Extremal Vector Research Articles

The question of embedding theorems in domains of the space En

The problem of formulating embedding theorems for main spaces of differentiable functions defined in a domain of the Euclidean space En is connected to the problem of finding an extremal (in a certain sense) n -dimensional vector satisfying a number of linear constraints determined by three groups of parameters: parameters of the original class of functions, parameters characterizing the embedding, and parameters characterizing some properties of the domain of the functions. Determination of the extremal vector is reduced to a linear programming problem. The vector thus obtained determines the form of the inequalities which determine the embedding under given relations between parameters. In those cases where this vector does not coincide with the extremal vector found for the same values of parameters of the original class of functions but for the domain G=En there is a kind of saturation of properties of the type of embedding theorems. An example is given, in which the linear programming problem is solved by the simplex method.

Observations on Hilbert's Independence Theorem and Born's Quantization of Field Equations

Born recently proposed a quantization of the field equations which is based upon Hilbert's independence theorem of the calculus of ${\mathrm{variations}.}^{1}$ My intention here is to give, in the first purely mathematical Part A, a formulation as simple and explicit as possible of the independence theorem. The agreement between the principle of variation and the independence theorem, complete in the case of one independent variable and one unknown function, fails in two respects in the case of several variables and functions; the independence theorem specializes the extremal vector field on the one hand and it discards the assumption of integrability on the other ${\mathrm{hand}.}^{2}$ In Part B, I first suggest a modification of Born's scheme without which it would be in disagreement with ordinary quantum mechanics even in the one-dimensional case. After the modification, a comparison with Heisenberg-Pauli's quantization becomes possible under the simplest circumstances. Born's scheme proves to be too narrow. Finally I raise the principal objection that the quantum-mechanical equation should not be of the form: four-dimensional divergence of $\ensuremath{\psi}$ equals $H\ensuremath{\psi}$ with a scalar operator of action $H$ but that it should rather consist of four components stating that differentiation of $\ensuremath{\psi}$ with respect to the four space-time coordinates is performed by means of the operators: energy and momentum.