Recent content by dismo

Let F: H ->H be a map of a Hilbert plane into itself. For any point A, denote F(A) by A`. Assume that AB is congruent to A'B' for any two points A,B. How can I prove that this map is in fact a bijection? In a arbitrary Hilbert plane, one can not be certain that square roots exist, so...

Working in a Hilbert plane, show that any rigid motion that fixes at least three noncollinear points must be the identity. I am certain that I can claim that: (i) any translation of the plane will fix none of the points (ii) any rotation will fix a single point (iii) any reflection will...

Problem: Let x(t), y(t) e a solution of dx/dt=y+x^2 dydt=x+y^2 with x(t0) NOT = y(t0) Show that x(t) NOT = y(t) for all t Attempt: I feel like the easiest way to show this would be to show that x=y is an orbit of the system and then simply use the fact that orbits may not cross due to the...

I believe that I mean if y1(x)=0 and y1(z)=0 then y2(a) must be equal to zero with x<a<z. In other words, y1 and y2 must alternate zeros as t varies.

Supose y1 and y2 are a fundamental set of solutions to a second order ODE on the interval -infinity<t<infinity. How can I show that there is one and only one zero of y1 between consecutive zeros of y2. I really don't even know how to egin approaching this problem. Any direction would be...