Uniform Continuity and Br\'ezis-Lieb Type Splitting for Superposition Operators in Sobolev Space

Using concentration-compactness arguments we prove a variant of the Brezis-Lieb-Lemma under weaker assumptions on the nonlinearity than known before. An intermediate result on the uniform continuity of superposition operators in Sobolev space is of independent interest

An Approach to Studying Quasiconformal Mappings on Generalized Grushin Planes

We demonstrate that the complex plane and a class of generalized Grushin planes $G_r$, where $r$ is a function satisfying specific requirements, are quasisymmetrically equivalent. Then using conjugation we are able to develop an analytic definition of quasisymmetry for homeomorphisms on $G_r$ spaces. In the last section we show our analytic definition of quasisymmetry is consistent with earlier notions of conformal mappings on the Grushin plane. This leads to several characterizations of conformal mappings on the generalized Grushin planes

External Price Benchmarking vs. Price Negotiation for Pharmaceuticals

External price benchmarking imposes a price cap for pharmaceuticals based on prices of identical products in other countries. Suppose that a regulatory agency can either directly negotiate drug prices with pharmaceutical manufacturers or implement a benchmarking regime based on foreign prices. Using a model where two countries differ only in their market size, we show that a country prefers benchmarking if its agency has considerably less bargaining power compared to the agency in the other country. Assuming that bargaining power is positively correlated to country size, we find that only small countries might have an incentive to engage in external price benchmarking. This incentive shrinks if population size grows.Pharmaceuticals; price negotiation; administered prices; external reference pricing

The generalized Lichnerowicz formula and analysis of Dirac operators

We study Dirac operators acting on sections of a Clifford module ${\cal E}$\ over a Riemannian manifold $M$. We prove the intrinsic decomposition formula for their square, which is the generalisation of the well-known formula due to Lichnerowicz [L]. This formula enables us to distinguish Dirac operators of simple type. For each Dirac operator of this natural class the local Atiyah-Singer index theorem holds. Furthermore, if $M$\ is compact and {{\petit \rm dim}\;M=2n\ge 4}, we derive an expression for the Wodzicki function $W_{\cal E}$, which is defined via the non-commutative residue on the space of all Dirac operators ${\cal D}({\cal E})$. We calculate this function for certain Dirac operators explicitly. From a physical point of view this provides a method to derive gravity, resp. combined gravity/Yang-Mills actions from the Dirac operators in question.Comment: 25 pages, plain te

Unification of Gravity and Yang-Mills-Higgs Gauge Theories

In this letter we show how the action functional of the standard model and of gravity can be derived from a specific Dirac operator. Far from being exotic this particular Dirac operator turns out to be structurally determined by the Yukawa coupling term. The main feature of our approach is that it naturally unifies the action of the standard model with gravity.Comment: 8 pages, late

On the Convergence Time of the Best Response Dynamics in Player-specific Congestion Games

We study the convergence time of the best response dynamics in player-specific singleton congestion games. It is well known that this dynamics can cycle, although from every state a short sequence of best responses to a Nash equilibrium exists. Thus, the random best response dynamics, which selects the next player to play a best response uniformly at random, terminates in a Nash equilibrium with probability one. In this paper, we are interested in the expected number of best responses until the random best response dynamics terminates. As a first step towards this goal, we consider games in which each player can choose between only two resources. These games have a natural representation as (multi-)graphs by identifying nodes with resources and edges with players. For the class of games that can be represented as trees, we show that the best-response dynamics cannot cycle and that it terminates after O(n^2) steps where n denotes the number of resources. For the class of games represented as cycles, we show that the best response dynamics can cycle. However, we also show that the random best response dynamics terminates after O(n^2) steps in expectation. Additionally, we conjecture that in general player-specific singleton congestion games there exists no polynomial upper bound on the expected number of steps until the random best response dynamics terminates. We support our conjecture by presenting a family of games for which simulations indicate a super-polynomial convergence time

A generalized Lichnerowicz formula, the Wodzicki Residue and Gravity

We prove a generalized version of the well-known Lichnerowicz formula for the square of the most general Dirac operator $\widetilde{D}$\ on an even-dimensional spin manifold associated to a metric connection $\widetilde{\nabla}$. We use this formula to compute the subleading term $\Phi_1(x,x, \widetilde{D}^2)$\ of the heat-kernel expansion of $\widetilde{D}^2$. The trace of this term plays a key-r\hat {\petit\rm o}le in the definition of a (euclidian) gravity action in the context of non-commutative geometry. We show that this gravity action can be interpreted as defining a modified euclidian Einstein-Cartan theory.Comment: 10 pages, plain te