Linearly dependent

Some variables often have dependencies, such as reservoir porosity and permeability (a positive correlation) or the capital cost of a specific equipment item and its lifetime maintenance cost (a negative correlation). We can test the linear dependency of two variables (say x and y) by calculating the covariance between the two variables (o ) and the correlation coefficient (r) ... 

The linear dependence of C witii temperahire agrees well with experiment, but the pre-factor can differ by a factor of two or more from the free electron value. The origin of the difference is thought to arise from several factors the electrons are not tndy free, they interact with each other and with the crystal lattice, and the dynamical behaviour the electrons interacting witii the lattice results in an effective mass which differs from the free electron mass. For example, as the electron moves tlirough tiie lattice, the lattice can distort and exert a dragging force. 

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... 

If the tr ansformation matr ix is orthogonal, then the tr ansformation is orthogonal. If the elements of A are numbers (as distinct from functions), the transformation is linear. One important characteristic of an orthogonal matrix is that none of its columns is linearly dependent on any other column. If the transfomiation matrix is orthogonal, A exists and is equal to the transpose of A. Because A = A ... 

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... 

Jorgensen, W. L. Gao, J. Ravimohan, C. J. Phys. Chem. 1985, 89, 3470 Recently a similar linear dependence of the heat capacity change upon solvation has been observed Madan, B. Sharp, K. J. Phys. Chem. B 1997, 707, 11237... 

In order to obtain more insight into the local environment for the catalysed reaction, we investigated the influence of substituents on the rate of this process in micellar solution and compared this influence to the correspondirg effect in different aqueous and organic solvents. Plots of the logarithms of the rate constants versus the Hammett -value show good linear dependences for all... 

When k, S k the S z process will show a linear dependence of... 

Let us derive a condition of nonpenetrating in general case (see Fig. 1.3). The Kirchhoff-Love hypothesis provides the linear dependence of the shell horizontal displacements on a distance from the mid-surface, namely... 

Conversely, the linear dependence of (1.52) on guarantees fulfilment of (1.52) provided that (1.53) holds. 

The Kirchhoff-Love model of the plate is characterized by the linear dependence of the horizontal displacements on the distance from the mid-surface, that is... 

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. 

The effect of pressure on the solubility of chlorine ia hydrochloric acid has been reported for pressures varying from about 100 to 6500 kPa (1—6.5 atm) (20). At pressures above 200 kPa, there is a linear dependence of pressure on the solubility in the acid concentration range of 0.1—5.0 N. 

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,... 

The equality holds if, and only if, the vec tors a, b are linearly dependent (i.e., one vec tor is scalar times the other vector). 

Linear dependence of current of additional peak 1 on concentration of Zr(IV) can be used for elaboration sensitive and selective determination of zirconium with detection limit of 1.7x10 mol/1. 

The worked out soi ption-photometric method of NIS determination calls preliminary sorption concentration of NIS microamounts from aqueous solutions on silica L5/40. The concentrate obtained is put in a solution with precise concentration of bromthymol-blue (BTB) anionic dye and BaCl, excess. As a result the ionic associate 1 1 is formed and is kept comparatively strongly on a surface. The BTB excess remains in an aqueous phase and it is easy to determinate it photometrically. The linear dependence of optical density of BTB solutions after soi ption on NIS concentration in an interval ITO - 2,5T0 M is observed. The indirect way of the given method is caused by the fact the calibration plot does not come from a zero point of coordinates, and NIS zero concentration corresponds to initial BTB concentration in a solution. 

A linear dependence was established between tga and the concenb ation of 2,4-dinitrophenol ... 

The mixture of acetonitrile/water (1 1, v/v) was selected as most effective mobile phase. The optimum conditions for chromatography were the velocity of mobile phase utilization - 0,6 ml/min, the wave length in spectrophotometric detector - 254 nm. The linear dependence of the height of peack in chromathography from the TM concentration was observed in the range of 1-12.0 p.g/mL. 

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. 

The insulating properties of polyethylene compare favourably with those of any other dielectric material. As it is a non-polar material, properties such as power factor and dielectric constant are almost independent of temperature and frequency. Dielectric constant is linearly dependent on density and a reduction of density on heating leads to a small reduction in dielectric constant. Some typical data are given in Table 10.6. 

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). 

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