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We establish a twistor correspondence between a cuspidal cubic curve in a complex projective plane, and a co-calibrated homogeneous G sub(2) structure on the seven-dimensional parameter space of such cubics. Imposing the Riemannian reality conditions leads to an explicit co-calibrated G sub(2) structure on SU(2, 1)/U(1). This is an example of an SO(3) structure in seven dimensions. Cuspidal cubics and their higher degree analogues with constant projective curvature are characterised as integral curves of certain seventh order ODEs. Projective orbits of such curves are shown to be analytic continuations of Aloff-Wallach manifolds, and it is shown that only cubics lift to a complete family of contact rational curves in a projectivised cotangent bundle to a projective plane.

We investigate monopoly profit under a new online tying strategy, namely sequential bundling. This method allows customers to buy additional products at discounted prices immediately after purchasing one or some of the available products. This strategy has been practiced by Walmart and others but has not been modelled to date. We use microeconomics analysis to conduct a comparison of the gained profit with the three commonly used bundling strategies: no bundling, pure bundling and mixed bundling. The main result shows that the sequential bundling strategy yields higher profits in comparison to all three classic strategies. In particular, for the symmetric cost case, our model provides a useful tool for practitioners such as common online shops.

We analyze mixed bundling in two-sided markets where installed base effects are present and find that the pricing structure deviates from traditional bundling as well as the standard two-sided markets literature-we determine prices on both sides fall with bundling. Mixed bundling acts as a price discrimination tool segmenting the market more efficiently. Consequently, as a by-product of this price discrimination, the two sides are better coordinated, and social welfare is enhanced. We show unambiguously that platform participations increase on both sides of the market. After theoretically evaluating the impact mixed bundling has on prices and welfare, we take the model predictions to data from the portable video game console market. We find empirical support for all theoretical predictions. This paper was accepted by J. Miguel Villas-Boas, marketing.

The Chow–Mumford (CM) line bundle is a functorial line bundle on the base of any family of klt Fano varieties. It is conjectured that it yields a polarization on the moduli space of K -poly-stable klt Fano varieties. Proving ampleness of the CM line bundle boils down to showing semi-positivity/positivity statements about the CM-line bundle for families with K -semi-stable/ K -polystable fibers. We prove the necessary semi-positivity statements in the K -semi-stable situation, and the necessary positivity statements in the uniform K -stable situation, including in both cases variants assuming K -stability only for general fibers. Our statements work in the most general singular situation (klt singularities), and the proofs are algebraic, except the computation of the limit of a sequence of real numbers via the central limit theorem of probability theory. We also present an application to the classification of Fano varieties. Additionally, our semi-positivity statements work in general for log-Fano pairs.

We consider a seller who owns two capacity-constrained resources and markets two products (components) corresponding to these resources as well as a bundle comprising the two components. In an environment where all customers agree that one of the two components is of higher quality than the other and that the bundle is of the highest quality, we derive the seller's optimal bundling strategy. We demonstrate that the optimal solution depends on the absolute and relative availabilities of the two resources as well as upon the extent of subadditivity of the quality of the products. The possible strategies that can arise as equilibrium behavior include a pure components strategy, a partial- or full-spectrum mixed bundling strategy, and a pure bundling strategy, where the latter strategy is optimal when capacities are unconstrained. These conclusions are contrary to findings in the prior literature on bundling that demonstrated the unambiguous dominance of the full-spectrum mixed bundling strategy. Thus, our work expands the frontier of bundling to an environment with vertically differentiated components and limited resources. We also explore how the bundling strategies change as we introduce an element of horizontal differentiation wherein different types of customers value the available components differently.

The paper studies product bundling in a duopoly supply chain network under the influence of different power-balance structures, bundling decisions and advertising efforts on total supply chain profit. Mathematical models comprising two manufacturers and a single retailer are developed to capture the impact of bundling policy and advertisement strategy under three power-balance structures, namely Manufacturer Stackelberg, Retailer Stackelberg and Vertical Nash. Following game theory models and numerical examples, the study found that the total profit of the supply chain is undifferentiated under the manufacturer Stackelberg and Vertical Nash case in the manufacturer bundling and retailer bundling strategies. However, total supply chain profit under manufacturer bundling strongly dominates under retailer bundling in Retailer Stackelberg and Vertical Nash, and remains valid under multiple settings of market size, price elasticity and advertising elasticity. It is also found that manufacturer bundling is significantly affected by advertising effort compared to retailer bundling. The study contributes to the literature interfacing supply chain and marketing by studying bundling policy and advertising strategy simultaneously for homogenous products, under various power-balance structures and price competition.

We introduce and evaluate the effectiveness of temptation bundling-a method for simultaneously tackling two types of self-control problems by harnessing consumption complementarities. We describe a field experiment measuring the impact of bundling instantly gratifying but guilt-inducing \"want\" experiences (enjoying page-turner audiobooks) with valuable \"should\" behaviors providing delayed rewards (exercising). We explore whether such bundles increase should behaviors and whether people would pay to create these restrictive bundles. Participants were randomly assigned to a full treatment condition with gym-only access to tempting audio novels, an intermediate treatment involving encouragement to restrict audiobook enjoyment to the gym, or a control condition. Initially, full and intermediate treatment participants visited the gym 51% and 29% more frequently, respectively, than control participants, but treatment effects declined over time (particularly following Thanksgiving). After the study, 61% of participants opted to pay to have gym-only access to iPods containing tempting audiobooks, suggesting demand for this commitment device. This paper was accepted by Uri Gneezy, behavioral economics.

Tractography based on non-invasive diffusion imaging is central to the study of human brain connectivity. To date, the approach has not been systematically validated in ground truth studies. Based on a simulated human brain data set with ground truth tracts, we organized an open international tractography challenge, which resulted in 96 distinct submissions from 20 research groups. Here, we report the encouraging finding that most state-of-the-art algorithms produce tractograms containing 90% of the ground truth bundles (to at least some extent). However, the same tractograms contain many more invalid than valid bundles, and half of these invalid bundles occur systematically across research groups. Taken together, our results demonstrate and confirm fundamental ambiguities inherent in tract reconstruction based on orientation information alone, which need to be considered when interpreting tractography and connectivity results. Our approach provides a novel framework for estimating reliability of tractography and encourages innovation to address its current limitations.

We prove the Topological Mirror Symmetry Conjecture by Hausel–Thaddeus for smooth moduli spaces of Higgs bundles of type S L n and P G L n . More precisely, we establish an equality of stringy Hodge numbers for certain pairs of algebraic orbifolds generically fibred into dual abelian varieties. Our proof utilises p-adic integration relative to the fibres, and interprets canonical gerbes present on these moduli spaces as characters on the Hitchin fibres using Tate duality. Furthermore, we prove for d prime to n , that the number of rank n Higgs bundles of degree d over a fixed curve defined over a finite field, is independent of d . This proves a conjecture by Mozgovoy–Schiffmann in the coprime case.

Target identification and contact selection are known contributors to variability in efficacy across different clinical indications of deep brain stimulation surgery. A retrospective analysis of responders to subcallosal cingulate deep brain stimulation (SCC DBS) for depression demonstrated the common impact of the electrical stimulation on a stereotypic connectome of converging white matter bundles (forceps minor, uncinate fasciculus, cingulum and fronto-striatal fibers). To test the utility of a prospective connectomic approach for SCC DBS surgery, this pilot study used the four-bundle tractography 'connectome blueprint' to plan surgical targeting in 11 participants with treatment-resistant depression. Before surgery, targets were selected individually using deterministic tractography. Selection of contacts for chronic stimulation was made by matching the post-operative probabilistic tractography map to the pre-surgical deterministic tractography map for each subject. Intraoperative behavioral responses were used as a secondary verification of location. A probabilistic tract map of all participants demonstrated inclusion of the four bundles as intended, matching the connectome blueprint previously defined. Eight of 11 patients (72.7%) were responders and 5 were remitters after 6 months of open-label stimulation. At one year, 9 of 11 patients (81.8%) were responders, with 6 of them in remission. These results support the utility of a group probabilistic tractography map as a connectome blueprint for individualized, patient-specific, deterministic tractography targeting, confirming retrospective findings previously published. This new method represents a connectomic approach to guide future SCC DBS studies.