12-267/Homework Assignment 3

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This assignment is due at the tutorial on Tuesday October 9. Here and everywhere, neatness counts!! You may be brilliant and you may mean just the right things, but if your readers have a hard time deciphering your work they will give up and assume it is wrong.

Task 0. Identify yourself in the Class Photo!

Task 1. Let \phi_n\colon X\to{\mathbb R} be a sequence of functions defined on some set X, and suppose that some sequence c_n of non-negative reals is given such that for every x\in X, |\phi_n(x)-\phi_{n+1}(x)|\leq c_n. Suppose also that \sum_{n=1}^\infty c_n is finite. Prove that the sequence \phi_n is uniformly convergent.

Task 2. Find the extrema of the following functionals:

  1. y\mapsto \int_0^1y'dx subject to y(0)=0 and y(1)=1.
  2. y\mapsto \int_0^1yy'dx subject to y(0)=0 and y(1)=1.
  3. y\mapsto \int_0^1xyy'dx subject to y(0)=0 and y(1)=1.
  4. y\mapsto\int_a^b\frac{y'^2}{x^3}dx.
  5. y\mapsto\int_a^b(y^2+y'^2+2ye^x)dx.
  6. Postponed! y\mapsto\int_0^1(y'^2+x^2)dx subject to \int_0^1y^2dx=2 and y(0)=0 and y(1)=0.

Task 3. A roach R I once met was mortally afraid of walls, and so when it walked on my kitchen's floor, its speed was exactly proportional to its distance from the nearest wall (that is, very near a wall it crawled very slowly, while in the centre of the room it run around quickly and happily). As a step towards simplifying R's life, help it find the fastest path from one point in the upper half plane \{y>0\} to another point in the upper half plane, assuming there is only one wall around, built along the x-axis y=0.

Dror's notes above / Student's notes below

Solution to Task 1. --Twine 18:11, 24 October 2012 (EDT)

Let \sum_{n=1}^\infty c_n = S. We have that \lim_{n \rightarrow \infty} c_n = 0.

Let \epsilon > 0.

\exists N \in \mathbb{Z}^+ such that \sum_{n=1}^N c_n > S - \epsilon.

Choose such an N. Then \sum_{n=N}^{\infty} c_n < \epsilon.

Then \forall x \in X, \forall m > N, n>N, we have |\Phi_n(x) - \Phi_m(x)| < \epsilon.

As this result is independent of our choice of x, we have by the cauchy criterion that \Phi_n is uniformly convergent and \lim_{n \rightarrow \infty} \Phi_n(x) = \Phi(x) exists.

Solutions to HW3: Mathstudent