Ranked-Choice Voting as Reprieve from the Court-Ordered Map

Thus far, legal debates about the rise of ranked-choice voting have centered on whether legislatures can lawfully adopt the practice. This Note turns attention to the courts and the question of remedies. It proposes that courts impose ranked-choice voting as a redistricting remedy. Ranked-choice voting allows courts to cure redistricting violations without also requiring that they draw copious numbers of districts, a process the Supreme Court has described as a “political thicket.” By keeping courts away from the fact-specific, often arbitrary judgments involved in redistricting, ranked-choice voting makes for the redistricting remedy that best protects the integrity of the judicial role

Obstructions to embeddability into hyperquadrics and explicit examples

We give series of explicit examples of Levi-nondegenerate real-analytic hypersurfaces in complex spaces that are not transversally holomorphically embeddable into hyperquadrics of any dimension. For this, we construct invariants attached to a given hypersurface that serve as obstructions to embeddability. We further study the embeddability problem for real-analytic submanifolds of higher codimension and answer a question by Forstneri\v{c}.Comment: Revised version, appendix and references adde

Pluricomplex Green and Lempert functions for equally weighted poles

For $\Omega$ a domain in $\mathbb C^n$, the pluricomplex Green function with poles $a_1, ...,a_N \in \Omega$ is defined as $G(z):=\sup \{u(z): u\in PSH_-(\Omega), u(x)\le \log \|x-a_j\|+C_j \text{when} x \to a_j, j=1,...,N \}$. When there is only one pole, or two poles in the unit ball, it turns out to be equal to the Lempert function defined from analytic disks into $\Omega$ by $L_S (z) :=\inf \{\sum^N_{j=1}\nu_j\log|\zeta_j|: \exists \phi\in \mathcal {O}(\mathbb D,\Omega), \phi(0)=z, \phi(\zeta_j)=a_j, j=1,...,N \}$. It is known that we always have $L_S (z) \ge G_S(z)$. In the more general case where we allow weighted poles, there is a counterexample to equality due to Carlehed and Wiegerinck, with $\Omega$ equal to the bidisk. Here we exhibit a counterexample using only four distinct equally weighted poles in the bidisk. In order to do so, we first define a more general notion of Lempert function "with multiplicities", analogous to the generalized Green functions of Lelong and Rashkovskii, then we show how in some examples this can be realized as a limit of regular Lempert functions when the poles tend to each other. Finally, from an example where $L_S (z) > G_S(z)$ in the case of multiple poles, we deduce that distinct (but close enough) equally weighted poles will provide an example of the same inequality. Open questions are pointed out about the limits of Green and Lempert functions when poles tend to each other.Comment: 25 page

Convergence and multiplicities for the Lempert function

Given a domain $\Omega \subset \mathbb C$, the Lempert function is a functional on the space Hol (\D,\Omega) of analytic disks with values in $\Omega$, depending on a set of poles in $\Omega$. We generalize its definition to the case where poles have multiplicities given by local indicators (in the sense of Rashkovskii's work) to obtain a function which still dominates the corresponding Green function, behaves relatively well under limits, and is monotonic with respect to the indicators. In particular, this is an improvement over the previous generalization used by the same authors to find an example of a set of poles in the bidisk so that the (usual) Green and Lempert functions differ.Comment: 24 pages; many typos corrected thanks to the referee of Arkiv for Matemati

Complex zeros of real ergodic eigenfunctions

We determine the limit distribution (as $\lambda \to \infty$) of complex zeros for holomorphic continuations \phi_{\lambda}^{\C} to Grauert tubes of real eigenfunctions of the Laplacian on a real analytic compact Riemannian manifold $(M, g)$ with ergodic geodesic flow. If $\{\phi_{j_k} \}$ is an ergodic sequence of eigenfunctions, we prove the weak limit formula \frac{1}{\lambda_j} [Z_{\phi_{j_k}^{\C}}] \to \frac{i}{\pi} \bar{\partial} {\partial} |\xi|_g, where [Z_{\phi_{j_k}^{\C}}] is the current of integration over the complex zeros and where $\bar{\partial}$ is with respect to the adapted complex structure of Lempert-Sz\"oke and Guillemin-Stenzel.Comment: Added some examples and references. Also added a new Corollary, and corrected some typo

Regularity of Kobayashi metric

We review some recent results on existence and regularity of Monge-Amp\`ere exhaustions on the smoothly bounded strongly pseudoconvex domains, which admit at least one such exhaustion of sufficiently high regularity. A main consequence of our results is the fact that the Kobayashi pseudo-metric k on an appropriare open subset of each of the above domains is actually a smooth Finsler metric. The class of domains to which our result apply is very large. It includes for instance all smoothly bounded strongly pseudoconvex complete circular domains and all their sufficiently small deformations.Comment: 14 pages, 8 figures - The previously announced main result had a gap. In this new version the corrected statement is given. To appear on the volume "Geometric Complex Analysis - Proceedings of KSCV 12 Symposium

Coherent states for compact Lie groups and their large-N limits

The first two parts of this article surveys results related to the heat-kernel coherent states for a compact Lie group K. I begin by reviewing the definition of the coherent states, their resolution of the identity, and the associated Segal-Bargmann transform. I then describe related results including connections to geometric quantization and (1+1)-dimensional Yang--Mills theory, the associated coherent states on spheres, and applications to quantum gravity. The third part of this article summarizes recent work of mine with Driver and Kemp on the large-N limit of the Segal--Bargmann transform for the unitary group U(N). A key result is the identification of the leading-order large-N behavior of the Laplacian on "trace polynomials."Comment: Submitted to the proceeding of the CIRM conference, "Coherent states and their applications: A contemporary panorama.

NO PLIF Imaging in the CUBRC 48 Inch Shock Tunnel

Nitric Oxide Planar Laser-Induced Fluorescence (NO PLIF) imaging is demonstrated at a 10 kHz repetition rate in the Calspan-University at Buffalo Research Center s (CUBRC) 48-inch Mach 9 hypervelocity shock tunnel using a pulse burst laser-based high frame rate imaging system. Sequences of up to ten images are obtained internal to a supersonic combustor model, located within the shock tunnel, during a single approx.10-millisecond duration run of the ground test facility. This represents over an order of magnitude improvement in data rate from previous PLIF-based diagnostic approaches. Comparison with a preliminary CFD simulation shows good overall qualitative agreement between the prediction of the mean NO density field and the observed PLIF image intensity, averaged over forty individual images obtained during several facility runs

Accelerated Sizing of a Power Split Electrified Powertrain

Component sizing generally represents a demanding and time-consuming task in the development process of electrified powertrains. A couple of processes are available in literature for sizing the hybrid electric vehicle (HEV) components. These processes employ either time-consuming global optimization techniques like dynamic programming (DP) or near-optimal techniques that require iterative and uncertain tuning of evaluation parameters like the Pontryagin's minimum principle (PMP). Recently, a novel near-optimal technique has been devised for rapidly predicting the optimal fuel economy benchmark of design options for electrified powertrains. This method, named slope-weighted energy-based rapid control analysis (SERCA), has been demonstrated producing results comparable to DP, while limiting the associated computational time by near two orders of magnitude. In this paper, sizing parameters for a power split electrified powertrain are considered that include the internal combustion engine size, the two electric motor/generator sizes, the transmission ratios, and the final drive ratio. The SERCA approach is adopted to rapidly evaluate the fuel economy capabilities of each sizing option in various driving missions considering both type-approval drive cycles and real-world driving profiles. While screening out for optimal sizing options, the implemented methodology includes drivability criteria along with fuel economy potential. Obtained results will demonstrate the agility of the developed sizing tool in identifying optimal sizing options compared to state-of-the-art sizing tools for electrified powertrains

High-Speed PLIF Imaging of Hypersonic Transition over Discrete Cylindrical Roughness

In two separate test entries, advanced laser-based instrumentation has been developed and applied to visualize the hypersonic flow over cylindrical protrusions on a flat plate. Upstream of these trips, trace quantities of nitric oxide (NO) were seeded into the boundary layer. The protuberances were sized to force laminar-to-turbulent boundary layer transition. In the first test, a 10-Hz nitric oxide planar laser-induced fluorescence (NO PLIF) flow visualization system was used to provide wide-field-of-view, high-resolution images of the flowfield. The images had sub-microsecond time resolution. However these images, obtained with a time separation of 0.1 sec, were uncorrelated with each other. Fluorescent oil-flow visualizations were also obtained during this test. In the second experiment, a laser and camera system capable of acquiring NO PLIF measurements at 1 million frames per second (1 MHz) was used. This system had lower spatial resolution, and a smaller field of view, but the images were time correlated so that the development of the flow structures could be observed in time