A delay-decomposition approach for stability of neural network with time-varying delay

This paper studies delay-dependent asymptotical stability problems for the neural system with time-varying delay. By dividing the whole interval into multiple segments such that each segment has a different Lyapunov matrix, some improved delay-dependent stability conditions are derived by employing an integral equality technique. A numerical example is given to demonstrate the effectiveness and less conservativeness of the proposed methods.     

A new approach to stability analysis of neural networks with time-varying delay via novel Lyapunov--Krasovskii functional

In this paper,new delay-dependent stability criteria for asymptotic stability of neural networks with time-varying delays are derived.The stability conditions are represented in terms of linear matrix inequalities(LMIs) by constructing new Lyapunov-Krasovskii functional.The proposed functional has an augmented quadratic form with states as well as the nonlinear function to consider the sector and the slope constraints.The less conservativeness of the proposed stability criteria can be guaranteed by using convex properties of the nonlinear function which satisfies the sector and slope bound.Numerical examples are presented to show the effectiveness of the proposed method.     

Three-loop vacuum integral with four-propagators using hypergeometry

A hypergeometric function is proposed to calculate the scalar integrals of Feynman diagrams.In this study,we verify the equivalence between the Feynman parametrization and the hypergeometric technique for the scalar integral of the three-loop vacuum diagram with four propagators.The result can be described in terms of generalized hypergeometric functions of triple variables.Based on the triple hypergeometric functions,we establish the systems of homogeneous linear partial differential equations(PDEs)satisfied by the scalar integral of three-loop vacuum diagram with four propagators.The continuation of the scalar integral from its convergent regions to entire kinematic domains can be achieved numerically through homogeneous linear PDEs by applying the element method.     

Stability analysis of linear dynamical systems with saturation nonlinearities and a short time delay

A class of linear dynamical systems subject to saturation nonlinearities and a short time delay were approximated by singular perturbation dynamical systems with saturation nonlinearities based on the notion of Pade approximation. The stability region of the approximate systems was proved to be decomposed and a convex Linear Matrix Inequality (LMI) optimization model was introduced to estimate the decomposed stability region with least degree of conservativeness.     

Towards a Theory of Conservative Computing

We extend the notion of conservativeness, given by Fredkin and Toffoli in 1982, to generic gates whose input and output lines may assume a finite number d of truth values. A physical interpretation of conservativeness in terms of conservation of the energy associated to the data used during the computation is given. Moreover, we define conservative computations, and we show that they naturally induce a new NP-complete decision problem. Finally, we present a framework that can be used to explicit the movement of energy occurring during a computation, and we provide a quantum implementation of the primitives of such framework using creation and annihilation operators on the Hilbert space ℂ^d, where d is the number of energy levels considered in the framework.     

Robust H_∞ control for uncertain Markovian jump systems with mixed delays

We scrutinize the problem of robust H∞control for a class of Markovian jump uncertain systems with interval timevarying and distributed delays. The Markovian jumping parameters are modeled as a continuous-time finite-state Markov chain. The main aim is to design a delay-dependent robust H∞control synthesis which ensures the mean-square asymptotic stability of the equilibrium point. By constructing a suitable Lyapunov–Krasovskii functional(LKF), sufficient conditions for delay-dependent robust H∞control criteria are obtained in terms of linear matrix inequalities(LMIs). The advantage of the proposed method is illustrated by numerical examples. The results are also compared with the existing results to show the less conservativeness.     

Electrostatic problem for a planar eccentric ring

The electrostatics problem for a concentric annular disk has been studied fairly comprehensively (see Ref. 1 and the literature cited). An eccentric ring is considered for the first time using an assumption known in the literature as the Kelvin transformation. This transformation allowed the initial problem to be reduced to that for a concentric ring with a point charge lying outside the ring in the same plane. This problem was solved using triple integral equations. An approximate solution for the charge density distribution at an eccentric annular disk is given as an expansion in terms of a small parameter. Zh. Tekh. Fiz. 68, 104–106 (December 1998)     

Nanoparticle plasmon resonances in the near-static limit

Localized surface plasmon resonances of metal nanoparticles of arbitrary shape are analyzed in the near-static limit with retardation included to the second order. Starting from the electrostatic approximation, the second-order correction to the resonant dielectric constant is expressed by means of a triple surface integral. For arbitrary nanoparticles with cylindrical symmetry we show how the triple surface integral can be significantly simplified, resulting in a computationally efficient scheme for evaluation of nanoparticle plasmon eigenresonances in the near-static limit. The approach allows for calculation of both dipolar and higher-order resonances.     

Stabilization strategy of a car-following model with multiple time delays of the drivers

An extended car-following model with multiple delays is constructed to describe driver's driving behavior. Through stability analysis, the stability condition of this uncontrolled model is given. To dampen the negative impact of the driver's multiple delays (i.e., stability condition is not satisfied), a novel control strategy is proposed to assist the driver in adjusting vehicle operation. The control strategy consists of two parts:the design of control term as well as the design of the parameters in the term. Bifurcation analysis is performed to illustrate the necessity of the design of parameters in control terms. In the course of the design of parameters in the control term, we improve the definite integral stability method to reduce the iterations by incorporating the characteristics of bifurcation, which can determine the appropriate parameters in the control terms more quickly. Finally, in the case study, we validate the control strategy by utilizing measured data and configuring scenario, which is closer to the actual traffic. The results of validation show that the control strategy can effectively stabilize the unstable traffic flow caused by driver's delays.     

The triple integral equations method for solving heat conduction equation

A new type of triple integral equation was used to determine a solution of nonstationary heat equation in axially symmetric cylindrical coordinates under mixed discontinuous boundary of the first and second kind conditions acted on the level surface of solid cylinder, with the aid of a Laplace transform, the solution of the given triple equations is introduced to a singular integral equation of the second kind.     

The stability of triple junctions during the grain boundary diffusion process

The grain boundary diffusion in a system with triple junctions is considered in such a geometry, in which the flows of diffusing atoms meet at the triple line. The solutions of the diffusion equation is given in the frameworks of Fisher's model and under the assumption of quasi-stationary distribution of the diffusing atoms along the grain boundaries. The change of the mechanical equilibrium at the triple junction due to the increase of the concentration of solute atoms is considered. It is shown that under some circumstances the triple junction looses its stability with respect to migration in the direction to the diffusion source. The stability diagrams in the segregation-diffusivity parameter space are plotted.     

Exponential synchronization of complex dynamical networks with Markovian jumping parameters using sampled-data and mode-dependent probabilistic time-varying delays

In this paper, the problem of exponential synchronization of complex dynamical networks with Markovian jumping parameters using sampled-data and Mode-dependent probabilistic time-varying coupling delays is investigated. The sam- pling period is assumed to be time-varying and bounded. The information of probability distribution of the time-varying delay is considered and transformed into parameter matrices of the transferred complex dynamical network model. Based on the condition, the design method of the desired sampled data controller is proposed. By constructing a new Lyapunov functional with triple integral terms, delay-distribution-dependent exponential synchronization criteria are derived in the form of linear matrix inequalities. Finally, two numerical examples are given to illustrate the effectiveness of the proposed methods.     

Coherent State Path Integral for the Harmonic Oscillator and a Spin Particle in a Constant Magnetic field

Definition and formulas for harmonic oscillator coherent states and spin coherent states are reviewed in detail. The path integral formalism and its relation with the partition function of a system are also reviewed. The harmonic oscillator coherent state path integral is evaluated exactly at the discrete level and then used to find its continuum limit using various regularizations. The computation of the path integral for a particle of spin s put in a constant magnetic field is carried out using harmonic oscillator coherent states and spin coherent states, with a careful analysis of infinitesimal terms (in 1/N where N is the number of time slices) appearing in the Lagrangian. A mapping of the spin system into a CP¹ model is shown explicitly. The theory of a spinless particle in the field of a magnetic monopole and its relation with the spin system are explained. The equivalence of these two models is established up to infinitesimal order by the introduction of an external field correction. This gives a new representation of a coherent state path integral in terms of a more familiar Feynman path integral.     

Three-fold photoelectron counting statistics for gaussian light

The three-fold generating function Q(λ₁, λ₂, λ₃) for photocount statistics is evaluated directly from the gap integral equation in the short counting time regime. The triple joint probabilities and the triple photoelectron correlation function are obtained via Q(λ₁, λ₂, λ₃) in terms of the field correlation function of the incident quasimonochromatic radiation. Three-fold moments are also evaluated via another approach which permits an extension to arbitrary counting times.     

Novel delay-dependent stability criteria for neural networks with interval time-varying delay

The problem of delay-dependent asymptotic stability for neural networks with interval time-varying delay is investigated. Based on the idea of delay decomposition method, a new type of Lyapunov-Krasovskii functional is constructed. Several novel delay-dependent stability criteria are presented in terms of linear matrix inequality by using the Jensen integral inequality and a new convex combination technique. Numerical examples are given to demonstrate that the proposed method is effective and less conservative.     

A simple model for the diffractive overlap integral

A model is introduced for the diffractive overlap integral in which the overlap is represented by a set of nonforward triple Regge diagrams. The formalism provides a clear framework within which to discuss questions concerning the peripherality of diffractive processes.     

Eigenvalue problems modelling the stability of a plane-parallel shear in a two-layer viscous composite

We present the derivation of a precise statement of a linearized eigenvalue problem that models the stability of a plane-parallel shear flow in a composite consisting of two different layers of heavy Newtonian media. The peculiarities of the influence on the stability of five dimensionless criteria participating in the equations and boundary conditions are discussed. The specific feature of the power conditions of contact is that a spectral parameter ?? enters these conditions nonlinearly. As applied to the composite structure, we develop the apparatus of integral relations, which makes it possible to derive the lower integral bounds for the stability parameter (the real part of ??), and hence also upper bounds of combinations of critical numbers.     

Determination of unsteady heat release distribution from acoustic pressure measurements: a reformulation of the inverse problem

An integral method is developed to solve the inverse problem of determining the oscillatory heat release distribution from the knowledge of the acoustic pressure field within a combustor. Unlike earlier approaches, in which the problem is formulated in terms of Fredholm integral equation, the inverse problem is reformulated in terms of Volterra integral equation. This reformulation, valid for low Mach numbers (M2 < 1), facilitates the recovery of heat release at all frequencies. The resulting Volterra integral equation is solved using both direct numerical method and implicit least-squares method. The results show that the implicit least-squares method is superior to the direct numerical method and yields accurate determination of heat release at all frequencies.     

Dyadic Green‘s function for an unbounded gyroelectric chiral medium in cylindrical coordinates

The dyadic Green‘s function for an unbounded gyroelectric chiral medium is formulated in the Fourier domain in cylindrical coordinates.After a triple Fourier integral of the dyadic Green‘s function has been obtained,the integral is reduced by performing the integration over the transverse Fourier variable.     

The elastic stress field produced by a point force in a cubic crystal

Fredholm in 1900 gave a solution in principle for the displacements generated by a point force at the origin in an infinite elastic solid. Eshelby in 1955 gave an approximate solution of the dilation for the triple force at the origin with no moment. Eshelby and Leibried gave expressions which converge only when The present calculation uses Fredholm's work to provide a Fourier expansion of displacement which converges for all stable cubic crystals.

The convergence of the series giving the displacement due to a point force is very slow for lithium which is on the verge of a phase change Detailed calculations of stress fields have been made for aluminium and copper. It is noted that for Cu a contribution of additional terms besides that given by Eshelby appears in the angular dependence of dilatation for the triple double force but for Al the dilatation is in close agreement with Eshelby's result. The elastic interaction energies of two triple double force interstitials and of two single double force interstitials with no moment are calculated. The calculated stresses are accurate to about 15 and 8% of their value for Cu and Al, respectively. Fifteen terms for Cu and six terms for Al were used in the series for stresses. Better accuracy could be obtained by including higher order terms in the expansions.