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A strip of square stamps can be folded in many ways such that all of the stamps are stacked in a single pile in the folded state. The stamp folding problem asks for the number of such foldings and has previously been studied extensively. We consider this problem with the additional restriction of fixing the mountain-valley assignment of each crease in the stamp pattern. We provide a closed form for counting the number of legal foldings on specific patterns of mountain-valley assignments, including a surprising appearance of the Catalan numbers. We describe results on upper and lower bounds for the number of ways to fold a given mountain-valley assignment on the strip of stamps, provide experimental evidence suggesting more general results, and include conjectures and open problems. Journal version 

The field of rigid origami concerns the folding of stiff, inelastic plates of material along crease lines that act like hinges and form a straight-line planar graph, called the crease pattern of the origami. Crease pattern vertices in the interior of the folded material and that are adjacent to four crease lines, i.e. degree-4 vertices, have a single degree of freedom and can be chained together to make flexible polyhedral surfaces. Degree-4 vertices that can fold to a completely flat state have folding kinematics that are very well-understood, and thus they have been used in many engineering and physics applications. However, degree-4 vertices that are not flat-foldable or not folded from flat paper so that the vertex forms either an elliptic or hyperbolic cone, have folding angles at the creases that follow more complicated kinematic equations. In this work we present a new duality between general degree-4 rigid origami vertices, where dual vertices come in elliptic-hyperbolic pairs that have essentially equivalent kinematics. This reveals a mathematical structure in the space of degree-4 rigid origami vertices that can be leveraged in applications, for example in the construction of flexible 3D structures that possess metamaterial properties. 

We introduce VISTA, a clustering approach for multivariate and irregularly sampled time series based on a parametric state space mixture model. VISTA is specifically designed for the unsupervised identification of groups in datasets originating from healthcare and psychology where such sampling issues are commonplace. Our approach adapts linear Gaussian state space models (LGSSMs) to provide a flexible parametric framework for fitting a wide range of time series dynamics. The clustering approach itself is based on the assumption that the population can be represented as a mixture of a fixed number of LGSSMs. VISTA’s model formulation allows for an explicit derivation of the log-likelihood function, from which we develop an expectation-maximization scheme for fitting model parameters to the observed data samples. Our algorithmic implementation is designed to handle populations of multivariate time series that can exhibit large changes in sampling rate as well as irregular sampling. We evaluate the versatility and accuracy of our approach on simulated and real-world datasets, including demographic trends, wearable sensor data, epidemiological time series, and ecological momentary assessments. Our results indicate that VISTA outperforms most comparable standard times series clustering methods. We provide an open-source implementation of VISTA in Python. 

Abstract Neuropsychiatric disorders pose a high societal cost, but their treatment is hindered by lack of objective outcomes and fidelity metrics. AI technologies and specifically Natural Language Processing (NLP) have emerged as tools to study mental health interventions (MHI) at the level of their constituent conversations. However, NLP’s potential to address clinical and research challenges remains unclear. We therefore conducted a pre-registered systematic review of NLP-MHI studies using PRISMA guidelines (osf.io/s52jh) to evaluate their models, clinical applications, and to identify biases and gaps. Candidate studies (n = 19,756), including peer-reviewed AI conference manuscripts, were collected up to January 2023 through PubMed, PsycINFO, Scopus, Google Scholar, and ArXiv. A total of 102 articles were included to investigate their computational characteristics (NLP algorithms, audio features, machine learning pipelines, outcome metrics), clinical characteristics (clinical ground truths, study samples, clinical focus), and limitations. Results indicate a rapid growth of NLP MHI studies since 2019, characterized by increased sample sizes and use of large language models. Digital health platforms were the largest providers of MHI data. Ground truth for supervised learning models was based on clinician ratings (n = 31), patient self-report (n = 29) and annotations by raters (n = 26). Text-based features contributed more to model accuracy than audio markers. Patients’ clinical presentation (n = 34), response to intervention (n = 11), intervention monitoring (n = 20), providers’ characteristics (n = 12), relational dynamics (n = 14), and data preparation (n = 4) were commonly investigated clinical categories. Limitations of reviewed studies included lack of linguistic diversity, limited reproducibility, and population bias. A research framework is developed and validated (NLPxMHI) to assist computational and clinical researchers in addressing the remaining gaps in applying NLP to MHI, with the goal of improving clinical utility, data access, and fairness. 

Rigid origami, with applications ranging from nano-robots to unfolding solar sails in space, describes when a material is folded along straight crease line segments while keeping the regions between the creases planar. Prior work has found explicit equations for the folding angles of a flat-foldable degree-4 origami vertex and some cases of degree-6 vertices. We extend this work to generalized symmetries of the degree-6 vertex where all sector angles equal 60 ∘ . We enumerate the different viable rigid folding modes of these degree-6 crease patterns and then use second-order Taylor expansions and prior rigid folding techniques to find algebraic folding angle relationships between the creases. This allows us to explicitly compute the configuration space of these degree-6 vertices, and in the process we uncover new explanations for the effectiveness of Weierstrass substitutions in modelling rigid origami. These results expand the toolbox of rigid origami mechanisms that engineers and materials scientists may use in origami-inspired designs.