Stochastic Models, Information Theory, and Lie Groups, by Gregory S. Chirikjian (auth.)

By Gregory S. Chirikjian (auth.)

The topics of stochastic tactics, details thought, and Lie teams are typically taken care of individually from one another. This distinct two-volume set offers those themes in a unified environment, thereby development bridges among fields which are not often studied by means of an analogous humans. in contrast to the various very good formal remedies to be had for every of those topics separately, the emphasis in either one of those volumes is at the use of stochastic, geometric, and group-theoretic techniques within the modeling of actual phenomena.

Volume 2 builds at the basics provided in Volume 1, delving deeper into relationships between stochastic geometry, geometric facets of the idea of communications and coding, multivariate statistical research, and mistake propagation on Lie teams. large routines, motivating examples, and real-world functions make the paintings appropriate as a textbook to be used in classes that emphasize utilized stochastic approaches or differential geometry.

Stochastic versions, details concept, and Lie Groups might be of curiosity to complex undergraduate and graduate scholars, researchers, and practitioners operating in utilized arithmetic, the actual sciences, and engineering.

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Extra resources for Stochastic Models, Information Theory, and Lie Groups, Volume 2: Analytic Methods and Modern Applications

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10 Lie Groups I: Introduction and Examples The concept of a group was described briefly in Chapter 1. 1 Furthermore, when referring to Lie groups, what will be meant in the context of this book is matrix Lie groups, where each element of the group is a square invertible matrix. Other books focusing specifically on matrix groups include [3, 9, 19]. In a sense, matrix Lie groups are “the most like Rn ” of any mathematical structure other than vector spaces. Indeed, Rn together with the operation of addition is an example of a Lie group.

Z(X, Y ) = log(eX eY ). By introducing the function . log x = F (x) = x−1 ∞ k=0 (−1)k (x − 1)k k+1 and evaluating it with exponentiated adjoint operators (which are defined by Taylor series in analogy with the definition of the matrix exponential), it can be shown that [13] 1 F (exp(t ad(X)) exp(ad(Y )))Xdt. 3 Change of Basis in a Matrix Lie Algebra 23 Substituting into the Taylor series for F (x) and integrating term by term then gives 1 1 Z(X, Y ) = X + Y + [X, Y ] + ([X, [X, Y ]] + [Y, [Y, X]]) 2 12 1 + ([Y, [X, [Y, X]]] + [X, [Y, [Y, X]]]) + · · · .

1, only the permutation example and the matrix addition examples are groups. The vector cross product is not associative, there is no identity element, and there is no inverse. So it is “far from” being a group. The matrix multiplication example is “almost” a group. The problem is that, in general, matrices are not invertible under the operation of multiplication. In general, a groupoid that satisfies Properties 1 and 2 but not Property 3 is called a semi-group. In stochastic processes semi-groups play an important role.

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