Invariance in form

Covariance of equations shall denote their invariance in form under coordinate transformations. Covariance of Newton s first law, Xi = 0 (i = 1,2,3), under general Galilean transformations given by Eqs. (2.14) and (2.15) is obvious, since... 

The reader should compare this formulation with its nonrelativistic counterpart given in section 2.1.2. As a direct consequence of the covariance, i.e., invariance in form, of Maxwell s equations we obtain the desired result that the speed of light c is the same in all inertial frames of reference, cf. Eq (3.1). This immediately implies that the coordinate transformations relating the description of events in different inertial frames can no longer be the Galilei transformations but have to be replaced by more suitable coordinate transformations. 

It is the fundamental Eq. (4.16), which has to obey the principle of Einstein s theory of special relativity, namely of being invariant in form in different inertial frames of reference. Hence, the choices for the Hamiltonian operator H are further limited by the requirement of form invariance of the whole equation under Lorentz transformations which will be discussed in detail in chapter 5. 

One postulate that has not explicitly been formulated as a basic axiom of quantum mechanics in the last chapter, because this postulate is valid for any physical theory, is that the equations of quantum mechanics have to be valid and invariant in form in all intertial reference frames. In this chapter, we take the first step toward a relativistic electronic structure theory and start to derive the basic quantum mechanical equation of motion for a single, freely moving electron, which shall obey the principles of relativity outlined in chapter 3. We are looking for a Hamiltonian which keeps Eq. (4.16) invariant in form under Lorentz transformations. 

The most important requirement for truly fundamental physical equations is their invariance in form under Lorentz transformations (principle of relativity). To investigate the behavior of the Dirac equation in Eq. (5.23) under Lorentz transformations, we rewrite it as... 

Because symmetry operators eommute with the eleetronie Hamiltonian, the wavefunetions that are eigenstates of H ean be labeled by the symmetry of the point group of the moleeule (i.e., those operators that leave H invariant). It is for this reason that one eonstruets symmetry-adapted atomie basis orbitals to use in forming moleeular orbitals. 

Note that 0" < A< 60". The invariants A , and form a cylindrical coordinate system relative to the principal coordinates, with axial coordinate / A equally inclined to the principal coordinate axes, with radial coordinate /3t, and with angular coordinate The plane A" = 0 is called the II plane. Because the principal values can be ordered arbitrarily, the representation of A through its invariants in n plane coordinates has six-fold symmetry. 

Poly(dG-dC) poly(dG-dC) and its methylated analogue structures assume left-handed conformation (Z-DNA) in high molar sodium salt (Na", K" ), in low molar divalent cations (Ca", Mg", Ni ), micromolar concentrations of hexaamine cobalt chloride (Co(NH3)6)Cl3 and in millimolar concentrations of polyamines. In order to analyse the binding of berberine to Z-form DNA, Kumar et al. [186] reported that the Z-DNA structure of poly(dG-dC) poly(dG-dC) prepared in either a high salt concentration (4.0 M) or in 40 mM (Co(NH3)6)Cl3 remained invariant in the presence of berberine up to a nucleotide phosphate/alkaloid molar ratio of 0.8 and suggested that berberine neither bormd to Z-form DNA nor converted the Z-DNA to the... 

The example specified as II a in Fig. 19 reveals the pattern found invariably in the methanol, ethanol, and 2-propanol inclusions of /. It is characterized by a loop of H-bonds which always involves two guest molecules opposing each other through a center of symmetry and two carboxyl groups of two symmetry-related molecules of 1 thus having adverse chirality (Fig. 17 a). The loop of H-bonds seems to be formed with... 

Thus, the orbitals uk and vk satisfy Hartree-Fock equations which are identical in form and differ only in the numerical values of the constants X/Jt and Ajk respectively. But since the latter are unknowns in the equation, and since 7(p) is itself invariant as shown in Eq. (21), we can say that the Hartree-Fock self-consistent-field equations are invariant under the orbital transformation given by Eqs. (5) and (6). This means in effect, that the energy integral ( H "X11/0 is minimized by the vk s as well as by the uk s — a circumstance which is in agreement with the invariance of and ( 1 under the transformation (5). 

Figure 2.13. Building blocks of binary phase diagrams examples of three-phase (invariant) reactions. In the upper part the general appearance, inside a phase diagram, of the two types of invariant equilibria is presented, that is, the so-called 1 st class (or eutectic type) and the 2nd class (or peritectic type) equilibria. In the lower part the various invariant equilibria formed by selected binary alloys for well-defined values of temperature and composition are listed. In the Hf-Ru diagram, for instance, three 1 st class equilibria may be observed, 1 (pHf) — (aHf) + HfRu (eutectoid, three solid phases involved), 2 L — (3Hf + HfRu (eutectic), 3 L —> HfRu + (Ru) (eutectic).

The catalytic center is formed by residues from both lobes. Sequence comparisons, mutation experiments and biochemical studies indicate an essential fimction in catalysis of phosphate transfer for the conserved amino acids Lys72, Aspl66 and Aspl84 (numbering of PKA). However, the catalytic mechanism of phosphate transfer is not definitely established. It is generally assumed that Aspl66, which is invariant in all protein kinases, serves as a catalytic base for activation of the Ser/Thr hydroxyl and that the reaction takes place by an in-line attack of the Ser-OH at the y-phosphate. 

As is well known in classical electromagnetics, the fields described by the Maxwell equations can be derived from a vector potential and a scalar potential. However, there are various forms that are possible, all giving the same fields. This is referred to as gauge invariance. In making measurements at some point... 

Electron motion is more generally formulated in a form of the Schrodinger equation, including the spin in the presence of external fields known as the Pauli equation. This equation is gauge invariant in the sense that a transformation as in (5) also changes the quantum wavefunction as... 

So our choices of the two antennas is not unique for separately emphasizing the Lorenz vector and scalar potentials. All that is required is for the two to have the same exterior fields (say, electric dipole fields, or more general multipole fields) with different potentials (related by the gauge condition). In a classical electromagnetic sense, these antennas cannot be distinguished by exterior measurements. This is a classical nonuniqueness of sources. In a QED sense, the same is the case due to gauge invariance in its currently accepted form. 

MINEEALS OF TIN.—Tin is found almost invariably in the form of biDoxido of tin or tin-stone, and in some instances associated with iron and copper pyrites, forming what is called tin pyrites or bell-metal ore, but the quantity so existing is very small. 

In 77) the authors give dependencies of the maximum Newtonian viscosity upon amplitude of periodic strain velocity q0 = f(e) for polyethylene and polystyrene. It has been also revealed that the dependency of normalized viscosity upon the velocity of stationary shear T /r 0 = f(y q0) obtained at r. = 0 coincides with a similar dependency when acoustic treatment is employed, i.e., at e 0. In other words, the effect of shear vibrations and velocity of stationary shear upon valuer] can be divided, representing the role of the first factor in form of dependency q0(s0) and that of the second in form of dependency (n/q0) upon (y r 0) invariant in relation to e. 

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