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The two rightmost columns indicate which irreducible representations describe the symmetry transformations of the three Cartesian coordinates (x, y and z), rotations about those three coordinates (R x, R y and R z), and functions of the quadratic terms of the coordinates(x 2, y 2, z 2, xy, xz, and yz).
The reducible representation of the bonding of water with C 2v symmetry. An initial assumption is that the number of molecular orbitals is equal to the number of atomic orbitals included in the linear expansion. In a sense, n atomic orbitals combine to form n molecular orbitals, which can be numbered i = 1 to n and which
The irreducible complex characters of a finite group form a character table which encodes much useful information about the group G in a concise form. Each row is labelled by an irreducible character and the entries in the row are the values of that character on any representative of the respective conjugacy class of G (because characters are class functions).
Ammonia solution, also known as ammonia water, ammonium hydroxide, ammoniacal liquor, ammonia liquor, aqua ammonia, aqueous ammonia, or (inaccurately) ammonia, is a solution of ammonia in water. It can be denoted by the symbols NH 3 (aq). Although the name ammonium hydroxide suggests a salt with the composition [NH + 4][OH −
This page contains tables of azeotrope data for various binary and ternary mixtures of solvents. The data include the composition of a mixture by weight (in binary azeotropes, when only one fraction is given, it is the fraction of the second component), the boiling point (b.p.) of a component, the boiling point of a mixture, and the specific gravity of the mixture.
The Simplified Molecular Input Line Entry System (SMILES) is a specification in the form of a line notation for describing the structure of chemical species using short ASCII strings. SMILES strings can be imported by most molecule editors for conversion back into two-dimensional drawings or three-dimensional models of the molecules.
If the answer to this question is affirmative then is said to be reducible to. The study of reducibility notions is motivated by the study of decision problems. For many notions of reducibility, if any noncomputable set is reducible to a set then must also be noncomputable. This gives a powerful technique for proving that many sets are ...
Now Maschke's theorem says that any finite-dimensional representation of a finite group is a direct sum of simple representations (provided the characteristic of the base field does not divide the order of the group). So in the case of finite groups with this condition, every finite-dimensional representation is semi-simple.