A non-relativistic limit of M-theory and 11-dimensional membrane Newton-Cartan geometry

We consider a non-relativistic limit of the bosonic sector of eleven-dimensional supergravity, leading to a theory based on a covariant ‘membrane Newton-Cartan’ (MNC) geometry. The local tangent space is split into three ‘longitudinal’ and eight ‘transverse’ directions, related only by Galilean rather than Lorentzian symmetries. This generalises the ten-dimensional stringy Newton-Cartan (SNC) theory. In order to obtain a finite limit, the field strength of the eleven-dimensional four-form is required to obey a transverse self-duality constraint, ultimately due to the presence of the Chern-Simons term in eleven dimensions. The finite action then gives a set of equations that is invariant under longitudinal and transverse rotations, Galilean boosts and local dilatations. We supplement these equations with an extra Poisson equation, coming from the subleading action. Reduction along a longitudinal direction gives the known SNC theory with the addition of RR gauge fields, while reducing along a transverse direction yields a new non-relativistic theory associated to D2 branes. We further show that the MNC theory can be embedded in the U-duality symmetric formulation of exceptional field theory, demonstrating that it shares the same exceptional Lie algebraic symmetries as the relativistic supergravity, and providing an alternative derivation of the extra Poisson equation.

Lower limb motor function assessment based on TensorFlow convolutional neural network and kernel entropy component analysis–local tangent space alignment

Motor function assessment of patients and the elderly is crucial to gait assessment and gait rehabilitation. Accuracy of the assessment is affected by clinician’s experience. To solve the problem, this article proposes motor function assessment index to assess the motor function of patients. VICON system collects video of subjects when they are walking. And the original gait videos are pre-processed by the pixel-based adaptive segmenter and extracted by the convolutional neural network. The kernel entropy component analysis and local tangent space alignment reduced the dimensions of extracted features, and motor function assessment index is obtained. The Pearson correlation analysis shows that the motor function assessment index and modified gait abnormality rating scale are significantly correlated, and Pearson correlation coefficient is 0.92. These effectiveness results demonstrate that the proposed method has the considerable potential to promote the future design of automatic motor function assessment for clinical rehabilitation research.