Homogeneous function defined

For the special case for which n = 2, it can be shown that the linearization method defined above becomes identical to the Newton-Raphson method. The result may be generalized to apply to any homogeneous function of degree n. 

In homogeneous turbulence, the velocity spectrum tensor is related to the spatial correlation function defined in (2.20) through the following Fourier transform pair ... 

Much of the early work which would lead to the identification of proteins as defined chemical entities started from observations on enzymes, either those involved in fermentation or on the characterization of components in gastric secretions which powerfully catalyzed the hydrolysis of different foodstuffs. As well as the digestive enzymes, a number of relatively pure proteins could be isolated from natural sources where they made up the major component (Table 1). Because of the importance and difficulty of isolating pure proteins and demonstrating their homogeneity, functionally active and relatively abundant... 

Consider a crystal which is in equilibrium having n chemical components (k = 1,2,..., ). We can define (at any given P and T) a Gibbs function, G, as a homogeneous function that is first order in the amount of components... 

This is a special case of Euler s homogeneous functions, which more generally are defined to obey the transformation law... 

Since Eqs. (80-83) are homogeneous they define the pairs of functions gk, fj,1 and gk,fk only up to the normalization factors. From Eqs. (53, 56, 63, 64) we see that the overall normalization factor for the functions gl,/, gk,fk gets cancel out and thus we are left with one free parameter to be fixed by the requirements on the boundary conditions on resulting function g->k- In the superconducting region r>T the components of gok decay to zero due to our asymptotic requirements while in the paramagnetic region there are four equations (one for each component of g->k) that should be satisfied. Fortunately all these four equations can be shown to be linear dependent, thus we are left with one requirement for which we have one free parameter. 

A computational method was developed by Gillespie in the 1970s [381, 388] from premises that take explicit account of the fact that the time evolution of a spatially homogeneous process is a discrete, stochastic process instead of a continuous, deterministic process. This computational method, which is referred to as the stochastic simulation algorithm, offers an alternative to the Kolmogorov differential equations that is free of the difficulties mentioned above. The simulation algorithm is based on the reaction probability density function defined below. 

There are in fact two heat capacities in common use for homogeneous fluids although their names belie the fact, both are state functions, defined unambiguously in relation to other state functions ... 

The derivative of each of the curves 1, 2, 3, etc., where it crosses the Vt line is the derivative of the volume with respect to n, at constant v—i.e., it is the partial molal volume corresponding to G of Equation 8. The functions represented by curves 1, 2, and 3 are not homogeneous. The slope of V is the derivative of the volume at constant specific area. V, of course, was defined so that it is a homogeneous function of n. 

Properties like mass m and volume Vare defined by the system as a whole. Such properties are additive, and are called extensive properties. Separation of the total change for a species into the external and internal parts may be generalized to any extensive property. All extensive properties are homogeneous functions of the first order in the mass of the system. For example, doubting the mass of a system at constant composition doubles the internal energy. 

Euler theorem concerns the so-called homogeneous functions of order n, defined by the property... 

The second equality holds for a homogeneous and isotropic fluid. A related quantity is the locational pair correlation function, defined in terms of the locational pair distribution function, i.e.,... 

We say that G is an extensive function of the extensive variables A and n. By this, we mean that the electrochemical free energy depends linearly on the physical extent of the system. If we double t size of the system by doubling A and all the then G doubles. Mathematically, such behavior implies that G(A, n ) is a linear homogeneous function of A and n. The Euler theorem (5) applies generally to homogeneous functions and, for linear ones, it allows us to define the function itself in terms of derivatives and variables as in (13.1.7). 

In this appendix we present a discussion of a few mathematical techniques frequently utilized in thermodynamics. We treat several topics in the analysis of real functions of several real variables. We assume that the functions considered have the continuity properties necessary for the operations performed upon them to be meaningful. In Sec. A-1, we discuss some of the properties of partial derivatives. In Sec. A-2 we define homogeneous functions and derive a useful relation. In Sec. A-3, we treat linear differential forms. Line integrals are discussed in Sec. A-4. 

In a homogeneous system, the correlation function defined in (5.44) is a function of Vi, V2, and the scalar separation between the two locations 1 S2 — Si. The generalized pair correlation function in (5.44) can also be viewed as an ordinary pair correlation function between two species Vi and V2. This point of view will be discussed further in Section 5.7. 

If we define the vector potential of a homogeneous magnetic field B as A(r) = B X (r — Ro)/2, then the gauge origin Rq dependence can be understood as a gauge transformation with the gauge function defined by x = —(B X Ro) rl1 (compare section 2.4). This vector potential then produces terms that depend on the arbitrary position Rq- Only for exact wave functions do these terms vanish (complete basis set), while they carmot be neglected in any (small) finite one-electron basis set. 

The additivity property of entropy and internal energy of subsystems demands that both S and U of the subsystems be first-order homogeneous functions of extensive properties which define the subsystem. From the definition of the first-order homogeneous property for U, one... 

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