text:
\section*{Introduction}
Intricate spatial structures are a hallmark of many complex and natural systems. The movement of individuals through space is one example of a natural system where understanding the spatial (or spatio-temporal) structures is key for distilling mechanistic insights from complex data \cite{nathan2022big}.
Since complex systems tend to produce complicated and highly correlated data, a purely descriptive exploratory analysis offers only limited insights. Model-based analysis, by contrast, suffers from (i) a lack of reliably useful mathematical or computer models; and (ii) the statistical challenges in fitting such models against data and in choosing the most appropriate model~\cite{kirk2013model}.
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The analysis of models and data with topological data analysis (TDA) is gaining increasing traction. Persistent homology (PH), a prominent tool in TDA, computes a geometric summary of the structure and shape of complex systems, including spatio-temporal mathematical models \cite{edelsbrunner2010computational, ghrist2008barcodes, wasserman2018topological}. Persistent homology builds a collection of discrete approximations on the data called {\em simplicial complexes}, computes topological invariants, such as homology, which provides a summary of topological features (\textit{e.g.}~connected components, loops, and voids).
Crucially, the PH algorithm involves the construction of a so-called filtration of simplicial complexes from the data, which quantifies the persistence of topological features in the data across changes in scale and resolution \cite{edelsbrunner2010computational, ghrist2008barcodes, roadmap}. In this way, a threshold parameter value often required in data analysis is sidestepped.

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Persistent homology provides a multiscale analysis of connectivity inherent in data. The powerful algebraic representation that encodes the topological features in persistent homology can be visualised by a \textit{persistence diagram} (see Figure~\ref{fig:pipeline}(A)).
A persistence diagram is a multiset (a generalisation of the conventional set where elements can appear more than once). Each $(x,y)$ point in the persistence diagram corresponds to a topological feature in the dataset; the importance of a topological feature is measured by its \textit{persistence}, \textit{i.e.}~the norm $|y-x|$ of the corresponding point in the multiset. Much statistical machinery has been developed to vectorise persistence diagrams for statistical or machine learning \cite{pun2022persistent, townsend2020representation}. Recent work has studied persistence diagrams with simplicial complexes \cite{jaramillo2022barcode}. PH analysis has led to parameter inference, pattern detection and non-trivial associations in highly structured and complex data \cite{hiraoka2016hierarchical, mcguirl2020topological, stolz2022multiscale, thorne2022topological,townsend2020representation,  vipond2021multiparameter}.

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Since its inception some 20 years ago, topological data analysis and persistent homology have proven to be (often vastly) more informative than conventional summary statistics of complex data. However, it is often more important to identify the likely cause of certain topological features than to merely characterise the full topology. Recent progress in extracting \textit{generators} of persistence diagrams, \textit{i.e.}~vertices in the dataset that form a \textit{cycle} corresponding to a specific $(x,y)$ point in the persistence diagram, has inspired new research that utilises generators as geometric and spatial features of data \cite{benjamin2022homology, emmett2015multiscale, henselmanghristl6, kovacev2016using, LiMinimalCycle}. For example, generators can localise the specific subchain responsible for the entanglement that distinguishes knotted and unknotted homologous proteins \cite{benjamin2022homology}. But different choice of vertices for the same topological feature can lead to different descriptions and interpretations of data.
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Here, we extend topological data analysis from one generator to all generators. To overcome the potential complications arising from the non-uniqueness of generators we develop two complementary approaches: matroid generators, which are efficiently computed using matroid theory \cite{henselmanghristl6}, and minimal generators \cite{emmett2015multiscale, LiMinimalCycle}, which we tailor to sequential datasets. We encode each generator as a hyperedge which forms the \textit{PH-hypergraph}. The PH-hypergraph encodes both local (geodesic) and global (Euclidean) geometric information of the data \cite{emmett2015multiscale,  LiMinimalCycle}.

The PH-hypergraph can be analysed using the armoury of modern graph theory and network science. Rather than applying PH to study network data \cite{aktasPersistenceHomologyNetworks2019a, FengSpatial}, we use network theory to analyse persistent homology. We show that hypergraph centrality measures and community detection distil information and insights from persistent homology generators. We refer to the full pipeline as \textit{hyperTDA}, as it combines hypergraph theory and topological data analysis (TDA). We show that hyperTDA can detect higher-order topological structures that are not captured by conventional TDA; moreover, it is robust to noisy data and to different choices of generators.
\par
We showcase hyperTDA on a wide range of datasets and highlight its ability to detect and quantify aspects of complex spatio-temporal data.
Motivated by protein carbon backbone \cite{benjamin2022homology} and animal movement trajectories \cite{nathan2022big}, here we analyse curves in three-dimensional space. Each curve can be translated into data as a collection of ordered vertices in 3-space and the number of vertices is the length of the curve.
We first consider two synthetic datasets: (i) two ``hand-drawn" curves to illustrate the concepts presented
and then 800 \textit{random curves} of four different lengths.
Our next dataset, called the \textit{AnDi trajectories}, consists of curves generated by five anomalous diffusion models \cite{munoz2021objective, munoz2020anomalous}. Lastly, we examine \textit{movement tracks}, which is a collection of experimental movement trajectories from nematodes moving in an agar plate and from zebrafish embryos migrating towards a wound \cite{garriga2016expectation, liepe2016accurate}. Tracks of nematodes consist of vertices in two dimensions; therefore, we convert the data into vertices in three dimensions by encoding the time variable in the $z$-coordinate.

\section*{From persistence diagrams to hypergraphs}
Our goal is to systematically analyse the multiscale cycles present in the data. We compute persistent homology, encode all generators by creating a PH-hypergraph, and then analyse it with network theory methods.

\begin{figure}[!ht]
    \centering
    \includegraphics[width = 15cm]{figure_1.pdf}
    \captionsetup{margin=-0.8cm}

    \caption{\textbf{Schematic of hyperTDA pipeline} \textbf{(A)} Given a collection of vertices, we first
    compute the persistence diagram. The persistence diagram depends on a choice of filtration $f$ to build a nested sequence of simplicial complexes. We choose a way of computing generators $g$ and then compute them for each point in the persistence diagram \textbf{(B)} We encode the information present in the persistence diagram and generators by constructing a hypergraph whose nodes are the vertices and whose hyperedges are the generators. The hyperedges are weighted by the persistence of the corresponding generators. We analyse and interpret the higher-order relationships in the data by performing network theoretic methods, given by h. Specifically, we compute PH-centrality, which quantifies the intensity of interactions between vertices, and PH-community detection, which partitions the vertices into communities that share homology generators.}
    \label{fig:pipeline}
\end{figure}

\subsection*{From data to generators} We use persistent homology to compute a multiscale summary of cycles present in a curve~\cite{benjamin2022homology, emmett2015multiscale, stolz2022multiscale}. Given a spatial curve (\textit{i.e.}~a sequence $\mathcal{V}$ = $\{v_1 \ldots v_n\}$ of ordered vertices, where $v_i$ are points in 3-space), we choose a filtration $f$ of simplicial complexes that represents the connectivity among vertices at various distance parameters. Here, $f$ is the Vietoris-Rips filtration \cite{ghrist2008barcodes}, which is based on the pairwise Euclidian distances of the vertices $v_i$. We then compute the dimension-1 PH, which summarises the one-dimensional topological features (\textit{i.e.}~cycles) present in the data. The birth and death of cycles across the filtration are summarised in a persistence diagram (Figure~\ref{fig:pipeline}(A)). Each point in the persistence diagram represents a cycle among the vertices, with $x$ and $y$ coordinates corresponding to \textit{birth} and \textit{death} parameters of the filtration. The life span of each point (\textit{birth} parameter $-$ \textit{death} parameter) is called its \textit{persistence} and provides a measure of its significance~\cite{ghrist2008barcodes, zomorodian2005computing}.

Given a point in the persistence diagram, the corresponding cycle is represented by a collection of vertices called a \textit{generator}. Generators are not unique and are therefore often not analysed, even though they encode how different vertices are organised to form cyclic features. Here, we present two methods $g$ for assigning generators to a persistence diagram: the \textit{matroid generators} and the \textit{minimal generators}~(see~\nameref{sec:codeAvail} and Supplementary Information). Matroid generators are the generators returned by the Eirene software \cite{henselmanghristl6} (which relies on matroid theory). While the matroid generators are fast to obtain, the selection of generators is not canonical or optimal. In order to compute generators in a principled way, we designed the minimal generators to capture the geometry and structure of the underlying curve. The minimal generators are computed by first finding the length-minimising generators \cite{LiMinimalCycle} and then minimising the number of jumps between non-consecutive vertices. The matroid and minimal generators are similar to each other (Figure \ref{fig:validation1}(F)), and the hyperTDA outputs are highly correlated (Figure \ref{fig:validation1}(G)). Since the results are comparable and the matroid generators are computationally faster to obtain, we present the results using the matroid generators.

\subsection*{From generators to PH-hypergraphs}
Once a choice of generators $g$ is made, all generators of a persistence diagram are translated into a \textit{hypergraph}. A hypergraph is a generalisation of a graph, in which hyperedges are arbitrary non-empty subsets of nodes \cite{bick2021higher}. More precisely, a weighted hypergraph is given by a triple $(\mathcal{N},\mathcal{E},\mathcal{W})$, where $\mathcal{E}$ is a collection of non-empty subsets of $\mathcal{N}$, with hyperedge-weights $\mathcal{W} \subset \mathbb{R}$. We construct the  \textit{PH-hypergraph} $\mathcal{H} = (\mathcal{V},\mathcal{G}, \mathcal{P})$, where each hyperedge $g_i \in \mathcal{G} =  \{g_1, \cdots g_h \}$ corresponds to the vertices in a generator. The weight $p_i \in \mathcal{P} = \{p_1, \cdots p_h \}$ is the persistence of the corresponding point in the persistent diagram (Figure~\ref{fig:pipeline}(B)).

To interpret the structural information of the curve's shape provided by the PH-hypergraph, we require a method $h$ to analyse it. We perform two analyses. First, we compute the vertex centrality~\cite{tudisco2021node}, which returns a vector whose entries give the importance of each vertex in the curve. This vector of vertex values -- \textit{PH-centrality} -- provides a ranking of vertices in the hypergraph based on their hyperedge membership and weights (Figure~\ref{fig:pipeline}(B)). In other words, the importance of a vertex depends on the importance of its connections; so a vertex with high centrality is a member of generators with large persistence. The second method we employ to analyse the PH-hypergraph is community detection to find functional modules in the curve. Community detection methods find densely-connected groups of vertices, with sparse connections between groups (relative to what one would expect at random) \cite{python_louvain}. Each vertex $v_i$ is assigned to a unique community $c_j$ where $j = 1, \ldots, k$ and $k$ is the number of communities. The communities are encoded in a matrix of size $n \times k$, where the ${(i,j)}$-entry is $1$ if vertex $v_i$ belongs to community $c_j$ and $0$ otherwise. We refer to the partition matrix of $\mathcal{H}$ as the \textit{PH-community matrix}, where each column is a \textit{PH-community} that corresponds to a functional module induced by higher-order interactions among the vertices in the curve (Figure~\ref{fig:pipeline}(B)).

\section*{Interpretation and robustness of hyperTDA}
HyperTDA detects and quantifies spatial properties of data by encoding higher-order interactions that are often undetectable with standard methods. Here we validate and interpret the output of hyperTDA on the random curves dataset. Curves in this dataset have various lengths and are generated as self-avoiding random walks. Further, we demonstrate that hyperTDA is robust to perturbation of data and different choices of generators.

\subsection*{Validation and correlation of hyperTDA with geometric invariants}
We first apply the hyperTDA pipeline on two illustrative curves (Figure~\ref{fig:validation1}(A), left, flower and right, roller-coaster). In the flower curve, PH-centrality (top) is essentially homogeneous, thus adequately recognising each of the petals as equally important features; however, the PH-communities (bottom) identifies all the loops as distinct communities. Similarly, in the roller-coaster curve, PH-centrality highlights the most significant geometric and structural features of the curve, while the PH-communities detect the individual loops.

\begin{figure}[!ht]
\centering
\hspace*{-15pt}
    \includegraphics[width = 16cm]{figure_2.pdf}
    \captionsetup{margin=-0.8cm}
    \caption{\textbf{HyperTDA quantifies spatial complexity and is robust to noise.} \textbf{(A)} Two illustrative curves with vertices coloured according to PH-centrality (top) and to PH-communities (bottom) \textbf{(B)} (Top) Correlation between PH-centrality and curvature, torsion, and density vectors for random curves. (Bottom) Correlation between PH-community matrices and pairwise distance matrices for random curves. \textbf{(C)} Example random curve $C$ of length 200. The colour of the vertices represents centrality values (top) and community membership (bottom, light grey vertices are singleton).  \textbf{(D)} The upper triangle shows the pairwise-distance matrix of $C$, with lower values corresponding to lighter shades of blue. The lower triangle shows the PH-community matrix corresponding to $\mathcal{H}(C)$, where non-zero entries are coloured according to community membership. \textbf{(E)} The persistence diagram of $C$, with example points highlighted. \textbf{(F)} The matroid generators and minimal generators for the highlighted points in (E). \textbf{(G)} (Top) Correlation between PH-centrality from matroid generators and from minimal generators. (Bottom) Adjusted mutual information (AMI) score between the partition induced by PH-communities from matroid generators and from minimal generators. \textbf{(H)} Consistency of PH-centrality and PH-communities among random curves and perturbed/smooth curves; top row shows the correlation among PH-centrality and the bottom row shows AMI score between PH-community matrices. \textbf{(I)} AMI score plotted against correlation, for random curves of length 200. \textbf{(J)} The curve obtained by smoothing $C$. Vertices are coloured according to PH-centrality (top) and PH-communities (bottom). PH-centrality identifies the central loop as the predominant feature, and the PH-community matrix identifies the same loop as a community.}
    \label{fig:validation1}
\end{figure}

To extensively test and validate the effectiveness of hyperTDA, we apply the pipeline to a dataset of 800 randomly generated piecewise linear open curves in 3-space. The curves are four different lengths, given by 100, 200, 300 and 400 vertices in the curve.  We construct the PH-hypergraph and compute the PH-centrality values and the PH-community matrices. Figure~\ref{fig:validation1}(C) shows an example curve of length 200, with vertices coloured according to PH-centrality values and community membership, respectively. PH-centrality identifies the loop in the middle of the structure as the curve's dominant feature; PH-communities partition the vertices into six distinct modules, one of which (green vertices) corresponds to the central loop (vertices in light blue in Figure~\ref{fig:validation1}(C), bottom row, are singletons that do not belong to any community).

The pairwise distance matrix of a curve has entries equal to the pairwise distances between vertices, which coarsely describes the curve's 3D structure, but fails to encode higher-order features. For example, the dominant loop in Figure~\ref{fig:validation1}(C) detected by hyperTDA cannot be inferred from the pairwise distance matrix, see Figure~\ref{fig:validation1}(D). Recall that the PH-community matrix is the matrix whose ${(i,j)}$-entry is 1 if the \textit{i}$^\text{th}$ and \textit{j}$^\text{th}$ vertices belong to the same community and 0 otherwise; see Figure~\ref{fig:validation1}(D). We compute the correlation between the pairwise distance and PH-community matrices. As shown in Figure~\ref{fig:validation1}(B), we observe a clear negative correlation, with median values of approximately $-$0.5. The negative correlation indicates that the PH-communities recover the global spatial structure coming from the Euclidean metric while providing additional information (\textit{e.g.}~functional modules given by higher-order interactions, as the loop in Figure~\ref{fig:validation1}(C)). Indeed, each community represents clusters of higher-order interactions among vertices (\textit{e.g.}~loops) that are often overlooked by the Euclidean metric alone.

Similarly, PH-centrality quantifies the data in a way that complements Euclidean distances. We compute the correlation between PH-centrality and torsion, curvature, and density (computed point-wise as the fraction of vertices with distance less than 2). As shown in Figure~\ref{fig:validation1}(B, top), we notice a very weak positive and negative correlation with torsion and curvature (median of approximately 0.1 and -0.14, respectively), and a slightly stronger positive correlation with density (median of approximately 0.27).

\subsection*{HyperTDA outputs consistent results across different choice of generators}
We find that the minimal generators and the matroid generators are very similar, as shown in Figure~\ref{fig:validation1}(F). Both PH-centrality and PH-communities computed from minimal generators and the matroid generators are highly correlated as shown in Figure~\ref{fig:validation1}(G). To compute correlations of the partitions induced by PH-communities, we used adjusted mutual information (AMI) score.

\subsection*{HyperTDA is robust to noisy data}

One of fundamental advantages of PH is its robustness to noise, which is guaranteed by theoretical results~\cite{cohen2007stability}. These results, however, do not extend automatically to generators. To assess the robustness of generators to noisy data, we apply hyperTDA to the dataset obtained by adding random Gaussian noise of increasing intensity to each vertex (up to $\delta = 0.2$, which is a fifth of the distance between adjacent vertices). We call the resulting curves \textit{perturbed curves}. Further, we create \textit{smooth curves} via a smoothing algorithm. Smoothing strongly affects the point cloud positions while maintaining a reasonable amount of the curve's dominant features; see Figure~\ref{fig:validation1}(J). To check that the curve's overall structure is still recovered, we compute the AMI score between the PH-community matrix of the random and perturbed curves. We also compute the correlations between centralities of the random and perturbed curves. Both PH-centrality and PH-community matrices present high similarities (Figure~\ref{fig:validation1}(H-J)).

\subsection*{PH-communities encode Euclidean and geodesic metrics}
We investigate how well PH-communities align with the ambient data space (\textit{i.e.}~Euclidean metric) and the intrinsic metric (\textit{i.e.}~geodesic, which in this case is relative to a discrete space curve). We compute three quantities associated with the communities. First, we estimate the \textit{volume} filled by a community as the radius of gyration of the sub-curve spanned by its vertices. Then, we define the \textit{geodesic size} of a community as $v_{min} - v_{max}$, where $v_{min}$ is the first vertex in the ordered vertex set and $v_{max}$ is the largest vertex in the ordered vertex set. Lastly, we compute the \textit{size} of a PH-community as the number of its vertices (Figure \ref{fig:commanalysis}(A-B)).

\begin{figure}[!ht]
    \hspace{-7mm}
    \includegraphics[width = 14cm]{comm_analysis.pdf}
    \captionsetup{margin=-0.8cm}
    \caption{\textbf{Euclidean and geodesic analysis of PH-communities}  \textbf{(A)} A curve with vertices coloured according to PH-community membership. \textbf{(B)} The panel shows a schematic representation of each community vector, together with their sizes and geodesic sizes. \textbf{(C)} Four examples of curves exhibiting different ambient and intrinsic assortativites \textbf{(D)} Schematic of pairwise geodesic intersection between communities, with examples of pairs with low and high intersection values. \textbf{(E)} Pairwise geodesic intersection values for the communities C1, C2 and C3 in (A).}
    \label{fig:commanalysis}
\end{figure}

In network science, a partition is said to exhibit \textit{assortativity} if similar vertices are more likely to be in the same partition than vertices in different groups. We measure similarity of vertices using either the ambient or intrinsic metric.

\begin{enumerate}

\item We say that a PH-hypergraph exhibits \textit{ambient assortativity} if vertices in the same PH-community are likely to be very close in space. We define the ambient assortativity as $ \frac{1}{k} \cdot \sum_{i=1}^k  \frac{\text{Size}(c_i)}{ \text{Volume}(c_i)} $, the average ratios between sizes and volumes of all non-singletons communities $c_1, \dots c_k$ (Figure~\ref{fig:commanalysis}(C)). Curves with highly packed communities will then have high ambient assortativity (Figure \ref{fig:commanalysis}(C)).

\item Similarly, we say that a PH-hypergraph exhibits \textit{intrinsic assortativity} if vertices in the same PH-community are likely to be separated by a small geodesic size. We measure the intrinsic assortativity by $\frac{1}{k} \cdot \sum_{i=1}^k  \frac{\text{Size}(c_i)}{\text{Geodesic size}(c_i)}$, the average ratios between sizes and geodesic sizes of all communities $c_1, \dots c_k$ (Figure~\ref{fig:commanalysis}(C)). With this definition, curves with communities of small geodesic sizes have high intrinsic assortativity. (Figure \ref{fig:commanalysis}(C)).

\end{enumerate}

By construction, PH-communities define clusters of vertices highly interacting with each other. From a geodesic point of view; however, pairs of communities might appear more or less overlapping, depending on the curve's structure. For example, in Figure~\ref{fig:commanalysis}(D), C2 and C3 are completely separated. In contrast, C1 and C3, and C1 and C2 intersect geodesically. To quantify this phenomenon, we define the \textit{geodesic intersection} as the \textit{p}-value of the Mann-Whitney U-test between the sets of vertices in the communities. Choosing the \textit{p}-value is needed to avoid artefacts due to noise and isolated vertices. Figure~\ref{fig:commanalysis}(E) shows examples of different pairs of PH-communities with high and low intersection values.

\section*{Results}
We first demonstrate the ability of hyperTDA to distinguish diffusion behaviours. Specifically, we show that PH-communities and PH-centrality present statistically significant differences among different diffusion models. We then showcase hyperTDA in the analysis of species movements from experimental data.

\subsection*{HyperTDA distinguishes and selects anomalous diffusion models}
Anomalous diffusion is the transport behaviour of a particle in which the mean square displacement~(MSD) is proportional to~$t^{\alpha}$ for time $t$ and some parameter $\alpha \neq 1$. Anomalous diffusion phenomena have been observed across several fields, including molecular dynamics~\cite{di2018anomalous, krapf2015mechanisms,sabri2020elucidating}, ecology~\cite{humphries2012foraging}, and finance~\cite{plerou2000economic}. The ubiquitous presence of anomalous diffusion has inspired important research efforts to investigate and model its mechanisms; these efforts resulted in several different anomalous diffusion models~(ADMs).

Inferring the correct model for experimental trajectories is a difficult problem~\cite{munoz2021objective}. One reason making the model detection challenging is that the 3D structure of a trajectory is strongly influenced by its MSD (and thus it depends on the parameter $\alpha$). On the other hand, each ADM is defined for a range of values of $\alpha$ and thus produces trajectories exhibiting a range of different MSDs. We apply the hyperTDA pipeline to the anomalous random walks analysed in the 2020 AnDi challenge~\cite{munoz2021objective, munoz2020anomalous}, which has been the first synergistic effort to develop methods able to classify individual trajectories among several models of diffusion. These models are the annealed transient time motion~(ATTM), the continuous time random walk~(CTRW), the fractional Brownian motion~(FBM), the L\'evy walk~(LW), and the scaled Brownian motion~(SBM) (Figure~\ref{fig:diffusioncommunity}(A)). These ADMs have been selected by the challenge organisers due to their biological relevance~\cite{munoz2021objective}, as they model diffusion processes in living cells (CTRW, ATTM and FBM \cite{ krapf2015mechanisms, saxton2007biological, verdier2021learning}), animal foraging (LW \cite{humphries2012foraging, verdier2021learning}), and human white matter (SBM \cite{fieremans2016vivo}).

\begin{figure}[!ht]
    \hspace{-7mm}
    \includegraphics[width = 14cm]{figure_3.pdf}
    \captionsetup{margin=-0.8cm}
    \caption{\textbf{HyperTDA distinguishes anomalous diffusion models (ADMs)}  \textbf{(A)} A representative trajectory for each model, in the following order: ATTM, CTRW, FBM, LW, SBM.  On the top row, vertices are coloured by PH-centrality. On the bottom, vertices are coloured by community membership. \textbf{(B)} Auto-correlation of centrality distributions, by model; for each value $m$ in $\{1,\cdots{}, 6\}$ the boxplots show the distribution of correlation values between centrality vectors and their values shifted by a lag $m$. \textbf{(C)} Analysis of the PH-communities, by model. Box plots show the distributions of volume, size, geodesic size and geodesic intersection are shown on the right. \textbf{(D)} Bar plots showing by model ambient and intrinsic assortativities. }
    \label{fig:diffusioncommunity}
\end{figure}

We apply hyperTDA to trajectories arising from these five models. We aim to detect and interpret model-specific differences arising from hyperTDA analysis of trajectories. We simulate 1000 length-200 trajectories for each of the five ADMs and apply the hyperTDA pipeline (compute PH-centrality and PH-communities) to all trajectories. We find the pattern of PH-centrality values depends on the underlying model and detects model-specific differences (Figure~\ref{fig:diffusioncommunity}(B)).
Next we compare the different models by computing the PH-community size, volume, geodesic size, and pairwise geodesic intersections
(Figure~\ref{fig:diffusioncommunity}(C)). Independent of the scaling exponent~$\alpha$, the distribution of these PH-community analysis gives a statistically significant difference between models.

We use network theory to further quantify the model-specific structural features highlighted by the PH-communities. We define the \textit{model ambient assortativity} as the average of the ratios between PH-community sizes and volumes over all non-singleton communities in the model. Similarly, the \textit{model intrinsic assortativity} is the average of the ratios between PH-community sizes and geodesic sizes over all non-singleton communities in the model (Figure~\ref{fig:diffusioncommunity}(D)). Assortativity provides a refined interpretation of the model-specific structure. For example, CTRW trajectories exhibit the highest discrepancy between Euclidean and geodesic metrics, as they score the lowest intrinsic assortativity (Figure~\ref{fig:diffusioncommunity}(D)) and highest ambient assortativity. Thus, their structure is characterised by compact, highly interconnected communities that are spread out along the curve. We highlight that standard computations of PH are blind to these features (see Supplementary Information, Figure \ref{fig:landscapes}).

\begin{figure}[!ht]
    \centering
    \includegraphics[width = 13cm]{CNN_figure.pdf}
    \captionsetup{margin=-0.8cm}

    \caption{\textbf{HyperTDA + CNN successfully select ADMs.} \textbf{(A)} The confusion matrix summarising the classification accuracy between all the 5 models. \textbf{(B)} The confusion matrix summarising the classification accuracy between CTRW, LW and SBM models. \textbf{(C)} A schematic description of the CNN architecture given a single datapoint input, that is: a matrix H with dimensions $\#\mathrm{hyperedges}\times\#\mathrm{vertices}$, indicating hyperedge memberships, and vertex centrality values V. Arrows follow the flow of the forward pass. Rectangular boxes represent matrices and vectors, and stacking indicates a third dimension. All trainable parameters are convolutional except when otherwise indicated. The label ``max'' stands for \textit{maxpooling}. For simplicity, the schematic only shows two convolutional kernels with 3 filters in the first layers, and 4 filters in the last.
    }
    \label{fig:CNN}
\end{figure}

Given trajectory data, we next explore whether hyperTDA can predict the model that generated the data. To test this hypothesis, we construct a convolutional neural network (CNN) that takes as input PH-hyperedges and node centrality values. A schematic of the architecture is shown in Figure~\ref{fig:CNN}(C), and the classification results are summarised in Figure~\ref{fig:CNN}(AB). Despite some difficulties in distinguishing the ATM trajectories from FBM and SBM, we obtain an accuracy comparable to the performance of ranked participants of the AnDi challenge~\cite{munoz2021objective, munoz2020anomalous}. We emphasise that the classification relies only on the structural description of the trajectories, effectively disregarding any temporal information. Further, when we restrict to a comparison among CTRW, LW, and SBM, we achieve high accuracy of approximately 92\%.

\subsection*{HyperTDA distinguishes species movements}

The increased availability of technologies to record animal trajectories has inspired research efforts aimed at inferring information on animal behaviour from their tracked movements~\cite{garriga2016expectation,lewis2021learning,maekawa2020deep, nathan2022big}. These approaches have the potential to reveal insight into interactions between individuals, and between animals and their environment, with unprecedented details~\cite{nathan2022big}. One of the main challenges in movement research is to find efficient and unsupervised ways to fragment animal trajectories into units known as \textit{behavioural nodes}~\cite{nathan2008movement}. Similarly, the recent advances in cell imaging have been stimulating progress in the study of cell migration~\cite{liepe2016accurate,masuzzo2016taking,svensson2018untangling,yamada2019mechanisms}. Natural challenges in this context include quantifying homogeneity of  migration patterns in a population and the identification of biases and persistence (\textit{i.e.}~the time period during which a cell moves in the same direction) in the underlying random walks~\cite{beltman2009analysing,krause2014steering,svensson2018untangling}.

\subsubsection*{PH-centrality identifies behavioural nodes in nematode trajectories}
We apply hyperTDA to animal movement trajectory data. The first dataset consists of six distinct solitary nematodes moving in an agar plate \cite{garriga2016expectation}. PH-centrality clearly distinguishes different behaviours of the nematode (Figure \ref{fig:animalcell}(A)). Indeed, portions of the curve assigned low PH-centrality (Figure \ref{fig:animalcell}(A), blue) can be interpreted as a behaviour known as relocation. Conversely, regions of the curve with increased PH-centrality correspond to behaviours known as local search and looping; furthermore, higher PH-centrality values correspond to more localised and intense search and looping behaviour (Figure \ref{fig:animalcell}(A), yellow). These behavioural classification of vertices obtained from PH-centrality is consistent with those of previous studies~\cite{garriga2016expectation}, yet it is unsupervised and arguably more nuanced. We find that PH-communities identify individual loops (and possibly groups of intertwined loops) of the worm's trajectory as functional modules (see Supplementary Information).

\begin{figure}[!ht]
    \includegraphics[width = 14cm]{figure_5.pdf}
    \captionsetup{margin=-0.8cm}

    \caption{\textbf{Animal movement and cell migration trajectories} \textbf{(A)} Two 2-D trajectories, with vertices plotted by centrality value. Centrality distinguishes different behaviours in terms of intensity of local search, looping behaviour and relocation. \textbf{(B)} Radar chart given by the medians of volume, size, geodesic size and pairwise geodesic intersection distributions for the unbiased and biased simulated data, and for the zebrafish trajectories. Results are shown with minimal generators. \textbf{(C)} Heat-map of pairwise Kolmogorov-Smirnov test on the cumulative distributions obtained by concatenating geodesic intersection, volume, size and geodesic size distributions. Results are shown with minimal generators.}
    \label{fig:animalcell}
\end{figure}

\subsubsection*{HyperTDA reveals migration bias of zebrafish embryos in \textit{in vivo} tracks}
As a final application, we apply the pipeline to assess the bias in neutrophil cell tracks extracted from zebrafish embryos migrating towards a wound~\cite{liepe2016accurate}. Neutrophils, a type of white blood cell, tend to migrate toward wounds; thus, the expectation is that they follow a \textit{biased persistent random walk}~\cite{liepe2016accurate}. To quantify the presence of the bias, we perform a statistical comparison of PH-communities in zebrafish tracks with those of biased and unbiased persistent random walks on an ellipse; all data is taken from~\cite{liepe2016accurate}. All simulated trajectories have length 20 while the lengths of \textit{in vivo} tracks vary. We thus partition the latter into segments of length 20 before applying hyperTDA; results are shown in Figure~\ref{fig:animalcell}(BC). PH-communities of biased and unbiased simulated trajectories present statistically significant differences (Figure~\ref{fig:animalcell}(BC)). Indeed, as expected, the PH-communities of biased tracks are on average more compact, less overlapping, and have smaller geodesic size than the unbiased tracks. Further, we observe a striking similarity between the \textit{in vivo} and the biased trajectories (Figure~\ref{fig:animalcell}(BC)). Based on the pairwise Kolmogorov-Smirnov tests, we reject the hypothesis that trajectories in zebrafish are unbiased. Lastly, the ambient and intrinsic assortativities are higher for zebrafish and biased trajectories than the unbiased trajectories (see Supplementary Information). All combined, the analysis demonstrates the bias induced by approaching the wound in the experimental trajectories.

Cell tracks often lie on curved surfaces~\cite{liepe2016accurate}; for this reason, classical methods necessitate the pre-processing of data via complex manifold learning algorithms. In contrast, the data is not pre-processed and hyperTDA can be applied directly to the dataset.
 Note that the trajectories analysed are quite short and have very simple structures. Importantly, even in this case, both matroid (see Supplementary Information) and minimal generators (Figure~\ref{fig:animalcell}(BC)) are successful in recognising the differences and common patterns. Minimal generators, in particular, performed better in recognising similarities in biased and \textit{in vivo} trajectories (see also SI, Figures~\ref{fig:animalcellSI} and \ref{fig:animalcellSI2}). This suggests minimal generators as a preferable option when dealing with limited data.

\section*{Discussion}
We propose hyperTDA, a systematic and extensive analysis of persistent homology generators, to interpret complex spatial systems. We demonstrate the effectiveness of hyperTDA in capturing defining features of curves that are not detected with traditional methods. The versatility of  hyperTDA has natural direct applications in different contexts, ranging from the statistical inference of behaviour from species tracks to a nuanced and improved characterisation of polymer structures. We successfully employed this method to distinguish and predict anomalous diffusion models and to identify biases in species tracks. As a further example, we are currently working on an extension of this approach to the study of the 3D organisation of the human genome.

The pipeline immediately generalises to the study of any point cloud. Naturally, the results and their interpretation will depend on the choice of filtration used to compute PH. While the Vietoris-Rips filtration is a standard choice in persistent homology, different options might be preferable depending on the data. Similarly, variations in methods to analyse the PH-hypergraph allow for a range of possible flavours and interpretations.

The hyperTDA pipeline involves a choice for computing generators for points of the persistence diagram. We propose minimal generators and matroid generators as two viable options, and we show that the hyperTDA outcomes are strongly correlated. Data is often limited as a consequence of financial constraints or ethical concerns. Even under these conditions, hyperTDA with minimal generators captures the subtle signals revealing common patterns and structural differences. On the other hand, when analysing large and complex datasets, computational efficiency is an issue. In such cases, hyperTDA with matroid generators has the advantage of a fast computation, yet it provides similar hyperTDA results to that of minimal generators. One could adapt hyperTDA to other choices of generators~\cite{ChenGenerators, Dey2019, emmett2015multiscale, EscolarHiraoka, LiMinimalCycle, Obayashi2018VolumeOC, Ripser_involuted} depending on the data and the end goal; clearly the fledgling area of research into, and construction of, topology generators has opened up new mathematical avenues for mathematical exploration; but already they are proving to be immensely useful in topological data analysis.
\par
HyperTDA is flexible, general, and remarkably and demonstrably robust to noise in data. We can get hopelessly lost in the exploration of high-dimensional data. Here we have demonstrated that hyperTDA allows us to navigate such data, detect and quantify higher-order structures and make sense of complex systems.

\section*{Methods}
\label{sec:Methods}
\subsection*{Computing persistent homology}
Persistent diagrams in dimension one are computed using the Julia software Eirene \cite{henselmanghristl6}. Persistent homology is computed with $\mathbb{F}_2$ coefficients.

\subsection*{From PH to homology generators}
The first choice of generators is the matroid generators; these are returned by Eirene's default computations  \cite{henselmanghristl6}, which rely on matroid theory. The second choice of generators is called minimal generators which are computed as follows. For each point in the persistence diagram, we compute the length-minimal generator \cite{LiMinimalCycle} with rational coefficients. Nearly all solutions have coefficients in $\{0, \pm 1\}$, so we interpret the solution as having coefficients in $\mathbb{F}_2$. We then perform a jump-minimisation step where we iteratively search for homologous generators with shorter jumps \cite{emmett2015multiscale}. That is, given a generator whose vertices are given by $[ \dots, v_i, v_{i+1} \dots]$, if such generator is homologous to $[ \dots v_i, v_*, v_{i+1} \dots]$ for some $v_i < v_* < v_{i+1}$, then we choose the latter generator. Details can be found in Supplementary Information.

\subsection*{From generators to hypergraphs.}
Given a choice $g$ of generators, we construct the PH-hypergraph as follows. The set of nodes is given by the vertices of the curve. Each homology generator identifies a subset of nodes; for each such subset, we add a hyperedge connecting the corresponding vertices. Then, hyperedges are assigned a weight given by the persistence of the corresponding point in the persistence diagram.

\subsection*{Analysis of hypergraphs generators}
Centrality is computed using the software from \cite{tudisco2021node}, where a class of spectral centrality measures is defined and analysed. The model in \cite{tudisco2021node} depends on the choice of nonlinear functions, with different choices inducing different centrality \textit{flavours} that dictate how hyperedge weights influence node values. For our analysis, we focus on the \textit{max} centrality flavour, which assigns large values to nodes that are part of at least one \textit{important} edge (see~\nameref{sec:codeAvail}).

As community detection on hypergraphs is computationally challenging, we first flatten each PH-hypergraph into a graph, as follows. The set of nodes is maintained; for each hyperedge involving the subset of nodes N$_i$ we add the complete-subgraph on N$_i$. Once the graph is constructed, we perform community detection using the method implemented in Python's Louvain module \cite{python_louvain} with default parameters (see~\nameref{sec:codeAvail}).

Note that some trajectories in the AnDi and Zebrafish datasets have trivial PH. We removed such trajectories from the analysis.

\subsection*{Noisy Data}
To assess the robustness of hyperTDA, we apply the pipeline to successive perturbations of the randomly generated curves of lengths 100, 200, 300 and 400. These perturbations are obtained by adding Gaussian noise to each point in a curve. For each curve we apply the perturbation process 4 times, with intensity $\sigma$ increasing from 0.05 to 0.2 (a fifth of the distance between consecutive vertices in a curve). Further, we obtain smooth versions of each curve by applying the pyknotid \texttt{smooth} function \cite{pyknotid}~(see~\nameref{sec:codeAvail}).

\subsection*{Interpolation and uninterpolation}
The AnDi trajectories have a high variance in distance between consecutive vertices. We thus modify the curves by inserting vertices between consecutive vertices of the original curve before computing hyperTDA. Specifically, we interpolate the length 199 trajectories to make consecutive vertices approximately equidistant. The interpolation is achieved by adding a total of 301 linearly interpolated vertices between adjacent vertices. We first assign a single interpolated vertex to the segment with the largest length. We repeat the process for 301 interpolated vertices. At the end of the process, all the vertices assigned to a given segment are distributed evenly between the endpoints. The resulting interpolated trajectories have lengths of 500. We refer to them as the \textit{interpolated curves}.

The outputs of hyperTDA are reported on the original curve. To do so, before computing PH-community size, geodesic size, and assortativity values, we perform \textit{un}interpolation to obtain centrality values and community membership values on the original curve. This is done by multiplying the PH-community matrix $M$ (of size $500 \times k$, where $k$ is the number of communities) by a sparse matrix $D$ that maps interpolated vertices to \textit{un}interpolated vertices. We choose to construct $D$ with a rolling average weighting, where an interpolated vertex partly maps to both adjacent \textit{un}interpolated vertices, with more weight towards the closest of the two. For more details, see~\nameref{sec:codeAvail}.

\subsection*{HyperTDA analysis}
A detailed explanation of the methods and notebooks containing the relevant code can be found in the hyperTDA GitHub repository~(see~\nameref{sec:codeAvail}).

\section*{}
\subsection*{Acknowledgements}
The authors thank Gregory Henselman-Petrusek and Benjamin Schweinhart for helpful conversations.

\subsection*{Funding}
AB gratefully acknowledges funding through a MACSYS Centre Development initiative from the School of Mathematics \& Statistics, the Faculty of Science and the Deputy Vice Chancellor Research, University of Melbourne. HY gratefully acknowledges funding through the Mark Foundation for Cancer Research. CDM gratefully acknowledges funding through a University of Melbourne PhD studentship. MPHS is funded through the University of Melbourne DRM initiative, and acknowledges financial support from the Volkswagen Foundation through a ``Life?'' program grant.
This research was partially supported by The University of Melbourne's Research Computing Services and the Petascale Campus Initiative. This research was undertaken using the LIEF HPC-GPGPU Facility hosted at the University of Melbourne. This Facility was established with the assistance of LIEF Grant LE170100200. HAH gratefully acknowledges funding from EPSRC EP/R005125/1, the Royal Society RGF$\backslash$EA$\backslash$201074 and UF150238. AB, DOA and HAH are grateful to the EPSRC GCRF grant EP/T001968/1, part of the Abram Gannibal Project, for supporting collaborative meetings. HAH, AB and CDM are grateful to the support provided by the UK Centre for Topological Data Analysis EPSRC grant EP/R018472/1. For the  purpose of Open Access, the authors have applied a CC BY public copyright licence to any Author Accepted Manuscript (AAM) version arising from this submission.

\subsection*{Code availability}
\label{sec:codeAvail}
The code and datasets are available online at the project's GitHub repository: \href{https://github.com/degnbol/hyperTDA}{https://github.com/degnbol/hyperTDA}. A complete description of each dataset can be found in the Supplementary Information. The software used to compute minimal generators can be found in the following GitHub repository: \href{https://github.com/irishryoon/minimal_generators_curves}{https://github.com/irishryoon/minimal\_generators\_curves}, and a description of the method is in the Supplementary Information.

\subsection*{Author contribution statements}
AB, HY, and HAH conceived the research; AB, HY, HAH, MPHS designed the research; AB, HY, DOA, and CDM performed research; AB, HY, and CDM analysed data; and all the authors wrote the paper.

redpajama_set_name: RedPajamaArXiv


The volatility put a spotlight on doubts about Trump's ability to deliver on his promises with China - and exposed a clash between his policy goals in China and his efforts to crack down on technology companies and other actors.
Asian indices were mostly in positive territory on Friday after a calmer end to trading in NY while the FTSE was forecast to open nearly 2% higher - recovering much of the previous day's losses.
China's foreign ministry said neither the United States nor Canada have explained reasons for the arrest.
David Mulroney, a former Canadian ambassador to China, says that while he doesn't want to overstate the possibility of a Canadian being jailed, China will be looking for ways to strike back. Earlier this year, the US Commerce Department said ZTE violated the settlement and barred it from buying any US components - a move that all but halted many ZTE operations. "At any other time, if the Chinese were feeling more confident, if they were feeling their economy were more robust, the response would have been to cancel any pledges they made".
The founder and CEO of the company, Ren Zhengfei, is a communist party member and when he joined the party said it was expected that all "exceptional" people should do so. He also noted that she was "provisionally detained" in Canada while she was transferring flights.
The arrest was made at Washington's request as part of a USA investigation of an alleged scheme to use the global banking system to evade US sanctions against Iran, according to people familiar with the probe. The Justice Department is seeking Meng's extradition, according to a White House official.
Trump and Xi, who were in Argentina for a summit of the Group of 20 major economies, had agreed to set up negotiations to discuss U.S. concerns over China's trade barriers.
The arrest wasn't made public until late Wednesday.
Cybersecurity officials of relevant government agencies will likely agree in a meeting as early as Monday to block the Chinese firms from taking part in government procurement without explicitly naming the two companies in consideration of the potential impact on Tokyo's relations with Beijing, which have shown signs of improvement, the source said.
Patrick Chovanec, a well-known American China analyst and managing director of Silvercrest Asset Management, responded on Twitter: "If she broke the law, she should be arrested and tried".
But, he added, "from the Chinese perspective, this is going to be absolutely seen as part and parcel of our negotiating practice".
"So not respecting this particular arrest, but Huawei is one company we've been concerned about", he said.
No one answered the door when Postmedia News visited the Dunbar home Thursday afternoon, but neighbourhood residents said they recognized Meng from media reports, though they did not want to talk on the record. The Trump administration says they benefit from improper subsidies and market barriers.
Perhaps the most striking detail of the arrest of Huawei executive Meng Wanzhou in Canada was that it took place on the same day that leaders from the U.S. and China were engaged in delicate trade talks in Buenos Aires.
Is the arrest of Meng Wanzhou related to these security concerns? But he called it a judicial matter and said politics had nothing to do it. "We are working very closely with the broader security community". Canada released its long-scheduled Cyber Threat Assessment today, the first of its kind in the country. It's also making it harder for Chinese firms to invest in US companies or to buy American technology in such cutting-edge areas as robotics, artificial intelligence and virtual reality.
Trump's tariff hikes on Chinese imports stemmed from complaints Beijing steals or pressures foreign companies to hand over technology. ZTE eventually got a reprieve after Xi asked Trump for help.
Huawei was founded in 1987 and has grown to become the world's largest provider of telecom equipment, and the world's No. 2 seller of smartphones beating out Apple to sit behind Samsung.


These terms and conditions explain your rights and obligations in relation to Dress Ballynahinch and any goods purchased through the website www.dressballynahinch.com. Please read them carefully. We may update these 'Terms' from time to time, posting them online at www.dressballynahinch.com/terms.php. Your continued use of this site shall mean your agreement to any changes made and you are bound by such. You should check this page regularly to take notice of any changes we have applied. Should you proceed to purchase goods from this site you are confirming your acceptance of and compliance with all terms and conditions given.
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A commentary on the Myanmar Penal Code that describes and critically evaluates the general principles of criminal responsibility contained in the Code.
Comprehensive analysis of legislation, major cases, academic commentary and law reform perspectives on criminal law.
A condensed and concise summary of the essence of various Forensic Science Disciplines.
The Criminal Law in Malaysia and Singapore - A Casebook Companion, Second Edition book comprises a selection of some of the leading Malaysian and Singaporean cases which make compulsory reading for a standard criminal law course.
Understanding Criminal Law has been written to give the reader a clear explanation of the fundamental principles of criminal law and the criminal justice system.
Written by the Judges of the Subordinate Courts of Singapore, the third edition of this title is a double volume edition in the Practitioners' Library series and comes ten years after the second edition was published. The various offences, sentences, legislation and case law have been updated to reflect the latest developments in Singapore.


22 May 2010 VJ FEST ISTANBUL is an independent and international audio-visual performance festival to share knowledge that is open and free. It demonstrates the art of veejaying as a social act since 2008.
Since than more than 200 artists from 22 countries joined.
Four days and nights festival is inspired of new artistic ways of audiovisual performing and visualization of music by using analog or digital technics where light and sound intersects with each other for altering human perception.
2019.04.20 Projection Mapping | Somewhatsuper Concert.
2019.04.20 Какво е 3D projection mapping?


text:
City of Normandy, Missouri
The City of Normandy, Missouri is located in Mid St. Louis County, about 20 minutes west of Downtown St. Louis via Interstate 70. Important roads for the community are Florissant Road, Natural Bridge Road, Interstate 70, Bermuda Drive and others. The City was officially incorporated in 1945.
Normandy Schools
The residents of Normandy are served by two school districts, the Normandy School District and the Ferguson-Florissant School District. The Normandy School District has five elementary school, an early childhood center, Normandy Positive Alternative Learning Center, Normandy Middle School and Normandy High School. Bermuda Elementary from the Ferguson Florissant School District is located in Normandy. The University of Missouri-St. Louis (UMSL) campus is partially located with in Normandy. UMSL is home to the Blanche M. Touhill Performing Arts Center. Touhill is a classroom for the College of Fine Arts at UMSL as well as being an entertainment venue for a variety of performances.
Normandy Parks and Recreation
The City of Normandy maintains several parks. The Robert Hoelzel Memorial Park is the city main and largest park. It features baseball and softball fields, two picnic pavilions and more. The others, Belwood Park, Hoard's Grove, Parchester Park and Tear Drop Park are considered to be "pocket parks", tucked into neighborhoods. Also located within Normandy is St. Vincent County Park. Amenities that Normandy residents, as well as all county residents, enjoy in this park include the St. Vincent Community Center that features a gymnasium, weight room and a community room. The St Vincent Water Park features a large water "playground" structure, a zero depth beach entry, open water slide, an 80 foot flume ride and two 25 yard lap lanes. The Water Park also has a concession stand, patio with tables and a large deck area. The Water Park offers swimming lessons and aquatic exercise programs and can be rented for social gatherings. Other amenities at the St. Vincent Park are picnic pavilions, tennis courts, fitness trail, playgrounds, and indoor basketball courts.
Normandy History - Glen Echo Country Club and St. Vincent's
The oldest golf course west of the Mississippi River is the private Glen Echo Country Club located in Normandy. Glen Echo was established in 1901 and hosted the 1904 Olympic Golf Matches. Since then, Glen Echo has hosted several LPGA professional events and many amateur golf championships throughout the years. The Glen Echo club house is a St Louis County Historic Landmark. There are several other historic landmarks in Normandy. One of them is the Hardy House built in 1891 for John Mullanphy Cates. It was Thomas Walter Hardy, founder of the Hardy Salt Company and prominent St Louis businessman who lived the house the longest, between 1915-1966. St. Vincent's Hospital, now the Castle Park Apartments, was built in 1894 and designed by Harvey Ellis, the architect who also designed the St. Louis City Hall and the Compton Hill Water Tower. One other is the Wilson Price Hunt House, built in 1906 on land that Hunt's great grandfather, J.B.C Lucas, acquired during the days of the Louisiana Purchase which he had named "Normandy" after his native land, Normandy, France. The house is also on the National Register of Historic Places.
Normandy, Missouri Address and Phone Numbers
Normandy Missouri City Hall
Normandy Missouri Related Links:
City of Normandy Missouri Website
Normandy Missouri Photo Gallery
Normandy Missouri Streetview Link
Normandy Missouri City Sign Link
Normandy Missouri Link
Historic Landmark-St Vincent's Hospital, built 1894 Link
Touhill Performing Arts Center on the UMSL campus Link
Normandy High School Link
Hoelzel Park in Normandy Missouri Normandy Missouri
Normandy Missouri City Hall Link
University of Missouri-St Louis (UMSL) Link
University of Missouri-St Louis (UMSL) in Normandy MO Link

redpajama_set_name: RedPajamaCommonCrawl


Global travel. Learning new languages and experiencing cultures. These are just some of the reasons that people decide to join the Foreign Service at the Department of State.
The Foreign Service is a group of citizens that are the face of America internationally. Their mission is to strengthen peace and support prosperity by promoting our business interests and protecting American citizens throughout the world.
To be considered for the foreign service, you must be whip-smart and pass a rigorous application and interview process.
Take Charles A. Ray, Ambassador-Designate to Zimbabwe.
Ray started his career as a Foreign Service Officer in 1982 after retiring as a Major in the Army after 20-years service.
Before he retired, Ray wondered what his next step would be. A librarian at the Defense Language Institute in Monterey, CA suggested he look at the State Department.
Then dominated by Ivy-league graduates, Ray admits that the Foreign Service seemed an odd place for a kid that grew up in East Texas.
Successfully completing the application and interview process, he initially had a tough transition from giving orders in the Army to taking them as a junior Foreign Service Officer.
Since joining the Foreign Service, Charles A. Ray has served in China and Thailand, and was Deputy Chief of Mission in Sierra Leone. In 1998, he was appointed as the first U.S. Consul General in Ho Chi Minh City, Vietnam. In 2002, he was sworn in as the U.S. Ambassador to the Kingdom of Cambodia, where he served until 2005. Mr. Ray served as Diplomat in Residence at the University of Houston for the 2005-2006 academic year.
When asked his opinion about Generation Y and the next generation at the State Department, Ambassador Ray saw many similarities to those inspired by President Kennedy. However, he stressed that their enthusiasm can also lead to frustration when they are not automatically put into leadership positions.
"Be patient and learn," Ambassador Ray advised.
Learn more about becoming a Foreign Service Officer at the State Department.


text:
Q: Issue with SQL Output I'm confused on why this is happening. So if I leave or opn = '9111' in the query it will display all devices but the IN_USE column value will be incorrect.
If I add AND opn = '9111' to replace or opn = '9111' it will only display DXE/RESMETAL with the correct IN_USE value of 25.
or opn = '9111' OUTPUT Image:

AND opn = '9111'OUTPUT Image:

The reason why the or opn = '9111' query won't work is because it combines opn 9118 value to 9111 which will be incorrect.
EXAMPLE: Device DXE/RESMETAL should be 25 but it adds opn 9118 and it came out to value 50.
How can I fix this query to where it displays like image two but it lists all the devices and not just one? Thank you in advance.
SELECT
              device,
              Sum(in_qty) AS in_use
            FROM
              LOT_MOVE_AGE @XXX.ITG.COM
            WHERE
              facility = 'DP1DM5'
              AND device = ANY(
                SELECT
                  device
                FROM
                  dm_device_attributes @XXX.itg.com
                WHERE
                  facility = 'DP1DM5'
                  AND status = 'A'
                  AND family LIKE 'PILOT%E'
              )
              AND departure_dttm IS null
              AND latest = 'O'
              AND (
                lpt = ANY(
                  SELECT
                    lpt
                  FROM
                    pilot_lpts
                  WHERE
                    facility = 'DP1DM5'
                    AND MODULE = 'Plasma'
                    AND category = 'In Use'
                )
                AND opn = '9111'
              )
            GROUP BY
              device
            having
              device like 'DX%'

A: You can use opn in group by to separate the values.
SELECT
              device,
              Sum(in_qty) AS in_use
            FROM
              LOT_MOVE_AGE @XXX.ITG.COM
            WHERE
              facility = 'DP1DM5'
              AND device = ANY(
                SELECT
                  device
                FROM
                  dm_device_attributes @XXX.itg.com
                WHERE
                  facility = 'DP1DM5'
                  AND status = 'A'
                  AND family LIKE 'PILOT%E'
              )
              AND departure_dttm IS null
              AND latest = 'O'
              AND (
                lpt = ANY(
                  SELECT
                    lpt
                  FROM
                    pilot_lpts
                  WHERE
                    facility = 'DP1DM5'
                    AND MODULE = 'Plasma'
                    AND category = 'In Use'
                )
              OR opn = '9111' -- Used OR
              )
          AND device like 'DX%' -- moved this condition here
            GROUP BY
              device, opn; -- added opn in GROUP BY list

You can add opn in select list also so that you can get to know that count and device refers to which opn.
Cheers!!

redpajama_set_name: RedPajamaStackExchange


text:
'Cinderella' Trailer: Camila Cabello Starrer Flips The Outdated Narratives For A Feminist Version And We Stan
'Cinderella' also starring Billy Porter as the fairy godmother, is written for the screen and directed by Kay Cannon. It is coming to Amazon Prime Video on September 3.
By Sushri Sahu August 5, 2021
Fairy tales but with a feminist retelling! Who doesn't like the sound of that? And if you do, you're bound to like the trailer of Amazon Prime Video's Cinderella. Starring American pop-star and musician Camila Cabello, the period film has done away with all the outdated, orthodox and regressive narratives and has introduced us to a modern-day version of children's favourite fantasy character. In fact, even the God mother is played by American actor, singer, style and LGBTQIA icon Billy Porter. Yeah, it doesn't get better than that.
SEE ALSO: 'Cinderella' Teaser: Camila Cabello Is An Aspiring Seamstress In Retelling Of The Classic Fairytale
Oh wait, I jumped the gun- it totally does! We are talking about a pantsuit wearing Cinderella after all, one that doesn't need a prince charming to rescue her. Instead, she aspires to be a seamstress and be a career/business woman. A bosslady, if you may. And if you ask me, that's exactly the kind of stories and examples impressionable young, little girls all over the world need to see. The synopsis hints more about the same and reads, "Cinderella is a musically-driven bold new take on the traditional story you grew up with. Our heroine (Cabello) is an ambitious young woman whose dreams are bigger than the world will allow, but with the help of her Fab G (Billy Porter), she is able to persevere and make her dreams come true. Watch the trailer here.
Porter, in an interview to CBS News earlier, had said "It hit me when I was on the set last week, how profound it is that I am playing the Fairy Godmother — they call it the Fab G Magic has no gender. This is a classic, this is a classic fairytale for a new generation. I think that the new generation is really ready. The kids are ready. It's the grownups that are slowing stuff down,"
The film also stars an all-star cast including Frozen star Idina Menzel who plays the evil stepmother, Minnie Driver, Nicholas Galitzine and Pierce Brosnan. The film is written for the screen and is directed by Kay Cannon. Cannon even told EW, "I just felt like it was a great opportunity to show this loved, iconic character that is Cinderella in a way that's more relatable to what girls and young women in particular are going through, where they can really see themselves."
A post shared by Amazon Prime Video US (@amazonprimevideo)
The feminist retelling also features covers of songs written by some of the top-selling musical artists of all time. It is produced by James Corden, Leo Pearlman, Jonathan Kadin and Shannon McIntosh while Louise Rosner and Josephine Rose serve as executive producers.
Cinderella is coming to Amazon Prime Video on September 3.
Cover artwork by Bhavya Poonia/Mashable India
Topics: Entertainment, Trailer, Cinderella, Amazon Prime Video, Billy Porter, Camila Cabello, Fairytale, Intersectional Feminism
Netflix Confirms 'Squid Game' Season 2 And It's The Beginning Of An Extended Universe?
"The Squid Game universe has just begun," says Netflix.
'The Batman's Haunting Theme Music Is Perfect For The Dark Knight's Journey
Director Matt Reeves shared the incredible theme from 'The Batman' composed by Michael Giacchino and it will give you goosebumps.
Deepika Padukone's 'Gehraiyaan' Has Fans Freaking Out Over The Grey Romance
From the fan theories to hyping up Alisha, Deepika's flawed character, here's what the internet has to say about the film:
Meat Loaf, Legendary 'Bat Out Of Hell' Singer And Actor, Dies At 74
His death was confirmed by his agent.
'Gehraiyaan': Deepika Padukone Opens Up About Filming Intimate Scenes, Says "Intimacy Is Not Easy"
Deepika Padukone talked about her "raw and relatable" 'Gehraiyaan', revealing that she had to delve into her own experiences for the role.