Equivalent definition

From tlie definition of conditional probability in Eq. (19.5.1), one can deduce tlie logically equivalent definition diat event A and event B are independent if and only if... 

If we multiply AA = I from the left by A, then we have the equivalent definition... 

The variable s is a dummy variable in the sense that it does not enter die final result. Thus, if the exponential function in Eq. (94) is expanded in a power series in s, the coefficients of successive powers of s are just the Hermite polynomials divided by u . It is not too difficult to show that Eqs. (93) and (94) are equivalent definitions of the Hermite polynomials. 

We shall give two equivalent definitions of a block structured scheme and demonstrate their equivalence. Then we shall describe and illustrate a procedure for converting a scheme (or program) into block structure. The first definition is really a restriction on graphs. We shall need some purely graph oriented... 

In the statistical theory of fluid mixing presented in Chapter 3, well macromixed corresponds to the condition that the scalar means () are independent of position, and well micromixed corresponds to the condition that the scalar variances are null. An equivalent definition can be developed from the residence time distribution discussed below. 

A second, equivalent, definition is y is greater than yW if and only if ytp) Can be obtained from y by moving boxes from the plus to the minus part, and/or downward within each part. 

This chemical cross-reference has been compiled by extracting information from chemistry textbooks, industrial references, and chemical formularies back to the early 1800 s. Equivalents, definitions, and cross-reference entries were loaded into our computer, sorted into alphabetical order, and then merged into one master cross-reference which follows. 

In many texts an alternative (but equivalent) definition is given for an action ... 

The hyper-Kahler structure is a quaternionic version of the Kahler structure. However, there is no good definition of the integrability (i.e. the existence of local charts) for the almost hyper-complex structure. Hence we generalize the second equivalent definition explained in above. 

For some purposes, an alternative but equivalent definition of x is preferable. In general, the energy of the substance is altered by the presence of the field. If, for unit volume, the energy is W, we write... 

Equations (4)—(7), yield four equivalent definitions for the chemical potential of component i ... 

In deterministic theory we started with the definition of a compartment as a kinetically homogeneous amount of material. The equivalent definition in stochastic theory is that the probability of a unit participating in a particular transfer out of a compartment, at any time, is the same for all units in the compartment. 

The significance of this finding is in establishing the Lanczos dual representation fn), Qn(u) that enables the following equivalent definitions ... 

The pericondensed extreme-left benzenoids constitute a subclass of the strictly pericondensed benzenoids in the sense of Dias [12, 15, 19, 21, 55-57] they are defined by having all their internal vertices connected and no catacondensed appendages. Phenalene, C13H9, which has only one internal vertex, is reckoned among the strictly pericondensed benzenoids. An equivalent definition in a most succint form reads ... 

When four base units are used, they are taken to be the cm, g and s for length, mass and time, and the franklin1 (symbol Fr) for the esu of charge, 1 Fr being chosen to be of such a magnitude that e0(ir) = 1 Fr2/erg cm. An equivalent definition of the franklin is that two charges of 1 Fr, 1 cm apart in a vacuum, repel each other with a force of one dyne. Other units are then derived from these four by the usual rules for constructing a coherent set of units from a set of base units. 

When described in terms of four base units, they are taken as the cm, g, s, and the unit of electric current, which we call the (emu of current). This is chosen to be of such a magnitude that p 0(ir) = 1 cm g s 2 (emu of current)" 2. An equivalent definition of the emu of current is that the force between two parallel wires, 1 cm apart in a vacuum, each carrying 1 emu of current, is 2 dyn per cm of wire. Comparison with the definition of the ampere then shows that 1 (emu of current) = 10 A. Other units are derived from these four by the usual rules.2... 

There is of course a more familiar equivalent definition where mult is the only map mentioned as such. To simplify what follows, we have built the existence assertions into the structure, so that the only axioms needed are equations (commutative diagrams). 

Note that the number of millimoles per milliliter is equal to the number of moles per liter, and thus millimoles per milliliter is another way to define molarity. This equivalent definition makes many problems easier because quantities used in chemical laboratories are often measured in millimoles and milliliters. 

This is the definition of number average molecular weight M . Equivalent definitions follow from simple arithmetic. Since ... 

In engineering calculations the formation volume factor /3 is most commonly used to express the change in liquid volume with pressure. / is defined by equation 4 or by an equivalent definition as the volume... 

Stoichiometric equation (viii), the equivalent definitions of the reaction rate R have the form... 

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