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Correlation means that there is a relationship between two or more variables (such as ice cream consumption and crime), but this relationship does not necessarily imply cause and effect. When two variables are correlated, it simply means that as one variable changes, so does the other. We can measure correlation by calculating a statistic known as a correlation coefficient. A correlation coefficient is a number from -1 to +1 that indicates the strength and direction of the relationship between...
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The distribution law or Nernst's distribution law is the law that governs the distribution of a solute between two immiscible solvents. This law, also known as the partition law, states that if a solute is added to the mixture of two immiscible solvents at a constant temperature, the solute is distributed between the two solvents in such a way that the ratio of solute concentrations in the solvents remains constant at equilibrium.
For extracting a solute from an aqueous phase into an...
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Dimensional analysis simplifies complex physical problems and guides experimental investigations, but it does not provide complete solutions. It identifies the dimensionless groups that influence a phenomenon, but experimental data is needed to establish the specific relationships and validate theoretical predictions.
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Calibration Curves: Correlation Coefficient01:10

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In a linear calibration curve, there is a value called the calibration coefficient, denoted by 'r,' which measures the strength and the direction of association between two variables. The correlation coefficient value ranges from −1 to +1. A value of +1 indicates a perfect positive linear correlation, −1 denotes a perfect negative correlation, and 0 implies no correlation between the two variables. A positive correlation value establishes that as one variable increases, the...
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Coefficient of Correlation01:12

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The correlation coefficient, r, developed by Karl Pearson in the early 1900s, is numerical and provides a measure of strength and direction of the linear association between the independent variable x and the dependent variable y.
If you suspect a linear relationship between x and y, then r can measure how strong the linear relationship is.
What the VALUE of r tells us:
The value of r is always between –1 and +1: –1 ≤ r ≤ 1.
The size of the correlation r indicates the...
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Maximum Power Transfer01:16

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Numerous practical applications within engineering disciplines, such as telecommunications, necessitate optimizing power delivery to a connected load. This pursuit, however, entails inherent internal losses, which can either equal or exceed the power supplied to the load. The Thevenin equivalent circuit is helpful in finding the maximum power a linear circuit can deliver to a load. It is assumed in this context that the load resistance can be adjusted.
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Using Informational Connectivity to Measure the Synchronous Emergence of fMRI Multi-voxel Information Across Time
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Effective intermittency and cross correlations in the standard map.

G Datseris1, F K Diakonos1, P Schmelcher2

  • 1Department of Physics, University of Athens, GR-15771 Athens, Greece.

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|August 15, 2015
PubMed
Summary

New correlation functions reveal long-range cross correlations in dynamical systems. This occurs due to intermittent dynamics in phase-space structures, extending findings to Hamiltonian maps.

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Area of Science:

  • Nonlinear dynamics
  • Statistical mechanics
  • Chaos theory

Background:

  • Dynamical systems exhibit complex behaviors influenced by phase-space structures.
  • Autocorrelation and cross-correlation functions are key tools for analyzing system dynamics.
  • Local chaos and transitions to global chaos are important characteristics of dynamical systems.

Purpose of the Study:

  • To define novel auto- and cross-correlation functions for capturing dynamical characteristics from local phase-space structures.
  • To investigate the emergence of long-range cross correlations in the standard map.
  • To link these correlations to intermittent dynamics in two-dimensional Hamiltonian maps.

Main Methods:

  • Definition and calculation of auto- and cross-correlation functions.
  • Simulation of noninteracting particles in the standard map with varying nonlinearity parameter k.
  • Analysis of particle trajectories and phase-space regions.
  • Application of symbolic dynamics to identify underlying dynamics.

Main Results:

  • Long-range cross correlations emerge in the standard map for 0.6
  • These correlations are an ensemble property dependent on phase-space cell selection.
  • The emergence of correlations is linked to intermittent dynamics in specific phase-space regions.
  • The findings are observed in regions characterized by local chaos or the transition to global chaos.

Conclusions:

  • The study successfully defines correlation functions that capture dynamical characteristics from phase-space structures.
  • Long-range cross correlations are demonstrated in the standard map, particularly in regions exhibiting local or transitional chaos.
  • The findings provide evidence for the connection between intermittent dynamics and cross correlations in two-dimensional Hamiltonian maps, extending previous results from 1D maps.