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An extension of the finite and infinite Lie groups properties of complex numbers and functions of complex variable is proposed. This extension is performed exploiting hypercomplex number systems that follow the elementary algebra rules. In particular the functions of such systems satisfy a set of partial differential equations that defines an infinite Lie group. Emphasis is put on the functional transformations of a particular two-dimensional hypercomplex number system, capable of maintaining the wave equation as invariant and then the speed of light invariant too. These functional transformations describe accelerated frames and can be considered as a generalization of two dimensional Lorentz group of special relativity. As a first application the relativistic hyperbolic motion is obtained.
By analogy with complex numbers, a system of hyperbolic numbers can be introduced in the same way: z=x+h*y with h*h=1 and x,y real numbers. As complex numbers are linked to the Euclidean geometry, so this system of numbers is linked to the pseudo-Euc
In this note we consider non-equilibrium steady states of one-dimensional models of heat conduction (wealth exchange) which are coupled to some reservoirs creating currents. In particular we will give sufficient and necessary conditions which will de
We consider infinite-dimensional diffusions where the interaction between the coordinates has a finite extent both in space and time. In particular, it is not supposed to be smooth or Markov. The initial state of the system is Gibbs, given by a stron
We prove that random walks in random environments, that are exponentially mixing in space and time, are almost surely diffusive, in the sense that their scaling limit is given by the Wiener measure.
We consider the matrix representation of the Eisenstein numbers and in this context we discuss the theory of the Pseudo Hyperbolic Functions. We develop a geometrical interpretation and show the usefulness of the method in Physical problems related to the anomalous scattering of light by crystals