Conference rusure::math

    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.

>    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
   

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

	Off the top of my head, some of my favourite theorems omitted from
the list are:

		(1) insolvability of the quintic.

		(2) Schur's computation of the character tables for the
	symmetric groups.

		(3) the Conway-Sloane packing of spheres in 24-dimensional
	space (although I don't really understand it).

		(4) the identification of all finite simple groups. (No one
	individual understands it all, least of all me!)

		(5) int(-% -> +%) exp(-x�).

		(6) central limit theorem.

		(7) Ramsey's Theorem: a k-colouring of the r-tuples of an
	infinite set contains a monochromatic subset.

		(8) There are n^(n-2) trees on n labelled vertices.

		(9) Cayley-Hamilton: a matrix satisfies its own characteristic
	equation.

		(10) Cauchy's Theorem for analytic functions (loop integrals
	vanishing in the complex plane).

	Incidentally, Favourites Theorems have been discussed previously in
this notesfile somewhere, but I haven't hunted for it.

regards,
Andrew.

    May or may not be in the right category, but G�del's [in]completeness
    theorem deserves mention.
    
    There's a lovely theorem by Banach, which I recall as ff:
    
    Let f be continuous on [a,b].  Let V(f) be the "variation" of f,
    that is, the total amount of change in f(x) over the [a,b] interval.
    Not just max-min -- the sum of all y-changes (which can be rigorously
    defined as the limit sum of |f(X_<n>) - f(X_<n+1>)| for points X_<n>
    in [a,b] and n->infinity).
    
    V(constant) = 0; V(x) = |a| + |b|; V(|x|) = |a| + |b| also; etc.
    
    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.
    
    Given all that, and a nebulous-to-nonexistent relation between V(f) and
    N(y), it is a bit surprising to discover that N(y) is integrable and
    
    	  +inf            x=b
    	int    N(y) dy  =  V (f)
    	  -inf		  x=a
    
    At least I find that rather astounding.  Wish I could explain it better.
    
      John

�    Given all that, and a nebulous-to-nonexistent relation between V(f) and
�    N(y), it is a bit surprising to discover that N(y) is integrable and
�    
�    	  +inf            x=b
�    	int    N(y) dy  =  V (f)
�    	  -inf		  x=a
�    
�    At least I find that rather astounding.  Wish I could explain it better.

    I don't know if I'm missing something, but the theorem seems fairly
    obvious if you think of it in a naive sense, drawing a "graph" of
    a continuous f(x) - particularly if you turn the graph on its side.

    Rigorous proof may be delicate given the perverse things which *can*
    happen in real analysis (continuous, nowhere differentiable functions,
    etc.)

    A few I really like:

	o    The uniformization theorem from complex analysis
	     (the names of those who contributed to getting this right
	      make an impressive list: Riemann, Dirichlet, Klein,
	      Poincare, Koebe, etc.)

	o    The generalized Gauss-Bonnet theorem

	o    Gauss theorem on quadratic reciprocity

	o    E. Cartan's classification of semisimple Lie groups

	o    Fourier's theorem (on Fourier series, eigenfunction expansions)

    On could just go on and on...

    Actually, I don't find isolated theorems as attractive as I do
    certain major conceptualizations.

    - Jim

	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

Re: .0,

>    17.	Every number greater than 77 is the sum of integers the sum
>		of whose reciprocals is 1.

	I realize this was just a short discription of the result, but the
	critical thing is that that the numbers are the sum of *distinct*
	*positive* integers, whose reciprocals sum to one.

	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.

>    18.	The number of representations of an odd number as the sum
>		of 4 squares is 8 times the sum of its divisors, 24 times
>		the sum of its odd divisors.

	This is a peculiar statement.   Something is wrong.
		(1) *every* divisor of an odd number is odd.
		(2) the sum of divisor of an odd number is not necessarily a
		multiple of 3.
		(3) 1 = 0� + 0� + 0� + 1�.   I don't see any other solutions!

Re .7,	

>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.

	|b-a|


Regards,
Andrew.

	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

	re .8,

>> >    18.	The number of representations of an odd number as the sum
>> >		of 4 squares is 8 times the sum of its divisors, 24 times
>> >		the sum of its odd divisors.
>>
>> 	This is a peculiar statement.   Something is wrong.
>> 		(1) *every* divisor of an odd number is odd.
>> 		(2) the sum of divisor of an odd number is not necessarily a
>> 		multiple of 3.
>> 		(3) 1 = 0� + 0� + 0� + 1�.   I don't see any other solutions!

	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.

	Trial: the number of representations of 2 would be C(4,2) * 2^2
	(the number of ways of selecting two of four coordinates to be
	one of +/- 1, the other coordinates being 0) or 24.

	So it must be true. :-)

	Dan

>>>	>	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

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?

    re .12,
    
    Yep.
    
    Eugene

    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

	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.

        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

    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

>    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."