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Nuclear quantum effects (NQEs) are known to impact a number of features associated with chemical reactivity and physicochemical properties, particularly for light atoms and at low temperatures. In the imaginary time path integral formalism, each atom is mapped onto a “ring polymer” whose spread is related to the quantum mechanical uncertainty in the particle’s position, i.e., its thermal wavelength. A number of metrics have previously been used to investigate and characterize this spread and explain effects arising from quantum delocalization, zero-point energy, and tunneling. Many of these shape metrics consider just the instantaneous structure of the ring polymers. However, given the significant interest in methods such as centroid molecular dynamics and ring polymer molecular dynamics that link the molecular dynamics of these ring polymers to real time properties, there exists significant opportunity to exploit metrics that also allow for the study of the fluctuations of the atom delocalization in time. Here we consider the ring polymer delocalization from the perspective of computational topology, specifically persistent homology, which describes the 3-dimensional arrangement of point cloud data, (i.e. atomic positions). We employ the Betti sequence probability distribution to define the ensemble of shapes adopted by the ring polymer. The Wasserstein distances of Betti sequences adjacent in time are used to characterize fluctuations in shape, where the Fourier transform and associated principal components provides added information differentiating atoms with different NQEs based on their dynamic properties. We demonstrate this methodology on two representative systems, a glassy system consisting of two atom types with dramatically different de Broglie thermal wavelengths, and ab initio molecular dynamics simulation of an aqueous 4 M HCl solution where the H-atoms are differentiated based on their participation in proton transfer reactions. 

We propose an Euler transformation that transforms a given [Formula: see text]-dimensional cell complex [Formula: see text] for [Formula: see text] into a new [Formula: see text]-complex [Formula: see text] in which every vertex is part of the same even number of edges. Hence every vertex in the graph [Formula: see text] that is the [Formula: see text]-skeleton of [Formula: see text] has an even degree, which makes [Formula: see text] Eulerian, i.e., it is guaranteed to contain an Eulerian tour. Meshes whose edges admit Eulerian tours are crucial in coverage problems arising in several applications including 3D printing and robotics. For [Formula: see text]-complexes in [Formula: see text] ([Formula: see text]) under mild assumptions (that no two adjacent edges of a [Formula: see text]-cell in [Formula: see text] are boundary edges), we show that the Euler transformed [Formula: see text]-complex [Formula: see text] has a geometric realization in [Formula: see text], and that each vertex in its [Formula: see text]-skeleton has degree [Formula: see text]. We bound the numbers of vertices, edges, and [Formula: see text]-cells in [Formula: see text] as small scalar multiples of the corresponding numbers in [Formula: see text]. We prove corresponding results for [Formula: see text]-complexes in [Formula: see text] under an additional assumption that the degree of a vertex in each [Formula: see text]-cell containing it is [Formula: see text]. In this setting, every vertex in [Formula: see text] is shown to have a degree of [Formula: see text]. We also present bounds on parameters measuring geometric quality (aspect ratios, minimum edge length, and maximum angle of cells) of [Formula: see text] in terms of the corresponding parameters of [Formula: see text] for [Formula: see text]. Finally, we illustrate a direct application of the proposed Euler transformation in additive manufacturing. 

Aerial imagery has the potential to advance high-throughput phenotyping for agricultural field experiments. This potential is currently limited by the difficulties of identifying pixels of interest (POI) and performing plot segmentation due to the required intensive manual operations. We developed a Python package, GRID (GReenfield Image Decoder), to overcome this limitation. With pixel-wise K-means cluster analysis, users can specify the number of clusters and choose the clusters representing POI. The plot grid patterns are automatically recognized by the POI distribution. The local optima of POI are initialized as the plot centers, which can also be manually modified for deletion, addition, or relocation. The segmentation of POI around the plot centers is initialized by automated, intelligent agents to define plot boundaries. A plot intelligent agent negotiates with neighboring agents based on plot size and POI distributions. The negotiation can be refined by weighting more on either plot size or POI density. All adjustments are operated in a graphical user interface with real-time previews of outcomes so that users can refine segmentation results based on their knowledge of the fields. The final results are saved in text and image files. The text files include plot rows and columns, plot size, and total plot POI. The image files include displays of clusters, POI, and segmented plots. With GRID, users are completely liberated from the labor-intensive task of manually drawing plot lines or polygons. The supervised automation with GRID is expected to enhance the efficiency of agricultural field experiments.