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On the Lagrangian multiform structure of the extended lattice Boussinesq system
No full textThe lattice Boussinesq (lBSQ) equation is a member of the latticeGel'fand-Dikii (lGD) hierarchy, introduced in \cite{NijPapCapQui1992}, which isan infinite family of integrable systems of partial difference equationslabelled by an integer , where represents the lattice Korteweg-deVries (KdV) system, and the Boussinesq system. In \cite{Hiet2011} it wasshown that, written as three-component system, the lBSQ system allows for extraparameters which essentially amounts to building the lattice KdV inside thelBSQ. In this paper we show that, on the level of the Lagrangian structure,this boils down to a linear combination of Lagrangians from the members of thelGD hierarchy as was established in \cite{LobbNijGD2010}. The correspondingLagrangian multiform structure is shown to exhibit a `double zero' structure.Comment: 10 pages, 0 figures Accepted version, formatted in accordence with the publication templat
KdV hierarchies and quantum Novikov's equations
No full textThis paper begins with a review of the well-known KdV hierarchy, the -thNovikov equation, and its finite hierarchy in the classical commutative case.This finite hierarchy consists of compatible integrable polynomialdynamical systems in . We discuss a non-commutative version ofthe -th Novikov hierarchy defined on the finitely generated free associativealgebra with generators. Using the method ofquantisation ideals in , for , we obtain two-sidedhomogeneous ideals (quantisationideals) that are invariant with respect to the -th Novikov equation and suchthat the quotient algebra has a well-defined Poincare-Birkhoff-Witt basis. This allowsus to define the quantum -th Novikov equation and its hierarchy on. We derive commuting quantum first integrals(Hamiltonians) and represent the equations of the hierarchy in the Heisenbergform. Essential for our research is the concept of cyclic Frobenius algebras,which we introduced in our recent paper. In terms of the quadratic form thatdefines the structure of a cyclic Frobenius algebra, we explicitly express thefirst integrals of the -th Novikov hierarchy in the commutative, free, andquantum cases.Comment: This is a formatted version of the previous version with dedication to D. Levi. Published in the special issue of the OCNM
On k-wave solutions of quasilinear systems of partial differential equations
No full textIn this paper, we establish a relation between two seemingly unrelatedconcepts for solving first-order hyperbolic quasilinear systems of partialdifferential equations in many dimensions. These concepts are based on avariant of the conditional symmetry method and on the generalized method ofcharacteristics. We present the outline of recent results on multiple Riemannwave solutions of these systems. An auxiliary result concerning a modificationof the Frobenius theorem for integration is used. We apply this result in orderto show that the conditional symmetry method can deliver larger classes ofmultiple Riemann wave solutions, through a simpler procedure, than the oneobtained from the generalized method of characteristics. We demonstrate thatsolutions can be interpreted physically as a superposition of k single waves.These theoretical considerations are illustrated by examples ofhydrodynamic-type systems in (n+1) dimensions.Comment: 20 page
On Classifying Continuous Constraint Satisfaction Problems
No full textA continuous constraint satisfaction problem (CCSP) is a constraintsatisfaction problem (CSP) with an interval domain . Weengage in a systematic study to classify CCSPs that are complete of theExistential Theory of the Reals, i.e., ER-complete. To define this class, wefirst consider the problem ETR, which also stands for Existential Theory of theReals. In an instance of this problem we are given some sentence of the form, where is a well-formed quantifier-free formula consisting of the symbols , the goal is to check whether thissentence is true. Now the class ER is the family of all problems that admit apolynomial-time many-one reduction to ETR. It is known that NP ER PSPACE. We restrict our attention on CCSPs with addition constraints ()and some other mild technical conditions. Previously, it was shown thatmultiplication constraints (), squaring constraints (),or inversion constraints () are sufficient to establishER-completeness. We extend this in the strongest possible sense for equalityconstraints as follows. We show that CCSPs (with addition constraints and someother mild technical conditions) that have any one well-behaved curved equalityconstraint () are ER-complete. We further extend our results toinequality constraints. We show that any well-behaved convexly curved and anywell-behaved concavely curved inequality constraint ( and) imply ER-completeness on the class of such CCSPs.Comment: 54 pages, 7 figure
Perturbation results for distance-edge-monitoring numbers
No full textFoucaud et al. recently introduced and initiated the study of a newgraph-theoretic concept in the area of network monitoring. Given a graph, a set is a distance-edge-monitoring set iffor every edge , there is a vertex and a vertex y \inV(G) such that the edge belongs to all shortest paths between and .The smallest size of such a set in is denoted by .Denoted by (resp. ) the subgraph of obtained byremoving the edge from (resp. a vertex together with all itsincident edges from ). In this paper, we first show that for any graph andedge . Moreover, the bound is sharp. Next, we construct two graphs and to show that \operatorname{dem}(G)-\operatorname{dem}(G\setminusu) and can bearbitrarily large, where and . We also study therelation between and , where is a subgraph of . In the end, we give an algorithm to judge whether thedistance-edge monitoring set still remain in the resulting graph when any edgeof the graph is deleted
Categorical absorptions of singularities and degenerations
No full textWe introduce the notion of categorical absorption of singularities: anoperation that removes from the derived category of a singular variety a smalladmissible subcategory responsible for singularity and leaves a smooth andproper category. We construct (under appropriate assumptions) a categoricalabsorption for a projective variety with isolated ordinary double points.We further show that for any smoothing of over a smoothcurve , the smooth part of the derived category of extends to a smoothand proper over family of triangulated subcategories in the fibers of.Comment: 41 pages; v2, v3: minor improvements, v4: revised according to the referee's report, to appear in EPIGA, v5: published versio
Lie pairs
No full textExtending the theory of systems, we introduce a theory of Lie semialgebra``pairs'' which parallels the classical theory of Lie algebras, but with a``null set'' replacing . A selection of examples is given. These Lie pairscomprise two categories in addition to the universal algebraic definition, onewith ``weak Lie morphisms'' preserving null sums, and the other with``-morphisms'' preserving a surpassing relation thatreplaces equality. We provide versions of the PBW (Poincare-Birkhoff-Witt)Theorem in these three categories.Comment: 40 page
Nous entrerons dans la carrière quand nos aînés n'y seront plus. Difficultés de découverte, de datation et d'interprétation des carrières médiévales en Lorraine
No full textA land of stone, Lorraine is still marked by the imprint of open-cast quarries, the size of which reflects the impact of mechanisation on quarrying from the 19th century onwards. However, many older quarries still lurk in the landscape. Often very discreet and filled in, they are not always easy to identify, particularly those from the Middle Ages, when quarry sites were very specific to the opening of a building site. This review and investigation therefore takes us into the field of small-scale medieval quarrying, in an attempt to address both methodological and scientific issues.Terre de pierre, l'espace lorrain reste marqué par l'empreinte de carrières à ciel ouvert dont les dimensions reflètent les apports de la mécanisation au travail d'extraction à partir du XIXe siècle. Pour autant, nombre de carrières plus anciennes se cachent encore dans le paysage. Bien souvent très discrètes, remblayées, leur identification n'est pas toujours aisée, particulièrement celles du Moyen Âge, où les sites d'extractions sont très circonstanciés à l'ouverture d'un chantier de construction. Ce bilan et cette enquête nous mènent donc sur le terrain des petites exploitations médiévales, en tentant de répondre à des problématiques méthodologiques aussi bien que scientifiques
Depth lower bounds in Stabbing Planes for combinatorial principles
No full textStabbing Planes (also known as Branch and Cut) is a proof system introducedvery recently which, informally speaking, extends the DPLL method by branchingon integer linear inequalities instead of single variables. The techniquesknown so far to prove size and depth lower bounds for Stabbing Planes aregeneralizations of those used for the Cutting Planes proof system. For sizelower bounds these are established by monotone circuit arguments, while fordepth these are found via communication complexity and protection. As suchthese bounds apply for lifted versions of combinatorial statements. Rank lowerbounds for Cutting Planes are also obtained by geometric arguments calledprotection lemmas. In this work we introduce two new geometric approaches to prove size/depthlower bounds in Stabbing Planes working for any formula: (1) the antichainmethod, relying on Sperner's Theorem and (2) the covering method which usesresults on essential coverings of the boolean cube by linear polynomials, whichin turn relies on Alon's combinatorial Nullenstellensatz. We demonstrate their use on classes of combinatorial principles such as thePigeonhole principle, the Tseitin contradictions and the Linear OrderingPrinciple. By the first method we prove almost linear size lower bounds andoptimal logarithmic depth lower bounds for the Pigeonhole principle andanalogous lower bounds for the Tseitin contradictions over the complete graphand for the Linear Ordering Principle. By the covering method we obtain asuperlinear size lower bound and a logarithmic depth lower bound for StabbingPlanes proof of Tseitin contradictions over a grid graph
Ancrage industriel en bioéconomie : quel rôle joue la gouvernance logistique ?
No full textThis paper tries to identify the anchoring factors for emerging bioeconomy activities in some territories, based on a reading of the logistic constraints of this sector which generates important flows upstream. It is based on an identification of location factors for industries using biomass; a comparison of the petroeconomy/bioeconomy “line” logistics flows and the analysis of three cases of companies: the first one remained anchored after having been bought; the other one quickly relocated; and the third one recently established. This leads to a proposal for the role of logistics governance that can contribute to anchor such activities facing highly international competition, dependent on raw materials price fluctuations and on uneven international regulations.Cet article tente d’identifier les facteurs d’ancrage des activités de bioéconomie en émergence sur certains territoires, à partir d’une lecture des contraintes logistiques de ce secteur qui génère des flux importants en amont. Il est fondé sur une identification des facteurs de localisation des industries qui utilisent la biomasse, une comparaison des flux logistiques entre les voies pétrochimie et bioéconomie, et l’analyse de trois cas d’entreprises : l’une restée ancrée après avoir été rachetée, l’autre rapidement délocalisée, la troisième récemment implantée. Il en résulte une proposition du rôle de la gouvernance logistique pouvant contribuer à l’ancrage de telles activités fortement concurrencées à l’international, dépendantes des fluctuations des cours des matières et de réglementations internationales peu uniformisées