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The application of Riemannian geometry in the decoding of brain-computer interfaces (BCIs) has swiftly garnered attention because of its straightforwardness, precision, and resilience, along with its aptitude for transfer learning, which has been demonstrated through significant achievements in global BCI competitions. This paper presents a comprehensive review of recent advancements in the integration of deep learning with Riemannian geometry to enhance EEG signal decoding in BCIs. Our review updates the findings since the last major review in 2017, comparing modern approaches that utilize deep learning to improve the handling of non-Euclidean data structures inherent in EEG signals. We discuss how these approaches not only tackle the traditional challenges of noise sensitivity, non-stationarity, and lengthy calibration times but also introduce novel classification frameworks and signal processing techniques to reduce these limitations significantly. Furthermore, we identify current shortcomings and propose future research directions in manifold learning and riemannian-based classification, focusing on practical implementations and theoretical expansions, such as feature tracking on manifolds, multitask learning, feature extraction, and transfer learning. This review aims to bridge the gap between theoretical research and practical, real-world applications, making sophisticated mathematical approaches accessible and actionable for BCI enhancements.

This paper introduces the Pandemic PACT Advanced Categorisation Engine (PPACE) along with its associated dataset. PPACE is a fine-tuned model developed to automatically classify research abstracts from funded biomedical projects according to WHO-aligned research priorities. This task is crucial for monitoring research trends and identifying gaps in global health preparedness and response. Our approach builds on human-annotated projects, which are allocated one or more categories from a predefined list. A large language model is then used to generate `rationales' explaining the reasoning behind these annotations. This augmented data, comprising expert annotations and rationales, is subsequently used to fine-tune a smaller, more efficient model. Developed as part of the Pandemic PACT project, which aims to track and analyse research funding and clinical evidence for a wide range of diseases with outbreak potential, PPACE supports informed decision-making by research funders, policymakers, and independent researchers. We introduce and release both the trained model and the instruction-based dataset used for its training. Our evaluation shows that PPACE significantly outperforms its baselines. The release of PPACE and its associated dataset offers valuable resources for researchers in multilabel biomedical document classification and supports advancements in aligning biomedical research with key global health priorities.

This is the third installment in a series of papers on the subject of derived contact structures. In this paper, we formally introduce Legendrians in the derived context and give a natural example. We then present affine models and prove a Legendrian-Darboux-type theorem for the Legendrians in contact derived schemes., Comment: 23 pages. Comments welcome!

$C^*$-algebra-valued kernels could pave the way for the next generation of kernel machines. To further our fundamental understanding of learning with $C^*$-algebraic kernels, we propose a new class of positive definite kernels based on the spectral truncation. We focus on kernels whose inputs and outputs are vectors or functions and generalize typical kernels by introducing the noncommutativity of the products appearing in the kernels. The noncommutativity induces interactions along the data function domain. We show that it is a governing factor leading to performance enhancement: we can balance the representation power and the model complexity. We also propose a deep learning perspective to increase the representation capacity of spectral truncation kernels. The flexibility of the proposed class of kernels allows us to go beyond previous commutative kernels, addressing two of the foremost issues regarding learning in vector-valued RKHSs, namely the choice of the kernel and the computational cost.

We report the results of an extensive set of simulations exploring the sensitivity of the BlackCAT CubeSat to long-duration gamma-ray bursts (GRBs). BlackCAT is a NASA APRA-funded CubeSat mission for the detection and real-time sub-arcminute localization of high-redshift ($z\gtrsim 3.5$) GRBs. Thanks to their luminous and long-lived afterglow emissions, GRBs are uniquely valuable probes of high-redshift star-forming galaxies and the intergalactic medium. In addition, each detected GRB with a known redshift serves to localize a region of high-redshift star formation in three dimensions, enabling deep follow-on searches for host galaxies and associated local and large-scale structures. We explore two distinct models for the GRB redshift distribution and luminosity function, both consistent with Swift observations. We find that, for either model, BlackCAT is expected to detect a mean of 42 bursts per year on-orbit, with 6.7% to 10% of these at $z>3.5$. BlackCAT bursts will be localized to $r_{90} \lesssim 55^{\prime\prime}$ precision and reported to the community within seconds. Due to the mission orbit and pointing scheme, bursts will be located in the night sky and well-placed for deep multiwavelength follow-up observations. BlackCAT is on schedule to achieve launch readiness in 2025., Comment: 10 pages, 4 figures, 3 tables; Published in ApJ

Machine learning has a long collaborative tradition with several fields of mathematics, such as statistics, probability and linear algebra. We propose a new direction for machine learning research: $C^*$-algebraic ML $-$ a cross-fertilization between $C^*$-algebra and machine learning. The mathematical concept of $C^*$-algebra is a natural generalization of the space of complex numbers. It enables us to unify existing learning strategies, and construct a new framework for more diverse and information-rich data models. We explain why and how to use $C^*$-algebras in machine learning, and provide technical considerations that go into the design of $C^*$-algebraic learning models in the contexts of kernel methods and neural networks. Furthermore, we discuss open questions and challenges in $C^*$-algebraic ML and give our thoughts for future development and applications., Comment: position paper

In this study, we propose an approach for predicting rare events by exploiting time series in coevolution. Our approach involves a weighted autologistic regression model, where we leverage the temporal behavior of the data to enhance predictive capabilities. By addressing the issue of imbalanced datasets, we establish constraints leading to weight estimation and to improved performance. Evaluation on synthetic and real-world datasets confirms that our approach outperform state-of-the-art of predicting home equipment failure methods.

The Event Horizon Telescope (EHT) Collaboration recently published the first images of the supermassive black holes in the cores of the Messier 87 and Milky Way galaxies. These observations have provided a new means to study supermassive black holes and probe physical processes occurring in the strong-field regime. We review the prospects of future observations and theoretical studies of supermassive black hole systems with the next-generation Event Horizon Telescope (ngEHT), which will greatly enhance the capabilities of the existing EHT array. These enhancements will open up several previously inaccessible avenues of investigation, thereby providing important new insights into the properties of supermassive black holes and their environments. This review describes the current state of knowledge for five key science cases, summarising the unique challenges and opportunities for fundamental physics investigations that the ngEHT will enable., Comment: To be submitted to journal. Comments are welcome

Background and objectives: Dynamic handwriting analysis, due to its non-invasive and readily accessible nature, has recently emerged as a vital adjunctive method for the early diagnosis of Parkinson's disease. In this study, we design a compact and efficient network architecture to analyse the distinctive handwriting patterns of patients' dynamic handwriting signals, thereby providing an objective identification for the Parkinson's disease diagnosis. Methods: The proposed network is based on a hybrid deep learning approach that fully leverages the advantages of both long short-term memory (LSTM) and convolutional neural networks (CNNs). Specifically, the LSTM block is adopted to extract the time-varying features, while the CNN-based block is implemented using one-dimensional convolution for low computational cost. Moreover, the hybrid model architecture is continuously refined under ablation studies for superior performance. Finally, we evaluate the proposed method with its generalization under a five-fold cross-validation, which validates its efficiency and robustness. Results: The proposed network demonstrates its versatility by achieving impressive classification accuracies on both our new DraWritePD dataset ($96.2\%$) and the well-established PaHaW dataset ($90.7\%$). Moreover, the network architecture also stands out for its excellent lightweight design, occupying a mere $0.084$M of parameters, with a total of only $0.59$M floating-point operations. It also exhibits near real-time CPU inference performance, with inference times ranging from $0.106$ to $0.220$s. Conclusions: We present a series of experiments with extensive analysis, which systematically demonstrate the effectiveness and efficiency of the proposed hybrid neural network in extracting distinctive handwriting patterns for precise diagnosis of Parkinson's disease.

This is a sequel of our previous work, arXiv:2209.09686, on the development of derived contact geometry, in which we formally introduced shifted contact structures on derived stacks and proved some results for $k$-shifted contact derived schemes, with $k<0$. In this paper, we extend these results from derived schemes to derived Artin stacks. In brief, we first show that for $k<0$, every $k$-shifted contact derived Artin stack admits a contact Darboux atlas. Secondly, we canonically describe the symplectification of a derived Artin stack equipped with a $k$-shifted contact structure, where $k<0$. Lastly, we give several constructions of contact derived stacks using certain cotangent stacks and shifted prequantization structures., Comment: V02: 25 Pages; a new section on examples added; minor edits; Comments welcome!

We present a new catalog of Kepler planet candidates that prioritizes accuracy of planetary dispositions and properties over uniformity. This catalog contains 4376 transiting planet candidates, including 1791 residing within 709 multi-planet systems, and provides the best parameters available for a large sample of Kepler planet candidates. We also provide a second set of stellar and planetary properties for transiting candidates that are uniformly-derived for use in occurrence rates studies. Estimates of orbital periods have been improved, but as in previous catalogs, our tabulated values for period uncertainties do not fully account for transit timing variations (TTVs). We show that many planets are likely to have TTVs with long periodicities caused by various processes, including orbital precession, and that such TTVs imply that ephemerides of Kepler planets are not as accurate on multi-decadal timescales as predicted by the small formal errors (typically 1 part in $10^6$ and rarely $ > 10^{-5}$) in the planets' measured mean orbital periods during the Kepler epoch. Analysis of normalized transit durations implies that eccentricities of planets are anti-correlated with the number of companion transiting planets. Our primary catalog lists all known Kepler planet candidates that orbit and transit only one star; for completeness, we also provide an abbreviated listing of the properties of the two dozen non-transiting planets that have been identified around stars that host transiting planets discovered by Kepler., Comment: 62 pages, 32 figures, 5 tables. Published in The Planetary Science Journal, volume 5, #6 (2024). The first table appears in abbreviated form in the main manuscript and is provided in full as a separate file. A standard resolution version of Figure 32 appears within the manuscript; a high-resolution version of Figure 32 is also provided separate file

Random features have been introduced to scale up kernel methods via randomization techniques. In particular, random Fourier features and orthogonal random features were used to approximate the popular Gaussian kernel. The former is performed by a random Gaussian matrix and leads exactly to the Gaussian kernel after averaging. In this work, we analyze the bias and the variance of the kernel approximation based on orthogonal random features which makes use of Haar orthogonal matrices. We provide explicit expressions for these quantities using normalized Bessel functions and derive sharp exponential bounds supporting the view that orthogonal random features are more informative than random Fourier features.

Subject: In this article, convolutional networks of one, two, and three dimensions are compared with respect to their ability to distinguish between the drawing tests produced by Parkinson's disease patients and healthy control subjects. Motivation: The application of deep learning techniques for the analysis of drawing tests to support the diagnosis of Parkinson's disease has become a growing trend in the area of Artificial Intelligence. Method: The dynamic features of the handwriting signal are embedded in the static test data to generate one-dimensional time series, two-dimensional RGB images and three-dimensional voxelized point clouds, and then one-, two-, and three-dimensional CNN can be used to automatically extract features for effective diagnosis. Novelty: While there are many results that describe the application of two-dimensional convolutional models to the problem, to the best knowledge of the authors, there are no results based on the application of three-dimensional models and very few using one-dimensional models. Main result: The accuracy of the one-, two- and three-dimensional CNN models was 62.50%, 77.78% and 83.34% in the DraWritePD dataset (acquired by the authors) and 73.33%, 80.00% and 86.67% in the PaHaW dataset (well known from the literature), respectively. For these two data sets, the proposed three-dimensional convolutional classification method exhibits the best diagnostic performance.

The directed Oberwolfach problem OP$^\ast(m_1,\ldots,m_k)$ asks whether the complete symmetric digraph $K_n^\ast$, assuming $n=m_1+\ldots +m_k$, admits a decomposition into spanning subdigraphs, each a disjoint union of $k$ directed cycles of lengths $m_1,\ldots,m_k$. We hereby describe a method for constructing a solution to OP$^\ast(m_1,\ldots,m_k)$ given a solution to OP$^\ast(m_1,\ldots,m_\ell)$, for some $\ell

In this paper we study the Exponentiated Hypoexponential Distribution with different parameters. The distribution added a parameter to the n parameters of the Hypoexponenial distribution. We first derive a closed expression of the probability density function and the cumulative distribution function of Maximum Exponentiated Exponential distribution. These functions are used to determine an exact expression of the probability density function, the cumulative function, reliability function, and hazard function of our Exponentiated Hypoexponential Distribution. We discuss estimation of the parameters by maximum likelihood estimators. The distribution has been fitted to a real life data set and the fit has been found to be a serious competitor to the others., Comment: 26 pages, 5 figures