Homogeneous function generalized

The concept of homogeneity naturally extends to functions of more than one variable. For example, a generalized homogeneous function of two variables, f(x,y), can be written in the form... 

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. 

When f(t) = 0, the equation is called homogeneous. The general solution is the sum of the solution of the homogeneous equation and a complementary function which can be found by several means. The form of the homogeneous solution... 

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

Homogeneous Functions in N Variables in N+2 Iterations and Rapidly Minimizes General Functions"... 

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. 

In the present work, the general mathematical scheme of construction of the equilibrium statistical mechanics on the basis of an arbitrary definition of statistical entropy for two types of thermodynamic potential, the first and the second thermodynamic potentials, was proposed. As an example, we investigated the Tsallis and Boltzmann-Gibbs statistical entropies in the canonical and microcanonical ensembles. On the example of a nonrelativistic ideal gas, it was proven that the statistical mechanics based on the Tsallis entropy satisfies the requirements of the equilibrium thermodynamics only in the thermodynamic limit when the entropic index z is an extensive variable of state of the system. In this case the thermodynamic quantities of the Tsallis statistics belong to one of the classes of homogeneous functions of the first or zero orders. 

Is it possible to make the similarity transformation (7.62) for other collision mechanisms In general, when the collision frequency (v, v) is a homogeneous function of particle volume, the transformation to an ordinary integrodifferential equation can be made. The function ff(v,v) s said tobc/joHiogencoH.vof degree A.if (au,Qrii) = cit (t),5). However, even though the transformation is possible, a solution to the transformed equation may not exist that satisfies the boundary conditions and integral constraints. 

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 the thermodynamic description of multicomponent systems, a principal relation is the Gibbs-Duhem equation. Astarita [1] has shown that the Gibbs-Duhem equation is not merely a thermodynamic relation it is a general repercussion of the properties of homogeneous functions. Consider a multivariant function, such as... 

In this appendix, we present the generalized Euler theorem for homogeneous functions of order one. We first write the Euler theorem for a discrete quasi-component distribution function QCDF) and then generalize by analogy for a continuous QCDF. A more detailed proof is available. ... 

Generalization of Euler s theorem on homogeneous functions to functionals [24, 32] allows one to write for the extensive quantity z... 

The similarity hypothesis (with a ferromagnetic as an example) states that the Gibbs potential G is a generalized homogeneous function of e and H, i.e. there sure two similarity parameters a, and a//, for which... 

The last equation generally resembles the miliar Euler theorem for the homogeneous functionals. For the particular case of two subsystems consisting the... 

In the CVM(PPM), the level of the approximation is determined by the largest cluster involved in the free energy(path probability function). Generally, the larger the basic cluster is, the better results one can expect. On the other hand, however, the number of variables becomes intractable with increasing the size of the basic cluster. Hence, even in the highest approximation employed so far, the extension of the basic cluster is limited only to a few atomic distances, and most of the CVM and PPM calculations assume the homogeneous distribution of the cluster probabilities in the microstructural scale. 

The power-law profiles satisfy Eq. (16-19b), and are therefore homogeneous function profiles. Consequently, the above expression is consistent with the more general result of Eq. (16-20). In the case of the infinite linear profile ( = 1), Eq. (17-6) reduces to the exact eigenvalue equation of Eq. (16-29), apart from a change in the multiplicative constant on the ri t from 1.018 to 3/(2 t ) S 1.024, a relative error of 0.6%. This excellent agreement is independent of and and demonstrates the accuracy of the Gaussian approximation for arbitrary eccentricity. 

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