Differential calculus over double Lie algebroids

The notion of double Lie algebroid was defined by M. Van den Bergh and was illustrated by the double quasi Poisson case. We give new examples of double Lie algebroids and develop a differential calculus in that context. We recover the non commutative de Rham complex and the double Poisson-Lichnerowicz cohomology (Pichereau-vanWeyer) as particular cases of our construction.Comment: 18 page

Different Clusters of Text from Ancient China, Different Mathematical Ontologies

Sources attesting to mathematical activities in ancient China form at least four distinct clusters of texts, bespeaking at least four different—though overlapping—ways of practicing mathematics. I will focus on two such sets of documents: the canons that in the seventh century constituted one of the two curricula taught in the Imperial “School of Mathematics,” and manuscripts recently excavated from tombs sealed in the last centuries BCE. I will argue that these two sets of documents testify to two different ways of practicing mathematics, which related to different material practices. Accordingly, we can perceive that mathematical objects were shaped and explored in different ways, with significant consequences for the knowledge produced

Downstream Competition, Foreclosure, and Vertical Integration.

This paper analyses the impact of competition among downstream firms on an upstream firm's payoff and on its incentive to vertically integrate when firms on both segments negotiate optimal contracts. We argue that tougher competition decreases the downstream industry profit, but improves the upstream firm's negotiation position. In particular, the upstream firm is better off encouraging competition when the downstream firms have high bargaining power. We derive implications on the interplay between vertical integration and competition among the downstream firms. The mere possibility of vertical integration may constitute a barrier to entry and may trigger strategic horizontal spin-offs or mergers. We discuss the impact of upstream competition on our results.Foreclosure; Vertical integration; Bargaining; Competition; Contracts;

Hold-Up, Stakeholders and Takeover Threats.

We analyze the impact of takeover threats on long term relationships between the target owners and other stakeholders. In the absence of takeovers, stakeholders’ bargaining power increases their incentive to invest but reduces the owners’ incentive to invest. The threat of a takeover that would transfer value from the stakeholders reduces their ex ante investment. However, the stakeholders may appropriate ex post some value created by a takeover. This can prevent some value-enhancing takeovers. We examine extensions to the disciplinary role of takeovers, takeover defence mechanisms, and trade credit, and discuss empirical predictions.Finances et gouvernance des entreprises; Structure du capital et de la propriété; Mouvements financiers;

Takeovers and the dynamics of information flows.

This Paper analyses the effect of a possible takeover on information flows and on the terms of trade in business relationships. We consider a long-term relationship between a firm and a privately-informed stakeholder, a buyer for example. In our model, takeovers both increase the surplus from trade and enable the firm to extract a potentially higher share of the surplus from the buyer. The possibility of a takeover that leaves the buyer with a higher (lower) rent than the incumbent manager increases (decreases) the buyer's willingness to reveal their valuation. We suggest a number of testable predictions on the performance of takeover targets and trade credit.takeovers; information; price; value; disclosure; buyer;

From Many-Valued Consequence to Many-Valued Connectives

Given a consequence relation in many-valued logic, what connectives can be defined? For instance, does there always exist a conditional operator internalizing the consequence relation, and which form should it take? In this paper, we pose this question in a multi-premise multi-conclusion setting for the class of so-called intersective mixed consequence relations, which extends the class of Tarskian relations. Using computer-aided methods, we answer extensively for 3-valued and 4-valued logics, focusing not only on conditional operators, but on what we call Gentzen-regular connectives (including negation, conjunction, and disjunction). For arbitrary N-valued logics, we state necessary and sufficient conditions for the existence of such connectives in a multi-premise multi-conclusion setting. The results show that mixed consequence relations admit all classical connectives, and among them pure consequence relations are those that admit no other Gentzen-regular connectives. Conditionals can also be found for a broader class of intersective mixed consequence relations, but with the exclusion of order-theoretic consequence relations.Comment: Updated version [corrections of an incorrect claim in first version; two bib entries added

Duality functors for quantum groupoids

We present a formal algebraic language to deal with quantum deformations of Lie-Rinehart algebras - or Lie algebroids, in a geometrical setting. In particular, extending the ice-breaking ideas introduced by Xu in [Ping Xu, "Quantum groupoids", Comm. Math. Phys. 216 (2001), 539-581], we provide suitable notions of "quantum groupoids". For these objects, we detail somewhat in depth the formalism of linear duality; this yields several fundamental antiequivalences among (the categories of) the two basic kinds of "quantum groupoids". On the other hand, we develop a suitable version of a "quantum duality principle" for quantum groupoids, which extends the one for quantum groups - dealing with Hopf algebras - originally introduced by Drinfeld (cf. [V. G. Drinfeld, "Quantum groups", Proc. ICM (Berkeley, 1986), 1987, pp. 798-820], sec. 7) and later detailed in [F. Gavarini, "The quantum duality principle", Annales de l'Institut Fourier 53 (2002), 809-834].Comment: La-TeX file, 47 pages. Final version, after galley proofs correction, published in "Journal of Noncommutative Geometry". Compared with the previously posted version, we streamlined the whole presentation, we fixed a few details and we changed a bit the list of reference

Suszko's Problem: Mixed Consequence and Compositionality

Suszko's problem is the problem of finding the minimal number of truth values needed to semantically characterize a syntactic consequence relation. Suszko proved that every Tarskian consequence relation can be characterized using only two truth values. Malinowski showed that this number can equal three if some of Tarski's structural constraints are relaxed. By so doing, Malinowski introduced a case of so-called mixed consequence, allowing the notion of a designated value to vary between the premises and the conclusions of an argument. In this paper we give a more systematic perspective on Suszko's problem and on mixed consequence. First, we prove general representation theorems relating structural properties of a consequence relation to their semantic interpretation, uncovering the semantic counterpart of substitution-invariance, and establishing that (intersective) mixed consequence is fundamentally the semantic counterpart of the structural property of monotonicity. We use those to derive maximum-rank results proved recently in a different setting by French and Ripley, as well as by Blasio, Marcos and Wansing, for logics with various structural properties (reflexivity, transitivity, none, or both). We strengthen these results into exact rank results for non-permeable logics (roughly, those which distinguish the role of premises and conclusions). We discuss the underlying notion of rank, and the associated reduction proposed independently by Scott and Suszko. As emphasized by Suszko, that reduction fails to preserve compositionality in general, meaning that the resulting semantics is no longer truth-functional. We propose a modification of that notion of reduction, allowing us to prove that over compact logics with what we call regular connectives, rank results are maintained even if we request the preservation of truth-functionality and additional semantic properties.Comment: Keywords: Suszko's thesis; truth value; logical consequence; mixed consequence; compositionality; truth-functionality; many-valued logic; algebraic logic; substructural logics; regular connective