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Math Help - Notation and terminology

  1. #1
    Member garymarkhov's Avatar
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    Notation and terminology

    I am trying to decipher the following, but need help figuring out how to read this para aloud:

    Suppose we want to numerically calculate the global maximum of the function f : X \rightarrow R,
    with X \subset R and with X being a compact and convex interval. Assume that f is strictly
    concave on X. If we know that the global maximum lies in the interior of X...

    So far, I would translate it as "Suppose we want to numerically calculate the global maximum of the function f for which X is comprised of real numbers and X is a subset of real numbers....

    I don't know what the rest means or even if I've got the first part right.
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  2. #2
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    Maybe,

    "Suppose we want to numerically calculate the global maximum a real function f attains over a compact, convex subinterval of the real number line..."
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  3. #3
    Member garymarkhov's Avatar
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    Quote Originally Posted by Defunkt View Post
    Maybe,

    "Suppose we want to numerically calculate the global maximum a real function f attains over a compact, convex subinterval of the real number line..."
    Okay, what is a "compact, convex subinterval" of the real number line?
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  4. #4
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    Oh, didn't realize you want it to be in simple English :P

    Well then, in R compact means closed and bounded - that's easily explained. How do you want to explain convex? And isn't an interval in R always convex?
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  5. #5
    Member garymarkhov's Avatar
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    Quote Originally Posted by Defunkt View Post
    Oh, didn't realize you want it to be in simple English :P
    Yes please. My understanding of notation leaves a lot to be desired.

    Quote Originally Posted by Defunkt View Post
    Well then, in R compact means closed and bounded - that's easily explained. How do you want to explain convex? And isn't an interval in R always convex?
    Closed and bounded. Great!

    Any thoughts on "convex"?
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  6. #6
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    A set being convex in R means that for every two points you choose in it, the "line" between these points is a subset of the bigger set.
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  7. #7
    Member garymarkhov's Avatar
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    Quote Originally Posted by Defunkt View Post
    A set being convex in R means that for every two points you choose in it, the "line" between these points is a subset of the bigger set.
    So X being a convex interval means that for any two points you choose in the X interval, the line between these points is a subset of R?

    What does "f is strictly concave on X" mean?
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  8. #8
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    I meant is a subset of X :P
    Also, every interval in R is convex, so that definition isn't necessary here.

    f being strictly concave means, not formally, that its graph will have a concave shape (think of something like the back of a spoon). Another explanation is that for every 2 points, the graph of f will lie "above" the line joining those two points. You can see a good visualization of this here: Concave function - Wikipedia, the free encyclopedia
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  9. #9
    Member garymarkhov's Avatar
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    Quote Originally Posted by Defunkt View Post
    I meant is a subset of X :P
    Also, every interval in R is convex, so that definition isn't necessary here.

    f being strictly concave means, not formally, that its graph will have a concave shape (think of something like the back of a spoon). Another explanation is that for every 2 points, the graph of f will lie "above" the line joining those two points. You can see a good visualization of this here: Concave function - Wikipedia, the free encyclopedia
    So X being a convex interval means that for any two points you choose in the X interval, the line between these points is a subset of X. Can you explain why that might be an important thing to state? What is the alternative? I'm just not getting the intuition here.

    I'm comfortable with the concave function bit.
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  10. #10
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    You can read on some uses of convexity here: Convex set - Wikipedia, the free encyclopedia

    Other than that, I'm not very familiar with advanced concepts relying on convexity, so someone else can probably give you more information than I can.
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