the more general case of not defined can be something akin to having a gap in the function. No physical motion allows this and we never saw a classical object discontinously jump from one place to other or vanishing
However, using L Hospital rule, rationalisation, factorisation, I can arrive at a limiting value..and I would consider that as velocity. But that's wrong because velocity should simply be not defined.
Hi, a quick question: In this blog post about the spectral theorem by terrence tao in exercises 16 and 17 the symbol $\ll$ is used to compare eg $\|\mu_{f,g}\|_{TV}\ll \|f\|\,\|g\|$. Apprently $\ll$ is a weaker notion than $\leq$, what does $\ll$ stand for in this context?
(I mean that in physics there's no case of indeterminacy so it's alright. However in maths I am allowed to remove indeterminacy. Why/How do such methods work?)
Always remember to not conflate the limit of something with the function value itself. The limit can tend towards some value, but the point can remain undefined. In physics, velocity is a derivative of displacement and since we never have any experimental records of things suddenly vanishing and appearing before moving, we tend to treat velocity as continuous
@Astyx Anyway, I framed my question differently. I hope it would be easy to answer for you > In physics, the limiting/approaching values lead us to the actual values. however, in mathematics, limiting values always lead us to approaching values only, not to the exact value at a point.
yes, that's what continuous means, the limit agrees with the function value
In physics, discontinuous functions related to motions are not as common compared to continuous functions. In fact, they are often assumed to be smooth
and I never heard of any case where a velocity can be discontinuous as even if you have infinite acceleration, the best you can do is making a cusp on your velocity function
Eh, I have a stupid question. How do you enjoy formulas? I'm finishing my degree in computer science and I really enjoy the theoretical side of it. However with formulas I'm much worse than with worded reasoning and its holding me back.
For me, some formulae have algebraic patterns that 1) make pretty pictures, 2) form nice relations that connect many other related formula together 3) the meaning of each symbol in the formula and how they go together explains how it can describe a theory in a concise manner
[Me today in chemistry: ] The orbital calculations are finally working: To accidentally celebrate this, I read 3 journal articles in 2 hours, while sending 20+ calculations to the computing cluster
(though to be fair, each article has max of 6 pages)
@Secret that answers why one should find them useful and beautiful. however do you actually think about the 'meaning' of every formula you come across or do you just work with them formally?
do you manipulate formulas in your head or on paper? I do all the thinking in my head and write things down after that. however that doesn't really work with complicated formulas. but looking at them on paper doesnt seem to work for me either
@Semiclassical ok, a more practical question: how do I build up my skills with formulas the best? Just working through a hard book with lots of formulas in it?
Hmm, well to be sure, are you thinking of formulae in a given context, like math, or just in general? If, for example, you see mathematical formulae as more of a means to the end of science, knowing how they're proven is sorta one of those things which is good on general principle and can help you understand something about the nature of the things that use it
But becomes more of a, oh I guess knowing this or that is cool and maybe useful
But yeah @Fabian, in this case you'd spend less time thinking of the formula and more time thinking of the contexts in which the formula comes up (which, to be fair, can often provide some insight into the formula)
On the other hand, if you also take math as an end in itself, you definitely want to know how they're proven, so while you may additionally think about contexts in which some formula comes up, you also want to think of the formula on its own terms and why it's true
@Daminark I think you have misunderstood me a little bit, to verify the correctness of a proof is the absolute minimum I would demand from myself. Its about feeling like I could have come up with them myself. to see the intuiton that binds together a group of theorems
@Fabian Do keep in mind that there is a difference between knowing that a proof is correct and being able to see how one could come up with it if you didn't already know it
@FabianGerhardt Some of these have easy or graphical manipulations that once get used to them, you can mentally rotate the formula, so to speak, but generally I tend to be better at working on paper because then I see where I am going. To me, using a formula is like driving around in mathematical space and you are heading somewhere
one example. why does e^ix go in a circle? approach number 1: split the sum defining e in to halfs sin and cos respectively. approach number 2: take the derivative of e^ix, its ie^ix, i something turns it by 90 degrees, the slope always is the tangent of a circle!
I kinda like this way of thinking of it: $|e^{ix}(w-z)|=|w-z|$, which means multiplication by $e^{ix}$ is an isometry that fixes the origin - it's a rotation. (Cont'd)
(Well it could be a rotation but trying it on $1$ and $i$ rules that out)
There are some occassions I don't like formulas though, such as how most integrations and their substitutions seemed to come out at nowhere. This is the reason I started the Integral Project, in order to allow me to be able to appreciate these formulae
Thus, if you trace the function $x\mapsto e^{ix}$, you're essentially multiplying $1$ by $e^{i\epsilon}$ many times, or slowly rotating $1$ at a constant speed.
@AkivaWeinberger I don't understand what you are saying, but that seems like exactly the thing I'd like, some abstract definitions helping one to conceptually grasp the topic
R. S. ELLIS (1981). Large deviations and other limit theorems for a class of dependent random variables with applications to statistical mechanics. Univ. of Massachusetts preprint (unpublished).
As of now, I have yet to locate that paper and thus explain the fate of those methane complexes in the analysis of the first landmark paper in the series
What's funny is that one of the papers which cites Ellis 1981 (indicating that another's papers results rest on a theorem in that unpublished manuscript) indicates in the bibliography that it's "apparently a preliminary version of Ellis (1985)"
@s.harp equally WTF, it is even harder as you literally need to know the person (and provided he/she still remember back then what they are doing) to get that missing info
If the proof is essentially known to be due to John Doe and John Doe tells you the proof and you reproduce it, you better not claim it as a result (or claim it as folklore).
@TedShifrin Do you know something about the classification of Del Pezzo manifolds (Fano manifolds of coindex 2, i.e. Fano index n-1 where n is the complex dimension)? I am interested to know how many there are in dimension 5.
@LeakyNun Well, I am thinking about that real induction link seen in the main, whether we can do everything of size continuum by considering it as union of intervals
One of my Fellows asked me whether total induction is applicable to real numbers, too ( or at least all real numbers ≥ 0) . We only used that for natural numbers so far.
Of course you have to change some things in the inductive step, when you want to use it on real numbers.
I guess that using i...
One thing I am interested in is how to think about cardinals more than continuum. I know I can use bijective maps to compare between them (and do some cardinal arithmetic), but I am not sure if something more concrete can be said about them such as their usual topologies
Could someone explain the differentiation of $1+ln^3x$ w.r.t. x to me? Research effort: Search and tried to understand from the steps displayed by derivative-calculator.net and tried to apply chain rule but I don't think it's significant here.
I am not really seeing the connection, or why you would think that I guess. It seems the product would be easier topologically to study. Maybe this is part of some other conversation I missed
The long line is $\omega_1 \times [0,1)$, the above just make it "longer" by using $\omega_2$ instead. If something interesting happens in terms of limit points and other topological things, then it will tell me more about sets of higher cardinality that is not just about their sizes
$\omega_1$ is the first uncountable ordinal, and I took the long line (actually it should be called the closed long ray) definition from wikipedia en.wikipedia.org/wiki/Long_line_(topology)
Hm. Suppose $X$ is any set and $f:X\to X$ is a function. Let $Y=\{x\in X:\exists n,f^n(x)=x\}$ be the set of elements of $x$ that get sent on loops by repeated applications of $f$.
Let $X$ be a CW-complex of finite dimension.
Suppose that $\pi_k(S^n)$ is isomorphic to $\pi_k(X)$ for all $k\geq 0$ and $n>1$.
Certainly, $\dim X\geq n$. It is easy to show that $X$ has homotopy type of $S^n$ provided $\dim X \leq n+1$.
Question: is it true for any $X$ of finite dimension?...
Let $X$ be a CW-complex of finite dimension.
Suppose that $\pi_k(S^n)$ is isomorphic to $\pi_k(X)$ for all $k\geq 0$ and $n>1$.
Certainly, $\dim X\geq n$. It is easy to show that $X$ has homotopy type of $S^n$ provided $\dim X \leq n+1$.
Question: is it true for any $X$ of finite dimension?...
