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Re: Thread for Non-CA Academic Questions

Postby Bullet51 » March 20th, 2018, 11:58 pm

What's the optimal way to encode the following constraint in conjunctive normal form?
Exactly 3 out of n propositional variables is allowed to be true


Is it of this kind?
At least 3 out of n propositional variables is allowed to be true
At most 3 out of n propositional variables is allowed to be true
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Re: Thread for Non-CA Academic Questions

Postby gameoflifemaniac » March 30th, 2018, 5:42 am

Are the functions 0 and e^x the only functions that are its own derivative?
And are there any functions that are its own derivative's derivative?
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Re: Thread for Non-CA Academic Questions

Postby 77topaz » March 30th, 2018, 7:05 am

Any scalar multiple of e^x (like 2*e^x) is also its own derivative. Furthermore, these are all also their derivatives' derivatives. Examples of functions that are not their own derivative but that are their derivatives' derivatives would be the hyperbolic cosine and sine.
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Re: Thread for Non-CA Academic Questions

Postby BlinkerSpawn » March 30th, 2018, 8:35 am

More generally, functions that equal their n-th derivative will be composed of terms of the form Ce^kx, where C is a constant and k is an n-th root of unity.
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Re: Thread for Non-CA Academic Questions

Postby gameoflifemaniac » March 30th, 2018, 9:52 am

BlinkerSpawn wrote:More generally, functions that equal their n-th derivative will be composed of terms of the form Ce^kx, where C is a constant and k is an n-th root of unity.

So, that n-th root can be complex?
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Re: Thread for Non-CA Academic Questions

Postby A for awesome » March 30th, 2018, 9:56 am

gameoflifemaniac wrote:
BlinkerSpawn wrote:More generally, functions that equal their n-th derivative will be composed of terms of the form Ce^kx, where C is a constant and k is an n-th root of unity.

So, that n-th root can be complex?

Yes — z^n = 1 has exactly n solutions which lie evenly spaced around the unit circle. For example, the fourth roots of unity are 1, i, -1, and -i, which form the vertices of a square centered around 0; the third roots of unity are 1, -(1/2)+(√(3)/2)i, and -(1/2)-(√(3)/2)i, which form an equilateral triangle centered around 0.
x₁=ηx
V ⃰_η=c²√(Λη)
K=(Λu²)/2
Pₐ=1−1/(∫^∞_t₀(p(t)ˡ⁽ᵗ⁾)dt)

$$x_1=\eta x$$
$$V^*_\eta=c^2\sqrt{\Lambda\eta}$$
$$K=\frac{\Lambda u^2}2$$
$$P_a=1-\frac1{\int^\infty_{t_0}p(t)^{l(t)}dt}$$

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Re: Thread for Non-CA Academic Questions

Postby Macbi » March 30th, 2018, 9:59 am

gameoflifemaniac wrote:
BlinkerSpawn wrote:More generally, functions that equal their n-th derivative will be composed of terms of the form Ce^kx, where C is a constant and k is an n-th root of unity.

So, that n-th root can be complex?

Yes, in particular cos(x) = (1/2)e^(ix)+(1/2)e^(-ix) and sin(x) = (-i/2)e^(ix)+(i/2)e^(-ix).
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Re: Thread for Non-CA Academic Questions

Postby 77topaz » March 30th, 2018, 4:53 pm

Macbi wrote:Yes, in particular cos(x) = (1/2)e^(ix)+(1/2)e^(-ix) and sin(x) = (-i/2)e^(ix)+(i/2)e^(-ix).


Yes, and those are examples of functions that equal their fourth derivative (but not their first, second, or third).
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Re: Thread for Non-CA Academic Questions

Postby muzik » May 25th, 2018, 5:14 pm

What would the tilings {1,∞} and {∞,1} look like?
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Re: Thread for Non-CA Academic Questions

Postby BlinkerSpawn » May 25th, 2018, 8:25 pm

muzik wrote:What would the tilings {1,∞} and {∞,1} look like?

{∞,1} would be a half-plane (and not exactly a tiling, considering the {,1} means the tiling consists of a single shape)
The structure of {1,∞} is a more difficult question, because it can't be a dual tiling to {∞,1}. If you were to take, say, the edge graph of a {n,∞} tiling you would then have a pattern with an infinite number of segments (one-sided polygons) meeting at each vertex in the hyperbolic plane, but these don't constitute a tiling in and of themselves because the constituent (one-sided) polygons share no edges.
Under the strictest definition of a tiling, a {1,∞} tiling would be an infinite number of edges between two vertices (or the dual tiling to {∞,2} embedded in the Euclidean plane)
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Re: Thread for Non-CA Academic Questions

Postby Apple Bottom » May 30th, 2018, 1:17 pm

A silly question from a silly pony---

I was looking at the slides for Harrison's recent "[i]Let's make set theory great again!/i]" talk (more out of curiosity than anything else); on page 31 (that is 10.1), he talks about, among other things,

[o]ther convenient ‘magic’ like using symmetries, transferring results via isomorphisms, homotopy equivalence or elementary equivalence (Urban’s Ultraviolence Axiom) is done by theorem proving, not the foundations.


...Urban's Ultraviolence Axiom? (Ultraviolence? Ultraviolence? Or just autocorrect gone wild?)

I imagine that "Urban" here is Christian Urban, but Google knows nothing of said axiom. Neither does Arxiv, not that I checked very thoroughly. Any pointers, tips, links...?
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Re: Thread for Non-CA Academic Questions

Postby Bullet51 » May 31st, 2018, 12:45 am

Apple Bottom wrote:(Ultraviolence? Ultraviolence? Or just autocorrect gone wild?)


It may be "univalence".
Well, autocorrect has gone wild.
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Re: Thread for Non-CA Academic Questions

Postby Apple Bottom » May 31st, 2018, 2:24 pm

Bullet51 wrote:
Apple Bottom wrote:(Ultraviolence? Ultraviolence? Or just autocorrect gone wild?)


It may be "univalence".
Well, autocorrect has gone wild.


That makes sense. Thanks!
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Re: Thread for Non-CA Academic Questions

Postby gameoflifemaniac » August 18th, 2018, 5:24 am

What curve has the property that an object rolling on it has constant speed?
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Re: Thread for Non-CA Academic Questions

Postby 77topaz » August 18th, 2018, 6:06 am

gameoflifemaniac wrote:What curve has the property that an object rolling on it has constant speed?


Assuming there's no friction, simply a horizontal surface.
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Re: Thread for Non-CA Academic Questions

Postby Gamedziner » August 18th, 2018, 7:15 am

77topaz wrote:
gameoflifemaniac wrote:What curve has the property that an object rolling on it has constant speed?


Assuming there's no friction, simply a horizontal surface.


That is true as long as the object is a perfect sphere of uniform density. Other cases would require a different solution. For a given object to roll on a frictionless surface at a constant speed, you need a surface that keeps the center of mass a constant distance above the ground.
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Re: Thread for Non-CA Academic Questions

Postby gameoflifemaniac » August 18th, 2018, 9:34 am

77topaz wrote:
gameoflifemaniac wrote:What curve has the property that an object rolling on it has constant speed?


Assuming there's no friction, simply a horizontal surface.

But a sphere rolling on a horizontal surface with some initial velocity will slow down. I figured out the curve will have decreasing curvature. The curve would look roughly like this:
img.png
What the curve would roughly look like
img.png (2.77 KiB) Viewed 3819 times

But what curve is it exactly?
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Re: Thread for Non-CA Academic Questions

Postby Saka » August 18th, 2018, 9:47 am

gameoflifemaniac wrote:What curve has the property that an object rolling on it has constant speed?

gameoflifemaniac wrote:But a sphere rolling on a horizontal surface with some initial velocity will slow down. I figured out the curve will have decreasing curvature. The curve would look roughly like this:
img
But what curve is it exactly?

Just a wild, wild guess but...
Brachistochrone?
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sorry, I don't really know
this is just a wild guess
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x = 17, y = 10, rule = B3/S23
b2ob2obo5b2o$11b4obo$2bob3o2bo2b3o$bo3b2o4b2o$o2bo2bob2o3b4o$bob2obo5b
o2b2o$2b2o4bobo2b3o$bo3b5ob2obobo$2bo5bob2o$4bob2o2bobobo!

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Re: Thread for Non-CA Academic Questions

Postby Macbi » August 18th, 2018, 11:40 am

If we ignore air-resistance, friction generally doesn't depend on velocity. So you just need a constant downward slope so that the constant friction cancels out the constant force of gravity.

EDIT: I'm being stupid. It doesn't even matter if friction does depend on speed, because the thing is going at a constant speed anyway. So even with air-resistance the answer is going to be a constant slope.
Last edited by Macbi on August 18th, 2018, 12:57 pm, edited 1 time in total.
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Re: Thread for Non-CA Academic Questions

Postby Gamedziner » August 18th, 2018, 11:51 am

gameoflifemaniac wrote:What curve has the property that an object rolling on it has constant speed?

gameoflifemaniac wrote:But a sphere rolling on a horizontal surface with some initial velocity will slow down. I figured out the curve will have decreasing curvature.
But what curve is it exactly?

The only reasons spheres slow down while rolling on horizontal surfaces are friction and air resistance/drag (mostly friction). You can only maintain a constant speed by rolling the sphere down a line/curve with a slope that causes the speed gained by gravity to be equal to the speed lost by friction and drag, even if that loss is zero. If the forces of gravity, friction, and drag all stay constant, maintaining a given speed requires a perfectly straight line, as changing the slope at any point changes how much energy, and therefore force, is transferred away from the sphere to the surface. A curve has changes in slope, meaning you would need changes in gravity, friction, and/or drag to keep a constant speed using one.
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Re: Thread for Non-CA Academic Questions

Postby muzik » October 4th, 2018, 8:52 am

I've probably asked this before, but is there a name for the following type of geometry, and has it been studied in any detail?:

Take two points anywhere on a plane, labelled A and B. If the straightest possible path from A to B must intersect itself at least once, the plane can be said to be "XXXX".
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Re: Thread for Non-CA Academic Questions

Postby Macbi » October 4th, 2018, 9:03 am

muzik wrote:I've probably asked this before, but is there a name for the following type of geometry, and has it been studied in any detail?:

Take two points anywhere on a plane, labelled A and B. If the straightest possible path from A to B must intersect itself at least once, the plane can be said to be "XXXX".
You'll have to give your definition more precisely. I can't think of anything that could have a property like this. If a path from A to B intersects itself at C, then there is a shorter path that goes from A to C and then directly to B. So the shortest path from A to B can't intersect itself. In most kinds of geometry the shortest path is considered straight.
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Re: Thread for Non-CA Academic Questions

Postby Gamedziner » October 4th, 2018, 8:13 pm

muzik wrote:I've probably asked this before, but is there a name for the following type of geometry, and has it been studied in any detail?:

Take two points anywhere on a plane, labelled A and B. If the straightest possible path from A to B must intersect itself at least once, the plane can be said to be "XXXX".

The geometry would have to be limited in directional movement, like lines tracing the paths of photons reflecting off mirrors.
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Re: Thread for Non-CA Academic Questions

Postby Sarp » October 5th, 2018, 1:28 am

Gamedziner wrote:
muzik wrote:Take two points anywhere on a plane, labelled A and B. If the straightest possible path from A to B must intersect itself at least once, the plane can be said to be "XXXX".

The geometry would have to be limited in directional movement, like lines tracing the paths of photons reflecting off mirrors.

Or we could have negative length lines
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Re: Thread for Non-CA Academic Questions

Postby muzik » January 23rd, 2019, 4:36 pm

Is this identical to a sphere?

x = 120, y = 114, rule = B3/S23
51bo$52b2o$54b2o$56bo$57bo$48bo8bo$49b2o7bo$51b2o6bo$10b97o$10bo42bo7b
o44bo$10bo42bo6b2o44bo$10bo41b2o5bo46bo$10bo40bo5b2o47bo$10bo39bo5bo
49bo$10bo38bo5bo50bo$10bo37bo5bo51bo$10bo42b2o51bo$10bo95bo$10bo48b2o
45bo$10bo48bobo44bo$10bo48b2o45bo$10bo48bobo44bo$10bo48b2o45bo$10bo95b
o$10bo95bo$10bo95bo$10bo95bo$10bo95bo$10bo95bo$10bo95bo$10bo95bo$10bo
95bo$10bo95bo$10bo95bo$10bo95bo$10bo95bo$10bo95bo$10bo95bo$10bo95bo$
10bo95bo$o9bo95bo$bo8bo10b2o83bo$2bo7bo8b2o85bo$3bo6bo8bo86bo$4bo5bo8b
o86bo$4bo5bo7bo87bo$5b2o3bo6bo9b3o76bo$7b2obo5bo10bobo76bo$9b2o4bo11b
3o76b4o$10bo4bo11bobo76bo3bo$10b2o3bo11bobo60b2o13b2o4bo$10bobobo75bob
o11bobo5b2o$10bo2b2o75b2o11bo2bo7bo$10bo79bobo8b2o3bo8bo$10bo79b2o9bo
4bo8bo$10bo89bo5bo9b2o$10bo89bo5b3o8bo$10bo88bo5b2o2bo7bo$10bo87bo5bob
o2bo8bo$10bo87bo4bo2bo2bo9bo$10bo86bo5bo2bo3bo$10bo85bo5bo3bo4bo$10bo
95bo4bo$10bo95bo$10bo95bo$10bo95bo$10bo95bo$10bo95bo$10bo95bo$10bo95bo
$10bo95bo$10bo95bo$10bo95bo$10bo95bo$10bo95bo$10bo95bo$10bo95bo$10bo
95bo$10bo95bo$10bo95bo$10bo95bo$10bo95bo$10bo95bo$10bo95bo$10bo95bo$
10bo95bo$10bo95bo$10bo48b3o44bo$10bo48bobo44bo$10bo48b3o44bo$10bo48bob
o44bo$10bo48bobo44bo$10bo95bo$10bo95bo$10bo95bo$10bo53bo41bo$10bo52bo
42bo$10bo51bo43bo$10bo51bo43bo$10bo50bo44bo$10bo49bo45bo$10bo48bo46bo$
10bo48bo46bo$10bo47bo47bo$10b97o$58bo$58bo$57bo$57b2o$59bo$60bo$61b3o$
64b4o$67b2o!
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