skip to main content


Title: Wedge reversion antisymmetry and 41 types of physical quantities in arbitrary dimensions
It is shown that there are 41 types of multivectors representing physical quantities in non-relativistic physics in arbitrary dimensions within the formalism of Clifford algebra. The classification is based on the action of three symmetry operations on a general multivector: spatial inversion, 1 , time-reversal, 1′, and a third that is introduced here, namely wedge reversion, 1 † . It is shown that the traits of `axiality' and `chirality' are not good bases for extending the classification of multivectors into arbitrary dimensions, and that introducing 1 † would allow for such a classification. Since physical properties are typically expressed as tensors, and tensors can be expressed as multivectors, this classification also indirectly classifies tensors. Examples of these multivector types from non-relativistic physics are presented.  more » « less
Award ID(s):
1807768
NSF-PAR ID:
10179570
Author(s) / Creator(s):
Date Published:
Journal Name:
Acta Crystallographica Section A Foundations and Advances
Volume:
76
Issue:
3
ISSN:
2053-2733
Page Range / eLocation ID:
318 to 327
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. Abstract

    In this article, we review the mathematical foundations of convolutional neural nets (CNNs) with the goals of: (i) highlighting connections with techniques from statistics, signal processing, linear algebra, differential equations, and optimization, (ii) demystifying underlying computations, and (iii) identifying new types of applications. CNNs are powerful machine learning models that highlight features from grid data to make predictions (regression and classification). The grid data object can be represented as vectors (in 1D), matrices (in 2D), or tensors (in 3D or higher dimensions) and can incorporate multiple channels (thus providing high flexibility in the input data representation). CNNs highlight features from the grid data by performing convolution operations with different types of operators. The operators highlight different types of features (e.g., patterns, gradients, geometrical features) and are learned by using optimization techniques. In other words, CNNs seek to identify optimal operators that best map the input data to the output data. A common misconception is that CNNs are only capable of processing image or video data but their application scope is much wider; specifically, datasets encountered in diverse applications can be expressed as grid data. Here, we show how to apply CNNs to new types of applications such as optimal control, flow cytometry, multivariate process monitoring, and molecular simulations.

     
    more » « less
  2. Billinge, Simon (Ed.)
    Periodic space crystals are well established and widely used in physical sciences. Time crystals have been increasingly explored more recently, where time is disconnected from space. Periodic relativistic spacetime crystals on the other hand need to account for the mixing of space and time in special relativity through Lorentz transformation, and have been listed only in 2-dimensions. This work shows that there exists a transformation between the conventional Minkowski spacetime (MS) and what is referred to here as renormalized blended spacetime (RBS); they are shown to be equivalent descriptions of relativistic physics in flat spacetime. There are two elements to this reformulation of MS, namely, blending and renormalization. When observers in two inertial frames adopt each other’s clocks as their own, while retaining their original space coordinates; the observers become blended. This process reformulates the Lorentz boosts into Euclidean rotations while retaining the original spacetime hyperbola describing worldlines of constant spacetime length from the origin. By renormalizing the blended coordinates with an appropriate factor that is a function of the relative velocities between the various frames, the hyperbola is transformed into a Euclidean circle. With these two steps, one obtains the RBS coordinates complete with new light lines, but now with a Euclidean construction. One can now enumerate the RBS point and space groups in various dimensions with their mapping to the well-known space crystal groups. The RBS point group for flat isotropic RBS spacetime is identified to be that of cylinders in various dimensions: mm2 which is that of a rectangle in 2D, (∞⁄m)m which is that of a cylinder in 3D, and that of hypercylinder in 4D. An antisymmetry operation is introduced that can swap between space-like and time-like directions, leading to color spacetime groups. The formalism reveals RBS symmetries that are not readily apparent in the conventional MS formulation. Mathematica® script is provided for plotting the MS and RBS geometries discussed in the work. 
    more » « less
  3. The increasing uncertainty of distributed energy resources promotes the risks of transient events for power systems. To capture event dynamics, Phasor Measurement Unit (PMU) data is widely utilized due to its high resolutions. Notably, Machine Learning (ML) methods can process PMU data with feature learning techniques to identify events. However, existing ML-based methods face the following challenges due to salient characteristics from both the measurement and the label sides: (1) PMU streams have a large size with redundancy and correlations across temporal, spatial, and measurement type dimensions. Nevertheless, existing work cannot effectively uncover the structural correlations to remove redundancy and learn useful features. (2) The number of event labels is limited, but most models focus on learning with labeled data, suffering risks of non-robustness to different system conditions. To overcome the above issues, we propose an approach called Kernelized Tensor Decomposition and Classification with Semi-supervision (KTDC-Se). Firstly, we show that the key is to tensorize data storage, information filtering via decomposition, and discriminative feature learning via classification. This leads to an efficient exploration of structural correlations via high-dimensional tensors. Secondly, the proposed KTDC-Se can incorporate rich unlabeled data to seek decomposed tensors invariant to varying operational conditions. Thirdly, we make KTDC-Se a joint model of decomposition and classification so that there are no biased selections of the two steps. Finally, to boost the model accuracy, we add kernels for non-linear feature learning. We demonstrate the KTDC-Se superiority over the state-of-the-art methods for event identification using PMU data. 
    more » « less
  4. Abstract

    The classification of point gap topology in all local non-Hermitian (NH) symmetry classes has been recently established. However, many entries in the resulting periodic table have only been discussed in a formal setting and still lack a physical interpretation in terms of their bulk-boundary correspondence. Here, we derive the edge signatures of all two-dimensional phases with intrinsic point gap topology. While in one dimension point gap topology invariably leads to the NH skin effect, NH boundary physics is significantly richer in two dimensions. We find two broad classes of non-Hermitian edge states: (1)infernal points, where a skin effect occurs only at a single edge momentum, while all other edge momenta are devoid of edge states. Under semi-infinite boundary conditions, the point gap thereby closes completely, but only at a single edge momentum. (2) NH exceptional pointdispersions, where edge states persist at all edge momenta and furnish an anomalous number of symmetry-protected exceptional points. Surprisingly, the latter class of systems allows for a finite, non-extensive number of edge states with a well defined dispersion along all generic edge terminations. Concomitantly, the point gap only closes along the real and imaginary eigenvalue axes, realizing a novel form of NH spectral flow.

     
    more » « less
  5. A bstract We consider the entanglement entropy of an arbitrary subregion in a system of N non-relativistic fermions in 2+1 dimensions in Lowest Landau Level (LLL) states. Using the connection of these states to those of an auxiliary 1 + 1 dimensional fermionic system, we derive an expression for the leading large- N contribution in terms of the expectation value of the phase space density operator in 1 + 1 dimensions. For appropriate subregions the latter can replaced by its semiclassical Thomas-Fermi value, yielding expressions in terms of explicit integrals which can be evaluated analytically. We show that the leading term in the entanglement entropy is a perimeter law with a shape independent coefficient. Furthermore, we obtain analytic expressions for additional contributions from sharp corners on the entangling curve. Both the perimeter and the corner pieces are in good agreement with existing calculations for special subregions. Our results are relevant to the integer quantum Hall effect problem, and to the half-BPS sector of $$ \mathcal{N} $$ N = 4 Yang Mills theory on S 3 . In this latter context, the entanglement we consider is an entanglement in target space. We comment on possible implications to gauge-gravity duality. 
    more » « less