Linearity defined

Mercer [Mercer, 1993] presents results for the two models discussed. For the memoryless non-linearity a section of music from a recording of a brass band was passed through the non-linearity defined by ... 

The problem treated in this paper is complicated by the fact that the linear operator T is supposed to be linearly defined on a linear space A = F of all complex functions F = F(X) of a real composite variable X = jcl5 x2, , xN, of which the L2 Hilbert space is only a small subspace. We will refer to this space A = F as the definition space of the operator T. Since it contains all complex functions, it is stable under complex conjugation, and, according to Eq. (1.50) one has T F = (TF ), which means that also the complex conjugate operator T is defined on this space. 

Linearity Defines the ability of the method to obtain test results proportional to the concentration of analyte. 

Optimization problems in which objective function and/or constrains are not linear define nonlinear optimization problems. While linear optimization problems can be solved in polynomial time, generally nonlinear optimization problems are much more difficult to solve. In discrete optimization problems, variables are defined discrete and thus they are nmilinear optimization problems. 

Nonetheless, the methodological potential of simple QQSPR techniques obviously appears in the present case, as no classic QSPR procedure can provide (1) a relationship for stereoisomer systems, able to offer a clear parameter, describing the distinction between R and S nature, as the one found in the form of a QS Euclidean distance and (2) a simple QQSPR for truly heterogeneous molecules. The origin of the nonlinear QQSPR structure found might be hidden in the approximate linearly defined structure of the QQSPR operator. As shown in Equation 17.9, the QQSPR operator is not completely appropriate to handle the R-S problem, as evidenced by the computed polynomial relations. Nevertheless, it can be said that there exists a linear relation of a type similar to Equation 17.10 for enantiomers, where the operator Q needs the addition of a new nonlinear extra term correcting the expectation value, expression. 

The present paper may be considered an extension of other efforts of the writer [6-9, 12] to state the bridge flutter-buffeting problem analytically with linearly defined deterministic parameters. In particular, the work outlined below constitutes the extension to three dimensions of the theory given in Ref.[9], as was promised in that paper. The final result offers what may be a useful theoretical format for examining the roles of both structural and aerodynamic parameters in specific practical cases. 

The functional behavior at large shear rates on the same log-log plot in Fig. 2.16, being linear defines K and n directly. The slc of the straight line defines n - 1, and is the value of when if = 1 (provided the point satisfies the power-law equation). The best values of K and r — 1 can be computed easily by regressional analysis of the data at higher shear rates. 

The Newton-Raphson approach, being essentially a point-slope method, converges most rapidly for near linear objective functions. Thus it is helpful to note that tends to vary as 1/P and as exp(l/T). For bubble-point-temperature calculation, we can define an objective function... 

The relationship between the flowrate (Q) towards the well and the pressure drawdown is approximately linear, and is defined by the productivity index (PI). 

Linearity Differential and integral distortion Vd,i and homogeneity Fld,i are defined at various locations of the image converter input to be able to establish the linearity of the imaging system. 

The geometrical measurements previously extracted help the making decision system to decide for example whether the defect is linear or not. This defect discrimination into two categories is considered as a first attempt for defect classification. To this end, we define a linearity ratio (Ri) Rl =Length / width. If Rl is equal or near to "1", the defect is volumic, otherwise it is a linear defect. 

Objects which are not covered completely by the fan-beam, which is defined by the linear detector array and the source can be inspected in a special mode. In this case the usable width of the fan is doubled, by placing the turning centre of the object onto the two edges oft the fan. 

That is, the effect of a translational operation is detennined solely by the vector with components (kyj yj ) which defines the linear momenUim. 

Figure Bl.5.5 Schematic representation of the phenomenological model for second-order nonlinear optical effects at the interface between two centrosynnnetric media. Input waves at frequencies or and m2, witii corresponding wavevectors /Cj(co and k (o 2), are approaching the interface from medium 1. Nonlinear radiation at frequency co is emitted in directions described by the wavevectors /c Cco ) (reflected in medium 1) and /c2(k>3) (transmitted in medium 2). The linear dielectric constants of media 1, 2 and the interface are denoted by E2, and s, respectively. The figure shows the vz-plane (the plane of incidence) withz increasing from top to bottom and z = 0 defining the interface.

The linear and nonlinear optical responses for this problem are defined by e, 2, e and respectively, as indicated in figure Bl.5.5. In order to detemiine the nonlinear radiation, we need to introduce appropriate pump radiation fields E(m ) and (co2)- If these pump beams are well-collimated, they will give rise to well-collimated radiation emitted tlirough the surface nonlmear response. Because the nonlinear response is present only in a thin layer, phase matching [37] considerations are unimportant and nonlinear emission will be present in both transmitted and reflected directions. 

Alternatively, the electron can exchange parallel momentum with the lattice, but only in well defined amounts given by vectors that belong to the reciprocal lattice of the surface. That is, the vector is a linear combination of two reciprocal lattice vectors a and b, with integer coefficients. Thus, g = ha + kb, with arbitrary integers h and k (note that all the vectors a,b, a, b and g are parallel to the surface). The reciprocal lattice vectors a and are related to tire direct-space lattice vectors a and b through the following non-transparent definitions, which also use a vector n that is perpendicular to the surface plane, as well as vectorial dot and cross products ... 

Detailed derivations of the isothemi can be found in many textbooks and exploit either statistical themio-dynaniic methods [1] or independently consider the kinetics of adsorption and desorption in each layer and set these equal to define the equilibrium coverage as a function of pressure [14]. The most conmion fomi of BET isothemi is written as a linear equation and given by ... 

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