78 research outputs found
Schrijver graphs and projective quadrangulations
In a recent paper [J. Combin. Theory Ser. B}, 113 (2015), pp. 1-17], the
authors have extended the concept of quadrangulation of a surface to higher
dimension, and showed that every quadrangulation of the -dimensional
projective space is at least -chromatic, unless it is bipartite.
They conjectured that for any integers and , the
Schrijver graph contains a spanning subgraph which is a
quadrangulation of . The purpose of this paper is to prove the
conjecture
Irregular graph pyramids and representative cocycles of cohomology generators
Structural pattern recognition describes and classifies data based on the relationships of features and parts. Topological invariants, like the Euler number, characterize the structure of objects of any dimension. Cohomology can provide more refined algebraic invariants to a topological space than does homology. It assigns âquantitiesâ to the chains used in homology to characterize holes of any dimension. Graph pyramids can be used to describe subdivisions of the same object at multiple levels of detail. This paper presents cohomology in the context of structural pattern recognition and introduces an algorithm to efficiently compute representative cocycles (the basic elements of cohomology) in 2D using a graph pyramid. Extension to nD and application in the context of pattern recognition are discussed
Combinatorial Alexander Duality -- a Short and Elementary Proof
Let X be a simplicial complex with the ground set V. Define its Alexander
dual as a simplicial complex X* = {A \subset V: V \setminus A \notin X}. The
combinatorial Alexander duality states that the i-th reduced homology group of
X is isomorphic to the (|V|-i-3)-th reduced cohomology group of X* (over a
given commutative ring R). We give a self-contained proof.Comment: 7 pages, 2 figure; v3: the sign function was simplifie
Tight local approximation results for max-min linear programs
In a bipartite max-min LP, we are given a bipartite graph \myG = (V \cup I
\cup K, E), where each agent is adjacent to exactly one constraint
and exactly one objective . Each agent controls a
variable . For each we have a nonnegative linear constraint on
the variables of adjacent agents. For each we have a nonnegative
linear objective function of the variables of adjacent agents. The task is to
maximise the minimum of the objective functions. We study local algorithms
where each agent must choose based on input within its
constant-radius neighbourhood in \myG. We show that for every
there exists a local algorithm achieving the approximation ratio . We also show that this result is the best possible
-- no local algorithm can achieve the approximation ratio . Here is the maximum degree of a vertex , and
is the maximum degree of a vertex . As a methodological
contribution, we introduce the technique of graph unfolding for the design of
local approximation algorithms.Comment: 16 page
Towards digital cohomology
We propose a method for computing the Z 2âcohomology ring of a simplicial complex uniquely associated with a threeâdimensional digital binaryâvalued picture I. Binary digital pictures are represented on the standard grid Z 3, in which all grid points have integer coordinates. Considering a particular 14âneighbourhood system on this grid, we construct a unique simplicial complex K(I) topologically representing (up to isomorphisms of pictures) the picture I. We then compute the cohomology ring on I via the simplicial complex K(I). The usefulness of a simplicial description of the digital Z 2âcohomology ring of binary digital pictures is tested by means of a small program visualizing the different steps of our method. Some examples concerning topological thinning, the visualization of representative generators of cohomology classes and the computation of the cup product on the cohomology of simple 3D digital pictures are showed
Connectivity forests for homological analysis of digital volumes
In this paper, we provide a graph-based representation of the homology (information related to the different âholesâ the object has) of a binary digital volume. We analyze the digital volume AT-model representation [8] from this point of view and the cellular version of the AT-model [5] is precisely described here as three forests (connectivity forests), from which, for instance, we can straightforwardly determine representative curves of âtunnelsâ and âholesâ, classify cycles in the complex, computing higher (co)homology operations,... Depending of the order in which we gradually construct these trees, tools so important in Computer Vision and Digital Image Processing as Reeb graphs and topological skeletons appear as results of pruning these graphs
Fermionic Casimir effect with helix boundary condition
In this paper, we consider the fermionic Casimir effect under a new type of
space-time topology using the concept of quotient topology. The relation
between the new topology and that in Ref. \cite{Feng,Zhai3} is something like
that between a M\"obius strip and a cylindric. We obtain the exact results of
the Casimir energy and force for the massless and massive Dirac fields in the
()-dimensional space-time. For both massless and massive cases, there is a
symmetry for the Casimir energy. To see the effect of the mass, we
compare the result with that of the massless one and we found that the Casimir
force approaches the result of the force in the massless case when the mass
tends to zero and vanishes when the mass tends to infinity.Comment: 7 pages, 4 figures, published in Eur. Phys. J.
Determination of set-membership identifiability sets
International audienceThis paper concerns the concept of set-membership identifiability introduced in \cite{jauberthie}. Given a model, a set-membership identifiable set is a connected set in the parameter domain of the model such that its corresponding trajectories are distinct to trajectories arising from its complementary. For obtaining the so-called set-membership identifiable sets, we propose an algorithm based on interval analysis tools. The proposed algorithm is decomposed into three parts namely {\it mincing}, {\it evaluating} and {\it regularization} (\cite{jaulin2}). The latter step has been modified in order to obtain guaranteed set-membership identifiable sets. Our algorithm will be tested on two examples
Algebraic topological analysis of time-sequence of digital images
This paper introduces an algebraic framework for a topological analysis of time-varying 2D digital binaryâvalued images, each of them defined as 2D arrays of pixels. Our answer is based on an algebraic-topological coding, called ATâmodel, for a nD (n=2,3) digital binary-valued image I consisting simply in taking I together with an algebraic object depending on it. Considering ATâmodels for all the 2D digital images in a time sequence, it is possible to get an ATâmodel for the 3D digital image consisting in concatenating the successive 2D digital images in the sequence. If the frames are represented in a quadtree format, a similar positive result can be derived
Cup products on polyhedral approximations of 3D digital images
Let I be a 3D digital image, and let Q(I) be the associated cubical complex. In this paper we show how to simplify the combinatorial structure of Q(I) and obtain a homeomorphic cellular complex P(I) with fewer cells. We introduce formulas for a diagonal approximation on a general polygon and use it to compute cup products on the cohomology H *(P(I)). The cup product encodes important geometrical information not captured by the cohomology groups. Consequently, the ring structure of H *(P(I)) is a finer topological invariant. The algorithm proposed here can be applied to compute cup products on any polyhedral approximation of an object embedded in 3-space
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