Recent content by GSpeight

Anyone have any ideas?

The monotone convergence theorem and dominated convergence theorem form measure theory can often be used to pass the limit inside an integral, rephrasing convergence in terms of convergence of sequences when necessary.

Hi there, I'm starting revision for Stochastic Analysis and have a few questions relating to the notes I'm reading. I'd much appreciate any clarification as I'm not as up to speed as I'd like. 1) In the definition of classical Wiener space I have H=L_{0}^{2,1}([0,T]; \mathbb{R}^{n}) the...

Surely not? In your example f is measurable but but f^-1 isn't :D Thanks again for your help :)

Thanks a lot for the explanation and example! It's clear now. The link you gave mentions the usual calculus formula for arc length doesn't work even though the cantor function is a curve differentiable almost everywhere. Is it sufficient for the curve to be Lipschitz continuous in order for the...

I've been doing a little work with Borel measures and don't want to confuse Borel measurable functions with Lebesgue measurable functions for R^n -> R^m. I'm, of course, familiar with the definition that a function f:R->R is Lebesgue measurable if the preimage of intervals/open sets/closed...

Thanks for the replies and encouragement. There's only 4 weeks left in the term and I'm still following the vast majority of my lectures. Hopefully if I hang in there I might be able to catch up with example sheets and work on my project over christmas. I think a change in environment might be...

What do other mathematicians do if they get "in a rut" and feel like they're losing motivation? I'm a 4th year MMath student in the UK and this term I'm taking courses in Stochastic Analysis, Differential Geometry, Brownian Motion and PDEs. I'm also supervising 1st year students (4 hours a week...

What do other mathematicians do if they get "in a rut" and feel like they're losing motivation? I'm a 4th year MMath student in the UK and this term I'm taking courses in Stochastic Analysis, Differential Geometry, Brownian Motion and PDEs. I'm also supervising 1st year students (4 hours a...

Sorry for the slow reply. I've been busy revising a different topic and haven't really encountered the generalisation of trace to operators on Hilbert spaces. Firstly sorry if my order of writing terms in the inner product was confusing - for some reason my lecturer prefers the inner product to...

Thanks! Could anyone give me a hint as to how to prove the Hilbert-Schmidt norm, ||T||_{HS}=(\sum_{n\geq 1}||Te_{n}||^{2})^{1/2} is independent of the choice of orthonormal basis. I've tried taking another orthonormal basis w_{n}, writing e_{n}=\sum_{k=1}^{\infty}(w_{k},e_{n})w_{k} so that...

The taylor series expansion is going to be valid where the sum converges, or alternatively you can just write that the taylor series expansion for exp is valid everywhere if you've been told this fact.

Sorry if I'm being too vague. x^2<e^2x so x^2+e^2x<2e^2x which gives you another useful bound :)

The ratio test is useful for calculating when taylor series converge. Each of the terms of your sum is in terms of 'x' so the ratio test tells you when the sum converges in terms of x.

Fine so far. How would you compare x^2 and e^2x? (at least for large x).