Discrete phase-space structure of nn-qubit mutually unbiased bases

We work out the phase-space structure for a system of $n$ qubits. We replace the field of real numbers that label the axes of the continuous phase space by the finite field \Gal{2^n} and investigate the geometrical structures compatible with the notion of unbiasedness. These consist of bundles of discrete curves intersecting only at the origin and satisfying certain additional properties. We provide a simple classification of such curves and study in detail the four- and eight-dimensional cases, analyzing also the effect of local transformations. In this way, we provide a comprehensive phase-space approach to the construction of mutually unbiased bases for $n$ qubits.Comment: Title changed. Improved version. Accepted for publication in Annals of Physic

Complementarity Among Vertical Integration Decisions: Evidence from Automobile Product Development

This paper examines complementarity among vertical integration decisions in automobile product development. Though most research assumes that contracting choices are independent of each other, contracting complementarity arises when the returns to a single vertical integration decision are increasing in the level of vertical integration associated with other contracting choices. First, effective coordination may depend on the level of (non-contractible) effort on the part of each agent; contracting complementarity results if coordination efforts are interdependent and vertical integration facilitates a higher level of non-contractible effort. Second, effective coordination may require the disclosure of proprietary trade secrets, and the potential for expropriation by external suppliers may induce complementarity among vertical integration choices. We provide evidence for complementarity in product development contracting by taking advantage of a detailed dataset that includes the level of vertical integration and the contracting environment for individual automobile systems in the luxury automobile segment. Using an instrumental variables framework that distinguishes complementarity from unobserved firm-level factors, the evidence is consistent with the hypothesis that contracting complementarity is an important driver of vertical integration choices. The findings suggest that contracting complementarity may be particularly important when coordination is important to achieve but difficult to monitor.

Blast analysis of enclosure masonry walls using homogenization approaches

A simple rigid-plastic homogenization model for the analysis of enclosure masonry walls sub- jected to blast loads is presented. The model is characterized by a few material parameters, is numerically inexpensive and very stable, and allows full parametric studies of entire walls subject to blast pressures. With the aim of considering the actual brickwork strength along vertical and horizontal axes, masonry out-of-plane anisotropic failure surfaces are obtained by means of a compatible homogenized limit analysis approach. In the model, a 3D system of rigid infinitely strong bricks connected by joints reduced to interfaces is identified with a 2D Kirchhoff-Love plate. For the joints, which obey an associated flow rule, aMohr-Coulomb fail- ure criterion with a tension cutoff and a linearized elliptic compressive cap is considered. In this way, the macroscopic masonry failure surface is obtained as a function of the macroscopic bending, torque, and in-plane forces by means of a linear programming problem in which the internal power dissipated is minimized. Triangular Kirchhoff-Love elements with linear in- terpolation of the displacements field and constant moment within each element are used at a structural level. In this framework, a simple quadratic programming problem is obtained to analyze entire walls subjected to blast loads. The multiscale strategy presented is adopted to predict the behavior of a rectangular wall supported on three sides (left, bottom, and right) representing an envelope wall in a building and subjected to a standardized blast load. The top edge of the wall is assumed unconstrained due to an imperfect connection (often an inter- layer material is used to prevent damage in the in-fill wall). A comparison with a standard elastic-plastic heterogeneous 3D analysis conducted with a commercial FE code is also pro- vided for a preliminary verification of the procedure at a structural level. The good agreement found and the very limited computational effort required for the simulations conducted with the presented model indicate that the proposed simple tool can be used by practitioners for the safety assessment of out-of-plane loaded masonry panels subjected to blast loading. An ex- haustive parametric analysis is finally conducted with different wall thicknesses, joint tensile strengths, and dynamic pressures, corresponding to blast loads (in kilograms of TNT) ranging from small to large

Simple modeling of reinforced masonry arches for associated and non-associated heterogeneous limit analysis

This paper proposes a limit analysis heterogeneous model for masonry arches in presence of innovative strengthening, rigid blocks, and joints reduced to interfaces. The reinforcement is considered in a simplified but effective way in the context of limit analysis, suitably modifying the admissibility conditions of the constitutive law that governs the behavior of contact joints. First, the force resultants at the interface after the reinforcement is investigated. Based on that, the yield condition and flow rule in the standard heterogeneous limit analysis formulation are updated. This approach is applied to solve both associated and non-associated sliding cases. For benchmarking purposes, the collapse of a 2D arch with both-side Fiber Reinforced Polymer (FRP) reinforcement is analyzed, followed by several parametric studies and a cost-benefit study. Collapse analysis of a real arch tested in-situ is also presented for further verification. The results show that in some cases an incorrect collapse mechanism and an overestimated ultimate load would be obtained through associated limit analysis. Such overestimation may become very significant once the reinforcement is taken into account. The presented cases illustrate that the maximum overestimation of the load could reach in meaningful cases of technical interest 91% of the associated prediction. This suggests the use of a non-associated flow rule to accurately predict the collapse load increase of reinforced arches. According to the cost-benefit study, it is recommended to strengthen at least half of the joints to guarantee an acceptable effect of the strengthening. The simulation of the collapse of the in-situ tested arch further proves the reliability of the proposed approach

How Licensing Resolves Hold-Up: Evidence from a Dynamic Panel Data Model with Unobserved Heterogeneity

In a patent thicket licensing provides a mechanism to either avoid or resolve hold-up. Firms' R&D incentives will differ depending on how licensing is used. In this paper we study the choice between ex ante licensing to avoid hold-up and ex post licensing to resolve it. Building on a theoretical model of a patent portfolio race, firms' choices of licensing contracts are modelled. We derive several hypotheses from the model and find support for these using data from the semiconductor industry. The empirical results show that firms' relationships in product markets and technology space jointly determine the type of licensing contract chosen. Implications for the regulation of licensing are discussed. We estimate a dynamic panel data model with unobserved heterogeneity and a lagged dependent variable. A method suggested by Wooldridge (2005) is employed to estimate a random effects probit model using conditional maximum likelihood

A Class of Mathematical Programs with Equilibrium Constraints: A Smooth Algorithm and Applications to Contact Problems

We discuss a special mathematical programming problem with equilibrium constraints (MPEC), that arises in material and shape optimization problems involving the contact of a rod or a plate with a rigid obstacle. This MPEC can be reduced to a nonlinear programming problem with independent variables and some dependent variables implicity defined by the solution of a mixed linear complementarity problem (MLCP). A projected-gradient algorithm including a complementarity method is proposed to solve this optimization problem. Several numerical examples are reported to illustrate the efficiency of this methodology in practice