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Symmetries and Group Theory
Published in Mattias Blennow, Mathematical Methods for Physics and Engineering, 2018
Example 4.15 The infinite chain of points of separation ℓ, see Fig. 4.8, has a symmetry group containing translations by a multiple of ℓ and reflections. The group is generated by the translation Tℓ, for which Tℓx=x+ℓ, and the reflection σ, for which σx=−x. It also holds that σ2=e and that σTℓσTℓ=e. Any group that is isomorphic to this group may be defined by two generators obeying these relations. We also note that the last relation is equivalent to Tℓ−1=σTℓσ and it follows that Tℓ and Tℓ−1 are in the same conjugacy class. Since two generators are enough to write any element in this group, it is finitely generated.
L 0-convex compactness and its applications to random convex optimization and random variational inequalities
Published in Optimization, 2021
Tiexin Guo, Erxin Zhang, Yachao Wang, Mingzhi Wu
Let E be a finitely generated -module, e.g. let for n-fixed elements . Then there exists a finite partition of Ω to such that is a free module of rank i over the algebra for any such that in which case E has the direct sum decomposition as and each such is unique up to the almost sure equality.
Darboux theory of integrability for polynomial vector fields on 𝕊n
Published in Dynamical Systems, 2018
One of the best tools for searching for invariant algebraic hypersurfaces is the extactic polynomial of associated to W. To our knowledge, it was first mentioned in the work of Lagutinskii, see [30]. To define it let W be a finitely generated vector subspace of vector space . The extactic polynomial of associated to W is where {v1,… , vl} is a basis of W, l is the dimension of W, and . It is well known that one of the main properties of the extactic polynomial is that its definition does not dependent of the chosen basis of W.
Shadowing property for the free group acting in the circle
Published in Dynamical Systems, 2020
A group G is finitely generated if there exists a finite set such that for any there exist with . The set S is called finite generator of G. If S is a finite generator of G and for all we have that , then the set S is called a finite symmetric generator.