Conformal classical Yang–Baxter equation, S-equation and $${\mathcal {O}}$$-operators

O-operators of Loday Algebras and Analogues of the Classical Yang-baxter Equation

We introduce notions of O-operators of the Loday algebras including the dendriform algebras and quadri-algebras as a natural generalization of Rota-Baxter operators. The invertible O-operators give a sufficient and necessary condition on the existence of the 2 operations on an algebra with the 2 operations in an associative cluster. The analogues of the classical Yang-Baxter equation in these a...

An Equation-Based Classical Logic

We propose and study a logic able to state and reason about equational constraints, by combining aspects of classical propositional logic, equational logic, and quantifiers. The logic has a classical structure over an algebraic base, and a form of universal quantification distinguishing between local and global validity of equational constraints. We present a sound and complete axiomatization f...

The Minimal Representation of the Conformal Group and Classical Solutions to the Wave Equation

Let n be an integer ≥ 2. We consider the wave operator = −∂ t + Pn i=1 ∂ xi , where (t, x) = (t, x1, . . . , xn) are the canonical coordinates on Minkowski space R 1,n = R× R . Lie’s prolongation algorithm calculates the Lie algebra of infinitesimal symmetries of to be isomorphic to the conformal Lie algebra g = so(2, n+1) plus an infinite dimensional piece reflecting the fact that is linear. I...

An Integral Equation in Conformal Geometry

Pell ’ s Equation

An arbitrary quadratic diophantine equation with two unknowns can be reduced to a Pell-type equation. How can such equations be solved? Recall that the general solution of a linear diophantine equation is a linear function of some parameters. This does not happen with general quadratic diophantine equations. However, as we will see later, in the case of such equations with two unknowns there st...