Simon (Theoret. Comput. Sci. 72 (1990) 65–94) has proved that every morphism from a free semigroup to a $nite semigroup S admits a Ramseyan factorization forest of height at most 9|S|. In this paper, we prove the same result of Simon with an improved bound of 7|S|. We provide a simple algorithm for constructing a factorization forest. In addition, we show that the algorithm cannot be improved s...

We count the number of occurrences of certain patterns in given words. We choose these words to be the set of all finite approximations of a sequence generated by a morphism with certain restrictions. The patterns in our considerations are either classical patterns 1-2, 2-1, 1-1· · · -1, or arbitrary generalized patterns without internal dashes, in which repetitions of letters are allowed. In p...

The method of conjugate gradients provides a very effective way to optimize large, deterministic systems by gradient descent. In its standard form, however, it is not amenable to stochastic approximation of the gradient. Here we explore ideas from conjugate gradient in the stochastic (online) setting, using fast Hessian-gradient products to set up low-dimensional Krylov subspaces within individ...

The method of conjugate gradients provides a very effective way to optimize large, deterministic systems by gradient descent. In its standard form, however, it is not amenable to stochastic approximation of the gradient. Here we explore a number of ways to adopt ideas from conjugate gradient in the stochastic setting, using fast Hessian-vector products to obtain curvature information cheaply. I...

The method of conjugate directions provides a very effective way to optimize large, deterministic systems by gradient descent. In its standard form, however, it is not amenable to stochastic approximation of the gradient. Here we explore ideas from conjugate gradient in the stochastic (online) setting, using fast Hessian-gradient products to set up low-dimensional Krylov subspaces within indivi...

In their book Rapoport and Zink constructed rigid analytic period spaces F for Fontaine’s filtered isocrystals and period morphisms from moduli spaces of p-divisible groups to some of these period spaces. They conjectured the existence of an étale bijective morphism F → F of rigid analytic spaces and of interesting local systems of Qp-vector spaces on F. For those period spaces possessing perio...

We prove that any dominant morphism of algebraic varieties over a field k of characteristic zero can be transformed into a toroidal (hence monomial) morphism by projective birational modifications of source and target. This was previously proved by the first and third author when k is algebraically closed. Moreover we show that certain additional requirements can be satisfied.

We prove that any (real or complex) analytic horizontally conformal submersion from a three-dimensional conformal manifold (M, cM ) to a twodimensional conformal manifold (N, cN ) can be, locally, ‘extended’ to a unique harmonic morphism from the H(eaven)-space (H, g) of (M, cN ) to (N, cN ). Moreover, any positive harmonic morphism with two-dimensional fibres from (H, g) is obtained this way.