The paper constructs generalized discrete Markov spectra for the family of equations x² + y² + z² + k1 yz + k2 zx + k3 xy = (3 + k1 + k2 + k3) xyz, with each spectrum element realized as both a Lagrange constant of a quadratic irrational and a Markov constant of an indefinite binary quadratic form.
Cluster algebraic interpretation of generalized Markov numbers and their matrixizations
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Defines words ω_t via binary-tree recursion for each positive rational slope t; matrix evaluation of ω_t yields a Markov-monodromy matrix that encodes the generalized Markov number at t.
Modified c- and g-vectors of the Markov quiver exhibit a fractal structure via linear isomorphisms and admit recursive formulas parameterized by coprime integers, which classify the complement of the G-fan.
Multiple equivalent combinatorial expansion formulas are given for generalized cluster algebras from arcs on punctured orbifolds, generalizing prior surface and orbifold cases.
Generalized k-Markov numbers grow monotonically along more random lines as k increases, supporting a k-analog of Frobenius' uniqueness conjecture.
k-Markov numbers satisfy Aigner's conjectures on their orderings and uniqueness properties in positive integer solutions.
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Generalized discrete Markov spectra
The paper constructs generalized discrete Markov spectra for the family of equations x² + y² + z² + k1 yz + k2 zx + k3 xy = (3 + k1 + k2 + k3) xyz, with each spectrum element realized as both a Lagrange constant of a quadratic irrational and a Markov constant of an indefinite binary quadratic form.
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Words for generalized Markov numbers
Defines words ω_t via binary-tree recursion for each positive rational slope t; matrix evaluation of ω_t yields a Markov-monodromy matrix that encodes the generalized Markov number at t.
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Fractal phenomenon in $c$- and $g$-vectors of the Markov quiver
Modified c- and g-vectors of the Markov quiver exhibit a fractal structure via linear isomorphisms and admit recursive formulas parameterized by coprime integers, which classify the complement of the G-fan.
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Cluster Expansions from Punctured Orbifolds
Multiple equivalent combinatorial expansion formulas are given for generalized cluster algebras from arcs on punctured orbifolds, generalizing prior surface and orbifold cases.
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Orderings of Generalized k-Markov Numbers
Generalized k-Markov numbers grow monotonically along more random lines as k increases, supporting a k-analog of Frobenius' uniqueness conjecture.
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Orderings of k-Markov Numbers
k-Markov numbers satisfy Aigner's conjectures on their orderings and uniqueness properties in positive integer solutions.