On distribution of fractional parts of linear forms

Peres–Schlag’s approach is used to construct badly approximable numbers in various problems of Diophantine approximation.     

Entire functions, analytic continuation, and the fractional parts of a linear function

The main result of the paper is as follows.Theorem. Suppose that G(z) is an entire function satisfying the following conditions: 1) the Taylor coefficients of the function G(z) are nonnegative: 2) for some fixed C>0 and A>0 and for |z|>R₀, the following inequality holds:

On the distribution of the sequence of fractional parts of a slowly increasing exponential function

In this paper for a slowly increasing exponential function we study the degree of deviation of the distribution of the sequence of its fractional parts from the “ideal” uniform distribution. An estimate is given which can appropriately be called the measure of irregularity of the distribution of a given sequence. Translated fromMatematicheskie Zametki, Vol. 65, No. 1, pp. 148–152, January, 1999.     

The bilevel linear/linear fractional programming problem

Bilevel programming involves two optimization problems where the constraint region of the first level problem is implicitly determined by another optimization problem. In this paper we consider the bilevel linear/linear fractional programming problem in which the objective function of the first level is linear, the objective function of the second level is linear fractional and the feasible region is a polyhedron. For this problem we prove that an optimal solution can be found which is an extreme point of the polyhedron. Moreover, taking into account the relationship between feasible solutions to the problem and bases of the technological coefficient submatrix associated to variables of the second level, an enumerative algorithm is proposed that finds a global optimum to the problem.     

Ranking of vertices in the linear fractional programming problem

In this paper the problem of ranking vertices in the linear fractional programming problem is considered. It is shown that a class of vertex ranking algorithms for the linear programming problem can be used with only minor modifications.     

Constrained integer linear fractional programming problem

An integer linear fractional programming problem, whose integer solution is required to satisfy any h out of given n sets of constraints has been discussed in this paper. Method for ranking and scanning all integer points has also been developed and a numerical illustration is included in support of theory.     

Inhomogeneous diophantine approximations and distribution of fractional parts

A new estimate of the remainder term is obtained in the problem of the distribution of fractional parts of nα over an arbitrary interval.     

Restoration of efficiency in a goal programming problem with linear fractional criteria

The problem resulting from a goal programming problem with linear fractional criteria is not easy to solve due to the non-linear constraints inherent in its formulation. This paper introduces a simple and reliable test to establish whether a linear fractional goal programming problem has solutions that verify all goals and, if so, how to find them by solving a linear programming problem. This paper also outlines a new technique for restoring efficiency based on a minimax philosophy. An example is presented.     

Time variant multi-objective linear fractional interval-valued transportation problem

This paper studies a time-variant multi-objective linear fractional transportation problem. In reality, transported goods should reach in destinations within a specific time. Considering the importance of time, a time-variant multi-objective linear fractional transportation problem is formulated here. We take into account the parameters as cost, supply and demand are interval valued that involved in the proposed model, so we treat the model as a multi-objective linear fractional interval transportation problem. To solve the formulated model, we first convert it into a deterministic form using a new transformation technique and then apply fuzzy programming to solve it. The applicability of our proposed method is shown by considering two numerical examples. At last, conclusions and future research directions regarding our study is included.     

A simplex-based labelling algorithm for the linear fractional assignment problem

This paper constructs an algorithm to solve the fractional assignment problem. Algorithms that are currently used are mostly based on parametric approaches and must solve a sequence of optimization procedures. They also neglect the difficulties caused by degeneracy. The proposed algorithm performs optimization once and overcomes degeneracy. The main features of the algorithm are an effective initial heuristic approach, a simple labelling procedure and an implicit primal-dual schema. A numerical example is presented and demonstrates that the proposed algorithm is easy to apply. Computational results are compared with those from other developed methods. The results show that the proposed algorithm is efficient.     

Proper efficiency in a linear fractional vector maximum problem with generalized convex constraints

In the present paper, a vector maximum problem (VMP) with linear fractional objectives and generalized convex constraints is considered. A necessary and sufficient condition for an efficient solution of (VMP) is derived in Kuhn-Tucker's form. Moreover, it is proved that under a certain boundedness assumption an efficient solution is properly efficient. This extends the results of Choo [3] for a linear fractional vector maximum problem with linear constraints to generalized convex constraints. An example is given to illustrate the results.     

Generalized Dirichlet problem for a fractional linear differential equation with constant coefficients

For a linear ordinary differential equation with constant coefficients, we construct a solution of a boundary value problem. We prove the existence of a regular solution of this problem and derive a condition for its uniqueness.     

Optimization of a linear fractional function on a hypersphere of an affine space

In this paper, we consider the following nonlinear programming problem: