By Mark Ladd ; foreword by Lord Lewis.

Topic class Code: X500 (NAL topic code)

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**Example text**

An operator 0 is linear if, for any functionJ Okf= k(OJ), where k is a constant, and if o(ri +h1 = of;+ Of, d wheref; a n d h are two functions. Evidently, -( ) is a linear operator but In( ), for dx example, is not. 4) the parenthetical expresion may be calculated first, if appropriate. The product of two linear operators follows the rule 010f=OI(O2J). 1. L e t O I = - ( br = (oioz)03. x’ - 3. Then, from the foregoing: (a) ~ f = i3x2 - 2; (b) 0 1 kfi = k(O& = k(3x2 - 2) = 6x2 - 4; cfi + f2 ) = of; + O h = (3x2 - 2) + (4x) = 3x2 + 4x - 2; (d) (01+ 03s = Of; + Oji = (3x2 - 2) + 6~ = 3x2 + 6~ - 2; (e) 010ji= 01 (02fi)= 01 (xs - 2x3 + x z ) = 5x4 - 6xz + 2x.

There is a one-to-one correspondence between the members of the groups: A ~Z ~I B ~Z2 ~2 C ~Z3 ~O The groups are also Abelian; this nature is revealed through the symmetry across the principal diagonals of the bodies of the group multiplication tables. We note also that no member of a group is repeated among any row or column within a group table. 7) The law of combination is vector addition, and the zero vector (nl = n: = n3 = 0) represents the identity operation; the negative signs on n, introduce the inverse members of the group.

1 The thirty-two crystallographic point groups and their subgroups; thin lines indicate the subgroups that are invariant. 5 Symmetry classes and conjugates Subgroups provide one method for separating the members of a group into smaller sets, each constituting a group. An alternative procedure introduces the topic of symmetry classes. 4, we introduced the similarity transformation; we now use this concept to discuss symmetry classes. 8) then B is the similarity transformation of A by R, and A and B are said to be conjugate to each other.