Expansion Trees with Cut

Herbrand's theorem is one of the most fundamental insights in logic. From the syntactic point of view it suggests a compact representation of proofs in classical first- and higher-order logic by recording the information which instances have been chosen for which quantifiers, known in the literature as expansion trees. Such a representation is inherently analytic and hence corresponds to a cut-free sequent calculus proof. Recently several extensions of such proof representations to proofs with cut have been proposed. These extensions are based on graphical formalisms similar to proof nets and are limited to prenex formulas. In this paper we present a new approach that directly extends expansion trees by cuts and covers also non-prenex formulas. We describe a cut-elimination procedure for our expansion trees with cut that is based on the natural reduction steps. We prove that it is weakly normalizing using methods from the epsilon-calculus

All Mutually Unbiased Product Bases in Dimension Six

All mutually unbiased bases in dimension six consisting of product states only are constructed. Several continuous families of pairs and two triples of mutually unbiased product bases are found to exist but no quadruple. The exhaustive classification leads to a proof that a complete set of seven mutually unbiased bases, if it exists, cannot contain a triple of mutually unbiased product bases.Comment: 32 pages, 3 figures, identical to published versio

Mixing Times of Markov Chains on Degree Constrained Orientations of Planar Graphs

We study Markov chains for $\alpha$-orientations of plane graphs, these are orientations where the outdegree of each vertex is prescribed by the value of a given function $\alpha$. The set of $\alpha$-orientations of a plane graph has a natural distributive lattice structure. The moves of the up-down Markov chain on this distributive lattice corresponds to reversals of directed facial cycles in the $\alpha$-orientation. We have a positive and several negative results regarding the mixing time of such Markov chains. A 2-orientation of a plane quadrangulation is an orientation where every inner vertex has outdegree 2. We show that there is a class of plane quadrangulations such that the up-down Markov chain on the 2-orientations of these quadrangulations is slowly mixing. On the other hand the chain is rapidly mixing on 2-orientations of quadrangulations with maximum degree at most 4. Regarding examples for slow mixing we also revisit the case of 3-orientations of triangulations which has been studied before by Miracle et al.. Our examples for slow mixing are simpler and have a smaller maximum degree, Finally we present the first example of a function $\alpha$ and a class of plane triangulations of constant maximum degree such that the up-down Markov chain on the $\alpha$-orientations of these graphs is slowly mixing

Selling When Brand Image Matters

This paper studies profit-maximizing seller behavior when brand image affects consumer demand. We consider a seller facing a population of consumers with heterogeneous tastes regarding product quality and brand image. First, we analyze “active branding” by the seller through costly advertising. Our analysis shows that advertising, price and profits are all increasing in the average valuation of brand image in the population. Second, we examine the role of “passive branding” emanating from the population’s consumption of the product. We demonstrate that seller profits increase in the average degree of conformity in the opulation whereas the price remains unaffected.Quality; brand image; advertising; conformity; exclusivity

Persuading Consumers With Social Attitudes

This paper analyzes persuasive advertising and pricing in oligopoly if firms sell differentiated products and consumers have heterogenous social attitudes towards the consumption by others. Deriving product demand from primitives, we show that the demand-enhancing effect of persuasive advertising varies across consumers and increases in the average degree of conformity. In equilibrium, both quality and cost leaders choose higher advertising intensities and charge higher prices than their competitors. In addition, we show that an increase in the average degree of conformity among consumers reinforces asymmetries between firms.Advertising, social attitude, consumption externality, quality