Abstract:
In 1870, R. Clausius found the virial theorem which amounts to introduce the trace of the stress tensor when studying the foundations of thermodynamics, as a way to relate the absolute temperature of an ideal gas to the mean kinetic energy of its molecules. In 1901, H. Poincar{\'e} introduced a duality principle in analytical mechanics in order to study lagrangians invariant under the action of a Lie group of transformations. In 1909, the brothers E. and F. Cosserat discovered another approach for studying the same problem though using quite different equations. In 1916, H. Weyl considered again the same problem for the conformal group of transformations, obtaining at the same time the Maxwell equations and an additional specific equation also involving the trace of the impulsion-energy tensor. Finally, having in mind the space-time formulation of electromagnetism and the Maurer-Cartan equations for Lie groups, gauge theory has been created by C.N. Yang and R.L. Mills in 1954 as a way to introduce in physics the differential geometric methods available at that time, independently of any group action, contrary to all the previous approaches. The main purpose of this paper is to revisit the mathematical foundations of thermodynamics and gauge theory by using new differential geometric methods coming from the formal theory of systems of partial differential equations and Lie pseudogroups, mostly developped by D.C Spencer and coworkers around 1970. In particular, we justify and extend the virial theorem, showing that the Clausius/Cosserat/Maxwell/Weyl equations are nothing else but the formal adjoint of the Spencer operator appearing in the canonical Spencer sequence for the conformal group of space-time and are thus totally dependent on the group action. The duality principle also appeals to the formal adjoint of a linear differential operator used in differential geometry and to the extension modules used in homological algebra.

Abstract:
We study all five-, six-, and one eight-vertex type two-state solutions of the Yang-Baxter equations in the form $A_{12} B_{13} C_{23} = C_{23} B_{13} A_{12}$, and analyze the interplay of the `gauge' and `inversion' symmetries of these solution. Starting with algebraic solutions, whose parameters have no specific interpretation, and then using these symmetries we can construct a parametrization where we can identify global, color and spectral parameters. We show in particular how the distribution of these parameters may be changed by a change of gauge.

Abstract:
A formulation of the dynamics of a collection of connected simple 1-dimensional Cosserat continua and rigid bodies is presented in terms of sections of an SO(3) fibration over a 1-dimensional net. A large class of junction conditions is considered in a unified framework. All the equations of motion and junction conditions are derived as extrema of a constrained variational principle on the net and are analysed perturbatively for structures with Kirchhoff constitutive properties. The whole discussion is based on the notion of a Cosserat net and its contractions obtained by taking certain limits that transform Cosserat elements to rigid structures. Generalisations are briefly discussed within this framework.

Abstract:
The derivation of the non-relativistic Cosserat equations that was described in Part I of this series of papers is extended from the group of rigid motions in three-dimensional Euclidian space to the Poincar\'e group of four-dimensional Minkowski space. Examples of relativistic Cosserat media are then given in the form of the free Dirac electron and the Weyssenhoff fluid.

Abstract:
We consider a subsystem of the Special Cosserat Theory of Rods and construct an explicit form of its solution that depends on three arbitrary functions in (s,t) and three arbitrary functions in t. Assuming analyticity of the arbitrary functions in a domain under consideration, we prove that the obtained solution is analytic and general. The Special Cosserat Theory of Rods describes the dynamic equilibrium of 1-dimensional continua, i.e. slender structures like fibers, by means of a system of partial differential equations.

Abstract:
The Cosserat equations for equilibrium are derived by starting from the action of the group of smooth functions with values in the Lie group of rigid spatial motions on rigid frames in Euclidian space. The method of virtual work is employed, instead of the method of action functionals, and is defined using the methods of of jet manifolds. It is shown how the restriction to fundamental forms (which define the dynamical state of the Cosserat object) that are Euclidian is essential to obtaining the Cosserat equations.

Abstract:
We construct a two-field higher-order gradient micropolar model for Cosserat media on the basis of a square lattice of elements with rotational degrees of freedom. This model includes equations of single-field higher-order gradient micropolar theory, and additional ones, which allow modelling of short wavelength phenomena. We demonstrate an example of short wavelength spatially localized static deformations in a structural system, which could not be obtained in the classical single-field framework, but which are captured by the proposed two-field model.

Abstract:
A new algebraic object is introduced - recurrent fractions, which is an n-dimensional generalization of continued fractions. It is used to describe an algorithm for rational approximations of algebraic irrational numbers. Some parametrization for generalized Pell's equations is constructed.

Abstract:
We suggest an alternative mathematical model for the electron in dimension 1+2. We think of our (1+2)-dimensional spacetime as an elastic continuum whose material points can experience no displacements, only rotations. This framework is a special case of the Cosserat theory of elasticity. Rotations of material points are described mathematically by attaching to each geometric point an orthonormal basis which gives a field of orthonormal bases called the coframe. As the dynamical variables (unknowns) of our theory we choose a coframe and a density. We then add an extra (third) spatial dimension, extend our coframe and density into dimension 1+3, choose a conformally invariant Lagrangian proportional to axial torsion squared, roll up the extra dimension into a circle so as to incorporate mass and return to our original (1+2)-dimensional spacetime by separating out the extra coordinate. The main result of our paper is the theorem stating that our model is equivalent to the Dirac equation in dimension 1+2. In the process of analyzing our model we also establish an abstract result, identifying a class of nonlinear second order partial differential equations which reduce to pairs of linear first order equations.

Abstract:
In the description of vibrational properties of deformable bodies, it is usually assumed that the size of the oscillating particles is negligible in comparison with the average distance between them, so to describe the kinematics of such media only the displacement vector is used. In the majority of work is considered that when the independent rotational degrees of freedom are taken into account it become necessary to introduce the couple stress. Such models of continuous media are well known, for example, moment theory of elasticity or Cosserat media.A distinctive feature of the reduced Cosserat medium is that the stress tensor is asymmetric, and in static problems, this tensor becomes symmetric. Thus, in statics the reduced Cosserat media is indistinguishable from the the classical continuum in which the rotational degrees of freedom are not independent, as they are expressed in terms of displacement and the stress tensor is symmetric.In this paper we investigate the wave motion of a three-dimensional, isotropic, elastic reduced Cosserat medium, the characteristic velocities of wave propagation are finding, we also construct and analyze the dispersion curve for the dynamic equations.