# A partition theorem for Euclidean $n$-space

@inproceedings{Samelson1958APT, title={A partition theorem for Euclidean \$n\$-space}, author={Hans Samelson and Robert M. Thrall and Oscar Wesler}, year={1958} }

Let Vn be an n-dimensional vector space over the reals. Let s . . ' Sins 7)1, * * I, nbe 2n vectors in Vn such that every sequence of vectors { a,, , an }, where ai is either ti or vi, is a linearly independent set. Let (a,, * *, I n) denote the cone spanned by these a's, i.e. the set of all linlear combinations of the ai with non-negative coefficients. The 2n cones in Vn spanlned by the 2n such sequences of a's will be said to partition Vn if their union is all of Vn and if the intersection of…

## 191 Citations

On a characterization of P-matrices

- MathematicsMath. Program.
- 1973

Mur ty [5] refined the above character izat ion by proving tha t M is a P-matrix if and only (1) has a unique solut ion for every g ~ P.

Linear Complementarity and the Degree of Mappings

- Mathematics
- 1983

Let M be an n × n real matrix and q an n-vector. The problem: Find n-vectors x and y such that
1a
× − My = q
1b
× > 0, y > 0
1c
Either xi = 0 or yi = 0 for 1 < i < n…

Finite test sets and $P$-matrices

- Mathematics
- 1982

The class of matrices with all principal minors positive, known as P-matrices, has been characterized by Murty and Tamir using a finite set of test vectors for the linear complementarity problem.…

A Partial Characterization of a Class of Matrices Defined by Solutions to the Linear Complementarity Problem

- MathematicsMath. Oper. Res.
- 1982

A labelling requirement for the faces is shown necessary for M to be in Q 0, and with proper restrictions and nondegenerate assumptions sufficiency is also shown.

On Covering Smooth Manifolds with Overlapping Simplicies: An inductive Characterization of Q-matrices

- Mathematics
- 2022

This paper is concerned with a covering problem of smooth manifolds of dimension n − 1 by stitching 2 n n -simplices formed with 2 n -lists of points along their common ( n − 1)-facets. The n…

Degeneracy in linear complementarity problems: a survey

- MathematicsAnn. Oper. Res.
- 1993

The literature on the implications of degeneracy to the linear complementarity theory is reviewed, finding that ifLCP(0,M) has a nontrivial solution, a condition related to degeneracy, then unless certain other conditions are satisfied the algorithm may not be able to decide about the solvability of the given LCP(q, M).

Separation Properties, Principal Pivot Transforms, Classes of Matrices Deenition: Subcomplementary Sets of Column Vectors

- Mathematics

In this chapter we present the basic mathematical results on the LCP. Many of these results are used in later chapters to develop algorithms to solve LCPs, and to study the computational complexity…

Polyhedral sets having a least element

- MathematicsMath. Program.
- 1972

It is shown the linear complementarity problem always has a unique solution which is at the same time a least element of the corresponding polyhedron if and only if its matrix is square, Leontief, and has positive diagonals.

Unique sink orientations of cubes

- MathematicsProceedings 2001 IEEE International Conference on Cluster Computing
- 2001

New algorithms are presented, a deterministic O(1.61/sup n/) procedure and a randomized O((43/20)/sup n/2/)=O( 1.47/Sup n%) procedure for unique sink orientations, which believe that unique sink orientation have a rich structure, and there is ample space for improvement on the bounds given above.