This contribution describes relationship between fractions, projective representation, duality, linear algebra and geometry. Many problems lead to a system of linear equations and this paper presents equivalence of the cross–product operation and solution of a system of linear equations Ax = 0 or Ax = b using projective space representation and homogeneous coordinates. It leads to conclusion th...

A representation of an arbitrary system of strict linear inequalities in IR n as a system of points is proposed. The representation is obtained by using a so-called polarity. Based on this representation an algorithm for constructing a committee solution of an inconsistent plane system of linear inequalities is given. A solution of two problems on minimal committee of a plane system is proposed...

In this paper, we establish the equivalence between the half-space representation and the vertex representation of a smooth parametric semiclosed polyhedron. By virtue of the smooth representation result, we prove that the solution set of a smooth parametric piecewise linear program can be locally represented as a finite union of parametric semiclosed polyhedra generated by finite smooth functi...

There are three phases in the life of a decision problem, specification, solution, and representation of solution. The specification and solution phases are off-line, while the represention of solution often shall serve an on-line situation with rather tough constraints on time and space. One of the advantages of influence diagrams (IDs) is that for small decision problems, the distinction betw...

The fixed charge transportation problem (FCTP) is a deployment of the classical transportation problem in which a fixed cost is incurred, independent of the amount transported, along with a variable cost that is proportional to the amount shipped. Since the problem is considered as an NP-hard, the computational time grows exponentially as the size of the problem increases. In this pape...

The exact solution of the anti-symmetric quadratic truly nonlinear oscillator is derived from the first integral of the nonlinear differential equation which governs the behaviour of this oscillator. This exact solution is expressed as a piecewise function including Jacobi elliptic cosine functions. The Fourier series expansion of the exact solution is also analyzed and its coefficients are com...

: In this paper, it is tried to explain the concept of representation and architectural representation through a qualitative methodology, approach its procedure for gradual creation in architecture and then according to scholars and to obtain the effect of this concept in the process of architectural facts the concepts are presented. In addition, it is referred to theories and practical texts b...

This paper addresses a rollon–rolloff waste collection vehicle routing problem which involves large containers that collect huge amounts of waste from construction sites and shopping malls. In this problem, vehicles could only move one container at a time between customers’ locations, a depot, a disposal center, and a recycling center site. Other than a disposal center, a recycling ...

We obtain a finite-sum representation for the general solution of the equation ∆ (p(n− 1)∆u(n − 1)) + q(n)u(n) = λr(n)u(n) in terms of a nonvanishing solution corresponding to some fixed value of λ = λ0. Applications of this representation to some results on the boundedness of solutions are given as well as illustrating examples.