In Reciprocal Freedom, Weinrib offers a neat and powerful explanation of the relationship between private law, corrective justice, public law, and distributive justice. In the Kantian view, private law and corrective justice are conceptually prior to public law and distributive justice. The primary function of public law is to publicly determine and enforce private rights. Institutions of distributive justice are required to legitimize a system of private rights that creates the possibility of subordination. In this comment, I argue that Reciprocal Freedom is a justification for what I will call ‘orthodox private law’ that, because it neglects the place of distributive justice within private law, fails to secure genuine independence for all persons.

In this study we focus on the collision rate and contact time of finite-sized droplets in homogeneous, isotropic turbulence. Additionally, we concentrate on sub-Hinze–Kolmogorov droplet sizes to prevent fragmentation events. After reviewing previous studies, we theoretically establish the equivalence of spherical and cylindrical formulations of the collision rate. We also obtained a closed-form expression for the collision rate of inertial droplets under the assumption of inviscid interactions. We then perform droplet-resolved simulations using the Basilisk solver with a multi-field volume-of-fluid method to prevent numerical droplet coalescence, ensuring a constant number of droplets of the same size within the domain, thereby allowing for the accumulation of collision statistics. The collision statistics are studied from numerical simulations, varying parameters such as droplet volume fraction, droplet size relative to the dissipative scale, density ratio and viscosity ratio. Our results show that the contact time is finite, leading to non-binary droplet interactions at high volume fractions. Additionally, the contact duration is well predicted by the eddy turnover time. We also find that the radial distribution at contact is significantly smaller than that predicted by the hard-sphere model due to droplet deformation in close proximity. Furthermore, we show that for neutrally buoyant droplets, the mean relative velocity is similar to the mean relative velocity of the continuous phase, except when the droplets are close. Finally, we demonstrate that the collision rate obeys the appropriate theoretical law, although a numerical prefactor weakly varies as a function of the dimensionless parameters, which differs from the constant prefactor from theory.

Bounding energetic growth of gyrokinetic instabilities is a complementary approach to linear instability analyses involving normal eigenmodes. Previous work has focused on upper bounds which are valid linearly and nonlinearly. However, if an upper bound on linear instability growth is desired, these nonlinearly valid bounds may be a poor predictor of the growth of the most unstable eigenmode. This is most evident for the simplest of instabilities: the ion-temperature-gradient (ITG) mode in a slab geometry. In this work, we derive energetic upper bounds specifically for linear instability growth, focusing on the slab ITG. We show that there is no fundamental limitation on how tightly linear growth can be bounded by an energetic norm, with the tightest possible bound being given by a special energy comprising projection coefficients of the linear eigenmode basis. Additionally, we consider ‘constrained optimal modes’ that maximise energy growth subject to constraints that are also obeyed by the linear eigenmodes. This yields computationally efficient upper bounds that closely resemble the linear growth rate, capturing effects connected to the real frequency of instabilities, which have been absent in the energetic bounds considered thus far.

This paper extends our previous study of the gyro-emission by energetic electrons in the magnetospheres of rapidly rotating, magnetic massive stars, through a quantitative analysis of the role of cooling by Coulomb collisions with thermal electrons from stellar wind material trapped within the centrifugal magnetosphere (CM). For the standard, simple CM model of a dipole field with aligned magnetic and rotational axes, we show that both gyro-cooling along magnetic loops and Coulomb cooling in the CM layer have nearly the same dependence on the magnitude and radial variation of magnetic field, implying then that their ratio is a global parameter that is largely independent of the field. Analytic analysis shows that, for electrons introduced near the CM layer around a magnetic loop apex, collisional cooling is more important for electrons with high pitch angle, while more field-aligned electrons cool by gyro-emission near their mirror point close to the loop base. Numerical models that assume a gyrotropic initial deposition with a gaussian distribution in both radius and loop co-latitude show the residual gyro-emission is generally strongest near the loop base, with highly relativistic electrons suffering much lower collisional losses than lower-energy electrons that are only mildly relativistic. Even for cases in which the energy deposition is narrowly concentrated near the loop apex, the computed residual emission shows a surprisingly broad distribution with magnetic field strength, suggesting that associated observed radio spectra should generally have a similarly broad frequency distribution. Finally, we briefly discuss the potential applicability of this formalism to magnetic ultracool dwarfs (UCDs), for which Very Long Baseline Interferometry (VLBI) observations indicate incoherent radio emission to be concentrated around the magnetic equator, in contrast to our predictions here for magnetic hot stars. We suggest that this difference could be attributed to UCDs having either a lower ambient density of thermal electrons, or more highly relativistic non-thermal electrons, both of which would reduce the relative importance of the collisional cooling explored here.

This paper examines the effects of average inflation targeting (AIT) on social welfare and fiscal multipliers under varying averaging windows using a nonlinear New Keynesian model. While the existing literature highlights AIT’s advantages over Inflation Targeting(IT) and longer-window AIT over shorter-window AIT in terms of social welfare, these conclusions often rely on linearized models that fail to capture expectation effects arising from window lengths. By solving the model nonlinearly, we find that social welfare increases with AIT windows up to six years but declines for longer windows. The key driver is the differing expectation effects, where longer windows reduce the likelihood of the zero lower bound (ZLB) binding but may overshoot inflation targets, leading to lower output and welfare. Our results reveal that the optimal averaging window for AIT depends critically on the ZLB probability: higher ZLB risks favor longer windows, while lower risks make shorter windows sufficient. Moreover, we investigate the fiscal multiplier under AIT and show that it differs significantly from IT. In addition, the welfare-maximizing AIT window does not align with the window that maximizes fiscal multipliers, highlighting trade-offs between welfare and fiscal policy effectiveness. This study underscores the importance of nonlinear methods in evaluating AIT and provides practical insights into its calibration for modern monetary policy frameworks.

Thérèse Humbert was a lowly French peasant until she saved the life of an American millionaire who left her a vast inheritance, but after twenty years of litigation over said inheritance, a massive web of deception unraveled. The millionaire had never existed, his “nephews” were actually Humbert’s brothers, and Humbert herself had swindled Europe’s moneyed men and working-class laborers out of millions of francs. Overnight, Humbert became a celebrity in the American press, even after she was convicted and imprisoned for fraud. The French swindler’s onslaught of coverage in American newspapers shaped Gilded Age anxieties about money, credit, and the place of women in an ever-changing world. Gender ideologies concerning women’s place in the economy and turn-of-the-century financial instability made the Humbert swindle irresistible to the American press, who saw the story as an opportunity to moralize about women and finance. The sheer scale of Humbert’s fraud and its American coverage make the story remarkable today as an astonishing episode in the Gilded Age and Progressive Era, especially among cultural historians and those interested in the New History of Capitalism.