We study the extension of the Kechris-Solecki-Todorcevic dichotomy onanalytic graphs to dimensions higher than 2. We prove that the extension ispossible in any dimension, finite or infinite. The original proof works in thecase of the finite dimension. We first prove that the natural extension doesnot work in the case of the infinite dimension, for the notion of continuoushomomorphism used in the original theorem. Then we solve the problem in thecase of the infinite dimension. Finally, we prove that the natural extensionworks in the case of the infinite dimension, but for the notion ofBaire-measurable homomorphism.
We prove that, for each non null countable ordinal alpha, there exist someSigma^0_alpha-complete omega-powers, and some Pi^0_alpha-complete omega-powers,extending previous works on the topological complexity of omega-powers. Weprove effective versions of these results. In particular, for each non nullrecursive ordinal alpha, there exists a recursive finitary language A such thatA^omega is Sigma^0_alpha-complete (respectively, Pi^0_alpha-complete). To dothis, we prove effective versions of a result by Kuratowski, describing a Borelset as the range of a closed subset of the Baire space by a continuousbijection. This leads us to prove closure properties for the classesEffective-Pi^0_alpha and Effective-Sigma^0_alpha of the hyperarithmeticalhierarchy in arbitrary recursively presented Polish spaces. We apply ourexistence results to get better computations of the topological complexity ofsome sets of dictionaries considered by the second author in [Omega-Powers andDescriptive Set Theory, Journal of Symbolic Logic, Volume 70 (4), 2005, p.1210-1232].
Diagrammatic logics were introduced in 2002, with emphasis on the notions ofspecifications and models. In this paper we improve the description of theinference process, which is seen as a Yoneda functor on a bicategory offractions. A diagrammatic logic is defined from a morphism of limit sketches(called a propagator) which gives rise to an adjunction, which in turndetermines a bicategory of fractions. The propagator, the adjunction and thebicategory provide respectively the syntax, the models and the inferenceprocess for the logic. Then diagrammatic logics and their morphisms are appliedto the semantics of side effects in computer languages.
A bijection $\Phi$ is presented between plane bipolar orientations withprescribed numbers of vertices and faces, and non-intersecting triples ofupright lattice paths with prescribed extremities. This yields a combinatorialproof of the following formula due to R. Baxter for the number $\Theta_{ij}$ ofplane bipolar orientations with $i$ non-polar vertices and $j$ inner faces:$\Theta_{ij}=2\frac{(i+j)!(i+j+1)!(i+j+2)!}{i!(i+1)!(i+2)!j!(j+1)!(j+2)!}$. Inaddition, it is shown that $\Phi$ specializes into the bijection of Bernardiand Bonichon between Schnyder woods and non-crossing pairs of Dyck words.
A linear method for inverting noisy observations of the Radon transform isdeveloped based on decomposition systems (needlets) with rapidly decayingelements induced by the Radon transform SVD basis. Upper bounds of the risk ofthe estimator are established in $L^p$ ($1\le p\le \infty$) norms for functionswith Besov space smoothness. A practical implementation of the method is givenand several examples are discussed.
D. Calaque, K. Ebrahimi-Fard and D. Manchon have recently defined a Hopfalgebra by introducing a new coproduct on a commutative algebra of rootedforests. The space of primitive elements of the graded dual is endowed with aleft pre-Lie product defined in terms of insertion of a tree inside another. Inthis work we prove a ``derivation'' relation between this pre-Lie structure andthe left pre-Lie product defined by grafting.
Since the end of the XIXth century, we know that each birational map of thecomplex projective plane is the product of a finite number of quadraticbirational maps of the projective plane; this motivates our work whichessentially deals with these quadratic maps. We establish algebraic propertiessuch as the classification of one parameter groups of quadratic birational mapsor the smoothness of the set of quadratic birational maps in the set ofrational maps. We prove that a finite number of generic quadratic birationalmaps generates a free group. We show that if f is a quadratic birational map oran automorphism of the projective plane, the normal subgroup generated by f isthe full group of birational maps of the projective plane, which implies thatthis group is perfect. We study some dynamical properties: following an idea ofGuillot, we translate some invariants for foliations in our context, inparticular we obtain that if two generic quadratic birational maps arebirationally conjugated, then they are conjugated by an automorphism of theprojective plane. We are also interested in the presence of "invariantobjects": curves, foliations, fibrations. Then follows a more experimentalpart: we draw orbits of quadratic birational maps with real coefficients andsets analogous to Julia sets for polynomials of one variable. We studybirational maps of degree 3 and, by considering the different possibleconfigurations of the exceptional curves, we give the "classification" of thesemaps. We can deduce from this that the set of the birational maps of degree 3exactly is irreducible, in fact rationally connected.
A sequence of reversals that takes a signed permutation to the identity isperfect if at no step a common interval is broken. Determining a parsimoniousperfect sequence of reversals that sorts a signed permutation is NP-hard. Herewe show that, despite this worst-case analysis, with probability one, sortingcan be done in polynomial time. Further, we find asymptotic expressions for theaverage length and number of reversals in commuting permutations, aninteresting sub-class of signed permutations.
Most often, in a categorical semantics for a programming language, thesubstitution of terms is expressed by composition and finite products. Howeverthis does not deal with the order of evaluation of arguments, which may havemajor consequences when there are side-effects. In this paper Cartesian effectcategories are introduced for solving this issue, and they are compared withstrong monads, Freyd-categories and Haskell's Arrows. It is proved that aCartesian effect category is a Freyd-category where the premonoidal structureis provided by a kind of binary product, called the sequential product. Theuniversal property of the sequential product provides Cartesian effectcategories with a powerful tool for constructions and proofs. To our knowledge,both effect categories and sequential products are new notions.