World scalar

As a particle moves in ordinary space its corresponding point in 4-space describes a path known as the world line. The vector dxM represents the change in the position 4-vector for differential motion along a world line. The absolute magnitude of the 4-vector is described by the dot product with itself and defines the invariant world scalar... 

Four-vectors for which the square of the magnitude is greater than or equal to zero are called space-like when the squares of the magnitudes are negative they are known as time-like vectors. Since these characteristics arise from the dot products of the vectors with reference to themselves, which are world scalars, the designations are invariant under Lorentz transformation[17], A space-like 4-vector can always be transformed so that its fourth component vanishes. On the other hand, a time-like four-vector must always have a fourth component, but it can be transformed so that the first three vanish. The difference between two world points can be either space-like or time-like. Let be the difference vector... 

The central idea in both proposals is that signals between emitter and absorber, going respectively forward and backward in time, could serve to establish two-way contact and facilitate the exchange of radiant energy as a transmission rather than emission. Both authors emphasized the fact that on the relativistic light cone, the vanishing world scalar... 

Momentum and energy are therefore linked together in a manner like that which joins the concepts of space and time9. The world momentum scalar... 

Here we propose a new reduced-cost variant of W1 theory which we shall denote Wlc (for cheap ), with Wlch theory being derived analogously from Wlh theory. Specifically, the core correlation and scalar relativistic steps are replaced by the approximations outlined in the previous two sections, i.e. the MSFT bond additivity model for inner-shell correlation and scaled B3LYP/cc-pVTZuc+l Darwin and mass-velocity corrections. Representative results (for the W2-1 set) can be seen in Table 2.1 complete data for the molecules in the G2-1 and G2-2 sets are available through the World Wide Web as supplementary material [63] to the present paper. 

Chirality in the world of observables is characterized by pseudoscalar properties—properties that remain invariant under proper rotation but change sign under improper rotation. Enantiomers and, in general, enan-tiomorphous molecules, have identical scalar properties, such as melting points or dipole moments, and pseudoscalar properties that are identical in... 

All intermolecular interactions can be adequately described, at least in principle, by multidimensional scalar and vector fields representing the energetics of a molecular system as functions of both intermolecular distances and orientations as well as intramolecular structure data. The visualization of these fields, however, has to be based on a three-dimensional picture or a two-dimensional projection because human pattern recognition ability is strongly related to the two- and three-dimensional world. Consequently, the multidimensional field has to be reduced to a two- or three-dimensional representation. In molecular science this can be done in many different ways. 

As discussed in Chapter 1, most real-world engineering problems require the simultaneous optimization of several objectives (multi-objective optimization, MOO) that cannot be compared easily with one another, i.e., are non-commensurate. These cannot be combined into a single, meaningful scalar objective function. A simple, two-objective example, involving two decision (n = 2) variables, x (= [xi, X2, is... 

Vamvatsikos D, Cornell CA (2002) Incremental dynamic analysis. Earthq Eng Stract Dyn 31(3) 491-514 Vamvatsikos D, Cornell CA (2004) Applied incremental dynamic analysis. Earthq Spectra 20(2) 523-553 Vamvatsikos D, Cornell CA (2005) Developing efficient scalar and vector intensity measures for IDA capacity estimation by incorporating elastic spectral shape information. Earthq Eng Struct Dyn 34 1573-1600 Veletsos AS, Newmaik NM (1960) Effect of inelastic behavior on the resptmse of simple systems to earthquake motions. In Proceedings of the 2nd world conference on earthquake enginetaing, vol 2. Japan, PP 895-912... 

X(k) - normalized scalar parameter that describes the position and orientation of the rigid object at time kAt on the given object path in the world coordinate system, where k=0,l,2,-. X( )=0 at the starting position and orientation of the path and X( )=l at the final position and orientation of the path, p, O - 3x1 vectors of position and orientation, respectively. 

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