Well that's assuming you're not using purely integers.
$\exp(x)$ for some integer is rather trivial.
to get the mantissa, I can just multiply by some power of 10. I then end up with a large integer that has all the digits; I would just have to add the decimal point myself.
How would you easily compute $(\cos(1) + i\sin(1))^{x}$ efficiently to get something very accurate for real-time applications? That is, on the order of nanoseconds.
@JoeShmo well, I would, but how many terms do I need to get something like 22 decimal places of accuracy? 100? 1000?
So, ideally, I would have something that can compute $e^{ix}$ in constant time as a rational approximation for all real values of x.
And then ideally by improving the approximation of one or more transcendental constants, I could get more digits out of the circular functions as needed but in the same amount of time regardless of whether I need 1000 decimal places or only 12.
Does that make sense?
Or have I "gone off the deep end"?
More terms of the infinite series definition of $e^{ix}$ means more operations which means more time, not constant time. That's why I don't want to use it.
Yes, but it requires a different implementation than 1000 decimal places... I want a function that can give any accuracy or precision, including infinite precision were it possible to have a computer with infinite memory.
Yeah, but I conjecture that any irrational number can be approximated as the ratio of two integers.
Or at the very least, two reals.
To get more digits, I should think it is pretty straightforward... you just improve the approximation by changing only the constants themselves. The number of operations remains constant and therefore the irrational number we are approximating is computed in constant time.
@AMDG That's obvious, $\pi$ is approximated as a ratio of two integers by $3/1$, $31/10$,$314/100$ etc., and you can do the same for any irrational number. But the complexity of dividing two integers is a function of their length, not a constant
Also computing better approximations is not constant time, in my silly example computing better approximations is just as hard as computing the irrational number directly
$2^x$ is trivially implemented as bitshifts as 1 << x, and $log_2(e)$ is a transcendental constant, but I can just take out my calculator and compute $log_2(e)$ and choose some arbitrary number of digits that is sufficient to give an integer approximation of $\exp(x)$. In theory, this means I can compute $a^{b}$ in constant time since I now have $\exp(x)$.
@AlessandroCodenotti Sure, but it doesn't necessarily have to be a function like that which computes $\frac{a}{10^{b}}$. There might be some function which can generate the representation of those quantities one or more digits at a time.
Just like how computers primarily manipulate binary strings to perform real computations.
The other thing I'd like to point out is that if we don't need angles, sine and cosine are trivial. The reason is obvious, I'm sure.
Also, I don't know precisely how far I can get with this, but it would seem that I can get rational approximations of circular functions somehow by just dividing the area of a circle evenly into circular sectors and ensuring one of the radii is on an axis.
I don't know... I mean I could just settle for my cosine approximation that gets to 10^(-7) in accuracy and call it a day for my standard library, but I like the challenge, as well as the clear benefits, of finding such a definition for the circular functions as I have just described.
Everyone benefits! :D
Alternatively, we could develop screens that implement polar coordinates in an analog format instead of using square grids of pixels, in which case I wouldn't really care about circular functions.
Polar Coordinates master race. Euclidean plane go burr.
does anyone know of a more elementary argument to prove the following? Let $a,b,c,a',b',c' \in \mathbb{C}$ and suppose $f_1 = ax+by+c$ and $f_2 = a'x+b'x+c'$ are non-constant, then the zero locus of $f_1$ and $f_2$ are equal iff $(a,b,c) = \lambda (a',b',c')$ for some non-zero complex $\lambda$: 1. $f_1,f_2$ are irreducible, so by Hilbert's Nullstellensatz, they have the same zero loci iff they are associate
perhaps there is a way to do it by just thinking about row-rank of a certain matrix?
sorry, there is a typo in the above, $f_2 = a'x +b'y + c' $ is what it should be
okay, this is the argument I can come up with via thinking about a certain matrix, but I think it is a bit ugly: since the solution space of $\begin{matrix}a & b\\a'&b'\end{matrix}\right)$ where by that I mean that matrix times some vector $= (c,c')$ is such that its space of translations has dimension $ 1$, since $ax + bx + c = 0$ and $a'x + b'x + c' = 0$ have the same solutions
, and are non-constant, the rows in that matrix are multiples of each other and so $ax_0 + by_0 = -c = \lambda (a' x_0 + b'y_0) = -\lambda c'$
besides these two arguments is there anything simpler
@AMDG Actually I'm designing the language of the metacompiler right now which is directly related to my studies that I mentioned about composition and decomposition of algebraic expressions. Current idea that I intend to implement is defining behaviors implicitly by relating three or more states.
Huh, I did not know that pings me.... lol. I was just trying to provide a continuation.
It just means that when someone who is not Catholic wants to use my software, you just follow the laws of the Church strictly with regards to that software insofar as you are able which is informally summarized as "just using reason" and tersely as "do not use my software to do evil".
I forgot how to prove that $\mathbb Q (\sqrt[2]2, \sqrt[3]2, \ldots)$ is an algebraic extension over $\mathbb Q$. I tried to show that each element $\alpha$ belongs to some $\mathbb Q(\sqrt[n_1]2, \ldots, \sqrt[n_k] 2)$, but I didn't succeed. Any ideas?
If F and G are two sheaves over a top space X, what's the comparison between the fiber of Hom(F,P) at $p\in X$ and Hom($F_p$, $G_p$) (with $F_p$ and $G_p$ being the fibers at p of $F$ and $G$) ?
@AMDG Was just kidding I understand nothing about Galois. I am also more practical load my memory only with essential routines compiling takes longer though!
You know what you need to know based on what path God has placed you on. I am a programmer, so I need to know all about computers and software. God usually speaks with ideas, not words.
@BalarkaSen TempleOS is a cool concept, but it obviously isn't practical for anything beyond a hobby to mess with. I'm working on a metacompiler that works with "NAND trees" that represent patterns, and the language will compile to NAND trees.
You can represent anything and everything as a decision tree. That's the principle this whole thing works on. You give it an input pattern, a conversion pattern, and it spits out an output pattern. Simple yet powerful.
Hello everyone, I am stuck on an exercise in Group Theory and am looking to find how this thing is called: Let N be the normalizer of a finite group G, let K be an arbitrary subgroup of G. So then we have K*={Nk : k in K}. Is there a name for K*?
I tried multiplying everything by factor $u(t)$ and $u(y)$ but nothing I could solve appeared. Multiplying everything by $u(t,y)$ leads to a partial differential equation too hard to solve.
@LeakyNun Wait, I made a mistake in my definition: N is the normalizer of H which is a subgroup of a finite group G. K is an arbitrary subgroup of G. Then we have K*={Nk : k in K}.
@LeakyNun true, so I get your reasoning, but it's in the chapter on Homorphisms, so I shouldn't need to use Quotient groups to solve it. So I am wondering if the author had something else in mind
I mean either that or he put the exercise in the wrong chapter, lol.
There's a high possibility it's downvoted because someone has seen it's not a piece of "TeXed" mathematics and hasn't bothered to check that it's gap code
If $f:[0,1]\rightarrow \mathbb{R}$ is such borel measurable and that $f>0$ and $\int_{0}^{1}f^n dx= a$ for each n, for some $a\geq 0$, what can we conclude about the function f?
we're supposed to relate this to the notion of almost surely functions
