Results 1 to 4 of 4

Math Help - Invariance Principle Question - from Arthur Engels book

  1. #1
    Junior Member
    Joined
    Nov 2011
    From
    Karachi
    Posts
    33

    Invariance Principle Question - from Arthur Engels book

    Quoting from Arthur Engel's "Problem Solving Strategies":

    Invariance Principle Question - from Arthur Engels book-invariance1.png

    To the part in red, how is this possible? If x_{n} keeps increasing, y_{n} has to decrease in order to keep the product of x_{n} and y_{n} constant at ab.

    To the part in blue, I don't get this.

    To the part in green, isn't harmonic and arithmetic mean equal in magnitude always?

    Thanks. Also I don't get the inequality which he has generated. What is he trying to show by x_{n+1} - y_{n+1}?
    Attached Thumbnails Attached Thumbnails Invariance Principle Question - from Arthur Engels book-invariance1.png  
    Follow Math Help Forum on Facebook and Google+

  2. #2
    GJA
    GJA is offline
    Member
    Joined
    Jul 2012
    From
    USA
    Posts
    109
    Thanks
    29

    Re: Invariance Principle Question - from Arthur Engels book

    Hi, cosmonavt.

    If I'm reading this right, the red part is what he's proving with the blue and green lines, so it's not supposed to make perfect sense until those other parts are established.

    The blue part is how you find the midpoint between two real numbers. If x and y are real numbers, then (x+y)/2 is the midpoint between x & y.

    The green part comes from an inequality that is used in analysis frequently. The inequality states that for nonnegative numbers x and y:

    \sqrt{xy}\leq \frac{x+y}{2}.

    The left side of the inequality is known as the "Geometric mean" and the right hand side is known as the "Arithmetic mean."

    Note: There is a more general statement of the above inequality. If you're curious you can find it here Inequality of arithmetic and geometric means - Wikipedia, the free encyclopedia


    Does that clear things up? Let me know. Good luck!
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Nov 2011
    From
    Karachi
    Posts
    33

    Re: Invariance Principle Question - from Arthur Engels book

    Thanks for the reply but it doesn't clear anything. FIrstly, I know that x_{n+1} is halfway between x_{n} and y_{n} but how is it showing that y_{n} is always less than x_{n}?

    Secondly, he didn't say geometric mean, he said harmonic mean. Harmonic mean and geometric mean have the same magnitude, always.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    GJA
    GJA is offline
    Member
    Joined
    Jul 2012
    From
    USA
    Posts
    109
    Thanks
    29

    Re: Invariance Principle Question - from Arthur Engels book

    The statement

    Harmonic mean and geometric mean have the same magnitude, always.
    is incorrect. The relationship is actually that the harmonic mean is less than or equal to the geometric mean - see Theorem 2 of http://faculty.ccc.edu/mhidegkuti/sh...eans/means.pdf for a proof.

    Second, you're right, he does quote the harmonic mean. But justifying the inequality while I read it, I used a method using the arithmetic/geometric mean inequality; I left the steps for you to see how this would work. If you would like a more explicit hint, I would suggest forming the ratio \frac{y_{n+1}}{x_{n+1}} and seeing what you can do.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. homotopy invariance of singular cohomology
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: June 11th 2012, 12:56 PM
  2. Lebesgue measure and translation invariance
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: October 27th 2010, 10:23 AM
  3. Translation Invariance of Integral
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: March 14th 2009, 06:29 PM
  4. Invariance of curvature
    Posted in the Calculus Forum
    Replies: 0
    Last Post: February 17th 2009, 05:26 PM
  5. Invariance of Curvature
    Posted in the Calculus Forum
    Replies: 1
    Last Post: February 17th 2009, 03:16 PM

Search Tags


/mathhelpforum @mathhelpforum