Ordered pairs

For each component in a system, there is a distribution like Figure 2.7-2. Each bur may represent an ordered pair (Xj, P) where x is the value of the variable and P, is the probability of the... 

To generalize, let x, represent the nth ab.scis.sa of the first distribution and Pj the corresponding ordinate of a normalized distribution. Then c is formed of the ordered pairs (equation 2.7-.30). As an example, for x,... 

The present chapter is organized as follows. We focus first on a simple model of a nonuniform associating fluid with spherically symmetric associative forces between species. This model serves us to demonstrate the application of so-called first-order (singlet) and second-order (pair) integral equations for the density profile. Some examples of the solution of these equations for associating fluids in contact with structureless and crystalline solid surfaces are presented. Then we discuss one version of the density functional theory for a model of associating hard spheres. All aforementioned issues are discussed in Sec. II. 

The sum over Cl coefficients is an interference factor resulting from the fact that the full Cl pair energies converge faster than the second-order pair energies. 

A function is a set of ordered elements such that no two ordered pairs have the same first element, denoted as (x,y) where x is the independent variable and y is the dependent variable. A function is established when a condition exists that determines y for each x, the condition usually being defined by an equation such as y = f(x) [2]. 

Properties D5 and D6 are sometimes referred to as the DcMorgan Laws. Finally, we mention an important operation called the cartesian (or direct) product of two sets, A X B, which is the set of ordered pairs of the form (a, 6), where a A and b B ... 

The fact that we are discussing an abstract space means that we know only that its elements (vectors) have the postulated properties e.g., that a scalar product exists, but at this level of the discussion we do not know the numerical value of the scalar product. We may choose at random some familiar collection of elements, perhaps the set of all ordered pairs of real numbers (n,m) or the set of all differentiable functions of position on a line, etc., and ask whether or not they form a Hilbert space. If they do, then we can in fact evaluate the scalar... 

Green and Pimblott (1991) criticize the truncated distributions of Mozumder (1971) and of Dodelet and Freeman (1975) used to calculate the free-ion yield in a multiple ion-pair case. In place of the truncated distribution used by the earlier authors, Green and Pimblott introduce the marginal distribution for all ordered pairs, which is statistically the correct one (see Sect. 9.3 for a description of this distribution). 

A (finite) directed graph or digraph consists of a finite set of vertices and a set of ordered pairs of vertices called arcs. We denote by VG and Eg the set of vertices and arcs of the digraph G, respectively. Given an ordering of the vertices, the adjacency matrix of a digraph G on n vertices, denoted by AG, is the (0, l)-matrix where the ij-th element... 

This differential equation is analytically solvable in functions, but here it is solved numerically. Represent equation by the first order pair,... 

Norris, A. T. and S. B. Pope (1991). Turbulent mixing model based on ordered pairing. 

The x-coordinate axis for a graph of Cartesian coordinates [x,y or [x, f(x)] or the x-value for any [x,y] ordered pair. This corresponds to the [Substrate Concentration]-axis in v versus [S] plots or the ll[Substrate Concentra-fton]-axis in so-called double-reciprocal or Lineweaver-Burk plots. 

A plot of the reciprocal of the initial velocity as a function of inhibitor concentrations at different fixed concentrations of substrate. For a competitive inhibitor, the lines passing through the data will intersect at a common point in the second quadrant (whose coordinates are the ordered pair (-Kis, llVmax)- Here, Kis is the dissociation constant for the competitive inhibitor and Rmax is the maximum velocity. 

For such a series of ordered pairs, the vertical displacement of a sample value, say Xj,y, from the straight line y = a + bxvs, yj - a - bxj, and the sum of these squares is given as ... 

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. )... 

Next we must verify that the Ust is long enough. If we measure the z-spins of both particles, we must find one of the six listed states. Also, none of these states have multiplicities because the spin states of the two individual particles have no multiplicities. Hence the set of six ordered pairs above is a basis for the quantum system consisting of one spin-1/2 and one spin-1 particle. 

In graph theory, graphs are defined by an ordered pair consisting of two sets, V and R ... 

For the function y = /(x), each ordered pair of numbers, (x,y), can be used to define the coordinates of a point in a plane, and thus can be represented by a graphical plot, in which the origin, O, with coordinates (0,0), lies at the intersection of two perpendicular axes. A number on the horizontal x-axis is known as the abscissa, and defines the x-coordinate of a point in the plane likewise, a number on the (vertical) y-axis is known as the ordinate, and defines the y-coordinate of the point. Thus, an arbitrary point (x,y) in the plane lies at a perpendicular distance x from the y-axis and y from the x-axis. If x > 0, the point lies to the right of the y-axis if x < 0, it lies to the left. Similarly, if y > 0, the point lies above the x-axis, and if y < 0, it lies below (see Figure 2.1). 

Quaternions are thus seen to form a 4-D real linear space ft 3ft3, comprising the real linear space 3ft (basis 1) and a 3-D real linear space 3ft3 with basis qi, q2, q3. An ordered pair representation can be established for q by defining... 

Coordinate geometry is a form of geometrical operations in relation to a coordinate plane. A coordinate plane is a grid of square boxes divided into four quadrants by both a horizontal (x) and vertical (y) axis. These two axes intersect at one coordinate point—(0,0)—the origin. A coordinate pair, also called an ordered pair, is a specific point on the coordinate plane with the first number representing the horizontal placement and second number representing the vertical. Coordinate points are given in the form of (x,y). 

To graph ordered pairs, follow these guidelines ... 

The x-coordinate is listed first in the ordered pair and tells you how many units to move either to the left or to the right. If the x-coordinate is positive, move to the right. If the x-coordinate is negative, move to the left. 

Notice that the graph is broken into four quadrants with one point plotted in each one. Here is a chart to indicate which quadrants contain which ordered pairs, based on their signs ... 

S. Salomonson, P. Oster, Relativistic all-order pair functions from a discretized singleparticle dirac Hamiltonian, Phys. Rev. A 40 (1989) 5548. 

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