The eta invariant on two-step nilmanifolds

The eta invariant appears regularly in index theorems but is known to be directly computable from the spectrum only in certain examples of locally symmetric spaces of compact type. In this work, we derive some general formulas useful for calculating the eta invariant on closed manifolds. Specifically, we study the eta invariant on nilmanifolds by decomposing the spin Dirac operator using Kirillov theory. In particular, for general Heisenberg three-manifolds, the spectrum of the Dirac operator and the eta invariant are computed in terms of the metric, lattice, and spin structure data. There are continuous families of geometrically, spectrally different Heisenberg three-manifolds whose Dirac operators have constant eta invariant. In the appendix, some needed results of L. Richardson and C. C. Moore are extended from spaces of functions to spaces of spinors.Comment: 53 pages, corrected final version, to appear in Communications in Analysis and Geometr

Automated mapping of virtual environments with visual predictive coding

Humans construct internal cognitive maps of their environment directly from sensory inputs without access to a system of explicit coordinates or distance measurements. While machine learning algorithms like SLAM utilize specialized visual inference procedures to identify visual features and construct spatial maps from visual and odometry data, the general nature of cognitive maps in the brain suggests a unified mapping algorithmic strategy that can generalize to auditory, tactile, and linguistic inputs. Here, we demonstrate that predictive coding provides a natural and versatile neural network algorithm for constructing spatial maps using sensory data. We introduce a framework in which an agent navigates a virtual environment while engaging in visual predictive coding using a self-attention-equipped convolutional neural network. While learning a next image prediction task, the agent automatically constructs an internal representation of the environment that quantitatively reflects distances. The internal map enables the agent to pinpoint its location relative to landmarks using only visual information.The predictive coding network generates a vectorized encoding of the environment that supports vector navigation where individual latent space units delineate localized, overlapping neighborhoods in the environment. Broadly, our work introduces predictive coding as a unified algorithmic framework for constructing cognitive maps that can naturally extend to the mapping of auditory, sensorimotor, and linguistic inputs

Isospectral deformations of closed Riemannian manifolds with different scalar curvature

We construct the first examples of continuous families of isospectral Riemannian metrics that are not locally isometric on closed manifolds, more precisely, on $S^n\times T^m$, where $T^m$ is a torus of dimension $m\ge 2$ and $S^n$ is a sphere of dimension $n\ge 4$. These metrics are not locally homogeneous; in particular, the scalar curvature of each metric is nonconstant. For some of the deformations, the maximum scalar curvature changes during the deformation.Comment: amstex, 10 pages, no figure

Unified model for Mullins effect and high cycle fatigue life prediction of rubber materials

Proceedings of the 8th European Conference on constitutive models for rubbers (ECCMR VIII), San Sebastian, Spain, 25-28 June 2013International audienceThe study describes the basic principles of a general damage model (GDMF) for Mullins effect and high cycle fatigue loadings of rubber materials and demonstrates its prediction possibilities for simulating the complete fatigue failure phenomenon. The present paper focuses on stiffness modelling of rubber materials for uniaxial and multiaxial static and fatigue loadings with a minimal number of material parameters in order to ensure robustness of the identification. The proposed hyperplastic model is expressed in terms of classical independent strain invariants. Mullins effect and high cycle fatigue loadings are both modelled according to a continuum damage mechanics approach

Establishing a Gold Standard for Noninvasive Identification of Painful Lumbar Discs: Prospective Comparison of Magnetic Resonance Spectroscopy vs Low-Pressure Provocation Discography.

PURPOSE: Verifying lumbar disc pain can present a clinical challenge. Low-pressure provocative discography (PD) has served as the gold standard, although it is invasive and often a challenge to interpret. We reported that magnetic resonance spectroscopy (MRS) biomarkers accurately predict PD results in lumbar discs and improved outcomes for patients with surgery at positive MRS levels versus nonsurgery. To further substantiate MRS for diagnosing painful discs, we report a prospective comparison of 2 MRS-derived measures: NOCISCORE (pain) and SI-SCORE (degeneration severity). METHODS: Lumbar MRS and software-based postprocessing (NOCISCAN-LS, Aclarion Inc.) was performed in 44 discs in 14 patients (prospective cohort [PC]). PC data were compared to prior data used to establish the NOCISCORE (training cohort [TC]). The NOCISCORE was converted to an ordinal value (high/intermediate/low; NOCI+/mild/-) and compared against painful (P) versus nonpainful (NP) control diagnosis (PD) for 19 discs where PD was performed in the PC (12 NP; 7 P). Sensitivity, specificity, and positive and negative predictive values were calculated. The SI-SCORE was compared against MRI Pfirrmann Grades for 465 discs in 126 patients (PC plus TC). RESULTS: For the PC, MRS (NOCI+/-) compared to PD (P/NP) with an accuracy of 87% and sensitivity and specificity of 100%. The positive and negative predictive values of MRS were 100%. NOCISCOREs were significantly higher for PD+ versus PD- discs for PC and TC (P < 0.05), and the NOCISCORE distributions for PD+/- group were not statistically different between the PC and TC (P > 0.05). SI-SCORES differed between Pfirrmann Grades 1 and 2 (less degenerated) versus Grades 3 and 4 (more degenerated; P < 0.05), with a progressively decreasing trend with Pfirrmann Grades 1-5. CONCLUSION: These current data provide prospective confirmation of the predictive value of disc MRS for distinguishing painful discs and for assessing the disc structural integrity. CLINICAL RELEVANCE: NOCISCAN is an adoptable, noninvasive, and objectively quantitative test to improve management of low back pain patients

A theory of network alteration for the Mullins effect

International audienceThis paper reports on the development of a new network alteration theory to describe the Mullins effect. The stress-softening phenomenon that occurs in rubber-like materials during cyclic loading is analysed from a physical point of view. The Mullins effect is considered to be a consequence of the breakage of links inside the material. Both filler-matrix and chain interaction links are involved in the phenomenon. This new alteration theory is implemented by modifying the eight-chains constitutive equation of Arruda and Boyce (J. Mech. Phys. Solids 41 (2) (1993) 389). In the present method the parameters of the eight-chains model, denoted C-R and N in the bibliography, become functions of the maximum chain stretch ratio. The accuracy of the resulting constitutive equation is demonstrated on cyclic uniaxial experiments for both natural rubbers and synthetic elastomers