Optimal solution of a Monge-Kantorovitch transportation problem

Suppose that P and Q are probabilities on a separable Banach space $\text{E}$. It is known that if (P, Q) satisfies certain regularity conditions and a random variable X has law P, then there exists a function $\text{f :}\text{E}\text{→}\text{E}$, such that the function f(X) has the law Q and the random pair (X, f(X)) is an optimal coupling for the Monge-Kantorovitch problem. In this paper we provide an approximation of the function f when the law Q is discrete. Thenwe extend this main result to any law Q. The proofs are based on a relationship between optimal couplings and nonlinear equations.     

On the decentralized transportation problem

The decentralized transportation problem is under study where the customers act individually maximizing their own profits while the producer determines only the sequence of their service. The problem is shown to be NP-hard, and some effective approximation algorithm is suggested with a guaranteed approximation bound in the case of the same demand volumes.     

Critical solution for a Hill's type problem

We study the problem of two satellites attracted by a center of force. Assuming that the motion of the center of mass of the two satellites is a Keplerian circular orbit around the center of force, we regularize the collision between them using the Levi-Civita procedure. The existence of a constant of motion in the extended phase space allows us to study the stability of the solution, where the two satellites are tied together in their circular motion around the center of force. We call this solution the critical solution. A theorem of M. Kummer is applied to prove, in specific conditions, the existence of two one-parametric families of almost periodic orbits for the satellites motion that bifurcates from the critical solution.     

Positive solution to a special singular second-order boundary value problem

Let λ be a nonnegative parameter. The existence of a positive solution is studied for a semipositone second-order boundary value problem

where d>0,α≥0,β≥0,α+β>0, q(t)f(t,u,v)≥0 on a suitable subset of [0,1]×[0,+∞)×(−∞,+∞) and f(t,u,v) is allowed to be singular at t=0,t=1 and u=0. The proofs are based on the Leray–Schauder fixed point theorem and the localization method.     

On a smooth solution of a boundary-value problem

We study a periodic boundary-value problem for the quasilinear equation u _tt −u _xx =F[u, u _t , u _x ], u(x, 0)=u(x, π)=0, u(x + ω, t) = u(x, t), x ∈ ℝ t ∈ [0, π], and establish conditions that guarantee the validity of a theorem on unique solvability. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 9, pp. 1293–1296, September, 1998.     

On the Cauchy problem of transportation equations

In this paper, we prove the global existence of generalized solutions of Cauchy problem for transportation equations; moreover, we construct the solution by generalized potential.     

On a special type of Ma-Minda function

We introduce a primitive class of analytic functions,by specializing in many well-known classes,classify Ma-Minda functions based on its conditions and their interesting geomet-rical aspects.Further,study a newly defined subclass of starlike functions involving a special type of Ma-Minda function introduced here for obtaining inclusion and radius results.We also establish some majorization,Bloch function norms,and other related problems for the same class.     

On a problem of the Goldbach-waring type

LetE(X)=‖{N≤X;N≠p ₁ ² +p ₂ ³ +p ₃ ⁴ +p ₄ ⁵ for any primesp _i}‖. It is proved in this paper that there exists a positive constant δ>0 such that

On the trade-off between staff-decomposed and activity-decomposed column generation for a staff scheduling problem

In this paper a comparison is made between two decomposition techniques to solve a staff scheduling problem with column generation. In the first approach, decomposition takes place on the staff members, whereas in the second approach decomposition takes place on the activities that have to be performed by the staff members. The resulting master LP is respectively a set partitioning problem and a capacitated multi-commodity flow problem. Both approaches have been implemented in a branch-and-price algorithm. We show a trade-off between modeling power and computation times of both techniques.     

On the uniqueness of a solution of the fourier problem for a system of sobolev-gal’pern type

We establish conditions for the uniqueness of a solution of the problem for a system of equations unresolved with respect to the time derivative without initial conditions in a noncylindrical domain. The system considered, in particular, contains pseudoparabolic equations.     

On a smooth solution of a nonlinear periodic boundary-value problem

We establish conditions for the existence of a smooth solution of a quasilinear hyperbolic equationu _tt - u_xx = ƒ(x, t, u, u, u _x),u (0,t) = u (π,t) = 0,u (x, t+ T) = u (x, t), (x, t) ∈ [0, π] ×R, and prove a theorem on the existence and uniqueness of a solution. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 11, pp. 1574–1576, November, 1999.     

On quadratic fractional generalized solid bi-criterion transportation problem

In this paper bi-criterion quadratic fractional generalized solid transportation problem is studied. An algorithm is devoloped to obtain the time-cost trade-off pairs and hence identifies the optimum trade-off pairs giving the equal priority to both time and cost. A numerical example is illustrated to support the algorithm.