Difference between revisions of "Public vs Private Networks"

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2) All the points of S are made up of raycasted pixel points. Meaning the initial point at which an imaginary line drawn outwards from a pixel of the camera frame intersects an external, real-world object (an unprojection), for every camera/node in the network.
2) All the points of S are made up of raycasted pixel points. Meaning the initial point at which an imaginary line drawn outwards from a pixel of the camera frame intersects an external, real-world object (an unprojection), for every camera/node in the network.


3) And we denote any private network as S<sub>p</sub>, and any decentralized (public) network as S<sub>d</sub>.
3) We'll denote any private network as S<sub>p</sub>, and any decentralized (public) network as S<sub>d</sub>.


4) And if we define the domination of one set, S, by another set, S', to be whenever points (raycasted pixel points) in S can be approximated (substituted) to a higher degree by points (raycasted pixel points) in S' than the other way around. To put it another way, the probability that any given x of S is arbitrarily close to some y of S' is higher than the probability that any given y of S' is arbitrarily close to some x of S. Therefore, the number of points in S' is greater than the number of points in S (even if it's just one more camera/node)
4) And we'll define the "domination" of one set, S, by another set, S', to be whenever the points (raycasted pixel points) in S can be approximated (substituted) to a higher degree by points (raycasted pixel points) in S' than the other way around. To put it another way, the probability that any given x S is arbitrarily close to some y S' is higher than the probability that any given y S' is arbitrarily close to some x S. Therefore, the number of points in S' is greater than the number of points in S (even if it's just one more camera/node)




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We can also demonstrate this by finding an embedding of S into the "countably infinite" sets and the "uncountably infinite" sets. And showing the differences in cardinality between them. In the sense that some infinities are larger than others. The points of a private Grassland network would have to be embedded in the set of all Integers. It can theoretically stretch infinitely in both directions, [-infinity, ..., -3, -2, -1, 0, 1, 2, 3, ..., infinity], with an infinite amount of raycasted pixel points from an infinite amount of cameras. But it's still bound to certain values; S<sub>p</sub> has more restrictions than S<sub>d</sub> by definition. Therefore, a decentralized (public) Grassland network, because it doesn't have those restrictions, finds a complementary embedding into the Reals, [-infinity, ..., -Pi, ..., -1.001, ..., -0.001, ..., 0, ..., 0.001, ..., 1.001, ..., Pi, ..., infinity]. It's an infinity too but it's a much larger infinity because it can theoretically stretch to infinity even between the Integers.
We can also demonstrate this by finding an embedding of S into the "countably infinite" sets and the "uncountably infinite" sets. And showing the differences in cardinality between them. In the sense that some infinities are larger than others. The points of a private Grassland network would have to be embedded in the set of all Integers. It can theoretically stretch infinitely in both directions, [-infinity, ..., -3, -2, -1, 0, 1, 2, 3, ..., infinity], with an infinite amount of raycasted pixel points from an infinite amount of cameras. But it's still bound to certain values; S<sub>p</sub> has more restrictions than S<sub>d</sub> by definition. Therefore, a decentralized (public) Grassland network, because it doesn't have those restrictions, finds a complementary embedding into the Reals, [-infinity, ..., -Pi, ..., -1.001, ..., -0.001, ..., 0, ..., 0.001, ..., 1.001, ..., Pi, ..., infinity]. It's an infinity too but it's a much larger infinity because it can theoretically stretch to infinity even between the Integers.


And it therefore approximates the Integers. S<sub>p</sub>'s point at "1" can be approximated by an infinite number of points in S<sub>d</sub>. Moreover, "1" in S<sub>p</sub> can be approximated to an arbitrary degree of accuracy by points in S<sub>d</sub> from both sides, "0.9999...9" and from "1.0000...1". This is because y of S<sub>d</sub> has the freedom to approach any x of S<sub>p</sub> along every dimension of the space they're embedded into.
And it therefore approximates the Integers. S<sub>p</sub>'s point at "1" can be approximated by an infinite number of points in S<sub>d</sub>. Moreover, "1" in S<sub>p</sub> can be approximated to an arbitrary degree of accuracy by points in S<sub>d</sub> from both sides, "0.9999...9" and from "1.0000...1". This is because y S<sub>d</sub> has the freedom to approach any x S<sub>p</sub> along every dimension of the space they're embedded into.


=== Conclusion: ===
=== Conclusion: ===
So since the more points of S<sub>p</sub> can be approximated to an arbitrary degree of accuracy by points in S<sub>d</sub> than the other way around, and since S<sub>d</sub> therefore contains the limit points of S<sub>p</sub>, we can say that the set, S<sub>pn</sub>, comprised of the neighbourhoods of all points in S<sub>p</sub> is merely a smaller subset of the set, S<sub>dn</sub>, a set comprised of the neighbourhoods of all points in S<sub>d</sub>. And therefore S<sub>p</sub> is dominated by S<sub>d</sub> according to point 4 above.
So since the more points of S<sub>p</sub> can be approximated to an arbitrary degree of accuracy by points in S<sub>d</sub> than the other way around, and since S<sub>d</sub> therefore contains the limit points of S<sub>p</sub>, we can say that the set, S<sub>pn</sub>, comprised of the neighbourhoods of all points in S<sub>p</sub> is merely a smaller subset of the set, S<sub>dn</sub>, a set comprised of the neighbourhoods of all points in S<sub>d</sub>. And therefore S<sub>p</sub> is dominated by S<sub>d</sub> according to point 4 above.

Revision as of 17:26, 2 June 2020

A Proof That It's Impossible for a Private (restricted) Grassland Network to Exist That Can't Be "Dominated" by a Public One

As Topological Spaces:

We can define Grassland networks as topological spaces (Topological space). That is to say, a set of points along with a set of neighbourhoods (Topological Neighbourhood) for each point.

1) We'll define each network, as a 3D, topological set, S, upon and the surface of an earth shaped spheroid

2) All the points of S are made up of raycasted pixel points. Meaning the initial point at which an imaginary line drawn outwards from a pixel of the camera frame intersects an external, real-world object (an unprojection), for every camera/node in the network.

3) We'll denote any private network as Sp, and any decentralized (public) network as Sd.

4) And we'll define the "domination" of one set, S, by another set, S', to be whenever the points (raycasted pixel points) in S can be approximated (substituted) to a higher degree by points (raycasted pixel points) in S' than the other way around. To put it another way, the probability that any given x ∈ S is arbitrarily close to some y ∈ S' is higher than the probability that any given y ∈ S' is arbitrarily close to some x ∈ S. Therefore, the number of points in S' is greater than the number of points in S (even if it's just one more camera/node)


Since both Sp and Sd are on the surface of the earth, a spheroid having no edges, no set, S, can contain ALL its limit points (Limit Point) to the effect of keeping points of another set, S', from approximating points of S. And it goes without saying, that this is true along the vertical axis. So even continent's aren't topologically compact. They don't contain their limit points in regards to camera placement since along vertical axis the "sky's the limit". Which means there's always some way for points in an unrestricted Grassland network, Sd, to approximate more points in Sp. Topologically speaking, a greater number of points in Sp can be shown to be merely a point in a neighbourhood of some point in Sd.

As Infinite Sets:

We can also demonstrate this by finding an embedding of S into the "countably infinite" sets and the "uncountably infinite" sets. And showing the differences in cardinality between them. In the sense that some infinities are larger than others. The points of a private Grassland network would have to be embedded in the set of all Integers. It can theoretically stretch infinitely in both directions, [-infinity, ..., -3, -2, -1, 0, 1, 2, 3, ..., infinity], with an infinite amount of raycasted pixel points from an infinite amount of cameras. But it's still bound to certain values; Sp has more restrictions than Sd by definition. Therefore, a decentralized (public) Grassland network, because it doesn't have those restrictions, finds a complementary embedding into the Reals, [-infinity, ..., -Pi, ..., -1.001, ..., -0.001, ..., 0, ..., 0.001, ..., 1.001, ..., Pi, ..., infinity]. It's an infinity too but it's a much larger infinity because it can theoretically stretch to infinity even between the Integers.

And it therefore approximates the Integers. Sp's point at "1" can be approximated by an infinite number of points in Sd. Moreover, "1" in Sp can be approximated to an arbitrary degree of accuracy by points in Sd from both sides, "0.9999...9" and from "1.0000...1". This is because y ∈ Sd has the freedom to approach any x ∈ Sp along every dimension of the space they're embedded into.

Conclusion:

So since the more points of Sp can be approximated to an arbitrary degree of accuracy by points in Sd than the other way around, and since Sd therefore contains the limit points of Sp, we can say that the set, Spn, comprised of the neighbourhoods of all points in Sp is merely a smaller subset of the set, Sdn, a set comprised of the neighbourhoods of all points in Sd. And therefore Sp is dominated by Sd according to point 4 above.