@dmckee, thanks for your advice on how to check the stability of one of my state variables that apparently settled into some final value. I wrote some simple functions that stops the integrator, so that I can perturb the state variable a little bit, and then run the solver again, using the final values from the first call to the solver as the initial conditions for the second call to the solver.
Do you think I should perturb all state variables, or just the one state variable of interest? And, what's a "standard" amount to perturb the variable by, before concluding that the solutions are stable / unstable? Is there a common benchmark? @dmckee
don't know. Probably because I don't really like to eat fried ice. or because don't know how to fry. Actually I have never fried anything. I usually eat in eateries.
two problems with a lot of parts on each of them, one of them is on geometric optics in curved spacetime and the other is on a "perturbation about a curved background"
I'll tell about them if you want after we turn in the exam
sign issues are a realm of suffering unto themselves
it's bad enough when you're checking your own work. it's worse when you're checking someone else's published work, because you're having to constantly argue with yourself over whether they did it wrong or you're just going crazy
(calculations involving imaginary time versus real time are especially annoying in that regard, ugh.)
@0celo7 sorry for the latency - checking servers. What was in the fried rice?
I must admit I like fried rice.
One of the local chip shops sells fried rice, chips (French fries) and curry. And after an evenings drinking it is delicious, though I'm not sure you would want to try it sober.
@Sid we had a long argument a few months back about te difference between a biryani and a pulav. The impression I got was that the words mean different things in different parts of India, because we ended up with students from different parts of India arguing with each other about what a biryani was.
I decided to cook Abgoosht for lunch. I realized that I had soaked way too much chickpeas yesterday. Don't know what to do with the extra chickpeas now :/
@lılostafa that looks a nice recipe. There are loads of different varients of choley. If you take a base of chickpea and fried onion, ginger and garlic you can add pretty much any spices you want.
Is there a sort of intuitive way to understand why the BCS density of states has its characteristic coherence peaks at the edge of the gap? (inspirehep.net/record/1254739/files/cleanscDOS.png) Is it fair to sort of think of it as the 'missing' density in the gap being shoved into those peaks? Or can someone offer a better picture
@MathematicsAminPhysics Editing and deleting messages after the time limit is generally reserved for exceptional cases, like removing sensitive information, inappropriate choice of words or similar.
@MathematicsAminPhysics I don't quite understand what you mean by "add". If you mean whether we can retroactively add a message between two messages already posted, then no.
@GPhys I like Terry Tao's description of nonstandard analysis as a form of 'epsilon management' here
(though I'll confess I've never tried to understand the entirety of that post, since i have the luxury of never having to think about epsilon-delta arguments in what i do)
@Semiclassical I'm somewhat surprised the internal set theory approach to nonstandard analysis hasn't grown more popular. I suppose in some sense it's more of a commitment, but I like the advantage of how you think about things in your head
In internal set theory instead of constructing the new numbers you add axioms to form a conservative extension of ZFC that adds the new numbers to existing sets
so it lets you have infinitely large numbers in the set of natural numbers without changing the definition of natural numbers - you just added the axioms that gave you the language to talk about the numbers that were "always there" (in some sense)
the primary thing you sacrifice is specification becomes a more subtle thing so that "the set of infinitely large natural numbers" may not exist (and in fact it doesn't)
@0celo7 you say that, but judging from the Terry Tao article that's not entirely true
i particularly have the second paragraph in mind
and this remark later on: "s a concrete example from my own experience, in one of my PDE papers with the “I-team” (Colliander, Keel, Staffilani, Takaoka, and myself), we had a rather severe epsilon management problem in our “hard analysis” arguments, requiring in fact seven very different small quantities $1 \gg \eta_0 \gg \ldots \gg \eta_6 > 0$, with each $\eta_i$ extremely small compared with the previous one."
@Semiclassical Well...yeah, they tend to be disguised as other things. I was reading a paper where they had to choose a radius carefully to apply a noncollapsing theorem. But those arguments are pretty complicated and I don't think nonstandard analysis would help...
the impression i get is that nonstandard analysis can be used to make the arguments simpler but at the cost of the bounds being qualitative rather than quantitative.
@Semiclassical In the talk I'm giving in two weeks, the key argument is actually that one knows exactly what the smallest number $C$ is such that an inequality $||x||_1\le C||x||_2$ is always true.
@Semiclassical Well, I'm not quite sure. I can verify the actual Pohozaev identity that he needs for the virial theorem, but not what he wrote there. Getting the $X,\sigma,\tau$ into the form he needs requires more signs
So the end result is true, but I suspect he wants a minus sign in front of that integral
@Semiclassical It's all quite confusing because you integrate out from the center of the star but that's where stuff is highest so the integrals sometimes get minus signs
