Bounded Functions Research Articles

Global exponential bipartite synchronization for neutral memristive inertial coupling mixed time-varying delays neural networks with antagonistic interactions

In this paper, the global exponential bipartite synchronization (GEBS) for neutral-type memristive inertial coupling mixed time-varying delays neural networks (NTMI-CMTVDNNs) with synergistic and antagonistic interactions (SAI) is investigated. Firstly, the NTMI-CMTVDNNs with SAI between neurons are modeled by a signed graph. Inertial network is equivalently transformed into the first-order differential equations via variable substitution. Secondly, a novel lemma is proposed to deal with parameter mismatch of the above modeled network. By designing delay-dependent and delay-independent discontinuous control laws, the delay-dependent sufficient conditions under the bounded and unbounded activation functions are obtained to ensure the GEBS of the NTMI-CMTVDNNs with SAI with or without a leader neurons, respectively. Finally, two numerical simulation examples illustrate the validity of our theoretical results.

M-Potential Theory and m-Generalized Lelong Numbers Associated with m-Positive Supercurrents

In this study, we first define the local potential associated to a weakly positive closed supercurrent in analogy to the one investigated by Ben Messaoud and El Mir in the complex setting. Next, we study the definition and the continuity of the m-superHessian operator for unbounded m-convex functions. As an application, we generalize our previous work on Demailly-Lelong numbers and several related results in the superformalism setting. Furthermore, strongly inspired by the complex Hessian theory, we introduce the Cegrell-type classes as well as a generalization of some m-potential results in the class of m-convex functions.

Performance-constrained fault-tolerant DSC based on reinforcement learning for nonlinear systems with uncertain parameters

In this paper, a performance-constrained fault-tolerant dynamic surface control (DSC) algorithm based on reinforcement learning (RL) is proposed for nonlinear systems with unknown parameters and actuator failures. Considering the problem of multiple actuator failures, the bound for sum of the failure parameters are estimated rather than the parameters themselves, an infinite number of actuator failures can be handled. To improve the performance of the system, based on actor-critic neural networks (NNs) and optimized backstepping control (OBC), RL is introduced to optimize the tracking errors and inputs. By introducing an intermediate controller, the controllers derived from RL algorithm and the fault-tolerant controller are isolated, the difficulties of using RL in fault-tolerant control (FTC) are reduced. In addition, an initial unbounded boundary function is used so that the initial value of the error does not need to be within a prescribed range, not only the tracking error can be reduced to the prescribed accuracy, but also all closed-loop signals are bounded. Finally, the effectiveness and advantages of the proposed algorithm are verified by two examples.

시변지연과 미지 제어 방향과 비억제 성장율을 갖는 피드포워드 비선형 시스템의 안정화

In this study, we investigate the regulation problem of a feedforward nonlinear system with time-varying state delay, an unknown control direction, and an unbounded growth function. The proposed controller was mainly developed using the Nussbaum gain and gain scaling techniques, and a rigorous system analysis was carried out to show that the controller achieved global regulation of the considered system. Numerical simulation results are provided to demonstrate the effectiveness of the proposed controller.

Partially Explore the Differences and Similarities between Riemann-Liouville Integral and Mellin Transform

At present many researchers devote themselves to studying the relationship between continuous fractal functions and their fractional integral. But little attention is paid to the relationship between Mellin transform and fractional integral. This paper aims to partially explore the differences and similarities between Riemann-Liouville integral and Mellin transform, then a 1-dimensional continuous and unbounded variational function defined on the closed interval [0,1] needs to be constructed. Through describing the image of the constructed function and its transformed function and proving the relevant properties, we obtain that Box dimension of its Riemann–Liouville integral of arbitrary order and its Mellin transformed function are also one. The smoothness of its Riemann–Liouville integral can only be improved, and its Mellin transformed function is differentiable.

Zero-sum risk-sensitive stochastic games with unbounded payoff functions and varying discount factors

This paper is an attempt of studying zero-sum risk-sensitive stochastic games with unbounded payoff functions and varying discount factors. First, we obtain a logarithm growth condition to guarantee the finiteness of the expected risk-sensitive discounted payoffs, which is an extension of the bounded reward/cost functions in the existing literature. Second, an example with unbounded payoff functions is given, where the solution to the (associated) Shapley equation (SE) no longer has the boundaries as in the existing research. This motivates us to replace the boundaries of the SE with new ones. Third, using the new boundaries and novel arguments, we not only establish the existence of a solution to the SE, but also prove the existence of the value of the game and a Nash equilibrium. Furthermore, we develop an iteration algorithm for computing (at least approximating) the value and Nash equilibria of the games. Finally, we use an inventory system to illustrate our results and show the difference between our conditions and those in the existing articles.

Fractal dimensions of mixed Katugampola fractional integral associated with vector valued functions

The aim of this article is to study the fundamental properties of the mixed Katugampola fractional integral (K-integral) of vector-valued functions and fractal dimensional results. We show that the mixed K-integral preserves the basic properties such as boundedness, continuity, and bounded variation of vector-valued functions. We also estimate the Hausdorff dimension of the graph of the vector-valued function and the graph of the mixed K-integral on the rectangular region. Moreover, we prove that the upper bound of the box dimension of the graph of each coordinate function of mixed K-integral of vector-valued functions is 3−min{μ1,μ2}, where μ1 and μ2 are order of the fractional integral with 0<μ1<1,0<μ2<1. Moreover, we give an example of unbounded variational vector-valued functions. In the end, we discuss some problems for future direction.

Some achieved wedge products of positive currents

In this paper, we study the existence of the current gT for positive plurisubharmonic currents T and unbounded plurisubharmonic functions g.

Shape preserving approximation for unbounded multivariate functions

The purpose of this research is to present a new method for creating polynomials and spline approximation to unbounded multivariate functions in L ρ , β ([–1, 1] I ) using ordinary modulus of smoothness, Sendov-Popov modulus and Ditzian-Totik modulus.

Approximation of Functions in Lp, Cn (\U0001d54f)-space

This research focused on finding a mathematical space, extending to the Lebesgue spaces, through which it is possible to get the approximation for unbounded functions depending on the fundamental theory of approximation (Korvkein Theorem) using some linear operators Depending on the averaged modulus of smoothness of order k (τ-modulus).

Best co-proximal and Co-chebyshev of unbounded functions.

The aim of this paper, we introduced some result of best co-approximation specifications of co-proximal and co-chebyshev of unbounded functions in weighted spaces. Also, we discuss some properties of best co-proximal and Co-chebyshev of un bounded functions which gives basic definitions and necessary concepts are convexity, orthogonlity and closed which helps us in the proofs of main results. In addition to that verification, when the function is multiplied by a constant, it maintains its properties.

(∆mv , f)-lacunary statistical convergence of order α

In this paper, we define the space Sαθ (∆mv, f) of all (∆mv, f)-lacunary statistical convergent sequences of order α with the help of unbounded modulus function f, lacunary sequence (θ), generalized difference operator ∆ mv and real number α ∈ (0, 1]. We also introduce the space ωαθ (∆mv, f) of all strong (∆mv, f)-lacunary summable sequences of order α. Properties related to these spaces are studied. Inclusion relations between spaces Sαθ (∆mv, f) and ωα θ (∆mv, f) are established under certain conditions.

N- Monotone Approximation in Weighted Space

The purpose of this paper is to discuss the degree of best monotone multi- approximation of unbounded functions weighted space in terms the modules of smoothness by using some algebraic linear operators. In addition, introduced ,proofs some properties of the modulus of smoothness.

Piecewise Monotone Approximation of Unbounded Functions In Weighted Space LP,w([-,

In this paper, investigate the approximation of unbounded functions in weighted space, by using trigonometric polynomials considered. We introduced type of polynomials piecewise monotone having same local monotonicity as unbounded functions without affecting the order of huge error have a finite number of max. and min. unbounded functions that amount. In addition, we established not included any of extreme points of this functions, of and closed subset γ on closed intervals then there exist class of polynomials such that the best of approximation has high or order of and such that for sufficiently great of the polynomials and functions have the same monotonicity at each of γ.

Best One Sided Multiplier Approximation of Unbounded Functions by Polynomials Operators

The purpose of this paper is present some operators by using polynomials operators of type G_n (f,x),g_n (f,x),L_n (f) and M_n (f), to get the degree of best one- sided multiplier approximation of unbounded functions by algebraic polynomials in L_(p,φ_n ) (X),X=[0,1] , by τ〖(f,δ)〗_(p,ψ_n ).

Bipartite leader-following synchronization of delayed incommensurate fractional-order memristor-based neural networks under signed digraph via adaptive strategy

In this work, an adaptive controller, formulated as linear feedback controls plus nonlinear parts, is synthesized to achieve bipartite leader-following synchronization of delayed incommensurate fractional-order memristor-based neural networks (FMNNs), in which follower FMNNs are linearly coupled under a signed digraph. The salient features of this research lie in two aspects: (1) the assumption on time-varying delays is very weak, since it neither requires boundedness of delays nor restricts the differentiation of time delay functions; (2) the adaptive controller contains no delay term, and it is feasible for both bounded and unbounded activation functions. As the preparatory work for stability analysis of the controlled synchronization error system, the ready-made results on delayed incommensurate fractional-order linear positive systems, especially stability condition and comparison principle, are perfected by relaxing premises of time-varying delays. Besides, a group of differential inclusion inequalities are established as a powerful aid in scaling the bipartite synchronization error system. More relevantly, an algebraic synchronization criterion, formulated in terms of coupling strength, inner coupling matrix, Laplacian matrix and FMNN system parameters, is proved with the benefit of vector Lyapunov function and positive system theory. With the controller utilized, linear feedback controls can be exerted merely on partial neurons of some selected FMNNs, which is exemplified by numerical simulations.

Continuity of envelopes of unbounded plurisubharmonic functions

On bounded B-regular domains, we study envelopes of plurisubharmonic functions bounded from above by a function phi which is continuous in the extended reals on the closure of the domain. For phi satisfying certain additional criteria limiting its behavior at the singularities, we establish a set where the Perron–Bremermann envelope P phi is guaranteed to be continuous. This result is a generalization of a classic result in pluripotential theory due to J. B. Walsh. As an application, we show that the complex Monge–Ampère equation (ddcu)n=μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} (dd^cu)^n = \\mu \\end{aligned}$$\\end{document}being uniquely solvable for continuous boundary data implies that it is also uniquely solvable for a class of boundary values continuous in the extended reals and unbounded from above.

ON BOX DIMENSION OF HADAMARD FRACTIONAL INTEGRAL (PARTLY ANSWER FRACTAL CALCULUS CONJECTURE)

This paper probes into change of box dimension for an arbitrary fractal continuous function after Hadamard fractional integration. For classic calculus, we know that a function’s differentiability increases one-order after integration, and decreases one-order after differentiation. But for fractional integro-differentiation, this similar fundamental problem is still open [F. B. Tatom, The relationship between fractional calculus and fractals, Fractals 2 (1995) 217–229; M. Zähle and H. Ziezold, Fractional derivatives of Weierstrass-type functions, J. Comput. Appl. Math. 76 (1996) 265–275; Y. S. Liang and W. Y. Su, Riemann–Liouville fractional calculus of 1-dimensional continuous functions, Sci. Sin. Math. 4 (2016) 423–438 (in Chinese); Y. S. Liang, Progress on estimation of fractal dimensions of fractional calculus of continuous functions, Fractals 26 (2019), doi:10.1142/S0218348X19500841] (see conjecture 1.1), although it is believed that the roughness of a fractal function is increased or decreased accounts [Formula: see text] after [Formula: see text]-order fractional differentiation or integration. This paper partly answers this problem. Particularly, the estimation of box dimension of Hadamard fractional integral is of methodology for considering similar fractional integration. In this paper, it is proved that the upper box dimension of the graph of [Formula: see text] does not increase after [Formula: see text]-order Hadamard fractional integral [Formula: see text], i.e. [Formula: see text] Particularly, while [Formula: see text], for example, a function is of unbounded variation and/or infinite length, it holds [Formula: see text] which answers completely this conjecture for one-dimensional fractal function. The published paper could verify Conjecture 1.1 under Riemann–Liouville or Wyel fractional integral, for only constructed functions with one-dimensional fractal [Y. S. Liang and W. Y. Su, Riemann–Liouville fractional calculus of 1-dimensional continuous functions, Sci. Sin. Math. 4 (2016) 423–438 (in Chinese); Y. Li and W. Xiao, Fractal dimensions of Riemann–Liouville fractional integral of certain unbounded variational continuous function, Fractals 25 (2017), doi:10.1142/S0218348X17500475; Y. S. Liang, Fractal dimension of Riemann–Liouville fractional integral of 1-dimensional continuous functions, Fract. Calc. Appl. Anal. 21 (2018) 1651–1658; Q. Zhang, Some remarks on one-dimensional functions and their Riemann–Liouville fractional calculus, Acta Math. Sin. 3 (2014) 517–524; J. Wang and K. Yao, Construction and analysis of a special one-dimensional continuous function, Fractals 25 (2017), doi:10.1142/S0218348X17500207; X. Liu, J. Wang and H. L. Li, The classification of one-dimensional continuous functions and their fractional integral, Fractals 26 (2018), doi:10.1142/S0218348X18500639].

Sampled-data observer based event-triggered leader-follower consensus for uncertain nonlinear multi-agent systems☆

This paper addresses the event-triggered output feedback consensus problem for feedforward nonlinear multi-agent systems with a directed topology graph. To avoid continuous monitoring and communication among agents, the periodic sampled-based output information is adopted to construct the observer. The observer-based controller for every agent updates under a newly designed triggering mechanism, which relies on its output and the relative output information of its neighbour agents at sampling points. With the designed event-triggered control protocol, the leader-follower consensus is achieved and the Zeno behavior is excluded effectively. Moreover, the sampling period and state delay of the considered system can be arbitrary positive numbers. Finally, the consensus result is extended to a class of feedforward multi-agent systems with the system nonlinearities satisfying the increment ratio of an unbounded time-varying function. A simulation example with two cases is given to demonstrate the proposed event-triggered control protocols.

The Cauchy problem and distribution of local fluctuations of one Riesz gravitational field

In the class of discontinuous unbounded initial functions with an integrable singularity, we consider the Cauchy problem for the pseudodifferential equation of local action of moving objects in the corresponding Riesz gravitational field. The fundamental solution to this problem is the Cauchy probability distribution of the force of local interaction between these objects. An explicit form of this solution is obtained, and the correct solvability of this problem is established. In this case, the form of the classical solution of the Cauchy problem is found, and the properties of its smoothness and behavior at infinity are studied.