We consider a manifestly Lorentz invariant form L of the biquaternion algebra and its generalization to the case of curved manifold. The conditions of L-differentiability of L-functions are formulated and considered as the primary equations for fundamental fields modeled with such functions. The exact form of the effective affine connection induced by L-differentiability equations is obtained for the flat and curved cases. In the flat case, the integrability conditions of the latter leads to the self-duality of the corresponding curvature, thus ensuring that the source-free Maxwell and SL(2, C) Yang-Mills equations hold on the solutions of the L-differentiability equations.

We show that (specifically scaled) equations of shear-free null geodesic congruences on the Minkowski space-time possess intrinsic self-dual, restricted gauge and algebraic structures. The complex eikonal, Weyl 2-spinor, SL(2, C) Yang-Mills and complex Maxwell fields, the latter produced by integervalued electric charges ("elementary" for the Kerr-like congruences), can all be explicitly associated with any shear-free null geodesic congruence. Using twistor variables, we derive the general solution of the equations of the shear-free null geodesic congruence (as a modification of the Kerr theorem) and analyze the corresponding "particle-like" field distributions, with bounded singularities of the associated physical fields. These can be obtained in a straightforward algebraic way and exhibit non-trivial collective dynamics simulating physical interactions.

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Projectile motion is a constant theme in introductory-physics courses. It is often used to illustrate the application of differential and integral calculus. While most of the problems used for this purpose, such as maximizing the range, are kept at a fairly elementary level, some, such as determining the safe domain, involve not so elementary techniques, which can hardly be assumed of the targeted audience. In the literature, several attempts have been undertaken to avoid calculus altogether and keep the exposition entirely within the realm of algebra and/or geometry. In this paper, we propose yet another non-calculus approach which uses the projectile's travel times to shed new light on these problems and provide instructors with an alternate method to address them with their students.