By Khadiga Arwini

This quantity makes use of details geometry to offer a standard differential geometric framework for quite a lot of illustrative purposes together with amino acid series spacings, cryptology reviews, clustering of communications and galaxies, and cosmological voids.

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Nine 1. three Joint chance Density features . . . . . . . . . . . . . . . . . . . . . . . . nine 1. three. 1 Bivariate Gaussian Distributions . . . . . . . . . . . . . . . . . . . . 10 1. four details conception . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . eleven 1. four. 1 Gamma Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . thirteen 2 advent to Riemannian Geometry . . . . . . . . . . . . . . . . . . . . 2. zero. 2 Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. zero. three Tangent areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. zero. four Tensors and types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. zero. five Riemannian Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. zero. 6 Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. 1 Autoparallel and Geodesic Curves . . . . . . . . . . . . . . . . . . . . . . . . . 2. 2 common Connections and Curvature . . . . . . . . . . . . . . . . . . . . . . 19 20 20 22 25 26 29 29 three info Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . three. 1 Fisher info Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . three. 2 Exponential kinfolk of likelihood Density capabilities . . . . . . . . . three. three Statistical a-Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . three. four Affine Immersions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . three. four. 1 Weibull Distributions: no longer of Exponential kind . . . . . . . 31 32 33 34 35 36 VII VIII four Contents three. five Gamma 2-Manifold G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . three. five. 1 Gamma a-Connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . three. five. 2 Gamma a-Curvatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . three. five. three Gamma Manifold Geodesics . . . . . . . . . . . . . . . . . . . . . . . . three. five. four together twin Foliations . . . . . . . . . . . . . . . . . . . . . . . . . . . three. five. five Gamma Affine Immersion . . . . . . . . . . . . . . . . . . . . . . . . . . three. 6 Log-Gamma 2-Manifold L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . three. 6. 1 Log-Gamma Random Walks . . . . . . . . . . . . . . . . . . . . . . . . three. 7 Gaussian 2-Manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . three. 7. 1 Gaussian usual Coordinates . . . . . . . . . . . . . . . . . . . . . . three. 7. 2 Gaussian details Metric . . . . . . . . . . . . . . . . . . . . . . . . three. 7. three Gaussian jointly twin Foliations . . . . . . . . . . . . . . . . . . three. 7. four Gaussian Affine Immersions . . . . . . . . . . . . . . . . . . . . . . . . . three. eight Gaussian a-Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . three. eight. 1 Gaussian a-Connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . three. eight. 2 Gaussian a-Curvatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . three. nine Gaussian collectively twin Foliations . . . . . . . . . . . . . . . . . . . . . . . . three. 10 Gaussian Submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . three. 10. 1 valuable suggest Submanifold . . . . . . . . . . . . . . . . . . . . . . . . . three. 10. 2 Unit Variance Submanifold . . . . . . . . . . . . . . . . . . . . . . . . . three. 10. three Unit Coefficient of edition Submanifold . . . . . . . . . . . . three. eleven Gaussian Affine Immersions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . three. 12 Log-Gaussian Manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 38 39 forty forty two forty two forty two forty five forty five forty seven forty seven forty eight forty eight forty nine forty nine 50 50 fifty one fifty one fifty two fifty two fifty two fifty three info Geometry of Bivariate households . . . . . . . . . . . . . . . four. 1 McKay Bivariate Gamma 3-Manifold M . . . . . . . . . . . . . . . . . . . . four. 2 McKay Manifold Geometry in average Coordinates . . . . . . . . . . four. three McKay Densities Have Exponential kind . . . . . . . . . . . . . . . . . . . four. three. 1 McKay info Metric . . . . . . . . . . . . . . . . . . . . . . . . . four. four McKay a-Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . four. four. 1 McKay a-Connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . four. four. 2 McKay a-Curvatures . . . . .

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