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We introduce the quaternionic Mahler measure for non-commutative polynomials, extending the classical complex Mahler measure. We establish the existence of quaternionic Mahler measure for slice regular polynomials in one and two variables. We study the quaternionic Mahler measure for real and slice regular polynomials, and consider the associated Lehmer problem. Various formulas of quaternionic Mahler measures are proved.

We study the conditional upper bounds and extreme values of derivatives of the Riemann zeta function and Dirichlet $L$-functions near the 1-line. Let $\ell$ be a fixed natural number. We show that, if $|\sigma-1|\ll1/\log_2t$, then $|\zeta^{(\ell)}(\sigma+ it)|$ has the same maximal order (up to the leading coefficients) as $|\zeta^{(\ell)}(1+ it)|$ when $t\to\infty$. The range $1-\sigma\ll1/\log_2t$ is wide enough, since we also show that $(1-\sigma) \log_2t \to \infty\; (t \to \infty)$ implies $\limsup_{t\to\infty}|\zeta^{(\ell)}(\sigma+ it)| / (\log_2t)^{\ell+1} = \infty$. Similar results can be obtained for Dirichlet $L$-functions $L^{(\ell)}(\sigma,\chi)$ with $\chi\pmod q$., Comment: 18 pages

Atrial fibrillation (AF) is a prevalent cardiac arrhythmia associated with significant health ramifications, including an elevated susceptibility to ischemic stroke, heart disease, and heightened mortality. Photoplethysmography (PPG) has emerged as a promising technology for continuous AF monitoring for its cost-effectiveness and widespread integration into wearable devices. Our team previously conducted an exhaustive review on PPG-based AF detection before June 2019. However, since then, more advanced technologies have emerged in this field. This paper offers a comprehensive review of the latest advancements in PPG-based AF detection, utilizing digital health and artificial intelligence (AI) solutions, within the timeframe spanning from July 2019 to December 2022. Through extensive exploration of scientific databases, we have identified 59 pertinent studies. Our comprehensive review encompasses an in-depth assessment of the statistical methodologies, traditional machine learning techniques, and deep learning approaches employed in these studies. In addition, we address the challenges encountered in the domain of PPG-based AF detection. Furthermore, we maintain a dedicated website to curate the latest research in this area, with regular updates on a regular basis.

In this article, we investigate the behaviour of values of zeta sums $\sum_{n\le x}n^{it}$ when $t$ is large. We show some asymptotic behaviour and Omega results of zeta sums, which are analogous to previous results of large character sums $\sum_{n\le x}\chi(n)$., Comment: 11 pagess

Federated Learning (FL) has been an area of active research in recent years. There have been numerous studies in FL to make it more successful in the presence of data heterogeneity. However, despite the existence of many publications, the state of progress in the field is unknown. Many of the works use inconsistent experimental settings and there are no comprehensive studies on the effect of FL-specific experimental variables on the results and practical insights for a more comparable and consistent FL experimental setup. Furthermore, the existence of several benchmarks and confounding variables has further complicated the issue of inconsistency and ambiguity. In this work, we present the first comprehensive study on the effect of FL-specific experimental variables in relation to each other and performance results, bringing several insights and recommendations for designing a meaningful and well-incentivized FL experimental setup. We further aid the community by releasing FedZoo-Bench, an open-source library based on PyTorch with pre-implementation of 22 state-of-the-art methods, and a broad set of standardized and customizable features available at https://github.com/MMorafah/FedZoo-Bench. We also provide a comprehensive comparison of several state-of-the-art (SOTA) methods to better understand the current state of the field and existing limitations.

In this article, we study the distribution of large quadratic character sums. Based on the recent work of Lamzouri~\cite{La2022}, we obtain the structure results of quadratic characters with large character sums., Comment: 15 pages

This paper investigates the safety guaranteed problem in spacecraft inspection missions, considering multiple position obstacles and logical attitude forbidden zones. In order to address this issue, we propose a control strategy based on control barrier functions, summarized as "safety check on kinematics" and "velocity tracking on dynamics" approach. The proposed approach employs control barrier functions to describe the obstacles and to generate safe velocities via the solution of a quadratic programming problem. Subsequently, we design a proportional-like controller based on the generated velocity, which, despite its simplicity, can ensure safety even in the presence of velocity tracking errors. The stability and safety of the system are rigorously analyzed in this paper. Furthermore, to account for model uncertainties and external disturbances, we incorporate an immersion and invariance-based disturbance observer in our design. Finally, numerical simulations are performed to demonstrate the effectiveness of the proposed control strategy., Comment: 22 pages, 9 figures, submitted to JGCD

This paper investigates the boresight alignment control problem under safety constraints and performance requirements, involving pointing-forbidden constraint, attitude angular velocity limitation, and pointing accuracy requirement. Meanwhile, the parameter uncertainty issue is taken into account simultaneously. To address this problem, we propose a modified composite framework integrating the Artificial Potential Field (APF) methodology and the Prescribed Performance Control (PPC) scheme. The APF scheme ensures safety, while the PPC scheme is employed to realize an accuracy-guaranteed control. A Switched Prescribed Performance Function (SPPF) is proposed to facilitate the integration, which monitors various constraints and further establishes compatibility between safety and performance concerns by leveraging a special PPC freezing mechanism. To further address the parameter uncertainty, we introduce the Immersion-and-Invariance (I\&I) adaptive control technique to derive an adaptive APF-PPC composite controller, further guaranteeing the closed-loop system's asymptotic convergence. Finally, numerical simulations are carried out to validate the effectiveness of the proposed scheme., Comment: Submitted to T-AES

This paper focuses on the spacecraft attitude control problem with intermittent actuator activation, taking into account the attitude rotation rate limitation and input saturation issue simultaneously. To address this problem, we first propose a composite event-trigger mechanism, which composed of two state-dependent trigger that governing the activation and deactivation of actuators. Subsequently, by introducing the cascaded decomposition of Backstepping control philosophy, the designed trigger mechanism is then applied to the decomposed dynamical subsystem, providing a layered intermittent stabilization strategy. Further, the basic intermittent attitude controller is extended to a "constrained version" by introducing a strictly bounded virtual control law and an input saturation compensation auxiliary system. By analyzing the local boundedness of the system on each inter-event time interval, a uniformly, strictly decreasing upper boundary of the lumped system is further characterized, thereby completing the proof of the system's uniformly ultimately boundedness (UUB). Finally, numerical simulation results are illustrated to demonstrate the effectiveness of the proposed scheme., Comment: Submitted to T-AES

This paper focuses on the problem of spacecraft attitude control in the presence of time-varying parameter uncertainties and multiple constraints, accounting for angular velocity limitation, performance requirements, and input saturation. To tackle this problem, we propose a modified framework called Compatible Performance Control (CPC), which integrates the Prescribed Performance Control (PPC) scheme with a contradiction detection and alleviation strategy. Firstly, by introducing the Zeroing Barrier Function (ZBF) concept, we propose a detection strategy to yield judgment on the compatibility between the angular velocity constraint and the performance envelope constraint. Subsequently, we propose a projection operator-governed dynamical system with a varying upper bound to generate an appropriate bounded performance envelope-modification signal if a contradiction exists, thereby alleviating the contradiction and promoting compatibility within the system. Next, a dynamical filter technique is introduced to construct a bounded reference velocity signal to address the angular velocity limitation. Furthermore, we employ a time-varying gain technique to address the challenge posed by time-varying parameter uncertainties, further developing an adaptive strategy that exhibits robustness on disturbance rejection. By utilizing the proposed CPC scheme and time-varying gain adaptive strategy, we construct an adaptive CPC controller, which guarantees the ultimate boundedness of the system, and all constraints are satisfied simultaneously during the whole control process. Finally, numerical simulation results are presented to show the effectiveness of the proposed framework., Comment: Submitted to T-AES

Existing normal estimation methods for point clouds are often less robust to severe noise and complex geometric structures. Also, they usually ignore the contributions of different neighbouring points during normal estimation, which leads to less accurate results. In this paper, we introduce a weighted normal estimation method for 3D point cloud data. We innovate in two key points: 1) we develop a novel weighted normal regression technique that predicts point-wise weights from local point patches and use them for robust, feature-preserving normal regression; 2) we propose to conduct contrastive learning between point patches and the corresponding ground-truth normals of the patches' central points as a pre-training process to facilitate normal regression. Comprehensive experiments demonstrate that our method can robustly handle noisy and complex point clouds, achieving state-of-the-art performance on both synthetic and real-world datasets., Comment: Accepted by ICME 2023

In this paper, we investigate the order types of reflection orders on irreducible affine Weyl groups. We show that they are intimately related to the Catalan combinatorics. We explicitly describe all of those order types and show that these order types can be parameterized by deformed Dyck words, which can be enumerated by Catalan numbers., Comment: 21 pages; add some explanatory remarks and pictures to improve readability; add an appendix on the enumeration of deformed Dyck words

An oft-cited open problem of federated learning is the existence of data heterogeneity at the clients. One pathway to understanding the drastic accuracy drop in federated learning is by scrutinizing the behavior of the clients' deep models on data with different levels of "difficulty", which has been left unaddressed. In this paper, we investigate a different and rarely studied dimension of FL: ordered learning. Specifically, we aim to investigate how ordered learning principles can contribute to alleviating the heterogeneity effects in FL. We present theoretical analysis and conduct extensive empirical studies on the efficacy of orderings spanning three kinds of learning: curriculum, anti-curriculum, and random curriculum. We find that curriculum learning largely alleviates non-IIDness. Interestingly, the more disparate the data distributions across clients the more they benefit from ordered learning. We provide analysis explaining this phenomenon, specifically indicating how curriculum training appears to make the objective landscape progressively less convex, suggesting fast converging iterations at the beginning of the training procedure. We derive quantitative results of convergence for both convex and nonconvex objectives by modeling the curriculum training on federated devices as local SGD with locally biased stochastic gradients. Also, inspired by ordered learning, we propose a novel client selection technique that benefits from the real-world disparity in the clients. Our proposed approach to client selection has a synergic effect when applied together with ordered learning in FL.

This paper focuses on the issue of how to realize spacecraft attitude control with guaranteed performance while conspicuously reducing the actuator acting frequency simultaneously. The prescribed performance control (PPC) scheme is often employed for the control with guaranteed performance. However, conventional PPC controllers are designed from the perspective of continuous system, which contradicts the "discrete" control logic in actual spacecraft control system, and such a problem limited the application value of PPC scheme in actual applications. In order to significantly lower the actuator acting frequency while still maintaining the desired performance, a composite event-trigger mechanism is proposed for this issue, turning off the actuator and eliminating unnecessary control output under appropriate conditions. Further, the proposed composite event-trigger mechanism is combined with a prescribed performance control scheme, constructing a complete controller structure for solving the presented issue. Meanwhile, a special performance function is designed to provide mild transition process. Based on the proposed scheme, a specific backstepping controller is further developed as a validation. Finally, numerical simulation results are presented to validate the effectiveness of the proposed scheme., Comment: 13 pages

Meta-learning often referred to as learning-to-learn is a promising notion raised to mimic human learning by exploiting the knowledge of prior tasks but being able to adapt quickly to novel tasks. A plethora of models has emerged in this context and improved the learning efficiency, robustness, etc. The question that arises here is can we emulate other aspects of human learning and incorporate them into the existing meta learning algorithms? Inspired by the widely recognized finding in neuroscience that distinct parts of the brain are highly specialized for different types of tasks, we aim to improve the model performance of the current meta learning algorithms by selectively using only parts of the model conditioned on the input tasks. In this work, we describe an approach that investigates task-dependent dynamic neuron selection in deep convolutional neural networks (CNNs) by leveraging the scaling factor in the batch normalization (BN) layer associated with each convolutional layer. The problem is intriguing because the idea of helping different parts of the model to learn from different types of tasks may help us train better filters in CNNs, and improve the model generalization performance. We find that the proposed approach, neural routing in meta learning (NRML), outperforms one of the well-known existing meta learning baselines on few-shot classification tasks on the most widely used benchmark datasets.

In this article, we study the distribution of values of Dirichlet $L$-functions, the distribution of values of the random models for Dirichlet $L$-functions, and the discrepancy between these two kinds of distributions. For each question, we consider the cases of $\frac12<\RE s<1$ and $\RE s=1$ separately., Comment: 24 pages

Clustered federated learning (FL) has been shown to produce promising results by grouping clients into clusters. This is especially effective in scenarios where separate groups of clients have significant differences in the distributions of their local data. Existing clustered FL algorithms are essentially trying to group together clients with similar distributions so that clients in the same cluster can leverage each other's data to better perform federated learning. However, prior clustered FL algorithms attempt to learn these distribution similarities indirectly during training, which can be quite time consuming as many rounds of federated learning may be required until the formation of clusters is stabilized. In this paper, we propose a new approach to federated learning that directly aims to efficiently identify distribution similarities among clients by analyzing the principal angles between the client data subspaces. Each client applies a truncated singular value decomposition (SVD) step on its local data in a single-shot manner to derive a small set of principal vectors, which provides a signature that succinctly captures the main characteristics of the underlying distribution. This small set of principal vectors is provided to the server so that the server can directly identify distribution similarities among the clients to form clusters. This is achieved by comparing the similarities of the principal angles between the client data subspaces spanned by those principal vectors. The approach provides a simple, yet effective clustered FL framework that addresses a broad range of data heterogeneity issues beyond simpler forms of Non-IIDness like label skews. Our clustered FL approach also enables convergence guarantees for non-convex objectives. Our code is available at https://github.com/MMorafah/PACFL.

This paper focuses on the attitude pointing control problem under pointing-forbidden constraints and performance constraints. The spacecraft is expected to align its sensor's boresight to a desired direction, while the terminal control accuracy and the attitude adjustment rapidity should also be guaranteed simultaneously. To resolve this problem, a switching controller structure is proposed in this paper based on the reduced-attitude representation, fusing the artificial potential field (APF) methodology and the Prescribed Performance Control (PPC) scheme together. Firstly, a novel artificial potential field is presented, and a particular function is designed for the mollification of the switching process, aiming at providing a smooth transition for the system status. Subsequently, we propose a special performance function, which can freeze the PPC part when necessary. In this way, the intrinsic contradictory between the fast attitude maneuver and forbidden direction avoidance is tackled Further, an asynchronous switching strategy is designed, guarantees the system's stability. Based on these proposed issues, a switching backstepping controller is developed, and a tracking differentiator(TD) is employed to generate a smooth approximation of differential signals. Numerical simulation results are illustrated to show the effectiveness of the proposed scheme., Comment: 12pages, 15 figures

This paper considers the prescribed performance control (PPC) of spacecraft attitude tracking under multiple physical constraints, focusing on the robust issues. A novel Barrier Lyapunov function is proposed to realize the guaranteed-performance control under angular velocity constraint without singularity. Additionally, an adaptive strategy for the performance function is presented to soften the constraint and quickly re-stabilize the system after severe disturbances occur, providing strong robustness. Further, an auxiliary system is designed to handle the input saturation issue, incorporating the actuator limitation into the system. Based on the proposed structure, a backstepping controller is developed accordingly using a double-layer PPC framework. Numerical simulation results are presented to validate the proposed controller framework's efficiency and robustness.8 pa, Comment: 8 pages, 11 figures

Classical federated learning approaches yield significant performance degradation in the presence of Non-IID data distributions of participants. When the distribution of each local dataset is highly different from the global one, the local objective of each client will be inconsistent with the global optima which incur a drift in the local updates. This phenomenon highly impacts the performance of clients. This is while the primary incentive for clients to participate in federated learning is to obtain better personalized models. To address the above-mentioned issue, we present a new algorithm, FLIS, which groups the clients population in clusters with jointly trainable data distributions by leveraging the inference similarity of clients' models. This framework captures settings where different groups of users have their own objectives (learning tasks) but by aggregating their data with others in the same cluster (same learning task) to perform more efficient and personalized federated learning. We present experimental results to demonstrate the benefits of FLIS over the state-of-the-art benchmarks on CIFAR-100/10, SVHN, and FMNIST datasets. Our code is available at https://github.com/MMorafah/FLIS.

The attitude tracking problem with preassigned performance requirements has earned tremendous interest in recent years, and the Prescribed Performance Control (PPC) scheme is often adopted to tackle this problem. Nevertheless, traditional PPC schemes have inherent problems, which the solution still lacks, such as the singularity problem when the state constraint is violated and the potential over-control problem when the state trajectory approaches the constraint boundary. This paper proposes a Singularity-Avoidance Prescribed Performance Control scheme (SAPPC) to deal with these problems. A novel shear mapping-based error transformation is proposed to provide a globally non-singular error transformation procedure, while a time-varying constraint boundary is employed to exert appropriate constraint strength at different control stages, alleviating the potential instability caused by the over-control problem. Besides, a novel piece-wise reference performance function (RPF) is constructed to provide a relevant reference trajectory for the state responding signals, allowing precise control of the system's responding behavior. Based on the proposed SAPPC scheme, a backstepping controller is developed, with the predefined-time stability technique and the dynamic surface control technique employed to enhance the controller's robustness and performance. Finally, theoretical analysis and numerical simulation results are presented to validate the proposed control scheme's effectiveness and robustness., Comment: 18 pages,18 figures

We investigate the meromorphic quasi-modular forms and their $L$-functions. We study the space of meromorphic quasi-modular forms. Then we define their $L$-functions by using the technique of regularized integral. Moreover, we give an explicit formula for the $L$-functions. As an application, we obtain some vanishing results of special $L$-values of meromorphic quasi-modular forms.

In this note, we show that there is a one-to-one correspondence between the set of biclosed sets in an irreducible root system of type $A_n$ and the set of quasitrivial semigroup structures on a set with $n+1$ elements., Comment: fix a typo in 2.2