Homogeneity, space

Let us calculate, for future reference, the dimension of the complex vector space of homogeneous polynomials (with complex coefficients) of degree n on various Euclidean spaces. Homogeneous polynomials of degree n on the real line R are particularly simple. This complex vector space is onedimensional for each n. In fact, every element has the form ex for some c e C. In other words, the one-element set x" is a finite basis for the homogeneous polynomials of degree n on the real line. 

If we specify that the points determined through (8), by the assumed V, lie in this hyperplane, equations (8) or rather (1) produce an unambiguous mapping between the points of the hyperplane and the points of the tangent spaces. Homogeneous coordinates in this hyperplane are defined through (7). 

Thus, it turns out that invariance of the equation of motion with respect to an arbitrary translation in time (time homogeneity) results in the energy conservation principle with respect to translation in space (space homogeneity) gives the total momentum conservation principle and with respect to rotation in space (space isotropy) implies the total angular momentum conservation principle. 

This equation is very often given, not with the electric field, but with the electric potential V, and the utilized form assumes a permittivity independent from space (homogeneous medium) as will be demonstrated by the Formal Graph, which justifies the mention horn under the equal sign in the following equation ... 

For polyelectrolyte solutions the relation between the mean interparticle distance d and the concentration is of great interest. Assuming that polyions fill the space homogeneously due to their strong mutual repulsion, the ple relation... 

The strain in these equations should be considered to include a contribution due to thermal expansion of the material. If the temperature depends on position, the viscoelastic function will also, and space homogeneity is lost. Equation (1.7.8) remains valid in fact but the dynamical equations, which will be introduced in Sect. 1.8, are altered in a significant manner. 

Non-intersecting walks completely filling the space. Homogeneous interactions (proteins)... 

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