Friedman equation

From the Friedman equation, because p varies as R4 (radiation-dominated) or at least as R 3 (matter-dominated), 2 = 1 at early times, to a very good approximation. (If it is minutely below 1, it becomes very much less than 1 at later times, and conversely only if it is exactly 1 does it remain so.) With 2 = 1, Eq. (4.10) becomes... 

If you write the Friedman equation in one of several standard forms as... 

The equations by Friedman et al. [85] 24.9.i and Fraser et al. [87] 24.9.iii are almost the same. If one reduces the equations, the exponents on each of the terms work out to be the same, with the exception of nh. The Friedman equation states SMD is proportional to while the Fraser equatirai states that the SMD is... 

Unfortunately, the logarithm gives negative values for a cooling process, which is mathematically invalid, as demonstrated by Vyazovkia Friedman proposed equation which is used to avoid obtaining erroneous values. The Friedman equation can be... 

H.L, Friedman (12), in 1962. proposed that there is a higher-order electrostatic effect, or limiting law, even when mixing ions of the same charge, Pitzer presented Friedmans equations for Gibbs excess free energy in a molality basis and omitted terms occurring due to differences in concentration, molality, etc, as these are dealt with by Pitzer s virial coefficients ... 

Rasaiah J C and Friedman H L 1968 Integral equation methods in computations of equilibrium properties of ionic solutions J. Chem. Phys. 48 2742... 

Friedman [12] introduced a Bayesian approach the Bayes equation is given in Chapter 16. In the present context, a Bayesian approach can be described as finding a classification rule that minimizes the risk of misclassification, given the prior probabilities of belonging to a given class. These prior probabilities are estimated from the fraction of each class in the pooled sample ... 

In simple terms, the meaning of the equation is that by measuring the thickness of a hydration layer on the surface of a piece of obsidian or of an obsidian tool, it is possible to calculate when the surface was first created and became exposed to the environment (Stevenson et al. 2000 Friedman and Smith 1960). 

Of course, as was shown in Section V-A, this latter expression may also be derived starting from the hydrodynamical equations for the pair distribution and the Poisson equation it is also the final result of the theories developed independently by Falken-hagen and Ebeling,9 and by Friedman 12-13 in these two approaches, the starting point is a Liouville equation for the system of ions with an ad hoc stochastic term describing the interactions with the solvent. 

Friedman (1962) has used the cluster theory of Mayer (1950) to derive equations which give the thermodynamic properties of electrolyte solutions as the sum of convergent series. The first term in these series is identical to and thus confirms the Debye-Huckel limiting law. The second term is an I2.nl term whose coefficient is, like the coefficient in the Debye-Huckel limiting law equation, a function of the charge type of the salt and the properties of the solvent. From this theory, as well as from others referred to above, a higher order limiting law can be written as... 

In order to simplify the Hirschfelder solution, Friedman and Burke [8] modified the Arrhenius reaction rate equation so the rate was zero at T = T0, but their simplification also required numerical calculations. 

Thus even approximate analytical solutions are often more instructive than the more accurate numerical solutions. However considerable caution must be used in this approach, since some of the approximations, employed to make the equations tractable, can lead to erroneous answers. A number of approximate solution for the hot spot system (Eq 1) are reviewed by Merzhanov and their shortcomings are pointed out (Ref 14). More recently, Friedman (Ref 15) has developed approximate analytical solutions for a planar (semi-infinite slab) hot spot. These were discussed in Sec 4 of Heat Effects on p H39-R of this Vol. To compare Friedman s approximate solutions with the exact numerical solution of Merzhanov we computed r, the hot spot halfwidth, of a planar hot spot by both methods using the same thermal kinetic parameters in both calculations. Over a wide range of input variables, the numerical solution gives values of r which are 33 to 43% greater than the r s of the approximate solution. Thus it appears that the approximate solution, from which the effect of the process variables are much easier to discern than from the numerical solution, gives answers that differ from the exact numerical solution by a nearly constant factor... 

A. Friedman and K. Tintarev, Boundary asymptotics for solutions of the Poisson-Boltzmann equation, J. Differential Equations, 69 (1987), pp. 15-38. 

A. Kolmogoroff, op. cit. in LI A. Friedman, Stochastic Differential Equations and Applications I (Academic Press, New York 1975) ch. I. See, however, the footnote on p. 296 of A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York 1965). 

Such purely mathematical problems as the existence and uniqueness of solutions of parabolic partial differential equations subject to free boundary conditions will not be discussed. These questions have been fully answered in recent years by the contributions of Evans (E2), Friedman (Fo, F6, F7), Kyner (K8, K9), Miranker (M8), Miranker and Keller (M9), Rubinstein (R7, R8, R9), Sestini (S5), and others, principally by application of fixed-point theorems and Green s function techniques. Readers concerned with these aspects should consult these authors for further references. 

Fig. 4.3. Relaxation time of absorption of HTO vapour by raindrops. Experimental results of Booker, 1965 (O) and Friedman etal., 1962 ( ). Line is theoretical, derived from equations (4.7) and (4.11).

The paper is organized as follows in section 2, we briefly review the geometry and dynamics of the Universe and then give the Einstein-Friedman-Lemaitre (hereafter EEL) equations section 3 introduces some important quantities needed for observations in section 4, we rapidly present some solutions of the EEL equations, i.e. some cosmological models in section 5, the Standard Big Bang Nucleosynthesis Model is described while section 6 shows a statement of observations of primordial abundances in section 7, we confront the predictions of the Standard Big Bang Nucleosynthesis (hereafter SBBN) model to the observations of the primordial abundances a brief conclusion is... 

Partial molar entropies of ions can, for example, be calculated assuming S (H+) = 0. Alternatively, because K+ and Cl ions are isoelectronic and have similar radii, the ionic properties of these ions in solution can be equated, e.g. analysis of B-viscosity coefficients (Gurney, 1953). In other cases, a particular theoretical treatment which relates solvation parameters to ionic radii indicates how the subdivision could be made. For example, the Bom equation requires that AGf (ion) be proportional to the reciprocal of the ionic radius (Friedman and Krishnan, 1973b). However, this approach involves new problems associated with the definition of ionic radius (Stem and Amis, 1959). In another approach to this problem, the properties of a series of salts in solution are plotted in such a way that the value for a common ion is obtained as the intercept. For example, when the partial molar volumes of some alkylammonium iodides, V (R4N+I ) in water (Millero, 1971) are plotted against the relative molecular mass of the cation, M+, the intercept at M + = 0 is equated to Ve (I-) (Conway et al., 1966). This procedure has been used to... 

Addition Reaction. The double bond of dehydroalanine and e-methyl dehydroalanine formed by the e-elimination reaction (Equation 6) is very reactive with nucleophiles in the solution. These may be added nucleophiles such as sulfite (44). sulfide (42), cysteine and other sulfhydryl compounds (20,47), amines such as a-N-acetyl lysine (47 ) or ammonia (48). Or the nucleophiles may be contributed by the side chains of amino acid residues, such as lysine, cysteine, histidine or tryptophan, in the protein undergoing reaction in alkaline solution. Some of these reactions are shown in Figure 1. Friedman (38) has postulated a number of additional compounds, including stereo-isomers for those shown in Figure 1, as well as those compounds formed from the reaction of B-methyldehydroalanine (from 6 elimination of threonine). He has also suggested a systematic nomenclature for these new amino acid derivatives (38). As pointed out by Friedman the stereochemistry can be complicated because of the number of asymmetric carbon atoms (two to three depending on derivative) possible. 

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