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  1. #61
    Senior Member nemo's Avatar
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    Yeah, I'll get working on it.

    I'll try to write the technical explanations but sprinkle it with non-technical "what does this mean?" remarks so everyone can follow

    Can I upload .pdfs somewhere? It'll probably be easier to write it in latex.

    But this week is finals week for me, so it may have to postpone this for a few days.
    You can't wait for inspiration. You have to go after it with a club. - Jack London

  2. #62

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    No rush at all. I am sure many of us are in final exam or midterm time depending on if we are on the quarter or semester systems.

    Accept the past. Live for the present. Look forward to the future.
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    "[A] scientist looking at nonscientific problems is just as dumb as the next guy." Richard Feynman
    "[P]etabytes of [] data is not the same thing as understanding emergent mechanisms and structures." Jim Crutchfield

  3. #63
    Senior Member bluebell's Avatar
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    Quote Originally Posted by nemo View Post
    Yeah, I'll get working on it.

    I'll try to write the technical explanations but sprinkle it with non-technical "what does this mean?" remarks so everyone can follow

    Can I upload .pdfs somewhere? It'll probably be easier to write it in latex.

    But this week is finals week for me, so it may have to postpone this for a few days.
    I've got no idea about PDF upload sites. But... once you've typed it up in latex and compiled it, then you could do a screen shot and save it as a gif or jpg to upload. I assume that would work?

    I'd be interested to see it too. I probably won't be able to follow all the details, but I hopefully will be able to grok the general idea.

  4. #64

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    Quote Originally Posted by bluebell View Post
    I've got no idea about PDF upload sites. But... once you've typed it up in latex and compiled it, then you could do a screen shot and save it as a gif or jpg to upload. I assume that would work?

    I'd be interested to see it too. I probably won't be able to follow all the details, but I hopefully will be able to grok the general idea.
    I know .pdf's get big.... But I think it is acceptable as an attachment. I've never tried.

    I can see not wanting to take half your quota for 1 file tough...

    I did some Google-ing and found this.

    I've never tried it, and my "site adviser" cautions me that it distributes cracks an wares on the site as well (I guess people can upload ANYTHING).

    As long as you steer clear of doing something along those lines, I think it'll be OK.

    Accept the past. Live for the present. Look forward to the future.
    Robot Fusion
    "As our island of knowledge grows, so does the shore of our ignorance." John Wheeler
    "[A] scientist looking at nonscientific problems is just as dumb as the next guy." Richard Feynman
    "[P]etabytes of [] data is not the same thing as understanding emergent mechanisms and structures." Jim Crutchfield

  5. #65
    ish red no longer *sad* nightning's Avatar
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    Random info from mouse:

    Try this instead...

    YouSendIt - Send large files - transfer delivery - FTP Replacement

    Type in random email addresses for both "to" and "from" inputs. After it finishes uploading, it'll give you an link display on the page. It accepts upto 100MB file size and 100 download per file.

  6. #66

    Accept the past. Live for the present. Look forward to the future.
    Robot Fusion
    "As our island of knowledge grows, so does the shore of our ignorance." John Wheeler
    "[A] scientist looking at nonscientific problems is just as dumb as the next guy." Richard Feynman
    "[P]etabytes of [] data is not the same thing as understanding emergent mechanisms and structures." Jim Crutchfield

  7. #67
    Glowy Goopy Goodness The_Liquid_Laser's Avatar
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    Quote Originally Posted by nemo View Post
    So I was trying to explain the Cantor Set to a friend at a party/get-together/birthday thing tonight.

    It is an uncountable set with measure zero.

    It contains 0 and 1, so you'd think intuitively that it would at least have measure 1 - 0 = 1. But no.

    WTF

    It's also compact, contains no open subsets, and every element is a cluster point.

    That's freakin weird.

    Have any of you guys encountered the Cantor Set? How would you explain it in a non-technical way to a friend who has just basic single-variable calculus under his belt?

    When I'm not so intoxicated and sleep deprived I will attempt an explanation myslef.
    Generally my explanations of the Cantor Set (and set theory in general) focus mostly on the fact that Cantor ended up in an sanatorium.
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  8. #68
    no clinkz 'til brooklyn Nocapszy's Avatar
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    Sounds like more of an evasion than an explanation. Ha!
    we fukin won boys

  9. #69
    Senior Member snegledmaca's Avatar
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    Hey, could some of you nice math people tell me what's wrong with my thinking here. I was thinking about that whole 0.999... = 1 and wanted to show it through set theory. The wikipedia article had no proof through set theory and I needed it for somethign else.

    In any case, here's my logic, take an expression 0.x1x2x3...xn... where xn is a digit from the decimal system, 0 or 1 or 2 or 3 or 4 or 5 or 6 or 7 or 8 or 9. Make a set with all the possible expressions. Two expressions 0.x1x2x3...xn... and 0.y1y2y3...yn... differ if there is at least one digit that differs, or if there exists an xn such that xn != yn. Find the cardinality of the set. Which is the continuum. Which means that there is a bijection between this set and the real numbers. Which would mean that there is no such thing as a real number having multiple decimal representations.

    My thinking is that if every real number had more then one decimal expression joined to it, and these were not repeated, wouldn't that mean that the set of all decimal expressions had the cardinality greater then the continuum?

  10. #70
    Senior Member nemo's Avatar
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    Quote Originally Posted by snegledmaca View Post
    In any case, here's my logic, take an expression 0.x1x2x3...xn... where xn is a digit from the decimal system, 0 or 1 or 2 or 3 or 4 or 5 or 6 or 7 or 8 or 9. Make a set with all the possible expressions. Two expressions 0.x1x2x3...xn... and 0.y1y2y3...yn... differ if there is at least one digit that differs, or if there exists an xn such that xn != yn. Find the cardinality of the set. Which is the continuum. Which means that there is a bijection between this set and the real numbers. Which would mean that there is no such thing as a real number having multiple decimal representations.
    Only if the decimal expansion is finite.

    By definition, the decimal expansion of some number r is



    Where {a1, a2, a3, ...,} is some sequence. Notice that the set {a1, a2, a3, ...} is necessarily infinite. (Hence the number 1 can also be expanded as 1.000000...).

    The problem is that infinite sums are not actual sums.

    They are the limit of the sequence of partial sums -- in this case {a1/10, a1/10 + a2/100, a1/10 + a2/100 + a3/1000, ...}.

    So you can't just compare each of your xn's and yn's.

    It's totally legitimate that the sequences {a1, a2, a3, ...} and {b1, b2, b3, ...} might not be equal, but the limits of the sequence of their partial sums are the same. In this case the set of an's and bn's are, {1, 0.0, 0.00, 0.000, ...} and {0, .9, .09, .009, ...}.

    If you must provide a proof via set theory, I'd probably go after some argument using the suprema and infima of the respective sets of partial sums. That's really just an analysis argument, though.

    Also, I'm not exactly sure what you mean by "the cardinality of the set." Which set are you talking about? R, or your sequences {x1/10, x2/100, ... xn/10^n, ...} and {y1/10, ... yn/10^n, ...}?
    You can't wait for inspiration. You have to go after it with a club. - Jack London

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