Cycles in the burnt pancake graphs

The pancake graph $P_n$ is the Cayley graph of the symmetric group $S_n$ on $n$ elements generated by prefix reversals. $P_n$ has been shown to have properties that makes it a useful network scheme for parallel processors. For example, it is $(n-1)$-regular, vertex-transitive, and one can embed cycles in it of length $\ell$ with $6\leq\ell\leq n!$. The burnt pancake graph $BP_n$, which is the Cayley graph of the group of signed permutations $B_n$ using prefix reversals as generators, has similar properties. Indeed, $BP_n$ is $n$-regular and vertex-transitive. In this paper, we show that $BP_n$ has every cycle of length $\ell$ with $8\leq\ell\leq 2^n n!$. The proof given is a constructive one that utilizes the recursive structure of $BP_n$. We also present a complete characterization of all the $8$-cycles in $BP_n$ for $n \geq 2$, which are the smallest cycles embeddable in $BP_n$, by presenting their canonical forms as products of the prefix reversal generators.Comment: Added a reference, clarified some definitions, fixed some typos. 42 pages, 9 figures, 20 pages of appendice

On the number of pancake stacks requiring four flips to be sorted

Using existing classification results for the 7- and 8-cycles in the pancake graph, we determine the number of permutations that require 4 pancake flips (prefix reversals) to be sorted. A similar characterization of the 8-cycles in the burnt pancake graph, due to the authors, is used to derive a formula for the number of signed permutations requiring 4 (burnt) pancake flips to be sorted. We furthermore provide an analogous characterization of the 9-cycles in the burnt pancake graph. Finally we present numerical evidence that polynomial formulae exist giving the number of signed permutations that require $k$ flips to be sorted, with $5\leq k\leq9$.Comment: We have finalized for the paper for publication in DMTCS, updated a reference to its published version, moved the abstract to its proper location, and added a thank you to the referees. The paper has 27 pages, 6 figures, and 2 table

System Dynamic Analysis of a Wind Tunnel Model with Applications to Improve Aerodynamic Data Quality

The research investigates the effect of wind tunnel model system dynamics on measured aerodynamic data. During wind tunnel tests designed to obtain lift and drag data, the required aerodynamic measurements are the steady-state balance forces and moments, pressures, and model attitude. However, the wind tunnel model system can be subjected to unsteady aerodynamic and inertial loads which result in oscillatory translations and angular rotations. The steady-state force balance and inertial model attitude measurements are obtained by filtering and averaging data taken during conditions of high model vibrations. The main goals of this research are to characterize the effects of model system dynamics on the measured steady-state aerodynamic data and develop a correction technique to compensate for dynamically induced errors. Equations of motion are formulated for the dynamic response of the model system subjected to arbitrary aerodynamic and inertial inputs. The resulting modal model is examined to study the effects of the model system dynamic response on the aerodynamic data. In particular, the equations of motion are used to describe the effect of dynamics on the inertial model attitude, or angle of attack, measurement system that is used routinely at the NASA Langley Research Center and other wind tunnel facilities throughout the world. This activity was prompted by the inertial model attitude sensor response observed during high levels of model vibration while testing in the National Transonic Facility at the NASA Langley Research Center. The inertial attitude sensor cannot distinguish between the gravitational acceleration and centrifugal accelerations associated with wind tunnel model system vibration, which results in a model attitude measurement bias error. Bias errors over an order of magnitude greater than the required device accuracy were found in the inertial model attitude measurements during dynamic testing of two model systems. Based on a theoretical modal approach, a method using measured vibration amplitudes and measured or calculated modal characteristics of the model system is developed to correct for dynamic bias errors in the model attitude measurements. The correction method is verified through dynamic response tests on two model systems and actual wind tunnel test data

Fashion Meets Architecture: Inspiration InterchangeA Collaborative Project for Apparel Design and Interior Design

Research has shown early exposure to interdisciplinary activities may contribute to a professional’s appreciation of perspectives from diverse fields. This exposure is the founding principle behind a collaborative project developed for sophomore-level apparel design (AD) and interior (ID) design students. This presentation highlights the objectives, method, and outcomes of the project, and addresses the future of this collaboration. This presentation will be a foundation for discussion of the possibilities, challenges, and progression of interdisciplinary projects

Equilibrium drop surface profiles in electric fields

Electrowetting is becoming a more and more frequently used tool to manipulate liquids in various microfluidic applications. On the scale of the entire drop, the effect of electrowetting is to reduce the apparent contact angle of partially wetting conductive liquids upon application of an external voltage. Microscopically, however, strong electric fields in the vicinity of the three phase contact line give rise to local deformations of the drop surface. We determined the equilibrium surface profile using a combined numerical, analytical, and experimental approach. We find that the local contact angle in electrowetting is equal to Young's angle independent of the applied voltage. Only on the scale of the thickness of the insulator and beyond does the surface slope assume a value consistent with the voltage-dependent apparent contact angle. This behaviour is verified experimentally by determining equilibrium surface profiles for insulators of various thicknesses between 10 and 250 µm. Numerically and analytically, we find that the local surface curvature diverges algebraically upon approaching the contact line with an exponent −1<μ<0. We discuss the relevance of the local surface properties for dynamic aspects of the contact line motion

CFD Simulation on the J-2X Engine Exhaust in the Center-Body Diffuser and Spray Chamber at the B-2 Facility

A computational fluid dynamic (CFD) code is used to simulate the J-2X engine exhaust in the center-body diffuser and spray chamber at the Spacecraft Propulsion Facility (B-2). The CFD code is named as the space-time conservation element and solution element (CESE) Euler solver and is very robust at shock capturing. The CESE results are compared with independent analysis results obtained by using the National Combustion Code (NCC) and show excellent agreement

Vibroacoustic Response Data of Stiffened Panels and Cylinders

NASA has collected vibroacoustic response data on a variety of complex, aerospace structures to support research into numerical modeling of such structures. This data is being made available to the modeling community to promote the development and validation of analysis methods for these types of structures. Existing data from two structures is described, as well as plans for a data set from a third structure. The first structure is a 1.22 m by 1.22 m stiffened aluminum panel, typical of a commercial aircraft sidewall section. The second is an enclosed, stiffened aluminum cylinder, approximately 3.66 m long and 1.22 m in diameter, constructed to resemble a small aircraft fuselage with no windows and a periodic structure. The third structure is a filament-wound composite cylinder with composite stiffeners. Numerous combinations of excitation and response variables were measured on the structures, including: shaker excitation; diffuse acoustic field; velocity response from a laser vibrometer; intensity scans; and point acceleration

A preorder-free construction of the Kazhdan-Lusztig representations of SnS_n, with connections to the Clausen representations

We use the polynomial ring $\mathbb{C}[x_{1,1},\ldots,x_{n,n}]$ to modify the Kazhdan-Lusztig construction of irreducible $S_n$-modules. This modified construction produces exactly the same matrices as the original construction in [$\textit{Invent. Math}$ $\mathbf{53}$ (1979)], but does not employ the Kazhdan-Lusztig preorders. We also show that our modules are related by unitriangular transition matrices to those constructed by Clausen in [$\textit{J. Symbolic Comput.}$ $\textbf{11}$ (1991)]. This provides a $\mathbb{C}[x_{1,1},\ldots,x_{n,n}]$-analog of results of Garsia-McLarnan in [$\textit{Adv. Math.}$ $\textbf{69}$ (1988)]

SHAIL: Safety-Aware Hierarchical Adversarial Imitation Learning for Autonomous Driving in Urban Environments

Designing a safe and human-like decision-making system for an autonomous vehicle is a challenging task. Generative imitation learning is one possible approach for automating policy-building by leveraging both real-world and simulated decisions. Previous work that applies generative imitation learning to autonomous driving policies focuses on learning a low-level controller for simple settings. However, to scale to complex settings, many autonomous driving systems combine fixed, safe, optimization-based low-level controllers with high-level decision-making logic that selects the appropriate task and associated controller. In this paper, we attempt to bridge this gap in complexity by employing Safety-Aware Hierarchical Adversarial Imitation Learning (SHAIL), a method for learning a high-level policy that selects from a set of low-level controller instances in a way that imitates low-level driving data on-policy. We introduce an urban roundabout simulator that controls non-ego vehicles using real data from the Interaction dataset. We then show empirically that our approach can produce better behavior than previous approaches in driver imitation which have difficulty scaling to complex environments. Our implementation is available at https://github.com/sisl/InteractionImitation