Variables in mathematics

The changeover to thermodynamic activities is equivalent to a change of variables in mathematical equations. The relation between parameters and a. is unambiguous only when a definite value has been selected for the constant p. For solutes this constant is selected so that in highly dilute solutions where the system p approaches an ideal state, the activity will coincide with the concenttation (lim... 

Barnett, J. (1979). The study of syntax variables. In G. Goldin C. E. McClintock (Eds.), Task variables in mathematical problem solving (pp. 23-68). Columbus, OH Clearinghouse for Science, Mathematics, and Environmental Education. 

Its geometric form (function, straight Une) is given in Fig. 5.16. In this expression X (y) are referred to as the input (output) variable in the modehng methodology causal (resultant) variable in philosophy independent (dependent) variable in mathematics and predictand (prediction) variables in engineering. 

Macroscopic thermodynamic variables such as temperature, pressure, volume, density, entropy, energy, and so on, can be dependent or independent variables in mathematical functions. 

In mathematical terms, PCA transforms a number of correlated variables into a smaller number of uneorrelated variables, the so-called principal components. 

Very often in practice a relationship is found (or known) to exist between two or more variables. It is frequently desirable to express this relationship in mathematical form by determining an equation connecting the variables. 

There are several mathematical methods for producing new values of the variables in this iterative optimization process. The relation between a simulation and an optimization is depicted in Eigure 6. Mathematical methods that provide continual improvement of the objective function in the iterative... 

The natural laws in any scientific or technological field are not regarded as precise and definitive until they have been expressed in mathematical form. Such a form, often an equation, is a relation between the quantity of interest, say, product yield, and independent variables such as time and temperature upon which yield depends. When it happens that this equation involves, besides the function itself, one or more of its derivatives it is called a differential equation. 

If the state and control variables in equations (9.4) and (9.5) are squared, then the performance index become quadratic. The advantage of a quadratic performance index is that for a linear system it has a mathematical solution that yields a linear control law of the form... 

It follows that 1/T is the integrating factor of SQ. Now since SQ is a function of two variables (in the simple case of a homogeneous fluid), and since the integrating factor of such a magni-" tude is usually also a function of the same two variables, we must regard the proposition that the integrating factor of SQ is a function of one variable only as expressing a physical, not a mathematical, truth. 

The need for dimensional consistency imposes a restraint in respect of each of the fundamentals involved in the dimensions of the variables. This is apparent from the previous discussion in which a series of simultaneous equations was solved, one equation for each of the fundamentals. A generalisation of this statement is provided in Buckingham s n theorem(4) which states that the number of dimensionless groups is equal to the number of variables minus the number of fundamental dimensions. In mathematical terms, this can be expressed as follows ... 

While most climate models consider feedbacks as being dependent on temperature (usually Ts), there are many other dependent variables in the climate system that could be involved, for example solar irradiance at the ground or rainfall. However, it is customary to describe these mathematically as functions of Tg,... 

Ideally, a mathematical model would link yields and/or product properties with process variables in terms of fundamental process phenomena only. All model parameters would be taken from existing theories and there would be no need for adjusting parameters. Such models would be the most powerful at extrapolating results from small scale to a full process scale. The models with which we deal in practice do never reflect all the microscopic details of all phenomena composing the process. Therefore, experimental correlations for model parameters are used and/or parameters are evaluated by fitting the calculated process performance to that observed. 

They cannot be part of a mathematical model whose purpose would be to turn the classification into a continuous quantitative variable. In particular, the example of physical factors illustrates this. Whereas for the highest degree criteria are the same as those of the NFPA code, the simple fact of wanting to add in physical factors to these calculation models forced the originators of this technique to forget about the NFPA code. 

When we consider the multivariate situation, it is again evident that the discriminating power of the combined variables will be good when the centroids of the two sets of objects are sufficiently distant from each other and when the clusters are tight or dense. In mathematical terms this means that the between-class variance is large compared with the within-class variances. 

The main process variables in differential contacting devices vary continuously with respect to distance. Dynamic simulations therefore involve variations with respect to both time and position. Thus two independent variables, time and position, are now involved. Although the basic principles remain the same, the mathematical formulation, for the dynamic system, now results in the form of partial differential equations. As most digital simulation languages permit the use of only one independent variable, the second independent variable, either time or distance is normally eliminated by the use of a finite-differencing procedure. In this chapter, the approach is based very largely on that of Franks (1967), and the distance coordinate is treated by finite differencing. 

In implicit estimation rather than minimizing a weighted sum of squares of the residuals in the response variables, we minimize a suitable implicit function of the measured variables dictated by the model equations. Namely, if we substitute the actual measured variables in Equation 2.8, an error term arises always even if the mathematical model is exact. 

The limitation of transfer function representation becomes plain obvious as we tackle more complex problems. For complex systems with multiple inputs and outputs, transfer function matrices can become very clumsy. In the so-called modem control, the method of choice is state space or state variables in time domain—essentially a matrix representation of the model equations. The formulation allows us to make use of theories in linear algebra and differential equations. It is always a mistake to tackle modem control without a firm background in these mathematical topics. For this reason, we will not overreach by doing both the mathematical background and the control together. Without a formal mathematical framework, we will put the explanation in examples as much as possible. The actual state space control has to be delayed until after tackling classical transfer function feedback systems. 

From the information-theoretical point of view, calibration corresponds to the coding of the input quantity into the output quantity and, vice versa, the evaluation process corresponds to decoding of output data. From the mathematical viewpoint, qin is the independent quantity in the calibration step and qout the dependent one. In the evaluation step, the situation is reverse qout is the independent, and qin the dependent quantity. From the statistical standpoint, qout is a random variable both in calibration and evaluation whereas qin is a fixed variable in the calibration step and a random variable in the evaluation step. This rather complicated situation has some consequences on which will be returned in Sect. 6.1.2. 

Since the integral method described above is based on the premise that some rate function exists that will lead to a value of i/ Cj) that is linear in time, deviations from linearity (or curvature) indicate that further evaluation or interpretation of the rate data is necessary. Many mathematical functions are roughly linear over sufficiently small ranges of variables. In order to provide a challenging test of the linearity of the data, one should perform at least one experimental run in which data are... 

Equation 41-A3 can be checked by expanding the last term, collecting terms and verifying that all the terms of equation 41-A2 are regenerated. The third term in equation 41-A3 is a quantity called the covariance between A and B. The covariance is a quantity related to the correlation coefficient. Since the differences from the mean are randomly positive and negative, the product of the two differences from their respective means is also randomly positive and negative, and tend to cancel when summed. Therefore, for independent random variables the covariance is zero, since the correlation coefficient is zero for uncorrelated variables. In fact, the mathematical definition of uncorrelated is that this sum-of-cross-products term is zero. Therefore, since A and B are random, uncorrelated variables ... 

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