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Stochastic Analysis of the Time Continuum^{ †}

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. The Intuitionistic Mathematics

#### 2.2. The Time Operator

## 3. Results

#### 3.1. The Continuum of Reals

#### 3.1.1. Real Numbers

#### 3.1.2. The Continuum Structure

#### 3.2. Wavelets and Multiresolution Hierarchy

#### 3.2.1. The Signal Space

#### 3.2.2. Signal Ensembles

## 4. Discussion

#### 4.1. Self-Organization of the Time Continuum

#### 4.1.1. Local and Global Complexity

Go to the top of Highgate Hill on a clear summer morning at five o’clock, and look at Westminster Abbey. You will receive an impression of a building enriched with multitudinous vertical lines. Try to distinguish one of these lines all the way down from the one next to it: You cannot. Try to count them: You cannot. Try to make out the beginning or end of any of them: You cannot. Look at it generally, and it is all symmetry and arrangement. Look at it in its parts, and it is all inextricable confusion.

#### 4.1.2. Dynamical Identity

#### 4.2. The Measurement Problem

#### 4.2.1. Quantum Measurement

#### 4.2.2. The Euclidean Paradigm

#### 4.2.3. Psychophysical Parallelism

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Rectangular diagram of the Euclidean algorithm. Steps of the measurement process are dyed in different shades.

**Figure 2.**The Ford diagram of circles. The real number corresponds to the progression of a vertical line represented by the arrow.

**Figure 3.**The Minkowski question mark function that is an automorphism of the continuum transfiguring its representation from the continued fraction to the binary code.

**Figure 4.**Inserting new circle of the Ford diagram, whose designator is the mediant of the adjacent ones.

**Figure 5.**Isomorphic representations of the Ford diagram related through the question mark function.

**Figure 6.**The tree of detail coefficients. Each node at a scale has two successors at the next one in the hierarchy.

**Figure 7.**The semigroup action that concerns blurring of the signal, which is related to an expansion of the spatial domain.

**Figure 8.**The wavelet domain hidden Markov model. Black nodes represent the detail coefficients and blue ones the hidden states.

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Milovanović, M.; Vukmirović, S.; Saulig, N. Stochastic Analysis of the Time Continuum. *Mathematics* **2021**, *9*, 1452.
https://doi.org/10.3390/math9121452

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Milovanović M, Vukmirović S, Saulig N. Stochastic Analysis of the Time Continuum. *Mathematics*. 2021; 9(12):1452.
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Milovanović, Miloš, Srđan Vukmirović, and Nicoletta Saulig. 2021. "Stochastic Analysis of the Time Continuum" *Mathematics* 9, no. 12: 1452.
https://doi.org/10.3390/math9121452