Constructing Runge–Kutta methods with the use of artificial neural networks

A methodology that can generate the optimal coefficients of a numerical method with the use of an artificial neural network is presented in this work. The network can be designed to produce a finite difference algorithm that solves a specific system of ordinary differential equations numerically. The case we are examining here concerns an explicit two-stage Runge–Kutta method for the numerical solution of the two-body problem. Following the implementation of the network, the latter is trained to obtain the optimal values for the coefficients of the Runge–Kutta method. The comparison of the new method to others that are well known in the literature proves its efficiency and demonstrates the capability of the network to provide efficient algorithms for specific problems.     

The Heine–Borel Challenge Problem. In Honor of Woody Bledsoe

Woody Bledsoes last challenge problem is the analogical transfer of the Heine–Borel theorem for real intervals to the two-dimensional case. This could not be solved by the up-to-then-known techniques of analogical theorem proving. The Heine–Borel theorem is a widely known result in mathematics. It is usually stated in the field of real numbers R¹, and similar versions are also true in R², in topology, and in metric spaces. This article shows how analogy-driven proof plan construction is applicable to this genuinely mathematical problem. Our goal here was to use a source proof plan of HB1 (the Heine–Borel theorem in R¹) as a guide to automatically produce a proof plan of HB2 (the Heine–Borel theorem in R²). We were able to accomplish our goal by generating the target proof plan of HB2 by reformulation and analogical replay.     

N-D representation and generalised Kalman–Yakubivich–Popov lemma of spatially interconnected systems with interconnected chains

This paper presents a multidimensional (N-D) hybrid Roesser model for a class of spatially interconnected systems (SISs) with interconnected chains. A generalised Kalman–Yakubivich–Popov (KYP) lemma for N-D hybrid Roesser model is developed, which plays a significant role in the system control and synthesis. As the application of the developed generalised KYP lemma, robust finite-frequency H_∞ filter problem for SISs with interconnected chains is studied. Moreover, it is shown that the developed finite-frequency filter design methods are generally less conservative than the entire frequency ones. Finally, a practical example is provided to validate the effectiveness of the proposed method.     

Optimization of sequences in CDMA systems: A statistical–mechanics approach

A statistical–mechanics approach is useful not only in analyzing the macroscopic system performance of wireless communication systems, but also in discussing design problems of wireless communication systems. We discuss a design problem of spreading sequences in code-division multiple-access (CDMA) systems, as an example demonstrating the usefulness of the statistical–mechanics approach. We analyze, via the replica method, the average mutual information between inputs and outputs of a randomly spread CDMA channel, and discuss the optimization problem with the average mutual information as the measure of optimization. It has been shown that the average mutual information is maximized by orthogonally-invariant random Welch bound equality (WBE) spreading sequences. This finding is an extension of the optimality of WBE spreading sequences with Gaussian inputs for non-Gaussian inputs.     

A third Strang lemma and an Aubin–Nitsche trick for schemes in fully discrete formulation

In this work, we present an abstract error analysis framework for the approximation of linear partial differential equation problems in weak formulation. We consider approximation methods in fully discrete formulation, where the discrete and continuous spaces are possibly not embedded in a common space. A proper notion of consistency is designed, and, under a classical inf–sup condition, it is shown to bound the approximation error. This error estimate result is in the spirit of Strang’s first and second lemmas, but applicable in situations not covered by these lemmas (because of a fully discrete approximation space). An improved estimate is also established in a weaker norm, using the Aubin–Nitsche trick. We then apply these abstract estimates to an anisotropic heterogeneous diffusion model and two classical families of schemes for this model: virtual element and finite volume methods. For each of these methods, we show that the abstract results yield new error estimates with a precise and mild dependency on the local anisotropy ratio. A key intermediate step to derive such estimates for virtual element methods is proving optimal approximation properties of the oblique elliptic projector in weighted Sobolev seminorms. This is a result whose interest goes beyond the specific model and methods considered here. We also obtain, to our knowledge, the first clear notion of consistency for finite volume methods, which leads to a generic error estimate involving the fluxes and valid for a wide range of finite volume schemes. An important application is the first error estimate for multi-point flux approximation L and G methods.     

Stability properties of discrete-continuous feedback control systems with signal dependent sampling III. Stability properties†

The nature and existence of recurrent trajectories have been considered in this final paper on the stability properties of discrete-continuous systems with signal dependent sampling. Simulation results illustrate the complexity which may arise in recurrent trajectories and furthermore demonstrate the effect of ‘stability control’ in a signal dependent sampling system. These results lead to the concepts of ‘controllable stability’ and ‘controllable quality’ of which the former has been considered in some detail.     

A Leslie–Gower-type predator–prey model with sigmoid functional response

In this work, a continuous-time predator–prey model of Leslie–Gower type considering a sigmoid functional response is analysed. Using the MatLab package some simulations of the dynamics are shown. Conditions for the existence of equilibrium points, their nature and the existence of at least one limit cycle in phase plane are established. The existence of a separatrix curve dividing the behaviour of trajectories is proved. Thus, two closed trajectories can have different ω-limits being highly sensitive to initial conditions. Moreover, for a subset of parameter values, it can be possible to prove that the point (0,0) can be globally asymptotically stable. So, both populations can go to extinction, but simulations show that this situation is very difficult. According to our knowledge no previous work exists analysing the model presented here. A comparison of the model here studied with the May–Holling–Tanner model shows a difference on the quantity of limit cycles.     

A Stochastic Chartist–Fundamentalist Model with Time Delays

A stochastic chartist?Cfundamentalist model of speculative asset dynamics in financial markets is developed. The model is represented by a stochastic delay-differential equation (SDDE). The SDDE is then solved using approximation and numerical Monte Carlo methods. The results show that for large time delays, the SDDE generates market-like stock price dynamics that reflect the memory effects of the time delay. The resultant dynamics agree with the empirical observation of the tendency of stock markets to deviate from pure random walk.     

Dynamic properties of the Fisher–Kolmogorov–Petrovskii–Piscounov equation with the deviation of the spatial variable

We consider the problem of the density wave propagation of a logistic equation with the deviation of the spatial variable and diffusion (the Fisher–Kolmogorov equation with the deviation of the spatial variable). The Ginzburg–Landau equation was constructed in order to study the qualitative behavior of the solution near the equilibrium state. We analyzed the profile of the wave equation and found conditions for the appearance of oscillatory regimes. The numerical analysis of the wave propagation shows that, for a fairly small spatial deviation, this equation has a solution similar to that the classical Fisher–Kolmogorov equation. An increase in this spatial deviation leads to the existence of the oscillatory component in the spatial distribution of solutions. A further increase in the spatial deviation leads to the destruction of the traveling wave. This is expressed in the fact that undamped spatiotemporal fluctuations exist in a neighborhood of the initial perturbation. These fluctuations are close to the solution of the corresponding boundary value problem with periodic boundary conditions. Finally, when the spatial deviation is large enough we observe intensive spatiotemporal fluctuations in the whole area of wave propagation.     

A multi-echelon production–inventory system with supply disruption

The article investigates an integrated multi-layer supply chain model consisting of supplier, manufacturer and retailer while supply disruption, machine breakdown, safety stock, maintenance breakdown occur simultaneously. At beginning of the production, manufacturer keeps some raw materials in stock received from second supplier at high price, as safety stock due to supply disruption of first supplier. Corrective maintenance is done immediately to restore its normal stage when machine breakdown occurs. Stock out situations at manufacturer and retailer are considered due to disruption of production for machine breakdown. The integrated expected costs of the chain in centralized (collaborating) and decentralized (Stakelberg approach) system are compared. A numerical example and its sensitivity analysis are provided to test feasibility of the model.     

On some properties of Goursat–Darboux systems with distributed and boundary controls

In this paper, a Goursat–Darboux control system is considered. In the first part, the bang-bang principle and some approximation results concerning the piecewise constant controls, for a linear system with distributed and boundary controls, are proved. In the second part, an approximation result concerning the bang-bang controls, for a special case of nonlinear (in a state) system with distributed controls, is derived.     

An efficient modified Polak–Ribière–Polyak conjugate gradient method with global convergence properties

The conjugate gradient (CG) method is one of the most popular methods for solving large-scale unconstrained optimization problems. In this paper, a new modified version of the CG formula that was introduced by Polak, Ribière, and Polyak is proposed for problems that are bounded below and have a Lipschitz-continuous gradient. The new parameter provides global convergence properties when the strong Wolfe-Powell (SWP) line search or the weak Wolfe-Powell (WWP) line search is employed. A proof of a sufficient descent condition is provided for the SWP line search. Numerical comparisons between the proposed parameter and other recent CG modifications are made on a set of standard unconstrained optimization problems. The numerical results demonstrate the efficiency of the proposed CG parameter compared with the other CG parameters.     

Constructing exponential estimates in compartmental systems with distributed delays: an approach based on the hale–lunel inequality

This paper presents a method for constructing exponential estimates in a compartmental system with distributed delays on the basis of the Hale–Lunel inequality and its application. The practical importance of this study is illustrated by a pharmacokinetic model from anesthesiology.     

A duality principle for state estimation with partially noise-corrupted measurements. Volume 37, number 5, pp. 1039–1056

This paper deals with the design and evaluation of a variable-structure stabilizer (VSS) for a synchronous machine using variable-structure systems theory. The stabilizer design is based on a recently proposed geometric approach for finding out the switching hyperplanes for discontinuous control. The transient response of the variable-structure stabilizer is compared with those obtained using a power system stabilizer (PSS) based on a speed signal. The computer simulation results show that the VSS is more effective in improving system damping, transient stability and post-fault recovery of the terminal voltage. It is also shown that by operating the control system in a sliding mode, the performance of the controlled synchronous machine can be made insensitive to changes in the system parameters.     

A delayed diffusive predator–prey system with predator cannibalism

A diffusive predator–prey system with cannibalism and maturation delay in predator subject to Neumann boundary condition is studied in this paper. Instability and Hopf bifurcation induced by time delay are investigated. By the theory of normal form and center manifold method, the conditions for determining the bifurcation direction and the stability of the bifurcating periodic solution are derived. Some numerical simulations are given to support our results.     

A Pursuit–Evasion Differential Game with Slow Pursuers on the Edge Graph of a Simplex. I

Automation and Remote Control - We consider a differential game with several pursuing points and one evading point moving along the 1-skeleton (i.e., the edge graph) of an arbitrary simplex in...     

A computational study on the quantum transport properties of silicene–graphene nano-composites

Graphene has been conjugated with Silicene which is a 2D nanosheet of silicon crystal to analyze myriad physico-chemical properties. Upon intercalation of silicene between two graphene nanosheets, there has been a significant shift in the energy of electronic configuration at different isovalues from − 0.12 to + 0.12. Similarly, by analyzing the electronic energy states of silicene–graphene–silicene, a range of isovalues from − 0.08 to + 0.08 were observed. I–V curve exhibited a linear response for graphene–silicene–graphene sandwiched structure and a semiconducting like behavior for silicene–graphene–silicene structure. Band gap measurement in case of graphene–silicene–graphene system is reported to be ~ 0.18 eV, which is a narrow region. While in case of silicene–graphene–silicene, a band gap value of ~ 1.01 eV is calculated that appears to be a pretty broad region. Transmission spectrum also shows intensity in peaks for Gr–Si–Gr case as compared to Si–Gr–Si combinations. Silicon is widely perceived to exhibit outstanding semiconducting behavior and has already been used in devising various electronic devices. In this present work, we try to analyze the outcome of the silicene and graphene at the nanometer scale in various combinations in a bid to understand the potential interaction mechanism between the two nanosheets which would help in the fabrication of the silicene–graphene based optoelectronic devices.