Topology Optimization of Electric Machines based on Topological Sensitivity Analysis

Topological sensitivities are a very useful tool for determining optimal designs. The topological derivative of a domain-dependent functional represents the sensitivity with respect to the insertion of an infinitesimally small hole. In the gradient-based ON/OFF method, proposed by M. Ohtake, Y. Okamoto and N. Takahashi in 2005, sensitivities of the functional with respect to a local variation of the material coefficient are considered. We show that, in the case of a linear state equation, these two kinds of sensitivities coincide. For the sensitivities computed in the ON/OFF method, the generalization to the case of a nonlinear state equation is straightforward, whereas the computation of topological derivatives in the nonlinear case is ongoing work. We will show numerical results obtained by applying the ON/OFF method in the nonlinear case to the optimization of an electric motor.Comment: 20 pages, 7 figure

Robust measurement of wave function topology on NISQ quantum computers

Topological quantum phases of quantum materials are defined through their topological invariants. These topological invariants are quantities that characterize the global geometrical properties of the quantum wave functions and thus are immune to local noise. Here, we present a strategy to measure topological invariants on quantum computers. We show that our strategy can be easily integrated with the variational quantum eigensolver (VQE) so that the topological properties of generic quantum many-body states can be characterized on current quantum hardware. We demonstrate two explicit examples that show how the Chern number can be measured exactly; that is, it is immune to the noise of NISQ machines. This work shows that the robust nature of wave function topology allows NISQ machines to determine topological invariants accurately.Comment: 14 pages, 9 figures, 3 table

Bayesian Structural Inference for Hidden Processes

We introduce a Bayesian approach to discovering patterns in structurally complex processes. The proposed method of Bayesian Structural Inference (BSI) relies on a set of candidate unifilar HMM (uHMM) topologies for inference of process structure from a data series. We employ a recently developed exact enumeration of topological epsilon-machines. (A sequel then removes the topological restriction.) This subset of the uHMM topologies has the added benefit that inferred models are guaranteed to be epsilon-machines, irrespective of estimated transition probabilities. Properties of epsilon-machines and uHMMs allow for the derivation of analytic expressions for estimating transition probabilities, inferring start states, and comparing the posterior probability of candidate model topologies, despite process internal structure being only indirectly present in data. We demonstrate BSI's effectiveness in estimating a process's randomness, as reflected by the Shannon entropy rate, and its structure, as quantified by the statistical complexity. We also compare using the posterior distribution over candidate models and the single, maximum a posteriori model for point estimation and show that the former more accurately reflects uncertainty in estimated values. We apply BSI to in-class examples of finite- and infinite-order Markov processes, as well to an out-of-class, infinite-state hidden process.Comment: 20 pages, 11 figures, 1 table; supplementary materials, 15 pages, 11 figures, 6 tables; http://csc.ucdavis.edu/~cmg/compmech/pubs/bsihp.ht

Tunelling with a Negative Cosmological Constant

The point of this paper is see what light new results in hyperbolic geometry may throw on gravitational entropy and whether gravitational entropy is relevant for the quantum origin of the univeres. We introduce some new gravitational instantons which mediate the birth from nothing of closed universes containing wormholes and suggest that they may contribute to the density matrix of the universe. We also discuss the connection between their gravitational action and the topological and volumetric entropies introduced in hyperbolic geometry. These coincide for hyperbolic 4-manifolds, and increase with increasing topological complexity of the four manifold. We raise the questions of whether the action also increases with the topological complexity of the initial 3-geometry, measured either by its three volume or its Matveev complexity. We point out, in distinction to the non-supergravity case, that universes with domains of negative cosmological constant separated by supergravity domain walls cannot be born from nothing. Finally we point out that our wormholes provide examples of the type of Perpetual Motion machines envisaged by Frolov and Novikov.Comment: 36 pages, plain TE

Equivariant infinite loop space theory, I. The space level story

We rework and generalize equivariant infinite loop space theory, which shows how to construct G-spectra from G-spaces with suitable structure. There is a naive version which gives naive G-spectra for any topological group G, but our focus is on the construction of genuine G-spectra when G is finite. We give new information about the Segal and operadic equivariant infinite loop space machines, supplying many details that are missing from the literature, and we prove by direct comparison that the two machines give equivalent output when fed equivalent input. The proof of the corresponding nonequivariant uniqueness theorem, due to May and Thomason, works for naive G-spectra for general G but fails hopelessly for genuine G-spectra when G is finite. Even in the nonequivariant case, our comparison theorem is considerably more precise, giving a direct point-set level comparison. We have taken the opportunity to update this general area, equivariant and nonequivariant, giving many new proofs, filling in some gaps, and giving some corrections to results in the literature.Comment: 94 page