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DTSTART;TZID=America/New_York:20210924T130000
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SUMMARY:Mathematics Class of 1960s Speaker: Prof. Alex Iosevich\, University of Rochester
DESCRIPTION:Mathematics Class of 1960s Speaker: Prof. Alex Iosevich\, University of Rochester\, The Vapnik-Chervonenkis Dimension and the Structure of Point Configurations in Vector Spaces Over Finite Fields\, 1 – 2:00 pm\, Wachenheim 116 \nAbstract: Let X be a set and let H be a collection of functions from X to {0\,1}. We say that H shatters a finite subset C of X if the restriction of H yields every possible function from C to {0\,1}. The VC-dimension of H is the largest number d such that there exists a set of size d shattered by H\, and no set of size d + 1 is shattered by H. Vapnik and Chervonenkis introduced this idea in the early 70s in the context of learning theory\, and this idea has also had a significant impact on other areas of mathematics. In this paper\, we study the VC-dimension of a class of functions H defined on Fqd\, the d-dimensional vector space over the finite field with q elements. Define \nHtd = {hy(x) : y is in Fqd}\, \nwhere for x in Fqd\, hy(x) = 1 if ||x – y|| = t\, and 0 otherwise\, where here\, and throughout\, ||x|| = x12 + x22 + … + xd2. Here t is a nonzero element of Fq. Define Htd(E) the same way with respect to E\, a subset of Fqd. The learning task here is to find a sphere of radius t centered at some point y in E unknown to the learner. The learning process consists of taking random samples of elements of E of sufficiently large size. \nWe are going to prove that when d = 2\, and |E| is greater than or equal to Cq(15/8)\, the VC-dimension of Ht2(E) is equal to 3. This leads to an intricate configuration problem which is interesting in its own right and requires a new approach. \n
URL:https://events.williams.edu/event/mathematics-class-of-1960s-speaker-prof-alex-iosevich-university-of-rochester/
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