Explicit estimates for solutions of nonlinear radiation-type problems

We establish the existence of weak solutions of a nonlinear radiation-type boundary value problem for elliptic equation on divergence form with discontinuous leading coefficient. Quantitative estimates play a crucial role on the real applications. Our objective is the derivation of explicit expressions of the involved constants in the quantitative estimates, the so-called absolute or universal bounds. The dependence on the leading coefficient and on the size of the spatial domain is precise. This work shows that the expressions of those constants are not so elegant as we might expect.     

A Boolean satisfiability approach to the resource-constrained project scheduling problem

We formulate the resource-constrained project scheduling problem as a satisfiability problem and adapt a satisfiability solver for the specific domain of the problem. Our solver is lightweight and shows good performance both in finding feasible solutions and in proving lower bounds. Our numerical tests allowed us to close several benchmark instances of the RCPSP that have never been closed before by proving tighter lower bounds and by finding better feasible solutions. Using our method we solve optimally more instances of medium and large size from the benchmark library PSPLIB and do it faster compared to any other existing solver.     

Domain Structure of Bulk Ferromagnetic Crystals in Applied Fields Near Saturation

We investigate the ground state of a uniaxial ferromagnetic plate with perpendicular easy axis and subject to an applied magnetic field normal to the plate. Our interest is in the asymptotic behavior of the energy in macroscopically large samples near the saturation field. We establish the scaling of the critical value of the applied field strength below saturation at which the ground state changes from the uniform to a multidomain magnetization pattern and the leading order scaling behavior of the minimal energy. Furthermore, we derive a reduced sharp interface energy, giving the precise asymptotic behavior of the minimal energy in macroscopically large plates under a physically reasonable assumption of small deviations of the magnetization from the easy axis away from domain walls. On the basis of the reduced energy and by a formal asymptotic analysis near the transition, we derive the precise asymptotic values of the critical field strength at which non-trivial minimizers (either local or global) emerge. The non-trivial minimal energy scaling is achieved by magnetization patterns consisting of long slender needle-like domains of magnetization opposing the applied field.     

Bounds for the solution set of linear complementarity problems

We give here bounds for the feasible domain and the solution norm of Linear Complementarity Problems (LCP). These bounds are motivated by formulating the LCP as a global quadratic optimization problem and are characterized by the eigenstructure of the corresponding matrix. We prove boundedness of the feasible domain when the quadratic problem is concave, and give easily computable bounds for the solution norm for the convex case. We also obtain lower and upper bounds for the solution norm of the general nonconvex problem.     

A Problem with Nonlocal Conditions for Partial Differential Equations Unsolved with Respect to the Leading Derivative

In the domain that is the product of a segment and a p-dimensional torus, we investigate the well-posedness of a problem with nonlocal boundary conditions for a partial differential equation unsolved with respect to the leading derivative with respect to a selected variable. We establish conditions for the the classical well-posedness of the problem and prove metric theorems on the lower bounds of small denominators appearing in the course of its solution.     

Computing the partial word avoidability indices of ternary patterns

We study pattern avoidance in the context of partial words. The problem of classifying the avoidable binary patterns has been solved, so we move on to ternary and more general patterns. Our results, which are based on morphisms (iterated or not), determine all the ternary patternsʼ avoidability indices or at least give bounds for them.     

On a Stefan-type model arising in ecology

We discuss the dispersal of a chemical, emitted from a point source, into a reacting medium. The diffusion and reaction are such that a front emerges, so that at all times the chemical remains confined to a bounded domain around the source. We focus on the final stationary state which is a non-standard Stefan-type boundary value problem for an Emden–Fowler equation. We prove its existence and uniqueness and obtain, analytically as well as numerically, bounds for the location of the front and for the concentration profile.     

Decentralized Adaptive Robust Tracking and Model Following for Large-Scale Systems Including Delayed State Perturbations in the Interconnections

The problem of decentralized robust tracking and model following is considered for a class of uncertain large-scale systems including delayed state perturbations in the interconnections. In this paper, it is assumed that the upper bounds of the delayed state perturbations, uncertainties, and external disturbances are unknown. A modified adaptation law with σ-modification is introduced to estimate such unknown bounds, and on the basis of the updated values of these unknown bounds, a class of decentralized local memoryless state feedback controllers is constructed for robust tracking of dynamical signals. The proposed decentralized adaptive robust tracking controllers can guarantee that the tracking errors between each time-delay subsystem and the corresponding local reference model without time-delay decrease uniformly asymptotically to zero. Finally, a numerical example is given to demonstrate the validity of the results.     

Effect of errors in the spatial geometry for temperature dependent Stokes flow

A priori bounds are established for the solution to the problem of Stokes flow in a bounded domain, for a viscous, heat conducting, incompressible fluid, when changes in the spatial geometry are admitted. These bounds demonstrate how the velocity field and the temperature field depend on changes in the spatial geometry and also yield a convergence theorem in terms of boundary perturbations. The results have a direct bearing on an error analysis for a numerical approximation to non-isothermal Stokes flow when the boundary of a complicated domain is approximated by a simpler one, e.g., in the procedure of triangulation combined with finite elements.     

A non-isothermal Ginzburg–Landau model in superconductivity: Existence,uniqueness and asymptotic behaviour

In this paper we study a Ginzburg–Landau model which describes the behaviour of a superconducting material including thermal effects. We extend the traditional formulation of the problem, by introducing the temperature as an additional state variable. Accordingly, together with the Gor’kov–Eliashberg system, we introduce an evolution equation for the absolute temperature. We examine in detail the case which allows only variations of the concentration of superconducting electrons and of the temperature, neglecting the electromagnetic field. For this problem existence and uniqueness of the solution are shown. Finally we analyze the asymptotic behaviour of the solutions, proving that the system possesses a global attractor.     

A Shell Model for Optimal Mixing

What is the maximum mixing efficiency of an incompressible flow? To address this question we introduce a shell model—a reduced model mimicking the kinematics of advection and diffusion—to study the evolution of an initially inhomogeneous tracer concentration carried by a given incompressible fluid on a periodic spatial domain. We pose the mixing task as an optimization problem: Find the divergence-free velocity field (the control variable) that produces a well-mixed tracer concentration field (the state variable). We consider two alternative objectives: local-in-time optimization (maximize the instantaneous mixing rate) and global-in-time optimization (maximize mixing at a prescribed final time). Throughout, we use a shell-model analog of the \(H^{-1}\) mix-norm to measure mixing. In addition, lower bounds on the mix-norm are obtained and rule out perfect mixing in finite time in particular cases.     

On the optimal control of arrivals to a single queue with arbitrary feedback delay

We consider a problem of admission control to a single queue in discrete time. The controller has access to k step old queue lengths only, where k can be arbitrary. The problem is motivated, in particular, by recent advances in high-speed networking where information delays have become prominent. We formulate the problem in the framework of Completely Observable Controlled Markov Chains, in terms of a multi-dimensional state variable. Exploiting the structure of the problem, we show that under appropriate conditions, the multi-dimensional Dynamic Programming Equation (DPE) can be reduced to a unidimensional one. We then provide simple computable upper and lower bounds to the optimal value function corresponding to the reduced unidimensional DPE. These upper and lower bounds, along with a certain relationship among the parameters of the problem, enable us to deduce partially the structural features of the optimal policy. Our approach enables us to recover simply, in part, the recent results of Altman and Stidham, who have shown that a multiple-threshold-type policy is optimal for this problem. Further, under the same relationship among the parameters of the problem, we provide easily computable upper bounds to the multiple thresholds and show the existence of simple relationships among these upper bounds. These relationships allow us to gain very useful insights into the nature of the optimal policy. In particular, the insights obtained are of great importance for the problem of actually computing an optimal policy because they reduce the search space enormously. This revised version was published online in June 2006 with corrections to the Cover Date.     

Numerical analysis of a mean field model of superconducting vortices

A finite volume/element approximation of a mean field modelof superconducting vortices in one and two dimensions is presented.The solutions of these approximations are investigated. A finiteelement approximation of the free boundary problem which isa special case of the steady state solution of the model isalso studied. We present some computed results from these approximations. Received 3 December 1997. Accepted 17 May 2000.     

Finite element approximation of elliptic control problems with constraints on the gradient

We consider an elliptic optimal control problem with control constraints and pointwise bounds on the gradient of the state. We present a tailored finite element approximation to this optimal control problem, where the cost functional is approximated by a sequence of functionals which are obtained by discretizing the state equation with the help of the lowest order Raviart–Thomas mixed finite element. Pointwise bounds on the gradient variable are enforced in the elements of the triangulation. Controls are not discretized. Error bounds for control and state are obtained in two and three space dimensions. A numerical example confirms our analytical findings.     

Computable bounds for eigenvalues and eigenfunctions of elliptic differential operators

Summary We are concerned with bounds for the error between given approximations and the exact eigenvalues and eigenfunctions of self-adjoint operators in Hilbert spaces. The case is included where the approximations of the eigenfunctions don't belong to the domain of definition of the operator. For the eigenvalue problem with symmetric elliptic differential operators these bounds cover the case where the trial functions don't satisfy the boundary conditions of the problem. The error bounds suggest a certain defectminization method for solving the eigenvalue problems. The method is applied to the membrane problem.     

Conformal upper bounds for the eigenvalues of the Laplacian and Steklov problem

In this paper, we find upper bounds for the eigenvalues of the Laplacian in the conformal class of a compact Riemannian manifold (M,g). These upper bounds depend only on the dimension and a conformal invariant that we call “min-conformal volume”. Asymptotically, these bounds are consistent with the Weyl law and improve previous results by Korevaar and Yang and Yau. The proof relies on the construction of a suitable family of disjoint domains providing supports for a family of test functions. This method is interesting for itself and powerful. As a further application of the method we obtain an upper bound for the eigenvalues of the Steklov problem in a domain with C¹ boundary in a complete Riemannian manifold in terms of the isoperimetric ratio of the domain and the conformal invariant that we introduce.     

Lower Bounds on the Generalized Central Moments of the Optimal Alignments Score of Random Sequences

We present a general approach to the problem of determining tight asymptotic lower bounds for generalized central moments of the optimal alignment score of two independent sequences of i.i.d. random variables. At first, these are obtained under a main assumption for which sufficient conditions are provided. When the main assumption fails, we nevertheless develop a “uniform approximation” method leading to asymptotic lower bounds. Our general results are then applied to the length of the longest common subsequences of binary strings, in which case asymptotic lower bounds are obtained for the moments and the exponential moments of the optimal score. As a by-product, a local upper bound on the rate function associated with the length of the longest common subsequences of two binary strings is also obtained.     

Solution Bounds for the Discrete Riccati Equation and Its Applications

In this paper, we address the estimation problem for the solution of the discrete algebraic matrix Riccati equation. Both upper and lower bounds are measured. Compared to the majority of the approaches proposed in the literature, the present results are sharper. We also apply the results obtained to solve the robust stabilization problem of discrete time-delay systems. A robust stabilizability criterion and the corresponding state feedback control law are proposed. Furthermore, the tolerable bound of the delay term is also estimated. Finally, numerical examples are given to demonstrate the applications of the results.     

Superconductivity and Ferromagnetism in Nonadiabatic Systems with Magnetic Impurity: A Step Beyond the Migdal Theorem

The possibility of the ferromagnetic ordering of a paramagnetic impurity in nonadiabatic superconducting systems is investigated. The effect of the relative shift of the Fermi surface by the internal magnetic field, the exchange interaction of the impurity ions through the conductivity electrons, and the spin–orbit interaction of the nonmagnetic impurity are taken into account. The problem is solved in the linear approximation with respect to the nonadiabaticity by taking the vertex and crossing diagrams corresponding to the electron–phonon and the electron–impurity interactions into account. We obtain basic equations of the superconductivity theory for nonadiabatic systems with the ferromagnetic ordering of the impurity spins and show that the nonadiabaticity alters the superconducting transition temperature T _c and the critical impurity concentration. The behavior of the magnetic-ordering temperature T _C as a function of the impurity concentration c in the superconductive state of the nonadiabatic system is also investigated. We obtain the phase diagram (T,c) and show that the nonadiabaticity effects lead to the enlargement of the domain where the superconductivity and the ferromagnetism exist simultaneously.     

Heat Conduction Problem in a Dilated Pipe: Existence and Uniqueness Result

We study the heat conduction through a pipe filled with incompressible viscous fluid. The goal of this paper is to take into account the effects of the spipe’s dilatation due to the heating. In view of that, we assume that the longitudinal dilatation of the pipe is described by a linear heat expansion law. We prove the existence and uniqueness theorems for the corresponding boundary value problem. The main difficulty comes from the fact that the flow domain changes depending on the solution of the heat equation leading to a nonlinear coupled governing problem.