Bray—Liebhafsky Oscillations

Summary. A system describing an oscillating chemical reaction (known as a Bray—Liebhafsky oscillating reaction) is considered. It is shown that large amplitude oscillations arise through a homoclinic bifurcation and vanish through a subcritical Hopf bifurcation. An approximate locus of points corresponding to the homoclinic orbit in a parameter space is calculated using a variation of the Bogdanov—Takens—Carr method. A special feature of the problem is related to the fact that nonlinear terms in the equations contain square and cubic roots of expressions depending on the unknowns. For a particular model considered it is possible to obtain most of the results analytically. Received September 21, 1998; revised March 29, 1999; accepted June 17, 1999     

The Moment Problem Associated with the q -Laguerre Polynomials

   Abstract. We consider the indeterminate Stieltjes moment problem associated with the q -Laguerre polynomials. A transformation of the set of solutions, which has all the classical solutions as fixed points, is established and we present a method to construct, for instance, continuous singular solutions. The connection with the moment problem associated with the Stieltjes—Wigert polynomials is studied; we show how to come from q -Laguerre solutions to Stieltjes—Wigert solutions by letting the parameter α —> ∞ , and we explain how to lift a Stieltjes—Wigert solution to a q -Laguerre solution at the level of Pick functions. Based on two generating functions, expressions for the four entire functions from the Nevanlinna parametrization are obtained.     

Singular Hopf Bifurcation in Systems with Fast and Slow Variables

Summary. We study a general nonlinear ODE system with fast and slow variables, i.e., some of the derivatives are multiplied by a small parameter. The system depends on an additional bifurcation parameter. We derive a normal form for this system, valid close to equilibria where certain conditions on the derivatives hold. The most important condition concerns the presence of eigenvalues with singular imaginary parts, by which we mean that their imaginary part grows without bound as the small parameter tends to zero. We give a simple criterion to test for the possible presence of equilibria satisfying this condition. Using a center manifold reduction, we show the existence of Hopf bifurcation points, originating from the interaction of fast and slow variables, and we determine their nature. We apply the theory, developed here, to two examples: an extended Bonhoeffer—van der Pol system and a predator-prey model. Our theory is in good agreement with the numerical continuation experiments we carried out for the examples. Received October 24, 1996; revised October 31, 1997; accepted November 3, 1997     

On the normality a posteriori for exponential distributions,using the Bayesian estimation

Summary The Bayesian estimation problem for the parameter θ of an exponential probability distribution is considered, when it is assumed that θ has a natural conjugate prior density and a loss-function depending on the squared error is used. It is shown that, with probability one, the posterior density of the Bayesian—centered and scaled parameter converges pointwise to the normal probability density. The weak convergence of the posterior distributions to the normal distribution follows directly. Both correct and incorrect models are studied and the asymptotic normality is stated respectively.     

Stokeswaves with vorticity

The existence of periodic waves propagating downstream on the surface of a two-dimensional infinitely deep body of water under the force of gravity is established for a general class of vorticities. When reformulated as an elliptic boundary value problem in a fixed semi-infinite cylinder with a parameter, the operator describing the problem is nonlinear and non-Fredholm. A global connected set of nontrivial solutions is obtained via singular theory of bifurcation. The proof combines a generalized degree theory, global bifurcation theory, and Whyburn’s lemma in topology with the Schauder theory for elliptic problems and the maximum principle.     

Implicit functionals and eigenvalue problems

An approach to nonlinear, positively homogeneous eigenvalue problems based on the use of the spectral parameter as a functional of Euler type is suggested. It allows one to present the spectral-parameter domain as a bifurcation diagram of the problem. Fučik spectrum problems (classical and for p-Laplacians) and the problem with a nonlinear dependence of the weight function on the spectral parameter are considered. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 5, pp. 169–186, 2005.     

Stable Modulating Multipulse Solutions for Dissipative Systems with Resonant Spatially Periodic Forcing

Summary.    We show the existence and stability of modulating multipulse solutions for a class of bifurcation problems given by a dispersive Swift-Hohenberg type of equation with a spatially periodic forcing. Equations of this type arise as model problems for pattern formation over unbounded weakly oscillating domains and, more specifically, in laser optics. As associated modulation equation, one obtains a nonsymmetric Ginzburg-Landau equation which possesses exponentially stable stationary n—pulse solutions. The modulating multipulse solutions of the original equation then consist of a traveling pulselike envelope modulating a spatially oscillating wave train. They are constructed by means of spatial dynamics and center manifold theory. In order to show their stability, we use Floquet theory and combine the validity of the modulation equation with the exponential stability of the n—pulses in the modulation equation. The analysis is supplemented by a few numerical computations. In addition, we also show, in a different parameter regime, the existence of exponentially stable stationary periodic solutions for the class of systems under consideration. Received November 30, 1999; accepted December 4, 2000 Online publication March 23, 2001     

Relaxation techniques for strictly convex network problems

Gauss—Seidel type relaxation techniques are applied in the context of strictly convex pure networks with separable cost functions. The algorithm is an extension of the Bertsekas—Tseng approach for solving the linear network problem and its dual as a pair of monotropic programming problems. The method is extended to cover the class of generalized network problems. Alternative internal tactics for the dual problem are examined. Computational experiments —aimed at the improved efficiency of the algorithm — are presented. This research was supported in part by National Science Foundation Grant No. DCR-8401098-A0l.     

Non-Existence and Uniqueness Results for Supercritical Semilinear Elliptic Equations

Non-existence and uniqueness results are proved for several local and non-local supercritical bifurcation problems involving a semilinear elliptic equation depending on a parameter. The domain is star-shaped and such that a Poincaré inequality holds but no other symmetry assumption is required. Uniqueness holds when the bifurcation parameter is in a certain range. Our approach can be seen, in some cases, as an extension of non-existence results for non-trivial solutions. It is based on Rellich–Pohožaev type estimates. Semilinear elliptic equations naturally arise in many applications, for instance in astrophysics, hydrodynamics or thermodynamics. We simplify the proof of earlier results by K. Schmitt and R. Schaaf in the so-called local multiplicative case, extend them to the case of a non-local dependence on the bifurcation parameter and to the additive case, both in local and non-local settings.     

Computing a closest bifurcation instability in multidimensional parameter space

Summary Engineering and physical systems are often modeled as nonlinear differential equations with a vector λ of parameters and operated at a stable equilibrium. However, as the parameters λ vary from some nominal value λ₀, the stability of the equilibrium can be lost in a saddle-node or Hopf bifurcation. The spatial relation in parameter space of λ₀ to the critical set of parameters at which the stable equilibrium bifurcates determines the robustness of the system stability to parameter variations and is important in applications. We propose computing a parameter vector λ_* at which the stable equilibrium bifurcates which is locally closest in parameter space to the nominal parameters λ₀. Iterative and direct methods for computing these locally closest bifurcations are described. The methods are extensions of standard, one-parameter methods of computing bifurcations and are based on formulas for the normal vector to hypersurfaces of the bifurcation set. Conditions on the hypersurface curvature are given to ensure the local convergence of the iterative method and the regularity of solutions of the direct method. Formulas are derived for the curvature of the saddle node bifurcation set. The methods are extended to transcritical and pitchfork bifurcations and parametrized maps, and the sensitivity to λ₀ of the distance to a closest bifurcation is derived. The application of the methods is illustrated by computing the proximity to the closest voltage collapse instability of a simple electric power system.     

Pseudoholomorphic and $$\varepsilon $$ -Pseudoregular Solutions of Singularly Perturbed Problems

Differential Equations - For nonlinear evolution equations in a Banach space that depend in two ways—regularly and singularly—on a small parameter, we construct $$\varepsilon $$...     

Hamiltonian spatial structure for three-dimensional water waves in a moving frame of reference

Summary The governing equations for three-dimensional time-dependent water waves in a moving frame of reference are reformulated in terms of the energy and momentum flux. The novelty of this approach is that time-independent motions of the system—that is, motions that are steady in a moving frame of reference—satisfy a partial differential equation, which is shown to be Hamiltonian. The theory of Hamiltonian evolution equations (canonical variables, Poisson brackets, symplectic form, conservation laws) is applied to the spatial Hamiltonian system derived for pure gravity waves. The addition of surface tension changes the spatial Hamiltonian structure in such a way that the symplectic operator becomes degenerate, and the properties of this generalized Hamiltonian system are also studied. Hamiltonian bifurcation theory is applied to the linear spatial Hamiltonian system for capillary-gravity waves, showing how new waves can be found in this framework.     

Generalized Chaplygin’s transformation and explicit integration of a system with a spherical support

We discuss explicit integration and bifurcation analysis of two non-holonomic problems. One of them is the Chaplygin’s problem on no-slip rolling of a balanced dynamically non-symmetric ball on a horizontal plane. The other, first posed by Yu. N. Fedorov, deals with the motion of a rigid body in a spherical support. For Chaplygin’s problem we consider in detail the transformation that Chaplygin used to integrate the equations when the constant of areas is zero. We revisit Chaplygin’s approach to clarify the geometry of this very important transformation, because in the original paper the transformation looks a cumbersome collection of highly non-transparent analytic manipulations. Understanding its geometry seriously facilitate the extension of the transformation to the case of a rigid body in a spherical support — the problem where almost no progress has been made since Yu.N. Fedorov posed it in 1988. In this paper we show that extending the transformation to the case of a spherical support allows us to integrate the equations of motion explicitly in terms of quadratures, detect mostly remarkable critical trajectories and study their stability, and perform an exhaustive qualitative analysis of motion. Some of the results may find their application in various technical devices and robot design. We also show that adding a gyrostat with constant angular momentum to the spherical-support system does not affect its integrability.     

On the cutting stock problem under stochastic demand

This paper addresses the one-dimensional cutting stock problem when demand is a random variable. The problem is formulated as a two-stage stochastic nonlinear program with recourse. The first stage decision variables are the number of objects to be cut according to a cutting pattern. The second stage decision variables are the number of holding or backordering items due to the decisions made in the first stage. The problem’s objective is to minimize the total expected cost incurred in both stages, due to waste and holding or backordering penalties. A Simplex-based method with column generation is proposed for solving a linear relaxation of the resulting optimization problem. The proposed method is evaluated by using two well-known measures of uncertainty effects in stochastic programming: the value of stochastic solution—VSS—and the expected value of perfect information—EVPI. The optimal two-stage solution is shown to be more effective than the alternative wait-and-see and expected value approaches, even under small variations in the parameters of the problem.     

Efficient solution methods for covering tree problems

In recent years, interest has been shown in the optimal location of ‘extensive’ facilities in a network. Two such problems—the Maximal Direct and Indirect Covering Tree problems—were introduced by Hutson and ReVelle. Previous solution techniques are extended to provide an efficient algorithm for the Indirect Covering Tree problem and the generalization in which demand covered is attenuated by distance. It is also shown that the corresponding problem is NP-hard when the underlying network is not a tree.     

Stochastic vendor selection problem: chance-constrained model and genetic algorithms

We study a vendor selection problem in which the buyer allocates an order quantity for an item among a set of suppliers such that the required aggregate quality, service, and lead time requirements are achieved at minimum cost. Some or all of these characteristics can be stochastic and hence, we treat the aggregate quality and service as uncertain. We develop a class of special chance-constrained programming models and a genetic algorithm is designed for the vendor selection problem. The solution procedure is tested on randomly generated problems and our computational experience is reported. The results demonstrate that the suggested approach could provide managers a promising way for studying the stochastic vendor selection problem. The authors would like to thank the referees for providing constructive comments that led to an improved version of the paper. Also, this research was partially supported by grants from National Natural Science Foundation (60776825)—China, 863 Programs (2007AA11Z208)—China, Doctorate Foundation (20040004012)—China, Villanova University Research Sabbatical Fall 2006, and the National Science Foundation (0332490)—USA.     

Refined Orthotropic Plate Motion Equations for Acoustoelasticity Problem Statement

The formulation of the acoustoelasticity problem is given on the basis of refined motion equations of orthotropic plates. These equations are constructed in the first approximation by reducing the three-dimensional equations of the theory of elasticity to the two-dimensional equations of the theory of plates, where the approximation of the transverse tangential stresses and the transverse reduction stress is made with the help of trigonometric basis functions in the thickness direction. Wherein at the points of the boundary (front) surfaces, the static boundary conditions of the problem for tangential stresses are satisfied exactly and for transverse normal stress — approximately. Accounting for internal energy dissipation in the plate material is based on the Thompson—Kelvin—Voigt hysteresis model. In case of formulating problems on dynamic processes of plate deformation in vacuum, the equations are divided into two separate systems of equations. The first of these systems describes non-classical shear-free, longitudinal-transverse forms of movement, accompanied by a distortion of the flat form of cross sections, and the second system describes transverse bending-shear forms of movement. The latter are practically equivalent in quality and content to the analogous equations of the well-known variants of refined theories, but, unlike them, with a decrease in the relative thickness parameter, they lead to solutions according to the classical theory of plates. The motion of the surrounding the plate acoustic media is described by the generalized Helmholtz wave equations, constructed with account of energy dissipation by introducing into consideration the complex sound velocity according to Skudrzyk.     

Multi-objective Pareto-optimal control: an application to wastewater management

This work treats, within a multi-objective framework, of an economical-ecological problem related to the optimal management of a wastewater treatment system consisting of several purifying plants. The problem is formulated as a multi-objective parabolic optimal control problem and it is studied from a cooperative point of view, looking for Pareto-optimal solutions. The weighting method is used here to characterize the Pareto solutions of our problem. To obtain them, a numerical algorithm—based in a characteristics-Galerkin discretization—is proposed and numerical results for a real world situation in the estuary of Vigo (NW Spain) are also presented.     

Unitary integrals and related matrix models

We briefly review the basic properties of unitary matrix integrals, using three matrix models to analyze their properties: the ordinary unitary, the Brezin—Gross—Witten, and the Harish—Chandra—Itzykson—Zuber models. We especially emphasize the nontrivial aspects of the theory, from the De Witt’Hooft anomaly in unitary integrals to the problem of calculating correlators with the Itzykson-Zuber measure. We emphasize the method of character expansions as a technical tool. Unitary integrals are still insufficiently investigated, and many new results should be expected as this field attracts increased attention.