Relation equivalence

The unit of A. in the table is cm equiv h The entities to which the equivalent relates are given in the first column. 

Use of Operating Curve Frequently, it is not possible to assume that = 0 as in Example 2, owing to diffusional resistance in the liquid phase or to the accumulation of solute in the hquid stream. When the back pressure cannot be neglected, it is necessary to supplement the equations with a material balance representing the operating line or curve. In view of the countercurrent flows into and from the differential section of packing shown in Fig. 14-3, a steady-state material balance leads to the fohowing equivalent relations ... 

Let M, V be two vertices of a tree. We say they are similar if there is an automorphism of the tree which maps u onto v. This relation of similarity is an equivalence relation and partitions the p vertices of the tree into equivalence classes. Let p be the number of equivalence classes. Similarly we say that two edges of the tree are similar if there is an automorphism which maps one onto the other. Let q be the number of equivalence classes of edges under this relation. A symmetric edge in a tree is an edge, uv say, such that there is an automorphism of the tree which interchanges u and v. Let s be the number of symmetric edges in a tree it is easy to see that s can only be 0 or 1. We then have the following theorem. 

A final example is the concept of QM state. It is often stated that the wave function must be square integrable because the modulus square of the wave function is a probability distribution. States in QM are rays in Hilbert space, which are equivalence classes of wave functions. The equivalence relation between two wave functions is that one wave function is equal to the other multiplied by a complex number. The space of QM states is then a projective space, which by an infinite stereographic projection is isomorphic to a sphere in Hilbert space with any radius, conventionally chosen as one. Hence states can be identified with normalized wave functions as representatives from each equivalence class. This fact is important for the probability interpretation, but it is not a consequence of the probability interpretation. 

Let us briefly discuss the theoretical results providing the basis for the improved efficiency of branch-and-bound algorithms. Let F = [x g(.x) lower-bound test. Then, the set L, defined by L =Fr X%, contains all the partial solutions, which can be terminated only by an equivalence relation. Recall that, by definition, no node in X% can be terminated by a dominance rule. 

Two solutions can have different tails which lead to solutions with equal objective function values but which cannot produce equivalence relations because they have the same ancestor. 

These equivalence relations are not the sane. Obviously strong equivalence implies recursive equivalence and recursive equivalence implies finite equivalence. None of the reverse inclusions hold. We are already in a position to show that finite equivalence does not imply recursive equivalence. Consider the scheme of P of Example II-3 which halts for all finite interpretations but diverges for some infinite hit recursive interpretations (for the interpretation of P we saw that caused divergence is obviously recursive). We can diagram P as ... 

DEFINITION A relationship v between schemes is a reasonable equivalence relation if P Q always implies that P is weakly equivalent to Q and if whenever P and Q are strongly equivalent then P Q. ... 

COROLLARY 6.11 There is no exhaustive simplification procedure for any reasonable equivalence relation. 

Homochirality is an equivalence relation only when the ligands of all considered configurations of a skeletal class are ordered sequentially and is stated on the basis of this order. 

The MMI (mass moment of inertia), EXC (excitation factor), and ZPE (zero point energy) terms are defined on successive lines of Equation 4.145. For reactions involving heavier isotopes the effects are no longer concentrated in the ZPE term and it is convenient to apply the Teller-Redlich product rule (Section 3.5.1) and eliminate the moments of inertia by using Equations 4.79,4.79a, and 4.141, thus obtaining an equivalent relation... 

The underlying space ofVjlG is the set of closed G -orbits modulo the equivalence relation defined by x y if and only if (3.2) holds. 

From the relations given in the previous section a vast number of equivalent relations can be obtained, each enlightening a different aspect of the problem. Here, as a conclusion, we wish to discuss how the many-body effects are averaged into the 1-RDM. 

Integrating the system of partial differential equations under study within the equivalence relations above, we obtain a set of ansatzes containing those equivalent to the Poincare-invariant ansatzes obtained in the previous section. That is why we concentrate on essentially new (non-Lie) ansatzes. It so happens that our approach gives rise to non-Lie ansatzes, provided the functions (x), 0 ( ) within the equivalence relations (89) have the form... 

A note on terminology operations that survive the equivalence are sometimes called well defined on equivalence classes. A function on the original set S taking the same value on every element of an equivalence class is called an invariant of the equivalence relation. We will see an example of an invariant of an equivalence class in our introduction to tensor products in Section 2.6. 

Exercise 1.19 Define an arrow in R to be an ordered pair (p, pf), where Pi and p2 are each a triple of real numbers. (Think of pi as the initial point and p as the endpoint.) Define a relation on the set of arrows by (/ b pf) (qt q .) if and only if P2 — Pi = qi qi- Show that this is an equivalence relation. Now think of each arrow as a point in R. Does the usual addition in R survive the equivalence relation If so, is the resulting addition on equivalence classes of arrows the same as the addition of 3-vectors you learned in linear algebra What about scalar multiplication in R . Find an injective and surjective linear function from R / to R. (Hint it will help to introduce some notation for (r, r2, r, r4, r, r(,) e R in the equivalence class corresponding to (si, 52, 53) e R. )... 

Exercise 1.24 Find another example of a meaningful equivalence relation from your own experience. Define the relation rigorously and prove that it is an equivalence relation. Which relevant operations survive for equivalence classes ... 

Recall from Section 1.7 that the standard mathematical way to deal with irrelevant ambiguity is to define an equivalence relation and work with equivalence classes. In this case of tensors, the irrelevant ambiguity arises from the different ways of writing the same object as a linear combination of products. We will use this insight to define tensor products. Suppose V and W are complex vector spaces. Consider the complex vector space V W generated by the set... 

Note that, Z(z) e IV. We call 7( ) the coefficient afv in z. Since ui,..., U2 is a basis, 7(2) is well defined as a function of z. Notice that each of the computation rules defining our equivalence relation leaves the coefficient of vi unchanged for example, we have a(u, w) iav, w), and while making this substitution changes the computation of the Cj s and bj s, it leaves the products Cjbj unchanged. The reader should check the other computation rules. Thus if zi 22, the coefficient of ui in "i must equal the coefficient of Ui in Z2. Similarly, we can define the coefficient of Vi in z for any i from 1... 

We make tliis idea precise by defining an equivalence relation (see Sec-... 

Proof. [Sketch] We leave it to the reader to check the first two criteria of Definition 3.2. As for Criterion 3, positive definiteness follows directly from the definition of the integral, while nondegeneracy can be deduced from the theory of Lebesgue integration, using the first equivalence relation defined in Section 3.1. The interested reader can work out the details in Exercise 3.9 or consult Rudin [Ru74, Theorem 1.39]. ... 

Show that is an equivalence relation. If the action cr is clear from the context, then the quotient space S/ is often denoted S/G. 

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