Math Topic v2

The last topic got closed because someone posted questions directly from another website! Please don't do that.

Post unique questions or answers.

Problem 1: Define a group which has intermediate growth.

Problem 2: Let H and K be fractals with Minkowski dimension h and k respectively. Prove that the Minkowski dimension of H×K is h+k.

Problem 3: What is the most effective way to close a topic on a chess forum that is not related to chess?

This is the off-topic subforum, and the subject of math should be more than welcome here.

Problem 3: Prove that a 2^n by 2^n chess board with one square removed can always be tiled using l-shaped tiles (the l-shaped tiles cover three squares exactly).

Solution 3:

For the base case n=1, the statement is trivially true.

Assume for induction that it holds for some n, and consider the case for n+1. The 2^(n+1) by 2^(n+1) board can be split into four equally sized 2^n by 2^n boards. Without loss of generality, remove one square from one of those four boards. By the induction hypothesis, the 2^n by 2^n with one square removed can be tiled. Next, orient one l-shaped tile so that it covers exactly one square from the remaining three 2^n by 2^n boards. Again, these remaining boards can be tiled since they have one square removed, hence the full 2^(n+1) by 2^(n+1) board can be tiled, as desired.

Solution 1:

Consider an infinite binary tree with edges oriented in the same way (see ).

Define, a, an automorphism of this tree which maps the subtree 0 to 1 and 1 to 0 (a is an involution).

Next denote φ as the mapping from the semidirect product (₲×₲)⋊ℤ₂ to ₲, where ₲ is the full group of automorphisms of the infinite binary tree. Define phi in the following way: When σ is the identity (ε), we have that φ(π₁,π₂;σ) = φ(π₁,π₂), and when σ ≠ ε, we have that φ(π₁,π₂;σ) = φ(π₁,π₂)∘a.

Define the following automorphisms implicitly: b = φ(a, c), c = φ(a, d), d = φ(ε, b).

Then the group generated by a,b,c,d, has intermediate growth.

Problem 4: Find the time-1 map of the autonomous differential equation dy/dx = sin(y).

can someone help me simplify?

7i < 42 u/2