In this article the notions of semi weak orthogonality and semi weak factorization structure in a category $mathcal X$ are introduced. Then the relationship between semi weak factorization structures and quasi right (left) and weak factorization structures is given. The main result is a characterization of semi weak orthogonality, factorization of morphisms, and semi weak factorization structur...

We give a complete description of the dimension spectra of Birkhoff averages on a hyperbolic set of a surface diffeomorphism. The main novelty is that we are able to consider simultaneously Birkhoff averages into the future and into the past, i.e., both for positive and negative time. We emphasize that the description of these spectra is not a consequence of the available results in the case of...

A fundamental result in rewriting is : E |= s ≈ t ⇐⇒1 E ` s = t ⇐⇒2 s↔E t 1. is Birkhoff’s theorem expressing that semantic consequence (|=) coincides with syntactic consequence (`). 2. expresses that equality in equational logic (=) coincides with convertibility in rewriting (↔∗), a property known as logicality. We show that the result can be extended to hold for several sub-equational logics ...

One of the most fundamental mathematical contributions of Garrett Birkhoff is the HSP theorem, which implies that a finite algebra B satisfies all equations that hold in a finite algebra A of the same signature if and only if B is a homomorphic image of a subalgebra of a finite power of A. On the other hand, if A is infinite, then in general one needs to take an infinite power in order to obtai...

Let R be a real closed field. The Pierce-Birkhoff conjecture says that any piecewise polynomial function f on Rn can be obtained from the polynomial ring R[x1, . . . , xn] by iterating the operations of maximum and minimum. The purpose of this paper is twofold. First, we state a new conjecture, called the Connectedness conjecture, which asserts the existence of connected sets in the real spectr...

We show that pseudovarieties of finitely generated algebras, i.e., classes C of finitely generated algebras closed under finite products, homomorphic images, and subalgebras, can be described via a uniform structure UC on the free algebra for C: the members of C then are precisely those finitely generated algebras A for which the natural mapping from the free algebra onto the term clone of A is...

This contribution presents a review of results obtained from computations of approximate equations of motion for the Fermi-Pasta-Ulam lattice. These approximate equations are obtained as a finite-dimensional Birkhoff normal form. It turns out that in many cases, the Birkhoff normal form is suitable for application of the KAM theorem. In particular this proves Nishida’s 1971 conjecture stating t...

The Schubert cells eλ are the orbits of the Iwahori subgroup B̃, while the Birkhoff strata Sλ are the orbits of the opposite Iwahori subgroup B̃ . The cells and the strata are dual in the sense that Sλ ∩ eλ = {λ}, and the intersection is transverse. The closure of eλ is the affine Schubert variety Xλ. It has dimension l (λ), where l is the minimal length occuring in the coset λW̃I , and its cells ...

Let R be a real closed field. The Pierce-Birkhoff conjecture says that any piecewise polynomial function f on Rn can be obtained from the polynomial ring R[x1, . . . , xn] by iterating the operations of maximum and minimum. The purpose of this paper is threefold. First, we state a new conjecture, called the Connectedness conjecture, which asserts, for every pair of points α, β ∈ Sper R[x1, . . ...

— These notes are based on lectures held at the Lanzhou university (China) during a CIMPA summer school in july 2004 but benefit from recent devellopements. Our aim is to explain some normal form technics that allow to study the long time behaviour of the solutions of Hamiltonian perturbations of integrable systems. We are in particular interested with stability results. Our approach is centere...