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Exactly 3 out of n propositional variables is allowed to be true
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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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Exactly 3 out of n propositional variables is allowed to be true
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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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b4o25bo$o29bo$b3o3b3o2bob2o2bob2o2bo3bobo$4bobo3bob2o2bob2o2bobo3bobo$
4bobo3bobo5bo5bo3bobo$o3bobo3bobo5bo6b4o$b3o3b3o2bo5bo9bobo$24b4o!
So, that n-th root can be complex?BlinkerSpawn wrote:More generally, functions that equal their n-th derivative will be composed of terms of the form Ce^kx, where C is a constant and k is an n-th root of unity.
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b4o25bo$o29bo$b3o3b3o2bob2o2bob2o2bo3bobo$4bobo3bob2o2bob2o2bobo3bobo$
4bobo3bobo5bo5bo3bobo$o3bobo3bobo5bo6b4o$b3o3b3o2bo5bo9bobo$24b4o!
Yes — z^n = 1 has exactly n solutions which lie evenly spaced around the unit circle. For example, the fourth roots of unity are 1, i, -1, and -i, which form the vertices of a square centered around 0; the third roots of unity are 1, -(1/2)+(√(3)/2)i, and -(1/2)-(√(3)/2)i, which form an equilateral triangle centered around 0.gameoflifemaniac wrote:So, that n-th root can be complex?BlinkerSpawn wrote:More generally, functions that equal their n-th derivative will be composed of terms of the form Ce^kx, where C is a constant and k is an n-th root of unity.
Yes, in particular cos(x) = (1/2)e^(ix)+(1/2)e^(-ix) and sin(x) = (-i/2)e^(ix)+(i/2)e^(-ix).gameoflifemaniac wrote:So, that n-th root can be complex?BlinkerSpawn wrote:More generally, functions that equal their n-th derivative will be composed of terms of the form Ce^kx, where C is a constant and k is an n-th root of unity.
Yes, and those are examples of functions that equal their fourth derivative (but not their first, second, or third).Macbi wrote:Yes, in particular cos(x) = (1/2)e^(ix)+(1/2)e^(-ix) and sin(x) = (-i/2)e^(ix)+(i/2)e^(-ix).
{∞,1} would be a half-plane (and not exactly a tiling, considering the {,1} means the tiling consists of a single shape)muzik wrote:What would the tilings {1,∞} and {∞,1} look like?
[o]ther convenient ‘magic’ like using symmetries, transferring results via isomorphisms, homotopy equivalence or elementary equivalence (Urban’s Ultraviolence Axiom) is done by theorem proving, not the foundations.
It may be "univalence".Apple Bottom wrote:(Ultraviolence? Ultraviolence? Or just autocorrect gone wild?)
That makes sense. Thanks!Bullet51 wrote:It may be "univalence".Apple Bottom wrote:(Ultraviolence? Ultraviolence? Or just autocorrect gone wild?)
Well, autocorrect has gone wild.
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b4o25bo$o29bo$b3o3b3o2bob2o2bob2o2bo3bobo$4bobo3bob2o2bob2o2bobo3bobo$
4bobo3bobo5bo5bo3bobo$o3bobo3bobo5bo6b4o$b3o3b3o2bo5bo9bobo$24b4o!
Assuming there's no friction, simply a horizontal surface.gameoflifemaniac wrote:What curve has the property that an object rolling on it has constant speed?
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.77topaz wrote:Assuming there's no friction, simply a horizontal surface.gameoflifemaniac wrote:What curve has the property that an object rolling on it has constant speed?
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x = 81, y = 96, rule = LifeHistory
58.2A$58.2A3$59.2A17.2A$59.2A17.2A3$79.2A$79.2A2$57.A$56.A$56.3A4$27.
A$27.A.A$27.2A21$3.2A$3.2A2.2A$7.2A18$7.2A$7.2A2.2A$11.2A11$2A$2A2.2A
$4.2A18$4.2A$4.2A2.2A$8.2A!
But a sphere rolling on a horizontal surface with some initial velocity will slow down. I figured out the curve will have decreasing curvature. The curve would look roughly like this: But what curve is it exactly?77topaz wrote:Assuming there's no friction, simply a horizontal surface.gameoflifemaniac wrote:What curve has the property that an object rolling on it has constant speed?
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b4o25bo$o29bo$b3o3b3o2bob2o2bob2o2bo3bobo$4bobo3bob2o2bob2o2bobo3bobo$
4bobo3bobo5bo5bo3bobo$o3bobo3bobo5bo6b4o$b3o3b3o2bo5bo9bobo$24b4o!
gameoflifemaniac wrote:What curve has the property that an object rolling on it has constant speed?
Just a wild, wild guess but...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?
gameoflifemaniac wrote:What curve has the property that an object rolling on it has constant speed?
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.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?
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x = 81, y = 96, rule = LifeHistory
58.2A$58.2A3$59.2A17.2A$59.2A17.2A3$79.2A$79.2A2$57.A$56.A$56.3A4$27.
A$27.A.A$27.2A21$3.2A$3.2A2.2A$7.2A18$7.2A$7.2A2.2A$11.2A11$2A$2A2.2A
$4.2A18$4.2A$4.2A2.2A$8.2A!
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.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.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".
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x = 81, y = 96, rule = LifeHistory
58.2A$58.2A3$59.2A17.2A$59.2A17.2A3$79.2A$79.2A2$57.A$56.A$56.3A4$27.
A$27.A.A$27.2A21$3.2A$3.2A2.2A$7.2A18$7.2A$7.2A2.2A$11.2A11$2A$2A2.2A
$4.2A18$4.2A$4.2A2.2A$8.2A!
Or we could have negative length linesGamedziner wrote:The geometry would have to be limited in directional movement, like lines tracing the paths of photons reflecting off mirrors.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".
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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!