Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: A set of nonnegative matricesM ={M1,M2,...,Mk} is called primitive if there exist indices i1,i2,...,im such that Mi1Mi2 ...Mim is positive (i.e. has all its entries > 0). The length of the shortest such product is called the exponent ofM. The concept of primitive sets of matrices comes up in a number of problems within control theory, non-homogeneous Markov chains, automata theory etc. Recently, connections beten synchronizing automata and primitive sets of matrices re established. In the present paper,  significantly strengthen these links by providing equivalence results, both in terms of combinatorial characterization, and computational aspects.  study the maximal exponent among all primitive sets ofn◊n matrices, which  denote by exp(n).  prove that limn!1 log exp(n) n = log 3 3 , and moreover,  establish that this bound leads to a resolution of the Eern˝ problem for carefully synchronizing automata.  also study the set of matrices with no zero rows and columns, denoted by NZ, due to its intriguing connections to the Eern˝ conjecture and the recent generalization of Perron-Frobenius theory for this class.  characterize computational complexity of dierent problems related to the exponent of NZ matrix
['Balázs Gerencsér', 'Vladimir V. Gusev', 'Raphaël M. Jungers']
I am writing a research paper on a idea Primitive sets of nonnegative matrices and synchronizing automata
[{"from": "human", "value": "I am writing a research paper on a idea Primitive sets of nonnegative matrices and synchronizing automata"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: A set of nonnegative matricesM ={M1,M2,...,Mk} is called primitive if there exist indices i1,i2,...,im such that Mi1Mi2 ...Mim is positive (i.e. has all its entries > 0). The length of the shortest such product is called the exponent ofM. The concept of primitive sets of matrices comes up in a number of problems within control theory, non-homogeneous Markov chains, automata theory etc. Recently, connections beten synchronizing automata and primitive sets of matrices re established. In the present paper,  significantly strengthen these links by providing equivalence results, both in terms of combinatorial characterization, and computational aspects.  study the maximal exponent among all primitive sets ofn\u25can matrices, which  denote by exp(n).  prove that limn!1 log exp(n) n = log 3 3 , and moreover,  establish that this bound leads to a resolution of the Eern\u02dd problem for carefully synchronizing automata.  also study the set of matrices with no zero rows and columns, denoted by NZ, due to its intriguing connections to the Eern\u02dd conjecture and the recent generalization of Perron-Frobenius theory for this class.  characterize computational complexity of dierent problems related to the exponent of NZ matrix"}]