Cluster algebras of infinite rank as colimits

We formalize the way in which one can think about cluster algebras of infinite rank by showing that every rooted cluster algebra of infinite rank can be written as a colimit of rooted cluster algebras of finite rank. Relying on the proof of the posivity conjecture for skew-symmetric cluster algebras (of finite rank) by Lee and Schiffler, it follows as a direct consequence that the positivity conjecture holds for cluster algebras of infinite rank. Furthermore, we give a sufficient and necessary condition for a ring homomorphism between cluster algebras to give rise to a rooted cluster morphism without specializations. Assem, Dupont and Schiffler proposed the problem of a classification of ideal rooted cluster morphisms. We provide a partial solution by showing that every rooted cluster morphism without specializations is ideal, but in general rooted cluster morphisms are not ideal.Comment: Included cluster algebras of uncountable rank, fixed some typos. Results on the countable case unchanged, comments appreciate

Economic Analysis of Pay-for-delay Settlements and Their Legal Ruling

In this paper, we ask whether courts should continue to rule settlements in the context of pharmaceutical claims per se legal, when these settlements comprise payments from originator to generic companies, potentially delaying generic entry compared to the underlying litigation. Within a theoretical framework we compare consumer welfare under the rule of per se legality with that under alternative standards. We find that the rule of per se legality induces maximal collusion among settling companies. In comparison, the rule of per se illegality entirely prevents collusion and the rule of reason induces limited collusion when antitrust enforcement is subject to error. Contrary to intuition, limited collusion can be welfare enhancing as it increases companies' expected settlement profits and thus fosters generic entry. Generic companies obtain additional incentives to challenge probabilistic patents, which potentially leads to overall increased competition. We further show that generic entry is fostered more effectively by inducing limited collusion than by rewarding first generic entrants with an exclusivity right

Mutation of torsion pairs in cluster categories of Dynkin type DD

Mutation of torsion pairs in triangulated categories and its combinatorial interpretation for the cluster category of Dynkin type $A_n$ and of type $A_\infty$ have been studied by Zhou and Zhu. In this paper we present a combinatorial model for mutation of torsion pairs in the cluster category of Dynkin type $D_n$, using Ptolemy diagrams of Dynkin type $D_n$ which were introduced by Holm, J{\o}rgensen and Rubey.Comment: Corrected typos, some arguments made more concise, results unchange

Economic Analysis of Pay-for-delay Settlements and Their Legal Ruling

In this paper, we ask whether courts should continue to rule settlements in the context of pharmaceutical claims per se legal, when these settlements comprise payments from originator to generic companies, potentially delaying generic entry compared to the underlying litigation. Within a theoretical framework we compare consumer welfare under the rule of per se legality with that under alternative standards. We find that the rule of per se legality induces maximal collusion among settling companies. In comparison, the rule of per se illegality entirely prevents collusion and the rule of reason induces limited collusion when antitrust enforcement is subject to error. Contrary to intuition, limited collusion can be welfare enhancing as it increases companies' expected settlement profits and thus fosters generic entry. Generic companies obtain additional incentives to challenge probabilistic patents, which potentially leads to overall increased competition. We further show that generic entry is fostered more effectively by inducing limited collusion than by rewarding first generic entrants with an exclusivity right.antitrust and intellectual property law; patent settlements; collusion; per se rule; rule of reason; Hatch-Waxman Act

Can Naked Exclusion Be Procompetitive?

Antitrust scholars have argued that exclusive contracts have anticompetitive, or at best neutral effects, if no efficiencies are generated. In contrast, this paper shows that exclusive contracts can have procompetitive effects, provided buyers are imperfect downstream competitors and contract breach is feasible. In that case an efficient entrant is not necessarily foreclosed through exclusive contracting but induces buyers to breach. Because breaching buyers have to pay expectation damages to the incumbent, the downstream profits they obtain when breaching must be large enough. Therefore, the entrant needs to set a lower wholesale price than absent exclusive contracting, leading to lower final consumer prices and higher welfare

Homotopy invariants of singularity categories

We present a method for computing $\mathbb{A}^1$-homotopy invariants of singularity categories of rings admitting suitable gradings. Using this we describe any such invariant, e.g. homotopy K-theory, for the stable categories of self-injective algebras admitting a connected grading. A remark is also made concerning the vanishing of all such invariants for cluster categories of type $A_{2n}$ quivers.Comment: final revisio

Optical and TEM study of shock metamorphism from the Sedan test site

Thus far, detailed petrologic studies of shock metamorphism have been performed on samples recovered from laboratory experiments and on a few natural impactites. The loading history of these samples is quite different: In particular, laboratory experiments spend only a short time (less than 1 microsec) at peak pressure, whereas natural impactites may have stress pulses from 0.1 - 1 ms. On the other hand, laboratory experiments have known stress histories; natural impactites do not. Natural samples are also subjected to thousands or millions of years of postshock annealing and/or weathering. A useful intermediate case is that of nuclear detonation. Stress pulses for these events can reach 0.1 ms or higher, and samples are obtained in pristine condition. All three types of loading produce stresses of hundreds of kilobars. Samples studied were taken from the Sedan nulcear test site, and consist of a coarse-grained granodiorite containing quartz, K-feldspar, cordierite, and hornblende. Samples were studied optically in this section, then were thinned with an ion mill and studied by transmission electron microscopy (TEM). Optically, quartz and K-feldspar displayed numerous sets of planar deformation features (PDF's) identical to the nondecorated PDF's seen in laboratory samples and many natural impactites. TEM study showed that the PDF's in quartz and feldspar corresponded to densely packed wide transformation lamellae identical to those described in laboratory studies. The transformation lamellae in both minerals were amorphous, with no sign of high-pressure phases. In the case of K-feldspar only, narrow sublamellae extended outward from some wide lamellae. Quartz, which was more abundant and studied more extensively, contained no shock-induced dislocations. Some planar features were also seen in cordierite, but could not be identified due to rapid beam damage. No shock defects were seen in hornblende in TEM. The shock-induced defects present at the Sedan site are very similar to those seen in shock recovery experiments, and also to those present at certain natural events (e.g., Meteor Crater). This suggests that shock deformation in quartz is not strongly dependent on shock pulse duration, and that laboratory recovery experiments are useful simulations of natural impact events

Cluster tilting modules for mesh algebras

We study cluster tilting modules in mesh algebras of Dynkin type, providing a new proof for their existence. In all but one case, we show that these are precisely the maximal rigid modules, and that they are equivariant for a certain automorphism. We further study their mutation, providing an example of mutation in an abelian category which is not stably 2-Calabi-Yau, and explicitly describe the combinatorics.Comment: comments appreciated; the third version includes a discussion on the combinatorics of the mutation