makes sense. with sequences of this type, unless there is a clear link to something more well known (e.g. the prime factorization of the input) brute force might be the best approach. even if you establish that something is bounded below by an increasing function you might still have a large number of cases to check if the function grows slowly.
if $\sigma\in S_n$ (the symmetric group of degree $n$) is such that $\sigma$ sends $i$ to $j$ and let $\theta\in S_n$ be such that $\theta$ sends $i$ to $s$ and $j$ to $t$ then $\theta^{-1}\sigma \theta$ sends $s$ to $t$
theta^{-1} sends s to i. sigma sends i to j. theta sends j to t. if you write permutations on the left, this would be a computation that the permutation denoted $\theta \circ \sigma \circ \theta^{-1}$ sends s to t
it's fairly common outside of algebra to write function composition $f \circ g$ to indicate the function where $g$ is applied first, and then $f$. this parallels the common practice of writing functions to the left of their arguments
in algebra sometimes people adopt the mirror image of that, and apply functions on the right
Where I can find formal proof of there exists an equivalent parse tree for each derivation? There is a lot of informal proof of equivalency on the internet but I need formal proof to reference it in a paper.
@love_sodam Yes. The logic given in the accepted answer applies to any n dimensional ball. It's related to the fact that the ball is the shape that minimizes the surface area for a given volume.
it's substance-free, just a notational thing. particularly once you have a number of different things acting or operating on other things, you end up having to keep track of this no matter what choice you make
but the purpose of adopting a convention is usually to minimize the amount of brain cycles you have to spend in keeping track of it
Ah, I first read some Herstein's algebra where $fg$ was simply composition of functions (from R to L), so thought similar convention would be followed in Topics in algebra. But you're right. the convention being followed is from L to R.
we ought to do that some of the time. applying on the left does make sense for a lot of things. (maybe less sense for composition operators, as we noted the other day :))
@leslietownes A baby lamb for your daughter. Her name is Bella, she's from a small farm in Idaho, and she's 15 minutes old in this photo. I saw her on EweTube.
@Semiclassical Postfix notation, aka Reverse Polish Notation, is fun. Early HP scientific calculators used it. And it's been used in a few programming languages, eg Logo and PostScript. I did a bit of Logo a few decades ago, but I've written a lot of PostScript. It takes a little while for my brain to get into RPN mode when I do PostScript coding, but after a few minutes or so it feels very natural.
you see this a lot even when people aren't defining maps on bases. e.g. if some chemical quantity f of interest is a function of pressure P and volume V of a gas, an applied chemist might write f(P,V) or f(V,P) for the exact same formula, because they can tell the letters apart. this is squarely at odds with the formal convention for writing multivariable functions.
under which if i write f(P,V) = P + 2V that tells me f(2,3) = 2 + 2*3 = 8, whereas if i write f(V,P) = P + 2V that means f(2,3) = 3 + 2*2 = 7.
the classic example is "if $f(x,y)=x^2+y^2$, what is $f(r,\theta)$?"
in a physics context, you'd always read that as $f(r,\theta)=r^2$
because we'd interpret that as writing $f$ in polar coordinates vs Cartesian coordinates
the more defensible interpretation of that, to me, is that one really has $f=f_c\circ g_c=f_p\circ g_p$ where $f$ is a function on R^2, $g_c,g_p$ assign points in R^2 to their cartesian/polar forms, and $f_c,f_p$ are how you compute relative to those forms. of course you can just take $g_c=\text{id}$ and $f=f_c$.
a friend of mine who went to grad school in physics but had more of a math background used to joke about this stuff all the time. 'we have no problems of computing things that have no formal mathematical meaning whatsoever, and never discuss our notational conventions, but you can be damn sure that a prof is going to spend 20 minutes on a rigorous proof of the cauchy-schwarz inequality"
@Koro Well, it progresses in normal reading order, from left to right, so to compose two functions you'd write x f g. Whereas, $g(f(x))$ progresses from the innermost-nested outwards, so it goes in the reverse of the normal reading order. So with something like $g(f(x)) + h(f(x))$, you're going backwards & forwards. In postfix notation, that'd just be x f g x f h +
i think you have to envision the domain as a set with distinguishable symbolic 'variables' and you're precomposing with something that acts on 'variables' as opposed to entries of an n-tuple
so what's that domain, i dunno. formal expressions in x_1, x_2, x_3 made up of, precalculus expressions? with a proviso that you only use expressions that make sense when you evaluate these expressions at x_1, x_2, x_3 lying in some region of 3-space?
i also need to write this out in Dirac notation. one often writes $\psi(x)=\langle x|\psi\rangle$ for the position-space representation of $\psi$. (it's not really correct without appealing to a rigged Hilbert space, b/c otherwise $\langle x|$ isn't actually a meaningful Hilbert-space vector)
"Borel, then an unknown young man, discovered that his summation method gave the 'right' answer for many classical divergent series. He decided to make a pilgrimage to Stockholm to see Mittag-Leffler, who was the recognized lord of complex analysis. Mittag-Leffler listened politely to what Borel had to say and then, placing his hand upon the complete works by Weierstrass, his teacher, he said in Latin, 'The Master forbids it'."
it's the natural definition as far as I'm concerned, same thing as with local homeomorphism
but I can't quite make heads or tails of that discussion at the wikipedia article either
the existence of a local diffeomorphism is not the right notion for things looking the same locally
"For example, one can impose two different differentiable structures on {\displaystyle \mathbb {R} }\mathbb {R} that make {\displaystyle \mathbb {R} }\mathbb {R} into a differentiable manifold, but both structures are not locally diffeomorphic (see below)"
the remark about patching things together makes no sense when talking about local diffeomorphism
it reads like whoever wrote that discussion part of the article was themselves confused about the difference between "local diffeomorphism" and "diffeomorphism defined on an open subset"
@TedShifrin Wikipedia doesn't have enough quality control, which is what allowed the Scots Wikipedia fiasco to occur en.wikipedia.org/wiki/Scots_Wikipedia Fortunately, the science & mathematics parts tend to get better scrutiny, although severe errors can sit there for years.
The stuff about smooth structures in $\Bbb R$ is red herring. Nothing local versus global there. There’s a unique smooth structure globally up to diffeomorphism.
Waiting at a vet in Agoura Hills. Our dog has been off her food, and it’s gotten to the point that we are very worried. We had a hard time even finding one that would take us. Took us 90 minutes just to do that. 3½ hours later they took her in.
@TedShifrin probably both. However, getting vet appointments has been awfully hard recently. People have gotten more pets because of covid and vets are taking advantage of the situation.
If it weren’t Sunday, we’d have just gone to our own vet near home.
It was $600 for two xrays then $1000 for an ultrasound at another vet as a follow up. I made a claim on my pet insurance, which all went to deductible, then I got a bill for the next year at over double what it was the year before.
our cat saw an cat eye specialist four times, probably $800 total. her pupils were sometimes not the same size due to an imbalance of fluid pressure in the eye. it went away eventually.
I've noticed that in QFT literature, integrals are usually written as $\int \!dx ~f(x)$ instead of $\int f(x) dx$. Why?
