# Introductory Treatise on Lie's Theory of Finite Continuous Transformation Groups

@article{BIntroductoryTO, title={Introductory Treatise on Lie's Theory of Finite Continuous Transformation Groups}, author={H. F. B.}, journal={Nature}, volume={71}, pages={49-50} }

THE theory of continuous groups should appeal to all who are interested in mathematics; it is based on the fundamental ideas involved in cases of change of the algebraic notation, and as such is an illuminating synthesis of a large number of our elementary operations; and the principal notions of the theory, once laid bare, are so simple and admit of so many familiar applications that these should form an integral part of elementary teaching, particularly in analytical geometry and differential…

## 22 Citations

DR AF T Poincaré and complex function theory

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- 2010

theory for them, and they now turned to recap their earlier study of differential equations, upon which they went on to base their theory of elliptic functions. Not only did this free the theory of…

Poincaré and the idea of a group

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- 2012

In many different fields of mathematics and physics Poincare found many uses for the idea of a group, but not for group theory. He used the idea in his work on automorphic functions, in number…

Poincaré and complex function theory

- Mathematics
- 2010

Poincare is still well known for the mathematical work that first made his name : his discovery in 1880--1881 of automorphic functions. New documents and insights were added in [Gray and Walter…

Fixed points of Lie group actions on surfaces

- MathematicsErgodic Theory and Dynamical Systems
- 1986

Abstract Let G be a connected finite-dimensional Lie group and M a compact surface. We investigate whether, for a given G and M, every continuous action of G on M must have a fixed (stationary)…

Self-Invariant Contact Symmetries

- Mathematics
- 2004

Abstract Every smooth second-order scalar ordinary differential equation (ODE) that is solved for the highest derivative has an infinite-dimensional Lie group of contact symmetries. However,…

Realizations of the Witt and Virasoro Algebras and Integrable Equations

- MathematicsJournal of Nonlinear Mathematical Physics
- 2019

In this paper we study realizations of infinite-dimensional Witt and Virasoro algebras. We obtain a complete description of realizations of the Witt algebra by Lie vector fields of first-order…

Practical Guide to the Symbolic Computation of Symmetries of Differential Equations

- Mathematics, Computer Science
- 2014

A computational approach to finding symmetries and computer algebra programs to compute the usually very large system of determining partial differential equations and a computer algebra algorithm that at least automatically solves most of these equations and in simple cases provides a complete solution.

On realizations of the Witt algebra in $\mathbb{R}^3$

- Mathematics
- 2014

We obtain exhaustive classification of inequivalent realizations of the Witt and Virasoro algebras by Lie vector fields of differential operators in the space $\mathbb{R}^3$. Using this…

Group classification of nonlinear evolution equations: Semi-simple groups of contact transformations

- MathematicsCommun. Nonlinear Sci. Numer. Simul.
- 2015

Image analysis by wavelet-type transforms: Group theoretic approach

- MathematicsJournal of Mathematical Imaging and Vision
- 2005

A group theoretic approach to image representation and analysis and the concept of a wavelet transform is extended to incorporate different types of groups to find the invariance measure for the admissibility condition of a mother wavelet-type transform.