We show in this paper that the tangent bundle $TG$, of a Poisson-Lie group $G$ has a Poisson-Lie group structure given by the canonical lifting of that of $G$. We determine the dual group of $TG$, its Lie bialgebra and its double Lie algebra.

We also show that any Poisson action of $G$ on a Poisson manifold $P$ is lifted on a Poisson action of $TG$ on the tanget bundle $TP$.

We study some properties of multidimensional Hamiltonian systems in the adiabatic limit. Using the properties of the Poincare-Cartan invariant we show that in the integrable case conservation of action requires conditions on the frequencies together with conservation of the product of energy and period. In the ergodic case the most general conserved quantity is not volume but rather symplectic capacity; we prove that even in this case there are periodic orbits whose actions are conserved.

Applying the concept of a momentum map for supersymplectic supervectorspaces to the one-dimensional Bose-Fermi oscillator, we show that the largest symmetry group that admits a momentum map is the identity component of the intersection of the orthosymplectic group OSp(2|2) and the group of super-symplectic transformations. This gives a systematic characterization of a certain class of odd supersymmetry transformations that were originally introduced in an ad hoc way.

The paper treats in one dimensional mixed heat transfer problem of steady conduction and radiation in a wire with internal source. We are led to a Cauchy problem consisting of a second order nonlinear ordinary differential equation. A special integrable case with two non independent left boundary conditions requires a hyperelliptic integral, for which a representation theorem has been established through the Gauss hypergeometric function ${_2\!F_1}$. The relevant steady solution is then found to grow monotonically with the distance from boundary, up to a certain limiting position where it suddenly jumps unbounded.

Proper velocities are measured by proper time as opposed to coordinate velocities, which are measured by coordinate time. The standard Lorentz transformation group, in which each transformation is expressed by a coordinate velocity and an orientation between two inertial frames, is well known. In contrast, the equivalent proper-time Lorentz transformation group, in which each transformation is expressed by a proper velocity and an orientation between two inertial frames is unknown. The dignity of special relativity theory requires that every possible means be explored for the solution of a problem so elegant and so celebrated. Fortunately, a so called gyro-formalism approach to special relativity enables the elusive proper-time Lorentz transformation group to be uncovered.