Nonlinear control of a variable displacement axial piston pump

In this contribution a mathematical model of a variable displacement axial piston pump controlled by a solenoid valve is derived. For the purpose of a controller design the mathematical model is simplified using singular perturbation arguments. The goal of the controller design is to track prescribed trajectories in the load pressure for arbitrary unknown load conditions. The control concept being proposed comprises a feedforward controller and a nonlinear backstepping controller combined with a load estimator for the trajectory error system. The feasibility of this control concept is shown by means of measurements on a test-stand. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stability of rotation of a vane in a flow

The results of investigation of the stability of permanent rotation of a four-blade vane on a weightless rod in the flow of a homogeneous medium are discussed. The rod rotates about a fixed point where a spherical joint is situated. The vane rotates about the second joint fixed at the other end of the rod. The stability of permanent rotation of the vane is studied when the rod coincides with the dynamic symmetry axis of the vane. The results are compared with the one-joint case. It is shown that increasing the number of degrees of freedom leads to “diminishing” the stability domain projection onto the corresponding subspace of parameters. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 7, pp. 73–95, 2005.     

Modeling partial Greeks of variable annuities with dependence

Dynamic hedging used to mitigate the financial risks associated with large portfolios of variable annuities requires calculating partial dollar deltas on major market indices. Metamodeling approaches have been proposed in the past few years to address the computational issues related to the calculation of partial dollar deltas. In this paper, we investigate whether the additional complication of modeling the dependence between the partial dollar deltas improves the accuracy of the metamodeling approaches. We use several copulas to model the dependence structures of the partial dollar deltas and conduct numerical experiments to compare different metamodels. Despite the evidence of strong dependence in the estimated models, our numerical results show that modeling the dependence structures in the metamodels does not improve the accuracy of the estimations at the portfolio level. This is because the dependence between the partial dollar deltas is well captured by the covariates used in the marginal models. This finding suggests that we should focus more on marginal models than specifying the dependence structure explicitly.     

The nozzle guide vane problem: Partitioning a heterogeneous inventory

In this paper we describe a problem which is encountered in the maintenance of gas turbine engines used in commercial and military aircraft. In particular, we address the problem of selecting a set of nozzle guide vanes, from a heterogeneous inventory of vanes, for the nozzle assembly of an engine. We formulate this problem as a partitioning problem for a heterogeneous population. Given the combinatorial complexity of the problem we develop a heuristic algorithm which is shown to be very effective in obtaining the maximum number of partitions.     

Modeling the effects of variable external influences and demographic processes on innovation diffusion

In this paper, a nonlinear mathematical model for innovation diffusion is proposed and analyzed by considering the effects of variable external influences (cumulative marketing efforts) and human population (variable marketing potential) in a society. The change in the population density is caused by various demographic processes such as immigration, emigration, intrinsic growth rate, death rate, etc.Thus, the problem of innovation diffusion is governed by three dynamic variables, namely, non adopters’ density, adopters’ density and the cumulative density of external influences. The model is analyzed by using the stability theory of differential equations and computer simulation.The model analysis shows that the main effect of the increase in cumulative density of external influences is to make the adopter population density reach its equilibrium at a much faster rate. It further shows that the density of adopters’ population increases as the parameters related to increase in non adopters’ population density increase. The effects of various parameters in the model on the nature of existing single equilibrium have also been discussed by using numerical simulation. It is shown that parameters related to the growth of non adopters’ population density have stabilizing effects on the system.     

Multirate models for simulating a colpitts oscillator

A model based on multirate partial differential algebraic equations yields an efficient numerical simulation of electric circuits in radio frequency applications. Considering frequency modulation, free parameters of the model are determined appropriately by a minimisation strategy. We apply the multirate approach to simulate a modified version of a Colpitts oscillator, which exhibits frequency modulation at widely separated time scales. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Miscible displacement in a Hele-Shaw cell

We formulated a theory of simple mixtures of incompressible miscible liquids in terms of the mass averaged velocityu and the solenoidal volume averaged velocityW, We derived simplified equations for miscible displacement in a Hele-Shaw cell. We obtained a steady solution of these equations corresponding to displacement under gravity with prescribed values of concentration and mean normal stress at the inlet and exit of the cell. We studied the stability of this steady flow. This differs from previous works which treat the stability of unsteady miscible displacement using a quasi-static assumption and classical equations based on divu=0. In our problem, replacingu withW gives rise to a difference in the mean normal stress, which alters the pressure drop across the cell and changes the velocity of free fall. We found that the stability equations are the same in the two formulations, but the boundary conditions are slightly different; however the difference will be small if diffusion is slow or the thickness of the cell is small. The results show that steady miscible displacement in a Hele-Shaw cell is stable to long and short waves. Within certain ranges of parameters, the displacement of glycerin into water can be unstable. This instability is basically of a Rayleigh-Taylor type, regularized by diffusion. As the diffusion parameterS becomes smaller, the waves of disturbances become finer and are confined to an increasingly thin diffusion layer. Water displacing glycerin is always stable. This is due to the fact that the steady equilibrium profile is not steep enough to create a fingering instability.Dedicated to Prof. Klaus Kirchgässner on the occasion of his sixtieth birthday     

Bifurcation and chaos in a system simulating magnetic field configurations

In the present article, the behaviour of a nonlinear dynamical system has been analysed using the approach of bifurcation theory. The system is important due to the fact that it can simulate the magnetic field configurations in various situations. The nature of bifurcation has been explored in the parameter space with the help of continuation algorithm. The various limit and bifurcation points (BPs) are classified. In the second part, we have studied the temporal evolution of the system which also shows a chaotic behaviour. The system under consideration shows instability both with respect to parameter variation and evolution of time. Lastly, some mechanisms have been studied to control such chaotic scenario.     

Coisotropic displacement and small subsets of a symplectic manifold

We prove a coisotropic intersection result and deduce the following: (a) Lower bounds on the displacement energy of a subset of a symplectic manifold, in particular a sharp stable energy-Gromov-width inequality. (b) A stable non-squeezing result for neighborhoods of products of unit spheres. (c) Existence of a “badly squeezable” set in ${\mathbb R^{2n}}$ of Hausdorff dimension at most d, for every n ≥ 2 and d ≥ n. (d) Existence of a stably exotic symplectic form on ${\mathbb R^{2n}}$ , for every n ≥ 2. (e) Non-triviality of a new capacity, which is based on the minimal action of a regular coisotropic submanifold of dimension d.     

Clustering with a constraining variable

Clustering can be treated as an optimization problem over a set of feasible clusterings. This paper deals with a clustering problem where the set of feasible clusterings is determined by constraining the function of values of a given (constraining) variable in each cluster. It can be shown that agglomerative clustering methods are not suitable for solving problems with constraining variables. For solving clustering problems of this type, local optimization procedures can be adapted. In the study of the influence of constraints on the clustering, a special coefficient is defined. The proposed procedures of clustering with constraining variables are illustrated by clustering the Slovene communes on the basis of the socioeconomic indicators.     

Forecasting and simulating mortality tables

In this paper we suggest solutions to the actuaries, facing the problem of estimating future mortality tables, especially in cases where there is a lack of relevant data and where the tendencies are not easy to estimate directly. We propose the utilization of external sources of information in the form of other, published mortality tables and use formal statistical tests to decide among these possible candidates. The procedure can also be applied for checking e.g. the goodness of mortality selection factors. We suggest the use of parametric families in modelling; for example the simple 2-parameter Azbel model. We conclude the paper by a simulation study which allows for the quantification of the possible risks related to unforeseen changes in the mortality tables in the future. To calibrate the variances of these models, initial estimates are needed, which we get by the Lee–Carter method.     

Impulsive displacement of liquid in a pipe

The solution of a viscous liquid, originally at rest in a pipe, which is set impulsively into motion by a pressure gradient is well known. Analyzing this solution shows that the displacement time – the time it takes to completely displace the liquid originally at rest in the pipe – falls primarily within an inviscid window. Assuming this result remains essentially unchanged when the displaced and displacing liquids are different we apply an inviscid analysis to determine the displacement time as a function of the density ratio of the liquids and the pressure difference. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On adding a variable to a Frobenius manifold and generalizations

Let $\pi :V\rightarrow M$ be a (real or holomorphic) vector bundle whose base has an almost Frobenius structure $(\circ _{M},e_{M},g_{M})$ and typical fiber has the structure of a Frobenius algebra $(\circ _{V},e_{V},g_{V})$ . Using a connection $D$ on the bundle $\pi : V{\,\rightarrow \,}M$ and a morphism $\alpha :V\rightarrow TM$ , we construct an almost Frobenius structure $(\circ , e_{V},g)$ on the manifold $V$ and we study when it is Frobenius. In particular, we describe all (real) positive definite Frobenius structures on $V$ obtained in this way, when $M$ is a semisimple Frobenius manifold with non-vanishing rotation coefficients. In the holomorphic setting, we add a real structure $k_{M}$ on $M$ and a real structure $k_{V}$ on the bundle $\pi : V \rightarrow M$ . Using $k_{M}$ , $k_{V}$ and $D$ we define a real structure $k$ on the manifold $V$ . We study when $k$ , together with an almost Frobenius structure $(\circ , e_{V}, g) $ , satisfies the tt*- equations. Along the way, we prove various properties of adding variables to a Frobenius manifold, in connection with Legendre transformations and $tt^{*}$ -geometry.     

Convection in a variable gravity field

The problem of convection in a variable gravity field is studied by using methods of linear instability theory and nonlinear energy theory. It is shown that the decreasing or increasing of gravity in a specific direction can be stabilizing or destabilizing and it is further shown how to quantify this effect. Specific results are presented for the situation where gravity decreases linearly throughout a plane layer. The nonlinear results are found to be very close to the linear ones and define a small band where possible subcritical instabilities may arise.     

Nonlinear equations of a variable type

In the paper one gives a survey of the investigations regarding, basically one-dimensional, model equations of the Navier-Stokes type with a variable nonlinear viscosity, including sign-changing viscosity. For concrete dependences of the viscosity on the velocity gradient, one establishes a priori estimates, one carries out a numerical investigation and one formulates results regarding the uniqueness of the solution.     

Boundary element method for simulating the coupled motion of a fluid and a three-dimensional body

A boundary element method for potential flow problem coupled with the dynamics of rigid body was developed to determine numerically the resultant force and moment of force acting on an arbitrarily three-dimensional solid body and its motion in a current of an infinite fluid. An accurate integration method for singular integrands occurring in the boundary integral equations, a computational method for the tangential gradient of a velocity potential on a surface, and a method to properly treat the singularities appearing in the system of the dynamic equations of a rigid body, were proposed to complete the numerical solution of the problem. Several numerical examples were given to show the validity of the method.     

Existence and multiple solutions for a variable exponent system

In this paper, we study the following variable exponent system