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Thread: Mathematical beauties

  1. #1 Mathematical beauties 
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    One of those little mathematical beauties...


    ,

    where the s are the volume of the even-dimensional unit spheres.

    Not really hard to see since , but it's still quite nice and (I think) fun.

    Wanted to share that. Post what you consider to be a 'mathematical beauty.' I think this could be a very fun and interesting thread.


    "There is a kind of lazy pleasure in useless and out-of-the-way erudition." -Jorge Luis Borges
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  3. #2 Re: Mathematical beauties 
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    Quote Originally Posted by Chemboy
    One of those little mathematical beauties...


    ,

    where the s are the volume of the even-dimensional unit spheres.

    Not really hard to see since , but it's still quite nice and (I think) fun.

    Wanted to share that. Post what you consider to be a 'mathematical beauty.' I think this could be a very fun and interesting thread.
    ,


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  4. #3 Re: Mathematical beauties 
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    Quote Originally Posted by esbo
    Quote Originally Posted by Chemboy
    One of those little mathematical beauties...


    ,

    where the s are the volume of the even-dimensional unit spheres.

    Not really hard to see since , but it's still quite nice and (I think) fun.

    Wanted to share that. Post what you consider to be a 'mathematical beauty.' I think this could be a very fun and interesting thread.
    ,
    Your expression is meaningless. You use n for both the index and the upper limit of the index.
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  5. #4 Re: Mathematical beauties 
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    Quote Originally Posted by mathman
    Quote Originally Posted by esbo
    Quote Originally Posted by Chemboy
    One of those little mathematical beauties...


    ,

    where the s are the volume of the even-dimensional unit spheres.

    Not really hard to see since , but it's still quite nice and (I think) fun.

    Wanted to share that. Post what you consider to be a 'mathematical beauty.' I think this could be a very fun and interesting thread.
    ,
    Your expression is meaningless. You use n for both the index and the upper limit of the index.
    N is any number so if goes from n=0 to the 'n you think of'
    some people use a different notation, it's only meaningles if you don't undertand it
    I hope you can comprehend it now.
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  6. #5  
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    no, it's still meaningless. It would more properly be written and this alternatively becomes but wait... if n=0, then the entire thing becomes undefined... so you MUST mean n=1... wow... funny...
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    Ah yes I should have said the limit of n/n as n appraoches 0, well spotted, I wondered who would be the first to spot that.
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    that doesn't make any sense either, the limit as is completely meaningless, as you are taking a sum, not looking at a graph. The function, though, as n approaches 0, will still be 1. it will, in fact, be 1 everywhere EXCEPT at 0, where it's undefined. This whole concept has no meaning.
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    No it's the sum of the limit of n/n as n approaches n.

    That's wha I meant, it's very elegant don't you think?
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    Not really, It doesn't make any sense. Limits don't have sums.
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    I like the topic of this thread. Let's not derail it by endlessly debating whether every post makes sense.
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    hmmm... Fair enough. Any other great examples?
    Wise men speak because they have something to say; Fools, because they have to say something.
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    Here's one. Let be complex variables.

    Then

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    hmm, okay, I'm going to attempt to understand this one. could also be written as I know its redundant, but its just for my understanding. This would mean you are summing 'n' products, right?
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    Take in mind I do not know of a simple proof (although there may be one).
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    a numerical gem, that I rather love,
    Wise men speak because they have something to say; Fools, because they have to say something.
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    Quote Originally Posted by salsaonline
    Take in mind I do not know of a simple proof (although there may be one).
    It wasn't so much a proof I was looking for, but rather that I grasped the concept you had posted.
    Wise men speak because they have something to say; Fools, because they have to say something.
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    Quote Originally Posted by salsaonline
    I like the topic of this thread. Let's not derail it by endlessly debating whether every post makes sense.
    It's not so easy to derail a stationary train
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  19. #18  
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    Quote Originally Posted by salsaonline
    I like the topic of this thread. Let's not derail it by endlessly debating whether every post makes sense.
    Thank you!
    "There is a kind of lazy pleasure in useless and out-of-the-way erudition." -Jorge Luis Borges
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    How about this: the space of all lines in the 2-dim plane is the same as an open mobius band (the mobius band with its edge removed).
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    I like that one.
    Wise men speak because they have something to say; Fools, because they have to say something.
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  22. #21  
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    Quote Originally Posted by salsaonline
    How about this: the space of all lines in the 2-dim plane is the same as an open mobius band (the mobius band with its edge removed).
    Really? How does that mapping work?

    Can't such lines be mapped uniquely to a single point (that closest to the origin) as well?
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  23. #22  
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    a mobius band is essentially a one sided shape band, and by leaving it open, a line can continue on it forever in any direction.
    Wise men speak because they have something to say; Fools, because they have to say something.
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  24. #23  
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    I know what a mobius strip is, but I was wondering how the mapping between the lines in a 2D plane and the points in a mobius strip/band/plane worked.
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    Quote Originally Posted by MagiMaster
    I know what a mobius strip is, but I was wondering how the mapping between the lines in a 2D plane and the points in a mobius strip/band/plane worked.
    A line in the plane is described by an equation of the form

    ax+by+c = 0

    Note that for t non-zero, (ta, tb, tc) describes the same line as (a,b,c).

    So the set of all lines in the plane is the same as the set of non-zero vectors (a,b,c), where we identify two vectors if they are scalar multiples of each other. This is the space RP^2, the real projective plane.

    Now, I've told a minor fib. There's one nonzero vector (a,b,c) that doesn't define a line: that's the vector (0,0,1). So, in reality, the set of all lines is RP^2 with one point removed.

    It is well-known that RP^2 \ {pt} is the same as an open mobius band. That's something you'll need to look up on your own. It's the kind of thing that's easy to show if you draw a picture, but anyway, this is the idea of the proof.
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  26. #25  
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    A more elementary result:

    Let A be an nxn matrix. Then

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    Quote Originally Posted by Arcane_Mathematician
    a numerical gem, that I rather love,
    I saw that nobody replied to this one... but is it true?
    I computed it, and it gives something around 1.7, not 2.
    Am I reading it the wrong way around?
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  28. #27  
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    Quote Originally Posted by lavoisier
    Quote Originally Posted by Arcane_Mathematician
    a numerical gem, that I rather love,
    I saw that nobody replied to this one... but is it true?
    I computed it, and it gives something around 1.7, not 2.
    Am I reading it the wrong way around?
    is not true.

    Since truth would require that and that would in turn require that which is not true.

    On the other hand .
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  29. #28  
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    Quote Originally Posted by DrRocket
    Quote Originally Posted by lavoisier
    Quote Originally Posted by Arcane_Mathematician
    a numerical gem, that I rather love,
    I saw that nobody replied to this one... but is it true?
    I computed it, and it gives something around 1.7, not 2.
    Am I reading it the wrong way around?
    is not true.

    Since truth would require that and that would in turn require that which is not true.

    On the other hand .
    He means taking the power of sqrt(2) infinitely-many times. Namely, he's claiming that if you define



    and x_0 = sqrt{2}, then the limit as n--> infinity exists and equals 2.

    I think this is true actually, but it requires a little analysis. I remember seeing something like this in the "mathematical fallacies and flim-flams" book.
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  30. #29  
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    In different and simpler words, the infinite tetration of is equal to .

    Although actually, that's not the same thing as taking the power of infinitely many times.
    "There is a kind of lazy pleasure in useless and out-of-the-way erudition." -Jorge Luis Borges
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  31. #30  
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    Quote Originally Posted by salsaonline
    Quote Originally Posted by DrRocket
    Quote Originally Posted by lavoisier
    Quote Originally Posted by Arcane_Mathematician
    a numerical gem, that I rather love,
    I saw that nobody replied to this one... but is it true?
    I computed it, and it gives something around 1.7, not 2.
    Am I reading it the wrong way around?
    is not true.

    Since truth would require that and that would in turn require that which is not true.

    On the other hand .
    He means taking the power of sqrt(2) infinitely-many times. Namely, he's claiming that if you define



    and x_0 = sqrt{2}, then the limit as n--> infinity exists and equals 2.

    I think this is true actually, but it requires a little analysis. I remember seeing something like this in the "mathematical fallacies and flim-flams" book.
    I don't think the statement is true then.

    Let and Now note that . If then which is a contradiction.
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  32. #31  
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    So maybe the limit is 4 then. I couldn't remember which it was. It's something.
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  33. #32  
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    Quote Originally Posted by salsaonline
    So maybe the limit is 4 then. I couldn't remember which it was. It's something.
    That same proof shows that the limit cannot be for any . So either the limit does not exist or it is 1.

    But which shows that the sequence is monotone increasing. Hence the limit does not exist.

    So, unless I have made a mistake (a distinct possibility) the limit does not exist.

    I should have thought of this argument earlier, but I don't put much effort into these sorts of problems unless prodded.

    I like your formiulas much better but don't see a reasonable proof. I suppose that they are actually true for an arbitrary field, but that probably would require some tedious combinatorial argument, I would be much more interested in seeing a clever argument for the complex numbers based on some insight due to Witten. I suspect that there is something deep there.
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    First off, I made a mistake in the definition: Define

    with .

    Then the limit of this sequence IS 2, as I had remembered originally (just got the definition of the sequence wrong). See, for example, Ted Courant, "Tower of powers: a potent paradox", Mathematical Journal 3, (1993) 60-64.

    Or my source, Edward J. Barbeau, "Mathematical Fallacies, Flaws, and Flim-Flam," published by MAA.

    As for the weird expressions I posted: they are all variations on the Atiyah-Hirzebruch theorem that I posted on earlier. The first one is the equivariant index of the operator for the standard torus action on . The equivariant index of this operator is given by that formula according to the Atiyah-Bott-Lefschetz fixed-point formula. That this index is constant follows from the following proof:

    First, reduce the case to an S^1 action by finding a 1-param subgroup of the n-dim torus with isolated fixed points.

    Second, apply the same reasoning used in the Atiyah-Hirzebruch theorem to show this equivariant index is constant. Taking appropriately chosen limits, it's not hard to see what that constant is.

    Finally, the case for the whole torus follows from the fact that 1-param subgroups are dense.
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  35. #34  
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    I meant what dr rocket had, I just forgot parentheses. it's the value raised to a further so that it should look like
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    That works too. But you should check out the corrected version of what I thought you meant, cause that equals 2 too. And it's an interesting enough example that people have written about it over the years. The issue is this: convince yourself that the answer is 2 and not 4.
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    what that becomes is Hmmm... this makes sense, because the moment it equals 2, assuming it did before n became arbitrarily large, it becomes infinitely repeating. Nice way to look at it, makes me sad I didn't mean the infinite string.
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