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- 13:32, 24 July 2008 (diff | hist) . . (+11) . . A.3 Old Kiwi (current)
- 13:32, 24 July 2008 (diff | hist) . . (+11) . . A.3 OldKiwi (current)
- 13:30, 24 July 2008 (diff | hist) . . (+1,436) . . N A.3 Old Kiwi (New page: A.3) Suppose that <math>y>0, \ x \in (\frac{1}{2}, \frac{3}{2})</math> and <math>0<h<\frac{1}{4}</math>, say. Then<math> -\frac{e^{-(x+h)y}-e^{-xy}}{h} \frac{1}{y^3+1} =- \frac{d}{dx} ...)
- 13:30, 24 July 2008 (diff | hist) . . (+1,436) . . N A.3 OldKiwi (New page: A.3) Suppose that <math>y>0, \ x \in (\frac{1}{2}, \frac{3}{2})</math> and <math>0<h<\frac{1}{4}</math>, say. Then<math> -\frac{e^{-(x+h)y}-e^{-xy}}{h} \frac{1}{y^3+1} =- \frac{d}{dx} ...)
- 13:00, 24 July 2008 (diff | hist) . . (+36) . . Problem set 10: These problems were all given by Dr. Davis on an exam Old Kiwi
- 13:00, 24 July 2008 (diff | hist) . . (+35) . . Problem set 10: These problems were all given by Dr. Davis on an exam OldKiwi
- 12:33, 24 July 2008 (diff | hist) . . (+755) . . N A.1 Old Kiwi (New page: Let G={x | f is continuous at x} Pick some <math>x \in G.</math> Then <math>\exists \ \delta>0 \ s.t.\ |x-y|<\delta \Rightarrow |f(x)-f(y)| < \frac{1}{2}</math>. But since f is integer ...) (current)
- 12:33, 24 July 2008 (diff | hist) . . (+755) . . N A.1 OldKiwi (New page: Let G={x | f is continuous at x} Pick some <math>x \in G.</math> Then <math>\exists \ \delta>0 \ s.t.\ |x-y|<\delta \Rightarrow |f(x)-f(y)| < \frac{1}{2}</math>. But since f is integer ...) (current)
- 12:21, 24 July 2008 (diff | hist) . . (+35) . . Problem set 10: These problems were all given by Dr. Davis on an exam OldKiwi
- 12:21, 24 July 2008 (diff | hist) . . (+36) . . Problem set 10: These problems were all given by Dr. Davis on an exam Old Kiwi
- 12:15, 24 July 2008 (diff | hist) . . (+925) . . N B.1 Old Kiwi (New page: B.1 By egorov, <math>\forall \ k</math> we may pick a set <math>E_k</math> such that <math>m(E_k)<\frac{1}{2^k}</math> and <math>f_n \rightarrow 0</math> uniformly off of <math>E_k</math>...) (current)
- 12:15, 24 July 2008 (diff | hist) . . (+925) . . N B.1 OldKiwi (New page: B.1 By egorov, <math>\forall \ k</math> we may pick a set <math>E_k</math> such that <math>m(E_k)<\frac{1}{2^k}</math> and <math>f_n \rightarrow 0</math> uniformly off of <math>E_k</math>...) (current)
- 12:08, 24 July 2008 (diff | hist) . . (+36) . . Problem set 10: These problems were all given by Dr. Davis on an exam Old Kiwi
- 12:08, 24 July 2008 (diff | hist) . . (+35) . . Problem set 10: These problems were all given by Dr. Davis on an exam OldKiwi
- 16:04, 22 July 2008 (diff | hist) . . (+1) . . Exam 9.2 Old Kiwi (current)
- 16:04, 22 July 2008 (diff | hist) . . (+1) . . Exam 9.2 OldKiwi (current)
- 16:03, 22 July 2008 (diff | hist) . . (-7) . . Exam 9.2 Old Kiwi
- 16:03, 22 July 2008 (diff | hist) . . (-7) . . Exam 9.2 OldKiwi
- 15:57, 22 July 2008 (diff | hist) . . (+728) . . N Exam 9.2 Old Kiwi (New page: 2) Pick a partition P of [a,b] <math>a=x_0<x_1<...<x_n=b</math> Pick <math>c_i \in (x_0, x_1) \ i=1,...,N.</math> Let n=N Define <math>c_{n+1}=b</math> and <math>c_0=a</math>. Then ...)
- 15:57, 22 July 2008 (diff | hist) . . (+728) . . N Exam 9.2 OldKiwi (New page: 2) Pick a partition P of [a,b] <math>a=x_0<x_1<...<x_n=b</math> Pick <math>c_i \in (x_0, x_1) \ i=1,...,N.</math> Let n=N Define <math>c_{n+1}=b</math> and <math>c_0=a</math>. Then ...)
- 15:42, 22 July 2008 (diff | hist) . . (+37) . . Sharks' answers Old Kiwi (current)
- 15:42, 22 July 2008 (diff | hist) . . (+36) . . Sharks' answers OldKiwi (current)
- 15:20, 22 July 2008 (diff | hist) . . (+1,128) . . Exam 9.9 Old Kiwi (current)
- 15:20, 22 July 2008 (diff | hist) . . (+1,128) . . Exam 9.9 OldKiwi (current)
- 14:57, 22 July 2008 (diff | hist) . . (+1,336) . . N Exam 9.9 Old Kiwi (New page: 9b) Let <math>[a,b] \subset [0,1]</math>. Since F is continuous, <math>F([a,b])</math> is compact, thus <math>\exists \alpha , \beta \in [a,b]</math> such that <math>F(\alpha) \leq F(x) ...)
- 14:57, 22 July 2008 (diff | hist) . . (+1,336) . . N Exam 9.9 OldKiwi (New page: 9b) Let <math>[a,b] \subset [0,1]</math>. Since F is continuous, <math>F([a,b])</math> is compact, thus <math>\exists \alpha , \beta \in [a,b]</math> such that <math>F(\alpha) \leq F(x) ...)
- 14:05, 22 July 2008 (diff | hist) . . (+34) . . Sharks' answers Old Kiwi
- 14:05, 22 July 2008 (diff | hist) . . (+33) . . Sharks' answers OldKiwi
- 13:34, 22 July 2008 (diff | hist) . . (+362) . . Exam 9.4 Old Kiwi (current)
- 13:34, 22 July 2008 (diff | hist) . . (+362) . . Exam 9.4 OldKiwi (current)
- 13:25, 22 July 2008 (diff | hist) . . (+872) . . N Exam 9.4 Old Kiwi (New page: 4a) 4b) <math>lim_{n\rightarrow \infty} \int_1^{n^2} \frac{n cos (\frac{x}{n^2})}{1+nln(x)}dx = lim_{n\rightarrow \infty} \int_1^{\infty} \frac{n cos (\frac{x}{n^2})}{1+nln(x)}\chi_{(1,...)
- 13:25, 22 July 2008 (diff | hist) . . (+872) . . N Exam 9.4 OldKiwi (New page: 4a) 4b) <math>lim_{n\rightarrow \infty} \int_1^{n^2} \frac{n cos (\frac{x}{n^2})}{1+nln(x)}dx = lim_{n\rightarrow \infty} \int_1^{\infty} \frac{n cos (\frac{x}{n^2})}{1+nln(x)}\chi_{(1,...)
- 13:03, 22 July 2008 (diff | hist) . . (+40) . . Sharks' answers Old Kiwi
- 13:03, 22 July 2008 (diff | hist) . . (+39) . . Sharks' answers OldKiwi
- 22:12, 21 July 2008 (diff | hist) . . (+2) . . Exam 9.5 Old Kiwi
- 22:12, 21 July 2008 (diff | hist) . . (+2) . . Exam 9.5 OldKiwi
- 22:11, 21 July 2008 (diff | hist) . . (+98) . . Exam 9.5 Old Kiwi
- 22:11, 21 July 2008 (diff | hist) . . (+98) . . Exam 9.5 OldKiwi
- 21:56, 21 July 2008 (diff | hist) . . (+81) . . Exam 9.5 Old Kiwi
- 21:56, 21 July 2008 (diff | hist) . . (+81) . . Exam 9.5 OldKiwi
- 21:53, 21 July 2008 (diff | hist) . . (+28) . . Exam 9.5 Old Kiwi
- 21:53, 21 July 2008 (diff | hist) . . (+28) . . Exam 9.5 OldKiwi
- 21:50, 21 July 2008 (diff | hist) . . (+1,090) . . Exam 9.5 Old Kiwi
- 21:50, 21 July 2008 (diff | hist) . . (+1,090) . . Exam 9.5 OldKiwi
- 21:02, 21 July 2008 (diff | hist) . . (+257) . . Exam 9.5 Old Kiwi
- 21:02, 21 July 2008 (diff | hist) . . (+257) . . Exam 9.5 OldKiwi
- 20:41, 21 July 2008 (diff | hist) . . (+883) . . N Exam 9.5 Old Kiwi (New page: Lemma: If <math> f </math> is as described then <math> f(x)=0 \ \forall \ x \in [0, \frac{1}{2c}]</math>. Pf: Suppose <math>\int_0^\frac{1}{2c} f(t) dt \neq 0</math>. Let <math>\epsil...)
- 20:41, 21 July 2008 (diff | hist) . . (+883) . . N Exam 9.5 OldKiwi (New page: Lemma: If <math> f </math> is as described then <math> f(x)=0 \ \forall \ x \in [0, \frac{1}{2c}]</math>. Pf: Suppose <math>\int_0^\frac{1}{2c} f(t) dt \neq 0</math>. Let <math>\epsil...)
- 19:59, 21 July 2008 (diff | hist) . . (+33) . . N Sharks' answers Old Kiwi (New page: #5 Solution)
- 19:59, 21 July 2008 (diff | hist) . . (+32) . . N Sharks' answers OldKiwi (New page: #5 Solution)
- 19:58, 14 July 2008 (diff | hist) . . (0) . . 8.5 Old Kiwi
- 19:58, 14 July 2008 (diff | hist) . . (0) . . 8.5 OldKiwi
- 19:57, 14 July 2008 (diff | hist) . . (+1,112) . . 8.5 Old Kiwi
- 19:57, 14 July 2008 (diff | hist) . . (+1,112) . . 8.5 OldKiwi
- 19:28, 14 July 2008 (diff | hist) . . (+236) . . N 8.5 Old Kiwi (New page: 8a) Given such an <math>r<s</math> for any <math>p \in (r,s)</math> we have <math>\int_X |f|^p = \int_{\{x:|f|\leq 1 \}} |f|^p+\int_{\{x:|f|> 1 \}} |f|^p \leq \int_{\{x:|f|\leq 1 \}} |f|...)
- 19:28, 14 July 2008 (diff | hist) . . (+236) . . N 8.5 OldKiwi (New page: 8a) Given such an <math>r<s</math> for any <math>p \in (r,s)</math> we have <math>\int_X |f|^p = \int_{\{x:|f|\leq 1 \}} |f|^p+\int_{\{x:|f|> 1 \}} |f|^p \leq \int_{\{x:|f|\leq 1 \}} |f|...)
- 19:21, 14 July 2008 (diff | hist) . . (+30) . . Problem Set 8 Old Kiwi
- 19:21, 14 July 2008 (diff | hist) . . (+29) . . Problem Set 8 OldKiwi
- 15:59, 13 July 2008 (diff | hist) . . (+230) . . Problem Set 7 7.6 Old Kiwi (current)
- 15:59, 13 July 2008 (diff | hist) . . (+230) . . Problem Set 7 7.6 OldKiwi (current)
- 16:56, 10 July 2008 (diff | hist) . . (+2) . . Problem Set 7 7.6 Old Kiwi
- 16:56, 10 July 2008 (diff | hist) . . (+2) . . Problem Set 7 7.6 OldKiwi
- 16:56, 10 July 2008 (diff | hist) . . (+972) . . N Problem Set 7 7.6 Old Kiwi (New page: 6. First we show <math>f_n\rightarrow f </math> in <math> L^p</math>. Let <math>\delta > 0</math>. Let <math> \epsilon = \frac {\delta }{1+2^{p+1}} </math> Then by Egorov <math>\exists ...)
- 16:56, 10 July 2008 (diff | hist) . . (+972) . . N Problem Set 7 7.6 OldKiwi (New page: 6. First we show <math>f_n\rightarrow f </math> in <math> L^p</math>. Let <math>\delta > 0</math>. Let <math> \epsilon = \frac {\delta }{1+2^{p+1}} </math> Then by Egorov <math>\exists ...)
- 16:22, 10 July 2008 (diff | hist) . . (+44) . . Team 2: The Sharks awesome answers Old Kiwi
- 16:22, 10 July 2008 (diff | hist) . . (+43) . . Team 2: The Sharks awesome answers OldKiwi
- 16:37, 8 July 2008 (diff | hist) . . (+1,808) . . N Exam 3.5 Old Kiwi (New page: 5) Fix <math>\alpha > 0</math>. Define <math>f_{n_k}</math> inductively as follows. Let <math>f_{n_{0}}=f_1</math> Let <math>\epsilon_1 = 1</math> Let <math>A_1=[0,\alpha+1]</math>. ...) (current)
- 16:37, 8 July 2008 (diff | hist) . . (+1,808) . . N Exam 3.5 OldKiwi (New page: 5) Fix <math>\alpha > 0</math>. Define <math>f_{n_k}</math> inductively as follows. Let <math>f_{n_{0}}=f_1</math> Let <math>\epsilon_1 = 1</math> Let <math>A_1=[0,\alpha+1]</math>. ...) (current)
- 15:41, 8 July 2008 (diff | hist) . . (+35) . . Practice Exam 3 Old Kiwi
- 15:41, 8 July 2008 (diff | hist) . . (+34) . . Practice Exam 3 OldKiwi
- 15:39, 8 July 2008 (diff | hist) . . (+762) . . N Exam 3.4 Old Kiwi (New page: 4a) <math>0 \leq sin^n(x)\leq 1</math> on <math>[0, \pi]</math> and 1 is integrable on <math>[0,\pi]</math>, so by Leb. Dom. Conv. Thm.: <math> lim_n \int_0^{\pi}sin^n(x)dx= \int_0^{\pi...) (current)
- 15:39, 8 July 2008 (diff | hist) . . (+762) . . N Exam 3.4 OldKiwi (New page: 4a) <math>0 \leq sin^n(x)\leq 1</math> on <math>[0, \pi]</math> and 1 is integrable on <math>[0,\pi]</math>, so by Leb. Dom. Conv. Thm.: <math> lim_n \int_0^{\pi}sin^n(x)dx= \int_0^{\pi...) (current)
- 15:22, 8 July 2008 (diff | hist) . . (-1) . . Practice Exam 3 Old Kiwi
- 15:22, 8 July 2008 (diff | hist) . . (-1) . . Practice Exam 3 OldKiwi
- 15:22, 8 July 2008 (diff | hist) . . (+36) . . Practice Exam 3 Old Kiwi
- 15:22, 8 July 2008 (diff | hist) . . (+35) . . Practice Exam 3 OldKiwi
- 15:04, 8 July 2008 (diff | hist) . . (+583) . . Exam 3.3 Old Kiwi (current)
- 15:04, 8 July 2008 (diff | hist) . . (+583) . . Exam 3.3 OldKiwi (current)
- 14:49, 8 July 2008 (diff | hist) . . (+1,136) . . N Exam 3.3 Old Kiwi (New page: 3a) Suppose that there is such an <math>f</math>. Then we may choose <math>N</math> large enough such that <math>m(\{x:f^*<f\}\cap[-N,N])>0</math>. Call this set <math>A</math>. Let <m...)
- 14:49, 8 July 2008 (diff | hist) . . (+1,136) . . N Exam 3.3 OldKiwi (New page: 3a) Suppose that there is such an <math>f</math>. Then we may choose <math>N</math> large enough such that <math>m(\{x:f^*<f\}\cap[-N,N])>0</math>. Call this set <math>A</math>. Let <m...)
- 14:12, 8 July 2008 (diff | hist) . . (+35) . . Practice Exam 3 Old Kiwi
- 14:12, 8 July 2008 (diff | hist) . . (+34) . . Practice Exam 3 OldKiwi
- 14:10, 8 July 2008 (diff | hist) . . (+55) . . Exam3.2 Old Kiwi (current)
- 14:10, 8 July 2008 (diff | hist) . . (+55) . . Exam3.2 OldKiwi (current)
- 14:07, 8 July 2008 (diff | hist) . . (+348) . . Exam3.2 Old Kiwi
- 14:07, 8 July 2008 (diff | hist) . . (+348) . . Exam3.2 OldKiwi
- 13:59, 8 July 2008 (diff | hist) . . (+459) . . Exam3.2 Old Kiwi
- 13:59, 8 July 2008 (diff | hist) . . (+459) . . Exam3.2 OldKiwi
- 13:46, 8 July 2008 (diff | hist) . . (+920) . . Exam3.2 Old Kiwi
- 13:46, 8 July 2008 (diff | hist) . . (+920) . . Exam3.2 OldKiwi
- 13:15, 8 July 2008 (diff | hist) . . (+342) . . N Exam3.2 Old Kiwi (New page: 2a. <math>( \Rightarrow )</math> Say <math>f</math> is A.C. Then <math>f</math> is of bounded variation, and since <math>f</math> is clearly nondecreasing, <math>f</math> must be bounded...)
- 13:15, 8 July 2008 (diff | hist) . . (+342) . . N Exam3.2 OldKiwi (New page: 2a. <math>( \Rightarrow )</math> Say <math>f</math> is A.C. Then <math>f</math> is of bounded variation, and since <math>f</math> is clearly nondecreasing, <math>f</math> must be bounded...)
- 13:08, 8 July 2008 (diff | hist) . . (-6) . . Practice Exam 3 Old Kiwi
- 13:08, 8 July 2008 (diff | hist) . . (-6) . . Practice Exam 3 OldKiwi
- 13:07, 8 July 2008 (diff | hist) . . (+2) . . Practice Exam 3 Old Kiwi
- 13:07, 8 July 2008 (diff | hist) . . (+2) . . Practice Exam 3 OldKiwi
- 13:07, 8 July 2008 (diff | hist) . . (+38) . . Practice Exam 3 Old Kiwi
- 13:07, 8 July 2008 (diff | hist) . . (+37) . . Practice Exam 3 OldKiwi
- 13:19, 5 July 2008 (diff | hist) . . (+336) . . N Solution to number 3 OldKiwi (New page: 3) Define <math>f(x)=\sum_{n=1}^\infty n\chi_{[2^{-n}, 2^{-(n-1)})}(x)</math> Then <math>\forall p \in (0, \infty), \ \int_0^1 |f|^p=\sum_{n=1}^\infty \frac{n^p}{2^n}<\infty</math> by t...) (current)
- 13:19, 5 July 2008 (diff | hist) . . (+336) . . N Solution to number 3 Old Kiwi (New page: 3) Define <math>f(x)=\sum_{n=1}^\infty n\chi_{[2^{-n}, 2^{-(n-1)})}(x)</math> Then <math>\forall p \in (0, \infty), \ \int_0^1 |f|^p=\sum_{n=1}^\infty \frac{n^p}{2^n}<\infty</math> by t...) (current)
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