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On Shadow Casting

February 26th, 2016 2 comments

Well, time to shake things up a bit! I have decided to change a few things on my blog, including the theme.

The latex plugin broke with the new WP auto-installation features and I haven't been writing on here much, but I've kept going on new Theorems whenever I can.  One of my latest successes, of which I am very proud of, is Shadow Casting.  It is in this new version of my Compendium of Claims and Proofs.  I will be trying to fix how the equations display in the meantime...

Part I v25

On Calculating Entropy

June 10th, 2014 3 comments

When talking about Pasquali patches that converge and stationary surfaces that they converge to, we discussed that such would stabilize because it wouldn't loop in any shape or form nor would it diverge.  Such Claim we called Entropy.  We didn't provide a mechanism to calculate such. We do that now.

The way we propose we calculate the entropy is by measuring the deviation from the stationary surface.  Thus for the case where we have a Pasquali patch p_t(x,y), t \in \mathbb{Z}^+ we could measure entropy by:

 S(t) = \int_{[0,1]^2} \vert p_t(x,y) - p_\infty(x) \vert \, dA

that is, by taking the absolute deviation from the stationary surface (notice we use the area element dA).  Since in this case the time element is configured at integer times, rather than on continuous time, letting  t \in \mathbb{R} implies using the Pasqualian.  Thus we would have 

 S(t) = \int_{[0,1]^2} \vert p(x,y,t) - p_\infty(x) \vert \, dA

It seems fairly clear that 0 < S(t) < \infty , but here we must agree that 0 means that the system is stable and has the most entropy.

We may however, rather than the absolute difference from the stationary surface, choose to emphasize larger changes and (heavily) discount smaller ones.  Thus we could suggest

 S^{\square}(t) = \int_{[0,1]^2} \left( p(x,y,t) - p_\infty(x) \right)^2 \, dA

for t \in \mathbb{R}.

 

Latest version of Compendium

February 9th, 2014 No comments

I am including here the latest version of the Compendium (v17).

Part I v17

A Remark based on False Claim 1

January 26th, 2014 No comments

So I was thinking that, the statistical description of a dynamical system (as one described by a generator Pasquali patch) really does give us a lot of power in computing the probable "position" of a particle (photon, electron) moving in space at different (integer) time intervals.  If the quantum mechanical supposition of time having a minimum discreteness (Planck-time) is correct, we can find the "finest" Pasquali patch generator that will give a complete description of the dynamical system.  Any Pasquali patch generator descriptive of the system which is not this "first" will generate an accurate, yet less refined ("coarser") version of the system (this is what we mean by Claim 3 of the previous post, in that such Pasquali patch will be "contained" in the finest description, yet is not the finest), and in fact either system of course converges to the same steady state (this is what is meant by Claim 4).  If we are able to find a continuous description (like the Shrodinger equation, via a "Pasqualian") of such system then we are in luck (this description would be the finest, though non-discrete, description), and I speculate though I cannot be sure yet that either discrete descriptions will be contained in such.

Whatever the description of the dynamical system via a generator Pasquali patch (or a Pasqualian), each Pasquali patch represents the transition (position) probabilities of a particle (photon, electron) moving within that system.  If we suppose that the particle moves with same velocity (take photons in vacuum as an example), then each Pasquali patch power is descriptive of the transition-position probability at equally spaced spacial or distance intervals.  Though I've remarked about this before, if it were the case that, for a particle with a particular (steady) velocity, the Pasquali patch power is not exactly equally spaced in distance intervals, it must mean that the arrow of time is bent (time is moving faster for smaller-spaced intervals, slower for longer-spaced).  We have not yet described accelerating particles but at present that is not of our interest.

However, we can tell if time is passing so long as each Pasquali patch description is different at each (equal or unequal) interval.  If a single Pasquali patch were to describe the system at EACH distance interval, there is no way to know if time is moving at all.  Take for example the False Claim 1 of the previous post.  We had the collection \mathbb{Q} = \{Q^1, Q^2, \ldots, Q^k, \ldots\}_{k \in \mathbb{Z}^+} with Q = q(x,y) being an explicit function of y and converging to Q^\infty = 1.  We could track the time-distance interval via the Pasquali patch power, so that 1 was the first distance interval from start (we take it as given that time is not being bent, so that a fixed distance implies the passage of 1 unit of time), 2 was the second distance interval (2 units of time), and so on, and we could tell if time were bent if each power were descriptive of different distance intervals.  Furthermore, since each Q \in \mathbb{Q} is different, this implies each position transition probability is different and the system is in movement.

This is definitely not the case with the collection \mathbb{P} = \{P^1 = 1, P^2 = 1, \ldots, P^j, \ldots \}_{j \in \mathbb{Z}^+} which also converges to P^\infty = 1.  Since at each distance interval the movement probability is the same (uniform), one cannot be convinced that each power represents a distance interval equal to equally spaced time intervals or different-spaced time intervals.  Where we could with \mathbb{Q} ascertain that time was moving, we cannot with \mathbb{P}.  The statistical description cannot tell if the system is frozen.

When a system has reached the steady state (which, is the highest entropy state!), there is no way to tell if time flows, as the statistical description is and forever will be unchanging.  Recall that the steady state for a Pasquali patch is always a function of x alone, say p(x).  Furthermore recall that any power of p(x) is always p(x) itself (see Compendium).  We reach an impasse: is time flowing normally, faster than what is conventional, slower? At such a point it is impossible to say, at least from the statistical point of view.  We would have to track particles individually in order to ascertain if they have deviated their path at all (it could be the case that they shifted to all positions with equal probability or in the shape of p(x), e.g., but we cannot be sure of either situation).

On Scaled Patches and Time-Warping

May 5th, 2013 No comments

Suppose we have an idealized canal of width 1, on which a fluid flow has been established in some remote past. Let us focus solely on the dynamics of the surface.  Pick a spot along the canal which we will call t_0.  Next pick a spot y_0 \in [0,1] along the width of the canal, which we will monitor.  Pick a second spot t_1 down the canal, some distance from the original spot we picked.  Now let us assume that, up the canal at some remote point, a paper boat has been released.  We will only care about the boat if the boat passes through (t_0, y_0) which we have picked, and we will write down the resultant position at t_1.  Let us do this a number of times with any number of boats, and obtain a distribution of the position of the boat at t_1, saving it. Next let us repeat the experiment, this time focusing on y_0^\prime, save the resultant distribution at t_1, and so on and so forth, until we are comfortable having mapped the totality of positions at t_0.  Let us next put together (stack, respecting the natural order of y) all the distributions we obtained at t_1.  We now have a discrete surface which we can smooth to obtain a Pasquali patch.

Let us now look at position t_2 which is the same distance as t_1 is from t_0.  Having defined the dynamics of the system (from a single Pasquali patch), the dynamics at t_2 can be theoretically described by P^2.  We can therefore ascertain the probability that we will find the boat at t_2 along the width of the canal.  In fact, at t_n, n very large, we can ascertain the probability that the boat will be at any position along the width.  It should be close to P^\infty.  More importantly, a great distance from the origin (any distance, not necessarily a distance n \cdot \Delta t_n), the position probability is aptly described by P^\infty.  See Figure 1 and Figure 2.

We can experimentally create Pasquali patch and use it for prediction.  We can perform the measurement at an arbitrary distance.

We can experimentally create Pasquali patch and use it for prediction. We can perform the measurement at an arbitrary distance.

 

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We can then use Pasquali patch powers for position prediction down the canal, at position n times delta t down the origin.

We can then use Pasquali patch powers for position prediction down the canal, at position n times delta t down the origin.

This simple thought experiment brings about several questions. What if the dynamics of the surface system are described by the Pasquali patch, but at points which are not a distance \Delta t_n apart? In other words, what if the description is apt but at points that are not linear in distance? This curious situation suggests a time anomaly, and therefore a manner in which we can measure time warps (by measuring the actual time differences between Pasquali patches). See Figure 3.

In this schematic a Pasquali patch and its powers do describe the system, but at non-equidistant points.  The arrow of time is warped.

In this schematic a Pasquali patch and its powers do describe the system, but at non-equidistant points. The arrow of time is warped.

Next, we looked at the surface dynamics of the system. If we add a depth variable to the canal, we can in theory produce a Pasquali cube, which would measure the dynamics of any point on the [0,1] \times [0,1] cross-section a discrete distance down the canal (and any distance very far from our origin).

A third question arises when we consider the same canal, but whose width opens by a scalar (linear) amount a distance from our chosen origin.  There is no reason we cannot “renormalize” the width (set it equal to 1 again) at a point some set distance from our chosen origin, and proceed with our analysis as before.  See Figure 4.

In this schematic the width of the canal grows linearly, but Pasquali patch dynamics are conserved, suitably scaled.

In this schematic the width of the canal grows linearly, but Pasquali patch dynamics are conserved, suitably scaled.

 

In a subsequent post I'd like to reconsider the slit experiment under this new light, see where it takes us.