In this note, we revisit the envelope dimension reduction, which was first introduced for estimating a sufficient dimension reduction subspace without inverting the sample covariance. Motivated by the recent developments in envelope methods and algorithms, we refresh the envelope inverse regression as a flexible alternative to the existing inverse regression methods in dimension reduction. We discuss the versatility of the envelope approach and demonstrate the advantages of the envelope dimension reduction through simulation studies.

This paper investigates the adaptive tracking control problem for AutoRegressive Moving Average (ARMA) systems with quantized observations, explicitly focusing on reference signals composed of non-periodic sequences. We propose an adaptive tracking control scheme integrating an adaptive controller with a stochastic approximation-type estimation algorithm. Different from the control scheme for Finite Impulse Response (FIR) systems, the estimation part not only estimates the unknown system parameters but also the unknown system outputs. Next, based on the certainty equivalent principle, the adaptive controller is designed using the above two estimates instead of the actual parameters and system outputs. To tackle the inherent coupling between the two estimates, we introduce a novel approach that combines the Lyapunov function method with a backward-shifted polynomial method featuring time-varying coefficients. This approach assists in establishing the mean square convergence of the estimates with a convergence rate of $O\left(\frac{1}{k}\right)$ under suitable conditions of the step size coefficient. Additionally, this paper shows that the designed adaptive control law can achieve asymptotically optimal tracking of non-periodic reference signals in the mean square sense. Finally, a numerical simulation is presented to validate the theoretical results obtained in this paper.

The support vector machine, a widely used binary classification method, may expose sensitive information during training. To address this, we propose a personalized differential privacy method that extends differential privacy. Specifically, we introduce personalized differentially private support vector machines to meet different individuals' privacy requirements, using a reweighting strategy and the Laplace mechanism. Theoretical analysis demonstrates that our proposed methods simultaneously satisfy the requirements of personalized differential privacy and ensure model prediction accuracy at these privacy levels. Extensive experiments demonstrate that our proposed methods outperform the existing methods.

In this article, the authors explore the online updating estimation for general estimating equations (EEs) in heterogeneous streaming data settings. The framework is based on more conservative model assumptions, leading to more robust estimations and preventing misspecification. The authors establish the standard renewable estimation under blockwise heterogeneity assumption, which can correctly specify model in some sense. To mitigate heterogeneity and enhance estimation accuracy, the authors propose two novel online detection and fusion strategies, with corresponding algorithms provided. Theoretical properties of the proposed methods are demonstrated in the context of small block sizes. Extensive numerical experiments validate the theoretical findings. Real data analysis of the Ford Gobike docked bike-sharing dataset verifies the feasibility and robustness of the proposed methods.

Smartphone-based electrocardiograms (ECGs) are increasingly utilized for monitoring atrial fibrillation (AF) recurrence after catheter ablation (CA), referred to as smartphone AF burden (SMURDEN). The SMURDEN data often exhibit complex patterns of zero AF episodes, which may arise from either true AF-free status (structural zeros) or missed AF episodes due to intermittent monitoring (random zeros). Such a mixture of AF-free and at-risk patients can lead to zero-inflation in the data. We propose a novel zero-inflation test for binomial regression models to identify recurrence-free AF populations. Unlike traditional approaches requiring fully specified zero-inflated models, the proposed test utilizes a weighted average of the discrepancies between observed and expected zero proportions, with weights determined by binomial sizes. A closed-form test statistic is developed, and its asymptotic distribution is derived using estimating equations. Simulations demonstrate superior performance over existing methods, and real-world AF monitoring data validate the practical utility of our proposed test.

Cloud computing has become prevalent in the sharing of outsourced data due to its strong computing power and storage capacity. Ensuring data security is vitally important when sharing data in the cloud. Recently, numerous broadcast proxy re-encryption (BPRE) schemes have been designed to address the data security issues of such applications. However, there are no any BPRE schemes that have been designed to address the issue of updating the re-encryption key in a dynamic cloud environment. Therefore, we propose a lightweight dynamic broadcast proxy re-encryption scheme (LD-BPRE) to address this issue in dynamic settings where the data owner can dynamically change the set of data users and does not need to update the re-encryption key for the new set of data users. In other words, the proxy can reset a re-encryption ciphertext for the new set of data users using the original re-encryption key. This is significant in a dynamic cloud setting and provides convenience for cloud users. Our LD-BPRE is lightweight for users with low-power devices as most of the computing overhead is offloaded to the cloud. We formally define the LD-BPRE scheme and prove its security under a decision $n$-BDHE assumption in the standard model. Finally, extensive comparisons and experiments indicate that LD-BPRE is efficient and practical.

In recent years, artificial cilia have attracted widespread research interest due to their enormous application prospects in the fields of medicine and environmental therapy. Deformation is a key issue to consider in the design and preparation of artificial cilia, however the corresponding mathematical analysis is still lacking. This paper introduces a multi-agent model for the magnetic artificial cilium, where each agent denoting a bead is influenced by the external magnetic field and neighboring agents. Then, we provide the existence and uniqueness of the solution to our proposed model, and give a stability condition for avoiding magnetic chain breakage and collisions between adjacent magnetic beads. To our best knowledge, it is the first mathematical result on the stability of magnetic bead chain. Finally, simulations are conducted to verify our theoretical result.

In this paper, disturbance attenuation is considered for linear systems with partially modeled disturbance. The disturbance signal is composed of known signals and uncertain parameters that leads to some difficulties for solving the disturbance rejection problem. To overcome this issue, the original system is reformulated as a linear parameter-varying (LPV) system by absorbing the unknown parameters in disturbance. Then an adaptive state-disturbance-feedback controller relying on a dictionary of state-feedback gains and disturbance-feedback gains is designed to estimate the uncertain parameters in the LPV system. Moreover, the presence of multiple variables in the sufficient condition given to reject the external disturbance of the LPV system also brings challenges. To tackle this problem, the quadratic separation technology is applied into the sufficient condition, and the original unsolvable condition can be successfully transferred into a solvable one. Furthermore, by adding the known part of the disturbance signal into the feedback loop, more information of the whole system can be utilized. Meanwhile, the asymptotical stability of the closed-loop system can be achieved and the $H_\infty$ performance index of the closed-loop system is verified to be smaller. Numerical simulations are given to illustrate the merits of the proposed approach.

This paper studies the leader-following adaptive tracking control problem for multi-agent systems comprising a leader agent and $N$ follower agents with uncertain nonlinear dynamics. Specifically, a novel event-triggered communication based adaptive distributed observer is developed to enable each follower agent to estimate the leader's information. Then, new forms of adaptive control law and parameter update law are designed with the estimated leader's signals. The developed distributed adaptive control strategy has several characteristics: (i) With the introduced time-varying observer gain, the designed adaptive distributed observer eliminates the need for global graph information but ensures convergence of the estimates; (ii) By appropriately designing the event-triggered mechanism, the communication frequency among follower agents is reduced in the sense that the communication rate decays over time; (iii) The newly designed adaptive control law ensures a linear estimation error equation, facilitating the development of a stable parameter update law without requiring prior knowledge of uncertain system parameters. The stability of closed-loop system and leader-following asymptotical tracking are achieved. Simulation study demonstrates the theoretical results.

This paper is concerned with the $N$-player stochastic differential game of optimal switching. Both Nash equilibrium and social optima are studied and prove to have the same mean field limit. We prove the convergence of the value functions in the sense of viscosity solution, and show the limit of Nash equilibrium coincides with social optima as $N\to\infty$. In virtue of the weak formulation, the limit problem is characterized by a weak mean field equilibrium, which corresponds to approximate solutions of $N$-player game. Moreover, we provide example and simulation to illustrate the connection between weak mean field equilibrium and $N$-player game.

This paper aims to develop a unified Bayesian approach for clustered data analysis when observations are subject to missingness at random. We consider a general framework in which the parameters of interest are defined through estimating equations, and the probability of missingness follows a general parametric form. The generalized method of moments framework is employed to derive an optimal combination of inverse-probability-weighted estimating equations for the parameters of interest and score equations for propensity score. Using this framework, we develop a quasi-Bayesian analysis for clustered samples with missing values. A unified model selection approach is also proposed to compare models characterized by different moment conditions. We systematically evaluate the large-sample properties of the proposed quasi-posterior density with both fixed and shrinking priors and establish the selection consistency of the proposed model selection criterion. Our results are valid under very mild conditions and offer significant advantages for parameters defined through non-smooth estimating functions. Extensive numerical studies demonstrate that the proposed method performs exceptionally well in finite samples.

In the era of massive data, the study of distributed data is a significant topic. Model averaging can be effectively applied to distributed data by combining information from all machines. For linear models, the model averaging approach has been developed in the context of distributed data. However, further investigation is needed for more complex models. In this paper, we propose a distributed optimal model averaging approach based on multivariate additive models, which approximates unknown functions using B-splines allowing each machine to have a different smoothing degree. To utilize the information from the covariance matrix of dependent errors in multivariate multiple regressions, we use the Mahalanobis distance to construct a Mallows-type weight choice criterion. The criterion can be computed by transmitting information between the local machines and the center machine in two steps. We demonstrate the asymptotic optimality of the proposed model averaging estimator when the covariates are subject to uncertainty, and obtain the convergence rate of the weight vector to the theoretically optimal weights. Our results remain novel even for additive models with a single response variable. The numerical examples show that our proposed method yields good performance.

Synthesizing images or texts automatically becomes a useful research area in the artificial intelligence nowadays. Generative adversarial networks (GANs), proposed by Goodfellow et al in 2014, make this task to be done more efficiently by using deep neural networks (DNNs). We consider generating corresponding images from a single-sentence input text description using a GAN. Specifically, we analyze the GAN-CLS algorithm, which is a kind of advanced method of GAN proposed by Reed et al in 2016. In this paper we show the theoretical problem with this algorithm and correct it by modifying the objective function of the model. Experiments are performed on the Oxford-102 dataset and the CUB dataset to support our theoretical results. Since our modification can be seen as an idea which can be used to improve all such kind of GAN models, we try two models, GAN-CLS and AttnGAN_GPT. As a result, in both of the two models, our modified algorithm is more stable and can generate images which are more plausible than the original algorithm. Also, some of the generated images match the input texts better, and our modified algorithm has better performance on the quantitative indicators including FID and inception score. Finally, we propose some future application prospect of our modification idea, especially in the area of large language models.

In this paper, we investigate asymptotic stability of Markovian jump Boolean networks with random time delays. Initially, by utilizing the algebraic formulation of time-delay switched Boolean networks, the system is transformed into a high-dimensional Markovian jump Boolean networks, and an equivalent Markov chain is constructed. Then the addressed asymptotic stability problem is reformulated as the set stability problem. Through the state space decomposition of the Markov chain, the corresponding criteria for the asymptotic stability are derived. Furthermore, the asymptotic stability problem is solvable by using the breadth-first search algorithm. Finally, the validity of the obtained results is demonstrated through a biological example.

This paper investigates the optimal output regulation of switched Boolean control networks by using a dynamic programming method. The reference signal studied in this paper is generated by the output trajectory of a switched Boolean network. First, a per-step cost vector is proposed based on the largest control invariant set of the augmented auxiliary system. Then, a novel criterion is derived for determining the solvability of the output regulation. The proposed criterion transforms the solvability of output regulation of switched Boolean control networks into an optimization problem, providing a new perspective for addressing output regulation through optimal control. Based on this, an optimal state feedback control is proposed to enable the output trajectory of the original network to completely track the reference signal. An algorithm is presented to calculate the optimal feedback gain matrix and the optimal value for each state. Compared with existing results, the optimal state feedback control presented in this paper offers a generalized optimization principle and effectively reduces the computational complexity associated with designing state feedback control. Finally, an illustrative example is provided to validate the effectiveness of the results obtained.

The intra-industry risk spillover is crucial in transforming individual risk into systemic risk in both the financial and non-financial sectors. Using data from non-financial listed firms in China from 2007 to 2021, we explore the impact of rising economic policy uncertainty (EPU) on intra-industry risk spillovers, as well as the moderating effects of industry vulnerability and interconnectedness. Empirical results indicate that rising EPU enhances the intra-industry risk spillover effect in the non-financial sector. Industry vulnerability, including inadequate profitability, high leverage, low liquidity, or low financialization, can exacerbate intra-industry risk spillovers. Regarding the impact of interconnectedness, policy uncertainty shocks generate a more pronounced risk spillover effect in industries with low asset redeployability or weak competition. Our study highlights that, given the shock of rising EPU, intra-industry risk spillovers exist in China's non-financial sector, which are amplified through industry vulnerability and interconnectedness, necessitating close attention from investors.

This paper applies the Cheng projection to the support vector machine (SVM) in handling missing data. In the process of handling missing data, each sample with missing values is replaced by its Cheng projection in the original space. Additionally, two classification algorithms for handling linearly separable and nonlinearly separable datasets with missing data are presented. For linearly separable datasets with missing data, Cheng kernel function is introduced, and an SVM classification algorithm that improves the linear kernel function to the Cheng kernel function is proposed. For nonlinearly separable datasets, a generalized Gaussian Radial Basis Function kernel is introduced and an SVM classification algorithm for handling missing data is given. For both algorithms, two comparative experiments are conducted to demonstrate their effectiveness.

This article aims to establish a Bayesian Stackelberg game framework for analyzing the incomplete information demand response management with overlapping electricity sales areas, and further provide the corresponding equilibrium strategies. Considering that the satisfaction parameters of power users are private, a Bayesian game model is constructed among these power users, and a non-cooperative game model is established due to the price competition of microgrids. To ensure the sequential interactions of demand response, a Stackelberg game is developed by assuming that the microgrids are leaders and the power users are followers, and the Bayesian Nash equilibrium and Stackelberg equilibrium are proved to exist and are unique under some conditions. In addition, the Bayesian Nash equilibrium for power users is obtained using the fictitious play method in the symmetrical case, and an iterative algorithm is presented for determining the Stackelberg equilibrium. Finally, the numerical simulations are provided showing the effectiveness and convergence of the iterative algorithm, which indicates that our approach can enhance profits for microgrids while ensuring power supply and demand balance.

In this paper, we address the problem of almost sure polynomial stabilization for a class of highly nonlinear stochastic systems via sampled-data feedback. The considered systems fall within a general framework that includes two key features: (a) continuous-time irreducible Markov chain- we introduce a continuous-time irreducible Markov chain to describe systems that can undergo sudden alterations in their parameters and structures. This flexibility allows us to model real-world scenarios more accurately; (b) diffusion and drift coefficients with polynomial growth - unlike existing literature that primarily focuses on systems with bounded delays, we investigate the stabilization conditions for highly nonlinear stochastic systems with pantograph delay, an unbounded delay. Specifically, we analyze systems where the diffusion and drift coefficients satisfy a polynomial growth condition. To achieve our goal, we employ M-matrix theory and Lyapunov functions as basic tools. Our main results establish that the system can attain almost sure polynomial stabilization through a subtly and innovatively designed sampled-data feedback. We validate our theoretical findings with numerical simulations, demonstrating the effectiveness of our approach. This work contributes to the understanding of stabilization in highly nonlinear stochastic systems, particularly those with unbounded delays, and broadens the practical applicability of stochastic modeling.

This paper investigates the problem of event-triggered disturbance attenuation and fault-tolerant constrained consensus in multi-agent systems with a variable number of agents. First, an event-triggered design combining a disturbance observer and a fault-tolerant controller is proposed, which reduces network bandwidth usage while accurately estimating and compensating for disturbances and partial actuator failures, thereby improving system reliability. Next, a time-varying impulsive Lyapunov function related to the number of agents is introduced, and the communication matrix changes—resulting from variations in the communication structure—are transformed into additive uncertainties, thus addressing topology switching issues arising from changes in the number of agents. To overcome the limitation of traditional $H_\infty$ control, which assumes zero initial conditions, a performance index dependent on the initial state is proposed, along with a novel event-triggered disturbance-rejection fault-tolerant control protocol. Sufficient conditions ensuring the consistency of disturbance attenuation and fault-tolerance constraints are then provided. Numerical simulations demonstrate the effectiveness of the proposed method.

Distributed learning is a well-established method for estimation tasks over extensively distributed datasets. However, non-randomly stored data can introduce bias into local parameter estimates, leading to significant performance degradation in classical distributed algorithms. In this paper, we propose a novel Distributed Quasi-Newton Pilot (DQNP) method for distributed learning with non-randomly distributed data. The proposed approach accommodates both randomly and non-randomly distributed data settings and imposes no constraints on the uniformity of local sample sizes. Additionally, it avoids the need to transfer the Hessian matrix or compute its inversion, thereby greatly reducing computational and communication complexity. We theoretically demonstrate that the resulting estimator achieves statistical efficiency under mild conditions. Extensive numerical experiments on synthetic and real-world data validate the theoretical findings and illustrate the effectiveness of the proposed method.

Classical linear discriminant analysis (LDA) (Fisher, 1936) implicitly assumes the classification boundary depends on only one linear combination of the predictors. This restriction can lead to poor classification in applications where the decision boundary depends on multiple linear combinations of the predictors. To overcome this challenge, we first project the predictors onto an envelope central space and then perform LDA based on the sufficient predictor. The performance of the proposed method in improving classification accuracy is demonstrated in both synthetic data and real applications.

The composite anti-disturbance $ H_{\infty} $ control problem for switched descriptor systems with multiple disturbances is investigated in this paper. One disturbance is modeled by an exogenous system with perturbations, while the other is norm-bounded. Firstly, based on generalized Sylvester equations, a novel reduced-order disturbance observer is proposed to estimate the unmeasurable state and modeling disturbance. Meanwhile, a composite anti-disturbance controller, consisting of a disturbance compensator and an estimated state-based controller, is developed based on the outputs of the proposed reduced-order disturbance observer. Then, under multi-Lyapunov functions and mode-dependent average dwell time, sufficient conditions are presented to ensure that the closed-loop switched descriptor systems are regular, impulse-free, globally uniformly exponentially stable with a weighted $H_{\infty}$ performance. Further, the design method of reduced-order disturbance observer and anti-disturbance controller is proposed by an algorithm. Finally, the superiority and practicality of the developed results are demonstrated through a numerical example and a Boost converter circuit in wind power system.

In this paper, a finite-time adaptive neural networks relative event-triggered command-filtered tracking control for the stochastic nonlinear systems with the signals constraint is presented. Firstly, the unknown nonlinear functions are approximated using neural networks radial basis function, and the barrier Lyapunov function is utilized to ensure the output signal constraint within a predefined range. Secondly, a finite-time command-filtered approach is adopted in the controller design to achieve finite-time stability. Thirdly, a relative threshold event-triggered mechanism is introduced to reduce the communication costs, and the threshold parameters are dynamically adjusted in response to the actual tracking performance, thereby enhancing the adaptability and efficiency of the control strategy. Finally, simulation results demonstrate the effectiveness of the proposed control method.

This paper delves into the determination of the optimal safety loading that an insurer ought to incorporate into reinsurance pricing. We base our analysis on the assumption that the insurer utilizes a specific form of the loss function and is confronted with losses that adhere to a zero - corrected exponential distribution. This assumption is steered by the expected premium principle. By minimizing the Value at Risk (VaR) of the insurer's liabilities and the Conditional Tail Expectation (CTE) risk measures, our research investigates the optimal safety loading principle for reinsurance premiums. This approach aims to curtail the potential losses that are associated with the insurer's premiums. Our research outcomes reveal that the results obtained from the VaR and CTE risk measures bear substantial significance in the real - world insurance and reinsurance markets.

Monotonic optimization is a special class of global optimization with applications cross fields. It addresses problems in which the objective and constraint functions are increasing w.r.t. each of the variables. In this work, we extend to the case where the objective and constraint functions are monotonic. We present a general framework to address such problems, and especially propose a complete algorithm that is guaranteed to terminate in finitely many steps for problems in a special form. Different from traditional optimization algorithms based on gradient descent, our algorithm does not require closed-form expressions of the functions. As an important application, the functions involved in the parameter optimization problem of LWE-based encryption scheme exhibit monotonicity w.r.t. each of the variables (but may not be increasing), and certain functions involved have no closed-form expression. Inspired by the idea of mathematics mechanization, we formalize practical problems into mathematical models and provide a framework for developing automatic and systematic approaches to tackle the parameter optimization problems in lattice-based cryptography. As an illustrative example, we consider the parameter optimization of BGV scheme in the context of minimizing communication overhead, without considering homomorphic operations, and provide optimal parameters for it under specified security levels and correctness probabilities.

This paper examines whether the parametric regression model is correctly specified for both source and target data and whether the regression pattern in the source domain aligns with that of the target domain. This evaluation is a critical prerequisite for applying model-based transfer learning methods under covariate shift assumptions. Traditional regression model checks and two-sample regression tests are insufficient to address this issue. To overcome these limitations, we propose a novel adaptive-to-regression test statistic that is asymptotically distribution-free. Under the null hypothesis, the test follows a chi-square weak limit, preserving the significance level and enabling critical value determination without resampling techniques. Additionally, we systematically analyze the test's power performance, highlighting its sensitivity to different sub-local alternatives that deviate from the null hypothesis. Numerical studies, including simulations, assess finite-sample performance, and a real-world data example is provided for illustration.

This study revisits Hotelling's $T^2$ (HT) tests for one- and two-sample mean problems, introduces a family of scaled Hotelling's $T^2$-type tests and develops two omnibus tests, termed Omnibus Hotelling's $T^2$ (HT-O), for both cases. Furthermore, we analyze the powers of the HT-O tests through the asymptotic null distributions of the scaled Hotelling's $T^2$-type tests. Extensive simulation results demonstrate that the proposed HT-O tests effectively control Type I error and maintain high power under complex correlation structures, outperforming the classical HT tests in various scenarios. Applications to anti-depressant imipramine efficacy and $\alpha$-amylase activity further highlight the superior performance and practical utility of the HT-O tests.

Quantile regression (QR) has become an important tool to measure dependence of response variable's quantiles on a number of predictors for heterogeneous data, especially heavy-tailed data and outliers. However, it is quite challenging to make statistical inference on distributed high-dimensional QR with missing data due to the distributed nature, sparsity and missingness of data and non-differentiable quantile loss function. To overcome the challenge, this paper develops a communication-efficient method to select variables and estimate parameters by utilizing a smooth function to approximate the non-differentiable quantile loss function and incorporating the idea of the inverse probability weighting and the penalty function. The proposed approach has three merits. First, it is both computationally and communicationally efficient because only the first- and second-order information of the approximate objective function are communicated at each iteration. Second, the proposed estimators possess the oracle property after a limited number of iterations without constraint on the number of machines. Third, the proposed method simultaneously selects variables and estimates parameters within a distributed framework, ensuring robustness to the specified response probability or propensity score function of the missing data mechanism. Simulation studies and a real example are used to illustrate the effectiveness of the proposed methodologies.

This article investigates the fuzzy adaptive resilient leader-following consensus control problem for a class of nonlinear heterogeneous multi-agent systems (HMASs) under denial-of-service (DoS) attacks. Since the considered leader's system contains unknown nonlinear dynamics and HMASs are subject to DoS attacks, the leader's states and output are inaccessible to followers, a stable distributed estimator is developed to estimate them. To solve the virtual controller non-differentiable problem in backstepping control design technique caused by DoS attacks, the command-filter is introduced into the backstepping control design process, and formulate a resilient leader-following consensus controller. It is proven that the proposed control scheme can guarantee that the controlled HMASs are stable, and the consensus errors converge to a small neighborhood of zero. Finally, a simulation example on unmanned aerial vehicle-unmanned ground vehicles (UAV-UGVs) is provided to verify the feasibility of the proposed control scheme.

In this paper, the asynchronously practical control is studied for discrete time switched systems with singular perturbations. Firstly, a novel Lyapunov function is constructed including the singular perturbation parameter and quasi-time parameter matrices. And then, quasi-time dependent criteria are achieved to study the practical stability and asynchronous stabilization; an allowable upper bound is expected to be obtained for the singular perturbation parameters; the asynchronous sampling controller is designed with quasi-time gains, and the application range of this controller is further widened via some constraints relaxed. At last, two simulation examples are utilized to illustrate that the proposed results are less conservative and effective.

This paper first proposes a distributed continuous-time Newton-Raphson algorithm for heterogeneous linear multi-agent systems over unbalanced digraphs. Then this approach extends to cases where the local cost functions and Hessian matrices are unknown. While local exponential stability of the inverse Hessian matrix estimator has been established for single-agent systems, this paper proves local exponential stability in multi-agent systems, ensuring the stability of the proposed distributed Newton-Raphson extremum seeking algorithm. A numerical example demonstrates the effectiveness of the proposed algorithms.

In this paper, we stumble upon that the normal ordering expansion for $\left(x\frac{\mathrm{d}}{\mathrm{d}x}\right)^n$ is equivalent to the expansion of $(bD_G)^n$, where $G$ is the context-free grammar defined by $G=\{a\rightarrow a, b\rightarrow 1\}$. Motivated by this fact, we introduce the definition of grammatical basis. We then study several grammatical bases generated by $G=\{a\rightarrow 1, b\rightarrow 1\}$. Using grammatical bases, we give a classification of grammars. In particular, we provide new grammatical descriptions for Ward numbers, Hermite polynomials, Bessel polynomials, Chebyshev polynomials and logarithmic polynomials arising from an integral. We end this paper by giving some applications of grammatical bases. One can see that if two or more polynomials share a grammatical basis, then they share the same coefficients, and it might be helpful for the detection of intrinsic relationship among superficially different structures.

In this paper, we are concerned with the problem of achieving Nash equilibrium in noncooperative games over networks. We propose two types of distributed projected gradient dynamics with accelerated convergence rates. The first type is a variant of the commonly-known consensus-based gradient dynamics, where the consensual terms for determining the actions of each player are discarded to accelerate the learning process. The second type is formulated by introducing the Nesterov's accelerated method into the distributed projected gradient dynamics. We prove convergence of both algorithms with at least linear rates under the common assumption of Lipschitz continuity and strongly monotonicity. Simulation examples are presented to validate the outperformance of our proposed algorithms over the well-known consensus-based approach and augmented game based approach. It is shown that the required number of iterations to reach the Nash equilibrium is greatly reduced in our proposed algorithms. These results could be helpful to address the issue of long convergence time in partial-information Nash equilibrium seeking algorithms.

This paper studies the adaptive mean-square output consensus problem of heterogeneous multi-agent systems with different multiplicative noises under a directed graph. Specifically, due to the presence of packet losses, the optimal estimator is first derived for the continuous-time stochastic system through discretization to estimate each agent's state. Based on this, we design an edge-based distributed adaptive control protocol that is independent of global information of the communication topology. With the aid of the distributed feedforward control approach and stochastic stability theory, the sufficient condition for achieving mean-square output consensus is derived. Moreover, the convergence of the optimal estimator is analyzed through rigorous mathematical derivation. Finally, a numerical simulation verifies the validity of the obtained results.

This paper proposes a factor model for interval-valued panel data. We exploit that the first $r$ largest eigenvalues of the sample covariance matrix divided by $N$ (i.e., $\frac{\langle s_Y^{\prime},s_Y\rangle_K}{NT}$) of interval-valued response variables are $O_p(1)$, while the rest are $o_p(1)$. Then the eigenvalue ratio-type estimators of the number of factors are proposed. Under certain conditions, the proposed estimators are all proven to be consistent. Moreover, the estimators of interval-valued factors and the loadings can be obtained by the principal components method. Monte Carlo simulation studies show that the proposed estimators have the desired finite sample properties. A real example is analysed for illustrations.

In this paper, we address the bounded leader-following consensus problem for linear multi-agent systems connected via undirected graphs, specifically in the presence of non-consistent time-varying communication delays. The periodic event-triggered scheme is proposed to mitigate the adverse effects of these delays, which ensures discrete data transmission only occurs at specific event instants. By leveraging the nature of periodic event-triggered schemes, Zeno-freeness can be guaranteed by discretely event-checking. Additionally, the appropriate design of the threshold range prevents the event-triggered consensus from degrading into sampled-data consensus. The numerical simulation is presented in the end to show the effectiveness of the provided approach.

This article explores the dynamic event-triggered (DET) ${H_\infty }$ load frequency control (LFC) for networked power systems (NPSs) subject to deception attacks. Firstly, a novel DET mechanism is proposed, which aims to improve the system control performance under deception attacks and save more network resources effectively. Compared with the existing DET mechanisms, the proposed DET mechanism involves an adaptive rule, which can be utilized to dynamically adjust the event-triggered threshold based on the relative rate of change and absolute difference of system state and the frequency of deception attacks. Then, considering the complexity of the actual power systems, a new DET-based LFC stochastic model is formulated, which integrates actuator failure, network-induced delay and deception attacks. Subsequently, by using Lyapunov theory, the sufficient conditions guaranteeing the asymptotic mean-square stability of the NPSs are derived. Finally, some simulation results are presented to validate the superiority of the designed approach.

The broad goal of the research surveyed in this article is to develop methods for understanding the aggregate behavior of interconnected dynamical systems, as found in mathematical physics, neuroscience, economics, power systems and neural networks. Questions concern prediction of emergent (often unanticipated) phenomena, methods to formulate distributed control schemes to influence this behavior, and these topics prompt many other questions in the domain of learning. The area of mean field games, pioneered by Peter Caines, are well suited to addressing these topics. The approach is surveyed in the present paper within the context of controlled coupled oscillators.

This paper addresses boundary control to input-to-state stabilization for fractional convection-diffusion-reaction (FCDR) systems governed by coupled time fractional partial differential equations (TFPDEs) under matched and unmatched disturbances over actuator/sensor networks using output feedback, fractional sliding mode (FSM) algorithm and sampled-in-space sensing. Here it is assumed that sensors provide discrete in space measurements, i.e., spatially averaged measurements (SAMs), and a limited number of sensors are allocated in a spatial domain. A sampled-data observation problem is first in investigation, which contains to design a FSM observer against boundary disturbances and to prove input-to-state stability (ISS) of the proposed observer. Using this observer and backstepping approach, we develop an output feedback FSM controller and establish the reaching condition to FSM surface. Using the fractional Lyapunov method, ISS of the closed-loop dynamics is achieved. Theoretical results are verified by numerical simulations.