Mathematical modelling of a diesel common-rail system

In diesel common-rail systems, the exact knowledge of the injection pressure is important to accurately control the injected diesel mass and thus the combustion process. This paper focuses on the mathematical modelling of the hydraulic and mechanical components of a common-rail system in order to describe the dynamics of the diesel rail pressure. Based on this model, an average model is derived to reduce the model complexity and to allow for a fast calculation of the mass flow into the rail for different crank shaft revolution speeds and openings of the fuel metering unit. The main purpose of this average model is to serve as a basis for a model-based (non-linear) controller design. The stationary accuracy of the models is validated by means of measurement data.     

Coding for a multiple access OR channel: A survey

The signature coding for M active users out of T total users over a multiple access OR channel is considered. The mathematical problem is equivalent to the M-cover-free problem of extremal set theory. We survey the upper and lower bounds on the minimal code word length n(T,M), and present some code constructions. According to the current state of the theory, for 1MT

so there is a huge gap between the upper and lower bounds. Moreover, there is no known construction approaching the upper bound.     

Mathematical modelling and optimisation of a waste water treatment plant by combined oxygen electrode and biological waste water treatment model

Structured mathematical models of oxygen electrode and biological waste water treatment were combined to optimise the position of electrode in the bioreactor. Parameters sensibility testing of mathematical model has shown that the model was most sensitive to the change of these parameters (in decreasing order of significance): oxygen consumption per unit of BOD_5, specific consumption rate of adsorbed BOD₅, volumetric coefficient of oxygen transfer rate (k_La), and waste water inflow. Two positions of electrode were tested: in the bioreactor and in the bioreactors syphon outlet shaft. Results obtained for the exchange of the oxic/anoxic phase and sludge recycling plant indicated that a reduction of electrical power for the aeration could be achieved if O₂ electrode is placed in the bioreactor tank, compared to position in outlet shaft. Better positioning of electrode has resulted in shortening of the oxic/anoxic cycle by 13% and the daily working time of aerators for 0.5 h per day, respectively. In the same time, the number of oxic/anoxic cycles per day was increased by 29%. In addition, results obtained by mathematical modelling indicated no debasement of chemical composition in out coming purified water if new position will be used.     

Mathematical modelling for keyhole surgery simulations: a biomechanical model for spleen tissue

There is growing interest in the potential for computer simulationsto provide good training materials for abdominal endoscopicor ‘keyhole’ surgery. This paper describes a preliminarystudy of the mechanical properties of abdominal (spleen) tissue.We show how experimental force–displacement data can beused to derive a biomechanical model for the tissue as an incompressible,homogeneous, isotropic nonlinear elastic material with an exponentialstress–strain law. We also show how the model can be usedto predict the response of the tissue to a surgical probe. Thisinvolves solving a complicated nonlinear constrained boundary-valueproblem, and there is a good fit between the computed solutionand experimental data.     

Mathematical modelling and dynamic response of a multi-straight-line path tracing flexible robot manipulator with rotating-prismatic joint

In this study, mathematical modelling and dynamic response of a flexible robot manipulator with rotating-prismatic joint are investigated. The tip end of the flexible robot manipulator traces a multi-straight-line path under the action of an external driving torque and an axial force. Considered robot manipulator consists of a rotating prismatic joint and a sliding flexible arm with a tip mass. Flexible arm is assumed to be an Euler–Bernoulli beam carrying an end-mass. Equations of motion of the flexible manipulator are obtained by using Lagrange’s equation of motion. Effect of rotary inertia, axial shortening and gravitation is considered in the analysis. Equations of motion are solved by using fourth order Runge–Kutta method. Numerical simulations obtained by using a developed computer program are presented and physical trend of the results are discussed.     

Nonlinear modelling and control of a common rail injection system for diesel engines

Modern trends in designing mechatronic systems call for a synergic design of the separated subsystems (mechanic, electronic parts, control modules, etc.) concurring to the overall performance. Following this point of view, this paper presents a control oriented model and a nonlinear control design for a Common Rail injection system. First a model is developed, which is tuned in a virtual simulation environment, representing the injection system in details in a reliable replication of reality. Then a sliding mode control is developed. Both the model of the injection process and of the control law are validated by a virtual detailed simulation environment. The prediction capability of the model and the control efficiency are clearly shown.     

Design of LMI‐based global sliding mode controller for uncertain nonlinear systems with application to Genesio's chaotic system

Design of a novel global sliding mode control law for the stabilization of uncertain nonlinear systems is presented in this article. A sufficient condition is derived using the Lyapunov theorem and linear matrix inequality to guarantee the asymptotical stability of the states and to improve the stability of the system. Under the uncertainty and nonlinearity effects, the reaching phase is eliminated and the chattering is reduced effectively and then, the robustness and performance of the system are improved. Lastly, the proposed method is applied on Genesio's chaotic system and the simulation results demonstrate the effectiveness of this technique. © 2014 Wiley Periodicals, Inc. Complexity 21: 94–98, 2015     

Zombies: a simple discrete model of the apocalypse

A simple discrete-time two-dimensional dynamical system is constructed and analyzed numerically, with modelling motivations drawn from the zombie virus of popular horror fiction, and with suggestions for further exercises or extensions suitable for an introductory undergraduate course.     

Gaussian elimination is stable for the inverse of a diagonally dominant matrix

Let be a row diagonally dominant matrix, i.e.,

where with We show that no pivoting is necessary when Gaussian elimination is applied to Moreover, the growth factor for does not exceed The same results are true with row diagonal dominance being replaced by column diagonal dominance.

     

Global stability of a multiple infected compartments model for waterborne diseases

In this paper, mathematical analysis is carried out for a multiple infected compartments model for waterborne diseases, such as cholera, giardia, and rotavirus. The model accounts for both person-to-person and water-to-person transmission routes. Global stability of the equilibria is studied. In terms of the basic reproduction number $R_0$, we prove that, if $R_0⩽1$, then the disease-free equilibrium is globally asymptotically stable and the infection always disappears; whereas if $R_0>1$, there exists a unique endemic equilibrium which is globally asymptotically stable for the corresponding fast–slow system. Numerical simulations verify our theoretical results and present that the decay rate of waterborne pathogens has a significant impact on the epidemic growth rate. Also, we observe numerically that the unique endemic equilibrium is globally asymptotically stable for the whole system. This statement indicates that the present method need to be improved by other techniques.     

Generating facets for the cut polytope of a graph by triangular elimination

The cut polytope of a graph arises in many fields. Although much is known about facets of the cut polytope of the complete graph, very little is known for general graphs. The study of Bell inequalities in quantum information science requires knowledge of the facets of the cut polytope of the complete bipartite graph or, more generally, the complete k-partite graph. Lifting is a central tool to prove certain inequalities are facet inducing for the cut polytope. In this paper we introduce a lifting operation, named triangular elimination, applicable to the cut polytope of a wide range of graphs. Triangular elimination is a specific combination of zero-lifting and Fourier–Motzkin elimination using the triangle inequality. We prove sufficient conditions for the triangular elimination of facet inducing inequalities to be facet inducing. The proof is based on a variation of the lifting lemma adapted to general graphs. The result can be used to derive facet inducing inequalities of the cut polytope of various graphs from those of the complete graph. We also investigate the symmetry of facet inducing inequalities of the cut polytope of the complete bipartite graph derived by triangular elimination.      

Terminal sliding mode control without singular for a class of uncertain system

An improved nonsingular terminal sliding mode method is proposed for a class of nonlinear systems with unmodeled dynamics. The proposed method can effectively avoid the singularity problem. The stability of the proposed procedure which could guarantee the robustness against uncertain unmodeled dynamic and external disturbances is proven by using the Lyapunov theory in finite time. An example is given to show the proposed improved terminal sliding mode control law without singular effectively. © 2016 Wiley Periodicals, Inc. Complexity 21: 566–572, 2016     

Development of a model for a two-station serial transfer line subject to machine and buffer failure

In modern automated production lines, it is common to connect pairs of machines with mechanical storage devices in order to provide buffering between processing stations. Since these devices are mechanical, they are prone to failure. Previous research concerning the analytical modeling of a class of production lines, the serial transfer line, assumes that these buffers are completely reliable. The concept of an unreliable buffer is introduced and an analytic model of a two machine line with an unreliable buffer is developed. It is proposed that this model will form the foundation for an analytic model of the more complex K > 2 machine serial transfer line with unreliable buffers.     

A study on a two‐wave mode Kadomtsev–Petviashvili equation: conditions for multiple soliton solutions to exist

We establish a two‐wave mode equation for the integrable Kadomtsev–Petviashvili equation, which describes the propagation of two different wave modes in the same direction simultaneously. We determine the necessary conditions that make multiple soliton solutions exist for this new equation. The simplified Hirota's method will be used to conduct this work. We also use other techniques to obtain other set of periodic and singular solutions for the two‐mode Kadomtsev‐Petviashvili equation. Copyright © 2017 John Wiley & Sons, Ltd.     

A competing infection model for the spread of different viewpoints of a divisive idea

We develop a non-network, deterministic, competing infections model for the spread of two competing viewpoints of a divisive idea that incorporates external factors in addition to interpersonal interactions. We consider divisive ideas to have polarizing support, i.e. there are no “shades of grey.” The proposed model for the spread of the competing support and skepticism of such an idea within a population is based on both epidemiological and competing species models. The model is then analyzed qualitatively and quantitatively in a case study of the 2016 Republican primary polls. Parameter fitting to this data shows the proposed model is plausible for the spread of viewpoints of a divisive idea.     

Application of a stochastic model to the breakage of flocs in stirred tanks

A stochastic model is employed to account for the dynamic behaviour of floc breakage. More specifically, the evolution of the particle-size distribution over time is presented. The model is compared to experimental data obtained by exposing kaolin—iron(III) flocs to hydrodynamic stress generated by a turbine impeller. The agreement between the model and experimental data appears to be satisfactory.     

Mathematical analysis of a model for the transmission dynamics of bovine tuberculosis

A deterministic model for studying the transmission dynamics of bovine tuberculosis in a single cattle herd is presented and qualitatively analyzed. A notable feature of the model is that it allows for the importation of asymptomatically infected cattle (into the herd) because re‐stocking from outside sources. Rigorous analysis of the model shows that the model has a globally‐asymptotically stable disease‐free equilibrium whenever a certain epidemiological threshold, known as the reproduction number, is less than unity. In the absence of importation of asymptomatically infected cattle, the model has a unique endemic equilibrium whenever the reproduction number exceeds unity (this equilibrium is globally asymptotically stable for a special case). It is further shown that, for the case where asymptomatically infected cattle are imported into the herd, the model has a unique endemic equilibrium. This equilibrium is also shown to be globally asymptotically stable for a special case. Copyright © 2011 John Wiley & Sons, Ltd.