Generators, extremals and bases of max cones

Max cones are max-algebraic analogs of convex cones. In the present paper we develop a theory of generating sets and extremals of max cones in ${{\mathbb R}}_+^n$. This theory is based on the observation that extremals are minimal elements of max cones under suitable scalings of vectors. We give new proofs of existing results suitably generalizing, restating and refining them. Of these, it is important that any set of generators may be partitioned into the set of extremals and the set of redundant elements. We include results on properties of open and closed cones, on properties of totally dependent sets and on computational bounds for the problem of finding the (essentially unique) basis of a finitely generated cone.Comment: 15 pages, 1 figure; v2: new layout, several new references, renumbering of result

Projections in minimax algebra

An axiomatic theory of linear operators can be constructed for abstract spaces defined over (R, ⊕, ⊗), that is over the (extended) real numbersR with the binary operationsx ⊕ y = max (x,y) andx ⊗ y = x + y. Many of the features of conventional linear operator theory can be reproduced in this theory, although the proof techniques are quite different. Specialisation of the theory to spaces ofn-tuples provides techniques for analysing a number of well-known operational research problems, whilst specialisation to function spaces provides a natural formal framework for certain familiar problems of approximation, optimisation and duality

Max-plus definite matrix closures and their eigenspaces

In this paper we introduce the definite closure operation for max-plus matrices with finite permanent, reveal inner structures of definite eigenspaces, and establish some facts about Hilbert distances between these inner structures and the boundary of the definite eigenspaceComment: 20 pages,6 figures, v2: minor changes in figures and in the main tex

Complete solution of a constrained tropical optimization problem with application to location analysis

We present a multidimensional optimization problem that is formulated and solved in the tropical mathematics setting. The problem consists of minimizing a nonlinear objective function defined on vectors over an idempotent semifield by means of a conjugate transposition operator, subject to constraints in the form of linear vector inequalities. A complete direct solution to the problem under fairly general assumptions is given in a compact vector form suitable for both further analysis and practical implementation. We apply the result to solve a multidimensional minimax single facility location problem with Chebyshev distance and with inequality constraints imposed on the feasible location area.Comment: 20 pages, 3 figure

Soluble approximation of linear systems in max-plus algebra

summary:We propose an efficient method for finding a Chebyshev-best soluble approximation to an insoluble system of linear equations over max-plus algebra

Green's J-order and the rank of tropical matrices

We study Green's J-order and J-equivalence for the semigroup of all n-by-n matrices over the tropical semiring. We give an exact characterisation of the J-order, in terms of morphisms between tropical convex sets. We establish connections between the J-order, isometries of tropical convex sets, and various notions of rank for tropical matrices. We also study the relationship between the relations J and D; Izhakian and Margolis have observed that $D \neq J$ for the semigroup of all 3-by-3 matrices over the tropical semiring with $-\infty$, but in contrast, we show that $D = J$ for all full matrix semigroups over the finitary tropical semiring.Comment: 21 pages, exposition improve

An algorithm to describe the solution set of any tropical linear system A x=B x

An algorithm to give an explicit description of all the solutions to any tropical linear system A x=B x is presented. The given system is converted into a finite (rather small) number p of pairs (S,T) of classical linear systems: a system S of equations and a system T of inequalities. The notion, introduced here, that makes p small, is called compatibility. The particular feature of both S and T is that each item (equation or inequality) is bivariate, i.e., it involves exactly two variables; one variable with coefficient 1 and the other one with -1. S is solved by Gaussian elimination. We explain how to solve T by a method similar to Gaussian elimination. To achieve this, we introduce the notion of sub-special matrix. The procedure applied to T is, therefore, called sub-specialization

Equation with residuated functions

summary:The structure of solution-sets for the equation $F(x)=G(y)$ is discussed, where $F,G$ are given residuated functions mapping between partially-ordered sets. An algorithm is proposed which produces a solution in the event of finite termination: this solution is maximal relative to initial trial values of $x,y$. Properties are defined which are sufficient for finite termination. The particular case of max-based linear algebra is discussed, with application to the synchronisation problem for discrete-event systems; here, if data are rational, finite termination is assured. Numerical examples are given. For more general residuated real functions, lower semicontinuity is sufficient for convergence to a solution, if one exists

Cyclic projectors and separation theorems in idempotent convex geometry

Semimodules over idempotent semirings like the max-plus or tropical semiring have much in common with convex cones. This analogy is particularly apparent in the case of subsemimodules of the n-fold cartesian product of the max-plus semiring it is known that one can separate a vector from a closed subsemimodule that does not contain it. We establish here a more general separation theorem, which applies to any finite collection of closed semimodules with a trivial intersection. In order to prove this theorem, we investigate the spectral properties of certain nonlinear operators called here idempotent cyclic projectors. These are idempotent analogues of the cyclic nearest-point projections known in convex analysis. The spectrum of idempotent cyclic projectors is characterized in terms of a suitable extension of Hilbert's projective metric. We deduce as a corollary of our main results the idempotent analogue of Helly's theorem.Comment: 20 pages, 1 figur

Perceptions of childhood immunization in a minority community: Qualitative study

This is the author's accepted manuscript. The final published article is available from the link below. Published article copyright @ The Royal Society of Medicine.Objective - To assess reasons for low uptake of immunization amongst orthodox Jewish families. Design - Qualitative interviews with 25 orthodox Jewish mothers and 10 local health care workers. Setting - The orthodox Jewish community in North East London. Main outcome measures - Identification of views on immunization in the orthodox Jewish community. Results - In a community assumed to be relatively insulated from direct media influence, word of mouth is nevertheless a potent source of rumours about vaccination dangers. The origins of these may lie in media scares that contribute to anxieties about MMR. At the same time, close community cohesion leads to a sense of relative safety in relation to tuberculosis, with consequent low rates of BCG uptake. Thus low uptake of different immunizations arises from enhanced feelings of both safety and danger. Low uptake was not found to be due to the practical difficulties associated with large families, or to perceived insensitive cultural practices of health care providers. Conclusions - The views and practices of members of this community are not homogeneous and may change over time. It is important that assumptions concerning the role of religious beliefs do not act as an obstacle for providing clear messages concerning immunization, and community norms may be challenged by explicitly using its social networks to communicate more positive messages about immunization. The study provides a useful example of how social networks may reinforce or challenge misinformation about health and risk and the complex nature of decision making about children's health.City and Hackney Teaching Primary Care Trus