Two Tales on Resale

In some markets vertically integrated firms sell directly to final customers hut also to independent downstream firms with whom they then compete on the downstream market. It is often argued that resellers intensify competition and benefit consumers, in particular when wholesale prices are regulated. However, we show that (i) resale may increase prices and make consumers worse off and that (ii) standard "retail minus X regulation" may increase prices and harm consumers. Our analysis suggests that this is more likely if the number of integrated firms is small, the degree of product differentiation is low, and/or if competition is spatial

Two Tales on Resale

In some markets vertically integrated firms sell directly to final customers hut also to independent downstream firms with whom they then compete on the downstream market. It is often argued that resellers intensify competition and benefit consumers, in particular when wholesale prices are regulated. However, we show that (i) resale may increase prices and make consumers worse off and that (ii) standard "retail minus X regulation" may increase prices and harm consumers. Our analysis suggests that this is more likely if the number of integrated firms is small, the degree of product differentiation is low, and/or if competition is spatial.Resale regulation; wholesale; spatial product differentiation; non-spatial product differentiation; vertical restraints

Inelastic electron backscattering in a generic helical edge channel

We evaluate the low-temperature conductance of a weakly interacting one-dimensional helical liquid without axial spin symmetry. The lack of that symmetry allows for inelastic backscattering of a single electron, accompanied by forward-scattering of another. This joint effect of weak interactions and potential scattering off impurities results in a temperature-dependent deviation from the quantized conductance, $\delta G \propto T^4$. In addition, $\delta G$ is sensitive to the position of the Fermi level. We determine numerically the parameters entering our generic model for the Bernevig-Hughes-Zhang Hamiltonian of a HgTe/CdTe quantum well in the presence of Rashba spin-orbit coupling.Comment: 4+ pages, 3 figures, published versio

Crossover from Attractive to Repulsive Casimir Forces and Vice Versa

Systems described by an O(n) symmetrical $\phi^4$ Hamiltonian are considered in a $d$-dimensional film geometry at their bulk critical points. The critical Casimir forces between the film's boundary planes $\mathfrak{B}_j, j=1,2$, are investigated as functions of film thickness $L$ for generic symmetry-preserving boundary conditions $\partial_n\bm{\phi}=\mathring{c}_j\bm{\phi}$. The $L$-dependent part of the reduced excess free energy per cross-sectional area takes the scaling form $f_{\text{res}}\approx D(c_1L^{\Phi/\nu},c_2L^{\Phi/\nu})/L^{d-1}$ when $d<4$, where $c_i$ are scaling fields associated with the variables $\mathring{c}_i$, and $\Phi$ is a surface crossover exponent. Explicit two-loop renormalization group results for the function $D(\mathsf{c}_1,\mathsf{c}_2)$ at $d=4-\epsilon$ dimensions are presented. These show that (i) the Casimir force can have either sign, depending on $\mathsf{c}_1$ and $\mathsf{c}_2$, and (ii) for appropriate choices of the enhancements $\mathring{c}_j$, crossovers from attraction to repulsion and vice versa occur as $L$ increases.Comment: 4 RevTeX pages, 2 eps figures; minor misprints corrected and 3 references adde

The thermodynamic Casimir effect for non-symmetry-breaking and symmetry-breaking boundary conditions

Wenn makroskopische Körper sich in einem fluktuierenden Medium befinden und dieses in seiner räumlichen Ausdehnung begrenzen, so können daraus langreichweitige Kräfte zwischen diesen Körpern folgen. Das Spektrum der Fluktuationen wird modifiziert und die Energie des Systems ist dann abhängig von den Abständen zwischen den Körpern, woraus die Existenz einer Kraft zwischen diesen folgt. Die vorliegende Arbeit ist der Untersuchung des Einflusses der Randbedingungen auf den thermodynamischen Casimir-Effekt gewidmet. Hier sind thermische Fluktuationen in der Nähe eines kritischen Punktes ursächlich für diese Kräfte. Es werden $O(n)$-symmetrische $\phi^4$-Theorien in $d$-dimensionalen Filmen der Filmdicke $L$ untersucht. Berücksichtigt man zudem externe symmetriebrechende Felder, so lauten die generischen Randbedingungen für diese Theorien $\partial_{n}\phi-c_j\,\phi=-h_j$. Dabei sind die Parameter $c_j$ Oberflächenkopplungen, welche im Rahmen der Landau-Theorie die Rolle linear extrapolierter Eindringtiefen in die Oberflächen spielen, und $h_j$ Oberflächenfelder. Der Einfluss der Variablen $c_j$ auf die Casimir-Kraft wird im Rahmen der renormierungsgruppenverbesserten Störungstheorie in $d=4-\epsilon$ Dimensionen in Zweischleifennäherung am \emph{Bulk}-kritischen Punkt untersucht. Besondere Aufmerksamkeit kommt dabei dem Fall kritischer Verstärkung der Oberflächenkopplungen zu, da hier die gewöhnliche Schleifenentwicklung aufgrund einer Nullmode im Spektrum zusammenbricht und in der $\epsilon$-Entwicklung daher nicht-ganzzahlige Potenzen von $\epsilon$ auftreten. Diese störungstheoretischen Rechnungen sind auf die ungeordnete Phase mit $T\geq T_{\mathrm{c},\infty}$, $c_j\geq c_\mathrm{sp}$ und $h_j=0$ beschränkt. Um die Analyse für alle Temperaturen zu ermöglichen, wird der exakt handhabbare Limes $n\to\infty$ des dreidimensionalen $\phi^4$-Modells untersucht. Dazu werden selbstkonsistente Gleichungen für die freie Energie hergeleitet, welche numerisch exakt gelöst werden können. Für Dirichlet-Randbedingungen wird hier bei $h_j=0$ eine Temperaturabhängigkeit der Casimir-Kraft gefunden, welche qualitativ mit den Messungen in der Nähe des Lambda-Übergangs in suprafluidem Helium übereinstimmt. Insbesondere hat sie ein ausgeprägtes Extremum unterhalb der \emph{Bulk}-kritischen Temperatur. Zudem wird im \emph{Large}-$n$-Modell der Einfluss der Oberflächenfelder $h_j\in\mathbb{R}$ untersucht. Setzt man die Parameter $c_j$ und $h_j$ auf ihre Fixpunktwerte, so ergeben sich Kräfte, die bei $T_{\mathrm{c},\infty}$ mit einem Potenzgesetz abfallen und entweder attraktiv oder repulsiv sind. Ist das System samt seiner Randbedingungen spiegelsymmetrisch bezüglich einer Ebene, so wirken immer anziehende Kräfte. Brechen die Randbedingungen jedoch diese Spiegelsymmetrie, so können die Kräfte auch abstoßend sein. Es wird explizit gezeigt, dass Abweichungen von den Fixpunktwerten der Parameter zu Übergängen zwischen attraktivem und repulsivem Verhalten bei Variation der Filmdicke $L$ führen können.When macroscopic bodies are immersed in fluctuating media, long-range forces between these bodies may occur. The fluctuation's spectrum is modified resulting in a dependence of the system's energy on the separation between the objects, straightforwardly leading to the existence of a force between the bodies. This work is dedicated to the analysis of how boundary conditions affect the thermodynamic Casimir effect where thermal fluctuations near a critical point induce these forces. $O(n)$ symmetric $\phi^4$ theories in $d$-dimensional slab geometries of thickness $L$ are considered. When symmetry breaking external fields are present as well, the generic boundary conditions of these theories read $\partial_{n}\phi-c_j\,\phi=-h_j$ where the coefficients $c_j$ are surface couplings, serving as linearly extrapolated penetration depths into the surfaces in Landau theory, and $h_j$ are surface fields. The influence of the surface couplings $c_j$ on the Casimir force is investigated by means of the renormalization-group-improved perturbation theory in $d=4-\epsilon$ dimensions to two-loop order at the bulk critical point. Special attention is paid to the case of critical enhancement of the surface interactions which results in the existence of a zero mode leading to a breakdown of the usual loop expansion of the free energy and implicating the emergence of non-integer powers of $\epsilon$ in the $\epsilon$ expansion. These perturbative methods are restricted to the disordered phase with $T\geq T_{\mathrm{c},\infty}$, $c_j\geq c_\mathrm{sp}$, and $h_j=0$. In order to extend the analysis to the whole temperature axis, the exactly treatable limit $n\to\infty$ of the three-dimensional $\phi^4$ model is investigated. A set of self-consistent equations for the free energy is derived that can be solved numerically exact. Considering Dirichlet boundary conditions and vanishing external fields, one finds a temperature dependence of the Casimir force that exhibits the qualitative features of the experimentally measured force near the lambda transition in superfluid helium. In particular the existence of an extremum below the bulk critical point is reproduced. Additionally, the influence of the surface fields $h_j\in\mathbb{R}$ is investigated in the large-$n$ limit. If one fixes the coefficients $c_j$ and $h_j$ to their respective fixed-point values, the Casimir forces decay algebraically at $T_{\mathrm{c},\infty}$ being either attractive or repulsive. When the system --- including the boundary conditions --- is mirror symmetric with respect to a plane, these interactions are guaranteed to be attractive. This does not hold when boundary conditions break mirror symmetry. In this case repulsive forces may occur. It is explicitly shown that deviations of the coefficients from their fixed-points may lead to crossovers between attractive and repulsive Casimir forces when varying the film thickness $L$

Content Representation of Tactile Mental Imagery in Primary Somatosensory Cortex

The imagination of tactile stimulation has been shown to activate primary somatosensory cortex (S1) with a somatotopic specificity akin to that seen during the perception of tactile stimuli. Using fMRI and multivariate pattern analysis, we investigate whether this recruitment of sensory regions also reflects content-specific activation (i.e., whether the activation in S1 is specific to the mental content participants imagined). To this end, healthy volunteers (n = 21) either perceived or imagined three types of vibrotactile stimuli (mental content) while fMRI data were acquired. Independent of the content, during tactile mental imagery we found activation of frontoparietal regions, supplemented with activation in the contralateral BA2 subregion of S1, replicating previous reports. While the imagery of the three different stimuli did not reveal univariate activation differences, using multivariate pattern classification, we were able to decode the imagined stimulus type from BA2. Moreover, cross-classification revealed that tactile imagery elicits activation patterns similar to those evoked by the perception of the respective stimuli. These findings promote the idea that mental tactile imagery involves the recruitment of content-specific activation patterns in sensory cortices, namely in S1