Scalar four-dimensional

Note that the scalar product is formally the same as in the nonrela-tivistic case it is, however, now required to be invariant under all orthochronous inhomogeneous Lorentz transformations. The requirement of invariance under orthochronous inhomogeneous Lorentz transformations stems of course from the homogeneity and isotropy of space-time, send corresponds to the assertion that all origins and orientation of the four-dimensional space time manifold are fully equivalent for the description of physical phenomena. 

Dahm, W.J.A., Su, L.K., and Southerland, K.B., A scalar imaging velocimetry technique for fully resolved four-dimensional vector velocity field measurements in turbulent flows, Phys. Fluids, A4, 2191-2206 (1992). 

A four-dimensional scalar product can alternatively be written as... 

For later convenience we introduce the four-dimensional scalar product between any two 4-vectors a and b by... 

Since the four-dimensional scalar product, with the definition of the 4-momentum in Eq. (3.110), yields... 

Dahm WJA, Su LK, Southerland KB (1992) A Scalar Imaging Velocimetry Technique for Fully Resolved Four-dimensional Vec-te Velocity Field Measurements in Turbulent Hows. Phys Huid 4(10) 2191-2206... 

The postulated elementary units should not be confused with waves as perceived in three-dimensional space, but rather as undulations in four-dimensional space-time. The mathematical description of these two types of swaying is fundamentally different. A steady state that results from four-dimensionally balanced forces, as a function of a scalar potential 0, obeys an equation of the type... 

The Hessian matrix H(r) is defined as the symmetric matrix of the nine second derivatives 82p/8xt dxj. The eigenvectors of H(r), obtained by diagonalization of the matrix, are the principal axes of the curvature at r. The rank w of the curvature at a critical point is equal to the number of nonzero eigenvalues the signature o is the algebraic sum of the signs of the eigenvalues. The critical point is classified as (w, cr). There are four possible types of critical points in a three-dimensional scalar distribution ... 

As quaternions have disappeared from the common chemists mathematical background, we recall here the basic properties of this beautiful mathematical instrument. Quaternions are objects having four components. The first one can be treated as a real scalar, whereas the other three can be considered components of a three-dimensional vector. These objects are customarily represented in a form similar to that of complex numbers ... 

In the present chapter we have presented a new family of exponentially-fitted four-step methods for the numerical solution of the one-dimensional Schrodinger equation. For these methods we have examined the stability properties. The new methods satisfy the property of P-stability only in the case that the frequency of the exponential fitting is the same as the frequency of the scalar test equation (i.e. they are singularly P-stable methods). The new methods integrate also exactly every linear combination of the functions... 

If we assume, for instance, that Mj,..., Mj are four covariant vectors, then so are the scalars W . The coordinates obtained in this way we call scalar coordinates. We are however, free also to choose other transformation laws for the Mj. Even so we can also introduce general affine coordinates in the five-dimensional associated space. We do this through the transformation... 

Four decades ago, heteronuclear H X double-resonance CW experiments became possible, by which the NMR sensitivity of the H nucleus was exploited in order to gain information on X nuclei. The prerequisite was scalar X—coupling. The development of two-dimensional (2D) NMR methods aimed for X/ H correlations. In the beginning, for simplicity of the experiment, X resonances were observed and later on, in the so-called inverse experiments in H/X correlations, resonances were onserved with the encoded information on the Si nuclei. ... 

Two-dimensional NOESY allows detection of spatial proximity between protons that are separated by less than 4.5 A. By the correlated spectroscopy method (COSY) scalar couplings between protons, which are separated by at most three (in some exceptions four) chemical bonds, are used. In the case of ribooligonucleotides such couplings consequently can only be observed between the protons of the ribose rings and the C5 and C6 protons of the pyrimidine bases of a given nucleotide. The NOESY method, however, permits the detection of contacts between the protons of different nucleotides. 

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