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Functional Equations on Affine Groups
Published in Michael Ruzhansky, Hemen Dutta, Advanced Topics in Mathematical Analysis, 2019
In the subsequent paragraphs we shall consider the two standard cases V = ℝn and V = ℂn. First we suppose that V = ℝn and we let H = O(n). The resulting affine group Aff O(n)=O(n)⋉ℝn is the group of rigid Euclidean motions. The elements of O(n) represent rotations and the elements of ℝn represent translations. As we have seen above O(n) and ℝn are embedded in this affine group: O(n) is isomorphic to the subgroup {(k,0):k∈O(n)} and ℝn is isomorphic to the normal subgroup {(id,u):u∈ℝn}. In fact, these isomorphisms are topological in the sense that they are homeomorphisms with respect to the topological group structures. In particular, O(n) is isomorphic to a compact subgroup of Aff O(n). Clearly, O(n) is disconnected, e.g. O(1)≅ℤ2, the two element cyclic group.
Multiscale Investigations of RC Shear Wall Buildings
Published in Journal of Earthquake Engineering, 2022
Tong Tong, Mohammed Bazroun, In Ho Cho, Keith A. Porter
where is the evolving buckling length; is the initial buckling length of jth steel bar (denoted by meaning a line entity); BUm is the basic topological group consisting of a bar and its adjacent solid elements; k is the effective length factor. Damage to surrounding concrete (BUm) leads to a new , and consequently a new buckling initiation point (where buckling initiates according to (Dhakal and Maekawa 2002)):
Topological speedups of ℤd-actions
Published in Dynamical Systems, 2022
Aimee S. A. Johnson, David M. McClendon
For Cortez' construction, we begin by considering any decreasing sequence of subgroups of , where each has finite index in . For each , let be the quotient map. Then, define is a topological group (the topology is the product of the discrete topologies on each ); for each there is a natural coordinate map . More importantly, there is a minimal action given by where the sum in the component is taken mod .
C 0-stability for actions implies shadowing property
Published in Dynamical Systems, 2021
Given a topological group G and a metric space , a dynamical system is formally defined as a triplet , where is a continuous function with for all and for all . The map Φ is called an action of G on X. It is possible to associate to each element of a homeomorphism where . For every we define the orbit of x as . A non-empty set is called minimal for the action if for any .