Hodge decomposition with degenerate weights and the Gross-Pitaevskii energy

We solve a Hodge decomposition problem with respect to a weight function that vanishes on the boundary. This problem is present in several works on the Gross-Pitaevskii energy, and we use our solution to show that a Γ-convergence result that links these works is still valid for more general data than previously considered.     

Equimodular and linearity in modular spaces

In this paper, the concept of equimodular is introduced. It contains several generalizations of Mazur-Ulam's isometric theorem in modular spaces. Let X and Y be two real modular spaces and X with δ₁-midpoint shrinking whose modular ρ_X satisfies the Δ₂-condition. Assume that an operator T maps X onto Y in an δ₂-{t_i}-equimodular manner for all , where {t_i} is a null-sequence of nonnegative reals with the property that t₀=1, t₁?1/2, and t_i?t_i−1?2t_i for i?2. Then T is affine.     

Eigenvalues of the p-Laplacian in fractal strings with indefinite weights

In this paper we study the spectral counting function of the weighted p-Laplacian in fractal strings, where the weight is allowed to change sign. We obtain error estimates related to the interior Minkowski dimension of the boundary. We also find the asymptotic behavior of eigenvalues.     

Eliciting density ratio classes

The probability distributions of uncertain quantities needed for predictive modelling and decision support are frequently elicited from subject matter experts. However, experts are often uncertain about quantifying their beliefs using precise probability distributions. Therefore, it seems natural to describe their uncertain beliefs using sets of probability distributions. There are various possible structures, or classes, for defining set membership of continuous random variables. The Density Ratio Class has desirable properties, but there is no established procedure for eliciting this class. Thus, we propose a method for constructing Density Ratio Classes that builds on conventional quantile or probability elicitation, but allows the expert to state intervals for these quantities. Parametric shape functions, ideally also suggested by the expert, are then used to bound the nonparametric set of shapes of densities that belong to the class and are compatible with the stated intervals. This leads to a natural metric for the size of the class based on the ratio of the total areas under upper and lower bounding shape functions. This ratio will be determined by the characteristics of the shape functions, the scatter of the elicited values, and the explicit expert imprecision, as characterized by the width of the stated intervals. We provide some examples, both didactic and real, and conclude with recommendations for the further development and application of the Density Ratio Class.     

Local search in routing problems with time windows

We develop local search algorithms for routing problems with time windows. The presented algorithms are based on thek-interchange concept. The presence of time windows introduces feasibility constraints, the checking of which normally requires O(N) time. Our method reduces this checking effort to O(1) time. We also consider the problem of finding initial solutions. A complexity result is given and an insertion heuristic is described.     

Global dynamics of approximate solutions to an age-structured epidemic model with diffusion

We consider an age-dependent s-i-s epidemic model with diffusion whose mortality is unbounded. We approximate the solution using Galerkin methods in the space variable combined with backward Euler along the characteristic direction in the age and time variables. It is proven that the scheme is stable and convergent in optimal rate in l ^∞,2 (L ²) norm. To investigate the global behavior of the discrete solution resulting from the algorithm, we reformulate the resulting system into a monotone form. Positivity of the nonlocal birth process is proved using the positivity of the first eigenvalue of the resulting matrix system and using the fact that the positivity is preserved along the characteristics. The difference equation of the steady state coupled with nonlocal birth process is solved by developing monotone iterative schemes. The stability of the discrete solution of the steady state is then analyzed by constructing suitable positive subsolutions. Mathematics subject classifications (2000) 65M12, 65M25, 65M60, 92D25 M.-Y. Kim: This work was supported by Korea Research Foundation Grant (KRF-2001-041-D00037).     

Generalized vector quasi-equilibrium problems with applications

In this paper, we consider the generalized vector quasi-equilibrium problem with or without involving Φ-condensing maps and prove the existence of its solution by using known fixed point and maximal element theorems. As applications of our results, we derive some existence results for a solution to the vector quasi-optimization problem for nondifferentiable functions and vector quasi-saddle point problem.     

On global attractor for m-Laplacian parabolic equation with local and nonlocal nonlinearity

In this paper, we study the long-time behavior of solutions for m-Laplacian parabolic equation in Ω×(0,∞) with the initial data u(x,0)=u₀(x)∈Lq, q?1, and zero boundary condition in ∂Ω. Two cases for a(x)?a₀>0 and a(x)?0 are considered. We obtain the existence and Lp estimate of global attractor A in Lp, for any p?max{1,q}. The attractor A is in fact a bounded set in if a(x)?a₀>0 in Ω, and A is bounded in if a(x)?0 in Ω.     

Some observations on maximum weight stable sets in certain P5-free graphs

The maximum weight stable set problem (MWS) is the weighted version of the maximum stable set problem (MS), which is NP-hard. The class of P₅-free graphs – i.e., graphs with no induced path of five vertices – is the unique minimal class, defined by forbidding a single connected subgraph, for which the computational complexity of MS is an open question. At the same time, it is known that MS can be efficiently solved for (_P5,F)$(P_5\text{,}F)$-free graphs, where F is any graph of five vertices different to a C₅. In this paper we introduce some observations on P₅-free graphs, and apply them to introduce certain subclasses of such graphs for which one can efficiently solve MWS. That extends or improves some known results, and implies – together with other known results – that MWS can be efficiently solved for (_P5,F)$(P_5\text{,}F)$-free graphs where F is any graph of five vertices different to a C₅.     

Optimality conditions and duality for a minimax fractional programming with generalized convexity

We establish the sufficient conditions for generalized fractional programming from a viewpoint of the generalized convexity. When the sufficient conditions are utilized, the corresponding duality theorems are derived for two types of duals of the generalized fractional programming. We extend the corresponding results of several authors.     

The limiting problem of the drift-diffusion-Poisson model with discontinuous p-n-junctions

In this paper, the authors consider the limiting problem of the drift-diffusion-Poisson model for semiconductors. Different from previous papers, the model considered involve some special doping profiles D which have the property that the function is allowed to have a jump-discontinuity and sign changing property but D² is required to be Lipschitz continuous. The existence, uniqueness and large-time asymptotic behavior of the global (in time) solutions are given.     

Mathematical modelling and analysis of bulk waves in rotating generalized thermoelastic media with voids

The present paper deals with the propagation of body waves in a homogenous isotropic, rotating, generalized thermoelastic solid with voids. The complex quartic secular equation has been solved by employing Descartes’ algorithm and perturbation method to obtain phase velocities, attenuations and specific loss factors of four attenuating and dispersive waves, which are possible to exist in such media. These wave characteristics have been computed numerically for magnesium crystal and presented graphically. Statistical analysis has been performed to compare the obtained computer simulated result in order to have estimate on the suitability of the method to compute various characteristics of the waves. This work may find applications in geophysics and gyroscopic sensors.     

Positive solutions for singular m-point boundary value problems with positive parameter

In this paper, some new existence results for positive solutions for a class of singular m-point boundary value problems with parameter be obtained.     

A new method for solving linear multi-objective transportation problems with fuzzy parameters

There are several methods in the literature for solving transportation problems by representing the parameters as normal fuzzy numbers. Chiang [J. Chiang, The optimal solution of the transportation problem with fuzzy demand and fuzzy product, J. Inform. Sci. Eng. 21 (2005) 439-451] pointed out that it is better to represent the parameters as (λ, ρ) interval-valued fuzzy numbers instead of normal fuzzy numbers and proposed a method to find the optimal solution of single objective transportation problems by representing the availability and demand as (λ, ρ) interval-valued fuzzy numbers. In this paper, the shortcomings of the existing method are pointed out and to overcome these shortcomings, a new method is proposed to find solution of a linear multi-objective transportation problem by representing all the parameters as (λ, ρ) interval-valued fuzzy numbers. To illustrate the proposed method a numerical example is solved. The advantages of the proposed method over existing method are also discussed.     

Stochastic decomposition in production inventory with service time

We study an (s, S) production inventory system where the processing of inventory requires a positive random amount of time. As a consequence a queue of demands is formed. Demand process is assumed to be Poisson, duration of each service and time required to add an item to the inventory when the production is on, are independent, non-identically distributed exponential random variables. We assume that no customer joins the queue when the inventory level is zero. This assumption leads to an explicit product form solution for the steady state probability vector, using a simple approach. This is despite the fact that there is a strong correlation between the lead-time (the time required to add an item into the inventory) and the number of customers waiting in the system. The technique is: combine the steady state vector of the classical M/M/1 queue and the steady state vector of a production inventory system where the service is instantaneous and no backlogs are allowed. Using a similar technique, the expected length of a production cycle is also obtained explicitly. The optimal values of S and the production switching on level s have been studied for a cost function involving the steady state system performance measures. Since we have obtained explicit expressions for the performance measures, analytic expressions have been derived for calculating the optimal values of S and s.     

Positive solutions of the p-Laplace equation with singular nonlinearity

For a given bounded domain Ω in Rn with C1,? boundary for some 0<?<1, and a possibly singular nonlinearity f on Ω×(0,∞), we give sufficient conditions on f so that the p-Laplace equation −Δpu=f(x,u) admits a solution . On the basis of a comparison principle we will give a sufficient condition under which such a problem admits a unique solution.     

Multiple positive solutions to a three-point boundary value problem with p-Laplacian

Sufficient conditions are obtained that guarantee the existence of at least two positive solutions for the equation (g(u′(t)))′+a(t)f(u)=0 subject to boundary conditions, by a simple application of a new fixed-point theorem due to Avery and Henderson.     

Existence of positive solutions for 2nth-order singular sublinear boundary value problems

This paper investigates the existence of positive solutions for 2nth-order (n>1) singular sub-linear boundary value problems. A necessary and sufficient condition for the existence of C2n−2[0,1] as well as C2n−1[0,1] positive solutions is given by constructing lower and upper solutions and with the maximal theorem. Our nonlinearity f(t,x₁,x₂,…,xn) may be singular at xi=0, i=1,2,…,n, t=0 and/or t=1.