| Summary: | 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">H</mi></semantics></math></inline-formula>. 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>∈</mo><mi mathvariant="script">V</mi><mo>=</mo><mi mathvariant="double-struck">R</mi><mo>\</mo><mo>{</mo><mn>0</mn><mo>}</mo><mo>×</mo><mi mathvariant="double-struck">R</mi><mo>\</mo><mo>{</mo><mn>0</mn><mo>}</mo></mrow></semantics></math></inline-formula> has an associated Brauer configuration algebra <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mo>Λ</mo><msup><mi mathvariant="fraktur">B</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></msup></msub></semantics></math></inline-formula> induced by a Brauer configuration <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="fraktur">B</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></msup></semantics></math></inline-formula>. Additionally, the Brauer configuration algebras associated with points in a subset of the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mo>⌊</mo><mo>(</mo><mi>x</mi><mo>)</mo><mo>⌋</mo><mo>,</mo><mo>⌈</mo><mo>(</mo><mi>x</mi><mo>)</mo><mo>⌉</mo><mo>]</mo><mo>×</mo><mo>(</mo><mo>⌊</mo><mo>(</mo><mi>y</mi><mo>)</mo><mo>⌋</mo><mo>,</mo><mo>⌈</mo><mo>(</mo><mi>y</mi><mo>)</mo><mo>⌉</mo><mo>]</mo><mo>⊂</mo><mi mathvariant="script">V</mi></mrow></semantics></math></inline-formula> have the same dimension. We give an analysis of Cayley hash values associated with Brauer messages <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="fraktur">M</mi><mo>(</mo><msup><mi mathvariant="fraktur">B</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></msup><mo>)</mo></mrow></semantics></math></inline-formula> defined by a semigroup generated by some appropriated matrices <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>A</mi><mn>0</mn></msub><mo>,</mo><msub><mi>A</mi><mn>1</mn></msub><mo>,</mo><msub><mi>A</mi><mn>2</mn></msub><mo>∈</mo><mi>GL</mi><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mi mathvariant="script">R</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> over a commutative ring <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">R</mi></semantics></math></inline-formula>. As an application, we use Brauer messages <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="fraktur">M</mi><mo>(</mo><msup><mi mathvariant="fraktur">B</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></msup><mo>)</mo></mrow></semantics></math></inline-formula> to construct explicit solutions for systems of linear and nonlinear differential equations of the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>X</mi><mrow><mo>″</mo></mrow></msup><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>+</mo><mi>M</mi><mi>X</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>X</mi><mo>′</mo></msup><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>−</mo><msup><mi>X</mi><mn>2</mn></msup><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mi>N</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><mi>N</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> for some suitable square matrices, <i>M</i> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>N</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula>. Python routines to compute Cayley hash values of Brauer messages are also included.
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