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National Repository of Grey Literature 3 records found  Search took 0.01 seconds. 
An algorithmic approach to resolutions in representation theory
 Ivánek, Adam ; Šťovíček, Jan (advisor) ; Růžička, Pavel (referee)
In this thesis we describe an algorithm and implement a construction of a projective resolution and minimal projective resolution in the representation the- ory of finite-dimensional algebras. In this thesis finite-dimensional algebras are KQ /I where KQ is a path algebra and I is an admissible ideal. To implement the algorithm we use the package QPA [9] for GAP [2]. We use the theory of Gröbners basis of KQ-modules and the theory described in article Minimal Pro- jective Resolutions written by Green, Solberg a Zacharia [5]. First step is find a direct sum such that i∈Tn fn∗ i KQ = i∈Tn−1 fn−1 i KQ ∩ i∈Tn−2 fn−2 i I. Next important step to construct the minimal projective resolution is separate nontri- vial K-linear combinations in i∈Tn−1 fn−1 i I + i∈Tn fn i J from fn∗ i . The Modules of the minimal projective elements are i∈Tn (fn i KQ)/(fn i I). 1
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An algorithmic approach to resolutions in representation theory
 Ivánek, Adam ; Šťovíček, Jan (advisor) ; Růžička, Pavel (referee)
In this thesis we describe an algorithm and implement a construction of a projective resolution and minimal projective resolution in the representation the- ory of finite-dimensional algebras. In this thesis finite-dimensional algebras are KQ /I where KQ is a path algebra and I is an admissible ideal. To implement the algorithm we use the package QPA [9] for GAP [2]. We use the theory of Gröbners basis of KQ-modules and the theory described in article Minimal Pro- jective Resolutions written by Green, Solberg a Zacharia [5]. First step is find a direct sum such that i∈Tn fn∗ i KQ = i∈Tn−1 fn−1 i KQ ∩ i∈Tn−2 fn−2 i I. Next important step to construct the minimal projective resolution is separate nontri- vial K-linear combinations in i∈Tn−1 fn−1 i I + i∈Tn fn i J from fn∗ i . The Modules of the minimal projective elements are i∈Tn (fn i KQ)/(fn i I). 1
Detailed record
Synchronizing automata
 Ivánek, Adam ; Holub, Štěpán (advisor) ; Hojsík, Michal (referee)
In this thesis we study Trahtman's proof of Road coloring problem and related algorithm. For every strongly connected directed multigraph with outdegree d and period 1, there exists synchronizing coloring. Béal and Perrin prove that Trahtman's proof can be simply generalized for every period and k-synchronizing coloring. We show generalized proof too. Trahtman's proof is constructive and is based on finding coloring with nontrivial stable states. We prove if there is only one maximal tree in Pα then the coloring is with nontrivial stable states. Subgraph Pα contains all edges with same color. We show how to find such coloring. Then we describe algorithms for finding k-synchronizing coloring. First algorithm uses proposition from Trahtman's proof with time complexity O((n−k)dn2 ). Then we show Trahtman's reduction and Béal and Perrin's algorithm based on Trahtman's proof but time complexity is O((n − k)dn) where n is the number of vertices. 1
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