An Efficient Algorithm for Solving the Rectilinear Multifacility Location Problem

One of the efficient methods for solving large rectilinear multifacilitylocation problems is the Direct Search method. The only drawbackof this method lies in the following difficulty. In some situations,when t new facilities are located together at one point, thenumber of arithmetic operations which are needed to establishoptimality is proportional to t²2^t. Therefore the method needsa prohibitive amount of computation time whenever t exceeds,say, 20. This paper gives a simple remedy for this problem.The paper states and proves a new necessary and sufficient optimalitycondition. This condition transforms the problem of computinga descent direction into a constrained linear least-squaresproblem. The latter problem is solved by a relaxation methodthat takes advantage of its special structure. The new techniqueis incorporated into the Direct Search method. This yields animproved algorithm that handles efficiently very large clusters.Numerical results are included.     

Flow Through Symmetrically Constricted Tubes

Fluid morion through a tube is discussed when there is a moderateor severe symmetric constriction at the wall, and the oncomingflow is fully developed. The Reynolds number R is assumed large.During a moderate constriction, where the typical slope isO(R^–), upstream separation can be provoked due to thedownstream pressure being transmitted back, through the inviscidcore flow. Separation can also occur after a point of maximumconstriction. Computations and analysis indicate that the upstreamseparation point is pushed increasingly ahead as the slope israised. The implication for a severe constriction, where isO(1), is that the flow separates at a large distance O(a lnR) upstream (a being the tubewidth) and produces a shear layerwhich, on nearing the constriction, reaches O(a) distances fromthe wall.     

Projections of Bodies and Hereditary Properties of Hypergraphs

We prove that for every n-dimensional body K, there is a rectangularparallelepiped B of the same volume as K, such that the projectionof B onto any coordinate subspace is at most as large as thatof the corresponding projection of K. We apply this theorem to projections of finite set systems andto hereditary properties. In particular, we show that everyhereditary property of uniform hypergraphs has a limiting density.     

On Unsteady Boundary Layers in a Rotating Flow

An approximate analysis of the rotating flow of an incompressible,viscous fluid over a rotating, porous plate is carried out underthe following conditions: (a) plate oscillating about a constantmean, (b) variable suction velocity, (c) rigid body rotation.Approximate solutions to velocity components u, v are derivedand shown graphically. The effects of E (Eckman number), (frequency)and A (suction parameter) on the velocity field, skin-frictionand mass flux in the Eckman layer are discussed quantitatively.     

The use of linear programming duality in mixed integer programming

This paper is about the primal-dual relationship in a mixedinteger programming problem (MIP) in which integer variablesare binary. It shows how the primal-dual relationship of a linearprogramming problem (LP) can be used to advantage in MIPs. Thecentral idea is to look conceptually at the nature of all possibleLPs that arise from all possible settings for the discrete variablesin order to deduce general properties of the solution set. Afterdeveloping the relevant theory, we show the usefulness of thisaproach by applying it to three totally different problems.New results are derived for the method of least median of squaresin robust regression, the problem of rectilinear obnoxious-facilitylocation, and the problem of finding a fixed-size rectanglecontaining the minimum weight of points.     

An Ordered Flow Shop Sequeneing Problem With TWK Type Due Dates and Total Tardiness Criterion

A static n job m-machine ordered flow shop problem is considered.Prior research with such a problem has dealt with the makespanand the mean flow criterion. This paper considers minimizationof total tardiness as the objective. Due dates for all jobsare considered to be proportional to the sum of the processingtimes for each individual job. It is shown that by simply arrangingjobs according to an ascending order of processing time (i.e.by the SPT rule) one can obtain a minimum tardiness sequence.     

A Constructive Method for Shape Optimization: A Problem in Hydromechanics

Certain hydromechanical quantities associated with a floatingor a totally immersed body depend explicitly on the body's geometry.In this paper, the authors consider the problem of choosingthe shape of the body so that one such quantity, added mass,is optimized. In particular, a constructive method of penalizationtype is proposed which depends on the availability of a completefamily of solutions of the original boundary value problem andit is shown how such families may be generated.     

The Scenery Flow for Hyperbolic Julia Sets

We define the scenery flow space at a point z in the Julia setJ of a hyperbolic rational map T : C C with degree at least2, and more generally for T a conformal mixing repellor. We prove that, for hyperbolic rational maps, except for a fewexceptional cases listed below, the scenery flow is ergodic.We also prove ergodicity for almost all conformal mixing repellors;here the statement is that the scenery flow is ergodic for therepellors which are not linear nor contained in a finite unionof real-analytic curves, and furthermore that for the collectionof such maps based on a fixed open set U, the ergodic casesform a dense open subset of that collection. Scenery flow ergodicityimplies that one generates the same scenery flow by zoomingdown towards almost every z with respect to the Hausdorff measureH^d, where d is the dimension of J, and that the flow has a uniquemeasure of maximal entropy. For all conformal mixing repellors, the flow is loosely Bernoulliand has topological entropy at most d. Moreover the flow atalmost every point is the same up to a rotation, and so as acorollary, one has an analogue of the Lebesgue density theoremfor the fractal set, giving a different proof of a theorem ofFalconer. 2000 Mathematical Subject Classification: 37F15, 37F35, 37D20.     

Steepest Descent on Real Flag Manifolds

Real flag manifolds are the isotropy orbits of noncompact symmetricspaces G/K. Any such manifold M is acted on transitively bythe (noncompact) Lie group G, and it is embedded in euclideanspace as a taut submanifold. The aim of this paper is to showthat the gradient flow of any height function is a one-parametersubgroup of G, where the gradient is defined with respect toa suitable homogeneous metric s on M; this generalizes the Kählermetric on adjoint orbits (the so-called complex flag manifolds).2000 Mathematics Subject Classification 53C30, 53C35.     

Uniform Convergence of Variational Methods for Elliptic Problems in N Dimensions

Earlier work in variational methods in the plane is extendedto N dimensions. Smoothness theorems are constructed and usedot convert means convergence to uniform convergence, establishingpointwise error bounds. A constrained variational method isused so that a priori bounds for the derivatives of the solutionmay be imposed on the approximating functions. In the non-linearproblem of compressible flow such a bound is provided by anassumption that the solution is subsonic.     

Linear and Non-linear Stability of Heat Exchangers

The hydrodynamical problem of one-dimensional flow with a uniformheat input resulting in a change of phase is considered. Equationsof mass, momentum, energy and state representing the dynamicbehaviour of such a system are reduced to two coupled equationsfor the density p(x, t) and the inlet velocity U(t) on the assumptionthat the pressure drop applied between the inlet and the outletis "small". A linear stability analysis is carried out whichleads to the problem of computing the zeros of a complicatedanalytic function. A non-linear analysis is applied to the caseof weak instability to find the evolution of the slowly varyingamplitude of a small oscillation: in certain circumstances,a "burst" occurs, and in such cases no such small oscillationcan exist.     

R-trees and laminations for free groups III: currents and dual R-tree metrics

We study the map which associates to a current its support (whichis a lamination). We show that this map is Out(F_N)-equivariant,not injective, not surjective and not continuous. However itis semi-continuous and almost surjective in a suitable sense.Given an -tree T (with dense orbits) in the boundary of outerspace and a current µ carried by the dual lamination ofT, we define a dual pseudo-distance d_µ on T. When thetree and the current come from a measured geodesic laminationon a surface with boundary, the dual distance is the originaldistance of the tree T. In general, such a good correspondencedoes not occur. We prove that when the tree T is the attractivefixed point of a non-geometric irreducible, with irreduciblepowers, outer automorphism, the dual lamination of T is uniquelyergodic and the dual distance d_µ is either zero or infinitethroughout T.     

A Short Proof of the Harris-Kesten Theorem

We give a short proof of the fundamental result that the criticalprobability for bond percolation in the planar square latticeZ² is equal to 1/2. The lower bound was proved by Harris, whoshowed in 1960 that percolation does not occur at p = 1/2. Theother, more difficult, bound was proved by Kesten, who showedin 1980 that percolation does occur for any p > 1/2. 2000Mathematics Subject Classification 60K35, 82B43.     

Simultaneous Packing and Covering in Three-Dimensional Euclidean Space

It is proved that the simultaneous lattice packing and latticecovering constant of every three-dimensional centrally symmetricconvex body is less than or equal to 7/4. Consequently, forany three-dimensional convex body K there is a correspondinglattice packing in which no extra translate of K can be added.     

On rectilinear duals for vertex-weighted plane graphs

Let G=(V,E) be a plane triangulated graph where each vertex is assigned a positive weight. A rectilinear dual of G is a partition of a rectangle into |V| simple rectilinear regions, one for each vertex, such that two regions are adjacent if and only if the corresponding vertices are connected by an edge in E. A rectilinear dual is called a cartogram if the area of each region is equal to the weight of the corresponding vertex. We show that every vertex-weighted plane triangulated graph G admits a cartogram of constant complexity, that is, a cartogram where the number of vertices of each region is constant. Furthermore, such a rectilinear cartogram can be constructed in O(nlogn) time where n=|V|.     

The Canonical Decomposition of the Poset of a Hammock

In the Auslander-Reiten quiver of a representation-directedalgebra several hammocks occur naturally; they begin at theprojective cover of a simple module E and end in the correspondinginjective hull. It is known that hammocks are Auslander-Reitenquivers of posets, so there is a poset corresponding to eachsimple module; it describes the set of modules having E as acomposition factor. In this paper we show that this poset Sdecomposes canonically into a coideal S⁺ and an ideal S^–which can easily be described by vectorspace-categories correspondingto a one-point extension or a one-point coextension, respectively.In addition, we describe the simple modules for which S⁺ andS^– are not comparable, and also those for which S⁺ S^–. We also show how to use the results in order to prove for certainposets that they do not occur as posets corresponding to simplemodules.     

On the Radiation of Short Surface Waves By a Finite Dock

A flat plate of finite width and infinite length lies on thesurface of a body of deep water and vibrates in such a way thatthe velocity distribution over the dock varies smoothly acrossits width and is simple harmonic in time. The amplitudes ofthe wave-trains radiated towards infinity are investigated inthe limit of waves having length small compared with the widthof the dock, by means of a formula expressing the wave amplitudein terms of a Green's function evaluated over the dock. Thisfunction is estimated in the short wave limit by an asymptoticapproximation which is uniformly valid over the dock exceptclose to the edges. Suitable edge corrections are presented,these being essentially the potentials in the correspondingproblems for docks of semi-infinite width. By this means itis found possible to derive the first two or three terms inan asymptotic development for the wave amplitudes, the firstterm agreeing with previous work, and the next term dependingon the nature of the velocity distribution V(x) near the edgesx = ±a. Explicit results are given for the cases whenV(x) is analytic, and when V(x) = O(a–x) near the edgex = a.     

Nonlinear high-wavenumber Bnard convection

Weakly nonlinear two-dimensional roll cells in Bnard convectionare examined in the limit as the wavenumber a of the roll cellsbecomes large. In this limit the second harmonic contributionsto the solution become negligible, and a flow develops wherethe fundamental vortex terms and the correction to the meanare determined simultaneously, rather than sequentially as inthe weakly nonlinear case. Extension of this structure to Rayleighnumbers O(a³) above the neutral curve is shown to be possible,with the resulting flow field having a form very similar tothat for strongly nonlinear vortices in a centripetally unstableflow. The flow in this strongly nonlinear regime consists ofa core region, and boundary layers of thickness O(a^–1)at the walls. The core region occupies most of the thicknessof the fluid layer and only mean terms and cos az terms playa role in determining the flow; in the boundary layer all harmonicsof the vortex motion are present. Numerical solutions of thewall layer equations are presented and it is also shown thatthe heat transfer across the layer is significantly greaterthan in the conduction state.     

Analytical results for 2‐D non‐rectilinear waveguides based on a Green's function

We consider the problem of wave propagation for a 2‐D rectilinear optical waveguide which presents some perturbation. We construct a mathematical framework to study such a problem and prove the existence of a solution for the case of small imperfections. Our results are based on the knowledge of a Green's function for the rectilinear case. Copyright © 2008 John Wiley & Sons, Ltd.