Dependence vectors

The strategy for representing this differential equation geometrically is to expand both H and p in tenns of the tln-ee Pauli spin matrices, 02 and and then view the coefficients of these matrices as time-dependent vectors in three-dimensional space. We begin by writing die the two-level system Hamiltonian in the following general fomi. 

In the case of linearly dependent vectors, each of them can be expressed as a linear combination of the others. For example, the last of the three vectors below can be expressed in the form Zj = z. 

Although hie rank of is two, the linearly dependent vector is not formed by a linear mixture. We have again used MATLAB to perform die computations. 

We seek to describe the time-dependent behavior of a metabolic network that consists of m metabolic reactants (metabolites) interacting via a set of r biochemical reactions or interconversions. Each metabolite S, is characterized by its concentration 5,(f) > 0, usually measured in moles/volume. We distinguish between internal metabolites, whose concentrations are affected by interconversions and may change as a function of time, and external metabolites, whose concentrations are assumed to be constant. The latter are usually omitted from the m-dimensional time-dependent vector of concentrations S(t) and are treated as additional parameters. If multiple compartments are considered, metabolites that occur in more than one compartments are assigned to different subscripts within each compartment. 

According to (9.3) this equation can be interpreted as follows. Define a time-dependent vector qn(t) by stipulating that at t = 0 it has the components qn, and for t > 0 it evolves according to (9.1). Then the average of Q at time t equals the average of q (t) over the initial distribution ... 

Thus one has formally transferred the time dependence from the probability distribution onto the observed quantity - in analogy with the quantum mechanical transformation from the Schrodinger representation to the Heisenberg representation. Accordingly one may define a time-dependent vector Q(t) by setting... 

The following terminology is important The set ft = z,... xt of vectors x, 6 S is linearly dependent, iff there exists a set of scalars a,. ..at, not all zero, such that orixi + —h a = 0. If this is not possible, then the vectors are linearly independent. A vector x, for which a, 0 is one of the linearly dependent vectors. The set of vectors defines a vector subspace S, of S, called span(ft), which consists of all possible vectors z = aix, + —h atzt. This definition also provides a mapping from the array., a ) e Rk to the vector space span(ft). If ft is a linearly independent set, then the dimension of S, is k, and then the vectors constitutes a basis set in Si. If it is linearly dependent, then there is a subset fti 6 ft of size ki = card (ft,) which is linearly independent and spans the same space. Then ki is the dimension of S,. 

The quantity A appears in these equations and is the vector potential of electromagnetic theory. In a very elementary discussion of the static electric field we are introduced to the theory of Coulomb. It is demonstrated that the electric field can be written as the gradient of a scalar potential E = —Vc)>, constant term to this potential leaves the electric field invariant. Where you choose to set the potential to zero is purely arbitrary. In order to describe a time-varying electric field a time dependent vector potential must be introduced A. If one takes any scalar function % and uses it in the substitutions... 

Trl(o — At — Tc, but a continuum in the energy domain. Under such conditions the FIR band shape of the absorption due to dipole-dipole coupling between the time-dependent vector of the FIR electric field and the time-dependent vector of the molecular dipole moment can be analyzed by a generalized fluctuation-dissipation theorem (Ref. 1, Eq. 3.E23), since the molecular dipole operator follows a GLE (Ref. 1, Eq. 3.F8) and off-diagonal elements of the density operator must be taken into account (Ref. 1, Eq. 3.E3). 

Each rotational state is coupled to all other states through the potential matrix V defined in (3.22). Initial conditions Xj(I 0) are obtained by expanding — in analogy to (3.26) — the ground-state wavefunction multiplied by the transition dipole function in terms of the Yjo- The total of all one-dimensional wavepackets Xj (R t) forms an R- and i-dependent vector x whose propagation in space and time follows as described before for the two-dimensional wavepacket, with the exception that multiplication by the potential is replaced by a matrix multiplication Vx-The close-coupling equations become computationally more convenient if one makes an additional transformation to the so-called discrete variable representation (Bacic and Light 1986). The autocorrelation function is simply calculated from... 

Ehrhardt, A. and Kay, M. A. (2002). A new adenoviral helper-dependent vector results in long-term therapeutic levels of human coagulation factor IX at low doses in vivo. Blood 99, 3923-3930. 

The superscript EQC in the entries of Table 2.7 related to BE indicates that the origin-dependent quantities to which they are associated refer to the so-called effective quadrupolar centre [14], R(EQC co), a particular frequency-dependent vector in the coordinate space defined with respect to a given choice of origin of the coordinates, or . 

A significant advantage of this procedure is that nearly linearly dependent vectors can be eliminated at this stage, simply reducing the value of m. The resulting unit vectors define an n x in column matrix. Using these unit vectors, the algorithm solves anmxm system of linear equations in the e-space,... 

Difficulties arise in the band structure treatment for quasilinear periodic chains because the scalar dipole interaction potential is neither periodic nor bounded. These difficulties are overcome in the approach presented in [115] by using the time-dependent vector potential, A, instead of the scalar potential. In that formulation the momentum operator p is replaced by tt =p + (e/c A while the corresponding quasi-momentum Ic becomes k = lc + (e/c)A. Then, a proper treatment of the time-dependence of k, leads to the time-dependent self-consistent field Hartree-Fock (TDHF) equation [115] ... 

Therefore, m solutions of linear equations (with a perturbation-dependent vector like W on the right-hand side) can replace the 0 m ) solutions for Note that the situation, though similar, is not completely analogous to the case of the (2n -I- 1) rule. 

This form, involving the Fourier transform of time dependent vector operators, is specifically related to the main theme of this book namely, the vibrational spectroscopy of hydrogenous molecular crystals. Other forms, more appropriate to different disciplines and systems, can be found in the specialist literature (see also Appendix 2). 

If r t) is the time dependent atomic position vector, taken with respect to the origin of the crystallographic cell, it can be expressed as the sum of two terms, the molecular centre of mass, this is a time dependent vector, (0ext> that allows a description of the vibrations of the crystal, the phonons. The second term is the position vector of the atom, H(0int, given by a Cartesian coordinate system with its origin at the molecular centre of mass. 

The functions Tf r. R. O. t = 0) involve a product of the initial wavefunction and the internal coordinate dependent vector components of the transition dipole moment (see Ref. [43] and Appendix B of Ref. [33] ). As the total angular iiioirioii-tuni is a conserved quantity during the time propagation of the wavepackot, wo may divide up the initial wavepacket (Eq. (23)) into three components [43], one for each of the allowed values of J. Thus E(p (23) may be rewritten as ... 

This estimate should be accepted if the process is in a stable state. In order to check if a transition is occurring the time-dependent vector q(t) is inspected. This vector is constructed according to the considerations outlined above ... 

In this paper, we present a novel derivation of the London equation based on DP theory The application of a time-dependent vector potential, A(t) = A(0)oos(2nvt) along a fiber generates a time-dq)endent ground state vector which for small time and weak field has the form... 

Consider, for instance, the time-periodic, spatially dependent vector-potential of an incident electromagnetic field, propagating along the z-axis. The Floquet expression for the vector potential of the nth harmonic emitted in the incident field propagation direction (z) is given by... 

Result (A.19) proves the necessity of a Lemma for linear independent vectors (and in fact it transforms one basis y-y of p-dimensional vector space to another one by linear transformation Q). To prove this for the remaining linearly dependent vectors y, we express them through those which are independent... 

So far, the current density functional has attracted attention, not in the context of the response to a magnetic field, as mentioned above, but to an electric field. The time-dependent Kohn-Sham equation in Eq. (4.27) incorporating the time-dependent vector potential, Aeff, is written as... 

Vignale and Kohn proved that this time-dependent vector potential is Fourier-transformed (t u>) using the current density j, which is usually the paramagnetic current density jp, to... 

The dependence graph corresponding to the APP algorithm contains opposite dependence vectors for instance, node (3,2,2) depends on node (2,2,2) hence the dependence vector (1,0,0). But node (1,2,2) also depends upon node (2,2,2) hence the dependence vector (-1,0,0). The existence of these two opposite... 

The systolic answer to the problem is to localize the broadcasts in the DG before scheduling and mapping its nodes [15], so as to synthesize an architecture where all communications are made local. Such a derivation process is well understood. Some recent synthesis results are discussed in chapter 6. The natural idea is to replace the broadcast of a variable by its pipelined propagation along the direction of the dependence vector. In this way, we obtain the DG of figure 3. 

We would like to point out that the scope-6 broadcast enables us to parametrize the localization of the DG of figure 2. The parameter 6 can be viewed either as the maximal length of the dependence vectors, or as the maximum number of copies of a given variable (the fan-out of the array). Hence, the parameter 6 can be adjusted to cope with current integration constraints. Further results on the scope-6 broadcast transformation are available elsewhere [31, 8, 30]. 

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