# Biblio

To decouple the multi-axis motion in the 6 degrees of freedom magnetically levitated actuators (MLAs), this paper introduces a numerical method to model the force and torque distribution. Taking advantage of the Gaussian quadrature, the concept of coil node is developed to simplify the Lorentz integral into the summation of the interaction between each magnetic node in the remanence region and each coil node in the coil region. Utilizing the coordinate transformation in the numerical method, the computation burden is independent of the position and the rotation angle of the moving part. Finally, the experimental results prove that the force and torque predicted by the numerical model are rigidly consistent with the measurement, and the force and torque in all directions are decoupled properly based on the numerical solution. Compared with the harmonic model, the numerical wrench model is more suitable for the MLAs undertaking both the translational and rotational displacements.

Motivated by the study of matrix elimination orderings in combinatorial scientific computing, we utilize graph sketching and local sampling to give a data structure that provides access to approximate fill degrees of a matrix undergoing elimination in polylogarithmic time per elimination and query. We then study the problem of using this data structure in the minimum degree algorithm, which is a widely-used heuristic for producing elimination orderings for sparse matrices by repeatedly eliminating the vertex with (approximate) minimum fill degree. This leads to a nearly-linear time algorithm for generating approximate greedy minimum degree orderings. Despite extensive studies of algorithms for elimination orderings in combinatorial scientific computing, our result is the first rigorous incorporation of randomized tools in this setting, as well as the first nearly-linear time algorithm for producing elimination orderings with provable approximation guarantees. While our sketching data structure readily works in the oblivious adversary model, by repeatedly querying and greedily updating itself, it enters the adaptive adversarial model where the underlying sketches become prone to failure due to dependency issues with their internal randomness. We show how to use an additional sampling procedure to circumvent this problem and to create an independent access sequence. Our technique for decorrelating interleaved queries and updates to this randomized data structure may be of independent interest.