Let me add this explanation which provides an excellent transition from intuitive (see pasmith's very enlightening examples above) to formal interpretation of the essence of stopping times!
Thanks, that's verry illuminative example! Since my primary concern was to understand the slogan why the technical condition ## \{ \omega \in \Omega : T(\omega)=n\} \in \mathcal F_n ## translates on intuitive level into slogan that at time ##t=n## we know if ##T## has happened at ##n##, lets...
So if your question is why I started with "such abstract definition" the answer is simply because that this is exactly the standard definition of stopping time and my motivation is to understand it on intuitive level, what seems apparently to be possible as I explained above (... cp with...
I have a question about intuitive meaning of stopping time in stochastics. A random variable ##\tau: \Omega \to \mathbb {N} \cup \{ \infty \}## is called a stopping time with resp to a discrete filtration ##(\mathcal F_n)_{n \in \mathbb {N}_0}##of ##\Omega ## , if for any ##n \in \mathbb{N}##...
Thank you, I think the rough idea is clearer now, let me try to rephrase how I understood it up to now (...for sake of simplicity only for QED / photon field since I'm concerned only about rough idea):
We have integral## \int \mathcal{D}[A] \exp (i \int \mathrm d^4 x \left ( - \frac{1}{4}...
Does anybody know to which "Gauss embedding theorem" the speaker in this video talk at minute 14 (point 5. in the displayed notes) is refering too? Sounds to be a standard result in differential geometry, but after detailed googling I found nothing to which the speaker may refering too in the...
I see presumably. And in this whole procedure of adding this ghost term, should one also carry somewhere in the construction about implementing a gauge fixing term ( eg one proportionally to ##\partial^{\mu} A_{\mu}=0 ##), or is this issue with "adding a gauge fixing term" remedied automatically...
Just to clarify: Do you refer to the "contributions" of the ghost field to the action functional (in sense of path integral formalism) formed with resp to the free Lagrangian for field ##A_{\mu}## when we add additionally the "ghost term", right? Is that what you mean the contributions coming...
I have a question about following statement about ghost fields in found here :
It states that introducing some ghost field provides one way to remove the two unphysical degrees of freedom of four component vector potential ##A_{\mu}## usually used to describe the photon field, since physically...
Of course not, but let's keep things a bit more solution oriented. A well choosen model (...even thought every model is an idealization, the question is how "much" real information this idealization captures) could for sure give at least hints - when performing calculations "inside the...
Wait, so you suggest to apply the model from #3 for case (M1) , right? But then I not understand the choice of parameters you are suggesting to analyse the circuit with. Firsly, you assume ##V_1=V_2=12V## and both ##R_s=0,1## Ohm. But then performing Kirchhoff's voltage rule ##V_i+ V_s+V_R=0##...
Yes, that's the issue. I'm wondering what kind of model one should looking for "profound" enough to predict such behavior, not "too complicated" in order to be computable at all.
Okay, say we add too each battery a small resistence ##R_s##, so replace (M1) by
and apply Kirchhoff rules to...
I would like to do now (theoretically) a couple of rather naive things with two batteries with voltages ##V_1## and ##V_2## in parallel charged with an abstract load with resistance ##R##.
Usually, one learns in elementary electronics that the only "right" configuration with batteries in...
when you write ##y'=(\text{stuff})'##, I assume that you mean that the right hand side not depends on ##y##, right? Because, so far I understand the integrable system so 'easy' to solve, is that the goal is then in the next step to integrate independently(!) both sides and obtain ##y= \int...
I' m reading wiki article about Solitons and have some some troubles to understand the meaning of the following:
Question: In context of systems of differential equations, what means precisely "integrability of the equations"?
Is there any good intuition how to think about it? Has it some...