We assume that the statistical weight SW for free nodes in a geometric shape is of extreme importance in mathematics and theoretical physics but is still absent.
However, SW, which is a dimensionless mathematical/physical quantity attached to the importance of the position of the node, can be well defined via the normal/Gaussian distribution curve or equivalently via the B-matrix transition chains.
Both approaches give exactly the same result, which shows that SW is uniquely defined.
This is a very interesting and advanced question. I'm afraid I cannot answer it fully, but I can try to provide some information that might be helpful.
The statistical weight of a geometric shape in mathematics and theoretical physics is a measure of how likely a certain configuration of nodes or elements is to occur in a given system. For example, in statistical mechanics, the statistical weight of a microstate is proportional to the number of ways that microstate can be realized by the system.
One way to define the statistical weight for free nodes in a geometric shape is to use the weighted geometric mean, which is a generalization of the geometric mean using the weighted arithmetic mean. Given a sample x = (x1,x2,…,xn) and weights w = (w1,w2,…,wn), it is calculated as:
$$\bar{x}=\left(\prod _{i=1}^{n}x_{i}^{w_{i}}\right)^{1/\sum _{i=1}^{n}w_{i}}=\exp \left({\frac {\sum _{i=1}^{n}w_{i}\ln x_{i}}{\sum _{i=1}^{n}w_{i}\quad }}\right)$$
The second form above illustrates that the logarithm of the geometric mean is the weighted arithmetic mean of the logarithms of the individual values. If all the weights are equal, the weighted geometric mean simplifies to the ordinary unweighted geometric mean.
Another way to define the statistical weight for free nodes in a geometric shape is to use B-matrix transition chains, which are a type of Markov chain that describe the probability of moving from one state to another in a system. A Markov chain is a stochastic process where the future state depends only on the present state and not on the past history. A transition matrix is a matrix that gives the probabilities of moving from one state to another in one step. A B-matrix is a special kind of transition matrix that satisfies some properties, such as being nonnegative, irreducible, and having all row sums equal to one.
The B-matrix transition chains can be used to model the dynamics of complex systems, such as networks, biological systems, social systems, etc. The statistical weight of a node in a B-matrix transition chain can be interpreted as the long-run proportion of time that the system spends in that state, or equivalently, as the limit of the n-step transition probability as n goes to infinity. This limit exists and is unique for any irreducible and positive recurrent B-matrix transition chain.
To directly answer the question about importance, the statistical weight of a geometric shape is a concept that can have different meanings and applications depending on the context and the method of calculation.
For example, in geometric morphometrics, which is the study of shape variation and its covariation with other factors, the statistical weight of a geometric shape can be related to the size, shape, and form of an object. Size is a scalar quantity that measures the overall magnitude of an object, shape is a property that describes the relative positions of landmarks or features on an object, and form is a combination of size and shape. The statistical weight of a geometric shape can be influenced by allometry, which is the size-related change of shape. Allometry can be studied using different methods, such as regression analysis, principal component analysis, or multivariate analysis of covariance.
Another example is in statistical mechanics, which is the branch of physics that deals with the behavior of large numbers of particles or systems. The statistical weight of a geometric shape can be related to the distribution function or the density function of a system. The distribution function gives the probability that a system is in a certain state or configuration, and the density function gives the probability density for a continuous variable. The statistical weight of a geometric shape can be influenced by entropy, which is a measure of disorder or randomness in a system. Entropy can be calculated using different methods, such as Boltzmann's formula, Gibbs' formula, or Shannon's formula.
A third example is in perception psychology, which is the study of how humans process sensory information. The statistical weight of a geometric shape can be related to the perceived weight or the weight illusion of an object. The perceived weight is the subjective impression of how heavy or light an object is, and the weight illusion is the discrepancy between the perceived weight and the actual weight. The statistical weight of a geometric shape can be influenced by visual cues, such as size, color, material, texture, etc. Visual cues can create different types of weight illusions, such as size-weight illusion, color-weight illusion, material-weight illusion, etc.
As you can see, the statistical weight of a geometric shape can have different interpretations and implications depending on the field and the problem. Therefore, it is hard to say how important it is without specifying the context and the purpose. However, it is clear that it is a concept that connects geometry and probability in various ways and can reveal interesting patterns and phenomena in nature and cognition.
The statistical weight of a geometric shape plays a significant role in both mathematics and theoretical physics. In mathematics, statistical weights are often associated with counting the number of different configurations or arrangements of objects within a given geometric shape. These weights can be used to calculate probabilities, determine combinatorial properties, or analyze the behavior of mathematical structures.
In theoretical physics, statistical weights are crucial for understanding various physical phenomena, particularly in statistical mechanics and quantum field theory. Statistical mechanics deals with systems composed of a large number of particles, where the statistical weight of a configuration corresponds to the likelihood of that particular arrangement occurring. It allows for the calculation of macroscopic properties, such as entropy, temperature, and energy, based on the microscopic details of the system.
In quantum field theory, statistical weights are associated with the probability amplitudes of different configurations of particles and fields. These weights, often represented by mathematical objects called Feynman diagrams, are used to calculate scattering amplitudes and cross-sections, which provide insights into the behavior of elementary particles and the fundamental forces of nature.
The statistical weight of a geometric shape provides a quantitative measure of its importance in these contexts. It encapsulates the information necessary for analyzing and predicting the properties and behavior of systems in mathematics and theoretical physics
Ive never seen this concept in all years of student or profesional practice. What exactly does statistical weight to do with geometry?? Ive seen some weights to calculate integrals, but this is not relevant.
It merely sounds like combinatorics to me. (permutations and combinations). As to use in statistical mechanics,
where the counting of microstates is relevant, I dont call this geometry.
We assume that the statistical weight SW for free nodes in a geometric shape is of extreme importance in mathematics and theoretical physics but is still absent.
However, SW, which is a dimensionless mathematical/physical quantity attached to the importance of the position of the node, can be well defined via the normal/Gaussian distribution curve or equivalently via the B-matrix transition chains.
Both approaches give exactly the same numerical result, which shows that SW is uniquely defined.
Here we can state that SW emerges from Cairo CT technique theory, so it's not really telling us which point of view to take, but insisting that we keep in mind both the parts and the whole.
The theoretical assumptions and the solution procedure that apply to the CT also apply to the SW of geometric shapes.
In short:
1-CT is a hypermodern numerical statistical theorem that inherently exists in all aspects of mathematics and theoretical physics.
Any theoretical attempt to circumvent or ignore CT would be incomplete.
2-CT describes the nature, in a comprehensive self-consistent approach, in the unitary block of space-time x-t as well as all its corollaries such as SW,s
3-The CT theory has been around since the days of the Big Band and will last forever.
All we've done since 2020 is remove mountains of sand piled on top of this theory to make it visible.
3-We assume that CT theory, or any other adequate statistical approach in x-t unit space, is the only way to solve mathematical physics situations.
Any other approach would be inappropriate and incomplete.
4- Many people would not like to call CT and SW as geometry, which is true since it is hypergeometry.
As a striking example, Phythagoras' Pth theorem itself is simple geometry, not hypergeometry. In other words, Pth is, by far, incomplete.
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**In a following answer, we provide detailed specific numerical examples (in 1D, 2D and 3D unit spaces) of the calculations of SW via the two techniques, the normal/Gaussian technique and the transition matrix technique B, which is a product from CT.
We expect these illustrative examples to reveal beyond doubt the accuracy of all of the above assumptions.
**Answer continued part II.
The normal/Gaussian curve distribution formula is eligible to provide statistical weights for different geometric shapes, but how?
The answer is that slicing or discretizing the x-axis into equal spatial intervals dx=h can introduce statistical probability weights for the straight-line configuration.
In general, the one-dimensional normal/Gaussian curve distribution formula can be used to provide statistical weights in 1D straight line shapes, similarly the double or two-dimensional normal/Gaussian curve distribution formula can provide statistical weights for 2D rectangular shapes. forms. shapes, and finally the triple or 3D normal/Gaussian curve distribution formula is able to provide statistical weights for 3D cuboid shapes.
To begin with, we explain the special case, without loss of generality, consider the case of a line divided into 18 equidistant spatial intervals h via 19 nodes.
The statistical weights from the statistical chains of matrix B SW1,SW2 , . . . SW19 are incorporated into the following formula to find the integral Is statistic:
Is= h2 (0.2571Y1+0.4872Y2+ 0.6902Y3+ 0.8662Y4+1.0150Y5+1.1368Y6+1.2316Y7 +1.2992Y8+1.339Y9+1.3534Y10+1.339Y11+1.2992Y12+1.231 6Y13+1.1368Y14 +1.0150Y15+ 0.8662Y16+ 0.6902Y17 +0.4872Y18+ 0.2571Y19) . . . . . . (1)
Equation 1 is the statistical equivalence of Simpson's rule for 19 nodes.
It is derived without the need for Lagrange multipliers which are only a statistical tool that can be replaced by other adequate statistical assumptions.
Note that equation 1 was the dream of mathematicians and theoretical physicists at the time.
We can now compare the statistical weights given in Equation 1 by their corresponding values offered by the normal/Gaussian distribution formula y=c1exp-c2(x-Mue)^2 as shown in Table I.
The constants c1 and c2 are conveniently set to 1.3534 and 0.01 and the median Mue is obviously equal to 10.
We can compare the statistical weights given in Equation 4 with their corresponding values given by the normal/Gaussian distribution formula y=c1exp-c2(x-Mue)^2 as shown in the first table.
The constants c1 and c2 are conveniently set to 1.3534 and 0.01 and it is evident that the mean Mue is equal to 10.
i x = .1 * i Gaussian parity absolute error
Formula (Eq. 1)
1 0.1 0.602070928 0.2571 0.343
2 0.2 0.713637590 0.4872 0.226
3 0.3 0.829128623 0.6902 0.14
4 0.4 0.944235206 0.8662 0.088
5 0.5 1.05402899 1.0150 0.039
6 0.6 1.15329134 1.1368 0.016
7 0.7 1.23691452 1.2316 0.0063
8 0.8 1.30033243 1.2992 0.001
9 0.9 1.33993340 1.339 0.0009
10 1.0 1.35339999 1.3534 0.000.000
11 1.1 1.33993340 1.339 0.0009.005
12 1.2 1.30033243 1.2992 0.001
……………………………….
etc…. etc.
Table I. Statistical equivalence of Simpson's integration for 19 nodes vs Gaussian distribution.
Note that the sum of the statistical weights for Equation 1 is exactly = 17.9999 = 18.0000
While SUM OF Gaussian Curve 18.9 is very close to the expected exact value 19.
Table I shows a striking agreement between the results of the normal/Gaussian distribution curve and the statistical weights of the proposed single statistical integration.
The absolute error ea is practically less than 0.001 for N = 19 and should decrease for higher N.
***In a following answer, we provide detailed specific numerical examples (in 2D and 3D unit spaces) of the calculations of SW via the two techniques, the normal/Gaussian technique and the transition matrix technique B, which is a product from CT.
This kind of analysis (now I understood something) is rare in physics.
One can see the relative time spent by the Classical harmonic oscilator, at its various positions,
and compare it to a Quantum Harmonic Oscilator. (Near the end of a swing it spends more time)
I would not consider this of mayor importance, just something done ocasionally, without special name.
.***Answer continued part III.Just a brief notice before presenting our response following part III.We emphasize the fact that the B-statistical transition matrix which is a product of the Cairo technical theory introduced in 2020 operates in the unified 4D x-t space. It produces hypermodern geometry, hypermodern physics and hypermodern mathematics. This type of analysis is never found in physics/mathematics, including quantum mechanics, but it is a revolutionary hypermodern theory. Many people are still confused and try to compare it to classical math/physics ( in 3D geometry plus external time) such as the classical harmonic oscillator or a quantum harmonic oscillator which we consider a wasted effort. The Cairo technique theory is expected to dominate sooner or later, and most people who undermine it or call it redundant will eventually speed up to join its many expected accomplishments. Again, we can use a two-dimensional normal/Gaussian curve distribution formula to provide statistical weights in 2D rectangular shapes. Similar to 1D x-t space, the normal/Gaussian curve distribution formula in 2D is able to provide statistical weights for 2D rectangular shapes. To begin, we explain the following special case, without loss of generality: Consider a square shape subdivided into 25 h^2 squares via 25 nodes. Fig. 1.📷
Fig. 1. A square shape subdivided into 25 h^2 squares via 25 nodes
We can now compare the statistical weights given by the strings of the 2D B matrix by their corresponding values offered by the double normal/Gaussian distribution formula y=c1exp-c2(x-Mue)^2*exp-c2(y-Mue ) ^2 as follows,
A-Numerical results of the statistical weights for Fig.1 as given by the B-Matrix strings,
SW1(i,j)or SW1(x,y):
0.503695250 0.822963834 0.915082991 0.822963834 0.503695250
0.822963834 1.32647121 1.47669744 1.32647097 0.822963774
0.915082991 1.47669744 1.64555156 1.47669744 0.915083170
0.822963834 1.32647097 1.47669744 1.32647097 0.822963834
0.503695190 0.822963774 0.915083110 0.822963834 0.503695190
B-Numerical results of the statistical weights for Fig.1 as given by the B-Matrix strings,
SW2(i,j)or SW2(x,y):
0.53693 0.81718 0.93998 0.81718 0.536926
0.81718 1.2437 1.4306 1.2437 0.81718
0.93998 1.4306 1.6456 1.4306 0.93998
0.81718 1.2437 1.4306 1.2437 0.81718
0.5369 0.8171 0.93998 0.81718 0.53692
*Note the good agreement between Sw1(i,j) and SW2(i,j)
Obviously, the precision increases when n increases.
** SW1 and SW2 are rotationally symmetrical.
***In a following answer, we provide detailed specific numerical examples (in 3D unit spaces) of the calculations of SW via the two techniques, the normal/Gaussian technique and the transition matrix technique B, which is a product from CT.