The weakening measurement

Unlike the special theory of relativity that the superluminal-neutrinos fiasco sought to defy, Heisenberg's uncertainty principle presents very few, and equally iffy, measurement techniques to stand verified. While both Einstein's and Heisenberg's foundations are close to fundamental truths, the uncertainty principle has more guided than dictated applications that involved its consequences. Essentially, a defiance of Heisenberg is one for the statisticians.

And I'm pessimistic. Let's face it, who wouldn't be?

Anyway, the parameters involved in the experiment were:

1. The particles being measured


2. Weak measurement


3. The apparatus

The experimenters claim that a value of the photon's original polarization, X, was obtained upon a weak measurement. Then, a "stronger" measurement was made, yielding a value A. However, according to Heisenberg's principle, the observation should have changed the polarization from A to some fixed value A'.

Now, the conclusions they drew:

1. Obtaining X did not change A: X = A


2. A' - A < Limits set by Heisenberg

The terms of the weak measurement are understood with the following formula in mind:



(The bra-ket, or Dirac, notation signifies the dot-product between two vectors or vector-states.)

Here, φ(1,2) denote the pre- and post-selected states, A-hat the observable system, and Aw the value of the weak-measurement. Thus, when the pre-selected state tends toward becoming orthogonal to the post-selected state, the value of the weak measurement increases, becoming large, or "strong", enough to affect the being-measured value of A-hat.

In our case: Aw = A - X; φ(1) = A; φ(2) = A'.

As listed above, the sources of error are:

1. φ(1,2)


2. X

To prove that Heisenberg was miserly all along, Aw would have been increased until φ(1) • φ(2) equaled 0 (through multiple runs of the same experiment), and then φ(2) - φ(1), or A' - A, measured and compared to the different corresponding values of X. After determining the strength of the weak measurement thus, A' - X can be determined.

I am skeptical because X signifies the extent of coupling between the measuring device and the system being measured, and its standard deviation, in the case of this experiment, is dependent on the standard deviation of A' - A, which is in turn dependent on X.

Interactions between a planetary system and an FFP: A fuzzy approach

For many years, astronomers and cosmologists thought planets were always bound to stars, doomed to journey in one parabolic orbit until the star that held it in its place died. However, recent far-ranging forays into regions of space lying much beyond the solar system and observations of vast sweeps of space by farsighted telescopes have revealed a clutch of exoplanets that don't orbit any star. Instead, they wander the universe as nomadic bodies, owing no other body any allegiance.

Because of their non-aligned nature, interesting things happen when they get too close to a proper planetary system with a star at the centre. The free-floating planet (FFP) exerts its gravitational force on the system and distorts its gravitational force distribution. For the sake of simplicity, let's assume that the system consists of one sun-like star and one Jupiter-like planet going round it. As the FFP gets close enough, the force that holds the planet (BP, for binary planet) around the star will experience a pull because the newcomer will have a gravitational field that pulls the BP away from the star.

There are four possible outcomes when this encounter happens, and the probability of one outcome happening over another is determining by a set of two parameters. The first is d, the impact parameter, and the second is φ, which essentially is the position of the BP in its journey around the star. For reasons better left untold, the value of d is less than zero if the FFP is on one side of an appropriately chosen reference line, usually one that runs through the centre of the star and bifurcates the planetary system into two symmetrical mechanical systems. When the FFP is on the other side, d is positive.



The four possible outcomes are:

1. Fly-by - The newcomer just flies by, leaving the planetary system untouched and unperturbed


2. Exchange - The newcomer swaps its position with the BP, becoming the new journeyman and liberating the BP into nomadhood


3. Resonance - The newcomer joins the planetary system and becomes the second journeyman


4. Disruption - The newcomer distorts the incumbent gravitational field enough to break up the system and leave all three wandering

(The likelihood of each of these outcomes is more precisely determined by the energy of the system at its centre-of-gravity. However, the ergonomics of the system is dependent absolutely o the masses of the star, the BP and the FFP, the approach-velocity of the FFP, r₀, the distance between the BP and its star - all of which are fixed constants - and φ and d - both of which are the only controllable variables.)

After this point, it becomes incredibly difficult to assess the probability of each outcome because of the immense variability in the BP's position relative to the newcomer. So, three smart physicists at the University of Thessaloniki, with access to what I can only imagine to be a bare-minimum of computing resources, decided to tackle the problem statistically.

In statistical mechanics, there is a lack of predictability: there are no formulae that correlate the behaviour of millions of small events to the entire system. Instead, there may be a few well-established formulae that provide correlations between other aspects of the system and its state. And when comes to statistical mechanical modelling, these few formulae are used to simulate the system in different states and look for patterns. Then, with the help of such patterns, a conclusion can be gradually drawn about the influence of small-scale patterns on large-scale ones.

In this case, the physicists of Thessaloniki programmed the initial conditions of the system into a computer: velocity of the incoming FFP, φ, d, masses of the FFP, BP and the star, and r₀. Then, they brought the newcomer close enough to the planetary system to affect its mass distribution 350,000 different times, each cycle corresponding to a uniquely chosen value of φ and d. In doing this, they gained a statistical understanding of the system that solidified an "estimated" understanding of it into a perfectly understood one.

Each time they brought the planet closer to the system, the value of φ they picked had to lie between 0 and 360 (for obvious reasons). The value of d, on the other hand, had to lie between 7r₀ and -7r₀ because of the assumption that the BP would be Jupiter-like, and in being Jovian like that, that's the value of d at which disturbances would begin to be felt. Such conditions are called initial conditions and play a very important role in fuzzy logic, which is the encompassing branch of logic at work in this example.

[caption id="attachment_21228" align="aligncenter" width="450" caption="The gap between crisp logic and fuzzy logic is the same as between precision and significance: sometimes, all you need to know is what's going to make all the difference, and fuzzy logic is a good way to get that information."][/caption]

Fuzzy logic is the set of mathematical tools that work with the fact that given a set of ground rules and an initial state, the final state can be deciphered to a heartening degree of accuracy, a degree dependent only on the accuracy of the ground rules. This gives us a basis for exploring the nature of satellite concepts to those currently understood: by using what we already know as initial conditions and an initial state, we get a fuzzy idea of the nature of the new concept. Fuzzy logic is particularly powerful when studying very large systems wherein the interference of chance is great.

The abstract of the original paper can be found here.

Exploring a clustering technique using mathematical group theory

Step I: Identification of ICM

[caption id="attachment_20428" align="aligncenter" width="596" caption="A scatter plot"][/caption]

The x-axis, identified by the base marker, represents numerical values from 1 to 100 (integers). The y-axis, identified by the decimal marker, represents numerical values from 0 to 1 (decimal, 2 significant digits).

From all the data, two points are selected at random, marked in the image by a black rectangle around them. They are labelled as the initial cluster markers (ICM).

Step II: Geometric tagging

The Euclidean distance between each point in the scatter plot and the ICM is calculated. In this case, there are 48 points (after excluding the ICM): x₁, x_2, . . ., x₄₈. Therefore, the distances between x₁ and ICM₁ and ICM₂ are calculated; between x₂ and ICM₁ and ICM₂, and so on until x₄₈ and ICM₁ and ICM₂.

If the distance between some x_i and ICM_a is lower than x_i and ICM_b, then x_i is grouped with ICM_a and tagged as x_i^a. This gives rise to a clear demarcation; the data set is now split amongst x_i^a and x_j^b. Each section is identified as S{x_i,j^a,b}.

Step III: Averaging

Before any average is computed, it must be determined as to which value is to be considered. Three cases are presented below.

1. x-value: if the x-value of each data-point is to be averaged in each cluster, then the computed average will lie along a straight line x_i = µ_i


2. y-value: if the y-value of each data-point is to be averaged in each cluster, then the computed average will lie along a straight line y_i = µ_i


3. Alien value: consider the following table.

Base marker (1-50)	Decimal marker (0-1)	Anonymous marker (1-100)
4	0.71	20
13	0.61	38
14	0.98	7
18	0.03	4
11	0.68	55
37	0.26	67
30	0.04	46
21	0.22	57
37	0.48	90
47	0.94	43

The x- and y-axes represent the values in the first two columns while the third column shows a set of values not considered in the construction of the scatter plot. Since each row in this table is denoted as a point in the plot, each such point is also associated with a certain “anonymous value” as shown in the table.

These can be considered in the averaging process, whereby all the “anonymous” values of all the points in each cluster are averaged separately:

S(x_i^a): A(x_i^a)

S(x_j^b): A(x_j^b)

These new points, A_a and A_b, are plotted in their respective clusters.

Step IV: Convergence

A_a and A_b become the new ICM, and steps I to III are repeated.

The iterations stop when the new A_a and A_b converge with the A_a and A_b from the previous iteration.

Result

After the values have converged, the two S{x_i,j^a,b} now presented by the machine are two distinct groups of data, the emergence of which also signals that the machine has “learnt”. If a very large database is supplied as an input to the machine, there is an advantage as well as a disadvantage.

• Advantage: the machine learns better


• Disadvantage: data may not converge at all due to improper choice of clusters

In order to prevent the second possibility, a suitable number of clusters has to be determined for n, the number of data-points. As a rule of thumb,

k = (n / 2)^1/2

Here, ‘k’ is the number of clusters.

If a dataset of 50 points is input, then k = 5 clusters are suggested. Once the iterations have been performed and convergence has been attained in five separate clusters, the centroids of each cluster (or, the average value of the data-points in each cluster) can now be used to arrive at 2 super-clusters.

Exploring a clustering technique using mathematical group theory

Step I: Identification of ICM

[caption id="attachment_20428" align="aligncenter" width="596" caption="A scatter plot"][/caption]

The x-axis, identified by the base marker, represents numerical values from 1 to 100 (integers). The y-axis, identified by the decimal marker, represents numerical values from 0 to 1 (decimal, 2 significant digits).

From all the data, two points are selected at random, marked in the image by a black rectangle around them. They are labelled as the initial cluster markers (ICM).

Step II: Geometric tagging

The Euclidean distance between each point in the scatter plot and the ICM is calculated. In this case, there are 48 points (after excluding the ICM): x₁, x_2, . . ., x₄₈. Therefore, the distances between x₁ and ICM₁ and ICM₂ are calculated; between x₂ and ICM₁ and ICM₂, and so on until x₄₈ and ICM₁ and ICM₂.

If the distance between some x_i and ICM_a is lower than x_i and ICM_b, then x_i is grouped with ICM_a and tagged as x_i^a. This gives rise to a clear demarcation; the data set is now split amongst x_i^a and x_j^b. Each section is identified as S{x_i,j^a,b}.

Step III: Averaging

Before any average is computed, it must be determined as to which value is to be considered. Three cases are presented below.

1. x-value: if the x-value of each data-point is to be averaged in each cluster, then the computed average will lie along a straight line x_i = µ_i


2. y-value: if the y-value of each data-point is to be averaged in each cluster, then the computed average will lie along a straight line y_i = µ_i


3. Alien value: consider the following table.

Base marker (1-50)	Decimal marker (0-1)	Anonymous marker (1-100)
4	0.71	20
13	0.61	38
14	0.98	7
18	0.03	4
11	0.68	55
37	0.26	67
30	0.04	46
21	0.22	57
37	0.48	90
47	0.94	43

The x- and y-axes represent the values in the first two columns while the third column shows a set of values not considered in the construction of the scatter plot. Since each row in this table is denoted as a point in the plot, each such point is also associated with a certain “anonymous value” as shown in the table.

These can be considered in the averaging process, whereby all the “anonymous” values of all the points in each cluster are averaged separately:

S(x_i^a): A(x_i^a)

S(x_j^b): A(x_j^b)

These new points, A_a and A_b, are plotted in their respective clusters.

Step IV: Convergence

A_a and A_b become the new ICM, and steps I to III are repeated.

The iterations stop when the new A_a and A_b converge with the A_a and A_b from the previous iteration.

Result

After the values have converged, the two S{x_i,j^a,b} now presented by the machine are two distinct groups of data, the emergence of which also signals that the machine has “learnt”. If a very large database is supplied as an input to the machine, there is an advantage as well as a disadvantage.

• Advantage: the machine learns better


• Disadvantage: data may not converge at all due to improper choice of clusters

In order to prevent the second possibility, a suitable number of clusters has to be determined for n, the number of data-points. As a rule of thumb,

k = (n / 2)^1/2

Here, ‘k’ is the number of clusters.

If a dataset of 50 points is input, then k = 5 clusters are suggested. Once the iterations have been performed and convergence has been attained in five separate clusters, the centroids of each cluster (or, the average value of the data-points in each cluster) can now be used to arrive at 2 super-clusters.

A simple connect between feedback systems and fuzzy logic controllers

Click on the image for a larger view.

The following picture details the three ways to depict the transformation from fuzzy logic to crisp logic. The processual representation (the flowchart in the middle) shows the positive feedback loop that the fuzzy logic controller uses to make the dataset crisp ("p.c." in the chart stands for "position controller").



 

The algorithm, shown leftmost, delineates the logical statements that constitute the following flowchart. The purpose of the positive feedback loop is served by the nested if-then clusters that, going by the graphics on the first row, ensure that intuitive conclusions (as represented by the fuzzy set) are brought as close to the real thing (as represented by the crisp set) as possible using statistical data. The equation on the rightmost determines the mathematical convergence, i.e., the minimum number of line segments that would have to be perfectly aligned for any following segments to just fall in place after them (ref: German tank problem).

A simple connect between feedback systems and fuzzy logic controllers

Click on the image for a larger view.

The following picture details the three ways to depict the transformation from fuzzy logic to crisp logic. The processual representation (the flowchart in the middle) shows the positive feedback loop that the fuzzy logic controller uses to make the dataset crisp ("p.c." in the chart stands for "position controller").



 

The algorithm, shown leftmost, delineates the logical statements that constitute the following flowchart. The purpose of the positive feedback loop is served by the nested if-then clusters that, going by the graphics on the first row, ensure that intuitive conclusions (as represented by the fuzzy set) are brought as close to the real thing (as represented by the crisp set) as possible using statistical data. The equation on the rightmost determines the mathematical convergence, i.e., the minimum number of line segments that would have to be perfectly aligned for any following segments to just fall in place after them (ref: German tank problem).