Scalar nonlinearity

Some comments about nonlinearities in the Hamiltonian may be added here. The case we are considering here is called scalar nonlinearity (in the mathematical literature it is also called nonlocal nonlinearity ) [7] this means that the operators are of the form P(u) = (An, u)Bu where A, B are linear operators and<.,.>is the inner product in a Hilbert space. The literature on scalar nonlinearities applied to chemical problems is quite scarce (we cite here a few papers [2,8]) but the results justified by this approach are of universal use in solvation methods. 

This "self-scaling idea was further developed by Spedicato 15 who considered formulae which were invariant to a scalar nonlinear transformation of f(x), and this also generalizes other attempts to approximate f(x) using more general classes than quadratic functions 16,17,18,19. ... 

The scalar nonlinear Helmholtz equations governing the quadratic interactions of two linearly polarized plane waves at fundamental frequency (FF) co, and SH frequency 2 in a layered, 1-D, finite structure can be written as [11 - 15] ... 

First, we deduce the theorem of Cauchy about representation of scalar (nonlinear) functions of vectors [12,SqcL 11] Ifa scalar function a ofr vectors ya (a = 1,..., r) is isotropic, i.e., if... 

This is scalar nonlinear equation for A Ai. We approximate its solution using a linear approximation at Q thus we have... 

A scalar operator which considers simultaneously complex hysteresis effects, log(t)-type creep effects as well as saturation effects can be constructed by the parallel connection of a Prandtl-Ishlinskii hysteresis operator H and a Prandtl-Ishlinskii log(t)-type creep operator K followed by a concatenation with a memory-free scalar nonlinearity S. In this case the mapping T in (6.75) and (6.76) is given by a so-called modified Prandtl-Ishlinskii creep extension Mk. The corresponding reconstruction model is then given by (6.77) and (6.78) with the compensator Tg... 

Approximating the nonlinear force f(y) over a time step by a suitable constant vector leads to a scheme whose origins for scalar equations can be traced back to [10] ... 

In order to describe second-order nonlinear optical effects, it is not sufficient to treat (> and x<2) as a scalar quantity. Instead the second-order polarizability and susceptibility must be treated as a third-rank tensors 3p and Xp with 27 components and the dipole moment, polarization, and electric field as vectors. As such, the relations between the dipole moment (polarization) vector and the electric field vector can be defined as ... 

Propagation of non-stationary light beam in a nonlinear medium with material dispersion is described by the scalar wave equation for the linearly-polarized y-component of electrical field E x,z,t) ... 

Using a single laser to excite multiple species or phenomena allows multiple parameters to be imaged simultaneously with multiple cameras. A Nd-YAG pumped dye laser with nonlinear crystal doubling was tuned either on or close to (but off) the (1,0) A-X transition of OH radicals located near 283 nm and two cameras simultaneously monitored selected combinations of two of five scalars OH via PLIF, PAH (via PLIF), soot (via LII), fuel, and air (both via acetone... 

In order to realize molecular-vibration spectroscopy, coherent anti-Stokes Raman scattering (CARS) spectroscopy is employed, which is one of the most widely used nonlinear Raman spectroscopes (Shen 1984). CARS spectroscopy uses three incident fields including a pump field (< i), a Stokes field (0)2, 0 2 < 1) and a probe field (<0/ = <0i), and induces a nonlinear polarization at the frequency of <03 = 2<0i - <02 which is given in a scalar form by... 

Second harmonic generation (SHG) is one of the most intensively studied nonlinear optical effects that have ever been combined with near-held scanning optical microscopy (Shen et al. 2000 Zayats and Sandoghdar 2000 Zayats and Sandoghdar 2001 Takahashi and Zayats 2002). SHG, which is an even-order nonlinear process, is forbidden in centrosymmetric media under the dipole approximation (Shen 1984). Non-centrosymmetric molecules and lattices are allowed to exhibit SHG light. The second-order nonlinear polarization for SHG (T shg) is given in a scalar form by... 

Another class of problems requiring iteration is minimization or maximization of a nonlinear scalar valued function g which depends on one or ( variables x (ref. 4). A value r of the independent variables is a local minimum point if g(r) < g(x) for all x in a neighborhood of r. ... 

Fig. 15.5 Illustration of the full Newton and modified Newton algorithm on a scalar problem, F(y) = 0. The curve represents a nonlinear function F(y), and the solution is the value of y at which the function is zero.

The standard Maxwell equations are the exact linearization by change of variables (not by truncation) of a set of nonlinear equations referring to the scalars <[), 0. The fact that this change is not completely invertible produces a hidden nonlinearity, thanks to which the linearity of the Maxwell equations is compatible with the existence of topological constants of motion that are nonlinear in A 1 and F[IV. 

Note that (139) are highly nonlinear in the scalars but become exactly the linear Maxwell equations in the fields F v and F 1V. In this sense, the Maxwell equations are the exact linearization (by change of variables, not by truncation ) of a nonlinear theory with topological properties, in which the force lines... 

We have found a structure with two levels. At the deeper one, it is nonlinear since the scalars < ) and 0 obey highly nonlinear equations. However, the transformation T a —>, given by (9) and (12)... 

Nonlinear models described by nonlinear equations (these can be transcendental systems of scalar equations, ODEs, or PDEs). These models can generally only be solved numerically. 

In the form of Eq. (33) j3 is a scalar parameter called a transport coefficient. In the form of Eq. (34) B is a tensor, the elements of which are the transport coefficients. In either form the transport coefficients may be complex nonlinear functions of the scalar invariants of V0. 

In this work, we will most frequently use the symbols c = c(nr) when referring to both the time signals and the autocorrelation functions Cn = C(nr). The set exp (—ia>kt) is exceedingly nonorthogonal, and this property causes numerical difficulties in all methods for nonlinear fittings of experimental time signals c to the form (42) as discussed in Ref. [46]. If instead of the matrix element (4>0 US 0), we consider a more general scalar product... 

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