Tensor metric

The matrix gp, represents the components of a covariant second-order tensor called the metric tensor , because it defines distance measurement with respect to coordinates To illustrate the application of this definition in the... 

Al) Freezing of bonds and angles defonns the phase space of the molecule and perturbs the time averages. The MD results, therefore, require a complicated correction with the so-called metric tensor, which undermines any gain in efficiency due to elimination of variables [10,17-20]. 

Bl) The metrics effect is very significant in special theoretical examples, like a freely joined chain. In simulations of polymer solutions of alkanes, however, it only slightly affects the static ensemble properties even at high temperatures [21]. Its possible role in common biological applications of MD has not yet been studied. With the recently developed fast recursive algorithms for computing the metric tensor [22], such corrections became affordable, and comparative calculations will probably appear in the near future. 

We shall denote the space time coordinates by a (which as a four-vector is denoted by a light face x) with x° — t, x1 = x, af = y, xz = z x — ai0,x. We shall use a metric tensor grMV = gliV with components... 

Methods of projection, 61 Metric tensor, 491 Michel, L., 539 Minkowski theorem, 58 Minimization, 286 Minmax, 286,308 approximation, 96 regret or risk riile, 315 theorem, 310... 

The metric term Eq. (2.8) is important for all cases in which the manifold M has non-zero curvature and is thus nonlinear, e.g. in the cases of Time-Dependent Hartree-Fock (TDHF) and Time-Dependent Multi-Configurational Self-Consistent Field (TDMCSCF) c culations. In such situations the metric tensor varies from point to point and has a nontrivial effect on the time evolution. It plays the role of a time-dependent force (somewhat like the location-dependent gravitational force which arises in general relativity from the curvature of space-time). In the case of flat i.e. linear manifolds, as are found in Time-Dependent Configuration Interaction (TDCI) calculations, the metric is constant and does not have a significant effect on the dynamics. 

Approximations thus must be introduced that involve modeling both the XC potential and the metric tensor, and a truncation of the space within which to choose the unknown functions v, to finite dimension r < >. The modeling is based on the restt-icted ansatz chosen for the form of states used to determine paths that approximate D (p), D](p) and ). It can be carried... 

The symplectic metric tensor defined on the tangent spaces of M... 

Equations (56) and (57) give six constrains and define the BF-system uniquely. The internal coordinates qk(k = 1,2, , 21) are introduced so that the functions satisfy these equations at any qk- In the present calculations, 6 Cartesian coordinates (xi9,X29,xi8,Xn,X2i,X3i) from the triangle Og — H9 — Oi and 15 Cartesian coordinates of 5 atoms C2,C4,Ce,H3,Hy are taken. These 21 coordinates are denoted as qk- Their explicit numeration is immaterial. Equations (56) and (57) enable us to express the rest of the Cartesian coordinates (x39,X28,X38,r5) in terms of qk. With this definition, x, ( i, ,..., 21) are just linear functions of qk, which is convenient for constructing the metric tensor. Note also that the symmetry of the potential is easily established in terms of these internal coordinates. This naturally reduces the numerical effort to one-half. Constmction of the Hamiltonian for zero total angular momentum J = 0) is now straightforward. First, let us consider the metric. 

U = metric tensor of the space nc = normal vector in gas-phase direction... 

Fig. 2.5. Possible applications of a coupling parameter, A, in free energy calculations, (a) and (b) correspond, respectively, to simple and coupled modifications of torsional degrees of freedom, involved in the study of conformational equilibria (c) represents an intramolecular, end-to-end reaction coordinate that may be used, for instance, to model the folding of a short peptide (d) symbolizes the alteration of selected nonbonded interactions to estimate relative free energies, in the spirit of site-directed mutagenesis experiments (e) is a simple distance separating chemical species that can be employed in potential of mean force (PMF) calculations and (f) corresponds to the annihilation of selected nonbonded interactions for the estimation of e.g., free energies of solvation. In the examples (a), (b), and (e), the coupling parameter, A, is not independent of the Cartesian coordinates, x. Appropriate metric tensor correction should be considered through a relevant transformation into generalized coordinates...

The most important second order tensor is the metric tensor g, whose components in a Riemann space are defined by the relations... 

It is obvious that gab z) is independent both of the choice of inertial frame at z, with its corresponding natural coordinate system (v), and the choice of curve x(X). The elements of g are known as the components of the metric tensor in this coordinate system. Expression (39) is the required generalization that allows evaluation of 4> at all points in terms of gab %) and the curve x(A). 

The concept of metric tensor becomes central whenever distances and projections are considered, particularly when least-square criterion are used, a point that will be discussed in Chapter 5. Let us ask the frequently raised question of how to find an expression in terms of old coordinates (e.g., oxide proportions) for a projection made in the non-Euclidian space. This could be the case for finding oxide abundances of a basalt composition projected in the Yoder and Tilley tetrahedron, or the oxide abundance of a metamorphic rock composition projected into an ACF diagram assuming that quartz is present. 

Third, the metric tensor is determined by the variables 4>, //, A. On the other hand and v never appear in Eqs.(6)-(9) (reflecting the fact that x° and x5 constant dilatations are always possible without harming the commutator relations for the Killing motions), so these equations are of first order on 4>, / and A. However, the equations can be rearranged resulting in the following symbolic structure ... 

The integration must go until the fluid-vacuum interface at rs, where an exterior vacuum solution continues the interior one. The details of matching conditions can be found in Ref. [14]. Applying them on the present problem we get the continuity of certain derivatives of the metric tensor, and those for the energy-momentum tensor result in the single equation,... 

The generalized Fisher theorems derived in this section are statements about the space variation of the vectors of the relative and absolute space-specific rates of growth. These vectors have a simple natural (biological, chemical, physical) interpretation They express the capacity of a species of type u to fill out space in genetic language, they are space-specific fitness functions. In addition, the covariance matrix of the vector of the relative space-specific rates of growth, gap, [Eq. (25)] is a Riemannian metric tensor that enters the expression of a Fisher information metric [Eqs. (24) and (26)]. These results may serve as a basis for solving inverse problems for reaction transport systems. 

Patriciu, A., Chirikjian, G.S., Pappu, R.V. Analysis of the conformational dependence of mass-metric tensor determinants in serial polymers with constraints. J. Chem. Phys. 2004,121,12708-20. 

A covariant metric tensor g p and contravariant inverse metric tensor in the full space are given by... 

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