Movable Points Research Articles

On reduction of a two-dimensional generalized Toda lattice to an ordinary differential equations system

Reduction of a two-dimensional generalized Toda lattice to an ordinary differential equations system, which defines the functional dependence of Toda functions on the corresponding separable solutions, is given. It is shown, in particular, that between all the equations of the form ∂2ρ/∂x ∂t=φ(∂) only Liouville (L), sine-Gordon (SG) and Bullough-Dodd (BD) equations, which are associated with simple Lie algebras of a finite growth of rank 1, lead to the third Painleve equation (P3). The last circumstance allows one, probably, to assume the existence of a deep relation between the complete integrability condition for the corresponding class of dynamical systems and the criterion of the absence of movable critical points in the solutions of an ordinary differential equation system of the second order. If this is so, it means that Painlee's equations and transcendents can be generalized for multi-component cases.

Simplified theory of dislocation damping including point-defect drag. II. Superposition of continuous and pinning-point-drag effects

The combined effects which occur when two kinds of drag, one acting continuously along the dislocations and other at discrete points (point-defect drag), are derived. It is found that the damping and modulus change can always be expressed in good approximation as the sum of two Debye relaxations. For widely separated relaxation times, one corresponds to the point-defect drag and the other to continuous drag acting on the in-between loops. Expressions are given for the relaxation times and strengths for each of the two components for any number of movable pinning points. It is shown that the characteristic pinning-point-drag effects have nothing to do with whether the drag acts continuously or at discrete points, but depend on the fact that the movable pinners exert restoring and viscous forces which act only on a part of the dislocation displacement.

A note on the cylindrically symmetric, static solutions of the weak field equations

The general solution of the weak field equations of the nonsymmetric unified field theory in the magnetic case is reduced to the solution of a second-order differential equation with movable critical points. Some special solutions are discussed.

Integration of the science of occlusion

The status of the science of occlusion is not satisfactory. Therefore, a revision of the principles of occlusion was made. This revision revealed that the theories had not been proved but were applied to practice without experimental verification. A series of facts are known, but they are not correctly related to each other by a general law. Therefore, this is not a science. Knowledge of a single fact not related to any other or of many facts with no mutual relation does not attain the meaning of science. Science is a number of systematic judgments that are not contradictory to each other and are experimentally verified. Judgment is the correlation between a question and an answer. All judgments must form a system of concepts that will not contradict but will complement each other. These concepts must be related by a general law that will permit us to understand all facts in our possession. The general law must be a law of causality. In the reproduction of the circumstances under which the act is taking place, the conditions must be faithfully reproduced. This is obtained by establishing a symmetery of causes. Therefore, symmetry of causes will persist in the effects. Observation and analysis of the mandibular function show that one is dealing with movable points and paths produced by these moving points. All of the movable points are in the lower part or mandible, and all of the paths are in the upper part or maxillae and are static. These elements must be logically represented in the articulator. Therefore, all of the paths must be placed in the upper member and all of the movable points must be in the lower member of the articulator. If this relation is altered, the articulator will not produce the same movements as are made by the mandible.