**4. 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.

#### **4.1 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).

10 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>�(*uβ*|*J*) = *PH*(*uβ*|*J*) *<sup>M</sup>*�*J*<sup>1</sup>|*J*<sup>0</sup>�(*vβ*|*J*) = *PH*(*vβ*|*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

All matrices in this chapter use this configuration for the matrix pattern to represent 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

(11)

(12)

elements generated as a matrix by C code

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

**3.7 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

**3.7.2 Representation patterns of matrices**

histograms arranged as a 4 × 4 matrix.

elements.

**4. Simulation results**

**4.1 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.

**3.7.1 Matrix and Its elements**

Representation patterns are illustrated in Fig 3(I). Eight 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 probability histograms are shown Fig 3(III) for *N* = 13 to represent four anti-symmetry

<sup>383</sup> From Local Interactive Measurements to Global Matrix Representations on Variant

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

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 essential to analyze differences among 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 D-P. Compared with distributions in (a-b) , it is feasible to identify the same components from

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

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

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

Construction – A Particle Model of Quantum Interactions for Double Path Experiments

and also the relevant histogram will be put on the position.

**5.1 Statistical distributions for a given function**

that are significantly different from the original components.

**5.1.1 Symmetry groups for a function**

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

histograms into a

groups respectively.

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

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

configurations.

respectively.

function.

D-W respectively.

original inputs.

**5. Analysis of results**

22*n*−<sup>1</sup>

**4.2 Global matrix representations**

matrix.

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 probability histograms; (III) *N* = {13}, *J* = 3 Two groups of eight probability histograms

Representation patterns are illustrated in Fig 3(I). Eight 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 probability histograms are shown Fig 3(III) for *N* = 13 to represent four anti-symmetry groups respectively.

### **4.2 Global matrix representations**

12 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 probability histograms; (III) *N* = {13}, *J* = 3 Two groups of eight probability

histograms

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

Applying the C code configuration, a given signal of a function determines an element on a matrix 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 also the relevant histogram will be put on the position.

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.

### **5. Analysis of results**

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

#### **5.1 Statistical distributions for a given function**

It is essential to analyze differences among various statistical distributions for a given function.

#### **5.1.1 Symmetry groups for a function**

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 D-W respectively.

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 D-P. Compared with distributions in (a-b) , it is feasible to identify the same components from original inputs.

However, for *PH*(*u*1|*J*) under synchronous conditions and with the same Left and Right input signals, the simulation shows D-W exhibiting interferences among the output distributions that are significantly different from the original components.

Matrices for D-W in Fig 4-5 (h) show different properties. In general, only one peak can be observed for each element especially for the *J* ∈ {10, 12, 3, 5} condition. Spectra appear to be much simpler than the original distributions in Fig 4-5 (e-f), and significant interference

<sup>385</sup> From Local Interactive Measurements to Global Matrix Representations on Variant

Construction – A Particle Model of Quantum Interactions for Double Path Experiments

Pairs of relationships can be checked on symmetry matrices in Figs 4-5 (a-d), four groups are

{*PH*(*u*+|*J*)} elements in Figs 4-5 (a) show (only) four distinct distributions. Each column contains only one distribution. Sixteen elements in the matrix can be classified into four vertical classes: {0, 2, 8, 10}, {4, 6, 12, 14}, {1, 3, 9, 11}, {5, 7, 13, 15} respectively. Four meta

{*PH*(*u*−|*J*)} elements in Figs 4-5 (b) show (a further) four distinct distributions. Each row contains only one distribution. Sixteen elements in the matrix can be classified into four horizontal classes: {0, 4, 1, 5}, {2, 6, 3, 7}, {8, 12, 9, 13}, {10, 14, 11, 15} respectively. Four meta

{*PH*(*u*0|*J*)} elements in Figs 4-5 (c) illustrate symmetry properties. There are six pairs of symmetry elements: {8 : 14}, {2 : 11}, {0 : 15}, {6:9}, {4 : 13}, {1:7}. In addition, four elements on anti-diagonals provide different distributions: {10, 12, 3, 5}. Under this condition,

{*PH*(*u*1|*J*)} elements in Figs 4-5 (d) illustrate symmetry properties. There are six pairs of symmetry elements: {8 : 14}, {2 : 11}, {0 : 15}, {6:9}, {4 : 13}, {1:7}. In addition, four elements on diagonal positions provide same distribution: {0, 6, 9, 15}. Two elements on anti-diagonals: {12, 3} have the same distribution in Fig 4 (d). Under this condition, nine or ten classes of different distributions can be identified for Fig 4 (d) and Fig 5 (d) respectively.

{*PH*(*v*+|*J*)} elements in Figs 4-5 (e) show that (only) four classes can be distinguished. Elements within these groups members are the same as for symmetry groups in Figs 4-5(a).

Figures 4-5 (e-h) represent anti-symmetry properties, four groups can be identified.

properties are observed.

identified.

**5.3 Four symmetry groups**

**5.3.1 Left: polarized vertical group**

distributions are given as {10, 14, 11, 15}.

**5.3.2 Right: polarized horizontal group**

distributions are given as {0, 2, 8, 10}.

ten classes of distributions are distinguished.

**5.3.3 D-P: particle group**

**5.3.4 D-W: wave group**

**5.4 Four anti-symmetry groups**

**5.4.1 Left: polarized vertical group**

Their distributions fall within the region [0.5, 1].

#### **5.1.2 Anti-symmetry groups for a function**

Four distributions are shown in Fig 3 (e-h) as asymmetry groups. A pair of equation *PH*(*v*+|*J*) = *PH*(1 − *v*−|*J*) shows that one distribution is a mirror image of another one. *PH*(*v*+|*J*) distribution is shown in Fig 3 (e) for Left signals and *PH*(*v*−|*J*) distribution is shown Fig 3 (f) for Right signals.

*PH*(*v*0|*J*) is shown in Fig 3 (g) for both paths open under asynchronous conditions to simulate D-P. Compared with (e-f) distributions, it is feasible to identify the same components from the original inputs.

However *PH*(*v*1|*J*) is shown in Fig 3 (h) under synchronous condition with both path signals as inputs to simulate D-W exhibiting interferences among the output distributions that are significantly different from the original components.

To show even and odd number's differences, *N* = 12 cases are shown in Fig 3 (II, a-h) and *N* = 13 cases are shown in Fig 3 (III, a-h) respectively.

#### **5.2 Global matrix representations**

Sixteen matrices are represented in Fig 4-5 (a-h) with eight signals generating two sets of 16 groups for *N* = {12, 13} respectively.

#### **5.2.1 Symmetry cases**

Matrices for the Left in Fig 4-5 (a) show elements in a column with the corresponding histogram showing polarized effects on the vertical.

Matrices for the Right in Fig 4-5 (b) show elements in a row with the corresponding histogram showing polarized effects on the horizontal.

Matrices for D-P in Fig 4-5 (c) provide asynchronous operations combined with both distributions from Fig 4-5 (a-b) to form a unified distribution. From each corresponding position, it is possible to identify each left and right component and the resulting shapes of the histogram.

Matrices for D-W in Fig 4-5 (d) provide synchronous operations combined with both distributions from Fig 4-5 (a-b) for each element. Compared with Fig 4-5 (c) and Fig 4-5 (d) respectively, distributions in Fig 4-5 (d) are much simpler with two original distributions especially on the anti-diagonal positions: *J* ∈ {10, 12, 3, 5}. Only less than half the number of spectrum lines are identified.

#### **5.2.2 Anti-symmetry cases**

In a similar manner to the symmetry conditions, four anti-symmetry effects can be identified in Fig 4-5 (e-h). Matrices in Fig 4-5 (e) are Left operations for different functions, elements are polarized on the vertical and matrices in Fig 4-5 (f) are Right operations, elements are polarized on the horizontal. Spectrum lines in Fig 4-5 (e) appear in the right half and spectrum lines in Fig 4-5 (f) are appeared in the left half respectively.

Matrices for D-P in Fig 4-5 (g) show additional effects for each distribution according to the relevant position with components that can be identified as corresponding to identifiable inputs in many cases. Anti-symmetry signals are generated in merging conditions.

Matrices for D-W in Fig 4-5 (h) show different properties. In general, only one peak can be observed for each element especially for the *J* ∈ {10, 12, 3, 5} condition. Spectra appear to be much simpler than the original distributions in Fig 4-5 (e-f), and significant interference properties are observed.

#### **5.3 Four symmetry groups**

14 Measurement Systems

Four distributions are shown in Fig 3 (e-h) as asymmetry groups. A pair of equation *PH*(*v*+|*J*) = *PH*(1 − *v*−|*J*) shows that one distribution is a mirror image of another one. *PH*(*v*+|*J*) distribution is shown in Fig 3 (e) for Left signals and *PH*(*v*−|*J*) distribution is shown

*PH*(*v*0|*J*) is shown in Fig 3 (g) for both paths open under asynchronous conditions to simulate D-P. Compared with (e-f) distributions, it is feasible to identify the same components from the

However *PH*(*v*1|*J*) is shown in Fig 3 (h) under synchronous condition with both path signals as inputs to simulate D-W exhibiting interferences among the output distributions that are

To show even and odd number's differences, *N* = 12 cases are shown in Fig 3 (II, a-h) and

Sixteen matrices are represented in Fig 4-5 (a-h) with eight signals generating two sets of 16

Matrices for the Left in Fig 4-5 (a) show elements in a column with the corresponding

Matrices for the Right in Fig 4-5 (b) show elements in a row with the corresponding histogram

Matrices for D-P in Fig 4-5 (c) provide asynchronous operations combined with both distributions from Fig 4-5 (a-b) to form a unified distribution. From each corresponding position, it is possible to identify each left and right component and the resulting shapes of

Matrices for D-W in Fig 4-5 (d) provide synchronous operations combined with both distributions from Fig 4-5 (a-b) for each element. Compared with Fig 4-5 (c) and Fig 4-5 (d) respectively, distributions in Fig 4-5 (d) are much simpler with two original distributions especially on the anti-diagonal positions: *J* ∈ {10, 12, 3, 5}. Only less than half the number of

In a similar manner to the symmetry conditions, four anti-symmetry effects can be identified in Fig 4-5 (e-h). Matrices in Fig 4-5 (e) are Left operations for different functions, elements are polarized on the vertical and matrices in Fig 4-5 (f) are Right operations, elements are polarized on the horizontal. Spectrum lines in Fig 4-5 (e) appear in the right half and spectrum

Matrices for D-P in Fig 4-5 (g) show additional effects for each distribution according to the relevant position with components that can be identified as corresponding to identifiable

inputs in many cases. Anti-symmetry signals are generated in merging conditions.

**5.1.2 Anti-symmetry groups for a function**

significantly different from the original components.

*N* = 13 cases are shown in Fig 3 (III, a-h) respectively.

histogram showing polarized effects on the vertical.

lines in Fig 4-5 (f) are appeared in the left half respectively.

showing polarized effects on the horizontal.

Fig 3 (f) for Right signals.

**5.2 Global matrix representations**

**5.2.1 Symmetry cases**

the histogram.

spectrum lines are identified.

**5.2.2 Anti-symmetry cases**

groups for *N* = {12, 13} respectively.

original inputs.

Pairs of relationships can be checked on symmetry matrices in Figs 4-5 (a-d), four groups are identified.

#### **5.3.1 Left: polarized vertical group**

{*PH*(*u*+|*J*)} elements in Figs 4-5 (a) show (only) four distinct distributions. Each column contains only one distribution. Sixteen elements in the matrix can be classified into four vertical classes: {0, 2, 8, 10}, {4, 6, 12, 14}, {1, 3, 9, 11}, {5, 7, 13, 15} respectively. Four meta distributions are given as {10, 14, 11, 15}.
