Scalars

The scalar t is in the range 0 to 1 and provides step-limiting, or damping, when required to obtain a convergent iteration. 

To go over to projection data of scalar invariant, use for NDTO sounding set of US waves,... 

The approximation of Fresnel is scalar approximation. Let u(, r],0-0) be the scalar wave function of the laser beam falling onto the optical element, and u( X,y,Cl) will the be scalar wave function in the plane Z = Cl. Then [3,4]... 

Finally, the standard draft provides a detailed model of the acquisition data, which intends to describe all the possible shapes which can be taken by NDE data OD (scalar or complex), ID (sampled - cf ultrasonics A-scans - or unsampled - ef ultrasonics time/amplitude data), 2D (images) or 3D (volumes). 

The dipole polarizability tensor characterizes the lowest-order dipole moment induced by a unifonu field. The a tensor is syimnetric and has no more than six independent components, less if tire molecule has some synnnetry. The scalar or mean dipole polarizability... 

The standard analytic treatment of the Ising model is due to Landau (1937). Here we follow the presentation by Landau and Lifschitz [H], which casts the problem in temis of the order-disorder solid, but this is substantially the same as the magnetic problem if the vectors are replaced by scalars (as the Ising model assumes). The themiodynamic... 

If the scalar order parameter of the Ising model is replaced by a two-component vector n = 2), the XY model results. All important example that satisfies this model is the 3-transition in helium, from superfiuid helium-II... 

Onsager relation implies that measurement of one of these effects is sufficient to detemiine the coupling for both. The coefficient L is proportional to the heat conductivity coefficient and is a single scalar quantity in... 

Once again, these fluxes are not all independent and some care must be taken to rewrite everything so that syimnetry is preserved [12]. Wlien this is done, the Curie principle decouples the vectorial forces from the scalar fluxes and vice versa [9]. Nevertheless, the reaction temis lead to additional reciprocal relations because... 

Here we shall consider two simple cases one in which the order parameter is a non-conserved scalar variable and another in which it is a conserved scalar variable. The latter is exemplified by the binary mixture phase separation, and is treated here at much greater length. The fonner occurs in a variety of examples, including some order-disorder transitions and antrferromagnets. The example of the para-ferro transition is one in which the magnetization is a conserved quantity in the absence of an external magnetic field, but becomes non-conserved in its presence. 

A completely difierent approach to scattering involves writing down an expression that can be used to obtain S directly from the wavefunction, and which is stationary with respect to small errors in die waveftmction. In this case one can obtain the scattering matrix element by variational theory. A recent review of this topic has been given by Miller [32]. There are many different expressions that give S as a ftmctional of the wavefunction and, therefore, there are many different variational theories. This section describes the Kohn variational theory, which has proven particularly useftil in many applications in chemical reaction dynamics. To keep the derivation as simple as possible, we restrict our consideration to potentials of die type plotted in figure A3.11.1(c) where the waveftmcfton vanishes in the limit of v -oo, and where the Smatrix is a scalar property so we can drop the matrix notation. 

Note that for potentials that depend only on the scalar distance r between the colliding particles, the amplitude y (9) does not depend on the azimuthal angle associated with the direction of observation. 

Since the potential depends only upon the scalar r, this equation, in spherical coordinates, can be separated into two equations, one depending only on r and one depending on 9 and ( ). The wave equation for the r-dependent part of the solution, R(r), is... 

Nuclear spin relaxation is caused by fluctuating interactions involving nuclear spins. We write the corresponding Hamiltonians (which act as perturbations to the static or time-averaged Hamiltonian, detemiming the energy level structure) in tenns of a scalar contraction of spherical tensors ... 

Scalar coupling 0 Relaxation of the coupled spin or exchange Can be Important for T2 Further reading... 

Hyperfine Interaction (dipolar and scalar) 2,0 Electron relaxation, may be complicated Paramagnetic systems and Impurities [17-191... 

Di Bari L, Kowalewski J and Bodenhausen G 1990 Magnetization transfer modes in scalar-coupled spin systems investigated by selective 2-dimensional nuclear magnetic resonance exchange experiments J. Chem. Rhys. 93 7698-705... 

The coupling constants of the hyperfme and the electron Zeeman interactions are scalar as long as radicals in isotropic solution are considered, leading to the Hamiltonian... 

Forbes M D E 1993 The effect of localized unsaturation on the scalar exchange coupling in flexible biradicals J. Phys. Chem. 97 3390-5... 

Taking advantage of the synnnetry of the crystal structure, one can list the positions of surface atoms within a certain distance from the projectile. The atoms are sorted in ascending order of the scalar product of the interatomic vector from the atom to the projectile with the unit velocity vector of the projectile. If the collision partner has larger impact parameter than a predefined maximum impact parameter discarded. If a... 

Almost everyone has a concept of pressure from weather reports of tlie pressure of the atmosphere around us. In this context, high pressure is a sign of good weather while very low pressures occur at the eyes of cyclones and hurricanes. In elementary discussions of mechanics, hydrostatics of fluids and the gas laws, most scientists leam to compute pressures in static systems as force per unit area, often treated as a scalar quantity. They also leam that unbalanced pressures cause fluids to flow. Winds are the flow of the atmosphere from regions of high to low... 

Figure B2.4.1 shows the lineshape for intennediate chemical exchange between two equally populated sites without scalar coupling. For more complicated spin systems, the lineshapes are more complicated as well, since a spin may retain its coupling infonnation even though its chemical shift changes in the exchange.

In a selective-inversion experiment, it is the relaxation of the z magnetizations that is being studied. For a system without scalar coupling, this is straightforward a simple pulse will convert the z magnetizations directly into observable signals. For a coupled spur system, this relation between the z magnetizations and the observable transitions is much more complex [22]. 

Nicolaides D and Bruce A D 1988 Universal configurational structure in two-dimensional scalar models J. 

This wave equation is tire basis of all wave optics and defines tire fimdamental stmcture of electromagnetic tlieory witli tire scalar function U representing any of tire components of tire vector functions E and H. (Note tliat equation (C2.15.5) can be easily derived by taking tire curl of equation (C2.15.1) and equation (C2.15.2) and substituting relations (C2.15.3) and (C2.15.4) into tire results.)... 

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