Colouring Lines in Projective Space

Let $V$ be a vector space of dimension $v$ over a field of order $q$. The $q$-Kneser graph has the $k$-dimensional subspaces of $V$ as its vertices, where two subspaces $\alpha$ and $\beta$ are adjacent if and only if $\alpha\cap\beta$ is the zero subspace. This paper is motivated by the problem of determining the chromatic numbers of these graphs. This problem is trivial when $k=1$ (and the graphs are complete) or when $v<2k$ (and the graphs are empty). We establish some basic theory in the general case. Then specializing to the case $k=2$, we show that the chromatic number is $q^2+q$ when $v=4$ and $(q^{v-1}-1)/(q-1)$ when $v > 4$. In both cases we characterise the minimal colourings.Comment: 19 pages; to appear in J. Combinatorial Theory, Series

Colouring quadrangulations of projective spaces

A graph embedded in a surface with all faces of size 4 is known as a quadrangulation. We extend the definition of quadrangulation to higher dimensions, and prove that any graph G which embeds as a quadrangulation in the real projective space P^n has chromatic number n+2 or higher, unless G is bipartite. For n=2 this was proved by Youngs [J. Graph Theory 21 (1996), 219-227]. The family of quadrangulations of projective spaces includes all complete graphs, all Mycielski graphs, and certain graphs homomorphic to Schrijver graphs. As a corollary, we obtain a new proof of the Lovasz-Kneser theorem

Noncontextuality, Finite Precision Measurement and the Kochen-Specker Theorem

Meyer recently queried whether non-contextual hidden variable models can, despite the Kochen-Specker theorem, simulate the predictions of quantum mechanics to within any fixed finite experimental precision. Clifton and Kent have presented constructions of non-contextual hidden variable theories which, they argued, indeed simulate quantum mechanics in this way. These arguments have evoked some controversy. One aim of this paper is to respond to and rebut criticisms of the MCK papers. We thus elaborate in a little more detail how the CK models can reproduce the predictions of quantum mechanics to arbitrary precision. We analyse in more detail the relationship between classicality, finite precision measurement and contextuality, and defend the claims that the CK models are both essentially classical and non-contextual. We also examine in more detail the senses in which a theory can be said to be contextual or non-contextual, and in which an experiment can be said to provide evidence on the point. In particular, we criticise the suggestion that a decisive experimental verification of contextuality is possible, arguing that the idea rests on a conceptual confusion.Comment: 27 pages; published version; minor changes from previous versio

A computational model of the referential semantics of projective prepositions

In this paper we present a framework for interpreting locative expressions containing the prepositions in front of and behind. These prepositions have different semantics in the viewer-centred and intrinsic frames of reference (Vandeloise, 1991). We define a model of their semantics in each frame of reference. The basis of these models is a novel parameterized continuum function that creates a 3-D spatial template. In the intrinsic frame of reference the origin used by the continuum function is assumed to be known a priori and object occlusion does not impact on the applicability rating of a point in the spatial template. In the viewer-centred frame the location of the spatial template’s origin is dependent on the user’s perception of the landmark at the time of the utterance and object occlusion is integrated into the model. Where there is an ambiguity with respect to the intended frame of reference, we define an algorithm for merging the spatial templates from the competing frames of reference, based on psycholinguistic observations in (Carlson-Radvansky, 1997)

A Method of Areas for Manipulating the Entanglement Properties of One Copy of a Two-Particle Pure State

We consider the problem of how to manipulate the entanglement properties of a general two-particle pure state, shared between Alice and Bob, by using only local operations at each end and classical communication between Alice and Bob. A method is developed in which this type of problem is found to be equivalent to a problem involving the cutting and pasting of certain shapes along with a certain colouring problem. We consider two problems. Firstly we find the most general way of manipulating the state to obtain maximally entangled states. After such a manipulation the entangled state |11>+|22>+....|mm> is obtained with probability p_m. We obtain an expression for the optimal average entanglement. Also, some results of Lo and Popescu pertaining to this problem are given simple geometric proofs. Secondly, we consider how to manipulate one two particle entangled pure state to another with certainty. We derive Nielsen's theorem (which states the necessary and sufficient condition for this to be possible) using the method of areas.Comment: 29 pages, 9 figures. Section 2.4 clarified. Error in second colouring theorem (section 3.2) corrected. Some other minor change