Proper and improper

Analytical Method Committee, Uses (proper and improper) of correlation coefficient, Analyst, 1988, 113, 1469-1471. 

Fig. 5.14 Presentation of the proper and improper way of bending a reed switch. (Supporting the switch lead while bending is a must.)...

Figure 6.54 Diagram illustrating several types of order parameters involved in proper and improper ferroics. (After Newnham Cross, 1981.)...

A hexagonal representation of proper and improper primary ferroics as proposed by Newnham Cross (1981) is given in Fig. 6.54. The order parameter for proper ferroics appears on the diagonals of the hexagon, while the sides of the hexagon represent improper ferroics. They indicate the cross-coupled origin of ferroic phenomena. An improper primary ferroic in this classification is distinguished from a true secondary... 

Uses (Proper and Improper) of Correlation Coefficients , Analyst London, 1988, 113, 1469. 

All of the matrices we have just worked out, as well as all others which describe the transformations of a set of orthogonal coordinates by proper and improper rotations, are called orthogonal matrices. They have the convenient property that their inverses are obtained merely by transposing rows and columns. Thus, for example, the inverse of the matrix... 

It follows from Exercise 2.1-3(a) and Example 2.1-1 that the only necessary point symmetry operations are proper and improper rotations. Nevertheless, it is usually convenient to make use of reflections as well. However, if one can prove some result for R and IR, it will hold for all point symmetry operators. 

For crystals, the point group must be compatible with translational symmetry, and this requirement limits n to 2,3,4, or 6. (This restriction applies to both proper and improper axes.) Thus the crystallographic point groups are restricted to ten proper point groups and a total of... 

This is a most useful result since we often need to calculate the inverse of a 3 x3 MR of a symmetry operator R. Equation (10) shows that when T(R) is real, I R)-1 is just the transpose of T(R). A matrix with this property is an orthogonal matrix. In configuration space the basis and the components of vectors are real, so that proper and improper rotations which leave all lengths and angles invariant are therefore represented by 3x3 real orthogonal matrices. Proper and improper rotations in configuration space may be distinguished by det T(R),... 

The group 0(3) comprises all proper and improper rotations in configuration space 9t3. It is... 

A quantity T that is invariant under all proper and improper rotations (that is, under all orthogonal transformations) so that T = T, is a scalar, or tensor of rank 0, written 7(0). If T is invariant under proper rotations but changes sign on inversion, then it is a pseudoscalar. 

Students will observe and evaluate the proper and improper ways to handle chemical materials used in art projects. 

A. The teacher should demonstrate the proper and improper use of chemicals in photo development ... 

In order to simplify the discussion, a benzenoid or coronoid system is to be drawn so that a pair of bonds of each hexagon lie in parallel with the vertical line. Let the sets of the circularly arranged three double bonds as shown below in a given Kekule pattern be called, respectively, proper and improper sextets. 

For a given benzenoid or coronoid system one can draw the set of all the K(G) Kekule patterns as exemplified in Fig. 1 for benzanthracene, XVI. The pair of patterns kj and k2 differ only in the arrangement of proper and improper sextets in a certain hexagon. According to Clar one may draw a circle representing an aromatic sextet in that hexagon as below. The sextet pattern... 

Using Fractions to Represent Real Situations Proper and Improper Fractions Like and Unlike Fractions Comparing Fractions Equivalent Fractions Simplifying Fractions... 

A benzenoid of hexagonal symmetry belongs to one of the symmetry groups D6h and C6h. For obvious reasons these systems are called snowflakes. Sometimes it is distinguished between the D6h and C6h groups by means of the terms proper and improper snowflakes, respectively. The proper snowflakes are also said to have regular hexagonal symmetry. Snowflakes, both of D6h and C6h, occur for... 

Table 31 shows the numbers of snowflakes, which supplement the few, small numbers found in Table 26. Tables 32 and 33 show the numbers of proper and improper snowflakes, respectively, classified into the neo categories. The data were...

All-benzenoids with hexagonal (D6h or C6h) symmetry have been referred to as all-flakes [129]. In other words, an all-flake is an all-benzenoid snowflake. We may also speak about proper (D6h) and improper (C6h) all-flakes as subclasses of the proper and improper snowflakes, respectively. 

A snowflake in the present sense is a benzenoid of hexagonal symmetry [8, 57, 58], belonging to D6h or C6h. Sometimes the symmetries D6h and C6h, are associated to proper- and improper snowflakes [59], respectively. 

The operators of discrete rotational groups, best described in terms of both proper and improper symmetry axes, have the special property that they leave one point in space unmoved hence the term point group. Proper rotations, like translation, do not affect the internal symmetry of an asymmetric motif on which they operate and are referred to as operators of the first kind. The three-dimensional operators of improper rotation are often subdivided into inversion axes, mirror planes and centres of symmetry. These operators of the second kind have the distinctive property of inverting the handedness of an asymmetric unit. This means that the equivalent units of the resulting composite object, called left and right, cannot be brought into coincidence by symmetry operations of the first kind. This inherent handedness is called chirality. 

Analytical Methods Committee. Uses (proper and improper)... 

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