• An Extra Illustration of J. E. Bottcher's Proof of the Pythagorean Theorem

    Updated: 2012-02-18 03:02:38
    An Extra Illustration of J. E. Bottcher's Proof of the Pythagorean Theorem: The applet below is intended to provide a motivating illustration for a proof of the Pythagorean Theorem by J. E. Bottcher that is listed as #36 in the collection. The proof procedes in three steps to observe which you are required to press the "Step" button

  • 254B, Notes 6: Non-concentration in subgroups

    Updated: 2012-02-14 00:51:02
    In the last three notes, we discussed the Bourgain-Gamburd expansion machine and two of its three ingredients, namely quasirandomness and product theorems, leaving only the non-concentration ingredient to discuss. We can summarise the results of the last three notes, in the case of fields of prime order, as the following theorem. Theorem 1 (Non-concentration implies [...]

  • 254B, Notes 5: Product theorems, pivot arguments, and the Larsen-Pink non-concentration inequality

    Updated: 2012-02-05 19:32:41
    In the previous set of notes, we saw that one could derive expansion of Cayley graphs from three ingredients: non-concentration, product theorems, and quasirandomness. Quasirandomness was discussed in Notes 3. In the current set of notes, we discuss product theorems. Roughly speaking, these theorems assert that in certain circumstances, a finite subset of a group [...]

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