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Statistical Mechanics of Macromolecules
Published in Gregory S. Chirikjian, Alexander B. Kyatkin, Engineering Applications of Noncommutative Harmonic Analysis, 2021
Gregory S. Chirikjian, Alexander B. Kyatkin
Recall that ℝ3 denotes 3-dimensional Euclidean space, and SE(3) denotes the group of rigid-body motions, x e ℝ3 is a position vector, and g = (R, r) ∈ SE (3) is a rigid body transformation (or frame of reference). The terminology SE(3) stands for “special Euclidean” group of 3-dimensional space. This group is the semi-direct product of the rotation group S O (3) and the translation group (ℝ3, + ). The group law is g1 ∘ g2 = (R1R2, R1r2 + r1) and geometrically represents the concatenation of rigid-body motions, or equivalently, a sequential change of reference frames. The inverse of any element g ∈ SE(3) and the group identity element are given respectively as g−1 = (RT, − RTr) and e=(I,0) where I is the 3 × 3 identity. The action of elements of SE (3) on elements of ℝ3 is defined as g ∘ x = Rx+ r.
Functional Equations on Affine Groups
Published in Michael Ruzhansky, Hemen Dutta, Advanced Topics in Mathematical Analysis, 2019
A special case of this semidirect product can be obtained if the automorphisms in H are inner automorphisms, i.e. conjugations. More exactly, we may have the following situation: let G be a group, and let I, N be subgroups of G with the property that N is a normal subgroup, I∩N={e}, the identity element, and G=IN, i.e. every element of G can be represented in the form of h n with h in I and n in N. In this case this representation is unique, as it is easy to see. Now we consider the set H of inner automorphisms of G corresponding to the elements of I: H={hi:i∈I,hi(g)=i−1gi for g∈G}.
Wave Equation with Random Boundary, Robots in a Fluid, Cosmology with Stochastic Perturbations, Interaction Between Graviton, Photon and Electron- Positron Fields, Evans-Hudson Flows for Quantization of Stochastic Evolutions
Published in Harish Parthasarathy, Electromagnetics, Control and Robotics, 2023
p(R, a) dR da where da is the Lebergue measure on ℝ3 ad dℝ is the Haar measure on SO(3). Note that the rotation translation group is SO(3) S ℝ3 (semidirect product)
On the possibility of frozen nuclei
Published in Molecular Physics, 2021
Using the MS group of benzene allows for a systematic analysis of all equivalent configurations. This group decomposes according to where is isomorphic to the point group ; see also Appendix A.2 for more explanations on the relation of point group operations and operations of the MS group. The group defines the permutation subgroup of ; it contains all feasible permutations of identical nuclei and, for benzene in its equilibrium structure, it is isomorphic to the point group [53]. For our later deliberations, it is interesting to note that with ⋊ defining the semi-direct product of the groups and . Both and are cyclic and their presentations can be written as [73] The irreducible representations of , , and are all known and tabulated in the literature [53, 74].