Vector functionals and

In formulating physical problems it is often necessary to associate with every point (x, y, z) of a region R of space some vector a(x, y, z). It is usual to call a(x,y,z) a vector function and to say that a vector field exists in II. If a scalar x, y, z) is defined at every point of R then a scalar field is said to exist in R. 

The operator V- may be applied to any vector function and the result is called the divergence, e.g. 

There is no constraint among the three functions Qa except on the boundary. Hence inside "F, 6 Qa is an independent vector function, and Eq. (29) leads to the vector equation... 

The Laplacian measures the local depletion or concentration of the function. The two other combinations produce a vector from a vector function and are used less commonly. 

The first commutator has been neglected in in both Eqs. (131) and (132), whereas the remaining two commutators were neglected only in the T2 equation. The removal of entire commutators assures the eize-extensivity of the CCSD(F12) energy [5, 41]. The Eqs. (130)-(132) are of a general form that is not yet suitable for the implementation. In the present work very often the expressions vector function and residual are used. They always refer to the many-index quantity, defined by the right hand sites of these equations. The working expressions of the coupled-cluster Ti, T2 and T2/ residuals are discussed in next subsections. 

The operators act on the space of smooth vector functions, and the Lie algebra Afn i is realized by square zero-trace matrices. The matrices a and b are diagonal with distrinct diagonal elements. According to the finite-zoned integration theory (see [77]), the commutativity equations [Lai Aa] = 0 are integrated by means of the theta-functions of the Riemann surface of the algebraic curve Q W A) = det(lV — X - Aa) = 0. 

The main steps of the described approach are (i) the use of modal analysis to decouple the equation of motion (ii) the determination, in state variable, of the evolutionary frequency response vector functions and of the evolutionary power spectral density function matrix of the structural response and (iii) the evaluatiOTi of the nongeometric spectral moments as weU as the spectral characteristics of the stochastic response of linear systems subjected to stationary... 

Here 2 ° (t " ) is the vector function and 2 (t " ) the Jacobian matrix calculated from the amplitudes... 

In (1.38)-(1,39), the electromagnetic fields are expressed in terms of the unknown scalar functions T>a and V/3, while in (1.41) and (1.42), the electromagnetic fields are expressed in terms of the unknown expansion coefficients Cmn and dmn These unknowns will be determined from the boundary conditions for each specific scattering problem. The vector functions and can be regarded as a generalization of the regular vector spherical wave functions and For isotropic media, we have eXfSfs = 1, = 0 and... 

This requirement implies that the same number of basis functions must be used to approximate the surface fields on each layer. For concentrically layered spheres, this requirement is not problematic because the basis functions are orthogonal on spherical surfaces. For nonspherical layered particles, we approximate the surface fields by a complete system of vector functions and it is natural to use fewer basis functions for smaller layer surfaces. However, for convergence tests it is simpler to consider a single trimcation index [181,248]. 

The function (p is called the potential of an irrotational vector field and the function is called the potential of a solenoidal vector field. 

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.)... 

The effective nuclear kinetic energy operator due to the vector potential is formulated by multiplying the adiabatic eigenfunction of the system, t t(/ , r) with the HLH phase exp(i/2ai ctan(r/R)), and operating with T R,r), as defined in Eq. fl), on the product function and after little algebraic simplification, one can obtain the following effective kinetic energy operator. 

The non-adiabatic coupling matrix t will be defined in a way similar to that in the Section V.A [see Eq. (51)], namely, as a product between a vector-function t(i) and a constant antisymmetric matrix g written in the form... 

For a multidimensional function, the variable x is replaced by the vector x and matrices are used for the various derivatives. Thus if the potential energy is a function of 3N... 

If A, S and C are differentiable vector functions of scalar t and is a differentiable function of / then... 

If V is the volume bounded by a closed surface S and A is a vector function of position with continuous derivatives, then... 

Figure 3.14 General representations of stress and strain out of phase by amount 5 (a) represented by oscillating functions and (b) represented by vectors.

P is a vector of inputs and T a vector of target (desired) values. The command newff creates the feed-forward network, defines the activation functions and the training method. The default is Fevenberg-Marquardt back-propagation training since it is fast, but it does require a lot of memory. The train command trains the network, and in this case, the network is trained for 50 epochs. The results before and after training are plotted. 

The state of any particle at any instant is given by its position vector q and its linear momentum vector p, and we say that the state of a particle can be described by giving its location in phase space. For a system of N atoms, this space has 6iV dimensions three components of p and the three components of q for each atom. If we use the symbol F to denote a particular point in this six-dimensional phase space (just as we would use the vector r to denote a point in three-dimensional coordinate space) then the value of a particular property A (such as the mutual potential energy, the pressure and so on) will be a function of r and is often written as A(F). As the system evolves in time then F will change and so will A(F). 

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