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Math Help - What is the definition of Single-Peaked function?

  1. #1
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    What is the definition of Single-Peaked function?

    Dear all,

    I have some confusion on the definition of Single-Peaked function. Can anybody tell me the official definition of it?

    It is intuitive to understand "Single-Peaked". However, what if the function has a flat part with its level lower than the peak? Can we say this function still Single-Peaked?

    In other words, do we require the strict monotonicity on both sides of the peak for it to be "Single-Peaked"?

    Thanks in advance.
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  2. #2
    No one in Particular VonNemo19's Avatar
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    Quote Originally Posted by analysismath View Post
    "Single-Peaked"?

    I've never heard this before...
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    My guess would be that a single peaked function would be one of which has only 1 maximum value. An example of this would be an inverted quadratic, a cubic with no points of inflection or even an inverted quartic with a point of inflection. This could extend further when looking at the function that generates the bell curve for a normal distribution.

    Maybe google knows?
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    Thank you very much for your informative reply!

    Along the line of your answer, my question is:

    Given the function has only one global maximum, if it has a flat part somewhere else other than the only "peak" point, would we still count it as the "Single-Peaked"?

    Equivalently, do we require strict monotonicity on both sides of the unique global maximum?
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  5. #5
    No one in Particular VonNemo19's Avatar
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    Quote Originally Posted by analysismath View Post

    Equivalently, do we require strict monotonicity on both sides of the unique global maximum?
    Are we concerned with the entire number line?
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  6. #6
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    Quote Originally Posted by analysismath View Post

    Given the function has only one global maximum, if it has a flat part somewhere else other than the only "peak" point, would we still count it as the "Single-Peaked"?
    I would say yes given the definition of a maximum is a point that has a positive gradient preceeding it and a negative gradient afterwards if the flat part did not adhere to these conditions itself.
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