Unear independence

Using the property of Unear independence of Fock space vectors in Eq. (31), and comparing Eqs. (31) and (24), we can see that Eq. (30) really represents the matrix elements of the parallel-transport operator. For closed paths, x(f ) = x(t") = x, Eq. (30) gives the holonomy operator Uki(x) and Ukk is the Wilson loop. Interestingly, the Wilson loop, which is supposed to describe a quark-antiquark interaction, is represented by a true quark and antiquark field, z and z, respectively. So, the mathematical trick can be interpreted physically. ... 

It is clear that propagators and scalar products are not Unearly independent, so that one can express some of the scalar products in terms of the propagators which are present in the graph (this procedure is sometines calles trivial tensor reduction). To make a simple example, if Kl and Epl are present, one can write the obvious identity... 

It is assumed that these model functions are Unearly independent and that they span the entire model space. There is, therefore, a one-to-one correspondence between the m model functions and m exact solutions of the Schrddinger equation. 

FIGURE 5. Representative graphs Q for an independent electron model based on Unear combinations of two-centre ir-orbitals nIL. Examples 1,3-butadiene 2, f3 )-hexa-1,3,5-triene 161, heptafulvene and... 

A chiral catalyst is not necessarily in enantiopure form. Deviation from the Unear relationship, namely non-linear effect , is sometimes observed between the enantiomeric purity of the chiral catalysts and the optical yields of the products (Figure 8C.2). The convex deviation—which Kagan [35a] andMikami [36) independently refer to as positive non-linear effect (abbreviated as (+)-NLE) and what Oguni refers to as [35c] asymmetric amplification —is currently attracting much attention for achieving a higher level of asymmetric induction that exceeds the enantiopurity of the non-racemic (partially resolved) catalysts. In turn, (-)-NLE stands for the opposite phenomenon of concave deviation, namely negative non-linear effect. 

MLR is most suited to the modeling of simple physical properties for which the contributions of the descriptors are more likely to be independent than cooperative. It is also the preferred technique when the amount of available data is extremely limited. MLR is less suited for modeling complex physical or biological processes, as these tend to be non-Unear in nature. 

This estimate is independent of a particular location in phase space. This implies that all points of phase space, in particular the period-1 points, are linearly unstable. Thus we have proved that the positively kicked hydrogen atom does indeed not possess any first order eUiptic islands in phase space. It is possible to extend this proof to period-iV points and to show that all period-N points, N integer, are Unearly unstable (Bliimel (1993c)). This implies that the positively kicked hydrogen atom is completely chaotic. We emphasize that, as far as we know, the kicked hydrogen atom is the only model for a physically realizable system where a numerically motivated chaos conjecture was followed up by an analytical proof. In this sense the kicked hydrogen atom is a most remarkable system. 

The data set comprises pairs of measurements of an independent variable x (concentration) and a dependent variable y (absorbance) and it is required to fit the data using a Unear model with the well known form. 

Generalized Unear mixed models (GLMMs) provide another type of extension of LME models aimed at non-Gaussian responses, such as binary and count data. In these models, conditional on the random effects, the responses are assumed independent and with distribution in the exponential family (e.g., binomial and Poisson) (8). As with NLME models, exact likelihood methods are not available for GLMMs because they do not allow closed form expressions for the marginal distribution of the responses. QuasUikelihood (9) and approximate likelihood methods have been proposed instead for these models. 

Linear refers to the response of the spin system to the incident microwave power. If the latter is sufficiently low that microwave power saturation is negligible, the spin system is said to respond linearly. This implies that the ESR signal has an intensity proportional to where Pq is the incident power, and a lineshape that is independent of power. If the spin system response shows evidence of saturation effects, it is said to respond non-Unearly. 

SPME is a multiphase equilibrium technique and, therefore, the analytes are not completely extracted from the matrix. Nevertheless, the method is useful for quantitative work and excellent precision and Unearity have been demonstrated. An extraction is complete when the concentration of analytes has reached distribution equilibrium between the sample and coating. This means that once the equihbrium is achieved, the amount extracted is independent of further increase in extraction time. If extraction is terminated before the equihbrium is reached, good precision and reproducibihty is still obtained if incubation temperature, sample agitation, sample pH and ionic strength, sample and headspace volume, extraction and desorption time are kept constant. The theory of the thermodynamic, kinetic and mass transfer processes underlying direct immersion and HS-SPME has been extensively discussed by Pawhszyn [2]. The sensitivity and time required to reach adsorption equilibriiun depends on the partition coefficients between the fiber and the analytes, and the thickness of the phase. Limits of detection and quantitation are often below 1 ppb. 

In electromagnetic theory the Faraday effect can be explained as follows. When the medium magnetization has non-zero projection on the wave vector ko of the incident radiation, two independent fundamental Maxwell equations solutions are circular polarized waves with different refractive indexes n+M n, respectively. At the output of the magnetic medium these waves gain phase shift and when added give Unearly polarized wave with rotated polarization plane. That is why Faraday effect is also called magnetic circular birefringence [26, 27]. 

In a pure quadrupolar field, the radial and axial components of an ion s motion are independent of each other. Also, the magnitude of the electric field varies linearly with distance from the center of the trap in both the axial and radial coordinate. Hence, a quadrupolar field is sometimes referred to as being Unear . 

MCs present an alternative to more conventional contactor systems. The main advantages of MC systems over conventional column contactors are the reduction in size, the operational flexibility, elevated mass transfer rate and Unear scale-up, and the possibiUty of integration. Independent of their... 

This is also called the linear-diffusion equation, which in its most elementary form is a linear second-order partial differential equation (PDE). The assmnption of a concentration-independent diffusion coefficient is generally true for diffusion in gases, hquids, and solutions. Polymers above the glass-transition temperature and, especially, rubbers such as PDMS can be expected to behave like liquids for small molecular diffusants. Analytical solutions to the Unear-diffusion equation, eq 2, for various... 

The system of the equations for the componentwise discrepancies of the material balance in the feed cross-section is solved for the set value of CT = R/Rmm. determining the more precise values of little concentrations of the non-key impurity components in the separation products and of tray numbers in the column sections. The difference of that from the corresponding step of the algorithm for intermediate splits consists of the fact that tray numbers in the sections are included into the number of independent variables besides the concentrations of the non-key impurity components in the separation products. In accordance with that, it is accepted that the concentration of each component in the feed cross-section is a Unear function not only of the little concentrations of the non-key impurity components in the corresponding product, but also of the tray numbers in the corresponding section ... 

The computational model is time independent and thus a stationary solver can be used. The energy equation is solved with the iterative Unear solver GMRES. 

Muitipie Linear Regression n A form of regression where the dependent variable is assumed to be a linear function of more than one independent variable. There are many methods for calculating the constants of the single variable linear function. Least squares method is the most widely used. The multiple Unear regression model function has the form ... 

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