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The purpose of this paper is to give a coordinate free pre-metric formulation of charge free electrodynamics, appropriate, in our view, for non-linearization of Maxwell equations in order to obtain pre-metric description of spatially finite electromagnetic field objects of photon-like nature. First we introduce some formal relations from multilinear algebra and differential geometry to be used further. Then we recall and appropriately modify the existing pre-metric formulation of linear charge free electrodynamics in pre-relativistic and relativistic forms as preparation to turn to corresponding pre-metric non-linearization. Then after some preliminary examples and notes on non-linearization, we motivate our view for existence and explicit formulation of time stable subsystems of the physical objects considered. Section 5 presents the formal results of our approach on the pre-metric nonlinear formulations in static case, in time-dependent case, and in space-time formulation. In the Conclusion we give our general view on “why and how to non-linearize”
We demonstrate that every non-tubular channel linear Weingarten surface in Euclidean space is a surface of revolution, hence parallel to a catenoid or a rotational surface of non-zero constant Gauss curvature. We provide explicit parametrizations and deduce existence of complete hyperbolic linear Weingarten surfaces.
We analyze the mechanics of planar networks of extensible fibers, for which we derive the general form of the mechanical energy. We consider especially networks made of two sets of non orthogonal and non equivalent fibers, called the parallelogram structure, with variants obtained as specific patterns called the square, rectangular, and rhombic structures. The fibers of the network are assumed to obey Bernoulli kinematics. A second order gradient continuum is obtained. The arguments of the energy of these four patterns are obtained based on the material symmetry group of the considered structures.
Given an associative supercommutative algebra equipped with an odd derivation, one considers the space of vector fields it defines, and show, under suitable hypothesis, they form a Jordan superalgebra; in contrast with the Lie superalgebras of Virasoro type constructed from even derivations. Relations with Anti Lie algebras studied by Ovsienko and collaborators are then shown.