DERIVED TERMS WITHOUT DERIVATION A SHIFTED PERSPECTIVE ON THE DERIVED-TERM AUTOMATON

We present here a construction for the derived term automaton (aka partial derivative, or Antimirov, automaton) of a rational (or regular) expression based on a sole induction on the depth of the expression and without making reference to an operation of derivation of the expression. It is particularly well-suited to the case of weighted rational expressions.

Equivalence checking for weak bi-Kleene algebra

Pomset automata are an operational model of weak bi-Kleene algebra, which describes programs that can fork an execution into parallel threads, upon completion of which execution can join to resume as a single thread. We characterize a fragment of pomset automata that admits a decision procedure for language equivalence. Furthermore, we prove that this fragment corresponds precisely to series-rational expressions, i.e., rational expressions with an additional operator for bounded parallelism. As a consequence, we obtain a new proof that equivalence of series-rational expressions is decidable.

Scattering in Quantum Dots via Noncommutative Rational Functions

In the customary random matrix model for transport in quantum dots with M internal degrees of freedom coupled to a chaotic environment via $$N\ll M$$ N ≪ M channels, the density $$\rho $$ ρ of transmission eigenvalues is computed from a specific invariant ensemble for which explicit formula for the joint probability density of all eigenvalues is available. We revisit this problem in the large N regime allowing for (i) arbitrary ratio $$\phi := N/M\le 1$$ ϕ : = N / M ≤ 1 ; and (ii) general distributions for the matrix elements of the Hamiltonian of the quantum dot. In the limit $$\phi \rightarrow 0$$ ϕ → 0 , we recover the formula for the density $$\rho $$ ρ that Beenakker (Rev Mod Phys 69:731–808, 1997) has derived for a special matrix ensemble. We also prove that the inverse square root singularity of the density at zero and full transmission in Beenakker’s formula persists for any $$\phi <1$$ ϕ < 1 but in the borderline case $$\phi =1$$ ϕ = 1 an anomalous $$\lambda ^{-2/3}$$ λ - 2 / 3 singularity arises at zero. To access this level of generality, we develop the theory of global and local laws on the spectral density of a large class of noncommutative rational expressions in large random matrices with i.i.d. entries.

Proinov type contractions on dislocated b-metric spaces

In this paper, we improve the Proinov theorem by adding certain rational expressions to the definition of the corresponding contractions. After that, we prove fixed point theorems for these modified Proinov contractions in the framework of dislocated b-metric spaces. We show some illustrative examples to indicate the validity of the main results.

A fixed point theorem involving rational expressions without using Picard iteration

In this paper, we consider a certain fixed point theorem that contains some rational expressions. The main aim of this paper is to prove a fixed point theorem without using the Picard iteration.