Recent Advances in Applied Probability

The range of books reviewed is wide, covering theory and applications in operations research, statistics, management science, econometrics, mathematics, computers, and information systems (no software is reviewed). In addition, we include books in other fields that emphasize technical applications. Publishers who wish to have their books reviewed should send them to Professor Benjamin Lev. We list the books received; not all books received can be reviewed because space is limited. Those who would like to review books are urged to send me their names, addresses, and specific areas of expertise. We commission all reviews and do not accept unsolicited book reviews. Readers are encouraged to suggest books that might be reviewed or to ask publishers to send me copies of such books. The authors or editors of the books reviewed in this issue are Tito A. Ciriani, Giorgio Fasano, Stefano Gliozzi, Roberto Tadei, Elliott Mendelson, John L. Nazareth, Narayan S. Rau, Philippe Robert, Stefan Seuring, Martin Muller, Maria Goldbach, and Uwe Schneidewind.

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Solutions of the Yang–Baxter Equation and Automaticity Related to Kronecker Modules

Cañadas

Espinosa

Ballester‐Bolinches

2023

Computation

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The Kronecker algebra K is the path algebra induced by the quiver with two parallel arrows, one source and one sink (i.e., a quiver with two vertices and two arrows going in the same direction). Modules over K are said to be Kronecker modules. The classification of these modules can be obtained by solving a well-known tame matrix problem. Such a classification deals with solving systems of differential equations of the form Ax=Bx′, where A and B are m×n, F-matrices with F an algebraically closed field. On the other hand, researching the Yang–Baxter equation (YBE) is a topic of great interest in several science fields. It has allowed advances in physics, knot theory, quantum computing, cryptography, quantum groups, non-associative algebras, Hopf algebras, etc. It is worth noting that giving a complete classification of the YBE solutions is still an open problem. This paper proves that some indecomposable modules over K called pre-injective Kronecker modules give rise to some algebraic structures called skew braces which allow the solutions of the YBE. Since preprojective Kronecker modules categorize some integer sequences via some appropriated snake graphs, we prove that such modules are automatic and that they induce the automatic sequences of continued fractions.

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Recent Advances in Applied Probability

Cited by 4 publications

References 261 publications

Identification of the Optimal Parameters for Forecasting the State of Technical Objects Based on the Canonical Random Sequence Decomposition

Identification of the Optimal Parameters for Forecasting the State of Technical Objects Based on the Canonical Random Sequence Decomposition

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Solutions of the Yang–Baxter Equation and Automaticity Related to Kronecker Modules

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