Poland's ‘fifth problem’

Semidynamical Systems and Hilbert’s Fifth Problem

g : [0,∞)×X → X and g(t, x) = T (t)x, t ≥ 0, x ∈ X, then g is continuous. For many semigroups strong continuity implies joint continuity. A function gx as in (1) is called a trajectory of T . If T has domain all of R instead of just [0,∞) we refer to T as a ‘group’. In his Fifth Problem, Hilbert asked whether assumptions of differentiability that Sophus Lie made were actually a consequence of L...

Hilbert’s Fifth Problem for Local Groups

We solve Hilbert’s fifth problem for local groups: every locally euclidean local group is locally isomorphic to a Lie group. Jacoby claimed a proof of this in 1957, but this proof is seriously flawed. We use methods from nonstandard analysis and model our solution after a treatment of Hilbert’s fifth problem for global groups by Hirschfeld.

Hilbert ’ s fifth problem and related topics

The State of the Second Part of Hilbert's Fifth Problem

"Moreover, we are thus led to the wide and interesting field of functional equations which have been heretofore investigated usually only under the assumption of the differentiability of the functions involved. In particular the functional equations treated by Abel (Oeuvres, vol. 1, pp. 1,61, 389) with so much ingenuity...and other equations occurring in the literature of mathematics, do not di...

Fifth Force from Fifth Dimension

Using a novel coordinate system, we rederive the eld equations and equations of motion for 5-dimensional relativity in the general case where the metric can depend on all 5 coordinates (i.e., Kaluza-Klein theory without the cylinder restriction). We show that in general the fth dimension produces a new dynamical force in 4-dimensional spacetime. This timelike fth force is proportional to the 4-...