Mean Value Theorem Research Articles

Stochastic asymptotic stability for stochastic inertial Cohen-Grossberg neural networks with time-varying delay

This paper studies stochastic asymptotic stability for stochastic inertial Cohen-Grossberg neural networks with time-varying delay. Firstly, the second-order differential equation is converted into the first-order differential equation by appropriate variable substitution. Secondly, the existence of the equilibrium point is derived by using homeomorphic mapping, finite increment formula of Lagrange mean value theorem and linear matrix inequality. The sufficient conditions for the stochastic asymptotic stability of the equilibrium point of the system are derived by defining the appropriate operator, and constructing the appropriate positive Lyapunov function and positive-definite matrix. Thirdly, a numerical example illustrates the correctness of these theorems.

Transient Nonlinear Heat Conduction in Concrete Structures: A Semi-Analytical Approach.

Thermal loading, especially in fire scenarios, challenges the safety and long-term durability of concrete structures. The resulting heat propagation within the structure is governed by the heat conduction equation, which can be difficult to solve analytically because of the nonlinearity related to the thermophysical properties of concrete. A semi-analytical approach for the transient nonlinear heat conduction problem in concrete structures was established in the present work. The nonlinearity related to the temperature-dependent thermal conductivity, mass density, and specific heat capacity of heated concrete was taken into consideration. A Taylor series approximate solution was first established within a small neighborhood, employing the Boltzmann transformation in combination with the mean value theorem. Thereafter, it was extended to the whole domain by utilizing the Bernstein polynomial. The semi-analytical approach was validated by comparing it with the numerical results of two independent Finite Element simulations of nonlinear heat conduction along concrete plates, subjected to either moderate or fierce thermal loading. Absolute values of the relative errors are smaller than 5%. The validated semi-analytical approach was further applied to prediction of the temporal evolution of the temperature field of a scaled model of a subway station, subjected to fire disaster. The nonlinearities, related to the time-dependent surface temperature and the temperature-dependent thermophysical properties of concrete, were taken into consideration. The predictions agree well with the experimental measurements. The established semi-analytical approach exhibits good accuracy and stability, providing insight into the interaction between the thermophysical properties of concrete in the heat conduction process.

Balanced harvesting of dynamical discrete Ricker & Beverton–Holt system

In this paper, we discuss a dynamic version of the Ricker & Beverton–Holt (R&B–H)model, which states a two populations commercial fishery, where stock-recruitment effects cannot be neglected. We investigate a stable and balanced fish catches strategy that local breeding stocks are preserved abundant and stable. The Ricker & Beverton–Holt model in commercial fishery, based on the Allee effect, is combined by Beverton–Holt Logistic equation and Ricker model, yn+1=azn1+pzne−yn,zn+1=byn1+qyne−zn. where the parameters a,b,p,q, are positive real numbers and the initial values y0,z0 are arbitrary nonnegative real numbers. Using the mean value theorem and Lyapunov functional skills, we obtain the sufficient conditions to guarantee the bounded and persistent solution of the Ricker & Beverton–Holt model, and global asymptotic stability of the equilibrium. Moreover, three numerical examples are given to elaborate on the results.

A Mean Value Theorem for Tangentially Convex Functions

The main result is an equality type mean value theorem for tangentially convex functions in terms of tangential subdifferentials, which generalizes the classical one for differentiable functions, as well as Wegge theorem for convex functions. The new mean value theorem is then applied, analogously to what is done in the classical case, to characterize, in the tangentially convex context, Lipschitz functions, increasingness with respect to the ordering induced by a closed convex cone, convexity, and quasiconvexity.

Exploring problem-solving through the intervention of technology and Realistic Mathematics Education in the Calculus content course

In Thai mathematics classrooms, there is a lack of attention to support students’ mathematical problem-solving skills by working from real-world contexts that make sense to students. This study aimed to investigate how pre-service mathematics teachers’ problem solving can be explored in their content course, intervening with technology and Realistic Mathematics Education (RME) through the Mean Value Theorem (MVT) lesson. The study included nine pre-service mathematics teachers purposively selected from a public university in Thailand who attended a Calculus course. Data was collected from classroom artifacts, observation notes, and interviews. It was found in this study that the intervention of technology and RME in pre-service mathematics teachers’ content courses has the potential to build pre-service mathematics teachers’ problem-solving abilities. It was also discussed that the intervention could use RME to conceptualize mathematics theorem and cultivate Polya’s problem-solving steps. The findings provide light on the efficacy of using technology and RME in enhancing problem-solving skills among other content courses and could be used to inform the creation of mathematics curricula and instructional strategies in undergraduate content courses for mathematics education programs.

Position Measurement Based Slave Torque Feedback Control for Teleoperation Systems With Time-Varying Communication Delays

Bilateral teleoperation system is referred to as a promising technology to extend human actions and intelligence to manipulating objects remotely. For the tracking control of teleoperation systems, velocity measurements are necessary to provide feedback information. However, due to hardware technology and cost constraints, the velocity measurements are not always available. In addition, the time-varying communication delay makes it challenging to achieve tracking task. This paper provides a solution to the issue of real-time tracking for teleoperation systems, subjected to unavailable velocity signals and time-varying communication delays. In order to estimate the velocity information, immersion and invariance (I&I) technique is employed to develop an exponential stability velocity observer. For the proposed velocity observer, a linear relationship between position and observation state is constructed, through which the need of solving partial differential and certain integral equations can be avoided. Meanwhile, the mean value theorem is exploited to separate the observation error terms, and hence, all functions in our observer can be analytically expressed. With the estimated velocity information, a slave-torque feedback control law is presented. A novel Lyapunov-Krasovskii functional is constructed to establish asymptotic tracking conditions. In particular, the relationship between the controller design parameters and the allowable maximum delay values is provided. Finally, simulation and experimental results reveal that the proposed velocity observer and controller can guarantee that the observation errors and tracking error converge to zero.

Adaptive Synchronization for Delayed Chaotic Memristor-Based Neural Networks.

This article considers the adaptive synchronization problem of delayed chaotic memristor-based neural networks (MNNs). Note that MNNs are modeled as continuous systems in the flux-voltage-time (ϕ,x,t) domain where memristors are viewed as continuous systems based on HP memristors. New adaptive controllers of MNNs are proposed, where controllers are both on memristors in the flux-time (ϕ,t) domain and neurons in the voltage-time (x,t) domain. Based on the Lyapunov method, Barbalat's lemma, differential mean value Theorem, and other inequality techniques, completed synchronization criteria for delayed chaotic MNNs are derived. In the end, two examples are given to demonstrate the validity of the derived results.

A mixed nonlinear time-fractional Rayleigh-Stokes equation

This paper investigates a nonlinear time-fractional Rayleigh-Stokes equation with mixed nonlinearity containing a power-type function, a logarithmic function and an inverse time-forcing term. Applying Lagrange's mean value theorem and the compactness of the Sobolev embeddings, we estimate the complex Lipschitz property of mixed nonlinearity. We investigate the local well-posed results (local existence, regularity estimate, continuation) of the solutions in Hilbert scales space. Moreover, the global existence theory affiliated to the finite-time blow-up is considered.

Parameterization of kernels of the Volterra series for systems given by nonlinear differential equations

The presented article is devoted on an issue regarding to the transformation of nonlinear models of a certain class to the Volterra functional series. The new identification method based on analytical input and output of a system was developed. The key task of representing kernels was achieved by Frechet functional derivative. Explicit calculation of the functional derivative for the output was solved using mean-value theorem, while the sifting property of the Dirac function was used in order to solve derivative of input. Attention is given to the calculation of the second-order functional derivative. The procedure for adding linear kernels to the composition of a quadratic kernel is described. All techniques of the method to other components of the model are described in detail. The method's outcome is differential equation, which allows representation of kernels. An illustrative example of the transformation of the nonlinear differential Riccati equation is considered. The form of a subsystem consisting of linear and quadratic kernels for adding to a complex system is shown. Linear and quadratic kernels were parameterized using operational calculus within the example. The agreement of the obtained analytical results, with the frequency characteristics, which was obtained by the test signals method, is shown.

Output Feedback Control of TS Fuzzy Systems with Unmeasured Nonlinearities via a Local Separation Principle

This paper presents a fuzzy observer-based control framework for TS fuzzy models with nonlinear consequents (N-TS). The N-TS fuzzy model is constructed such that only measured nonlinearities are considered to obtain the fuzzy membership functions, and all unmeasured nonlinearities are modeled as local consequent parts. Based on the differential mean value theorem, the closed-loop system is rewritten in an adequate format, which allows proving the separation principle for asymptotic TS fuzzy observer-based control design. Sufficient design conditions, expressed in terms of linear matrix inequalities, are derived via delayed nonquadratic Lyapunov functions for the synthesis of both stabilizing fuzzy controller and Luenberger-like fuzzy observer. Using Lyapunov-based arguments of a local separation principle, a constructive procedure is provided to maximize the closed-loop domain of attraction (DoA) estimation, which is guaranteed to be included inside the predefined fuzzy modeling region. Numerical examples are given to illustrate the effectiveness of the proposed approach.

New criteria on global Mittag-Leffler synchronization for Caputo-type delayed Cohen-Grossberg Inertial Neural Networks

<abstract><p>Our focus of this paper is on global Mittag-Leffler synchronization (GMLS) of the Caputo-type Inertial Cohen-Grossberg Neural Networks (ICGNNs) with discrete and distributed delays. This model takes into account the inertial term as well as the two types of delays, which greatly reduces the conservatism with respect to the model. A change of variables transforms the $ 2\beta $ order inertial frame into $ \beta $ order ordinary frame in order to deal with the effect of the inertial term. In the following steps, two novel types of delay controllers are designed for the purpose of reaching the GMLS. In conjunction with the novel controllers, utilizing differential mean-value theorem and inequality techniques, several criteria are derived to determine the GMLS of ICGNNs within the framework of Caputo-type derivative and calculus properties. At length, the feasibility of the results is further demonstrated by two simulation examples.</p></abstract>

Generalization of Lagrange’s Mean Value Theorem

This paper deals with the mean value theorem of differential calculus due to the Lagrange’s mean value theorem which has appeared in Lagrange’s book Theories functions analytiques in 1797 as an extension of Rolle’s result from 1619. We state and prove a theorem which is similar to the generalized mean value theorem for the single variable function. Firstly, we review the mean value theorems. Moreover, the theorem is illustrated with some examples. The main propose of this paper is to establish the different forms of Lagrange’s Mean Value Theorem, as an extended forms.

An Alternative Numerical Scheme to Approximate the Early Exercise Boundary of American Options

This paper deals with a new numerical method for the approximation of the early exercise boundary in the American option pricing problem. In more detail, using the mean-value theorem for integrals, we provide a flexible algorithm that allows for reaching a more accurate numerical solution with fewer calculations rather than other previously described methods.

A class of elliptic inclusion in fractional Orlicz–Sobolev spaces

ABSTRACT We consider an elliptic inclusion that is driven by ( − Δ ) a ( ⋅ ) s called fractional a ( ⋅ ) -Laplacian operator, with Dirichlet-type boundary conditions. We take two different assumptions on the Carathéodory function f . One of them is where f satisfies the localy Lipchitz condition and the other one is when f does not satisfy an assumption of growth. With a regular Clarke subdifferential ∂ C and Lebourg's mean value theorem, by applying a variational approach together with the critical point theory, we obtain a weak solution to the inclusion problem in appropriate fractional Orlicz–Sobolev spaces.

A fractional SVIR-B epidemic model for Cholera with imperfect vaccination and saturated treatment.

Given the limited medical resources in most cholera endemic countries, in this paper, a nonlinear fractional SVIR-B (Susceptible-Vaccinated-Infected-Recovered, Bacterial) cholera epidemic model with imperfect vaccination and saturated treatment is proposed and investigated. The model performs well-posed in both epidemiologically and mathematically, especially, we demonstrate the positivity and boundedness of all solutions using the generalized mean value theorem. The control reproduction number is derived using the next generation matrix method and both local and global stability analyses for the disease-free equilibrium is performed by analyzing the characteristic equation and using Lyapunov functional. Then, the existence and stability of endemic equilibria are further addressed. Sufficient conditions for the existence of backward bifurcation and Hopf bifurcation are also derived. Subsequently, an optimal control problem with vaccination, media coverage, treatment, and sanitation as control strategies is proposed and analyzed, providing a rationale for cholera control and prevention. In addition, several numerical examples are employed to illustrate our theoretical results.

Stability analysis and system properties of Nipah virus transmission: A fractional calculus case study

In this paper, we establish a Caputo-type fractional model to study the Nipah virus transmission dynamics. The model describes the impact of unsafe contact with an infectious corpse as a possible way to transmit this virus. The corresponding area to the system properties, including the positivity and boundedness of the solution, is explored by using the generalized fractional mean value theorem. Also, we investigate sufficient conditions for the local and global stability of the disease-free and the endemic steady-states based on the basic reproduction number R0. To show these important stability features, we employ fractional Routh–Hurwitz criterion and LaSalle’s invariability principle. For the implementation of this epidemic model, we also use the Adams–Bashforth–Moulton numerical method in a fractional sense. Finally, in addition to compare the fractional and classical results, as one of the main goals of this research, we demonstrate the usefulness of minimal unsafe touch with the infectious corpse. Simulation and comparative results verify the theoretical discussions.

Lattice packings through division algebras

In this text, we will show the existence of lattice packings in a family of dimensions by employing division algebras. This construction is a generalization of Venkatesh’s lattice packing result Venkatesh (Int Math Res Notices 2013(7): 1628–1642, 2013). In our construction, we replace the appearance of the cyclotomic number field with a division algebra over the rational field. We employ a probabilistic argument to show the existence of lattices in certain dimensions with good packing densities. The approach improves the best known lower bounds on the lattice packing problem for certain dimensions.We work with a moduli space of lattices that are invariant under the action of a finite group, one that can be embedded inside a division algebra. To obtain our existence result, we prove a division algebra variant of the Siegel’s mean value theorem. In order to establish this, we describe a useful description of the Haar measure on our moduli space and a coarse fundamental domain to perform the integration.

Mean value theorem for a real continuous function with several real variables

In this paper, we establish a mean value theorem for a real continuous function with several real variables, using the Frechet semi-differentials of the function.

DMVT-based observer design for general affine class of fractional-order nonlinear MIMO systems

In this article, a new observer design technique is proposed for the general affine class of fractional-order nonlinear multi-input/multi-output (MIMO) systems. This study is presented based on the differential mean value theorem. One significant characteristic of the suggested approach is that the nonlinear dynamic of observer error is converted into a linear parameter-varying system. Stability and convergence of observation error are shown using the Lyapunov direct method leading to feasibility and existence of a solution for some linear matrix inequalities. The performance and efficacy of the proposed method are evaluated through some illustrated simulations.

A New conformable fractional derivative and applications

The motivation for this paper comes from other papers treating the fractional derivatives. We introducea new definition of fractional derivative which obeys classical properties including linearity, product rule, quotient rule, power rule, chain rule, Rolle’s theorem, mean value theorem and Taylor series. Usage of thedefined derivative is given in the example section which shows how our derivative can be used in solving differential equations. Comparison of our derivative with the derivative defined by Abdejjawad and overall conclusions are given in the conclusion section.