Mathematical statement

One feature of this inequality warrants special attention. In the previous paragraph it was shown that the precise measurement of A made possible when v is an eigenfiinction of A necessarily results in some uncertainty in a simultaneous measurement of B when the operators /land fido not conmuite. However, the mathematical statement of the uncertainty principle tells us that measurement of B is in fact completely uncertain one can say nothing at all about B apart from the fact that any and all values of B are equally probable A specific example is provided by associating A and B with the position and momentum of a particle moving along the v-axis. It is rather easy to demonstrate that [p, x]=- ih, so that If... 

The goal of approximate and numerical methods is to provide convenient techniques for obtaining useful information from mathematical formulations of physical problems. Often this mathematical statement is not solvable by analytical means. Or perhaps analytic solutions are available but in a form that is inconvenient for direct interpretation... 

Apply a suitable optimization technique to the mathematical statement of the problem. 

The theory is initially presented in the context of small deformations in Section 5.2. A set of internal state variables are introduced as primitive quantities, collectively represented by the symbol k. Qualitative concepts of inelastic deformation are rendered into precise mathematical statements regarding an elastic range bounded by an elastic limit surface, a stress-strain relation, and an evolution equation for the internal state variables. While these qualitative ideas lead in a natural way to the formulation of an elastic limit surface in strain space, an elastic limit surface in stress space arises as a consequence. An assumption that the external work done in small closed cycles of deformation should be nonnegative leads to the existence of an elastic potential and a normality condition. 

Step 3 Refer to the Process Flow Sheets. All material balances are logical. The process flow sheets are the basis for rationalizing the mathematical statement form of material balances. 

A calculation of the flammability limits of complex gas mixtures is carried out by the application of the mixture rule. Stated simply, the mixture rule is that if two limit mixUires of different gases are added together, the resulting mixmre also will be a limit mixture. The mathematical statement of this law is as follows ... 

Therefore, iiweknowthecompactedbulkdensity,thenitispossibletocomputethemass in thebed using the mathematical statement for the conservation of mass. In this case the reactoranditsphysical dimensions define thecontrol volume. The rateofcatalyst delivery is a constant that we will call min The rate of mass flow out of the reactor is zero, that is,... 

The requirement that x x = xlixll is the mathematical statement that light propagates with the same speed with respect to both 0 and O, if they are equivalent observers. In (9-8) the transformation coefficients A"v are all real. The condition (9-10) requires that... 

A mathematical statement of the observation that a linear relationship effectively exists between and Ah ... 

As we have seen earlier, the mathematical statement of the First Law of Thermodynamics in terms of the universe can be written as... 

Figure 3. Mathematical statement of Solutal Model (SM) for microscopic solidification.

In Boyle s work the pressure was subsequently plotted as a function of the reciprocal of the volume, as calculated here in the third column of Thble 1. The graph of P vs. l/V is shown in Fig. lb. This result provided convincing evidence of the relation given by Eq. (3), the mathematical statement of Boyle s law. Clearly, the slope of the straight tine given in Fig. 1 b yields a value of C(T) at die temperature of the measurements [Eq. (3)] and hence a value of the gas constant 17. However, the significance of the temperature was not understood at the time of Boyle s observations. 

In isotropic media 0 and S are related by = < , where the scalar parameter a is now referred to as the permittivity. In the international (SI) system it is given by s = erso. where o is the permittivity of vacuum (see Appendix fl) and e, is a dimensionless permittivity that characterizes the medium. Furthermore, according to Ohm s law the current is given by 7 = cr< , where a is the electrical conductivity. The relation V S3 = 0 is a mathematical statement of the observation that isolated magnetic poles do not exist. 

Fick s first law is a concise mathematical statement however, it is not directly applicable to solutions of most pharmaceutical problems. Fick s second law presents a more general and useful equation in resolving most diffusion problems. Fick s second law can be derived from Fick s first law. 

This expression is a mathematical statement of the combined (or general) gas law. In words, the volume of a given sample of gas is inversely proportional to its pressure and directly proportional to its absolute temperature. 

These bounds originate from the systematic errors (biases) due to the finite sampling in free energy simulations and they differ from other inequalities such as those based on mathematical statements or the second law of thermodynamics. The bounds become tighter with more sampling. It can be shown that, statistically, in a forward calculation AA(M) < AA(N) for sample sizes M and N and M > N. In a reverse calculation, AA(M) > AA(N). In addition, one can show that the inequality (6.27) presents a tighter bound than that of the second law of thermodynamics... 

Hamilton s principle is equivalent to the statement that for the actual motion the average kinetic energy approaches the average potential energy as closely as possible. If the potential energy is a function of position only the equivalent mathematical statement is... 

Solution. How should we start to convert the words of the problem into mathematical statements First, let us define the variables. There will be four of them (tAVtA2, tBl, and tB2, designated as a set by the vector t) representing, respectively, the number of days per year each plant operates on each material as indicated by the subscripts. 

Step 4 suggests that the mathematical statement of the problem be simplified as much as possible without losing the essence of the problem. First, you might... 

Formulate a complete mathematical statement of the problem, and label each individual part, identifying the objective function and constraints with the correct units (, days, etc.). Make a list of the variables by names and symbol plus units. Do not solve. 

The formulation of objective functions is one of the crucial steps in the application of optimization to a practical problem. As discussed in Chapter 1, you must be able to translate a verbal statement or concept of the desired objective into mathematical terms. In the chemical industries, the objective function often is expressed in units of currency (e.g., U.S. dollars) because the goal of the enterprise is to minimize costs or maximize profits subject to a variety of constraints. In other cases the problem to be solved is the maximization of the yield of a component in a reactor, or minimization of the use of utilities in a heat exchanger network, or minimization of the volume of a packed column, or minimizing the differences between a model and some data, and so on. Keep in mind that when formulating the mathematical statement of the objective, functions that are more complex or more nonlinear are more difficult to solve in optimization. Fortunately, modem optimization software has improved to the point that problems involving many highly nonlinear functions can be solved. 

In order to determine the values of Fa,Fb, and Fc that maximize the daily profit, prepare a mathematical statement of this problem as a linear programming problem. Do not solve it. 

We now develop a mathematical statement for model predictive control with a quadratic objective function for each sampling instant k and linear process model in Equation 16.1 ... 

The most general mathematical statement of an optimization problem is... 

In this mathematical statement of Boyle s law, if you know any three quantities, you can calculate the fourth. 

The basic operations of real numbers include addition, subtraction, multiplication, division, and exponentiation (discussed in Chapter 7 of this book). Often, in expressions, there are grouping symbols—usually shown as parentheses—which are used to make a mathematical statement clear. In math, there is a pre-defined order in which you perform operations. This agreed-upon order that must be used is known as the order of operations. 

It is mathematically interesting that the gradient at the surface will always be a boundary condition to the mathematical statement of the problem. Thus, the mathematical solution is necessary simply to evaluate the boundary condition. 

Equation (6.16), which includes Equation (6.6), is a mathematical statement of Carnot s theorem ... 

Equation (6.87) is a condensed mathematical statement of the second law the inequality applies to any real process, which is necessarily irreversible, and the equality applies to the limiting case of the reversible process. 

We can obtain an explicit equation for the entropy of an ideal gas from the mathematical statements of the two laws of thermodynamics. It is convenient to derive this equation for reversible changes in the gas. However, the final result will be perfectly general because entropy is a state function. 

Bennett etal. have presented a model for gaseous pollution sorption by plants. The model includes all the known factors that might have a significant effect on pollution sorption by plant leaves, including gas concentration (ambient air and internal leaf), gas fluxes (external and internal), resistance to flow (leaf boundary layer, stomatal, and internal), nature of leaf surfaces (stomatal presence, cutin, and surface properties), importance of gas solubility and thus solute concentration within the leaf, and ability of the plant to metabolize pollutants (decontaminate itself). They mentioned the reactivity of ozone as another factor to consider. They believe that surface sorption may be important, at least over short periods. They presented a possible mathematical representation of these factors, which they suggested is equivalent to the mathematical statement of Ohm s law. This material is well int ated in the review by Bennett and Hill. ... 

Ions or molecules flowing down their concentration gradients is one aspect of a very general statement known as the Second Law of Thermodynamics. The Second Law is a mathematical statement to the effect that all real processes increase the disorder, captured in a quantity known as entropy, of the universe. Entropy is a measure of disorder or randomness and may be thought of as negative information. 

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