Recent content by Reb

(don't have an answer yet) We say that two sequences f,g are f=O(g) if-f there is a c>0 such that |f(n)|<c|g(n)| uniformly as n tends to infinity. If g(n)>2, does f=O(g) imply lnf=O(ln(g))?

There's also a different side to doing math. Reading from S. Ulam's "Adventures of a Mathematician": "In many cases, mathematics is an escape from reality. The mathematician finds his own monastic niche and happiness in pursuits that are disconnected from external affairs. Some...

Well, if I was to transform the question into the Euclidean setting, I would ask why would someone need to bother with what a plane is, in order to define straight lines! What I mean is that there is a minimum of insight required in order to go into the theory of geodesic distance, and this...

That's a bit funny to ask, since a Riemannian manifold must come along with a metric. Yes there is. That's what enables one to define the Levi-Civita connection in the first place. You can look it up in every good book about Differentiable Manifolds (like Do Carmo's book)...

I'd be surprised. http://www.math.hmc.edu/~gu/curves_and_surfaces/surfaces/moebius.html" page calculates for us that, for the standard parametrization,

On a manifold, that is... I know http://www.cambridge.org/US/catalogue/catalogue.asp?isbn=9780521409971" book that deal with qualitative properties of the solution of the heat equation on a manifold (which, since the Laplacian depends on the metric, becomes non-constant coefficient...

Not really. Under the given conditions, the implicit function theorem guarantees that the expression f(p)=f(p_1,p_2)=0 defines a function y=y(x), x\in (p_1-\epsilon, p_1+\epsilon)=I (or x=x(y), but choose the former for convenience) . So an extremum for g on f^{-1}(0) is in fact the extremum of...

Yeah, but they also don't. Anyway, I'm not a vector, I'm a human being (dammit). I'm not linear, and I'm surely not stable.

...Twisted?

How about ...you re right? I mistakingly assumed that every solution has to meet the y axis.

Ok let's see... For 1), you are given the ODE with the parameter shifted by a small \epsilon, and you are required to show that this new ODE with \epsilon will have an equilibrium, which is "close" to the original one. Since f_a has nonzero gradient, continuity implies that f'_{a+\epsilon}...

Three constants? That would be the case if the ODE was linear. There is another problem here. The ODE is autonomous, and all functions of the form y=const. are solutions, plus the solutions cannot meet each other. So this ODE has only trivial solutions.

The http://en.wikipedia.org/wiki/G%C3%A2teaux_derivative" derivative. Integration on Banach spaces is a more delicate subject. There is however an approach parallel to the construction of the Lebesque integral. A taste of it http://www.worldscibooks.com/etextbook/5905/5905_chap1.pdf"...

This is called a curve of 'constant inclination', as t/k=1=cotan(a), where a is a constant angle formed by the tangent to the curve and a specific fixed direction. This will make integration easier. Ps. There is (free) software on the internet, which will draw a curve on the input of...

They are the same notion under a different name: The Euler-Lagrange equations for the variation of geodesic length, over the space of geodesics with fixed ends.