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Fuzzy rough set (FRS) theory is generally used to measure the uncertainty of data. However, this theory cannot work well when the class density of a data distribution differs greatly. In this work, a relative distance measure is first proposed to fit the mentioned data distribution. Based on the measure, a relative FRS model is introduced to remedy the mentioned imperfection of classical FRSs. Then, the positive region, negative region, and boundary region are defined to measure the uncertainty of data with the relative FRSs. Besides, a relative fuzzy dependency is defined to evaluate the importance of features to decision. With the proposed feature evaluation, we propose a feature selection algorithm and design a classifier based on the maximal positive region. The classification principle is that an unlabeled sample will be classified into the class corresponding to the maximal degree of the positive region. Experimental results show the relative fuzzy dependency is an effective and efficient measure for evaluating features, and the proposed feature selection algorithm presents better performance than some classical algorithms. Besides, it also shows the proposed classifier can achieve slightly better performance than the KNN classifier, which demonstrates that the maximal positive region-based classifier is effective and feasible.

This brief investigates the problem of state estimation of quaternion-valued neural networks (QVNNs) with time-varying delays. First, by extending the Jensen inequality to quaternion domain, an extended Jensen inequality with quaternion term is derived. Second, a class of proportional-integral state estimator (PISE) with exponential decay term is proposed. Then, by constructing a suitable Lyapunov-Krasovskii functional (LKF), some sufficient conditions are obtained to ensure the existence of the designed PISE and the gain matrices of the designed PISE can be directly computed. Simulations are given to illustrate the advantage of the proposed method.

For more than two decades, proving or refuting the following statement has remained a challenging open problem in the theory of secret sharing schemes (SSSs): every ideal access structure admits an ideal perfect multi-linear SSS. The class of group-characterizable (GC) SSSs include the multi-linear ones. Hence, if the above statement is true, then so is the following weaker statement: every ideal access structure admits an ideal perfect GC SSS. One contribution of this paper is to show that ideal SSSs are not necessarily GC. Our second contribution is to study the above two statements with respect to several variations of weakly-ideal access structures. Recently, Mejia and Montoya studied ideal access structures that admit ideal multi-linear schemes and provided a classification-like theorem for them. We additionally present some tools that are useful to extend their result.

In this article, we define a new class of functions on Riemannian manifolds, called geodesic \begin{document}$ \mathcal{E} $\end{document} -prequasi-invex functions. By a suitable example it has been shown that it is more generalized class of convex functions. Some of its characteristics are studied on a nonlinear programming problem. We also define a new class of sets, named geodesic slack invex set. Furthermore, a sufficient optimality condition is obtained for a nonlinear programming problem defined on a geodesic local \begin{document}$ \mathcal{E} $\end{document} -invex set.

Subfield codes of linear codes over finite fields have recently received a lot of attention, as some of these codes are optimal and have applications in secrete sharing, authentication codes and association schemes. In this paper, a class of binary subfield codes is constructed from a special family of MDS codes, and their parameters are explicitly determined. The parameters of their dual codes are also studied. Some of the codes presented in this paper are optimal or almost optimal.

Absorbing sets are combinatorial structures in the Tanner graphs of low-density parity-check (LDPC) codes that have been shown to inhibit the high signal-to-noise ratio performance of iterative decoders over many communication channels. Absorbing sets of minimum size are the most likely to cause errors, and thus have been the focus of much research. In this paper, we determine the sizes of absorbing sets that can occur in general and left-regular LDPC code graphs, with emphasis on the range of \begin{document}$ b $\end{document} for a given \begin{document}$ a $\end{document} for which an \begin{document}$ (a,b) $\end{document} -absorbing set may exist. We identify certain cases of extremal absorbing sets that are elementary, a particularly harmful class of absorbing sets, and also introduce the notion of minimal absorbing sets which will help in designing absorbing set removal algorithms.

In this paper, we first introduce a class of quartic Boolean functions. And then, the construction of weightwise perfectly balanced Boolean functions on \begin{document}$ 2^m $\end{document} variables are given by modifying the support of the quartic functions, where \begin{document}$ m $\end{document} is a positive integer. The algebraic degree, the weightwise nonlinearity, and the algebraic immunity of the newly constructed weightwise perfectly balanced functions are discussed at the end of this paper.

In this work, multi-variable derivative-free optimization algorithms for unconstrained optimization problems are developed. A novel procedure for approximating the gradient of multi-variable objective functions based on non-commutative maps is introduced. The procedure is based on the construction of an exploration sequence to specify where the objective function is evaluated and the definition of so-called gradient generating functions which are composed with the objective function such that the procedure mimics a gradient descent algorithm. Various theoretical properties of the proposed class of algorithms are investigated and numerical examples are presented.

This article studies the control problem for a class of stochastic nonlinear time-delay systems with uncertain output functions. Under the appropriate assumptions, a stabilization controller is explicitly constructed by applying the adding a power integrator method. Then, using the Lyapunov-Krasovskii functionals to address time-delay, it is proven that the designed controller can guarantee the closed-loop system to be globally asymptotically stable (GAS) in probability. Finally, two simulations show that the control strategy is effective and can be applied to the actual system.

This paper studied new class of integral equation called the Korteweg-de Vries-Kadomtsev-Petviashvili (KdV-KP) equation. This equation consist of the well-known fifth-order KdV equation in the context of the Kadomtsev-Petviashvili equation. The newly gathered class of sixth-order KdV-KP equation is studied using the sub-equation method to obtain several soliton-type solutions which consist of trigonometric, hyperbolic, and rational solutions. The application of the sub-equation approach in this work draws attention to the outstanding characteristics of the suggested method and its ability to handle completely integrable equations. Furthermore, the obtained solutions have not been reported in the previous literature and might have significant impact on future research.

Hole-twins – graphs that arise when a vertex is added to a hole in such a way to form a twin with some vertex of the hole – were discussed in a recent paper by Dai, Foley, and Hoang where it was shown that there is a polynomial time algorithm to color ( c l a w , 4 K 1 , hole-twin)-free graphs. We continue our investigation of hole-twin-free graphs and show the coloring problem for ( c l a w , P 7 , hole-twin)-free graphs is also polynomial-time solvable.

Power of cycle graphs C n k have been extensively studied with respect to coloring problems, being both the vertex and the edge-coloring problems already solved in the class. The total coloring problem (of determining the minimum number of colors needed to color the vertices and the edges of a graph in a manner that no two adjacent elements are colored the same), however, is still open for power of cycle graphs. Actually, despite partial results for specific values of n and k , not even the well-known Total Coloring Conjecture is settled in the class. A remarkable conjecture by Campos and Mello of 2007 states that if C n k is neither a cycle nor a complete graph, then it has total chromatic number χ T = Δ + 2 if n is odd and n 3 ( k + 1 ) , and χ T = Δ + 1 otherwise. We provide strong evidences for this conjecture: we settle a dichotomy for all power of cycle graphs with respect to conformability (a well-known necessary condition for a graph to have χ T = Δ + 1 ) and we develop a framework which may be used to prove that, for any fixed k , the number of C n k graphs with χ T ≠ Δ + 1 is finite. Moreover, we prove this finiteness for any even k and for k ∈ { 3 , 5 , 7 } . We also use our composition technique to provide a proof of Campos and Mello’s conjecture for all C n k graphs with k ∈ { 3 , 4 } .

Observability of a hybrid system is defined as the ability to determine the continuous state of the system. Whether a hybrid system is observable or not depends on which events can be disabled, which events can be forced, and the connectivity of the discrete states, as well as its continuous dynamics. We model a hybrid system using a hybrid machine that takes into consideration both continuous variables and discrete events. We classify hybrid systems into four classes based on their discrete-event parts. For each class, conditions are derived to check observability. If a hybrid system is not observable, then we check if a weaker version of observability, called B-observability, is satisfied. B-observability requires that a hybrid system become observable after some finite occurrences of events. Conditions are derived to check B-observability. These conditions involve both the discrete-event and continuous-variable parts of hybrid systems. If the continuous-variable part of a system has a constant A matrix, then the conditions for the continuous-variable part can be simplified. We illustrate the results by an example of a battery management system of an electric vehicle.

This paper focuses on the problem of prescribedtime control for a class of uncertain nonlinear systems. First, a prescribed-time stability theorem is proposed by following the adaptive technology for the first time. Based on this theorem, a new state feedback control strategy is put forward by using the backstepping method for high-order nonlinear systems with unknown parameters to ensure the prescribed-time convergence. Moreover, the prescribed-time controller is obtained in the form of continuous time-varying feedback, which can render all system states converge to zero within the prescribed-time. It should be noted that the prescribed-time is independent of system initial conditions, which means that the prescribed-time can be set arbitrarily within the physical limitations. Finally, two simulation examples are provided to illustrate the effectiveness of our proposed algorithm.

There are several approaches for evaluating conflict within belief functions. In this paper, we develop one of them based on axioms and show its connections to the decomposition approach. We describe a class of conflict measures satisfying this system of axioms and show that measuring conflict can be realized through the clusterization of a body of evidence. We also show that well-known conflict measures like the auto-conflict measure and the measure of dissonance do not satisfy the proposed system of axioms. We also tackle the problem of simplifying a body of evidence based on clusterization and show the application of developed theoretical constructions for data processing.

Sierpinski triangle graphs S n have often been mistaken for Sierpinski graphs S 3 n . Whereas the latter’s metric properties are by now well understood, the former graphs were mostly just considered as a pictorial representation of approximations to the Sierpinski triangle fractal. Therefore, we present here a new labeling for them which shows the relation, but also the differences to the more famous Sierpinski graphs proper. On the base of this labeling we describe an algorithm to obtain individual distances between vertices. This type of algorithm can then be extended to base- p Sierpinski triangle graphs S p n which are related to the class of classical Sierpinski graphs S p n , p ≥ 2 . Some of the metric properties of S p n can now be investigated for S p n as well; e.g., we characterize center and periphery of S p n .

Similar to the construction of ordinal sums of overlap functions, Baczynski et al. introduced two new kinds of ordinal sums of fuzzy implications without additional restrictions on summands in 2017. In this paper, based on the ordinal sum of fuzzy implications with the first method, we discuss its basic properties, such as iterative Boolean law, right ordering property and strong boundary condition. Meanwhile, we give characterizations of such ordinal sum of fuzzy implications that is a QL-implication constructed from tuples ( O , G , N ⊤ ) or a D-implication derived from grouping function G, and show some conclusions about its relations with ( G , N ) - and R O -implications. Moreover, we study the distributivity of such ordinal sum of fuzzy implications over a class of overlap and grouping functions. More specifically, we give necessary and sufficient conditions under which this ordinal sum of fuzzy implications is distributive over overlap and grouping functions satisfying some conditions.

Let G = ( V , E ) be a simple undirected graph. A set D ⊆ V is called a vertex-edge dominating set of G if for each edge e = u v ∈ E , either u or v is in D or one vertex from their neighbor is in D . Simply, a vertex v ∈ V , vertex-edge dominates every edge u v , as well as every edge adjacent to these edges. The objective of the vertex-edge domination problem is to find a minimum size vertex-edge dominating set of G . Herein, we study the vertex-edge domination problem in unit disk graphs and prove that the decision version of the problem belongs to the NP-complete class. We also show that the problem admits a simple polynomial-time 4-factor approximation algorithm and a polynomial-time approximation scheme (PTAS) in unit disk graphs.

Bitprobe complexity of the static membership problem has been widely investigated since Burhman et al. (2000) studied the worst-case bounds for a whole range of membership problems. Though tremendous progress has been made in recent years, as is evident from the remarkable papers due to Alon and Feige (2009) [1] , Viola (2012) [8] , and Lewenstein et al. (2014), among others, Nicholson et al. (2013) [5] in their survey of the state of the art of the area noted that even the first non-trivial scenario of two probe adaptive schemes storing two elements of the universe is not completely settled, in that the lower bound for this problem is still open. We propose a proof (Kesh and Sharma, 2019) [3] for the lower bound for the same problem, but for a restricted class of schemes. We believe that this proof makes progress over the ideas proposed by Radhakrishnan et al. (2001, 2010) towards the proof. It is our belief that the single restriction imposed on the general class of schemes towards our proof faithfully captures the dependencies among the elements that share the same bit in the datastructure, and we hope that the general lower bound proof can be developed using similar ideas.

The loss count distributions whose probabilities ultimately satisfy Panjer’s recursion were classified between 1981 and 2002; they split into six types, which look quite diverse. Yet, the distributions are closely related – we show that their probabilities emerge out of one formula: the binomial series. We propose a parameter change that leads to a unified, practical and intuitive, representation of the Panjer distributions and their parameter space. We determine the subsets of the parameter space where the probabilities are continuous functions of the parameters. Finally, we give an inventory of parameterisations used for Panjer distributions.

In this paper, we study a class of matrix functions of the combinatorial Laplacian that preserve its structure, i.e. that define matrices which are positive semidefinite, and which have zero row-sum and non-positive off-diagonal entries. This formulation has the merit of presenting different incarnations of the Laplacian matrix appeared in the recent literature, in a unified framework. For the first time, we apply this family of Laplacian functions to consensus theory, and we show that they leave the agreement value unchanged and offer distinctive advantages in terms of performance and design flexibility. The theory is illustrated via worked examples and numerical experiments featuring four representative Laplacian functions in a shape-based distributed formation control strategy for single-integrator robots.

We determine those imaginary quadratic number fields whose 2-class groups have rank 3 and 4-rank ≤2 and such that the 2-class groups of their Hilbert 2-class fields have rank 2. This result, along with previous work, gives a complete determination of all complex quadratic number fields for which the 2-class groups of their 2-class fields have rank at most 2.

For a given odd positive integer $n$ and an odd prime $p$, we construct an infinite family of quadruples of imaginary quadratic fields $\mathbb{Q}(\sqrt{d})$, $\mathbb{Q}(\sqrt{d+1})$, $\mathbb{Q}(\sqrt{d+4})$ and $\mathbb{Q}(\sqrt{d+4p^2})$ with $d\in \mathbb{Z}$ such that the class number of each of them is divisible by $n$. Subsequently, we show that there is an infinite family of quintuples of imaginary quadratic fields $\mathbb{Q}(\sqrt{d})$, $\mathbb{Q}(\sqrt{d+1})$, $\mathbb{Q}(\sqrt{d+4})$, $\mathbb{Q}(\sqrt{d+36})$ and $\mathbb{Q}(\sqrt{d+100})$ with $d\in \mathbb{Z}$ whose class numbers are all divisible by $n$. Our results provide a complete proof of Iizuka's conjecture (in fact a generalization of it) for the case $m=1$. Our results also affirmatively answer a weaker version of (a generalization of) Iizuka's conjecture for $m\geq 4$., Comment: 10 pages; some minor typos have been fixed as per the suggestions from the referee; to appear in 'Journal of Number Theory'

We study combinatorial auctions with interdependent valuations. In such settings, each agent $i$ has a private signal $s_i$ that captures her private information, and the valuation function of every agent depends on the entire signal profile, ${\bf s}=(s_1,\ldots,s_n)$. The literature in economics shows that the interdependent model gives rise to strong impossibility results, and identifies assumptions under which optimal solutions can be attained. The computer science literature provides approximation results for simple single-parameter settings (mostly single item auctions, or matroid feasibility constraints). Both bodies of literature focus largely on valuations satisfying a technical condition termed {\em single crossing} (or variants thereof). We consider the class of {\em submodular over signals} (SOS) valuations (without imposing any single-crossing type assumption), and provide the first welfare approximation guarantees for multi-dimensional combinatorial auctions, achieved by universally ex-post IC-IR mechanisms. Our main results are: $(i)$ 4-approximation for any single-parameter downward-closed setting with single-dimensional signals and SOS valuations; $(ii)$ 4-approximation for any combinatorial auction with multi-dimensional signals and {\em separable}-SOS valuations; and $(iii)$ $(k+3)$- and $(2\log(k)+4)$-approximation for any combinatorial auction with single-dimensional signals, with $k$-sized signal space, for SOS and strong-SOS valuations, respectively. All of our results extend to a parameterized version of SOS, $d$-SOS, while losing a factor that depends on $d$., To appear in the 20th ACM conference on Economics and Computation (ACM EC '19)

Here we introduce the idea of using rational expectations, a core concept in economics and finance, as a tool to predict the optimal failure time for a wide class of weighted k-out-of-n reliability systems. We illustrate the concept by applying it to systems which have components with heterogeneous failure times. Depending on the heterogeneous distributions of component failure, we find different measures to be optimal for predicting the failure time of the total system. We give examples of how, as a given system deteriorates over time, one can issue different optimal predictions of system failure by choosing among a set of time-dependent measures., Comment: 28 pages, 7 figures

We study nonlinear parabolic stochastic partial differential equations with Wick-power and Wick-polynomial type nonlinearities set in the framework of white noise analysis. These equations include the stochastic Fujita equation, the stochastic Fisher-KPP equation and the stochastic FitzHugh-Nagumo equation among many others. By implementing the theory of $C_0-$semigroups and evolution systems into the chaos expansion theory in infinite dimensional spaces, we prove existence and uniqueness of solutions for this class of SPDEs. In particular, we also treat the linear nonautonomous case and provide several applications featured as stochastic reaction-diffusion equations that arise in biology, medicine and physics.

The law of importation between fuzzy implications and conjunctions is an important property in both the theory and the application of fuzzy logic. Although many interesting results have been reported in the literature, the related open problem of finding all possible pairs of implications and conjunctions satisfying the law of importation has not been fully solved. As a continuation of two articles S. Massanet et al. (2018) [23] ; S. Massanet et al. (2018) [24] , this article aims to characterize fuzzy implication solutions with a continuous natural negation to the law of importation with respect to a fixed conjunctive uninorm having continuous underlying operators in several cases. Such solutions for the cases where the continuous underlying triangular norm and triangular conorm of a prefixed uninorm are idempotent or Archimedean are completely characterized, and the solutions for the cases where the continuous underlying operators are given by ordinal sums are characterized under some additional assumptions. Finally, some comments on relations to recent independent work W.-H. Li and F. Qin (2021) [25] are given. These two independent articles will go a further step towards answering the open problem regarding the law of importation.

Inspired by the notion of quasi-homogenous copulas, we introduce a new class of functions with a given curved section. The convexity of curved sections plays a key role in characterizing the corresponding copulas. This class of copulas generalizes the class of quasi-homogenous copulas.

This paper has two main goals: (a) establish several statistical properties—consistency, asymptotic distributions, and convergence rates—of stationary solutions and values of a class of coupled nonconvex and nonsmooth empirical risk-minimization problems and (b) validate these properties by a noisy amplitude-based phase-retrieval problem, the latter being of much topical interest. Derived from available data via sampling, these empirical risk-minimization problems are the computational workhorse of a population risk model that involves the minimization of an expected value of a random functional. When these minimization problems are nonconvex, the computation of their globally optimal solutions is elusive. Together with the fact that the expectation operator cannot be evaluated for general probability distributions, it becomes necessary to justify whether the stationary solutions of the empirical problems are practical approximations of the stationary solution of the population problem. When these two features, general distribution and nonconvexity, are coupled with nondifferentiability that often renders the problems “non-Clarke regular,” the task of the justification becomes challenging. Our work aims to address such a challenge within an algorithm-free setting. The resulting analysis is, therefore, different from much of the analysis in the recent literature that is based on local search algorithms. Furthermore, supplementing the classical global minimizer-centric analysis, our results offer a promising step to close the gap between computational optimization and asymptotic analysis of coupled, nonconvex, nonsmooth statistical estimation problems, expanding the former with statistical properties of the practically obtained solution and providing the latter with a more practical focus pertaining to computational tractability.

The existing results of Lagrange stability for neural networks with distributed time delays are scale-free, which introduces conservativeness naturally. A class of Takagi-Sugeno fuzzy memrisive neural networks (FMNNs) on time scales with discrete time-varying and infinite distributed delays is brought in this paper. First, a new scale-limited Halanay inequality is demonstrated by timescale theory. Next, on the basis of inequality techniques on time scales, some new scale-limited algebraic criteria and linear matrix inequality criteria of Lagrange stability are obtained by comparison strategy and generalized Halanay inequality. All scale-limited sufficient criteria of Lagrange stability for FMNNs not only apply to continuous-time FMNNs and their discrete-time analogues, but also could deal with the arbitrary combination of them. Finally, two numerical simulations are given to verify the validity of the obtained theoretical results.

Several operators have emerged in the framework of Choquet integral with the purpose of simultaneously generalizing weighted means and ordered weighted averaging (OWA) operators. However, on many occasions, not enough attention has been paid to whether the constructed operators behaved similarly to the weighted means and OWA operators that have been generalized. In this sense, it seems necessary that these new operators preserve the weights assigned to the information sources (which are established through the weighting vector associated with the weighted mean) and that they are able to rule out extreme values (which is an important characteristic of OWA operators). In this paper we analyze a family of operators recently introduced in the literature through the Crescent Method. First, we introduce a broad class of weighting vectors that allow us to guarantee that the games generated with the Crescent Method are capacities. Next we analyze the conjunctive/disjunctive character of the Choquet integrals associated with these capacities and we also give closed-form expressions of some tolerance and importance indices such as k-conjunctiveness/disjunctiveness indices, the veto and favor indices, and the Shapley values. Finally, we give two examples to show the usefulness of the results obtained.

The issues of complete synchronization (CS) and finite-time synchronization (F-TS) for a class of fractional-order fuzzy neural networks are addressed based on nonlinear feedback control in this paper. First, a fractional-order finite-time convergence principle is established by virtue of fractional calculus basic theory and reduction to absurdity. Next, two novel nonlinear controllers, namely the adaptive nonlinear controller and discontinuous nonlinear controller, are designed. Then some easily validated criteria to guarantee CS and F-TS are derived with the help of some useful analysis techniques and our newly established convergence principle. Moreover, the settling time of F-TS is effectively estimated. Finally, some numerical results are presented to show the validity of derived theoretical results.

We study the problem of learning the structure of an optimal Bayesian network when additional constraints are posed on the network or on its moralized graph. More precisely, we consider the constraint that the network or its moralized graph are close, in terms of vertex or edge deletions, to a sparse graph class $Π$. For example, we show that learning an optimal network whose moralized graph has vertex deletion distance at most $k$ from a graph with maximum degree 1 can be computed in polynomial time when $k$ is constant. This extends previous work that gave an algorithm with such a running time for the vertex deletion distance to edgeless graphs [Korhonen & Parviainen, NIPS 2015]. We then show that further extensions or improvements are presumably impossible. For example, we show that learning optimal networks where the network or its moralized graph have maximum degree $2$ or connected components of size at most $c$, $c\ge 3$, is NP-hard. Finally, we show that learning an optimal network with at most $k$ edges in the moralized graph presumably has no $f(k)\cdot |I|^{O(1)}$-time algorithm and that, in contrast, an optimal network with at most $k$ arcs can be computed in $2^{O(k)}\cdot |I|^{O(1)}$ time where $|I|$ is the total input size., 42 pages

This paper presents a new optimization-based approach to determine the class of stabilizing control structures with the necessary set of feedback links for interconnected systems. The proposed approach relies on a graph theoretic interpretation and its equivalence in terms of binary linear programs (BLP). To carry out the primary goal, first, the stabilizability of an LTI system under the decentralized control structure is presented in terms of a BLP. Next, two graph-based criteria are proposed to characterize stabilizing control structures with the required feedback links. Finally, all possible stabilizing control structures with the necessary feedback links are derived via solving a set of BLPs. In addition to the analysis of stabilizing control structures, the proposed graph-theoretic approach offers a versatile set of tools to find the simplest control structures capable of assigning the closed-loop spectrum in a totally arbitrary desired region. Simulation results illustrate the assessment of stabilizing control structures, the required additional feedback links, and the elegant generalization of the proposed approach to regional pole-placement.

For all Eichler orders with a same squarefree level in a definite quaternion algebra over the field of rational numbers, we prove that a weighted sum of Jacobi theta series associated to these orders is a Jacobi Eisenstein series. Multiply the Fourier coefficients of the latter by a power of 2 to get our modified Hurwitz class numbers. As its corollaries, the modified Hurwitz class numbers can be used to calculate the traces of Brandt matrices and the type number of the Eichler orders.

As one class of binary continuous aggregation functions which play an important role in practical applications, overlap functions defined on the unit closed interval have been developed rapidly in the past decade. At the same time, Paiva et al. recently extended the concept of overlap functions on unit closed interval to the lattice-valued status and called them quasi-overlap functions on bounded lattices. In this paper, we mainly study the extension methods of quasi-overlap functions and their three generalized forms on bounded partially ordered sets. More concretely, first, we show some new extension methods of quasi-overlap functions, 0 P -quasi-overlap functions, 1 P -quasi-overlap functions and 0 P , 1 P -quasi-overlap functions on any bounded partially ordered set P by the so-called ⋄-operators and 0,1-homomorphisms, 1-⋄-operators and 1-homomorphisms, 0-⋄-operators and 0-homomorphisms, and 0,1-⋄-operators and ord-homomorphisms, respectively, which are different from the extension methods obtained by Qiao lately. And then, as an application of the new extension methods, some concrete quasi-overlap functions, 0 P -quasi-overlap functions, 1 P -quasi-overlap functions and 0 P , 1 P -quasi-overlap functions on some certain bounded partially ordered set P are constructed. Finally, we prove that these extensions maintain idempotent and Archimedean property of the known quasi-overlap functions, 0 P -quasi-overlap functions, 1 P -quasi-overlap functions and 0 P , 1 P -quasi-overlap functions on any bounded partially ordered set P.

In this article, the problem of adaptive tracking control is tackled for a class of high-order nonlinear systems. In contrast to existing results, the considered system contains not only unknown nonlinear functions but also unknown rational powers. By utilizing the fuzzy approximation approach together with the barrier Lyapunov functions (BLFs), we present a new adaptive tracking control strategy. Remarkably, the BLFs are employed to determine a priori the compact set for maintaining the validity of fuzzy approximation. The primary advantage of this article is that the developed controller is independent of the powers and can be capable of ensuring global stability. Finally, two illustrative examples are given to verify the effectiveness of the theoretical findings.

A static output feedback based event-triggered control mechanism for a class of switched nonlinear systems with a passivity property is proposed in this paper. Each subsystem of such a switched system does not necessarily have passivity in the classical sense, while maintaining the passivity property only on the active time intervals. Moreover, passivity is based on multiple storage functions and the storage functions are allowed to increase at switching instants. The proposed event-triggered output feedback controller can not only maintain asymptotic stability but also exclude Zeno behavior under some conditions. Finally, a numerical example is presented to illustrate the effectiveness of the method.

Previous research proposed the uniform mutation inside the sphere as a new mutation operator for evolution strategies (continuous evolutionary algorithms), with case study the elitist algorithm on the SPHERE. For that landscape, one-step success probability and expected progress were estimated analytically, and further proved to converge, as space dimension increases, to the corresponding asymptotics of the algorithm with normal mutation. This paper takes the analysis further by considering the RIDGE, an asymmetric landscape almost uncovered in literature. For the elitist algorithm, estimates of expected progress along the radial and longitudinal axes are derived, then tested numerically against the real behavior of the algorithm on several functions from this class. The global behavior of the algorithm is predicted correctly by iterating the one-step analytical formulas. Moreover, experiments show identical mean value dynamics for the algorithms with uniform and normal mutation, which implies that the derived formulas apply also to the normal case. Essential to the whole analysis is θ, the inclination angle of the RIDGE. The behavior of the algorithm on the SPHERE and HYPERPLANE is also obtained, at the limits of the θ interval (0∘,90∘.

In this paper, we consider the dynamical modeling of a class of quantum network systems consisting of qubits. Qubit probes are employed to measure a set of selected nodes of the quantum network systems. For a variety of applications, a state space model is a useful way to model the system dynamics. To construct a state space model for a quantum network system, the major task is to find an accessible set containing all of the operators coupled to the measurement operators. This paper focuses on the generation of a proper accessible set for a given system and measurement scheme. We provide analytic results on simplifying the process of generating accessible sets for systems with a time-independent Hamiltonian. Since the order of elements in the accessible set determines the form of state space matrices, guidance is provided to effectively arrange the ordering of elements in the state vector. Defining a system state according to the accessible set, one can develop a state space model with a special pattern inherited from the system structure. As a demonstration, we specifically consider a typical 1D-chain system with several common measurements, and employ the propo

In multilabel learning, each training example is represented by a single instance, which is relevant to multiple class labels simultaneously. Generally, all relevant labels are considered to be available for labeled data. However, instances with a full label set are difficult to obtain in real-world applications, thus leading to the weakly multilabel learning problem, that is, relevant labels of training data are partially known and many relevant labels are missing, and even abundant training data are associated with an empty label set. To address the problem, we propose a new multilabel method to learn from weakly labeled data. To be specific, an optimization framework is constructed based on the manifold regularized sparse model, in which the correlations among labels and feature structure are considered to model global and local label correlations, thereby achieving discriminative feature analysis for mapping training data to ground-truth label space. Moreover, the proposed method has an excellent mechanism to conduct semisupervised multilabel learning by exploiting training data with the predicted label set of the unlabeled. Experiments on various real-world tasks reveal that the proposed method outperforms some state-of-the-art methods.

By using the Banach fixed point theorem and Schauder fixed-point theorem for semilinear spaces, we study the existence of solutions to some class of boundary value problems for interval-valued differential equations on unbounded domains. Some sufficient conditions are provided in order to deduce the existence of solutions without switching points, and also for mixed solutions with a unique switching point. The influences of the range of the parameter in the boundary value condition has on the existence of solutions is also discussed. Finally, two examples are given to demonstrate the feasibility of the theorems.

This article tackles the global exponential stability for a class of delayed complex-valued inertial neural networks in a discrete-time form. It is assumed that the activation function can be separated explicitly into the real part and imaginary part. Two methods are employed to deal with the stability issue. One is based on the reduced-order method. Two exponential stability criteria are obtained for the equivalent reduced-order network with the generalized matrix-measure concept. The other is directly based on the original second-order system. The main theoretical results complement each other. Some comparisons with the existing works show that the results in this article are less conservative. Two numerical examples are given to illustrate the validity of the main results.

The law of importation, given by the equality (x^y)→z ≈ (x→(y→z)), is a tautology in classical logic and has been proved to be widely used in approximate reasoning and image processing. Some open problems of fuzzy implication dealing with the law of importation were suggested on 8th international conference on Fuzzy Set Theory and Applications (FSTA 2006). In this paper, we partially solve one open problem associated with this property. Specifically, we mainly devote ourselves to solving the general form of the law of importation I(U(x, y), z) = I(x, I(y, z)), where I is a fuzzy implication and U is a conjunctive uninorm with a continuous underlying tnorm and a continuous underlying t-conorm. Along this study, given a fixed uninorm with continuous underlying operators, all fuzzy implications that satisfy the law of importation with respect to this uninorm, and having an α-section that is a continuous negation, are characterized.

We study a class of linear-quadratic stochastic differential games in which each player interacts directly only with its nearest neighbors in a given graph. We find a semiexplicit Markovian equilibrium for any transitive graph, in terms of the empirical eigenvalue distribution of the graph’s normalized Laplacian matrix. This facilitates large-population asymptotics for various graph sequences, with several sparse and dense examples discussed in detail. In particular, the mean field game is the correct limit only in the dense graph case, that is, when the degrees diverge in a suitable sense. Although equilibrium strategies are nonlocal, depending on the behavior of all players, we use a correlation decay estimate to prove a propagation of chaos result in both the dense and sparse regimes, with the sparse case owing to the large distances between typical vertices. Without assuming the graphs are transitive, we show also that the mean field game solution can be used to construct decentralized approximate equilibria on any sufficiently dense graph sequence.

The new model class of mixtures of generalised nonlinear models (GNMs) is introduced. The model is specified, identifiability issues discussed, the fitting in a maximum likelihood framework using the expectation-maximisation (EM) algorithm outlined and an appropriate computational implementation introduced. The new model class is applied to capture cross-country heterogeneity when considering the augmented Solow model including human capital accumulation as underlying model structure. The inherent heterogeneity is attributed to multiple regimes being present within the selected country data set. The results highlight that country-specific differences lead to distinct components. Countries belonging to the same component exhibit convergence to a homogeneous steady state. The components differ in the initial technological endowment and the contribution of the economic variables to economic growth.