A geometric approach to an equation of J. D’Alembert

By using a geometric framework of PDE's we prove that the set of solutions of the D'Alembert equation $(*)\hskip 3pt {{\partial ^2\lg f}\over{\partial x\partial y}}=0$ is larger than the set of smooth functions of two variables $f(x,y)$ of the form $(**)\hskip 3pt f(x,y)=h(x).g(y)$. The set of $2$-dimensional integral manifolds of PDE $(*)$ properly contains the ones representable by graphs of $2$-jet-derivatives of functions $f(x,y)$ expressed in the form $(**)$.} A generalization of this result to functions of more than two variables is sketched too

Asymptotic stability of the Cauchy and Jensen functional equations

The aim of this note is to investigate the asymptotic stability behaviour of the Cauchy and Jensen functional equations. Our main results show that if these equations hold for large arguments with small error, then they are also valid everywhere with a new error term which is a constant multiple of the original error term. As consequences, we also obtain results of hyperstability character for these two functional equations

An Introduction to Hyperbolic Barycentric Coordinates and their Applications

Barycentric coordinates are commonly used in Euclidean geometry. The adaptation of barycentric coordinates for use in hyperbolic geometry gives rise to hyperbolic barycentric coordinates, known as gyrobarycentric coordinates. The aim of this article is to present the road from Einstein's velocity addition law of relativistically admissible velocities to hyperbolic barycentric coordinates along with applications.Comment: 66 pages, 3 figure

Gyrations: The Missing Link Between Classical Mechanics with its Underlying Euclidean Geometry and Relativistic Mechanics with its Underlying Hyperbolic Geometry

Being neither commutative nor associative, Einstein velocity addition of relativistically admissible velocities gives rise to gyrations. Gyrations, in turn, measure the extent to which Einstein addition deviates from commutativity and from associativity. Gyrations are geometric automorphisms abstracted from the relativistic mechanical effect known as Thomas precession