Eulers theorem

Using Eq. (31) in Eq. (35) and using Euler theorem again, since is a homogeneous function of the first degree in Nx and N2, gives... 

The main properties of a polyhedral surface are undoubtedly related to the topology of the underlying manifold that it decorates. This topology can be completely described by two properties the Euler characteristic and the orientability of the surface. The former can easily be calculated from the celebrated Euler theorem [9] which states that the number of vertices, edges, and faces, denoted as V, E, and F respectively, obey the following mle ... 

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... 

The extended Euler theorem can also be used to give a version of equation (15) in which vertex and face, but not edge, terms appear. 

Applying the Euler theorem to the second term on the right-hand side of Equation (1.107) gives ... 

Alternatively, by substituting the strain in Eq. (11.66) sls a. linear function of the stress, the free energy can be represented as a quadratic function of Uy. Applying once more the Euler theorem, the corresponding counterpart of Eq. (17.64) in terms of the stress is obtained as... 

The formalism of the statistical mechanics agrees with the requirements of the equilibrium thermodynamics if the thermodynamic potential, which contains all information about the physical system, in the thermodynamic limit is a homogeneous function of the first order with respect to the extensive variables of state of the system [14, 6-7]. It was proved that for the Tsallis and Boltzmann-Gibbs statistics [6, 7], the Renyi statistics [10], and the incomplete nonextensive statistics [12], this property of thermodynamic potential provides the zeroth law of thermodynamics, the principle of additivity, the Euler theorem, and the Gibbs-Duhem relation if the entropic index z is an extensive variable of state. The scaling properties of the entropic index z and its relation to the thermodynamic limit for the Tsallis statistics were first discussed in the papers [16,17],... 

To prove the homogeneity properties of the thermodynamic quantities and the Euler theorem for the Tsallis statistics in the canonical ensemble, we will consider, as an example, the exact analytical results for the nonrelativistic ideal gas. 

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

The generalization of (3.132) and (3.133) for 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 (3.133). More details can be found in Appendix B. 

From the Euler theorem for homogenous functions of order one, one gets for any extensive quantity, the identity... 

It is also natural to require that the number of faces (films) F, edges (the Plateau borders) B, and nodes N shall satisfy the fundamental topological Euler theorem about polyhedra [101,479] ... 

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 fullerenes the pyramidalization of the sp -C-atoms is provided by the introduction of 12 pentagons (Eulers theorem). In a graphitic sheet which consists exclusively of a network of hexagons pyramidalization could in principle be... 

Klochko, M.A. Analogy between phase rule and the Euler theorem for polyhedrons. Izvest. SectoraFiz.-Khim. Anal., Inst. Obshei. i Neoig. Khim., Akadd. NaukSSSR 19,82-88 (1949) Levin, I.J. The phase rule and topology. J. Chem. Educ. 23,183-185 (1946)... 

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. ... 

It can be shown by the Euler theorem (see Appendix C, B4) that M = rijm = Hifni or m = zimi... 

B4. Relationship between Partial Molar Property and State Variable (Euler Theorem)... 

The Euler theorem may be considered as the dimensional form of this theorem, which states that the alternating sum of the characters of the induced representations under the unit element, E, is equal to 2, but the present theorem extends this character equality to all the operations of the group. The theorem silently implies that irreps can be added and subtracted. In the example of the tetrahedron, the theorem is expressed as ... 

Ceulemans, A., Fowler, P.W. Extension of Euler theorem to the symmetry properties of poly-hedra. Nature 353, 52 (1991)... 

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