Recent content by alexfloo

As HallsOfIvy brings up, frequently in differential equations, especially with nonlinear or even chaotic systems, we're really not interested in specific solutions at all. Rather, we're interested in the effects of parameters, or general long-term behaviors ("Will the two species live in stable...

The elements of the quotient space V/W are each hyperplanes which lie parallel to and have equal dimension to W. Since W is the only one passing through the origin, it is the zero of the space. For instance, ℝ2/Y, where Y is the y-axis, is essentially the x-axis. This is because each line...

My point is that this is a deep area of mathematics that's been studied for thousands of years. The last couple of centuries have seen incredible advances, and they have usually required extremely complicated and advanced theoretical techniques from complex analysis, modern algebra, and a whole...

http://www.claymath.org/millennium/Riemann_Hypothesis/1859_manuscript/EZeta.pdf You may be interested in this paper. Riemann was a very stark writer, and leaves little explanation. Most of his work was in geometry, but his single foray into analytic number theory, the paper I have posted a...

The sin(nx) are eigenfunctions of that equation. This doesn't mean they are spanning. In fact, just sinx is an eigenfunction, but just sinx is not enough. They point is that we need all the eigenfunctions, so we need all the sin(nx) and all the cos(nx). The whole point of these decompositions...

That worked perfectly: F(z,t) = \frac{tf(z) + (1-t)g(z)}{|tf(z) + (1-t)g(z)|} is my homotopy. Thanks a lot!

If homotopies between corresponding points in paths constituted a homotopy of those paths, then every path-connected space would be simply connected.

A homotopy between which two points? Having a homotopy between f(x) and g(x) for each fixed x doesn't give me a homotopy between f and g, since those homotopies need not vary continuously with x.

"Let f,g:Sn→Sn be maps so that f(x) and g(x) are not antipodal for any x. Show that f and g are homotopic." Here's my initial approach: I figured it would be easier to work in In instead, so I note that Sn is the quotient of the n-cube with its boundary. Therefore, each map Sn→Sn can be...

Nope! Note that singletons are in fact closed intervals. Therefore, if one of the closed intervals is a singleton than it can be. That's not necessary, though. Consider In=[-1/n,1/n]. Zero is in every one of those intervals, and every nonzero number fails to be in one of the intervals...

The Cantor set is *not* a subset of the rationals. Remember, we start with a real interval. We only remove intervals, and every nonempty nonsingleton interval contains both rational and irrational numbers. This doesn't in itself prove that its not a subset of the rationals, but hopefully it...

Another interpretation of your question is "Why do we include this additional axiom f(1)=1 explicitly?" If that's what you meant, then the answer is essentially the same. It's because it's not implied by the other ones.When we're talking about a concrete category of algebraic objects, a full...

I'm still not 100% certain what your question is, so first I'll give my interpretation of the question. I believe you are asking why the axioms of general ring homomorphism (f(x)f(y)=f(xy) and f(x)+f(y)=f(x+y)) imply the additional axiom of ring homomorphism for unital rings (f(1)=1), in the...

Epsen, epimorphism is a term primarily used within the context of a category where the meaning can be expressed otherwise. For instance we might alternatively choose to say that a map is "an epimorphism to Y" but not an epimorphism to some larger set Y'. This would support the interpretation of...

In category theory, a relatively new field, the prefix co- commonly signals the corresponding concept in reverse. Indeed, the codomain in the category Set is exactly the domain of the corresponding morphism in the dual category Set*. This strongly suggests that the term codomain originated in...