An Anomalous Case of Southern Sympathy: New Jersey\u27s Civil War Stance

A popular narrative of the Civil War assumes that all Northern states stood united behind President Abraham Lincoln in their loyalty to the Union. However, the case of New Jersey suggests that this narrative of devotion is simply a myth. The agrarian economy of New Jersey kept the state firmly opposed to universal emancipation, and New Jersey behaved more like a border state than its geographic neighbors of Pennsylvania and New York. By examining New Jersey\u27s response to the release of the Emancipation Proclamation and the Election of 1864, the myth of Northern unity is broken by understanding persistent state-level economic factors

Theoretical fan velocity distortions due to inlets and nozzles

Nonuniform velocity profiles imposed on the propulsion system fan can cause fan blade stresses and thrust losses. A theoretical parametric study of the effects of inlets with 0 deg and 90 deg nozzle deflection on the velocity profile at a hypothetical fan is presented. The parameters investigated are fan-to-nozzle spacing and inlet centerline offset. The interaction between the inlet and nozzle is also investigated. The study is made using a two-dimensional analysis

A Kakeya maximal function estimate in four dimensions using planebrushes

We obtain an improved Kakeya maximal function estimate in $\mathbb{R}^4$ using a new geometric argument called the planebrush. A planebrush is a higher dimensional analogue of Wolff's hairbrush, which gives effective control on the size of Besicovitch sets when the lines through a typical point concentrate into a plane. When Besicovitch sets do not have this property, the existing trilinear estimates of Guth-Zahl can be used to bound the size of a Besicovitch set. In particular, we establish a maximal function estimate in $\mathbb{R}^4$ at dimension $3.059$. As a consequence, every Besicovitch set in $\mathbb{R}^4$ must have Hausdorff dimension at least $3.059$.Comment: 40 pages 2 figures. v2: revised based on referee's comments. In v1, the Nikishin-Pisier-Stein factorization theorem was stated (and used) incorrectly. This version corrects the problem by introducing several new arguments. The new argument leads to a Kakeya maximal function estimate at dimension 3.059, which is slightly worse than the previously claimed exponent 3.085