Binary operator

Other binary operations like the -operator (bitwise inclusive OR), and the right and left shift, and [Pg.569]

The sot R, together with the binary operations addition and multiplication, is a ring TZ = ( R,o,+) if it satisfies the following postulates ... 

For example, consider trying to prove that one solution, x, will dominate another solution, y, if they both have scheduled the same batches, but the end-times of each machine are earlier in the partial schedule (partial solution), X, than in y. The general theory will not be couched in these terms, but more abstractly, in terms of the properties of binary operators. 

In mathematical terms a field F is defined as a set with elements (a, 6, c,...) for which two binary operations are defined and which satisfies the following conditions ... 

In order to frame the QCT method within the general expression (3), we need to give an expression for the weight operator u ci. From the above discussion, it is clear that this operator will depend on r and that it is a binary operator, so its value is either 0 or 1. For a given atom A, the operator vanishes at every point in space, except within the basin of A, where it is equal to one. This way all atomic basins are indeed mutually exclusive. 

The binary operation ( multiplication ) in the Lie algebra is that of taking the commutator. As usual, we denote the commutator by square brackets, [A, fi] = AB- BA. A set of operators A is a Lie algebra when it is closed under commutation. That is, for every operator X in the algebra G (which we write as X e G)... 

The notion of a group is a natural mathematical abstraction of physical symmetry. Because quantum mechanical state spaces are linear, symmetries in quantum mechanics have the additional structure of group representations. Formally, a group is a set with a binary operation that satisfies certain criteria, and a representation is a natural function from a group to a set of linear operators. 

Consider a set of elements Q together with a binary operation represented by o, such that two elements of the set can be combined to form a new quantity... 

One way of characterizing a group is by its multiplication table. Consider, for example, the set E,A,B,C,D,F and a binary operation whose results are represented by Table 1.1. 

The averages of additive and binary operators, respectively, read... 

For every pair of elements A and B, there exists a binary operation that yields the product AB belonging to the set. 

The number of two-polymer, multipolymer, and multimonomer systems reported in the scientific and patent literature continues to rise without an adequate nomenclature to describe the several materials. This chapter is divided into three parts. (1) A proposed nomenclature system which uses a short list of elements (polymers or polymer reaction products). These elements are reacted together in specific ways by binary operations which join the two polymers to form blends, grafts, blocks, crosslinked systems, or more complex combinations. (2) The relationship between the proposed nomenclature and the mathematics of ring theory (a form of the new math9 ) is discussed. (3) A few experimental examples now in the literature are mentioned to show how the new nomenclature scheme already has been used to discover new multipolymer systems. 

The following nomenclature scheme uses a short list of elements in Table I (polymers and polymer-reaction products) which are reacted together in specific ways by binary operations (a joining of two elements), Table II. A series of numerical subscripts are used on the elements to allow an arbitrarily large number of different polymers to be designated conveniently. 

Reaction Examples. The binary operation symbol used between any two elements reacts them together in the required manner. (The reader is reminded that all combinations, mathematical or chemical, are binary operations. The advent of modern computers has focused much attention on this often neglected fact.) The first formed element appears on the left and the reaction-time sequence (when required) proceeds from left to right. 

The above examples use the abstract element symbols Pi, Gi2, etc. Briefly let us show how this system will operate with real monomers, polymers, and multipolymer combinations, using chemical notation instead of the elements but retaining the binary operation notation. 

Tables. The value of the proposed system lies principally in its capability of depicting very complex multipolymer combinations. These tables, unlimited in size, join combinations of elements systematically. Tables IV and V provide examples. Each binary operation has a table and all of the elements can appear in any table. The rows are reacted with the columns in that order. For example, in Table V, Pi at the left of the row is reacted with P2 at the top of the column to synthesize Gi2.

These elements are combined by any of the binary operations shown in Table II. While Om forms the addition mode, all others constitute modes of multiplication. Simple examples of modes of combination are shown in Figure 1. 

Thus the series of rings will be shown to each contain the Om binary operation, combined with a different one of the multiplicative binary operations. For brevity, the combinations generated by Og will be examined in detail. The extension to the other tables is obvious in many cases. 

Six Requirements of a Ring. The following discussion is based on establishing that Laws One through Six of McCoy (30, p. 23) hold for the binary operations Om and Og. A small finite segment of the addition and multiplication tables generated by Om and Og are shown in Table IV and V, respectively. 

Law Three There exists an element 0 of R (set of all elements) such that a + 0 = a for every element of R (existence of a zero). Under the binary operation Om, it means that nothing was added. Pi Om 0 = P ... 

The above discussion shows that with (i) a zero, (ii) inverse additive elements, and (iii) coefficients, the set defined by R and the binary operations Om and Og form a ring. According to Laws Five and Six, only one chemical isomer is permitted. However the legal isomer element is either isomorphic or homomorphic with the other chemical possibilities, which may be formed into alternate tables according to simple chemical rules. 

Rings Involving Oc and Ob Operations. Let us now examine the tables generated by the Oc and Ob binary operations. Each element in these two tables is isomorphic, i.e., has a bijection, with its corresponding element in the 0G-generated table. Laws One, Two, Three, and Four apply only to the 0M-generated table. The applicability of Laws Five and Six needs to be established. 

Laws Two, Three, and Four form the basis for a group, McCoy (30, p. 135). Since all three apply to the elements under the binary operation Om, it follows that the polymer blends constitute an additive group. In fact, Om generates a free abelian group on (Pi, P2, P3,. . . , Pn) since its elements (plus inverses and zero) are commutative under Om ... 

Because Law One is followed (P2 Om Pi = Pi Om P2), a corollary can be written the elements of the table generated by 0M-binary operations are symmetric about the diagonal. 

Summary of Ring Notation. In summary, the addition of a zero, additive inverses, and coefficients allows for Laws One through Six to be followed. The polymer blend, graft, and IPN nomenclature scheme forms three rings, and the fact that the binary operation Om on R = (Pi, P2, P3,. . . , Pn) constitutes a group puts the system on an improved mathematical ground. Similar relationships can be developed for the other operations. 

In the proceeding, the notions of elements and binary operations were developed. An important part of the mathematics of ring theory involves the concept of functions. 

A group in which the binary operation is commutative, that is, ab=ba for all elements a abd b in the group, abscissa... 

A binary operation is an operation that involves two operands. For example, addition and subtraction are binary operations, bijection... 

A mathematical system consisting of elements from a set G and a binary operation such that... 

A special case of cell-based methods is a diversity measure proposed for binary fingerprints. Unlike continuous descriptors, binary descriptors such as structural keys and hashed fingerprints can be compared using fast binary operations to give rapid estimates of molecular similarity, diversity, and complementarity. The most common example of a diversity measure applied to binary descriptors is the binary union (inclusive or ). This can be exploited in a number of different ways elegant examples can be found in the following references. ... 

Daily clients demands - The forecast of the demands is provided in a daily basis, which is also implemented in the model formulation. For this purpose, it is built a binary operator that transforms discrete time information in to continuous time. 

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