Local or Uniform Lipschitz Constants
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Determine if the following functions satisfy local or uniform Lipschitz condition.
1). te^y
My work:
I found d/dy (te^y) = te^y, and this is not bounded above for any value of y, so this made me conclude that it has locally Lipschitz condition since the Lipschitz constant here changes as the reagion changes? Am I right? I used the equation
| f(t,y_1) - f(t, y_2) | = d/dy(f(t,y)) + | y_1 - y_2|
If my work is incorrect, provide the correct answer and approach.
2). y t^2/ (1 + y^2)
I used the same approach here and d/dy (f ) = t^2 - 2y + y^2/ ( 1 + y^2)^2, which is clearly could be bounded above by a constant but this constant changes as the reagion changes so it is local lipschitz. Please check my answer, and if it is incorrect provide the right answer.
P.S. Since these problems are extremely easy to people used to work with DE problems, I will only give 1 credit for it, 5 mins per each one is enough time I believe. Thanks in advance.
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Solution Summary
Local or Uniform Lipschitz Constants are investigated. The solution is detailed and well presented.
Education
- BSc , Wuhan Univ. China
- MA, Shandong Univ.
Recent Feedback
- "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
- "excellent work"
- "Thank you so much for all of your help!!! I will be posting another assignment. Please let me know (once posted), if the credits I'm offering is enough or you ! Thanks again!"
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