Constant, fundamental invariance

Basic physical theories and their application to other fields of science and technology always involve certain fundamental invariant quantities, called briefly fundamental constants. Well-known examples of such constants are the speed of light in vacuum, the elementary charge (electron charge), the mass of the electron, and so on. It is important to know the numerical values of the fundamental constants with the highest possible accuracy, because the attained accuracy determines the accuracy of the quantitative predictions of fundamental theories. Moreover, the accurate numerical values of the fundamental constants test the overall consistency and correctness of those theories. 

As mentioned earlier, heavy polar diatomic molecules, such as BaF, YbF, T1F, and PbO, are the prime experimental probes for the search of the violation of space inversion symmetry (P) and time reversal invariance (T). The experimental detection of these effects has important consequences [37, 38] for the theory of fundamental interactions or for physics beyond the standard model [39, 40]. For instance, a series of experiments on T1F [41] have already been reported, which provide the tightest limit available on the tensor coupling constant Cj, proton electric dipole moment (EDM) dp, and so on. Experiments on the YbF and BaF molecules are also of fundamental significance for the study of symmetry violation in nature, as these experiments have the potential to detect effects due to the electron EDM de. Accurate theoretical calculations are also absolutely necessary to interpret these ongoing (and perhaps forthcoming) experimental outcomes. For example, knowledge of the effective electric field E (characterized by Wd) on the unpaired electron is required to link the experimentally determined P,T-odd frequency shift with the electron s EDM de in the ground (X2X /2) state of YbF and BaF. 

Conversely, the relationship (7.2) expresses a time-scale invariance (selfsimilarity or fractal scaling property) of the power-law function. Mathematically, it has the same structure as (1.7), defining the capacity dimension dc of a fractal object. Thus, a is the capacity dimension of the profiles following the power-law form that obeys the fundamental property of a fractal self-similarity. A fractal decay process is therefore one for which the rate of decay decreases by some exact proportion for some chosen proportional increase in time the self-similarity requirement is fulfilled whenever the exact proportion, a, remains unchanged, independent of the moment of the segment of the data set selected to measure the proportionality constant. 

Any of Eqs. [8], [9] and [10] allow the determination of C and also of relevant rate constants in the simplest case defined above. Complications arise, when these rates are time-or polymer yield-dependent. C is seldom time-invariable formation and deactivation of active centers are quite often difficult to express by simple kinetic laws. Nevertheless, the fundamental Eq. [7] should be valid, and Ivanov et al. 35) and Ermakov and Zakharov 6> suggested its modified version applicable to the non-stationary kinetics of polymerization, if only transfer reactions occur. Supposing that kp, ktr, [M], and [X] are time-independent and substituting C by Rp/kp[M], equation ... 

Although glass transition is conventionally defined by the thermodynamics and kinetic properties of the structural a-relaxation, a fundamental role is played by its precursor, the Johari-Goldstein (JG) secondary relaxation. The JG relaxation time, xjg, like the dispersion of the a-relaxation, is invariant to changes in the temperature and pressure combinations while keeping xa constant in the equilibrium liquid state of a glass-former. For any fixed xa, the ratio, T/G/Ta, is exclusively determined by the dispersion of the a-relaxation or by the fractional exponent, 1 — n, of the Kohlrausch function that fits the dispersion. There is remarkable similarity in properties between the JG relaxation time and the a-relaxation time. Conventional theories and models of glass transition do not account for these nontrivial connections between the JG relaxation and the a-relaxation. For completeness, these theories and models have to be extended to address the JG relaxation and its remarkable properties. 

As with the trends previously mentioned, proposals have been promulgated for internal and external constraints. At first pass, it is tempting to account for relations between life history variables almost purely on the basis of fundamental allometric constraints. Metabolic rate, lifespan, fecundity, age at maturity, and maternal investment all vary with body mass as power functions. In fact, relations are invariant between some of these variables. For example, lifespan scales with body mass by a 1/4 power, and heart rate (or the rate of ATP synthesis) scales with body mass by a — 1/4 power. The product yields an approximately constant number of metabolic events in mammal species, independent of body mass or lifespan. Age at maturity / lifespan, and annual maternal investment / lifespan (for indeterminate growers), are also invariant ratios (Chamov, 1993 Chamov et al., 2001 Steams, 1992). West and Brown (2004) point out that invariant ratios, and universal quarter-power allometric trends in general, suggest underlying physical first principles. They employ their model to explain these life history relations (Enquist et al., 1999 Niklas and Enquist, 2001 West et al., 2001). 

Previously, it had been shown for many glassformers that the frequency dispersion of the a-relaxation (or n) is invariant to changes of T and P if is kept constant. From this added feature of the JG p-relaxation, we have coinvariance of three quantities, r , n, and Tjq, to widely different T and P combinations involving large variations of specific volume and entropy. This remarkable relation between and Tjg is another strong evidence that the JG p-relaxation has fundamental significance and its relation to the a-relaxation must be taken into account. However, none of the theories cited in the NY Times article paid any attention to it. 

Implications. These results have an important implication concerning the use of Fourier analysis of DC transients in polymeric materials to extract the frequency-dependence of the dielectric response (12)- In order for the principle of superposition to apply the electric field inside the material being measured must be time- and space-invariant. This critical condition may not be met in polymers which contain mobile ionic impurities or injected electrons. Experimentally, we can fix only the average of the electric field. Moreover, our calculations demonstrate that the bulk field is not constant in either time or space. Thus, the technique of extracting the dielectric response from the Fourier components of the transient response is fundamentally flawed because the contribution due to the formation of ionic and electronic space-charge to the apparent frequency-dependent dielectric response can not generally be separated from the dipole contribution. 

Galilean invariance (Rothman Zalesky, 1997) is a fundamental tenet of Newtonian mechanics. It is invariance under the transformation x = x - wt, where w is the constant velocity of a moving frame of reference, and embodies the concept that only the relative velocities and positions of two bodies determine their interaction. Galilean invariance is lost in lattice gas simulations because every particle has only one possible speed. This loss is an artifact that can be eliminated for incompressible fluids by re-scaling the velocity. According to Boghosian (1993), more sophisticated lattice gas models overcome this problem. Appropriate application of lattice gas models also requires certain restrictions on the mean free path of a particle (Rothman, 1988). 

Another fundamental mathematical concept important in pharmacokinetics is the difference between a variable and a constant. For the purposes of pharmacokinetics, a variable is something that changes over time. Conversely, a constant is time invariant. Box 2.1 presents some examples of variables and constants as well as rules showing whether an expression containing variables and/ or constants will give rise to a variable or a constant. 

Progress in precision studies and shortage of data on possible extension of the Standard Model of weak, electromagnetic and strong interactions have produced a situation where a number of experiments to search for so-called new physics have been performed or planned in atomic physics. Among such experiments are a search for an electric dipole moment of an electron and a neutron, search for variation of fundamental constants and violation of Lorentz invariance, etc. 

An example of such a problem is a relation of possible time or space variation of fundamental constants and a basic relativistic principle of local position/time invariance (LPI/LTI). Some scientists consider possible variation of constants as violation of LTI. However, that is not correct. 

The simplest issue is an observational one. The variations are long-term changes in values of fundamental constants, while a violation of Lorentz invariance could produce periodic effects because of the Earth rotation and its motion around the Sun (more precisely both motions should be considered with respect to the remote stars ). That can be resolved experimentally. 

The fundamental equations above describing the operation of stirred-flow reactors are valid whether the reaction takes place at constant volume or not. It is important to distinguish here carefully between the volume of the physical enclosure in which the system reacts and the volume V occupied by a given mass of the reacting system. Both are not necessarily equal. Furthermore, while it is clear that F, is practically invariant, F almost always varies with extent of reaction in an isothermal system, except if the reaction mixture is an ideal gas contained in a batch reactor or passed through a flow reactor (provided that in the latter case the reaction is not accompanied by a change in number of moles). 

For tests of Lorentz invariance and time invariance of fundamental constants [3,108]... 

On the basis of this expected invariance of the fundamental constants in space and time, it appears reasonable to relate the units of measurement for physical quanti-... 

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