Equivalent object

The authority havingjurisdiction may waive specific requirements in this Code or permit alternate methods where it is assured that equivalent objectives can be achieved by establishing and maintaining effective safety. 

Equivalence" is also used as a synonym for "equality with respect to"- This phrase implies that two such equivalent objects are not equal in 1 rejects, but equal with respect to a property of interest-For example, the well known aromaticity 4n+2 rule classifies even-polycyclic conjugated benzenoids into two classes- Under such a relation molecules such as naphthalene and anthracene are considered equivalent-... 

A pictorial illustration of equivalent objects is depicted below where the individual elements occupy the edges of a complete graph ... 

Another classification is based on the presence or absence of translation in a symmetry element or operation. Symmetry elements containing a translational component, such as a simple translation, screw axis or glide plane, produce infinite numbers of symmetrically equivalent objects, and therefore, these are called infinite symmetry elements. For example, the lattice is infinite because of the presence of translations. All other symmetry elements that do not contain translations always produce a finite number of objects and they are called finite symmetry elements. Center of inversion, mirror plane, rotation and roto-inversion axes are all finite symmetry elements. Finite symmetry elements and operations are used to describe the symmetry of finite objects, e.g. molecules, clusters, polyhedra, crystal forms, unit cell shape, and any non-crystallographic finite objects, for example, the human body. Both finite and infinite symmetry elements are necessary to describe the symmetry of infinite or continuous structures, such as a crystal structure, two-dimensional wall patterns, and others. We will begin the detailed analysis of crystallographic symmetry from simpler finite symmetry elements, followed by the consideration of more complex infinite symmetry elements. 

Except for the center of inversion, which results in two objects, and three-fold inversion axis, which produces six symmetrically equivalent objects. See section 1.20.4 for an algebraic definition of the order of a symmetry element. 

The mirror plane (two-fold inversion axis) reflects a clear pyramid in a plane to yield the shaded pyramid and vice versa, as shown in Figure 1.12 on the right. The equivalent symmetry element, i.e. the two-fold inversion axis, rotates an object by 180" as shown by the dotted image of a pyramid with its apex down in Figure 1.12, right, but the simultaneous inversion through the point from this intermediate position results in the shaded pyramid. The mirror plane is used to describe this operation rather than the two-fold inversion axis because of its simplicity and a better graphical representation of the reflection operation versus the roto-inversion. The mirror plane also results in two symmetrically equivalent objects. 

It is easy to see that the six symmetrically equivalent objects are related to one another by both the simple three-fold rotation axis and the center of inversion. Hence, the three-fold inversion axis is not only the result of two simultaneous operations (3 and 1), Iwt it is also the result of two independent operations. In other words, 3 is identical to 3 then 1. 

The four-fold rotation axis Figure 1.14, left) results in four symmetrically equivalent objects by rotating the original object around the axis by 90°, 180°, 270° and 360°. 

The four-fold inversion axis Figure 1.14, right) also produces four symmetrically equivalent objects. The original object, e.g. any of the two clear p)ramids with apex up, is rotated by 90° in any direction and then it is immediately inverted from this intermediate position through the center of inversion. This transformation results in a shaded pyramid with its apex down in the position next to the original pyramid but in the direction opposite to the direction of rotation. By applying the same transformation to this shaded pyramid, the third symmetrically equivalent object would be a clear pyramid next to the shaded pyramid in the direction opposite to the direction of rotation. The fourth object is obtained in the same fashion. Unlike in the case of the three-fold inversion axis (see above), this combination of four objects cannot be produced by appl)dng the four-fold rotation axis and the center of inversion separately, and therefore, this is a unique symmetry element. As can be seen from Figure 1.14, both four-fold axes also contain a two-fold rotation axis (180° rotations) as a sub-element. 

The six-fold inversion axis Figure 1.15, right) also produces six symmetrically equivalent objects. Similar to the three-fold inversion axis, this symmetry element can be represented by two independent simple symmetry elements the first one is the three-fold rotation axis, which connects pyramids 1-3-5 and 2-4-6, and the second one is the mirror plane perpendicular to the three-fold rotation axis, which connects pyramids 1-4, 2-5, and 3-6. As an exercise, try to obtain all six symmetrically equivalent pyramids starting from the pyramid 1 as the original object by applying 60° rotations followed by immediate inversions. Keep in mind that objects are not retained in the intermediate positions because the six-fold rotation and inversion act simultaneously. 

The mirror plane is, therefore, a derivative of the two-fold rotation axis and the center of inversion located on the axis. The derivative mirror plane is perpendicular to the axis and intersects the axis in a way that the center of inversion also belongs to the plane. If we start from the same pyramid A and apply the center of inversion first (this results in pyramid D) and the twofold axis second (i.e. A -> B and D C), the resulting combination of four symmetrically equivalent objects and the derivative mirror plane remain the same. 

This example not only explains how the two symmetry elements interact, but it also serves as an illustration to a broader conclusion deduced above any two symmetry operations applied in sequence to the same object create a third symmetry operation, which applies to all symmetrically equivalent objects. Note, that if the second operation is the inverse of the first, then the resulting third operation is unity (the one-fold rotation axis, 1). For example, when a mirror plane, a center of inversion, or a two-fold rotation axis are applied twice, all result in a one-fold rotation axis. 

As established above, the interaction between a pair of symmetry elements (or symmetry operations) results in another symmetry element (or symmetry operation). The former may be new or already present within a given combination of symmetrically equivalent objects. If no new symmetry element(s) appear, and when interactions between all pairs of the existing ones are examined, the generation of all symmetry elements is completed. The complete set of symmetry elements is called a symmetry group. 

Because of this, for example, glide plane, a, can be perpendicular to either b or c, but it cannot be perpendicular to a. Similarly, glide plane, b, cannot be perpendicular to b, and glide plane, c, cannot be perpendicular to c. Since the translation is always by 1/2 of the corresponding basis vector, these planes produce two symmetrically equivalent objects within one full length of the corresponding basis vector (and within one unit cell), i.e. their order is 2. 

Figure 1.29 illustrates how the two-fold screw axis generates an infinite number of symmetrically equivalent objects via rotations by 180° around the axis with the simultaneous translations along the axis by 1/2 of the length of the basis vector to which the axis is parallel. 

In the chapter of Briiggemann and Carlsen some concepts introduced in the chapter of El-Basil are revitalized and explained in the context of the multivariate aspect. Basic concepts, like chain, anti-chain, hierarchies, levels, etc., as well as more sophisticated ones, like sensitivity studies, dimension theory, linear extensions and some basic elements of probability concepts are at the heart of this chapter. The difficult problem of equivalent objects, which lead to the items object sets vs. quotient sets are explained and exemplified. 

Equivalent objects in Hasse diagrams Different objects that have the same data with respect to a given set of attributes. Equality with respect to a given set of attributes defines an equivalence relation, 91 . Objects having the same values of all their attributes form disjoint subsets of A, the equivalence classes. An equivalence class with only one object is called a singleton and is called trivial. The equivalence classes can be considered as elements of a set, the quotient set El9E Usually the partial order is based on the quotient set and -if necessary- the equivalent elements are associated with that vertex, where a representative element out of the equivalence... 

Fig. 10. Example, how to assign the levels. If one vertex contains several equivalent objects, than these objects belong all to the same level. The vertical arrow symbolizes the order induced by the vertical arrangement of the vertices...

Fig. 8. Complete order sets for the inorganic (A), the organic compounds (C), and a synopsis of both (B), achieved by the calculation of averaged ranks. Note that there are equivalent objects (see text)...

All three diagrams also show some similarities. They all have 4 levels and no equivalent objects. The number of comparabilities decreases with the number of attributes. For the number of incomparabilities the situation is the other way round. In other words The more attributes the data-matrix encompasses, the less comparabilities are found. 

All VE applications are founded on the generation, perception, and manipulation of naturalistic or abstract virtual worlds without any physical equivalent Objects existing within virtual worlds can possess various qualities and behaviors. Examples are graphics, sound, and force feedback. By multiple addressing of the human senses, the attempt is made to generate the greatest possible intuitiveness of virtual environments VE can be experienced through visualization, meuked out by 3D object representations and real-time-orientated interaction modes. 

In this problem, T is the control (u). The equivalent objective is to minimize the functional... 

Reactor CGU of HDS is used as a case study to illustrate the fault propagation model. The simplified P ID of the reactor CGU is shown in figure 3-21. The equivalent object-oriented model representation of the... 

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