Bolza quaternion order and asymptotics of systoles along congruence subgroups

We give a detailed description of the arithmetic Fuchsian group of the Bolza surface and the associated quaternion order. This description enables us to show that the corresponding principal congruence covers satisfy the bound sys(X) > 4/3 log g(X) on the systole, where g is the genus. We also exhibit the Bolza group as a congruence subgroup, and calculate out a few examples of "Bolza twins" (using magma). Like the Hurwitz triplets, these correspond to the factoring of certain rational primes in the ring of integers of the invariant trace field of the surface. We exploit random sampling combined with the Reidemeister-Schreier algorithm as implemented in magma to generate these surfaces.Comment: 35 pages, to appear in Experimental Mathematic

What do we know about interchange fees and what does it mean for public policy? : commentary on Evans and Schmalensee

Debit cards ; Credit cards

Competition or Predation? Schumpeterian Rivalry in Network Markets

We explore the logic of predation and rules designed to prevent it in markets subject to network effects. Although, as many have informally argued, predatory behavior is plausibly more likely to succeed in such markets, we find that it is particularly hard to intervene in network markets in ways that improve welfare. We find that imposition of the leading proposals for rules against predatory pricing may lower or raise consumer welfare, depending on conditions that may be difficult to identify in practice.

Morse theory of harmonic forms

We consider the problem of whether it is possible to improve the Novikov inequalities for closed 1-forms, or any other inequalities of a similar nature, if we assume, additionally, that the given 1-form is harmonic with respect to some Riemannian metric. We show that, under suitable assumptions, it is impossible. We use, in an essential way, a theorem of E.Calabi characterizing 1-forms which are harmonic with respect to some metric. We also study some interesting examples illustrating our results.Comment: 16 pages, AMSTex, 12 figure

Dynamic trapping near a quantum critical point

The study of dynamics in closed quantum systems has recently been revitalized by the emergence of experimental systems that are well-isolated from their environment. In this paper, we consider the closed-system dynamics of an archetypal model: spins near a second order quantum critical point, which are traditionally described by the Kibble-Zurek mechanism. Imbuing the driving field with Newtonian dynamics, we find that the full closed system exhibits a robust new phenomenon -- dynamic critical trapping -- in which the system is self-trapped near the critical point due to efficient absorption of field kinetic energy by heating the quantum spins. We quantify limits in which this phenomenon can be observed and generalize these results by developing a Kibble-Zurek scaling theory that incorporates the dynamic field. Our findings can potentially be interesting in the context of early universe physics, where the role of the driving field is played by the inflaton or a modulus.Comment: 4 pages, 3 figures + 5 page supplemen

Innovation, Rent Extraction, and Integration in Systems Markets

We consider innovation incentives in markets where final goods comprise two strictly complementary components, one of which is monopolized. We focus on the case in which the complementary component is competitively supplied, and in which innovation is important. We explore ways in which the monopoly may have incentives to confiscate efficiency rents in the competitive sector, thus weakening or destroying incentives for independent innovation. We discuss how these problems are affected if the monopolist integrates into the competitive sector.

The Economics of Product-Line Restrictions With an Application to the Network Neutrality Debate

We examine the welfare effects of product-line restrictions, such as those called for by some proponents of network neutrality regulation. We consider a platform that brings together households and application providers. We find that restricting a monopoly platform to a single product has the following effects: (a) application providers that would otherwise have purchased a low-quality variant are excluded from the market; (b) applications 'in the middle' of the market utilize a higher and more efficient quality; and (c) applications at the top utilize a lower and less efficient quality than otherwise. Total surplus may rise or fall, although the analysis suggests to us that harm to welfare is likely. We also examine a duopoly model and find that the welfare effects are similar.

Innovation, Rent Extraction, and Integration in Systems Markets

We consider innovation incentives in markets where final goods comprise two strictly complementary components, one of which is monopolized. We focus on the case in which the complementary component is competitively supplied, and in which innovation is important. We explore ways in which the monopoly may have incentives to confiscate efficiency rents in the competitive sector, thus weakening or destroying incentives for independent innovation. We discuss how these problems are affected if the monopolist integrates into the competitive sector.