3,767 research outputs found
Line problems in nonlinear computational geometry
We first review some topics in the classical computational geometry of lines,
in particular the O(n^{3+\epsilon}) bounds for the combinatorial complexity of
the set of lines in R^3 interacting with objects of fixed description
complexity. The main part of this survey is recent work on a core algebraic
problem--studying the lines tangent to k spheres that also meet 4-k fixed
lines. We give an example of four disjoint spheres with 12 common real
tangents.Comment: 22 pages, 13 color figure
Characteristic Length and Clustering
We explore relations between various variational problems for graphs like
Euler characteristic chi(G), characteristic length mu(G), mean clustering
nu(G), inductive dimension iota(G), edge density epsilon(G), scale measure
sigma(G), Hilbert action eta(G) and spectral complexity xi(G). A new insight in
this note is that the local cluster coefficient C(x) in a finite simple graph
can be written as a relative characteristic length L(x) of the unit sphere S(x)
within the unit ball B(x) of a vertex. This relation L(x) = 2-C(x) will allow
to study clustering in more general metric spaces like Riemannian manifolds or
fractals. If eta is the average of scalar curvature s(x), a formula mu ~
1+log(epsilon)/log(eta) of Newman, Watts and Strogatz relates mu with the edge
density epsilon and average scalar curvature eta telling that large curvature
correlates with small characteristic length. Experiments show that the
statistical relation mu ~ log(1/nu) holds for random or deterministic
constructed networks, indicating that small clustering is often associated to
large characteristic lengths and lambda=mu/log(nu) can converge in some graph
limits of networks. Mean clustering nu, edge density epsilon and curvature
average eta therefore can relate with characteristic length mu on a statistical
level. We also discovered experimentally that inductive dimension iota and
cluster-length ratio lambda correlate strongly on Erdos-Renyi probability
spaces.Comment: 20 pages 13 figure
Lectures on the three--dimensional non--commutative spheres
These are expanded notes for a short course given at the Universidad Nacional
de La Plata. They aim at giving a self-contained account of the results of
Alain Connes and Michel Dubois--Violette.Comment: 17 page
Link Homotopy in S^n x R^{m-n} and Higher Order mu-invariants
Given a suitable link map f into a manifold M, we constructed, in [10], link
homotopy invariants kappa(f) and mu(f). In the present paper we study the case
M=S^n x R^{m - n} in detail. Here mu(f) turns out to be the starting term of a
whole sequence mu^(s)(f), s = 0, 1, ..., of higher mu-invariants which together
capture all the information contained in kappa(f). We discuss the geometric
significance of these new invariants. In several instances we obtain complete
classification results. A central ingredient of our approach is the homotopy
theory of wedges of spheres
Symmetry in Sphere-based Assembly Configuration Spaces
Many remarkably robust, rapid and spontaneous self-assembly phenomena in
nature can be modeled geometrically starting from a collection of rigid bunches
of spheres. This paper highlights the role of symmetry in sphere-based assembly
processes. Since spheres within bunches could be identical and bunches could be
identical as well, the underlying symmetry groups could be of large order that
grows with the number of participating spheres and bunches. Thus, understanding
symmetries and associated isomorphism classes of microstates correspond to
various types of macrostates can significantly reduce the complexity of
computing entropy and free energy, as well as paths and kinetics, in high
dimensional configuration spaces. In addition, a precise understanding of
symmetries is crucial for giving provable guarantees of algorithmic accuracy
and efficiency in such computations. In particular, this may aid in predicting
crucial assembly-driving interactions.
This is a primarily expository paper that develops a novel, original
framework for dealing with symmetries in configuration spaces of assembling
spheres with the following goals. (1) We give new, formal definitions of
various concepts relevant to sphere-based assembly that occur in previous work,
and in turn, formal definitions of their relevant symmetry groups leading to
the main theorem concerning their symmetries. These previously developed
concepts include, for example, (a) assembly configuration spaces, (b)
stratification of assembly configuration space into regions defined by active
constraint graphs, (c) paths through the configurational regions, and (d)
coarse assembly pathways. (2) We demonstrate the new symmetry concepts to
compute sizes and numbers of orbits in two example settings appearing in
previous work. (3) We give formal statements of a variety of open problems and
challenges using the new conceptual definitions
Volume-minimizing foliations on spheres
The volume of a k-dimensional foliation in a Riemannian
manifold is defined as the mass of image of the Gauss map, which is a
map from M to the Grassmann bundle of k-planes in the tangent bundle.
Generalizing a construction by Gluck and Ziller, "singular" foliations by
3-spheres are constructed on round spheres , as well as a singular
foliation by 7-spheres on , which minimize volume within their
respective relative homology classes. These singular examples provide lower
bounds for volumes of regular 3-dimensional foliations of and
regular 7-dimensional foliations of .Comment: 12 pages, no figure
Topological field theories, string backgrounds and homotopy algebras
String backgrounds are described as purely geometric objects related to
moduli spaces of Riemann surfaces, in the spirit of Segal's definition of a
conformal field theory. Relations with conformal field theory, topological
field theory and topological gravity are studied. For each field theory, an
algebraic counterpart, the (homotopy) algebra satisfied by the tree level
correlators, is constructed.Comment: 12 page
Helly numbers of acyclic families
The Helly number of a family of sets with empty intersection is the size of
its largest inclusion-wise minimal sub-family with empty intersection. Let F be
a finite family of open subsets of an arbitrary locally arc-wise connected
topological space Gamma. Assume that for every sub-family G of F the
intersection of the elements of G has at most r connected components, each of
which is a Q-homology cell. We show that the Helly number of F is at most
r(d_Gamma+1), where d_Gamma is the smallest integer j such that every open set
of Gamma has trivial Q-homology in dimension j and higher. (In particular
d_{R^d} = d). This bound is best possible. We prove, in fact, a stronger
theorem where small sub-families may have more than r connected components,
each possibly with nontrivial homology in low dimension. As an application, we
obtain several explicit bounds on Helly numbers in geometric transversal theory
for which only ad hoc geometric proofs were previously known; in certain cases,
the bound we obtain is better than what was previously known.Comment: Minor change
The many faces of modern combinatorics
This is a survey of recent developments in combinatorics. The goal is to give
a big picture of its many interactions with other areas of mathematics, such
as: group theory, representation theory, commutative algebra, geometry
(including algebraic geometry), topology, probability theory, and theoretical
computer science
Genus expansion for real Wishart matrices
We present an exact formula for moments and cumulants of several real
compound Wishart matrices in terms of an Euler characteristic expansion,
similar to the genus expansion for complex random matrices. We consider their
asymptotic values in the large matrix limit: as in a genus expansion, the terms
which survive in the large matrix limit are those with the greatest Euler
characteristic, that is, either spheres or collections of spheres. This
topological construction motivates an algebraic expression for the moments and
cumulants in terms of the symmetric group. We examine the combinatorial
properties distinguishing the leading order terms. By considering higher
cumulants, we give a central limit-type theorem for the asymptotic distribution
around the expected value
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