The Generalized Euler Theorem

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.  

Let E be any extensive thermodynamic quantity expressed as a function of the variables T, P, and N (where N is the total number of molecules in the system). Viewing the system as a mixture of quasi-components, we can express as a function of the new set of variables T, P, and N. For concreteness, consider the QCDF based on the concept of CN. The two possible functions mentioned above are then 

For the sake of simplicity, we henceforth use N(K) in place of Ncn(K) so that the treatment will be valid for any discrete 

3) and (E.4) we have stressed the fact that the partial molar quantities depend on the whole vector N. 

The generalization of (E.3) and (E.4) to the case of a continuous QCDF requires the application of the technique of functional differentiation. We introduce the generalized Euler theorem by way of analogy with (E.3). 

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In the previous section, we reinterpreted relations (5.45) and (5.46) as special cases of the generalized Euler theorem, i.e.,... 

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

Euler s Theorem Rotation is the general movement of a rigid body in space with a single point fixed. 

Scalar counting relations for sets of structural components can seen as expressions for characters under the identity operation of more general relations between representations of those sets. For example, the Euler relation in topology can be generalised to connect not only the numbers of edges, vertices and faces of a polyhedron, but also various symmetries associated with the structural features. The well-known Euler theorem... 

ORDINARY DIFFERENTIAL EQUATIONS, I.G. Petrovski. Covers basic concepts, some differential equations and such aspects of the general theory as Euler lines, Ariel s theorem, Beano s existence theorem, Osgood s uniqueness theorem, more. 45 figures. Problems. Bibliography. Index, xi + 232pp. 5X 8H. 

The general necessity for twelve pentagons, with a (nearly) arbitrary number of hexagons follows from Euler s theorem a network of trivalent nodes comprising P pentagons and H hexagons contains n = Euler characteristic n — b + a must be 2, which implies P = 12, with no explicit constraint on H. (In fact, polyhedra exist for any... 

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). 

We present the general form of Euler s homogeneity theorem. A function is homogeneous of order k, when... 

Theorem 5.1.3 (see [61], [161]). If a heavy rigid body of the general form is dynamically nonsymmetric then its equations of motion v = sgradlT (where the Hamiltonian H = Hq + eH is a small perturbation of the Hamiltonian Hq of the integrable Euler case) do not have a formal additional fourth integral which has analytic coelBcients on the four-dimensional level surface M23 and is independent of the function H = Hq -h eHi. 

The above parts show the minimum principle for vector processes in the frame of the generalized Onsager constitutive theory by the directions of Onsager s last dissip>ation of energy principle. We had seen above that in case of source-free balances, this principle is equivalent with the principle of minimal entropy production. The equivalence of the two theorems in the frame of the linear constitutive theory was proven by Gyarmati [2] first. Furthermore, we showed that in case when the principle of minimal entropy production is used for the determination of the possible forms of constitutive equations, the results are similar to the linear theory in the frame of the Onsager s constitutive theory, where the dissipation potentials are homogeneous Euler s functions. 

To discretize the continuity, the differential equation is first integrated over the scalar grid cell volume. This is the same grid cell volume as was used for the generalized scalar quantity in Sect. 12.8. By use of the first order implicit Euler time discretization scheme and the basic theorem of integration for the convective fluxes, a discrete form of (12.195) can be expressed as ... 

When nanotubes are synthesized,8,26,27 they are typically closed (capped) at both ends. For the (9,0) and the (5,5), the caps can be fullerene hemispheres 6 in general, however, caps can have various structures, shapes and degrees of curvature.8 One requirement that they do have to satisfy is imposed by Euler s theorem, 8,26,27 according to which the closure of any hexagonal framework can be achieved only by the introduction of exactly twelve pentagons. Thus, each cap must have six. 

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

An additional limitation of the index theorem is that in higher dimensions, there may often be different time scales corresponding to fast and slow reactions, so that the dynamics rapidly relax to a manifold that is embedded in the N-sphere. In this case, the Euler-Poincare characteristic of the submanifold may be different from that of the iV-sphere and (14) would have to be modified appropriately. A very general mathematical approach in which the geometric and topological properties chemical kinetic equations is treated has recently been developed. Although the formulation is in principle capable... 

Theorem 4.2.6. Let p G G be a linear operator, on a semisimple Lie algebra G, self-conjugate with respect to the Killing form. The Euler equation X = [X, pX] is Hamiltonian simultaneously with respect to both Poisson brackets (the element a is a covector of general position), and, a if and only if p [Pg.217]

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