**5. Simulation results**

For ease of illustration, as different signals have intrinsic random properties, only statistical distributions and global matrix representations are selected in this section.

#### **5.1 Statistical distributions**

The simulation provides a series of output results. In this section, *N* = {12, 13}, *n* = 2, {*J* = 3, *J*<sup>+</sup> = 11, *J*<sup>−</sup> = 2} are selected. Corresponding to Left path (Left), Right path (Right), Double path for Particles (D-P) and Double path for Waves (D-W) under symmetry and anti-symmetry conditions respectively.

From a given function, a set of histograms can be generated as two groups of eight probability histograms. To show their refined properties, it is necessary to represent them in both odd and even numbers. A total of sixteen histograms are required. For convenience of comparison, sample cases are shown in Figures 3(I-III).

12 Measurement Systems

After local interactive measurements and statistical process are undertaken for a given function *J*, eight histograms are generated. The Global Matrix Representation GMR component performs its operations into two stages. In the first stage, exhausting all possible

scheme. Here, we can see Left and Right path reactions polarized into Horizontal and Vertical

*<sup>M</sup>*�*J*<sup>1</sup>|*J*<sup>0</sup>�(*u*˜*β*|*J*) = *PH*(*u*˜*β*|*J*) *<sup>M</sup>*�*J*<sup>1</sup>|*J*<sup>0</sup>�(*v*˜*β*|*J*) = *PH*(*v*˜*β*|*J*) *<sup>J</sup>* <sup>∈</sup> *<sup>B</sup>*2*<sup>n</sup>*

For example, using *n* = 2, *P* = (3102), Δ = (1111) conditions, a C code case contains sixteen

0 4 1 5 2 6 3 7 8 12 9 13 10 14 11 15

All matrices in this chapter use this configuration for the matrix pattern representing their

For ease of illustration, as different signals have intrinsic random properties, only statistical

The simulation provides a series of output results. In this section, *N* = {12, 13}, *n* = 2, {*J* = 3, *J*<sup>+</sup> = 11, *J*<sup>−</sup> = 2} are selected. Corresponding to Left path (Left), Right path (Right), Double path for Particles (D-P) and Double path for Waves (D-W) under symmetry and

From a given function, a set of histograms can be generated as two groups of eight probability histograms. To show their refined properties, it is necessary to represent them in both odd and even numbers. A total of sixteen histograms are required. For convenience of comparison,

distributions and global matrix representations are selected in this section.

<sup>2</sup> ; *<sup>J</sup>*1, *<sup>J</sup>*<sup>0</sup> <sup>∈</sup> *<sup>B</sup>*2*n*−<sup>1</sup>

2

elements and each element

(14)

(15)

elements generated as a matrix by C code

<sup>2</sup> to generate eight sets, each set contains 22*<sup>n</sup>*

**4.7 Global Matrix Representations**

is a histogram. In the second stage, arranging all 22*<sup>n</sup>*

For a given C scheme, let *<sup>C</sup>*(*J*) = �*J*1|*J*0�, each element

**4.7.2 Representation patterns of matrices**

histograms arranged as a 4 × 4 matrix.

elements.

**5. Simulation results**

**5.1 Statistical distributions**

anti-symmetry conditions respectively.

sample cases are shown in Figures 3(I-III).

⎧ ⎪⎨

⎪⎩

functions for <sup>∀</sup>*<sup>J</sup>* <sup>∈</sup> *<sup>B</sup>*2*<sup>n</sup>*

relationships respectively.

**4.7.1 Matrix and its elements**

(I) Representative patterns of Histograms for function *J* (a-d) symmetric cases; (e-h) antisymmetric cases

Representation patterns are illustrated in Fig 3(I). Eight conditional probability histograms of *PH*(*u*˜+|*J*) = *PH*(*u*˜+|*J*) are shown in Fig 3(II) for *N* = 12 to represent four symmetry groups and another eight conditional probability histograms are shown Fig 3(III) for *N* = 13

<sup>353</sup> From Conditional Probability Measurements to Global Matrix Representations on Variant

Applying the C code configuration, any given signal of a function determines a matrix element to represent its histogram. There is one to one correspondence among different configurations. Using this measurement mechanism, eight types of statistical histograms are systematically illustrated. Each element in the matrix is numbered to indicate its corresponding function

For *n* = 2 cases, sixteen matrices are shown in Figs 5-6 (a-h). Figs 5-6 (a-d) represent Symmetry groups and Figs 5-6 (e-h) represent Anti-symmetry groups. To show odd and even number configurations, Fig 5 (a-h) shows *N* = 12 cases and Fig 6 (a-h) shows *N* = 13 cases

In the previous section, results of different statistical distributions and their global matrix representations were presented. In this section, plain language is used to explain what various

It is necessary to analyze the differences among the various statistical distributions for a given

For the selected function *J* = 3, four distributions in symmetry groups are shown in Fig 3 (a-d). (a) *PH*(*u*˜+|*J*) for Left; (b) *PH*(*u*˜−|*J*) for Right; (c) *PH*(*u*˜0|*J*) for D-P; and (d) *PH*(*u*˜1|*J*) for

Under Symmetry conditions, *PH*(*u*˜+|*J*) = *PH*(*u*˜−|*J*), both Left and Right distributions are the same. *PH*(*u*˜0|*J*) generated with both paths open under asynchronous conditions simulates a D-P. Compared with distributions in (a-b) , it is possible to identify the components from

However, for *PH*(*u*˜1|*J*) under synchronous conditions and with the same Left and Right input signals, the simulation shows a D-W exhibiting interferences among the output distributions

visual effects might be illustrated and to discuss local and global arrangements.

into a matrix and a C code scheme of variant logic is applied to organize a set of 22*<sup>n</sup>*

Construction – A Particle Model of Intrinsic Quantum Waves for Double Path Experiments

functions are applied. It is convenient to arrange all the histograms generated

histograms

to represent four anti-symmetry groups respectively.

matrix.

**6.1 Statistical distributions for a given function**

that are significantly different from the original components.

**6.1.1 Symmetry groups for a function**

**5.2 Global matrix representations**

and the relevant histogram is shown.

<sup>×</sup> 22*<sup>n</sup>*−<sup>1</sup>

All possible 22*<sup>n</sup>*

into a 22*<sup>n</sup>*−<sup>1</sup>

respectively.

function.

D-W respectively.

original inputs.

**6. Analysis of results**

Fig. 3. (I-III) *N* = {12, 13}, *J* = 3 Simulation results ; (I) Representative Patterns for *PH*(*u*˜+|*J*) = *PH*(*u*˜−|*J*) and *PH*(*v*˜+|*J*) = *PH*(1 − *v*˜−|*J*) conditions; (II) *N* = {12}, *J* = 3 Two groups of eight histograms on conditional probability; (III) *N* = {13}, *J* = 3 Two groups of eight histograms on conditional probability

Representation patterns are illustrated in Fig 3(I). Eight conditional probability histograms of *PH*(*u*˜+|*J*) = *PH*(*u*˜+|*J*) are shown in Fig 3(II) for *N* = 12 to represent four symmetry groups and another eight conditional probability histograms are shown Fig 3(III) for *N* = 13 to represent four anti-symmetry groups respectively.

### **5.2 Global matrix representations**

14 Measurement Systems

(a) Left (b) Right

(c) D-P (d) D-W

(e) Left (f) Right

(g) D-P (h) D-W (III) *N* = {13}, *J* = 3 Two groups of results in eight histograms

Fig. 3. (I-III) *N* = {12, 13}, *J* = 3 Simulation results ; (I) Representative Patterns for *PH*(*u*˜+|*J*) = *PH*(*u*˜−|*J*) and *PH*(*v*˜+|*J*) = *PH*(1 − *v*˜−|*J*) conditions; (II) *N* = {12}, *J* = 3 Two groups of eight histograms on conditional probability; (III) *N* = {13}, *J* = 3 Two groups of

eight histograms on conditional probability

All possible 22*<sup>n</sup>* functions are applied. It is convenient to arrange all the histograms generated into a matrix and a C code scheme of variant logic is applied to organize a set of 22*<sup>n</sup>* histograms into a 22*<sup>n</sup>*−<sup>1</sup> <sup>×</sup> 22*<sup>n</sup>*−<sup>1</sup> matrix.

Applying the C code configuration, any given signal of a function determines a matrix element to represent its histogram. There is one to one correspondence among different configurations.

Using this measurement mechanism, eight types of statistical histograms are systematically illustrated. Each element in the matrix is numbered to indicate its corresponding function and the relevant histogram is shown.

For *n* = 2 cases, sixteen matrices are shown in Figs 5-6 (a-h). Figs 5-6 (a-d) represent Symmetry groups and Figs 5-6 (e-h) represent Anti-symmetry groups. To show odd and even number configurations, Fig 5 (a-h) shows *N* = 12 cases and Fig 6 (a-h) shows *N* = 13 cases respectively.
