This paper explores the application of noncooperative game theory together with the concept of Nash equilibrium to the investigation of some basic problems on multi-scale structure, especially the meso-scale structure in the multi-phase complex systems in chemical engineering. The basis of this work is the energy-minimization-multi-scale (EMMS) model proposed by Li and Kwauk (1994) and Li, et al. (2013) which identifies the multi-scale structure as a result of ‘compromise-in-competition between dominant mechanisms’ and tries to solve a multi-objective optimization problem. However, the existing methods often integrate it into a problem of single objective optimization, which does not clearly reflect the ‘compromise-in-competition’ mechanism and causes heavy computation burden as well as uncertainty in choosing suitable weighting factors. This paper will formulate the compromise in competition mechanism in EMMS model as a noncooperative game with constraints, and will describe the desired stable system state as a generalized Nash equilibrium. Then the authors will investigate the game theoretical approach for two typical systems in chemical engineering, the gas-solid fluidization (GSF) system and turbulent flow in pipe. Two different cases for generalized Nash equilibrium in such systems will be well defined and distinguished. The generalize Nash equilibrium will be solved accurately for the GSF system and a feasible method will be given for turbulent flow in pipe. These results coincide with the existing computational results and show the feasibility of this approach, which overcomes the disadvantages of the existing methods and provides deep insight into the mechanisms of multi-scale structure in the multi-phase complex systems in chemical engineering.

Stanley Milgram’s small world experiment presents “six degrees of separation” of our world. One phenomenon of the experiment still puzzling us is that how individuals operating with the social network information with their characteristics can be very adept at finding the short chains. The previous works on this issue focus whether on the methods of navigation in a given network structure, or on the effects of additional information to the searching process. In this paper, the authors emphasize that the growth and shape of network architecture is tightly related to the individuals’ attributes. The authors introduce a method to reconstruct nodes’ intimacy degree based on local interaction. Then we provide an intimacy based approach for orientation in networks. The authors find that the basic reason of efficient search in social networks is that the degree of “intimacy” of each pair of nodes decays with the length of their shortest path exponentially. Meanwhile, the model can explain the hubs limitation which was observed in real-world experiment.

In this paper, the leader-following consensus for discrete-time multi-agent systems with parameter uncertainties is investigated based on the event-triggered strategy. And the parameter uncertainty is assumed to be norm-bounded. A consensus protocol is designed based on the event-triggered strategy to make the multi-agent systems achieve consensus without continuous communication among agents. Each agent only needs to observe its own state to determine its own triggering instants under the triggering function in this paper. In addition, a sufficient condition for the existence of the eventtriggered consensus protocol is derived and presented in terms of the linear matrix inequality. Finally, a numerical example is given to illustrate to efficiency of the event-triggered consensus protocol proposed in this paper.

Complex cyber-physical network refers to a new generation of complex networks whose normal functioning significantly relies on tight interactions between its physical and cyber components. Many modern critical infrastructures can be appropriately modelled as complex cyber-physical networks. Typical examples of such infrastructures are electrical power grids, WWW, public transportation systems, state financial networks, and the Internet. These critical facilities play important roles in ensuring the stability of society as well as the development of economy. Advances in information and communication technology open opportunities for malicious attackers to launch coordinated attacks on cyber-physical critical facilities in networked infrastructures from any Internet-accessible place. Cybersecurity of complex cyber-physical networks has emerged as a hot topic within this context. In practice, it is also very crucial to understand the interplay between the evolution of underlying network structures and the collective dynamics on these complex networks and consequently to design efficient security control strategies to protect the evolution of these networks. In this paper, cybersecurity of complex cyber-physical networks is first outlined and then some security enhancing techniques, with particular emphasis on safety communications, attack detection and fault-tolerant control, are suggested. Furthermore, a new class of efficient secure control strategies are proposed for guaranteeing the achievement of desirable pinning synchronization behaviors in complex cyber-physical networks against malicious attacks on nodes. The authors hope that this paper motivates to design enhanced security strategies for complex cyber-physical network systems, to realize resilient and secure critical infrastructures.

Recently, the robust output regulation problem for continuous-time linear systems with both input and communication time-delays was studied. This paper will further present the results on the robust output regulation problem for discrete-time linear systems with input and communication delays. The motivation of this paper comes from two aspects. First, it is known that the solvability of the output regulation problem for linear systems is dictated by two matrix equations. While, for delay-free systems, these two matrix equations are same for both continuous-time systems and discretetime systems, they are different for continuous-time time-delay systems and discrete-time time-delay systems. Second, the stabilization methods for continuous-time time-delay systems and discrete-time time-delay systems are also somehow different. Thus, an independent treatment of the robust output regulation problem for discrete-time time-delay systems will be useful and necessary.

Optimal control problem with partial derivative equation (PDE) constraint is a numericalwise difficult problem because the optimality conditions lead to PDEs with mixed types of boundary values. The authors provide a new approach to solve this type of problem by space discretization and transform it into a standard optimal control for a multi-agent system. This resulting problem is formulated from a microscopic perspective while the solution only needs limited the macroscopic measurement due to the approach of Hamilton-Jacobi-Bellman (HJB) equation approximation. For solving the problem, only an HJB equation (a PDE with only terminal boundary condition) needs to be solved, although the dimension of that PDE is increased as a drawback. A pollutant elimination problem is considered as an example and solved by this approach. A numerical method for solving the HJB equation is proposed and a simulation is carried out.

The genetic models are greatly important in the analysis of genetic epidemiologic studies and many of the studies are conducted using the trend test under the additive model. However, for many complex diseases and traits, the underlying genetic model for a genetic locus is usually uncertain. So a robust test free of genetic model is appropriate. In this paper, the authors propose a model-embedded trend test by incorporating Hardy-Weinberg equilibrium information and obtain the explicit formula to calculate its statistical significance. Extensive simulation studies show the proposed test is more robust than the existing procedures. Finally, a real application is further analyzed to show the performance of the proposed test.

Existing estimators of the central mean space are known to have uneven performances across different types of link functions. By combining the strength of the ordinary least squares and the principal Hessian directions, the authors propose a new hybrid estimator that successfully recovers the central mean space for a wide range of link functions. Based on the new hybrid estimator, the authors further study the order determination procedure and the marginal coordinate test. The superior performance of the hybrid estimator over existing methods is demonstrated in extensive simulation studies.

Structural equation model (SEM) is a multivariate analysis tool that has been widely applied to many fields such as biomedical and social sciences. In the traditional SEM, it is often assumed that random errors and explanatory latent variables follow the normal distribution, and the effect of explanatory latent variables on outcomes can be formulated by a mean regression-type structural equation. But this SEM may be inappropriate in some cases where random errors or latent variables are highly nonnormal. The authors develop a new SEM, called as quantile SEM (QSEM), by allowing for a quantile regression-type structural equation and without distribution assumption of random errors and latent variables. A Bayesian empirical likelihood (BEL) method is developed to simultaneously estimate parameters and latent variables based on the estimating equation method. A hybrid algorithm combining the Gibbs sampler and Metropolis-Hastings algorithm is presented to sample observations required for statistical inference. Latent variables are imputed by the estimated density function and the linear interpolation method. A simulation study and an example are presented to investigate the performance of the proposed methodologies.

The generalized linear model is an indispensable tool for analyzing non-Gaussian response data, with both canonical and non-canonical link functions comprehensively used. When missing values are present, many existing methods in the literature heavily depend on an unverifiable assumption of the missing data mechanism, and they fail when the assumption is violated. This paper proposes a missing data mechanism that is as generally applicable as possible, which includes both ignorable and nonignorable missing data cases, as well as both scenarios of missing values in response and covariate. Under this general missing data mechanism, the authors adopt an approximate conditional likelihood method to estimate unknown parameters. The authors rigorously establish the regularity conditions under which the unknown parameters are identifiable under the approximate conditional likelihood approach. For parameters that are identifiable, the authors prove the asymptotic normality of the estimators obtained by maximizing the approximate conditional likelihood. Some simulation studies are conducted to evaluate finite sample performance of the proposed estimators as well as estimators from some existing methods. Finally, the authors present a biomarker analysis in prostate cancer study to illustrate the proposed method.

Creative telescoping is the method of choice for obtaining information about definite sums or integrals. It has been intensively studied since the early 1990s, and can now be considered as a classical technique in computer algebra. At the same time, it is still a subject of ongoing research. This paper presents a selection of open problems in this context. The authors would be curious to hear about any substantial progress on any of these problems.

In this paper, the concept of toric difference varieties is defined and four equivalent descriptions for toric difference varieties are presented in terms of difference rational parametrization, difference coordinate rings, toric difference ideals, and group actions by difference tori. Connections between toric difference varieties and affine N[x]-semimodules are established by proving the one-to-one correspondence between irreducible invariant difference subvarieties and faces of N[x]-semimodules and the orbit-face correspondence. Finally, an algorithm is given to decide whether a binomial difference ideal represented by a Z[x]-lattice defines a toric difference variety.

Gr¨obner basis theory for parametric polynomial ideals is explored with the main objective of mimicking the Gr¨obner basis theory for ideals. Given a parametric polynomial ideal, its basis is a comprehensive Gr¨obner basis if and only if for every specialization of its parameters in a given field, the specialization of the basis is a Gr¨obner basis of the associated specialized polynomial ideal. For various specializations of parameters, structure of specialized ideals becomes qualitatively different even though there are significant relationships as well because of finiteness properties. Key concepts foundational to Gr¨obner basis theory are reexamined and/or further developed for the parametric case: (i) Definition of a comprehensive Gr¨obner basis, (ii) test for a comprehensive Gr¨obner basis, (iii) parameterized rewriting, (iv) S-polynomials among parametric polynomials, (v) completion algorithm for directly computing a comprehensive Gr¨obner basis from a given basis of a parametric ideal. Elegant properties of Gr¨obner bases in the classical ideal theory, such as for a fixed admissible term ordering, a unique Gr¨obner basis can be associated with every polynomial ideal as well as that such a basis can be computed from any Gr¨obner basis of an ideal, turn out to be a major challenge to generalize for parametric ideals; issues related to these investigations are explored. A prototype implementation of the algorithm has been successfully tried on many examples from the literature.

Hybrid systems are dynamical systems with interacting discrete computation and continuous physical processes, which have become more common, more indispensable, and more complicated in our modern life. Particularly, many of them are safety-critical, and therefore are required to meet a critical safety standard. Invariant generation plays a central role in the verification and synthesis of hybrid systems. In the previous work, the fourth author and his coauthors gave a necessary and sufficient condition for a semi-algebraic set being an invariant of a polynomial autonomous dynamical system, which gave a confirmative answer to the open problem. In addition, based on which a complete algorithm for generating all semi-algebraic invariants of a given polynomial autonomous hybrid system with the given shape was proposed. This paper considers how to extend their work to non-autonomous dynamical and hybrid systems. Non-autonomous dynamical and hybrid systems are with inputs, which are very common in practice; in contrast, autonomous ones are without inputs. Furthermore, the authors present a sound and complete algorithm to verify semi-algebraic invariants for non-autonomous polynomial hybrid systems. Based on which, the authors propose a sound and complete algorithm to generate all invariants with a pre-defined template.

The McKean-Vlasov filtering problem is a special kind of filtering problem, with the state and/or observation processes governed by McKean-Vlasov stochastic differential equations, which has extensive applications in various scenarios. In this paper, the authors will propose a novel numerical algorithm to solve the McKean-Vlasov filtering problem based on the Hermite spectral method under the framework of Yau-Yau algorithm. As the first approach to numerically solving the Duncan-Mortensen-Zakai equation associated with the McKean-Vlasov filtering problem, the proposed algorithm can provide accurate estimations of the conditional expectation and conditional probability density of the state process with a reasonable online computational complexity. The efficiency of the proposed algorithm is verified both theoretically and numerically in this paper.

With the development and applications of the Smart Court System (SCS) in China, the reliability and accuracy of legal artificial intelligence have become focal points in recent years. Notably, criminal sentencing prediction, a significant component of the SCS, has also garnered widespread attention. According to the Chinese criminal law, actual sentencing data exhibits a saturated property due to statutory penalty ranges, but this mechanism has been ignored by most existing studies. Given this, the authors propose a sentencing prediction model that combines judicial sentencing mechanisms including saturated outputs and floating boundaries with neural networks. Building on the saturated structure of our model, a more effective adaptive prediction algorithm will be constructed based on the fusion of several key ideas and techniques that include the utilization of the $L_1$ loss together with the corresponding gradient update strategy, a data pre-processing method based on large language model to extract semantically complex sentencing elements using prior legal knowledge, the choice of appropriate initial conditions for the learning algorithm and the construction of a double-hidden-layer network structure. An empirical study on the crime of disguising or concealing proceeds of crime demonstrates that our method can achieve superior sentencing prediction accuracy and significantly outperform common baseline methods.

Rosenblatt and Rosenblatt-Volterra processes are two families of stochastic processes that are described by double Wiener-Itô integrals with singular kernels. The Rosenblatt processes have exponential singular kernels and the Rosenblatt-Volterra processes have singular Volterra kernels for the Wiener-Itô integrals. Empirical evidence shows that for many control systems the assumption of Gaussian noise is not appropriate so Rosenblatt and Rosenblatt-Volterra processes are some generalizations of Gaussian processes that can provide natural alternatives to Gaussian probability laws. Furthermore, the results for Rosenblatt and Rosenblatt-Volterra processes are tractable for some applications. These results can be compared to prediction for Gaussian processes and Gauss-Volterra processes.

Ever since Brockett and Clark (1980), Brockett (1981) and Mitter (1980) introduced the estimation algebra method, it becomes a powerful tool to classify finite-dimensional filtering systems. In this paper, the authors investigate estimation algebra on state dimension $n$ and linear rank $n-1$, especially the case of $n=4$. Mitter conjecture is always a key question on classification of estimation algebra. A weak form of Mitter conjecture states that observation functions in finite dimensional filters are affine functions. In this paper, the authors shall focus on the weak form of Mitter conjecture. In the first part, it will be shown that partially constant structure of $\varOmega$ is a sufficient condition for weak form Mitter conjecture to be true. In the second part, the authors shall prove partially constant structure of $\varOmega$ for $n = 4$ which implies the weak form Mitter conjecture for this case.

The authors consider the problem of reaching consensus over a communication network via asynchronous interaction between pairs of agents. A well-known method is the linear gossip algorithm due to Tsitsiklis (1984). Extension of this, allowing the selection of a strictly stationary sequence of communicating pairs, was given in Picci and Taylor (2013). Extension of the linear gossip algorithm to directed communication networks, retaining the linear dynamics, was proposed by Cai and Ishii (2012), later extended by Silvestre, et al. (2018). A definite novelty of these algorithms is that $L_2$-convergence with exponential rate can be established. The authors attend the above issues, extending the result of Picci and Taylor (2013) motivated by features of algorithms for directed networks. The authors present and discuss the algorithm of Silvestre, et al. (2018), together with systematic simulation results based on 5M randomly chosen parameter settings. The core of the proposed mathematical technology is a set of simple observations, presented with a tutorial aspect, by which the authors can conveniently establish various results on the almost sure convergence of products of strictly stationary sequences of matrices to a rank-1 matrix.

Evanescent random walks are instances of stochastic processes that terminate at a specific rate. They have proved relevant in modeling diverse behaviors of complex systems from protein degradation in gene networks (Ali and Brewster (2022), Ghusinga, et al. (2017), and Ham, et al. (2024)) and "nonprocessive" motor proteins (Kolomeisky and Fisher (2007)) to decay of diffusive radioactive matter (Zoia (2008)). The present work aims to extend a well-established estimation and control problem, the so-called Schrödinger's bridge problem, to evanescent diffusion processes. Specifically, the authors seek the most likely law on the path space that restores consistency with two marginal densities—One is the initial probability density of the flow, and the other is a density of killed particles. The Schrödinger's bridge problem can be interpreted as an estimation problem but also as a control problem to steer the stochastic particles so as to match specified marginals. The focus of previous work in Eldesoukey, et al. (2024) has been to tackle Schrödinger's problem involving a constraint on the spatio-temporal density of killed particles, which the authors revisit here. The authors then expand on two related problems that instead separately constrain the temporal and the spatial marginal densities of killed particles. The authors derive corresponding Schrödinger systems that contain coupled partial differential equations that solve such problems. The authors also discuss Fortet-Sinkhorn-like algorithms that can be used to construct the sought bridges numerically.

The authors consider the property of detectability of discrete event systems in the presence of sensor attacks in the context of cyber-security. The authors model the system using an automaton and study the general notion of detectability where a given set of state pairs needs to be (eventually or periodically) distinguished in any estimate of the state of the system. The authors adopt the ALTER sensor attack model from previous work and formulate four notions of CA-detectability in the context of this attack model based on the following attributes: strong or weak; eventual or periodic. The authors present verification methods for strong CA-detectability and weak CA-detectability. The authors present definitions of strong and weak periodic CA-detectability that are based on the construction of a verifier automaton called the augmented CA-observer. The development also resulted in relaxing assumptions in prior results on D-detectability, which is a special case of CA-detectability.