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    embedding

    Embedding.

    It has to do with topology.

    The concept is so vitally important, it's impossible to overstate its importance.

    I will illustrate by example.

    The quantum model is a DISCRETE space embedded into a CONTINUOUS space.

    Energy is discrete because it is quantized. And we imagine SpaceTime to be entirely continuous.

    Question: what new feature or phenomenon does this embedding confer?

    The answer: COUNTABILITY. You can "count" the quanta.

    When we "measure" something, the very first thing we have to do is overlay a number system onto whatever space embeds the object were trying to measure. In continuous space, we arbitrarily assign a unit, could be an inch, a millimeter, or a mile. But in the discrete space the units are chosen for us, one Quantum is indivisible.

    Consider in the abstract, the idea of embedding a discrete space into a continuous space. If we count discrete items 1, 2, 3 and so on, what is "between" the points? The answer is, nothing. That is to say, more precisely, that our discrete assignment of numbers has not completely "filled" (the technical term is "covered") the underlying space. In our example, the discreet embedding only covers the integers, and there are no negative numbers since you cannot count Less Than Zero.

    In quantum mechanics we can envision a PROCESS (a "generator", as in random stochastic process), that embeds the discrete system into the continuous space. In this example, it's as simple as placing the points, so like, you're saying "number one goes here, and number two goes there" and etc. In the process of so doing, you are "filling space". Not "all" of the space, just the points corresponding to the integers.

    And, how "much" of the space is that? How much of a continuous space will a discrete integer embedding cover? We know it must be greater than 0%, and less than 100%. It is some fraction, maybe a third, maybe 1/5 or 1/10.

    It turns out we can calculate this. Topological embeddings conform to the algebra of group Theory, and we can predict which symmetries will work in which contexts and exactly how they will work.

    The Brilliance of the embedding concept, has to do with what the calculus people call "transforms". For example, if you have a function that's a figure 8, it is undifferentiable in at least two places, the two places where the slope is vertical. However what you can do is you can "lift" the Figure 8 into three dimensions, by multiplying it by another function, which makes it differentiable everywhere in three dimensions (because by doing this you've expanded the figure 8 into a helix) - and you can then do the math, and project the results back down into two dimensions. The only reason the original function was undifferentiable, is because of the choice that was made in terms of embedding it into the continuous analytical space. In some other space, the function becomes perfectly differentiable.

    The same is true of stochastic generators. Generally speaking, they can be integrated and differentiated just like any other functions. The math for doing this is called Ito's calculus. The math for calculating coverage is simple fractal geometry.

    Underneath all this, is the simple concept of countability. The attribute of countability, surprisingly enough, cannot be directly derived from the topology. It's something independent, something different.
    Last edited by nonsqtr; 03-09-2018 at 12:20 AM.

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    Is this what you think of in the middle of the night? Nuttin else to do?

    Man, you are much more brighter than I. Do you and Stephen Hawking hang out at night and cruise for chicks?

    I only say this because I know nothing of what you talk about.
    Last edited by Rickity Plumber; 03-09-2018 at 04:36 AM.



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    Quote Originally Posted by Rickity Plumber View Post
    Is this what you think of in the middle of the night? Nuttin else to do?

    Man, you are much more brighter than I. Do you and Stephen Hawking hang out at night and cruise for chicks?

    I only say this because I know nothing of what you talk about.
    He's talking about flying to Australia on Quantas Airlines and counting the number of people on the plane.
    There should be no space or empty seats.

    A young boy approached @nonsqtr on the plane with his baseball and greeted him with a hearty good morning.
    As he tossed his ball up and down, he asked mr. Nonsqtr in his opinion what motion the ball was taking. Mr. Nonsqtr said , "obviously, up and down son".

    The young boy looked mr. Nosqtr in the eyes and exclaimed , "yes, this is correct".
    HOWEVER, the young lad told him, that is only a trivial part of the ball's movement. The ball is also traveling forward at 600mph as it is also going up and down.

    Mr. Nonsqtr then looked the boy straight in the eyes and borrowed a line from obama telling the young lad;
    "If i had a son........".
    Last edited by HawkTheSlayer; 03-09-2018 at 09:02 AM.
    Today we live. Tomorrow we die.
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    The point is, the countability becomes a "physical" property of the system. It will ALWAYS happen, it's a mathematical consequence.

    It does NOT require an observer from the outside, to overlay some preconceived metric onto SpaceTime. The countability happens from the bottom up, not the top down.

    This is SYMMETRY we're talking about. How many ways can you flip things around or turn them upside down, and still retain the same object? What are the "invariances"? When you measure something one way, will it still be the same as when you measure it a different way?

    Anyone who's taking a Calculus class, knows that you can unfold a point singularity, in other words you can make a non differentiable function differentiable, simply by applying the mathematical trick of pre multiplying it by "some other function".

    In this case, YOU choose the preconceived function because you have a preconceived goal in mind ("differentiability"). But it is equally likely that a function could be chosen automatically on the basis of some constraints, like lowest energy state or whatever.

    The concept of EMBEDDING becomes particularly acute when we think of changes of basis. Think of the simple thing - the concept of "now". That's a change of basis, because now is egocentric, and there is no such Concept in the physical universe. "Now" requires a preconception of something "other than now", in other words, if you overlay a numbering system on time t, you have already made important assumptions about its behavior and characteristics.

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    Okay, let's do this differently. It's easy to embed a two dimensional piece of paper into three space. We do it every day. And you can play games with the different ways of doing it, like you can put a Twist in the piece of paper to create a Moebius strip and etc.

    But what about the other way? Could we embed 3 space into a two dimensional piece of paper?

    Intuition says "no". There are too many dimensions, to fit into a piece of paper.

    HOWEVER - it turns out we don't have to embed "all" of 3 space, only the points were interested in. So for example, let's say we have an object, or a shape, or an area that were interested in. It doesn't even have to be all in the same place, it could be a series of disconnected points. In general, call it a "Graph", some representation of a function whose image is the area of Interest.

    It turns out, that "how much" of three space we can represent in two dimensions, depends on "how much" of those two Dimensions is actually being filled by our coordinate system. It intuitively makes sense that you could represent "less" in an airy mesh, then you could in a solid piece of paper.

    When considered in this way, the differential projection becomes almost like an "encoding" or "encryption" of the source. Data compression, is what we're doing.

    Graph embedding - Wikipedia
    Last edited by nonsqtr; 03-12-2018 at 09:44 PM.

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