Generalized Euler theorem

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. 

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. ... 

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

ChaC82 Chao, C-Y. Generalizations of theorems of Wilson, Fermat and Euler. J. Number Theory 15 (1982) 95-114. 

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... 

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). 

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. 

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. 

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. 

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

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

---