An official website of the United States government
This content will become publicly available on January 1, 2028
Codimension sequence, grade, and generating degree: an operadic approach
We study several classes of operadic ideals of the unital associative algebra operad uAss. As an application, we classify quotient operads of uAss of GK-dimension no more than 6. This corresponds to a classification of all T-ideals of codimension growth no more than 5 (or equivalently, varieties of grade no more than 5).
more »
« less
- PAR ID:
- 10683540
- Publisher / Repository:
- The Hebrew University Magnes Press
- Date Published:
- Journal Name:
- Israel Journal of Mathematics
- ISSN:
- 0021-2172
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
null (Ed.)This article extends the notion of a Frobenius power of an ideal in prime characteristic to allow arbitrary nonnegative real exponents. These generalized Frobenius powers are closely related to test ideals in prime characteristic, and multiplier ideals over fields of characteristic zero. For instance, like these well-known families of ideals, Frobenius powers also give rise to jumping exponents that we call critical Frobenius exponents. In fact, the Frobenius powers of a principal ideal coincide with its test ideals, but Frobenius powers appear to be a more refined measure of singularities than test ideals in general. Herein, we develop the theory of Frobenius powers in regular domains, and apply it to study singularities, especially those of generic hypersurfaces. These applications illustrate one way in which multiplier ideals behave more like Frobenius powers than like test ideals.more » « less
-
Urban Air Mobility, the scenario where hundreds of manned and Unmanned Aircraft Systems (UASs) carry out a wide variety of missions (e.g., moving humans and goods within the city), is gaining acceptance as a transportation solution of the future. One of the key requirements for this to happen is safely managing the air traffic in these urban airspaces. Due to the expected density of the airspace, this requires fast autonomous solutions that can be deployed online. We propose Learning-‘N-Flying (LNF), a multi-UAS Collision Avoidance (CA) framework. It is decentralized, works on the fly, and allows autonomous Unmanned Aircraft System (UAS)s managed by different operators to safely carry out complex missions, represented using Signal Temporal Logic, in a shared airspace. We initially formulate the problem of predictive collision avoidance for two UASs as a mixed-integer linear program, and show that it is intractable to solve online. Instead, we first develop Learning-to-Fly (L2F) by combining (1) learning-based decision-making and (2) decentralized convex optimization-based control. LNF extends L2F to cases where there are more than two UASs on a collision path. Through extensive simulations, we show that our method can run online (computation time in the order of milliseconds) and under certain assumptions has failure rates of less than 1% in the worst case, improving to near 0% in more relaxed operations. We show the applicability of our scheme to a wide variety of settings through multiple case studies.more » « less
-
We present the implementation and validation of Sketch, an algorithm that uses two Unpiloted Aerial Systems (UASs) to trace the boundary of volcanic plumes. Sketch guarantees asymptotically optimal flight distance and turning by maintaining a sandwich invariant where one UAS stays inside the plume boundary (defined by a CO2 concentration threshold) and the other UAS stays outside. The UASs adjust their flight paths based on real-time CO2measurements to maintain this invariant. This paper details the implementation of Sketchon a real-world UAS platform, the Dragonfly drone. We evaluate the efficacy of Sketch through extensive testing in physics-based simulations and real-world outdoor environments using virtual plumes. The algorithm is compared to a single-UAS baseline algorithm called Zigzag. Results show that Sketch meets the expectations set by theory, and it is more efficient than Zigzag, achieving shorter flight paths, less turning, and faster mapping times. While Zigzag exhibits slightly higher accuracy in estimating plume area and boundary, Sketch offers a more efficient real-time volcanic plume monitoring approach, especially in time-sensitive situations. These results demonstrate the feasibility and efficacy of Sketchfor real-time volcanic plume monitoring, paving the way for accurate CO2 emission estimation in hazardous and challenging environments.more » « less
-
null (Ed.)Several independent approaches exist for state estimation and control of multirotor unmanned aerial systems (UASs) that address specific and constrained operational conditions. This work presents a complete end-to-end pipeline that enables precise, aggressive and agile maneuvers for multirotor UASs under real and challenging outdoor environments. We leverage state-of-the-art optimal methods from the literature for trajectory planning and control, such that designing and executing dynamic paths is fast, robust and easy to customize for a particular application. The complete pipeline, built entirely using commercially available components, is made open-source and fully documented to facilitate adoption. We demonstrate its performance in a variety of operational settings, such as hovering at a spot under dynamic wind speeds of up to 5–6 m/s (12–15 mi/h) while staying within 12 cm of 3D error. We also characterize its capabilities in flying high-speed trajectories outdoors, and enabling fast aerial docking with a moving target with planning and interception occurring in under 8 s.more » « less
