Showing posts with label Beauty of the Math World. Show all posts
Showing posts with label Beauty of the Math World. Show all posts
Sunday, June 21, 2009
Friday, May 15, 2009
Normal distribution and dependence
Normally distributed and uncorrelated does not imply independent
http://en.wikipedia.org/wiki/Normally_distributed_and_uncorrelated_does_not_imply_independent
A counterexample
The fact that two random variables X and Y both have a normal distribution does not imply that the pair (X, Y) has a joint normal distribution. A simple example is one in which Y = X if X > 1 and Y = −X if X < 1.
http://en.wikipedia.org/wiki/Multivariate_normal_distribution
http://en.wikipedia.org/wiki/Normally_distributed_and_uncorrelated_does_not_imply_independent
A counterexample
The fact that two random variables X and Y both have a normal distribution does not imply that the pair (X, Y) has a joint normal distribution. A simple example is one in which Y = X if X > 1 and Y = −X if X < 1.
http://en.wikipedia.org/wiki/Multivariate_normal_distribution
Perturbation methods
Chapter II: Introduction to perturbation methods by Johan Byström, Lars-Erik Persson, and Fredrik Strömberg
Introduction to regular perturbation theory by Eric Vanden-Eijnden (PDF)
Duality in Perturbation Theory
Perturbation Method of Multiple Scales
Retrieved from "http://en.wikipedia.org/wiki/Perturbation_theory"
Introduction to regular perturbation theory by Eric Vanden-Eijnden (PDF)
Duality in Perturbation Theory
Perturbation Method of Multiple Scales
Retrieved from "http://en.wikipedia.org/wiki/Perturbation_theory"
Saturday, May 2, 2009
Seashells: the Plainness and Beauty of Their Mathematical Description
by Jorge Picado
Departamento de Matemática
Universidade de Coimbra
picado@mat.uc.pt
Fulltext
Abstract:
One might at first tend to think that the growth of plants and animals, because of their elaborate forms, are ruled by highly complex laws. However, this is surprisingly not always true: many aspects of the growth of plants and animals may be described by remarkably simple mathematical laws. An obvious example of this are the seashells and snails, as we show here: with a very simple model it is possible to describe and generate any of the many types of seashells that one may find classified in a good seashell bookguide. The fact that the animal which lives at the open edge of the shell places new shell material always in that edge, and faster on one side than the other, makes the shell to grow in a spiral. The rates at which shell material is secreted at different points of the open edge are presumably determined by the anatomy of the animal. And, surprisingly, even fairly small changes in such rates can have quite tremendous effects on the overall shape of the shell, which is in the origin of the existence of a great diversity of shells.
Departamento de Matemática
Universidade de Coimbra
picado@mat.uc.pt
Fulltext
Abstract:
One might at first tend to think that the growth of plants and animals, because of their elaborate forms, are ruled by highly complex laws. However, this is surprisingly not always true: many aspects of the growth of plants and animals may be described by remarkably simple mathematical laws. An obvious example of this are the seashells and snails, as we show here: with a very simple model it is possible to describe and generate any of the many types of seashells that one may find classified in a good seashell bookguide. The fact that the animal which lives at the open edge of the shell places new shell material always in that edge, and faster on one side than the other, makes the shell to grow in a spiral. The rates at which shell material is secreted at different points of the open edge are presumably determined by the anatomy of the animal. And, surprisingly, even fairly small changes in such rates can have quite tremendous effects on the overall shape of the shell, which is in the origin of the existence of a great diversity of shells.
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