# Geometrical Information Flow Regulated by Time Lengths: An Initial Approach

## Abstract

**:**

## 1. Introduction

^{2}of area, 51 water utilities, and 2274 water distribution points (schools and administrative units). In this way, public schools present a large variety of features (variables, Figure 1) [2] that make difficult for managers to set a unique method to determine how to manage water consumption or other natural resource types that is of use in other public provision services. Therefore, the proposed methodology in the article [5] of knowing in large samples how water consumption in schools occurs through linear time series of analysis and coefficient of determination-${R}^{2}$, trying to extract universal indicators that can serve as a reference for the whole State, was shown to be limited, due to impossibility of evaluating and predict for future time how consumption behavior will be expressed.

## 2. Methodology

#### 2.1. Theoretical Framework of Experiment

#### 2.1.1. Information Flow and Ergodic Properties

**Theorem**

**1.**

**Lemma**

**1.**

**Proof**

**of Lemma 1.**

_{k}passes, there is a growth of the variable ${x}_{1}$ and ${x}_{2}$ revealing binary sequences that repeat cumulatively and asymmetrically on time length $\left({T}_{k}\to \infty \right)$, according to Table 1.

**Proof**

**of Theorem 1.**

**Theorem**

**2.**

_{k}is equal to the variables ${x}_{1}$ and ${x}_{2}$ in its probabilistic expressions as follows:

**Proof**

**of Theorem 2.**

#### 2.1.2. Information Flow and Time as the Cause of Oscillations

## 3. Results

#### Non-Ergodicity

## 4. Discussion

- -
- -
- vector based diseases [35];
- -
- -
- -
- -
- logistics aspects of administration and commerce flow [36];
- -
- -
- engineering of materials [37];
- -
- -
- language processing regarding space and time accelerations or other biological aspects [22];
- -
- -
- -
- duality based and quantum based phenomena, such as economics and particles and other fields of knowledge related to the issue [30]; and,
- -
- computing, networking, and communications [38].

- (a)
- proportionality between/among variable’s distribution (for open/closed system)—linearity
- -
- General management of deterministic flows methods; and,

- (b)
- disproportionality between/among variable’s distribution (open system)—nonlinearity
- -
- Time regulated dynamics.

## 5. Conclusions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Variables affecting water consumption at the public schools of Secretary of State for Education of Paraná, Brazil [2].

**Figure 2.**Water usage at public schools. The data consist of 149 schools at different regions of the Paraná State with a population of $u=133.783$ individuals. Using R

^{2}(determination coefficient) for both data, the linear function presented for the chosen population data can’t be equally found in the water usage behavior.

**Figure 3.**Flowchart of Section 2. Methodology, showing theoretical experiment of information regulated by time length’s effects on water consumption and school’s population.

**Figure 6.**Nonlinearity effects caused by time according to Figure 7.

**Figure 7.**Information flow at geometric variables (I) and not geometric variables (II). A: Resources and B: Population.

**Figure 8.**Oscillation’s quantitative aspects due to information distribution regulated by time lengths. (

**a**) Equal amount of variables A and B $\therefore Y\to \infty $, where $\uparrow Y\uparrow B\uparrow {X}_{1}\uparrow {X}_{2}\downarrow A$ for optimal resources distribution. (

**b**) Time length for event where amount of variable A is not used fully by B caused by finite time $\therefore Y<\infty $, where in this case, $<Y<B<{X}_{1}<{X}_{2}$ AND $\uparrow A$, being A not consumed in time given. Resources wrongly distributed. (

**c**) Time length for event where amount of variable A is limited for use caused by B variable ${X}_{2}$ presence $\therefore Y<\infty $ where $\uparrow B\uparrow {X}_{2}\uparrow A\uparrow {X}_{1}\downarrow A\downarrow {X}_{1}\downarrow B\downarrow {X}_{2}$. Resources containment and population-resource chaotic regulation.

**Figure 9.**Water consumption and individual repetition. Observation: the lines are colored for the benefit of graph visualization.

**Figure 10.**Representation of population-resource ratio analyzed by continuous time length. Iterations order can’t affect the system if time is continuous. Another effect would be expected if a time interval interrupts the flow of variables, leading the system to an insufficient distribution of resources as described in item (b) and (c). Observation: the lines are colored for the benefit of graph visualization.

**Figure 11.**Flow of water consumption and population size regulated by finite time interval. Consider in the graph a maximum time of permanence in the place (lower resource trajectories) in which the population size demands more time to obtain full correspondence between population and resource ratio. The flow of resources is not reached to full system size due to the time given. Observation: the lines are colored for the benefit of graph visualization.

**Figure 12.**Deterministic to chaotic behavior of variables regulated by time lengths. Observation: the lines are colored for the benefit of graph visualization.

**Figure 13.**Scheme for resources and population dynamics regulated by time lengths. Possible results obtained through iteration, frequency and time over variables ${x}_{1}$ and ${x}_{2}$. It is theoretically postulated that time lengths have specific effects over the event, causing specific phase space’s trajectories.

**Figure 14.**Constant binary distribution of population-resource ratios. Imagine a restaurant where the brown line represents the queue of individuals. Blue line, the variable ${x}_{2}$ and orange line, resources. As time passes, individuals at queue start getting access to resources, as waiting time stay relatively constant and resources are consumed in the same proportion of individuals in the queue.

**Figure 15.**Time regulated dynamics of population-resource ratio. It is possible to observe the oscillations of variables in the system as time passes. Blue line, the variable ${x}_{2}$, orange line, resources and light brown line, population.

**Figure 16.**Time series of population variables ${x}_{1}$ and ${x}_{2}$ expressing proportionality for population-resource ratio.

**Figure 17.**Time series of population variables ${x}_{1}$ and ${x}_{2}$ expressing disproportionality for population-resource ratio. The asymmetrical pattern shows recurrence at original state (indicated arrows) and time regulation equilibrium.

**Figure 18.**To the left, the representation of Figure 16 and to the right, Figure 17. In this view, it is possible to see specific phases of event of Figure 17 that are marked with color circles. Red circle, event start. Light green, time regulation. Blue circle, ${x}_{2}$ saturation. Yellow, chaotic phase. Purple, population growth and variable variable ${x}_{2}$ reduction. Dark green, recurrence of variable ${x}_{2}$.

**Figure 19.**Evolution of system dynamics. Population and resource ratio are represented in two possible pathways. (1) Variables and resources recurrence to the original state. The amount of resource available at pathway 1 is proportional to the population previous aspects. In this case, resource amount is higher than the original state amount. (2) Pathway 2 leads to the end of the event. It is expected for the resource time series at bottom of Figure 19, a constant reduction of information flow until it reaches 0 (zero for both variables interaction).

Time. | T_{1} | T_{2} | T_{3} | T_{4} | T_{5} | T_{6} | T_{7} | T_{8} | $\text{}\cdots $ |
---|---|---|---|---|---|---|---|---|---|

Variables | X_{1} | X_{2} | X_{1} | X_{2} | X_{1} | X_{2} | X_{1} | X_{2} | $\text{}\cdots $ |

Bits | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | $\cdots $ |

0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | $\cdots $ |

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Roberto Telles, C.
Geometrical Information Flow Regulated by Time Lengths: An Initial Approach. *Symmetry* **2018**, *10*, 645.
https://doi.org/10.3390/sym10110645

**AMA Style**

Roberto Telles C.
Geometrical Information Flow Regulated by Time Lengths: An Initial Approach. *Symmetry*. 2018; 10(11):645.
https://doi.org/10.3390/sym10110645

**Chicago/Turabian Style**

Roberto Telles, Charles.
2018. "Geometrical Information Flow Regulated by Time Lengths: An Initial Approach" *Symmetry* 10, no. 11: 645.
https://doi.org/10.3390/sym10110645