in my opinion doing old exams is good preparation for the exam since it gets you used to the type of exercises your prof gives, it's not necessarily a good way to learn a topic though
@KajHansen does walking on the street while reading what you learn count? :d (or more like: pump some weights, then read something, then pump more, then try to reproduce)
That's definitely true. The study most fresh in my mind showed that one's ability to learn and retain new information is improved within a few hours of exercise.
I'm sure walking is better than not walking @Null. I don't know what's "best" and for what purpose though. I.e. cardio might have different effects than anaerobic exercise.
I don't sleep a lot, 6-7 hours, but I have a regular schedule so it's not that bad? Anyway it's an holiday tomorrow here so I can sleep in a bit if needed
Hello I have a question about mathematics: when I have a variable x, if I have a statement: $\forall x P(x) \land \forall x Q(x)$, is it implied that I am testing the same x in both P(x) and Q(x)?
I've bountied someone @Ted. It raises the visibility of the question immensely. The one I bountied was somewhat ignored until I did, then it got 2 or 3 responses.
Let's say I have a contour integral on a non-closed contour with starting point $z_0$ and ending point $z_1$. Am I allowed to do a substitution like this? And under what assumptions?
$$\displaystyle \int_{z_0}^{z_1} f (z) \, \mathrm d z = \int_{u^{-1}(z_0)}^{u^{-1}(z_1)} f (u (z))u'(z) \, \math...
Once I get to 20k, I'll do it more often. I think it's a courtesy to someone who's put effort into an interesting question and that question isn't getting much visibility
@GFauxPas: Part of the problem is that your question is really a bit muddled. There's too much going on and confusing you. Branch cuts, homotopic deformation of the curve, and then substitution (which is a change of variables that keeps the curve the same). Just too much.
@Null keeps getting bogged down with the fact that $\Bbb R$ does not "canonically" sit inside $\Bbb R^2$ or $\Bbb R^3$. But there is a canonical $\Bbb R$ sitting inside $\Bbb C$ (by convention).
@GFauxPas: In your question, you also did not specify the original contour/path (or maybe I missed it). I assume you mean the line segment joining the two points, but you really need to specify it.
because you're thinking about the vector space structure, so you're talking about an isomorphism of vector spaces @null
$\mathbb{C}$ and $\mathbb{R}$ are isomorphic if you only think about the additive group structure but not if you think about them as vector spaces over $\mathbb{R}$
@TedShifrin, it took me a while, but I finally saw the error in my reasoning regarding that $x^8 - 2$ Galois group the other day. The "rotation" generator I was taking for granted wasn't sending $\sqrt[8]{2}$ to $\sqrt[8]{2} \omega$, with $\omega$ a primitive root.
He introduced the most basic properties that I never even thought of proving , I was taking them a priori, then I realized that those are axioms, but when doing this having to declare most basic things to be true in order to move forward and get more results, aren't we making Math separate from the physical universe ? Theoretical ? @TedShifrin
@Mahmoud: This is a philosophy question, not a math question. But I think the physical universe is perfectly fine with the axioms for the real numbers.
@Mahmoud, math has always been separate from the physical universe. The idea is that math, for whatever reason, is very apt at constructing predictive models of the universe. It is on us, however, to choose our axioms appropriately to give rise to useful / accurate predictive models.
If we are defining concepts to be independent from the physical universe, for example $\pi$ is the ratio of the circumference of a circle to it's radius, but a physical circle can't have an infinitely complex irrational circumference, isn't math not quite describing reality as it is ?
@WDUK If I'm not wrong ( I usually am ) you can't do that because you dont know if the terms are possitive or negative, so you have to evaluate each case
Exaclty @Mahmoud. Even "good" current models might break down in real-universe pathological scenarios because we like to model everything with continuous functions. But things like mass and energy are discrete and quantized.
One thing that concerns me about modelling the physical universe with continuous functions is that there are a number of things that have a property for all $n$ it holds, but the property breaks down with $n \rightarrow \infty$
Let's say I have a contour integral on a non-closed contour with starting point $z_0$ and ending point $z_1$. Am I allowed to do a substitution like this? And under what assumptions?
$$\displaystyle \int_{z_0}^{z_1} f (z) \, \mathrm d z = \int_{u^{-1}(z_0)}^{u^{-1}(z_1)} f (u (z))u'(z) \, \math...
