Yeah it's a random side effect of multiplying by the null operator. It requires knowing the position of a particle over time, so the probability that it is multiplied by $3$ varies in proportion to the probability density function describing the location of that particle in space, and you don't know if it was multiplied by $3$ or not until you evaluate the expression.
@LearningCHelpMeV2 the idea is fine, the exposition could be improved. for example, instead of saying the "numerator equals zero" (what numerator) i would have said something like "multiplying both sides by $a_{k-1}$..."
@BAYMAX is there some origin for this or just a formula that popped into your head?
@BAYMAX if you take $y=1$, $u=0$ and $x=2$ you can choose $a,b,c,d,e,f$ to be very small positive numbers (essentially zero) and you have essentially $2 < 1$.
but why would you think it true in the first place???
@AMDG Here's the error plot for the 10th degree Chebyshev polynomial for $2^x$, i.e., the curve shows $cheb(x)-2^x$, where cheb(x) is the poly. The minimax poly is very similar. As you can see the maximum absolute error is ~2e-16, so around 1 bit for standard 53 bit floats. The blue curve is for the poly computed using 160 bit real coefficients, the green curve is the approximation using standard floats.
@PM2Ring I also just realized that since my function for quotients is actually valid for positive rational inputs as well (which is deduced from the definition of ratios by definition), it can be used to approximate $2^x$ directly which is in the integer portion since it is of the form $Q = 2^x + \sum$. Therefore, $Q - \sum = 2^x$.
So for the infinite sum (we'll call it $I$), we have $I\equiv Q - 2^x = \frac{2^{x +\lfloor\log_2(y)\rfloor + 1}}{y} - 2^x$.
Now, being that we restrict our domain to $x,y\in \Bbb{N}$, we apply argument reduction to obtain an approximation of the quotient and then set equal to the RHS and solve for $2^x$, then substitute back into the above equation and solve once again for $2^x$ to get an identity of $2^x$ which is suitable to be approximated efficiently.
So technically 80-bit floats are what x86 supports, but that never seems to be seen unless of course those floats are in intermediate registers with undocumented register names that I can access.
Theirs uses 64 whole bits of significand and the rest is sign and exponent.
Sage is very flexible. It's very easy to specify whatever precision you want. I often use 80 bits for general work. To compute Chebyshev coefficients, it's nice to use at least double the precision of what you want for the coefficients. And for minimax, the Remez algorithm often gets flaky if you don't use a lot of precision. It's not unusual for me to use 240 or 320 bits, or even more.
FWIW, I posted an integer sqrt algorithm in Python that uses no division, only bitshifts. stackoverflow.com/a/70841843/4014959 Of course, it's not as fast as implementations in C, but I mostly wrote it to illustrate the algorithm. Think of it as runnable pseudocode. ;) It'd be easy to adapt it to C, if you have code for basic arbitrary precision integer arithmetic.
Yes, that's the one thing I like about Python and which I intend to solve with Project Aquinas.
Python is great if you just want to get something up and running, i.e. prototyping, due to the interpreted nature and the fact that it can execute statements and code instantly in a shell.
Of course the shell, performance, and readability are then the bottlenecks with Python for such a usecase.
That's the issue though. A language shouldn't allow for writing "cryptic one-liners", and of course, every language is readable to its users who use it day in and day out.
It's just a design flaw to have too many degrees of freedom without necessity.
That's also why I don't like shader nodes. They're ugly and cringe compared to the ordered and structured nature of the linear, 2D text plane of programming languages. I would like them more if the degree of freedom were restricted to fixed directed graphs on a low-precision grid.
I don't think a general-purpose language can avoid the ability to write cryptic code. The users of the language need to adopt and promote good practices.
@PM2Ring I agree on the latter, and it is impossible to remove all possibility of cryptic code, but at the same time, the language itself should facilitate that and not make it easy to make a mess.
But this is the maths chat, not a programming chat. We should only talk about programming stuff incidentally, preferably in relation to something mathematical.
This isn't it, but cell 37 is essentially what happens when we simplify the identity of $\frac{2^x}{y}$ by removing the common factor of $2^x$. desmos.com/calculator/sy5jp6biq1
And "integral ideal" is the same as fractional ideal?
uhh, yeah no I'm still sorting out these definitions
So divisorial ideals are special fractional ideals
It sounds like I'm supposed to conclude that a principal divisorial ideal for the algebraic integers is already contained in the algebraic integers (i.e. is 'integral')?
@LeakyNun Frankly I don't see that you presented an entire argument. But from the last comment I do see one: principal ideals are divisorial, and the algebraic integers does not satisify the ACC on principal ideals, so it is not Mori.
You aren't eliding "fractional" from any of those, are you? Obviously principal fractional ideals are divisorial but I'm still looking at the claim that principal (integral) ideals are divisorial...
if you've got the field of fractions K for a domain D, the fractional ideals are the D submodules of K
including the integral ideals
principal integral ideals are therefore principal fractional
ok, that makes sense then, and is simpler than i was worreid about! thank you
@LeakyNun Part of the reason I'm asking is that someone was looking for "really badly behaved integral domains" as in, "one without any of the other nicer conditions"
From the data I store, that woud involve avoiding Mori, N-1, atomic, maybe also Noetherian
@LeakyNun Wait... this also means that all more domains satisfy the ACC on principal ideals... I wonder why i haven't seen that before
waitaminit. The wiki says " ascending chain condition on integral divisorial ideals."?
I thought it was just "ascending chain condition on divisorial ideals."
To find oblique asymptotes of this curve $f(x) = \frac{x(x-2)(x-a)}{x^2-1}$ do you just consider when $x$ gets very large $f(x) \approx \frac{x^3}{x^2} \approx x$?
@TedShifrin Ohhh. You'd just express it in the form $Q(x) + \frac{R(x)}{x^2-1}$ Second term would approach $0$ as $x$ approaches infinity and the oblique asymptote would be $Q(x)$
Is it just me or is $\frac{\ln(2)}{x}$ an asymptotically good approximation to $2^{\frac{1}{x}}$?
For $x>0$
It would make sense, logically, now that I think about it. If you already knew the $x$th root of $2$, you could just divide by the root raised to an integer power such that the quotient is $2^{\frac{1}{x}}$.
@Xander That is not unusual behavior. We find plenty of examples in this chat of people who "forget" to read things we have just told them seconds before.
@TedShifrin The post there is not about the main page, but I commented on that post that it is showing up on our main questions page and linked to your question.
so leave your question on math.meta for now.
Davide Cervone will look at it. He is responsive (unlike the new responsive interface).
I have recollections of interacting with Davide when I was deeply ensconced in WeBWork (and wrote close to 1000 questions for my multivariable course).
We're also mathematical relatives — he's my nephew, I guess. :)
@robjohn @leslie Here is a more interesting question for you. Offhand, I don't see in my head that that norm is comparable to the usual Sobolev norm, but maybe I'm missing something.
I think that bounding the first derivative and knowing the value at a point, should give an equivalent norm, but the function has to be continuous to use $f(0)$.
@TedShifrin Regarding this bug report, we can mark meta questions with the tag status-review in order to put it into the debugging queue at a network level. Thus bug reports on meta contribute to the bug tracking system. Marking a bug as a duplicate (even if there isn't an answer) is a way of indicating that the bug has been appropriately reported, and that someone will get to it.
