Terms Of Integrals Research Articles

Ab initio extended Hubbard model of short polyenes for efficient quantum computing.

We propose introducing an extended Hubbard Hamiltonian derived via the abinitio downfolding method, which was originally formulated for periodic materials, toward efficient quantum computing of molecular electronic structure calculations. By utilizing this method, the first-principles Hamiltonian of chemical systems can be coarse-grained by eliminating the electronic degrees of freedom in higher energy space and reducing the number of terms of electron repulsion integral from O(N4) to O(N2). Our approach is validated numerically on the vertical excitation energies and excitation characters of ethylene, butadiene, and hexatriene. The dynamical electron correlation is incorporated within the framework of the constrained random phase approximation in advance of quantum computations, and the constructed models capture the trend of experimental and high-level quantum chemical calculation results. As expected, the L1-normof the fermion-to-qubit mapped model Hamiltonians is significantly lower than that of conventional abinitio Hamiltonians, suggesting improved scalability of quantum computing. Those numerical outcomes and the results of the simulation of excited-state sampling demonstrate that the abinitio extended Hubbard Hamiltonian may hold significant potential for quantum chemical calculations using quantum computers.

An existence result for integro-differential degenerate sweeping process with convex sets.

In this paper, we study the well-posedness in the sense of existence and uniqueness of a solution of integrally perturbed degenerate sweeping processes, involving convex sets in Hilbert spaces. The degenerate sweeping process is perturbed by a sum of a singlevalued map satisfying a Lipschitz condition and an integral forcing term. The integral perturbation depends on two time-variables, by using a semi-discretization method. Unlike the previous works, the Cauchy’s criterion of the approximate solutions is obtained without any new Gronwall’s like inequality.

Particle trajectories in the KP-II equation

In the current work, we consider particle trajectories beneath traveling soliton solutions described by the Kadomtsev–Petiviashvili-II (KP-II) equation, which is a model for small-amplitude water waves in shallow water. The KP-II equation has a large number of exact solutions describing crossing line solitons. Here, we describe the particle drift induced by the single line soliton and various two- and three-soliton solutions on a two-dimensional horizontal domain.First, the derivation of the KP-II equation is used to exhibit expressions for the three components of the fluid velocity vector induced by a surface wave profile given by the KP-II equation. These velocity components can be used to specify a coupled system of ordinary differential equations (ODEs) which describe the motion of a fluid particle. Given an initial position inside the fluid for a specific particle, as well as continuously providing time-dependent values for the surface deflection, the ODE system can be solved numerically to find the particle path induced by the passage of a wave. A special numerical integration grid tracking the particle is used to handle integral terms occurring in the fluid velocity expressions.

Bioprinting of osteochondral tissues: A perspective on current gaps and future trends.

Osteochondral tissue regeneration has remained a critical challenge in orthopaedic surgery, especially due to complications of arthritic degeneration arising out of mechanical dislocations of joints. The common gold standard of autografting has several limitations in presenting tissue engineering strategies to solve the unmet clinical need. However, due to the complexity of joint anatomy, and tissue heterogeneity at the interface, the conventional tissue engineering strategies have certain limitations. The advent of bioprinting has now provided new opportunities for osteochondral tissue engineering. Bioprinting can uniquely mimic the heterogeneous cellular composition and anisotropic extra-cellular matrix (ECM) organization, while allowing for targeted gene delivery to achieve heterotypic differentiation. In this perspective, we discuss the current advances made towards bioprinting of composite osteochondral tissues and present an account of challenges—in terms of tissue integration, long-term survival, and mechanical strength at the time of implantation—required to be addressed for effective clinical translation of bioprinted tissues. Finally, we highlight some of the future trends related to osteochondral bioprinting with the hope of in-clinical translation.

Random telegraph processes with nonlocal memory.

We study two-state (dichotomous, telegraph) random ergodic continuous-time processes with dynamics depending on their past. We take into account the history of the process in an explicit form by introducing integral nonlocal memory term into conditional probability function. We start from an expression for the conditional transition probability function describing additive multistep binary random chain and show that the telegraph processes can be considered as continuous-time interpolations of discrete-time dichotomous random sequences. An equationinvolving the memory function and the two-point correlation function of the telegraph process is analytically obtained. This integral equationdefines the correlation properties of the processes with given memory functions. It also serves as a tool for solving the inverse problem, namely for generation of a telegraph process with a prescribed pair correlation function. We obtain analytically the correlation functions of the telegraph processes with two exactly solvable examples of memory functions and support these results by numerical simulations of the corresponding telegraph processes.

Neuro-adaptive path following control of autonomous ground vehicles with input deadzone

This article investigates the path-following control problem of an autonomous ground vehicle (AGV) with unknown external disturbances and input deadzones. Neural networks are used to estimate unknown external disturbances, dead zones, and nonlinear functions. The minimum learning parameter scheme is employed to adjust the neural network to reduce the computational load. A backstepping control is proposed to facilitate the tracking of the target path. The steady-state path-following error is decreased by adding an integral error term to the backstepping controller. Command filtering is employed to address the explosion of the complexity issue of the conventional backstepping approach, and the filtering error is compensated via an auxiliary signal. Lyapunov stability study indicates that the AGV closed-loop system is bounded by the proposed control with reasonable accuracy. At last, simulations are given to demonstrate the potential of the proposed scheme in path-following control.

Observer-based guaranteed-cost bipartite time-varying formation tracking control for multiagent systems subject to communication delays

The paper centers on the guaranteed-cost bipartite time-varying formation tracking control for multiagent systems with nonuniform time-varying communication delays under the framework of a distributed observer. An integral quadratic cost function is presented, founded on formation tracking and observation error. Firstly, distributed state observers, using the relative measurement output between neighbors, are established so that the leader and follower can accurately estimate each agent's unmeasurable state. Subsequently, a bipartite formation tracking protocol is proposed, incorporating estimation information and nonuniform time-varying communication delays. This protocol aims to achieve bipartite time-varying formation tracking for multiagent systems. Next, a new augmented Lyapunov-Krasovskii functional with delay-product and triple integral terms is constructed. By integrating the second-order Bessel-Legendre inequality technique, the delay-dependent reciprocally convex combination inequality method, and the free weight matrix technique, a new guaranteed-cost bipartite time-varying formation tracking criterion is developed, offering less conservatism. Compared to certain existing delay conditions, the stability conditions assure a greater allowable upper bound of delay. Finally, a numerical example is conducted to confirm the validity and superiority of the theoretical findings.

Differential image motion in astrometric observations with very large seeing-limited telescopes

We investigate how to quantitatively model the observed differential image motion (DIM) in relative astrometric observations. As a test bed we used differential astrometric observations from the FORS2 camera of the Very Large Telescope (VLT) obtained during 2010--2019 under several programs of observations of southern brown dwarfs . The measured image motion was compared to models that decompose atmospheric turbulence in frequency space and translate the vertical turbulence profile into DIM amplitude. This approach accounts for the spatial filtering by the telescope's entrance pupil and the observation parameters (field size, zenith angle, reference star brightness and distribution, and exposure time), and it aggregates that information into a newly defined metric integral term. We demonstrate excellent agreement (within 1<!PCT!>) between the model parameters derived from the DIM variance and determined by the observations. For a 30 s exposure of a typical 1 field close to the Galactic plane, image motion limits astrometric precision to sim 60 mu as when sixth-order transformation polynomial is applicable. We confirm that the measured image motion variance is well described by Kolmogorov-type turbulence with exponent 11/3 dependence on the field size at effective altitudes of 16--18 km, where the best part of the DIM is generated. Extrapolation to observations with extremely large telescopes enables the estimation of the astrometric precision limit for seeing-limited observations of sim 5 mu as, which has a variety of exciting scientific applications.

Finite-time H∞ synchronization of semi-Markov jump neural networks with two delay components with stochastic sampled-data control

This article investigates the finite-time H∞ synchronization for semi-Markov jump neural networks with two delay components based on stochastic sampled data control. Additionally, the parametric uncertainties are randomly varying which follows the Bernoulli distributed sequences. In the stochastic sampled data control, the sampling interval ′m′ is supposed to be two different values in the time-varying component with given probability conditions. By constructing triple and quadruple integral term in the Lyapunov-Krasovskii functional (LKF) a new integral inequality technique is addressed to derive the main results. Dissimilar from previous literature, involving the new integral inequality, a delay dependent finite-time H∞ synchronization requirements are acquired with regard to linear matrix inequalities (LMIs). In the end, the effectiveness of the considered stochastic sampled data control finite time synchronization scheme is highlighted by numerical examples.

Investigation of the thermal conductivity of SiO2 glass using molecular dynamics simulations

AbstractA recent study showed that generating local crystalline paths within a glassy matrix provides only a moderate increase in thermal conductivity until the path fills most of the total volume. It was indicated that percolation theory governs the thermal conductivity in such crystalline‐implanted glass composites. In this study, we use computer simulations to investigate the effect of partial vitrification by breaking structural order of silica crystals by local vitrification. Such locally vitrified crystal structures are made by the bond‐transposition method, where the extent of vitrification can be tuned. Also, we test melt‐quenching of three SiO2 crystalline structures (α‐quartz, coesite, and stishovite) to make silica glasses with different densities. The cooling rates and pressures were varied during the melting process in order to obtain glass structures with varied degrees of order. Our results show that the thermal conductivity of silica glasses calculated using Green–Kubo relation decreases rapidly as soon as the crystallinity is disturbed by bond transition or generated local glassy phase. This implies the phonon mean free path rapidly shrinks with even a small number of scatterers, regardless of the original polymorph. Through the investigation of the pressure effect, a strong proportional relation between the thermal conductivity and the density is found among all silica samples. Such correlation can be accounted by assuming a constant integral term in the Green–Kubo relation without the volume scaling, which is reasonable based on a weak dependence of κV on the density of silica glass except with a slight increase at low quench pressures. We hypothesize that when the quench pressure increases moderately, an increase of characteristic ring size tends to generate a more flexible SiO2 structure which results more phonon scattering. Green–Kubo modal analysis further shows that the contribution from modes at low frequencies is suppressed when SiO2 is melt‐quenched at higher pressure and vice versa.

Development of an Adaptive Fuzzy Integral-Derivative Line-of-Sight Method for Bathymetric LiDAR Onboard Unmanned Surface Vessel

Previous control methods developed by our research team cannot satisfy the high accuracy requirements of unmanned surface vessel (USV) path-tracking during bathymetric mapping because of the excessive overshoot and slow convergence speed. For this reason, this study developed an adaptive fuzzy integral-derivative line-of-sight (AFIDLOS) method for USV path-tracking control. Integral and derivative terms were added to counteract the effect of the sideslip angle with which the USV could be quickly guided to converge to the planned path for bathymetric mapping. To obtain high accuracy of the look-ahead distance, a fuzzy control method was proposed. The proposed method was verified using simulations and outdoor experiments. The results demonstrate that the AFIDLOS method can reduce the overshoot by 79.85%, shorten the settling time by 55.32% in simulation experiments, reduce the average cross-track error by 10.91% and can ensure a 30% overlap of neighboring strips of bathymetric LiDAR outdoor mapping when compared with the traditional guidance law.

Takagi-Sugeno fuzzy sampled-data observer design for nonlinear permanent magnet synchronous generator-based wind turbine systems using auxiliary function-based integral inequality technique

This study focuses on designing the fuzzy sampled-data observer (FSDO) scheme for nonlinear permanent magnet synchronous generator (PMSG)-based wind turbine systems (WTS). The primary objective of this problem is to solve an unmeasurable state problem under the mismatched premise variables conditions among system, observer, and controller, respectively. To do this, firstly, the proposed wind turbine model is described in a Takagi-Sugeno fuzzy model due to the complex nonlinearities in PMSG-based WTS. Next, based on the measured output signals and sampled estimated states, the FSDO is designed to understand some unmeasurable state variables of the studied system. Then, a novel auxiliary function-based integral inequality (AFBII) is presented to approximate the integral quadratic terms related to sampling information. Thanks to the proposed AFBII technique and the introduction of slack variables, several new and less conservative stability requirements are derived in the formulations of linear matrix inequalities (LMIs) using the looped Lyapunov function. Finally, the simulation studies for the considered wind turbine model are validated numerically, and then some comparative results are provided to show the superiority and applicability of the theoretical observations.

Asymptotical convergence of solutions of boundary value problems for singularly perturbed higher-order integro-differential equations

The article considers a two-point boundary value problem for a linear integro-differential equation of the n + m order with small parameters for m higher derivatives, provided that the roots of the additional characteristic equation are negative. The aim of the study is to obtain asymptotic estimates of the solution, to find out the asymptotic behavior of solutions in the vicinity of points where additional conditions are set, as well as to construct a degenerate problem, the solution of which tend to the solution of the initial perturbed boundary value problem. The Cauchy function and boundary functions of a boundary value problem for a singularly perturbed homogeneous differential equation are constructed, and their asymptotic estimates are obtained. Using the Cauchy functions and boundary functions, an analytical formula for solutions to the boundary value problem is obtained. A theorem on an asymptotic estimate for the solution of the considered boundary value problem is proved. The asymptotic behavior of the solution with respect to a small parameter and the order of growth of its derivatives are established. It is shown that the solution of the boundary value problem under consideration at the left end of this segment has the phenomenon of an initial jump and the order of this jump is determined. A modified degenerate boundary value problem containing initial jumps of the solution and the integral term is constructed.

Robust stability and H∞$$ {H}_{\infty } $$ performance evaluation for time‐varying‐delayed uncertain linear systems based on variable‐augmented‐based free‐weighting matrices

AbstractThe problem of the robust stability and performance evaluation of time‐varying‐delayed uncertain linear systems (TVDULSs) is studied in this article. At first, to obtain a modified robust delay‐dependent condition of TVDULSs, a novel augmented Lyapunov‐Krasovskii functional (LKF) is established, and a variable‐augmented‐based free‐weighting matrices (VAFWMs) technique is tailored to evaluate the functional derivative. Compared with the existing robust stability methods, the constructed functional takes fully time delay information of the system into consideration and provides more delay‐cross terms, where the corresponding matrices are used to relax the constraints of linear matrix inequalities in the stability condition. Also, the VAFWMs method fully considers the linear relation of some delayed integral terms in the functional derivative, which avoids the usage of a quadratic order or higher order terms of the time delay. Then, an improved performance analysis criterion of TVDULSs is obtained against the disturbance on the basis of the proposed robust stability condition. The admissible maximal delay upper bounds that the uncertain linear system can tolerate are provided by proposed criterion under a given performance level. At last, two numerical examples are presented to verify the effectiveness and the superiority of the methods obtained.

Numerical approximation of a generalized time fractional partial integro-differential equation of Volterra type based on a meshless method

The moving least squares (MLS) approximation is used in this study to solve time-fractional partial integro-differential equations (TFPIDE). In our approach, we approximate the time Caputo fractional derivative term and the integral term by using a finite difference scheme and trapezoidal method, respectively, which yields an order of accuracy of O(τ+τ2−α), where 0<α<1. We thoroughly investigate the applicability and validity of the proposed method and provide an error estimate for its performance. Additionally, we solve various numerical problems to demonstrate the accuracy and computational efficiency of our approach, confirming the theoretical findings.

A three-dimensional meshless fluid–shell interaction framework based on smoothed particle hydrodynamics coupled with semi-meshless thin shell

Meshless methods are suitable for fluid–structure interaction simulations due to its Lagrangian feature and capability of handling large deformations. A three-dimensional meshless framework for the fluid–structure interaction simulation with shell structures is proposed in this study. The weakly compressible smoothed particle hydrodynamics is deployed in the fluid domain, where the boundary integral terms are incorporated to handle the one-layer shell boundary. On the other hand, a semi-meshless finite volume modeling method based on the Reissner–Mindlin shell is deployed in the one-layer shell domain. A one-way coupling method is employed for the interactive forcing between the fluid and shell particles near the interface, where the boundary integral method is employed to approximate the gradient and Laplacian operators in the fluid governing equations. Regarding the no-slip boundary, the original boundary integral method always leads to the artificial slipping phenomenon between the fluid and shell interface. To this end, a velocity penalty term with a relaxation factor is proposed to enforce the no-slip boundary condition, which substantially improves the accuracy of the original boundary integral method. In order to reduce computational costs, a multi-resolution strategy for the meshless framework is employed. A set of challenging cases with low and high Reynolds numbers are simulated by the present framework, and good accuracy is observed with the velocity penalty term in these cases. Overall, the new framework shows a satisfactory performance.

Finite Difference with Quintic B-Splines for Solving a system of nonlinear Volterra Integro-Differential Equations of integer order

This study presents a novel numerical approach for solving systems of nonlinear Volterra Integro-Differential Equations (VIDEs) of integer order by integrating the Finite Difference Method (FDM) with Quintic B-Splines for. The proposed method aims to address the challenges posed by the nonlinearity and integral terms inherent in VIDEs while providing enhanced accuracy and stability in the numerical solution. An error estimate of the approximate solution is proved. Finally, Some example are presented to illustrate the simplicity and the effectiveness of the propose method

A new inequality and its application in descriptor Markovian jump systems with time-varying delays

This paper is focused on the dissipativity analysis of descriptor Markovian jump systems with time-varying delays. For achieving this aim, an augmented mode-dependent Lyapunov-Krasovskii functional (LKF) is constructed on the basis of the state decomposition method. Next, a novel integral inequality is proposed through the generalized free-weighting-matrix (GFWM) method as well as zero-valued equalities to estimate single integral terms more precisely. Then, an ameliorative stochastic admissibility and dissipative criterion is derived by utilizing the proposed integral inequality, the augmented LKF and other bounding techniques. Finally, the effectiveness and superiority of the presented results are confirmed with some numerical examples.

Well-Posedness and Regularity of Solutions to Neural Field Problems with Dendritic Processing

We study solutions to a recently proposed neural field model in which dendrites are modelled as a continuum of vertical fibres stemming from a somatic layer. Since voltage propagates along the dendritic direction via a cable equation with nonlocal sources, the model features an anisotropic diffusion operator, as well as an integral term for synaptic coupling. The corresponding Cauchy problem is thus markedly different from classical neural field equations. We prove that the weak formulation of the problem admits a unique solution, with embedding estimates similar to the ones of nonlinear local reaction–diffusion equations. Our analysis relies on perturbing weak solutions to the diffusion-less problem, that is, a standard neural field, for which weak problems have not been studied to date. We find rigorous asymptotic estimates for the problem with and without diffusion, and prove that the solutions of the two models stay close, in a suitable norm, on finite time intervals. We provide numerical evidence of our perturbative results.

Improved results for T-S fuzzy systems with two additive time-varying delays via an extended binary quadratic function negative-determination lemma

This paper is concerned with the stability and stabilization problems for T-S fuzzy systems with two additive time-varying delays. Firstly, a novel Lyapounov-Krasovskii functional (LKF) is constructed by delay-product-type functional method together with the state vector augmentation. In order to handle time-delay square terms introduced into the derivative of LKF, an extended binary quadratic function negative-determination lemma is proposed. A less conservative delay-dependent stability condition is developed since not only some new bounding technologies are employed to deal with integral terms, but also the proposed lemma is employed to dispose nonlinear time-delay terms in derivative of constructed function. Then, the corresponding controller design method for closed-loop delayed fuzzy system is derived based on parallel distributed compensation scheme. Finally, four numerical examples are given to illustrate the superiority and effectiveness of the proposed criteria.