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Math 55 Problem Set 3 Neil Herriot with additions by Andrei Jorza 1. (i) d 1(f;f) = R 1 0 jf(x) f(x)jdx= R 1 0 0dx= 0. And, if f6= g, then F(x) = jf(x) g(x)jis not identically zero. Hence 9x 0 so F(x 0) = 2 > 0. And by continuity, 9 such that 8x;x 0 <x<x 0+ ;F(x) > . So, d 1(f;g) = R 1 0 F(x)dx= R x 0 0 F(x)dx+ x 0+ x 0 F(x)dx 1 x 0+ F(x)dx R x ...
- Math 55 - Harvard University
Worksheets/Solutions. Worksheet 1 and Solutions. Worksheet 2...
- Demystifying Math 55 - Harvard Math
Math 55 is difficult and it is purposefully structured that...
- Math 55a: Honors Advanced Calculus and Linear Algebra - Harvard University
1. [Contraction mapping theorem; cf. the last problem of the...
- Math 55 - Harvard University
Worksheets/Solutions. Worksheet 1 and Solutions. Worksheet 2 and Solutions. Worksheet 3 and Solutions. Worksheet 4 and Solutions. Worksheet 5 and Solutions. Worksheet 6 and Solutions. Worksheet 7 and Solutions. Worksheet 8 and Solutions.
This is Harvard College’s famous Math 55a, instructed by Dennis Gaitsgory. The formal name for this class is \Honors Abstract and Linear Algebra" but it generally goes by simply \Math 55a".
Math 55 is difficult and it is purposefully structured that way as it’s meant to help students mature as mathematicians rather than as simple course takers.
1. [Contraction mapping theorem; cf. the last problem of the Topology IV set.] A function ffrom a metric space Xto itself is said to be a contraction if there exists a constant c<1 such that d(f(x);f(y)) cd(x;y) for all x;y2X[i.e., f shrinks all distances by a factor of at least 1 : c].
About the class. We will cover the basic principles of logic, mathematical induction, sets, relations, and functions, and provide an introduction to graph theory, elementary number theory, combinatorics, algebraic structures, and discrete probability theory.
Prove that Q(k) ≡ 0 mod 3 for every integer k . ___ Q(0) = 0 and Q(k±1)–Q(k) = ±3(k2 ± k + 1) ≡ 0 mod 3 ; now use induction. Or Q(k) = k3 + 2k ≡ k + 2k ≡ 0 mod 3 by Fermat’s Little Theorem kp ≡ k mod p . Or Q(k) = k·(k2+2) ≡ k·(k2–1) ≡ (k+1)·k·(k–1) ≡ 0 mod 3 . _____ [2 pts.] See text p. 228 #18. 5.
