Linear dependence/independence

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. 

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. 

In ra-space any n + 1 of these are linearly dependent. But unless the matrix is rather special in form (derogatory), there exist vectors vx for which any n consecutive vectors are linearly independent (in possible contrast to the behavior in the limit). In fact, this is true of almost every vector vx. Hence, if... 

Where Br nucleophilically promotes the Br+/OTf- elimination to generate free Br2 and cyclohexene. This process requires that the rate of solvolysis of 4 be linearly dependent on [Br ]. However, control (ref. 15) kinetics experiments indicate that the rate constant for solvolysis of 4 in HOAc or MeOH are independent of Br" thus generation of free Br2 must occur after the rate limiting step. This nicely confirms the previous conclusion based upon the invariance of the n0a+10hV9h ratio on [Br]. 

It has been shown that the p columns of an nxp matrix X generate a pattern of p points in 5" which we call PP. The dimension of this pattern is called rank and is indicated by liPP). It is equal to the number of linearly independent vectors from which all p columns of X can be constructed as linear combinations. Hence, the rank of PP can be at most equal to p. Geometrically, the rank of P can be seen as the minimum number of dimensions that is required to represent the p points in the pattern together with the origin of space. Linear dependences among the p columns of X will cause coplanarity of some of the p vectors and hence reduce the minimum number of dimensions. 

We can now define the rank of the column-pattern as the number of linearly independent columns or rank of X. If all 50 points are coplanar, then we can reconstruct each of the 50 columns, by means of linear combinations of two independent ones. For example, if x, and Xj 2 linearly independent then we must have 48 linear dependences among the 50 columns of X ... 

If A is a symmetric positive definite matrix then we obtain that all eigenvalues are positive. As we have seen, this occurs when all columns (or rows) of the matrix A are linearly independent. Conversely, a linear dependence in the columns (or rows) of A will produce a zero eigenvalue. More generally, if A is symmetric and positive semi-definite of rank r[Pg.32]

We find that the rates of reaction for the various amines examined in (28) are governed by an extremely complex set of equilibria. For example, when R = n-Pr, n-Bu or s-Bu, the rate of reaction exhibit first order dependence on [EtjSiH] at constant amine concentration. However, the rate of reaction exhibits inverse non-linear dependence on [n-PrNH2] and [n-BuNH2], but positive non-linear dependence on [s-BuNH2] at constant [Et-jSiH]. Furthermore, if R t-Bu, then the rate of reaction is almost independent of both [t-BuNH2] and [EtjSiH]. Studies of the rate dependence on catalyst concentration for reaction (28) where R NH2 is n-BuNH2 reveal relative catalyst activities that are inversely dependent on [Ruo(C0) 2]. Similar studies with R NH2 = t-BuNH2 reveal that the rate or reaction is linearly dependent on [Ru3(CO) 2]. Piperidine is unreactive under the reaction conditions studied. 

Figure 8 reveals that the few data available for surfactant-laden bubbles do confirm the capillary-number dependence of the proposed theory in Equation 18. Careful examination of Figure 8, however, reveals that the regular perturbation analysis carried out to the linear dependence on the elasticity number is not adequate. More significant deviations are evident that cannot be predicted using only the linear term, especially for the SDBS surfactant. Clearly, more data are needed over wide ranges of capillary number and tube radius and for several more surfactant systems. Further, it will be necessary to obtain independent measurements of the surfactant properties that constitute the elasticity number before an adequate test of theory can be made. Finally, it is quite apparent that a more general solution of Equations 6 and 7 is needed, which is not restricted to small deviations of surfactant adsorption from equilibrium. 

The heat of fusion AHf (obtained from the area under the DSC melting curve) and percentage crystallinity calculated from AHf is found to be linearly dependent on butadiene content, and independent of the polymer architecture. This is shown in Figure 3. Also, the density of the block copolymers was found to be linearly dependent on butadiene content (see Figure 4). The linear additivity of density (specific volume) has been observed by other workers for incompatible block copolymers of styrene and butadiene indicating that very little change in density from that of pure components has occurred on forming the block copolymers.(32) While the above statement is somewhat plausible, these workers have utilized the small positive deviation from the linear additivity law to estimate the thickness of the boundary in SB block copolymers.(32)... 

XOD is one of the most complex flavoproteins and is composed of two identical and catalytically independent subunits each subunit contains one molybdenium center, two iron sulfur centers, and flavine adenine dinucleotide. The enzyme activity is due to a complicated interaction of FAD, molybdenium, iron, and labile sulfur moieties at or near the active site [260], It can be used to detect xanthine and hypoxanthine by immobilizing xanthine oxidase on a glassy carbon paste electrode [261], The elements are based on the chronoamperometric monitoring of the current that occurs due to the oxidation of the hydrogen peroxide which liberates during the enzymatic reaction. The biosensor showed linear dependence in the concentration range between 5.0 X 10 7 and 4.0 X 10-5M for xanthine and 2.0 X 10 5 and 8.0 X 10 5M for hypoxanthine, respectively. The detection limit values were estimated as 1.0 X 10 7 M for xanthine and 5.3 X 10-6M for hypoxanthine, respectively. Li used DNA to embed xanthine oxidase and obtained the electrochemical response of FAD and molybdenum center of xanthine oxidase [262], Moreover, the enzyme keeps its native catalytic activity to hypoxanthine in the DNA film. So the biosensor for hypoxanthine can be based on... 

In the meantime temperature-dependent stopped-flow measurements were conducted on the latter complex in order to determine the activation parameters of the N-N cleavage reaction (24). Plots of the absorption intensity at 418 nm vs. time at T — —35 to +15°C indicate biphasic kinetics with two rate constants 0bs(p and obs(2)> in analogy to our measurements of the tungsten complex. This time, however, both rates depended upon the acid concentration. Interestingly much smaller rate constants 0bs(i) and 0bs(2)> were found for all acid concentrations than given by Henderson et al. for his (single) rate constant kobs (up to 1 order of magnitude). Furthermore plots of 0bs(i) and kohs(2) vs. the acid concentration showed no saturation behavior but linear dependencies with slopes k and k and intercepts k und k, respectively (s — acid dependent and i — acid independent), Eq. (2) ... 

Rate equation (1) indicates that ku should be inversely proportional to the activity of water for solvolysis by the AAil mechanism and independent of it if the bimo-lecular processes (pathways (i) and (ii)) pertain. Fig. 12 illustrates that acid independent rate constants at different volume fractions of D20 in CD3CN, /cH, were linearly dependent upon the inverse of ud2o in CD3CN as determined from the corresponding activities of H20 in CH3CN.142 This is in accord with the AA]1 mechanism (pathway (iii), Scheme 6). 

As mentioned earlier, singular matrices have a determinant of zero value. This outcome occurs when a row or column contains all zeros or when a row (or column) in the matrix is linearly dependent on one or more of the other rows (or columns). It can be shown that for a square matrix, row dependence implies column dependence. By definition the columns of A, a, are linearly independent if... 

Conversely, linear dependence occurs when some nonzero set of values for dj satisfies Equation (A.22). The rank of a matrix is defined as the number of linearly independent columns n). 

Also, since each measurement corresponding to yi was redundant for the original system, each row of Ci was linearly dependent on some other rows of [ ]. But, any row in C must be linearly independent of any row in C2 otherwise y2 would be redundant. Hence, any row in Ci is dependent on the other rows in [ , ]. The dependency is unchanged by the transformation F thus, yi is also redundant in the system [ "]. ... 

It will be implicitly assumed (as is almost always the case) that the components of R() are linearly independent, e.g., Ri () 7 aRj () for all i j. If two components were linearly dependent, then file corresponding columns of T could be replaced by a single column, and file linearly dependent component of R() removed. Linear independence implies that file Jacobian matrix formed from R() will be full rank for arbitrary choices of... 

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