0) has the cardinality of the continuum.
Dan
|
1286.8 | 3 nits | HERON::BUCHANAN | combinatorial bomb squad | Fri Aug 17 1990 13:19 | 39 |
1286.9 | | GUESS::DERAMO | Dan D'Eramo | Fri Aug 17 1990 15:46 | 9 |
| re .-1,
>> > Are you sure it's not V(x) = V(|x|) = b - a?
>>
>> |b-a|
That's what I said :-) ... the context was functions on [a,b].
Dan
|
1286.10 | | GUESS::DERAMO | Dan D'Eramo | Fri Aug 17 1990 15:57 | 27 |
1286.11 | How does this look, nice person? | VMSDEV::HALLYB | The Smart Money was on Goliath | Fri Aug 17 1990 16:59 | 25 |
| >>> > Are you sure it's not V(x) = V(|x|) = b - a?
>>>
>>> |b-a|
No. Consider the case [-2,3]. I'll just plot the lattice points:
x |x|
3 * *
2 * * *
1 * * *
0 * *
-1 *
-2 *
0 0
In both cases the variation is 5. You can see that the integral of
N(y) dy is also 5.
Note that N(y) is the number of y's for which f(x) = y. Thus there
is no "continuum problem" -- you'd need a vertical line, hence not a
(continuous or otherwise) function, to have that big a value for N(y).
John
|
1286.12 | My favourite... | UTRUST::DEHARTOG | moduladaplisprologopsimulalgol | Sat Aug 18 1990 19:43 | 6 |
| What puzzled me since my teacher at highschool told me was the fact that
"you can't divide an angle in three equal angles with ruler and
compasses".
Or is there a more general theorem behind this?
|
1286.13 | | HPSTEK::XIA | In my beginning is my end. | Sat Aug 18 1990 20:09 | 5 |
| re .12,
Yep.
Eugene
|
1286.14 | O(Reals) > O(Rationals) | CHOVAX::YOUNG | Turf = Ownership - Accountability | Sun Aug 19 1990 05:07 | 8 |
| One thing that is unclear here, is whether it is the *Theorem* or the
*Proof* of that theorem that is being considered.
For instance I find the "Countability" theorem itself to be fairly
plain. But its proof is, to me, one of the most beautiful work in all
of mathematics.
-- Barry
|
1286.15 | rat-warren alert | HERON::BUCHANAN | combinatorial bomb squad | Sun Aug 19 1990 12:54 | 107 |
| Arrgh! There is a rapidly-increasing number of discussions, of
various degrees of gravity, going on here. Let me try and simplify them.
-------------------------------------------------------------------------------
Re my .8, I received mail from Dan saying that there was an interesting
ambiguity in the following:
> With swaps permitted, any number above 23 can be represented.
> With swaps and negative numbers permitted, any number can be
> represented, trivially.
> With negative numbers, but no swaps, I don't know which of the
> numbers -% to 77 now become possible: this is an interesting puzzle.
It concerns "swap". I meant "swap" to mean "duplicate copy", *not*
"transposition". In English English at least, this usage is OK: it derives
from bubblegum-card collecting, or stamp collecting. If I have n copies of the
same card, then n-1 of them are called "swaps" because I am prepared to swap
them with some other collector, for new cards that I don't have yet!
I guess to us computer weenies, the sense "transposition" should be
the first one to come to mind, because of the phrase "swapped out", but on
this occasion, for me, it was the other sense which occurred to me when I
picked the word.
The ambiguity derives special force because of the juxtaposition
of this discussionette with the sum-of-squares mini-conversazione, where
the duplicate-vs-transposition issue is at the heart of counting the
possibilities.
What I was saying on these two problems is still true, I believe.
Status: semantic nit dealt with, interesting puzzle is still open.
-------------------------------------------------------------------------------
Re: Dan's .9
>>> > Are you sure it's not V(x) = V(|x|) = b - a?
>>>
>>> |b-a|
>
> That's what I said :-) ... the context was functions on [a,b].
Yes, of course, I wasn't thinking straight.
But on this same topic, Re John's .11:
>>>> > Are you sure it's not V(x) = V(|x|) = b - a?
>>>>
>>>> |b-a|
>
> No. Consider the case [-2,3].
>
> In both cases the variation is 5. You can see that the integral of
> N(y) dy is also 5.
>
> Note that N(y) is the number of y's for which f(x) = y. Thus there
> is no "continuum problem" -- you'd need a vertical line, hence not a
> (continuous or otherwise) function, to have that big a value for N(y).
>
> John
3 - (-2) *is* 5, John, so you in fact agree with us!
Status: trivial errors hopefully removed, does anyone have anything
substantive to add to John's intro & Jim's reply?
-------------------------------------------------------------------------------
>Re: Dan's .10:
>
> I'm not too familiar with this theorem, though I read about it
> once in a number theory book (Hardy and Wright?). The
> "representations" being counted must be ordered quadruples of
> signed integers in order to get eight solutions for one:
> <-1,0,0,0>,<1,0,0,0>,<0,-1,0,0>,<0,1,0,0>,.... It is supposed
> to be true of all numbers, not just of all odd numbers. The
> 24x part must be about even numbers.
...and the 8x part must be about *odd* numbers (try n=4).
Status: so we probably know what the Theorem must have been now.
Re: Hans' .12
>What puzzled me since my teacher at highschool told me was the fact that
>
> "you can't divide an angle in three equal angles with ruler and
> compasses".
>
>Or is there a more general theorem behind this?
The Theory of Field Extensions is treated in some detail elsewhere in this
Notesfile (try dir /title=Galois).
Status: referenced to another Note.
-------------------------------------------------------------------------------
Re: Barry's .14
> One thing that is unclear here, is whether it is the *Theorem* or the
> *Proof* of that theorem that is being considered.
Yes, but this is too important a point to make in a
recreational topic. :-).
-------------------------------------------------------------------------------
Moral: it is tricky to have 24 simultaneous discussions in the same topic.
|
1286.16 | | GUESS::DERAMO | Dan D'Eramo | Mon Aug 20 1990 02:07 | 13 |
| Rereading .0, the subscribers were asked to rank 24
specific theorems, and produced the ordering shown in .0.
The subscribers weren't asked to name "their" most
"beautiful" theorems.
If I had put together the list, there would be more of a
slant towards topology, set theory, and perhaps number
theory. For example: Urysohn's theorem and Tychonoff's
theorem in topology, Zermelo's Well Ordering theorem.
There should also be a list for favorite conjectures.
Dan
|
1286.17 | | ALLVAX::JROTH | It's a bush recording... | Mon Aug 20 1990 16:01 | 14 |
| This is another instance of "see it once, see it again..."
I was looking at the current issue of the mathematical monthly
and Paul Halmos lists about a dozen major 20'th century ideas
in mathematics.
The list is pretty uneven - amongst truly impenetrable concepts
(like K-theory) there are simple applied things like the FFT.
There was stuff like deBranges proof of the Bieberbach conjecture,
the classification of finite simple groups, the 4 color map theorem,
chaos theory, fractals, etc.
- Jim
|
1286.18 | A (very) little help | CIVAGE::LYNN | Lynn Yarbrough @WNP DTN 427-5663 | Mon Aug 27 1990 16:43 | 19 |
| > 23. The maximum area of a quadrilateral with sides a,b,c,d
> is [(s-a)(s-b)(s-c)(s-d)]^(1/2), s is half the perimeter.
If any of a,b,c,d is zero, this reduces to Hero[n]'s formula for the area
of any triangle. It is also the formula for the area of a quadrilateral
inscribed in a circle. Does there exist a similar formula for the area of a
convex pentagon?
> 24.
> 5[(1 - x^5)(1 - x^10)(1 - x^15)...]^5
> ---------------------------------------- =
> [(1 - x)(1 - x^2)(1 - x^3)(1 - x^4)...]^6
>
> p(4) + p(9)x + p(14)x^2 + ...,
>
> where p(n) is the number of partitions of n.
I'm pretty sure this is Ramanujan's. To paraphrase (that is, misquote)
Hardy's comment on this and similar theorems, "It must be correct. No one
in his right mind would have conceived of it if it weren't."
|