Inequality. Symbols

Linear ineqmlities are solved in much the same way as simple equations. The most important difference is that when an inequality is multiplied or divided by a negative number, the inequality symbol changes direction. 

Because you are dividing by a negative number, the inequality symbol changes direction ... 

Divide each term by -5, changing the direction of both inequality symbols ... 

In Eq. (56) the inequality symbol refers to the case, when a contribution of one of subensembles is negligible the equality means that these contributions are commensurable. 

Fig. 10.5 Comparison of the mutagenic potency of nitro-aromatic compounds in the naphthalenic and benzylic series as the molecular dimensionality increases. Inequality symbols indicate the relative mutagenic potency of each compound. Figure adapted from [54]...

The brackets symbolize fiinction of, not multiplication.) Smce there are only two parameters, and a, in this expression, the homogeneity assumption means that all four exponents a, p, y and S must be fiinctions of these two hence the inequalities in section A2.5.4.5(e) must be equalities. Equations for the various other thennodynamic quantities, in particular the singidar part of the heat capacity Cy and the isothemial compressibility Kp may be derived from this equation for p. The behaviour of these quantities as tire critical point is approached can be satisfied only if... 

Here the less than or equal to symbol is used to indicate that the boundary values themselves are included in the set of possible values these variables may assume. If the boundary values are not included, less than symbols (<) are used. The presence of the additional equal to symbol in Equations 2.6 and 2.7 does not prevent the constraints from being inequalities. 

It is not difficult to show that the inequality fi < fl, which should be met for the sudden approximation to hold, is equivalent to (2.86) if we introduce the angle 2(p between the reactant and product valleys where tan tp = Cl2/C. The borders of the regions of validity of the sudden and adiabatic approximations in the (C, 0) plane are symbolically drawn in Figure 4.7. The only physically sensible parameters are those for which... 

Rational numbers are numbers that can be written as fractions (and decimals and repeating decimals). Similarly, rational equations are equations in fraction form. Rational inequalities are also in fraction form and use the symbols <,>,<, and > instead of =. 

These small-particle formulae have only symbolic validity in the zero-salt Km, nm - 0 limit. Why Because even though icm and nm need not appear explicitly, the results have been derived under the condition TV Fs [Pg.228]

The symbol ([A,B]) is just (T [A,B] T, a quantum-mechanical average over the quantum state Tq. The inequality is known as Heisenberg-Kennard-Robertson relationship, which has often been interpreted as the mathematical expression of the disturbance following measurement. Ballentine noted that this relationship does not seem to have any bearing on the issue of joint measurement instead, this relation can be traced back to the preparation process of an initial state, see Ref. [1,10]. 

The commodification of traditional knowledge poses problems that also transcend intellectual property considerations since, in indigenous communities, medicinal plants can have cultural, symbolic and ritual values that go beyond a simple medicinal or economic use. Thus, the cultivation of a plant outside its natural habitat and the capture of its medicinal properties into a commercial product can generate mistrust, inequality and betrayal because the loss of the cultural value is not addressed by any monetary compensation. These problems have been exemplified by the development of Hoodia gordo-nii,46 a sacred life force of the South African San, which was turned into a commercial slimming aid .47... 

The symbols are P for profit, / for equality constraints, g for inequality constraints, x for optimization variables, y for dependent variables, and / (constant) for updated parameters. The objective function is a scalar measure of plant profit it is usually the instantaneous profit ( /hr), because the optimization variables do not involve the time value of money. Typical equality constraints include material and energy balances, heat and mass transfer relationships, and thermodynamic and kinetic models, and typical inequality constraints include equipment limitations limit compressor horsepower, and distillation tray hydraulics. The optimization variables are flow rates, pressures, temperatures, and other variables that can be manipulated directly. The dependent variables involve intermediate values required for the detailed models for example, all distillation tray compositions, flow rates, and temperatures. Because of the fundamental models often used in RTO, the number of dependent variables can be quite large, on the order of hundreds of thousands. 

Inequalities contain the greater than, less than, greater than or equal to, or less than or equal to symbols. When you solve the inequalities for x, you are figuring out a range of numbers that your unknown... 

In eq 2.133 the superscript t refers to the total value of each property. The inequality of eq 2.133 applies for infinitely small changes between non-equilibrium states and the equality symbol holds for infinite small changes between equilibrium states, that is, a reversible processes. 

The inequality (A6.5) must apply to an arbitrary vector h = (h, hj > such, that IAII + 0. In practical cases, the symbol d, - is usually used instead of A . The relation (A6.5) is equivalent to the well-known Sylvester conditions, i.e. that the main minors of the matrix are positive. From relations (A6.1) and (A6.2) follows... 

If we further assume that there is only one dominant mode of coupling, the symbol Ijgj can be replaced by L gm, where is an effective degeneracy factor, and the subscript m denotes the effective phonon mode responsible for the radiationless process. Of the modes responsible for the decay, the highest energy optical phonon modes can be expected to make the greatest contribution. We further note that, for radiationless transitions in rare earth ions in crystals, the inequality... 

Some menu commands are instruments used to solve the problems. Thus, we can solve an analytical equation or inequality by the command Symbolic Variable Solve if we previously type the expression in a worksheet and mark the variable to... 

One method of verifying these conditions is to use the following procedure the technical details are left to the interested reader. Firstly, expand the integrands in (3.305) and (3.306) in terms of (3 and then integrate to first order in j3. This gives two equations in the two variables and 6. Secondly, solve these simultaneous equations to find the solutions for and b as functions of to first order. The desired results in equations (3.317) to (3.320) follow directly. This process can be carried out relatively swiftly using a symbolic computer package. Consequently, if either of the inequalities in (3.315) holds and also, from (3.317), if... 

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