where (ai, bj) ∈ R, ν is the commensurable differentiation order,mB and mA are respectively numerator and denominator degrees, with mA > mB for strictly causal systems. Stability of fractional differentiation systems is addressed in the following theorem. Theorem 1.1. (Stability Matignon (1998)). A commensurable transfer function with a commensurable order ν, as in (4), with T and R two coprime...

We compare two families of continued fractions algorithms, the symmetrized Rosen algorithm and the Veech algorithm. Each of these algorithms expands real numbers in terms of certain algebraic integers. We give explicit models of the natural extension of the maps associated with these algorithms; prove that these natural extensions are in fact conjugate to the first return map of the geodesic fl...

The word metric on a group is independent of the choice of generating set, so the quasiisometry type of a group is well defined. We say that two groups are (abstractly) commensurable if they are related to one another by any (finite) sequence of finite processes: taking finite-index subgroups or supergroups, finite quotients or extensions. Groups which are commensurable are quasi-isometric. The...

Suppose n > 3, let M1,M2 be n-dimensional connected complete finitevolume hyperbolic manifolds with non-empty geodesic boundary, and suppose that π1(M1) is quasi-isometric to π1(M2) (with respect to the word metric). Also suppose that if n = 3, then ∂M1 and ∂M2 are compact. We show that M1 is commensurable with M2. Moreover, we show that there exist homotopically equivalent hyperbolic 3-manifol...

The tendency for the period of the helically ordered moments in holmium to lock into values which are commensurable with the lattice is studied theoretically as a function of temperature and magnetic field. The commensurable effects are derived in the mean-field approximation from numerical calculations of the free energy of various commensurable structures, and the results are compared with th...

A. Reid [R] showed that if Γ1 and Γ2 are arithmetic lattices in G = PGL2(R) or in PGL2(C) which give rise to isospectral manifolds, then Γ1 and Γ2 are commensurable (after conjugation). We show that for d ≥ 3 and S = PGLd(R)/POd(R), or S = PGLd(C)/PUd(C), the situation is quite different: there are arbitrarily large finite families of isospectral non-commensurable compact manifolds covered by S...

Let G be a countable discrete group. Call two subgroups H1 and H2 of G commensurable if H1 ∩H2 has finite index in both H1 and H2. We say that an action of G on a discrete set X has noncommensurable stabilizers if the stabilizers of any two distinct points of X are not commensurable. We prove in this paper that the action of the mapping class group on the complex of curves has noncommensurable ...