Linearly dependent equations

LPRINT "LINEARLY DEPENDENT EQUATIONS, STATUS FLAG " ER GOTO 178 LPRINT... 

To set up stoichiometric equations for other relations between the fluxes is possible as well, but not additionally, as it would lead to linear dependent equations. The photosynthetic quotient turns out be... 

The problem of selecting the most reasonable pivot to detect real linearly dependent equations occurs not only for underdimensioned linear systems but also in the solution of singular square systems. It happens, for instance, when a Newton s method is adopted to solve a square nonlinear system and the resulting... 

With square systems with very ill-conditioned (or singular) matrices and underdimensioned systems, it is mandatory to swap the columns during pivoting. This is the only way to obtain a reasonable solution and identify the real linearly dependent equations. 

We regard Eq.(7.1.1) as a set of constraints the variable z has to obey, thus as (a part of) the model of some physical (technological) process. So if the condition (7.1.2) is not obeyed then the model is simply wrong, having no solution. Let thus the condition be satisfied. If we have AT < M then certain (M-M ) rows of matrix (C, c), thus certain scalar equations are linear combinations of the remaining M ones, and can be deleted. This done, the matrix of the new system is of full row rank. In what follows, let us suppose that linearly dependent equations have been deleted a priori, hence assume... 

As summarized at the end of Section 7.2, before Remarks, the canonical format determines certain invariants of the linear system (7.4.2). For the solvability in z (of Cz + c = 0) and for the regularity (7.4.1), see above (absence of linearly dependent equations). Then the invariants are the number H (degree... 

A linear dependence approximately describes the results in a range of extraction times between 1 ps and 50 ps, and this extrapolates to a value of Ws not far from that observed for the 100 ps extractions. However, for the simulations with extraction times, tg > 50 ps, the work decreases more rapidly with l/tg, which indicates that the 100 ps extractions still have a significant frictional contribution. As additional evidence for this, we cite the statistical error in the set of extractions from different starting points (Fig. 2). As was shown by one of us in the context of free energy calculations[12], and more recently again by others specifically for the extraction process [1], the statistical error in the work and the frictional component of the work, Wp are related. For a simple system obeying the Fokker-Planck equation, both friction and mean square deviation are proportional to the rate, and... 

The third equation above is implicit for but the linear dependency... 

Any linearly independent set of simultaneous homogeneous equations we can construct has only the zero vector as its solution set. This is not acceptable, for it means that the wave function vanishes, which is contrai y to hypothesis (the electron has to be somewhere). We are driven to the conclusion that the normal equations (6-38) must be linearly dependent. 

Linearly dependent sets of homogeneous simultaneous equations, for example. 

For the equation set to be linearly dependent, the secular determinant must be zero... 

Equations 11 and 12 caimot be used to predict the mass transfer coefficients directly, because is usually not known. The theory, however, predicts a linear dependence of the mass transfer coefficient on diffusivity. 

For many modeling purposes, Nhas been assumed to be 1 (42), resulting in a simplified equation, S = C, where is the linear distribution coefficient. This assumption usually works for hydrophobic polycycHc aromatic compounds sorbed on sediments, if the equdibrium solution concentration is <10 M (43). For many pesticides, the error introduced by the assumption of linearity depends on the deviation from linearity. 

Linear Free Energy—Linear Solvation Energy Relationships. Linear free energy (LFER) and linear solvation energy (LSER) relationships are used to develop correlations between selected properties of similar compounds. These are fundamentally a collection of techniques whereby properties can be predicted from other properties for which linear dependency has been observed. Linear relationships include not only simple y = rax + b relationships, but also more compHcated expressions such as the Hammett equation (254) which correlates equiUbrium constants for ben2enes,... 

Linear Differential Equations with Constant Coeffieients and Ri ht-Hand Member Zero (Homogeneous) The solution of y" + ay + by = 0 depends upon the nature of the roots of the characteristic equation nr + am + b = 0 obtained by substituting the trial solution y = in the equation. 

In the equation shown above, the first term—including p for density and the square of the linear velocity of u—is the inertial term that will dominate at high flows. The second term, including p. for viscosity and the linear velocity, is the viscous term that is important at low velocities or at high viscosities, such as in liquids. Both terms include an expression that depends on void fraction of the bed, and both change rapidly with small changes in e. Both terms are linearly dependent on a dimensionless bed depth of L/dp. 

Aris and Amundson (1958) solved the coupled, time-dependent material and energy balances, linearizing the equations about the operating point by a Taylor series expansion. This made the solution possible by the method of characteristic equations. The solution yielded two equations, one the slope condition and the other recognized by Gilles and Hofmann (1961) as the condition that sets the limits to avoid rate oscillation. This is called the... 

The ratio Db/Da is a so-called relative sensitivity factor D. This ratio is mostly determined by one element, e. g. the element for insulating samples, silicon, which is one of the main components of glasses. By use of the equation that the sum of the concentrations of all elements is equal to unity, the bulk concentrations can be determined directly from the measured intensities and the known D-factors, if all components of the sample are known. The linearity of the detected intensity and the flux of the sputtered neutrals in IBSCA and SNMS has been demonstrated for silicate glasses [4.253]. For SNMS the lower matrix dependence has been shown for a variety of samples [4.263]. Comparison of normalized SNMS and IBSCA signals for Na and Pb as prominent components of optical glasses shows that a fairly good linear dependence exists (Fig. 4.49). 

Equation (8.91) is singular since it has a zero determinant. Also the column vectors are linearly dependent since the second column is —5 times the first column and therefore the system is unobservable. 

The problems experienced in drying process calculations can be divided into two categories the boundary layer factors outside the material and humidity conditions, and the heat transfer problem inside the material. The latter are more difficult to solve mathematically, due mostly to the moving liquid by capillary flow. Capillary flow tends to balance the moisture differences inside the material during the drying process. The mathematical discussion of capillary flow requires consideration of the linear momentum equation for water and requires knowledge of the water pressure, its dependency on moisture content and temperature, and the flow resistance force between water and the material. Due to the complex nature of this, it is not considered here. 

The resulting finite difference equations constitute a set of nonho-mogeneous linear algebraic equations. Because there are three dependent variables, the number of equations in the set is three times the number of material points. Obviously, if a large number of points is required to accurately represent the continuous elastic body, a computer is essential. 

The low-coverage parts of the adsorption isotherms evaluated at different temperatures have shown a remarkable feature of linear dependence between the adsorption and the logarithm of gas pressure. This sort of behavior corresponds to the well-known Temkin equation of adsorption... 

This weighting procedure for the linearized Arrhenius equation depends upon the validity of Eq. (6-7) for estimating the variance of y = In k. It will be recalled that this equation is an approximation, achieved by truncating a Taylor s series expansion at the linear term. With poor precision in the data this approximation may not be acceptable. A better estimate may be obtained by truncating after the quadratic term the result is... 

Attack by alkali solution, hydrofluoric acid and phosphoric acid A common feature of these corrosive agents is their ability to disrupt the network. Equation 18.1 shows the nature of the attack in alkaline solution where unlimited numbers of OH ions are available. This process is not encumbered by the formation of porous layers and the amount of leached matter is linearly dependent on time. Consequently the extent of attack by strong alkali is usually far greater than either acid or water attack. 

The constant 607 is a combination of natural constants, including the Faraday constant it is slightly temperature-dependent and the value 607 is for 25 °C. The IlkoviC equation is important because it accounts quantitatively for the many factors which influence the diffusion current in particular, the linear dependence of the diffusion current upon n and C. Thus, with all the other factors remaining constant, the diffusion current is directly proportional to the concentration of the electro-active material — this is of great importance in quantitative polarographic analysis. 

In order for the inverse of [C CT] to exist, C must have at least as many columns as rows. Since C has one row for each component and one column for each sample, this means that we must have at least as many samples as components in order to be able to compute equation [33]. This would certainly seem to be a reasonable constraint. Also, if there is any linear dependence among the rows or columns of C, [C CT] will be singular and its inverse will not exist. One of the most common ways of introducing linear dependency is to construct a sample set by serial dilution. 

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