Rotor (mathematics)

Abstract: A method and apparatus for assembling a rotatable machine is provided. The machine includes a plurality of blades that extend radially outwardly from a rotor. The method includes determining a moment weight of each blade in a row of blades, determining a geometric parameter of each blade in the same row of blades, and determining a mapping order of each blade using the moment weight and the geometric parameter. The apparatus includes a computer system that includes a software product code segment for minimizing imbalance in a bladed rotor wherein the segment is configured to receive a moment weight value for each blade to be installed in said rotor, receive a geometric parameter value for each blade to be installed in said rotor, calculate a blade location on the rotor based on the received values, and generate a blade map based on the calculated location.

Abstract: We analyze the quantum penny flip game using geometric algebra and so determine all possible unitary transformations which enable the player Q to implement a winning strategy. Geometric algebra provides a clear visual picture of the quantum game and its strategies, as well as providing a simple and direct derivation of the winning transformation, which we demonstrate can be parametrized by two angles �;� . For comparison we derive the same general winning strategy by conventional means using density matrices.

Abstract: This paper exploits Geometric (Clifford) Algebra (GA) theory in order to devise and introduce a new adaptive filtering strategy. From a least-squares cost function, the gradient is calculated following results from Geometric Calculus (GC), the extension of GA to handle differential and integral calculus. The novel GA least-mean-squares (GA-LMS) adaptive filter, which inherits properties from standard adaptive filters and from GA, is developed to recursively estimate a rotor (multivector), a hypercomplex quantity able to describe rotations in any dimension. The adaptive filter (AF) performance is assessed via a 3D point-clouds registration problem, which contains a rotation estimation step. Calculating the AF computational complexity suggests that it can contribute to reduce the cost of a full-blown 3D registration algorithm, especially when the number of points to be processed grows. Moreover, the employed GA/GC framework allows for easily applying the resulting filter to estimating rotors in higher dimensions.

Abstract: Several problems in atomic physics, such as diamagnetism in high Rydberg states, or the electron-electron interaction in high doubly excited states, and analogous problems at high excitation in nuclear physics or in other branches of physics, display certain characteristic and common features. Among these are the existence of classes of sharply localized states formed through large superpositions of basis states that are degenerate in the absence of the interaction. The asymmetric rotor at high angular momentum J displays these same features: In particular, most of its eigenstates divide into two groups with different localizations (along the directions of minimum and maximum moments of inertia). A few states lie in a ``separatrix'' region in between, corresponding to localization along the axis with an intermediate moment of inertia. A 1:1 mapping is made from the atomic and nuclear problems into the asymmetric rotor, wherein the principal quantum number n and orbital angular momentum l of the former are put in correspondence, respectively, with J and M, the azimuthal projection of the rotor's angular momentum. For each given problem, the ``asymmetry parameter'' of the corresponding rotor is identified, and eigenvalues and eigenfunctions are presented. The concept of a conjugate rotor is also introduced. The key, common feature that underlies localization is the vanishing as l\ensuremath{\rightarrow}n of the off-diagonal coupling between the states in a degenerate manifold {nl} just as in the asymmetric rotor where the matrix elements of the step-up and step-down operators in the \ensuremath{\Vert}JM〉 basis vanish when M\ensuremath{\rightarrow}\ifmmode\pm\else\textpm\fi{}J.