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The conjectures of Artin-Tate and Birch-Swinnerton-Dyer
No full textWe provide two proofs that the conjecture of Artin-Tate for a fibered surfaceis equivalent to the conjecture of Birch-Swinnerton-Dyer for the Jacobian ofthe generic fibre. As a byproduct, we obtain a new proof of a theorem ofGeisser relating the orders of the Brauer group and the Tate-Shafarevich group.Comment: 18 pages, final versio
Restricted generating trees for weak orderings
No full textMotivated by the study of pattern avoidance in the context of permutationsand ordered partitions, we consider the enumeration of weak-ordering chainsobtained as leaves of certain restricted rooted trees. A tree of order isgenerated by inserting a new variable into each node at every step. A nodebecomes a leaf either after steps or when a certain stopping condition ismet. In this paper we focus on conditions of size 2 (, , or )and several conditions of size 3. Some of the cases considered here lead to thestudy of descent statistics of certain `almost' pattern-avoiding permutations.Comment: 16 pages. Final versio
-fixed Hilbert schemes on surfaces, modular forms, and eta products
No full textLet be a complex surface with an effective action of a group which preserves the holomorphic symplectic form. Let be the generating function for the Euler characteristics of the Hilbert schemesof -invariant length subschemes. We show that its reciprocal, is the Fourier expansion of a modular cusp form of weight for the congruence subgroup . We give anexplicit formula for in terms of the Dedekind eta function for all 82possible . The key intermediate result we prove is of independentinterest: it establishes an eta product identity for a certain shifted thetafunction of the root lattice of a simply laced root system. We extend ourresults to various refinements of the Euler characteristic, namely the Ellipticgenus, the Chi- genus, and the motivic class.Comment: Published version. Greatly simplified proof of Proposition 3.
On morphisms preserving palindromic richness
No full textIt is known that each word of length contains at most distinctpalindromes. A finite rich word is a word with maximal number of palindromicfactors. The definition of palindromic richness can be naturally extended toinfinite words. Sturmian words and Rote complementary symmetric sequences formtwo classes of binary rich words, while episturmian words and words codingsymmetric -interval exchange transformations give us other examples onlarger alphabets. In this paper we look for morphisms of the free monoid, whichallow us to construct new rich words from already known rich words. We focus onmorphisms in Class . This class contains morphisms injective on thealphabet and satisfying a particular palindromicity property: for everymorphism in the class there exists a palindrome such that is a first complete return word to for each letter . Wecharacterize morphisms which preserve richness over a binaryalphabet. We also study marked morphisms acting on alphabets withmore letters. In particular we show that every Arnoux-Rauzy morphism isconjugated to a morphism in Class and that it preserves richness
Graphs containing finite induced paths of unbounded length
No full textThe age of a graph (undirected and without loops) is thecollection of finite induced subgraphs of , considered up to isomorphy andordered by embeddability. It is well-quasi-ordered (wqo) for this order if itcontains no infinite antichain. A graph is \emph{path-minimal} if it containsfinite induced paths of unbounded length and every induced subgraph withthis property embeds . We construct path-minimal graphs whoseages are pairwise incomparable with set inclusion and which are wqo. Ourconstruction is based on uniformly recurrent sequences and lexicographical sumsof labelled graphs.Comment: 28 pages, 3 figure
Reachability and liveness in parametric timed automata
No full textWe study timed systems in which some timing features are unknown parameters.Parametric timed automata (PTAs) are a classical formalism for such systems butfor which most interesting problems are undecidable. Notably, the parametricreachability emptiness problem, i.e., the emptiness of the parameter valuationsset allowing to reach some given discrete state, is undecidable.Lower-bound/upper-bound parametric timed automata (L/U-PTAs) achievedecidability for reachability properties by enforcing a separation ofparameters used as upper bounds in the automaton constraints, and those used aslower bounds. In this paper, we first study reachability. We exhibit a subclass of PTAs(namely integer-points PTAs) with bounded rational-valued parameters for whichthe parametric reachability emptiness problem is decidable. Using this class,we present further results improving the boundary between decidability andundecidability for PTAs and their subclasses such as L/U-PTAs. We then study liveness. We prove that: (1) deciding the existence of at least one parameter valuation for whichthere exists an infinite run in an L/U-PTA is PSpace-complete; (2) the existence of a parameter valuation such that the system has adeadlock is however undecidable; (3) the problem of the existence of a valuation for which a run remains in agiven set of locations exhibits a very thin border between decidability andundecidability.Comment: This manuscript is an extended version of two conference papers published in the proceedings of ICFEM 2016 and ACSD 201
Linguistic Fingerprints on Translation's Lens
No full textWhat happens to the language fingerprints of a work when it is translated into another language? While translation studies has often prioritized concepts of equivalence (of form and function), and of textual function, digital humanities methodologies can provide a new analytical lens onto ways that stylistic traces of a text's source language can persist in a translated text. This paper presents initial findings of a project undertaken by the Stanford Literary Lab, which has identified distinctive grammatical features in short stories that have been translated into English. While the phenomenon of "translationese" has been well established particularly in corpus translation studies, we argue that digital humanities methods can be valuable for identifying specific traits for a vision of a world atlas of literary style
On Lie algebras associated with a spray
No full textThe Lie algebra of infinitesimal isometries of a Riemannian manifold containsat most two commutative ideals. One coming from the horizontal nullity space ofthe Nijenhuis tensor of the canonical connection, the other coming from theconstant vectors fields independent of the Riemannian metric
Bounded Reachability Problems are Decidable in FIFO Machines
No full textThe undecidability of basic decision problems for general FIFO machines suchas reachability and unboundedness is well-known. In this paper, we provide anunderapproximation for the general model by considering only runs that areinput-bounded (i.e. the sequence of messages sent through a particular channelbelongs to a given bounded language). We prove, by reducing this model to acounter machine with restricted zero tests, that the rational-reachabilityproblem (and by extension, control-state reachability, unboundedness, deadlock,etc.) is decidable. This class of machines subsumes input-letter-boundedmachines, flat machines, linear FIFO nets, and monogeneous machines, for whichsome of these problems were already shown to be decidable. These theoreticalresults can form the foundations to build a tool to verify general FIFOmachines based on the analysis of input-bounded machines
Upward-closed hereditary families in the dominance order
No full textThe majorization relation orders the degree sequences of simple graphs intoposets called dominance orders. As shown by Ruch and Gutman (1979) and Merris(2002), the degree sequences of threshold and split graphs form upward-closedsets within the dominance orders they belong to, i.e., any degree sequencemajorizing a split or threshold sequence must itself be split or threshold,respectively. Motivated by the fact that threshold graphs and split graphs havecharacterizations in terms of forbidden induced subgraphs, we define a class of graphs to be dominance monotone if whenever no realization of contains an element as an induced subgraph, and majorizes, then no realization of induces an element of . We presentconditions necessary for a set of graphs to be dominance monotone, and weidentify the dominance monotone sets of order at most 3.Comment: 19 pages, 5 figures. Final accepted versio