Linearity concepts

There is much confusion about the terms linear and linearity. Suppose a model is needed for relating x, (the predictor) to y, (the predictand), where i indexes the objects and runs from 1 to I. Such a model is linear in the parameters and in the variables if it can be written as 

It is useful to distinguish these two types of linearity. If in the chemical sciences the term linearity is loosely used, then often linearity in the parameters is meant. Hence, models in Equations (2.12) and (2.13) are linear in the parameters, because when x is fixed, y is a linear function of the parameters bo, b (and b2). 

The notion of linearity can be extended to bilinearity. Suppose that a model is needed for X(I x J), with elements Xy. A two-component PCA model of X (see Chapter 1 and for a more detailed explanation Chapter 3) is then  

There is some confusion in the second-order calibration literature [Sanchez Kowalski 1988] where the term bilinearity is reserved for equations of the form of Equation (2.14) with a one component PCA model for X. However, bilinearity is more general and holds for an arbitrary number of components [Kruskal 1984], 

The notion of bilinearity can be extended further. Consider a model of the three-way array X(7 x J x K), with elements Xp  

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More generally, our readers need to think vectorially and to envision matrices, linear concepts, and matrix and vector notation throughout this book and, we believe, in any other project that involves numerical computations. 

The time constant is a linear concept, which derives from the solution of a linear differential equation such as that used to model valve 1 in Section 2.2 ... 

While the time constant is strictly a linear concept, the basic idea can be transferred to nonlinear systems by linearizing about an operating point. Now the time constants will not be constant at all, but will depend, at any instant of time, on the values of the states... 

Equation (24.36) may be applied to any point (a p,j + otj, e ) on the curve of Figure 24.2 to obtain a value for cy. Bearing in mind the fact that we are invoking linear concepts to model a nonlinear response, experience has shown that a reasonable range of system behaviour is captured if we choose aj, = ajd where ej... 

Phytosociology is based on too many deterministic asp>ects, first of all the importance given to the linear concept of ecological succession (serai step ), not compatible with the reality, being in contrast with the new scientific paradigms. 

Now our mission will be to find how these could be calculated from the above curves. This will be clarified with the help of a few numerical examples. From previous discussion, it is clear that f t) is the failure density function over a time interval (tl and ti) (from piecewise linear concept). Therefore, it the ratio of the number of failures occurring in the interval n(t2) — n(ti) to the size of the original population (N), divided by the length of the time interval (say) t2- i ... 

B. Geometrical configuration is considered as a very small cube with linear changes of the variables considered along its sides. Consideration of very small cube size in the derivations implies the assumptions of homogeneity, uniformity and isotropy automatically. Such a combination of assumptions further implies the use of the arithmetic average and linearity concepts. 

A student or family s language, cultural set, or world-view may necessitate the need to implement additional strategies to reach for cultural competency. Cultural competence is not a linear concept so that once a practitioner or service system seemingly attains it, it always remains. In fact, the opposite has been argued there must be constant efforts to provide and maintain culturally and linguistically responsive and effective mental health services. 

LCAO method A method of calculation of molecular orbitals based upon the concept that the molecular orbital can be expressed as a linear combination of the atomic orbitals. 

It is possible to limit our choice for stochastic modeling by stationary, linear, nonlinear, and ergodic models in combination with deterministic function. In this case the following well studied models can be proposed for the accepted concept [1] ... 

A variety of experimental data has been found to fit the Langmuir equation reasonably well. Data are generally plotted according to the linear form, Eq. XVn-9, to obtain the constants b and n from the best fitting straight line. The specific surface area, E, can then be obtained from Eq. XVII-10. A widely used practice is to take to be the molecular area of the adsorbate, estimated from liquid or solid adsorbate densities. On the other hand, the Langmuir model is cast around the concept of adsorption sites, whose spacing one would suppose to be characteristic of the adsorbent. See Section XVII-5B for an additional discussion of the problem. 

The characteristic isotherm concept was elaborated by de Boer and coworkers [90]. By accepting a reference from a BET fit to a standard system and assuming a density for the adsorbed film, one may convert n/rim to film thickness t. The characteristic isotherm for a given adsorbate may then be plotted as t versus P/P. For any new system, one reads t from the standard r-curve and n from the new isotherm, for various P/P values. De Boer and co-work-ers t values are given in Table XVII-4. A plot of t versus n should be linear if the experimental isotherm has the same shape as the reference characteristic isotherm, and the slope gives E ... 

This basic LFER approach has later been extended to the more general concept of fragmentation. Molecules are dissected into substructures and each substructure is seen to contribute a constant inaement to the free-energy based property. The promise of strict linearity does not hold true in most cases, so corrections have to be applied in the majority of methods based on a fragmentation approach. Correction terms are often related to long range interactions such as resonance or steric effects. 

You can interpret results, including dipole moments and atomic charges, using the simple concepts and familiar vocabulary of the Linear Combination of Atomic Orbitals (LCAO)-molecular orbital (MO) theory. 

In Section 4.3.f it was shown that there are 3N — 5 normal vibrations in a linear molecule and 3N — 6 in a non-linear molecule, where N is the number of atoms in the molecule. There is a set of fairly simple rules for determining the number of vibrations belonging to each of the symmetry species of the point group to which the molecule belongs. These rules involve the concept of sets of equivalent nuclei. Nuclei form a set if they can be transformed into one another by any of the symmetry operations of the point group. For example, in the C2 point group there can be, as illustrated in Figure 6.18, four kinds of set ... 

The hnearity between M and makes the concept of absorbance so usehil that measurements made by sampling methods other than transmission are usually converted to a scale proportional to absorbance. The linearity between M and i is maintained only if the resolution of the spectrometer is adequate to eliminate contributions from wavelengths not absorbed by the species being measured. In addition, the apparent value of a is very dependent on resolution because a is 2l strong function of wavelength (30,31). 

The Stress-Rang e Concept. The solution of the problem of the rigid system is based on the linear relationship between stress and strain. This relationship allows the superposition of the effects of many iadividual forces and moments. If the relationship between stress and strain is nonlinear, an elementary problem, such as a siagle-plane two-member system, can be solved but only with considerable difficulty. Most practical piping systems do, ia fact, have stresses that are initially ia the nonlinear range. Using linear analysis ia an apparendy nonlinear problem is justified by the stress-range concept... 

This concept is explained by Figure 12 which shows the uniaxial stress— strain curve for a ductile material such as carbon steel. If the stress level is at the yield stress B or above, the problem is no longer a linear one. 

The concept of functionaUty and its relationship to polymer formation was first advanced by Carothers (15). Flory (16) gready expanded the theoretical consideration and mathematical treatment of polycondensation systems. Thus if a dibasic acid and a diol react to form a polyester, assumiag there is no possibihty of other side reactions to compHcate the issue, only linear polymer molecules are formed. When the reactants are present ia stoichiometric amouats, the average degree of polymerization, follows the equatioa ... 

Pressure. Most pressure measurements are based on the concept of translating the process pressure into a physical movement of a diaphragm, bellows, or a Bourdon element. For electronic transmission, these basic elements are coupled with an electronic device for transforming a physical movement associated with the element into an electronic signal proportional to the process pressure, eg, a strain gauge or a linear differential variable transformer (LDVT). 

C-Flex Concept linear EB polyisoprene medical appHcations contains... 

Stiffness The concept of stiffness is described for a system of linear equations. 

The concept of transfer functions facilitates the combination of linear elements. The rule is ... 

The importance of inherent flaws as sites of weakness for the nucleation of internal fracture seems almost intuitive. There is no need to dwell on theories of the strength of solids to recognize that material tensile strengths are orders of magnitude below theoretical limits. The Griffith theory of fracture in brittle material (Griflfith, 1920) is now a well-accepted part of linear-elastic fracture mechanics, and these concepts are readily extended to other material response laws. 

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