I show how one can modify the random-walk Metropolis MCMC method in such a way that a sequence of modified Metropolis updates takes little computation time when the rejection rate is outside a desired interval. This allows one to effectively adapt the scale of the Metropolis proposal distribution, by performing several such “short-cut” Metropolis sequences with varying proposal stepsizes. Unlik...

For a field K, rational function φ ∈ K(z) of degree at least two, and α ∈ P(K), we study the polynomials in K[z] whose roots are given by the solutions in K to φ(z) = α, where φ denotes the nth iterate of φ. When the number of irreducible factors of these polynomials stabilizes as n grows, the pair (φ,α) is called eventually stable over K. We conjecture that (φ, α) is eventually stable over K w...

We give a short and elementary proof of the theorem Lutz Weis that growth bound positive $C_0$-semigroup on $L_p(\mu)$ equals spectral its generator. In addition, we generalise result to case uniformly eventually semigroups.

We consider various decision problems for probabilistic finite automata (PFA)s with isolated cut-points. Recall that a cut-point x is said to be isolated for a PFA if the acceptance probability of all finite strings is bounded away from x. First we establish the exact level of undecidability of the problem of determining if a cut-point is isolated; we show this problem to be Σ2-complete. Next w...

We initiate a theory of locally eventually positive operator semigroups on Banach lattices. Intuitively this means: given initial datum, the solution corresponding Cauchy problem becomes (and stays) in part domain, after sufficiently large time. A drawback present C0-semigroups is that it applicable only when leading eigenvalue semigroup generator has strongly eigenvector. weaken requirement an...