230 research outputs found
Band Connectivity for Topological Quantum Chemistry: Band Structures As A Graph Theory Problem
The conventional theory of solids is well suited to describing band
structures locally near isolated points in momentum space, but struggles to
capture the full, global picture necessary for understanding topological
phenomena. In part of a recent paper [B. Bradlyn et al., Nature 547, 298
(2017)], we have introduced the way to overcome this difficulty by formulating
the problem of sewing together many disconnected local "k-dot-p" band
structures across the Brillouin zone in terms of graph theory. In the current
manuscript we give the details of our full theoretical construction. We show
that crystal symmetries strongly constrain the allowed connectivities of energy
bands, and we employ graph-theoretic techniques such as graph connectivity to
enumerate all the solutions to these constraints. The tools of graph theory
allow us to identify disconnected groups of bands in these solutions, and so
identify topologically distinct insulating phases.Comment: 19 pages. Companion paper to arXiv:1703.02050 and arXiv:1706.08529
v2: Accepted version, minor typos corrected and references added. Now
19+epsilon page
Q-spectral and L-spectral radius of subgroup graphs of dihedral group
Research on Q-spectral and L-spectral radius of graph has been attracted many attentions. In other hand, several graphs associated with group have been introduced. Based on the absence of research on Q-spectral and L-spectral radius of subgroup graph of dihedral group, we do this research. We compute Q-spectral and L-spectral radius of subgroup graph of dihedral group and their complement, for several normal subgroups. Q-spectrum and Lspectrum of these graphs are also observed and we conclude that all graphs we discussed in this paper are Q-integral dan L-integral
Large-eddy simulation of the flow in a lid-driven cubical cavity
Large-eddy simulations of the turbulent flow in a lid-driven cubical cavity
have been carried out at a Reynolds number of 12000 using spectral element
methods. Two distinct subgrid-scales models, namely a dynamic Smagorinsky model
and a dynamic mixed model, have been both implemented and used to perform
long-lasting simulations required by the relevant time scales of the flow. All
filtering levels make use of explicit filters applied in the physical space (on
an element-by-element approach) and spectral (modal) spaces. The two
subgrid-scales models are validated and compared to available experimental and
numerical reference results, showing very good agreement. Specific features of
lid-driven cavity flow in the turbulent regime, such as inhomogeneity of
turbulence, turbulence production near the downstream corner eddy, small-scales
localization and helical properties are investigated and discussed in the
large-eddy simulation framework. Time histories of quantities such as the total
energy, total turbulent kinetic energy or helicity exhibit different evolutions
but only after a relatively long transient period. However, the average values
remain extremely close
Learning Harmonic Molecular Representations on Riemannian Manifold
Molecular representation learning plays a crucial role in AI-assisted drug
discovery research. Encoding 3D molecular structures through Euclidean neural
networks has become the prevailing method in the geometric deep learning
community. However, the equivariance constraints and message passing in
Euclidean space may limit the network expressive power. In this work, we
propose a Harmonic Molecular Representation learning (HMR) framework, which
represents a molecule using the Laplace-Beltrami eigenfunctions of its
molecular surface. HMR offers a multi-resolution representation of molecular
geometric and chemical features on 2D Riemannian manifold. We also introduce a
harmonic message passing method to realize efficient spectral message passing
over the surface manifold for better molecular encoding. Our proposed method
shows comparable predictive power to current models in small molecule property
prediction, and outperforms the state-of-the-art deep learning models for
ligand-binding protein pocket classification and the rigid protein docking
challenge, demonstrating its versatility in molecular representation learning.Comment: 25 pages including Appendi
Local-Global Results on Discrete Structures
Local-global arguments, or those which glean global insights from local information, are central ideas in many areas of mathematics and computer science. For instance, in computer science a greedy algorithm makes locally optimal choices that are guaranteed to be consistent with a globally optimal solution. On the mathematical end, global information on Riemannian manifolds is often implied by (local) curvature lower bounds. Discrete notions of graph curvature have recently emerged, allowing ideas pioneered in Riemannian geometry to be extended to the discrete setting. Bakry- Ămery curvature has been one such successful notion of curvature. In this thesis we use combinatorial implications of Bakry- Ămery curvature on graphs to prove a sort of local discrepancy inequality. This then allows us to derive a number of results regarding the local structure of graphs, dependent only on a curvature lower bound. For instance, it turns out that a curvature lower bound implies a nontrivial lower bound on graph connectivity. We also use these results to consider the curvature of strongly regular graphs, a well studied and important class of graphs. In this regard, we give a partial solution to an open conjecture: all SRGs satisfy the curvature condition CD(â, 2). Finally we transition to consider a facility location problem motivated by using Unmanned Aerial Vehicles (UAVs) to guard a border. Here, we find a greedy algorithm, acting on local geometric information, which finds a near optimal placement of base stations for the guarding of UAVs
Graph embedding and geometric deep learning relevance to network biology and structural chemistry
Graphs are used as a model of complex relationships among data in biological science since the advent of systems biology in the early 2000. In particular, graph data analysis and graph data mining play an important role in biology interaction networks, where recent techniques of artificial intelligence, usually employed in other type of networks (e.g., social, citations, and trademark networks) aim to implement various data mining tasks including classification, clustering, recommendation, anomaly detection, and link prediction. The commitment and efforts of artificial intelligence research in network biology are motivated by the fact that machine learning techniques are often prohibitively computational demanding, low parallelizable, and ultimately inapplicable, since biological network of realistic size is a large system, which is characterised by a high density of interactions and often with a non-linear dynamics and a non-Euclidean latent geometry. Currently, graph embedding emerges as the new learning paradigm that shifts the tasks of building complex models for classification, clustering, and link prediction to learning an informative representation of the graph data in a vector space so that many graph mining and learning tasks can be more easily performed by employing efficient non-iterative traditional models (e.g., a linear support vector machine for the classification task). The great potential of graph embedding is the main reason of the flourishing of studies in this area and, in particular, the artificial intelligence learning techniques. In this mini review, we give a comprehensive summary of the main graph embedding algorithms in light of the recent burgeoning interest in geometric deep learning
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