Expectation Values and Variance Based on Lp-Norms
Abstract
:1. Introduction
2. The Generalized Formal Scheme of Means Characterization
2.1. The Means Characterization Based on Optimization Methods
2.2. Formal Scheme of Means Characterization
3. The Concept of -Expectation Values
3.1. The Non-Euclidean Norm Operator
- (i)
- The non-Euclidean mean of is the Euclidean mean of
- (ii)
- Zero-mean of ,
- (iii)
- (iv)
- In the Euclidean case, degenerates to the identity operator .
- (v)
- Linear operations: .Hence, , which reads Equation (8).
- (vi)
- Non-additivity of : .
- (vii)
- Inverse of non-Euclidean norm operator, , ,
3.2. The Non-Euclidean -Mean Estimator and Its Expectation Value
3.3. Examples
3.3.1. Gas at Thermal Equilibrium
3.3.2. Plasma Out of Thermal Equilibrium
3.3.3. D-Dimensional Quantum Harmonic Oscillator in Thermal Equilibrium
4. The Non-Euclidean -Variance of the -Expectation Value
4.1. Preliminaries: Formulations
4.2. Examples
4.2.1. Example 1: Gaussian distribution
4.2.2. Example 2: Generalized Gaussian distribution
4.3. Justification of the -Variance Expression
5. Further Analytical and Numerical Examples
5.1. Analytical Example: The Spectrum of the Means and Its Degeneration
5.2. Numerical Example: Earth’s Magnetic Field
6. Conclusions
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Livadiotis, G. Expectation Values and Variance Based on Lp-Norms. Entropy 2012, 14, 2375-2396. https://doi.org/10.3390/e14122375
Livadiotis G. Expectation Values and Variance Based on Lp-Norms. Entropy. 2012; 14(12):2375-2396. https://doi.org/10.3390/e14122375
Chicago/Turabian StyleLivadiotis, George. 2012. "Expectation Values and Variance Based on Lp-Norms" Entropy 14, no. 12: 2375-2396. https://doi.org/10.3390/e14122375
APA StyleLivadiotis, G. (2012). Expectation Values and Variance Based on Lp-Norms. Entropy, 14(12), 2375-2396. https://doi.org/10.3390/e14122375