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An Intensional Probability Theory: Investigating the Link between Classical and Quantum Probabilities^{ †}

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Fundamental Conception

#### 2.1. Intensionality of Geometrical Structuring

#### 2.2. Foundation of Probability Theory

#### 2.3. Geometrical Structure of Probability

## 3. Results

#### 3.1. Quantum Ensembles and Measurement

#### 3.2. Time Continuum and Intuitionism

- (i)
- annihilation $j<0\Rightarrow {\psi}_{j,k}=0$;
- (ii)
- periodization ${\psi}_{j,k}={\psi}_{j,k+{2}^{j}}$;
- (iii)
- translation ${\psi}_{j,k}\left(x-\frac{m}{{2}^{j}}\right)={\psi}_{j,k+m}\left(x\right)$;
- (iv)
- evolution ${U}^{\u2020}{\psi}_{j,k}=\frac{1}{\sqrt{2}}{\psi}_{j-1,k}$;
- (v)
- basicity $\varphi =A+{\sum}_{j\ge 0}^{}{\sum}_{1}^{{2}^{j}}{D}_{j,k}{\psi}_{j,k}$ for every variable $\varphi $ from ${L}_{\mu}^{2}$ and some coefficients of approximation A and details ${D}_{j,k}$;
- (vi)
- orthonormality ${\psi}_{j,k},1\le k\le {2}^{j}$ and the constant variable 1 constitute an orthonormal base of ${L}_{\mu}^{2}$.

#### 3.3. Wavelet-Domain Hidden Markov Model

## 4. Discussion

#### 4.1. The Art of Memory

#### 4.2. Postmodern Science

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The Last Supper.

**Left**: A modern painting that corresponds to the set of incident elements.

**Right**: A traditional icon that corresponds to a nested structure.

**Figure 2.**The double-slit experiment. A trace of the particle is mediated by two slits that might be opened or closed.

**Figure 3.**The binary tree of detail coefficients. Each node at a scale is inherited by two of them at the next one.

**Figure 4.**The wavelet-domain hidden Markov model. Black nodes correspond to detail coefficients and blue ones to hidden variables.

**Figure 5.**Temporal organization of the traditional iconography.

**Left**: The temporally based hierarchy which is represented by an erection from horizontal through semi-vertical to the vertical position in respect to Christ’s figure.

**Right**: Hierarchical structures of geometry, which have emerged in the von Koch curve.

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Milovanović, M.; Saulig, N. An Intensional Probability Theory: Investigating the Link between Classical and Quantum Probabilities. *Mathematics* **2022**, *10*, 4294.
https://doi.org/10.3390/math10224294

**AMA Style**

Milovanović M, Saulig N. An Intensional Probability Theory: Investigating the Link between Classical and Quantum Probabilities. *Mathematics*. 2022; 10(22):4294.
https://doi.org/10.3390/math10224294

**Chicago/Turabian Style**

Milovanović, Miloš, and Nicoletta Saulig. 2022. "An Intensional Probability Theory: Investigating the Link between Classical and Quantum Probabilities" *Mathematics* 10, no. 22: 4294.
https://doi.org/10.3390/math10224294