Related papers: Counting point configurations in projective space
A rank is a notion in descriptive set theory that describes ranks such as the Cantor-Bendixson rank on the set of closed subsets of a Polish space, differentiability ranks on the set of differentiable functions in $C[0,1]$ such as the…
In this note we exhibit the so-called Harbourne constants which capture and measure the Bounded Negativity on various birational models of an algebraic surface. We show an estimation for Harbourne constants for conic configurations on the…
We study the generalized and extended weight enumerator of the q-ary Simplex code and the q-ary first order Reed-Muller code. For our calculations we use that these codes correspond to a projective system containing all the points in a…
We give a geometric characterisation of plus-one generated projective line arrangements that are next-to-free. We present new succinct proofs, via associated line bundles, for some properties of plus-one generated projective line…
This thesis deals with the enumerative study of combinatorial maps, and its application to the enumeration of other combinatorial objects. Combinatorial maps, or simply maps, form a rich combinatorial model. They have an intuitive and…
A configuration of the triple $(\mathcal{P}, \mathcal{L}, \mathcal{I})$ on the incidence relation which holds the properties of "Any two points are incident with at most one line" and "Any two lines are incident with at most one point". In…
The use of geometric invariants has recently played an important role in the solution of classification problems in non-commutative ring theory. We construct geometric invariants of non-commutative projectivizations, a significant class of…
Compact sets in constructive mathematics capture our intuition of what computable subsets of the plane (or any other complete metric space) ought to be. A good representation of compact sets provides an efficient means of creating and…
We compute the integral cohomology ring of configuration spaces of two points on a given real projective space. Apart from an integral class, the resulting ring is a quotient of the known integral cohomology of the dihedral group of order 8…
We use a well known problem in discrete and computational geometry (partitions of measures by $k$-fans) as a motivation and as a point of departure to illustrate many aspects, both theoretical and computational, of the problem of…
The classification of shapes is of great interest in diverse areas ranging from medical imaging to computer vision and beyond. While many statistical frameworks have been developed for the classification problem, most are strongly tied to…
In semialgebraic geometry, projections play a prominent role. A definable choice is a semialgebraic selection of one point in every fiber of a projection. Definable choices exist by semialgebraic triviality, but their complexity depends…
We re-examine the notion of relative $(p,\eps)$-approximations, recently introduced in [CKMS06], and establish upper bounds on their size, in general range spaces of finite VC-dimension, using the sampling theory developed in [LLS01] and in…
This note is motivated by the Question 16 of http://cubics.wikidot.com: Which configurations of 15 points in the projective 3-space arise as eigenpoints of a cubic surface? We prove that a general eigenscheme in the projective n-space is…
There is no field with only one element, yet there is a well-defined notion of what projective geometry over such a field means. This notion is familiar to experts and plays an interesting role behind the scenes in combinatorics and…
Kim defined a very general combinatorial abstraction of the diameter of polytopes called subset partition graphs to study how certain combinatorial properties of such graphs may be achieved in lower bound constructions. Using Lov\'asz'…
Packing problems in discrete geometry can be modeled as finding independent sets in infinite graphs where one is interested in independent sets which are as large as possible. For finite graphs one popular way to compute upper bounds for…
This expository article describes applications of topological configuration spaces to the control of robotic systems. In particular, we review recent work by the authors on configuration spaces of graphs. These are lovely spaces: we show…
A locally-optimal structure is a combinatorial structure such as a maximal independent set that cannot be improved by certain (greedy) local moves, even though it may not be globally optimal. It is trivial to construct an independent set in…
The concept of graph compositions is related to several number theoretic concepts, including partitions of positive integers and the cardinality of the power set of finite sets. This paper examines graph compositions where the total number…