A short proof that N 3 is not a circle containment order

A partially ordered set P is called a circle containment order provided one can assign to each xP a circle C _x so that . We show that the infinite three-dimensional poset N ³ is not a circle containment order and note that it is still unknown whether or not [n]³ is such an order for arbitrarily large n.     

The many faces of circle orders

A finite partially ordered set P is called a circle order if one can assign to each x P a circular disk C _x so that xy iff C _x C _y . It is interesting to observe that many other classes of posets, such as space-time orders, parabola orders, the Loewner order for 2×2 Hermitian matrices, etc. turn out to be exactly circle orders (or their higher dimensional analogues). We give a global proof for these equivalences.Research supported in part by the Office of Naval Research, the Air Force Office of Scientific Research and by DIMACS.     

Automorphisms and isotone self-maps of ordered sets with top and bottom

For an ordered setP letP ^P denote the set of all isotone self-maps on P, that is, all mapsf fromP toP such thatxy impliesf(x)f(y), and let Aut (P) the set of all automorphisms onP, that is, all bijective isotone self-maps inP ^P . We establish an inequality relating ¦P ^P ¦ and ¦Aut(P)¦ in terms of the irreducibles ofP. As a straightforward corollary, we show that Rival and Rutkowski's automorphism conjecture is true for lattices. It is also true for ordered sets with top and bottom whose covering graphs are planar.Supported in part by NSERC (Grant no. A2507).Supported under an NSERC International Research Fellowship.     

Sphere orders

Ann-sphere order is a finite partially ordered set representable by containment ofn-spheres in Euclidean (n+1)-space. We present a sequence {P _i } of ordered sets such that eachP _i is ann-sphere order only forni; one consequence is that we are able to determine the dimension of a Euclidean space-time manifold from the finite suborders of its causality order.Research supported by ONR grant N00014 85-K-0769.     

Cube Orders

A finite partially ordered set P =(X, ) is a cube order in R ^m if there is a map C from X into cubes in R ^m such that, for all x and y in X, x y if and only if C(x) C(y). It is proved that every finite poset with order dimension 2 m is a cube order in R ^m .     

Degrees of freedom versus dimension for containment orders

Given a family of sets L, where the sets in L admit k degrees of freedom, we prove that not all (k+1)-dimensional posets are containment posets of sets in L. Our results depend on the following enumerative result of independent interest: Let P(n, k) denote the number of partially ordered sets on n labeled elements of dimension k. We show that log P(n, k)nk log n where k is fixed and n is large.Research supported in part by Allon Fellowship and by a grant from Bat Sheva de Rothschild Foundation.Research supported in part by the Office of Naval Research, contract number N00014-85-K0622.     

Linear Discrepancy and Weak Discrepancy of Partially Ordered Sets

The linear discrepancy of a partially ordered set P=(X,) is the least integer k for which there exists an injection f: XZ satisfying (i) if xy then f(x)<f(y) and (ii) if xy then |f(x)–f(y)|k. This concept is closely related to the weak discrepancy of P studied previously. We prove a number of properties of linear and weak discrepancies and relate them to other poset parameters. Both parameters have applications in ranking the elements of a partially ordered set so that the difference in rank of incomparable elements is minimized.     

Filtering on manifolds

We coasider a partially observable diffusion process (x _t,y_t)t0 whose unobservable componentx _t lives on a submanifold M ofR ⁿ . We present some general conditions under which the conditional law ofx _t, given the observationsy _s ,s [0,t], admits a density w.r.t. a given measure on M. We characterize the analytical properties of this density by using appropriate Sobolev spaces.Research supported by the Hungarian National Foundation of Scientific Research No. 2290.     

Lawless order

R. Baer asked whether the group operation of every (totally) ordered group can be redefined, keeping the same ordered set, so that the resulting structure is an Abelian ordered group. The answer is no. We construct an ordered set (G, ) which carries an ordered group (G, , ) but which islawless in the following sense. If (G, *, ) is an ordered group on the same carrier (G, ), then the group (G, *) satisfies no nontrivial equational law.Research partially supported by NSERC of Canada Grants #A4044 and A3040.Research partially supported by NSERC of Canada Grant #U0075.Research partially supported by a grant from the BSF.     

Angle orders and zeros

An angle order is a partially ordered set whose points can be mapped into unbounded angular regions in the plane such that x is less than y in the partial order if and only if x's angular region is properly included in y's. The zero augmentation of a partially ordered set adds one point to the set that is less than all original points. We prove that there are finite angle orders whose augmentations are not angle orders. The proof makes extensive use of Ramsey theory.     

Simplex pivots on the set packing polytope

This paper considers the set packing problem max{wx: Ax b, x 0 and integral}, whereA is anm × n 0–1 matrix,w is a 1 ×n weight vector of real numbers andb is anm × 1 vector of ones. In equality form, its linear programming relaxation is max{wx: (x, y) P(A)} whereP(A) = {(x, y):Ax +I _m y =b, x0,y0}. Letx ¹ be any feasible solution to the set packing problem that is not optimal and lety ¹ =b – Ax ¹; then (x ¹,y ¹) is an integral extreme point ofP(A). We show that there exists a sequence of simplex pivots from (x ¹,y ¹) to (x*,y*), wherex* is an optimal solution to the set packing problem andy* =b – Ax*, that satisfies the following properties. Each pivot column has positive reduced weight and each pivot element equals plus one. The number of pivots equals the number of components ofx* that are nonbasic in (x ¹,y ¹).This research was supported by NSF Grants ECS-8005360 and ECS-8307473 to Cornell University.     

On a class of least-element complementarity problems

The present paper studies the linear complementarity problem of finding vectorsx andy inR ₊ ⁿ such thatc + Dx + y 0,b – x 0 andx ^T (c + Dx + y) = y ^T (b – x) = 0 whereD is aZ-matrix andb > 0. Complementarity problems of this nature arise, for example, from the minimization of certain quadratic functions subject to upper and lower bounds on the variables. Two least-element characterizations of solutions to the above linear complementarity problem are established first. Next, a new and direct method to solve this class of problems, which depends on the idea of least-element solution is presented. Finally, applications and computational experience with its implementation are discussed.Research partially supported by the National Science Foundation Grant MCS 71-03341 A04 and the Air Force Office of Scientific Research Contract F 44620 14 C 0079.     

Varieties of lattice ordered groups that contain no non-abelian o-groups are solvable

It is shown that the variety _n of lattice ordered groups defined by the identity x ⁿ y ⁿ =y ⁿ x ⁿ , where n is the product of k (not necessarily distinct primes) is contained in the (k+1)st power A ^k+1 of the variety A of all Abelian lattice ordered groups. This implies, in particular, that _n is solvable class k + 1. It is further established that any variety V of lattice ordered groups which contains no non-Abelian totally ordered groups is necessarily contained in _n , for some positive integer n.This work was supported in part, by NSERC Grant A4044.     

Biconvex sets and optimization with biconvex functions: a survey and extensions

The problem of optimizing a biconvex function over a given (bi)convex or compact set frequently occurs in theory as well as in industrial applications, for example, in the field of multifacility location or medical image registration. Thereby, a function is called biconvex, if f(x,y) is convex in y for fixed x∈X, and f(x,y) is convex in x for fixed y∈Y. This paper presents a survey of existing results concerning the theory of biconvex sets and biconvex functions and gives some extensions. In particular, we focus on biconvex minimization problems and survey methods and algorithms for the constrained as well as for the unconstrained case. Furthermore, we state new theoretical results for the maximum of a biconvex function over biconvex sets. J. Gorski and K. Klamroth were partially supported by a grant of the German Research Foundation (DFG).     

On the Uniqueness ofL 2-solutions in half-space of certain differential equations

Square integrable solutions to the equation{– ²/y² + P(D_x)+b(y)–}u(x, y) = f(x, y) are considered in the half-spacey>0, x ⁿ , whereP(D _x) is a constant coefficient operator. Under suitable conditions on lim_y0u(x, y), b(y), f(x, y) and , it is shown that suppu = suppf. This generalizes a result due to Walter Littman.Research partially supported by USNSF Grant 79-02538-A02.     

An implicit function theorem

Suppose thatF:DR ⁿ×R^mRⁿ, withF(x ⁰,y ⁰)=0. The classical implicit function theorem requires thatF is differentiable with respect tox and moreover that ₁ F(x ⁰,y ⁰) is nonsingular. We strengthen this theorem by removing the nonsingularity and differentiability requirements and by replacing them with a one-to-one condition onF as a function ofx.     

Fractional dimension of partial orders

Given a partially ordered setP=(X, ), a collection of linear extensions {L ₁,L ₂,...,L _r } is arealizer if, for every incomparable pair of elementsx andy, we havex<y in someL _i (andy<x in someL _j ). For a positive integerk, we call a multiset {L ₁,L ₂,...,L _t } ak-fold realizer if for every incomparable pairx andy we havex<y in at leastk of theL _i 's. Lett(k) be the size of a smallestk-fold realizer ofP; we define thefractional dimension ofP, denoted fdim(P), to be the limit oft(k)/k ask. We prove various results about the fractional dimension of a poset.Research supported in part by the Office of Naval Research.     

Sieving by large prime factors

LetQ be a set of primes with density <1. An asymptotic is proved for the number of positive integers x which do not have a prime divisor which is >y and belongs toQ.     

Two-step fourth orderP-stable methods for second order differential equations

A family of two-step fourth order methods, which requires two function evaluations per step, is derived fory=f(x,y). We then show the existence of a sub-family of these methods which when applied toy=–k ² y,k real, areP-stable.     

On ordered-covering mappings and implicit differential inequalities

We define the set of ordered covering of a mapping that acts in partially ordered spaces; we suggest a method for finding the set of ordered covering of vector functions of several variables and the Nemytskii operator acting in Lebesgue spaces. We prove assertions on operator inequalities in arbitrary partially ordered spaces. We obtain conditions that use a set of ordered covering of the corresponding mapping and ensure that the existence of an element u such that f(u) ≥ y implies the solvability of the equation f(x) = y and the estimate x ≤ u for its solution. We study the problem on the existence of the minimal and least solutions. These results are used for the analysis of an implicit differential equation. For the Cauchy problem, we prove a theorem on an inequality of the Chaplygin type.