31,973 research outputs found
Axiomatic Digital Topology
The paper presents a new set of axioms of digital topology, which are easily
understandable for application developers. They define a class of locally
finite (LF) topological spaces. An important property of LF spaces satisfying
the axioms is that the neighborhood relation is antisymmetric and transitive.
Therefore any connected and non-trivial LF space is isomorphic to an abstract
cell complex. The paper demonstrates that in an n-dimensional digital space
only those of the (a, b)-adjacencies commonly used in computer imagery have
analogs among the LF spaces, in which a and b are different and one of the
adjacencies is the "maximal" one, corresponding to 3n\"i1 neighbors. Even these
(a, b)-adjacencies have important limitations and drawbacks. The most important
one is that they are applicable only to binary images. The way of easily using
LF spaces in computer imagery on standard orthogonal grids containing only
pixels or voxels and no cells of lower dimensions is suggested
Enumeration Reducibility in Closure Spaces with Applications to Logic and Algebra
In many instances in first order logic or computable algebra, classical
theorems show that many problems are undecidable for general structures, but
become decidable if some rigidity is imposed on the structure. For example, the
set of theorems in many finitely axiomatisable theories is nonrecursive, but
the set of theorems for any finitely axiomatisable complete theory is
recursive. Finitely presented groups might have an nonrecursive word problem,
but finitely presented simple groups have a recursive word problem. In this
article we introduce a topological framework based on closure spaces to show
that many of these proofs can be obtained in a similar setting. We will show in
particular that these statements can be generalized to cover arbitrary
structures, with no finite or recursive presentation/axiomatization. This
generalizes in particular work by Kuznetsov and others. Examples from first
order logic and symbolic dynamics will be discussed at length
Ramsey-type theorems for lines in 3-space
We prove geometric Ramsey-type statements on collections of lines in 3-space.
These statements give guarantees on the size of a clique or an independent set
in (hyper)graphs induced by incidence relations between lines, points, and
reguli in 3-space. Among other things, we prove that: (1) The intersection
graph of n lines in R^3 has a clique or independent set of size Omega(n^{1/3}).
(2) Every set of n lines in R^3 has a subset of n^{1/2} lines that are all
stabbed by one line, or a subset of Omega((n/log n)^{1/5}) such that no
6-subset is stabbed by one line. (3) Every set of n lines in general position
in R^3 has a subset of Omega(n^{2/3}) lines that all lie on a regulus, or a
subset of Omega(n^{1/3}) lines such that no 4-subset is contained in a regulus.
The proofs of these statements all follow from geometric incidence bounds --
such as the Guth-Katz bound on point-line incidences in R^3 -- combined with
Tur\'an-type results on independent sets in sparse graphs and hypergraphs.
Although similar Ramsey-type statements can be proved using existing generic
algebraic frameworks, the lower bounds we get are much larger than what can be
obtained with these methods. The proofs directly yield polynomial-time
algorithms for finding subsets of the claimed size.Comment: 18 pages including appendi
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