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The Group Theory
Published in Mikhail G. Brik, Chong-Geng Ma, Theoretical Spectroscopy of Transition Metal and Rare Earth Ions, 2019
Mikhail G. Brik, Chong-Geng Ma
Example II. All rational positive numbers form a group, if the operation of mathematical multiplication is taken as a group multiplication rule. The product of two rational numbers is another rational number.Multiplication is an associative operation: (A × B) × C = A × (B × C).Unity is an identity element: A × 1 = A.For each number A its reciprocal number “1/A” is an inverse element: A × 1/A = 1. The word “positive” is very important in the description of the numbers in this example, since it excludes zero from the considered set of numbers and allows each number to have its inverse.
Belief systems and ideological deep disagreement
Published in International Journal of General Systems, 2022
J. L. Usó-Doménech, J. A. Nescolarde-Selva, H. Gash
Any pair has the following logical properties: Closure: Associativity:Identity element:, Inverse element:Commutativity:
On the Symmetry of Blast Waves
Published in Nuclear Technology, 2021
where the double overbars represent transformed variables, is known as the group parameter, and , , and are constants to be determined. The transformations given by Eq. (32) are referred to a Lie group, where the infinite number of group elements are the transformations for the possible values of the continuous parameter ; moreover, they feature an associative binary operation, and the group contains an identity , an inverse element, and is closed under the binary composition operation.
Modelling and control of a spherical pendulum via a non–minimal state representation
Published in Mathematical and Computer Modelling of Dynamical Systems, 2021
Ricardo Campa, Israel Soto, Omar Martínez
The proof that the quaternion multiplication (6) is closed and associative relies in using the unit norm constraint (5) for each operand, and some of the above properties of the skew–symmetric operator. The identity element of the unit quaternion group is , and the inverse element of is , so that