Cayley Hash Values of Brauer Messages and Some of Their Applications in the Solutions of Systems of Differential Equations


Cayley hash values are defined by paths of some oriented graphs (quivers) called Cayley graphs, whose vertices and arrows are given by elements of a group $\mathfrak{H}$. On the other hand, Brauer messages are obtained by concatenating words associated with multisets constituting some configurations called Brauer configurations. These configurations define some oriented graphs named Brauer quivers which induce a particular class of bound quiver algebras named Brauer configuration algebras. Elements of multisets in Brauer configurations can be seen as letters of the Brauer messages. This paper proves that each point $(x,y)\in\mathscr{V}=\mathbb{R}\backslash{0} \times \mathbb{R}\backslash{0}$ has an associated Brauer configuration algebra $\Lambda_{\mathfrak{B}^{(x,y)}}$ induced by a Brauer configuration $\mathfrak{B}^{(x,y)}$. Additionally, the Brauer configuration algebras associated with points in a subset of the form $(\lfloor(x)\rfloor,\lceil (x) \rceil]\times (\lfloor(y)\rfloor,\lceil (y) \rceil]\subset\mathscr{V}$ have the same dimension. We give an analysis of Cayley hash values associated with Brauer messages $\mathfrak{M}(\mathfrak{B}^{(x,y)})$ defined by a semigroup generated by some appropriated matrices $A_{0}, A_{1}, A_{2}\in \mathrm{GL}(2, \mathscr{R})$ over a commutative ring $\mathscr{R}$. As an application, we use Brauer messages $\mathfrak{M}(\mathfrak{B}^{(x,y)})$ to construct explicit solutions for systems of linear and nonlinear differential equations of the form $X’'(t)+MX(t)=0$ and $X’(t)-X^{2}(t)N(t)=N(t)$ for some suitable square matrices, $M$ and $N(t)$. Python routines to compute Cayley hash values of Brauer messages are also included.

Computation, 10(9), Art. 164. doi:10.3390/computation10090164