Split graph

Split graph

Error-Oblivious Sample Preparation with DMFBs

Published in Sudip Poddar, Bhargab B. Bhattacharya, Error-Tolerant Biochemical Sample Preparation with Microfluidic Lab-on-Chip, 2022

Sudip Poddar, Bhargab B. Bhattacharya

Note that the depth of the mix-split graph/tree (also denoting the accuracy level n of a particular CF), is determined by a user-defined parameter “ τ” (error-tolerance limit), where 0<τ<1. In order to bound the concentration error in the target-CF within the error-tolerance limit (τ) = 12n+1 = 0.52n [131, 163], each CF is represented as an n-bit binary fractional number x2n, where x∈ℕ, 0 ≤x≤2n, and n∈ℕ [15]. However, for the sake of pictorial clarity, we will represent, each CF-value, for a given n, by its numerator only, i.e., we will write CF = x in all figures, instead of x2n. The value of each CF is represented as the label of the corresponding node in the mix-split graph/tree.

State splitting, strong shift equivalence and stable isomorphism of Cuntz–Krieger algebras

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Journal Information

Published in Dynamical Systems, 2019

Kengo Matsumoto

Next, we will recall the notion of in-splitting of a graph G following [16], Definition 2.4.7. For each state , let be a partition of into mutually disjoint sets. Denote by this partition of . Denote by the family We construct a new graph such that . For let j be the unique element in such that and let Define new edges such that for all . Let be the set of such edges. The resulting directed graph is called the in-split graph formed from G using . We note that if G is irreducible, then the graph is also irreducible ([16], Exercise 2.4.5). The converse procedure of constructing G from is called in-amalgamation (see [15]).

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Error-Oblivious Sample Preparation with DMFBs

State splitting, strong shift equivalence and stable isomorphism of Cuntz–Krieger algebras