Hello, I'm making a volumetric renderer. I'm trying to render the union of a rectangular plane and a square whose sides are equal to the maximum length of one side of the rectangle, but without using "if" in the GLSL shader code. Any ideas?
The rectangle and square are coplanar and the centers of both are aligned.
I'm also trying to calculate a vector using a sphere segment sitting on the square. The circular plane on the segment is coplanar with the square and the screen, and the width of the segment is equal to the width of the square.
Complex Analysis: To transform the plane with slit [−1,1] into the upper half plane, I have the following: $T_1(z)=z-1,T_2(z)=1/z,T_3(z)= ? , T_4(z)=\sqrt(z),T_5(z)=iz $ thus $T=T_5\circ T_4\circ T_3\circ T_2\circ T_1$ is the desired transformation. My question is how should $T_3$ be defined?
And since I can do this from big to small, I can generate a sequence of sums of truncations that limits to $\pi + e$. If eventually periodic phenomenon is observed, then we are settled
For those who don't know what I am doing, here's a brief summary:
The above shorthands allows the computation of sums from the largest digit to the smallest digit by taking cared of carryovers in a commutative and associative manner. The basic idea is as follows:
1. Write the binary number you are interested in say 10010.01
2. Write the positions, denoted by the negative of the exponent, where it is one. i.e. $[-4,-1,2]$
3. Do the same for another binary number e.g. $111.01 \implies [-2,-1,0,2]$
4. Combine the contents of the two lists together $[-4,-1,2]+[-2,-1,0,2] = [-4,-2,-1,-1,0,2,2]$
5. Use the following axioms to simplify the resulting list
I use negative of the exponent so that when I deal with $\pi + e$ I don't need to put minus signs everywhere which is annoying. If you want a more natural formulation, you can just write the positions as the exponents. Then you will use these rules instead:
This means, at least for this length, the probability that the digits of $e$ get shifted to the left by one is at least twice as high than those of $\pi$
In addition, the regions where there is a cluster of ones seemed to be uniformly spaced from each other, suggesting some kind of quasiperiodicity
It is unclear if as we make the binary expansion longer, it will tend towards periodicity
For comparision, here's the plot for the distributions of ones in $\pi$ and $e$
All 3 of them. Note how there are digits that are never populated by $\pi,e\,\pi +e$ at least for this truncation
Also note how similar is the distribution of ones between $e$ and $\pi +e$, except with more periodicity. This suggests that while mostly what happens in the addition is that the ones in $e$ preferentially get shifted to the left, the occasional shifting of the digits of $\pi$ lead to more periodicity
In order to investigate further, we need some tools to quantify this change in the distribution, in order to figure out whether in the limit of countable places, $\pi+e$ will be eventually periodic
Hey. Can anyone explain it to me? It is claimed that every Einstein solution with cosmological constant in vacuum is locally isomorphic to Minkowski, deSitter, or adS?
In this que https://math.stackexchange.com/questions/667998/show-that-ex-1-x-x2-2-cdots-xk-k-for-n-geq-0-x-0-by-in, 6005 says that $e^x > 1$ for $x > 0$; shouldn't that be $e^x = 1 + x$?
Leaky: Well if each Rambles get frozen due to inactivity and I had to make a new one, you can easily see how quickly those rooms will pile up. In fact, who's idea in SE to make chat rooms expire such that room owners cannot unlock them
@ShaVuklia Maybe this could help. :) I just Googled it. I don't know myself. There's a forum for LaTeX that also has a chat. You should search or ask there!
@MatsGranvik, I think I might be able to interest you in the last question I wrote... based on some previous questions I have seen you answer/engage with.
@Mason Looks a bit like Dirichlet series for logarithms. I know that such series converge if the sum of the period is zero but I don't know exactly why.
@Mason Looks like you are doing Hurwitz zeta functions or almost the same the generalized Riemann zeta function, according to the Wolfram Alpha output.
If we are going to chat about the question itself it should be somewhere that future readers can find it.
@lush. Hello.
@MatsGranvik. If you are correct about this (and I suspect you must be at least in some trivial ways: consider $s=1+i$ and $s=1-i$... these must relate to one another.) It should be a comment on the page itself so other readers can see it.
So much for the hope of algebra to find those intersections. What you can do is replace the logarithms with rational numbers but it will not get you anywhere before you understand this picture:
@user170039 I think they are equal, take $A \subseteq X$. Then $Cl_1(A)$ and $Cl_2(A)$ both represent the closure of $A$ in the topology on $X$ (since they induce the same topology) so $Cl_1(A) = Cl_2(A)$. This applies for every $A \in P(X)$ so $Cl_1 = Cl_2$.
For some things my YouTube lectures might help you. For basic proofs/reasoning, I'm fond of a book by a British guy named Kevin Houston. It's called How to Think Like a Mathematician.
I thought taking powers of numbers on the unit circle meant rotating them, so I can't really figure out how +i would ever result on something still on the unit circle
