768 research outputs found
Disjointly representing set systems
AbstractA family F of sets is s-disjointly representable if there is a family S of disjoint sets each of size s such that every F∈F contains some S∈S. Let f(r,s) be the minimum size of a family F of r-sets which is not s-disjointly representable. We give upper and lower bounds on f(r,s) which are within a constant factor when s is fixed
Homotopy versus isotopy: spheres with duals in 4-manifolds
David Gabai recently proved a smooth 4-dimensional "Light Bulb Theorem" in
the absence of 2-torsion in the fundamental group. We extend his result to
4-manifolds with arbitrary fundamental group by showing that an invariant of
Mike Freedman and Frank Quinn gives the complete obstruction to "homotopy
implies isotopy" for embedded 2-spheres which have a common geometric dual. The
invariant takes values in an Z/2Z-vector space generated by elements of order 2
in the fundamental group and has applications to unknotting numbers and
pseudo-isotopy classes of self-diffeomorphisms. Our methods also give an
alternative approach to Gabai's theorem using various maneuvers with Whitney
disks and a fundamental isotopy between surgeries along dual circles in an
orientable surface.Comment: Included into section 2 of this version is a proof that the operation
of `sliding a Whitney disk over itself' preserves the isotopy class of the
resulting Whitney move in the current setting. Some expository clarifications
have also been added. Main results and proofs are unchanged from the previous
version. 39 pages, 25 figure
The sleep cycle: a mathematical analysis from a global workspace perspective
Dretske's technique of invoking necessary conditions from information theory to describe mental process can be used to derive a version of Hobson's AIM treatment of the sleep/wake cycle from a mathematical formulation of Baars' Global Workspace model of consciousness. One implication of the analysis is that some sleep disorders may be recognizably similar to many other chronic, developmental dysfunctions, including autoimmune and coronary heart disease, obesity, hypertension, and anxiety disorder, in that these afflictions often have roots in utero or adverse early childhood experiences or exposures to systematic patterns of structured stress. Identification and alteration of such factors might have considerable impact on population-level patterns of sleep disorders, suggesting the possibility of a public health approach rather than current exorbitantly expensive case-by-case medical intervention
Improved Cheeger's Inequality: Analysis of Spectral Partitioning Algorithms through Higher Order Spectral Gap
Let \phi(G) be the minimum conductance of an undirected graph G, and let
0=\lambda_1 <= \lambda_2 <=... <= \lambda_n <= 2 be the eigenvalues of the
normalized Laplacian matrix of G. We prove that for any graph G and any k >= 2,
\phi(G) = O(k) \lambda_2 / \sqrt{\lambda_k}, and this performance guarantee
is achieved by the spectral partitioning algorithm. This improves Cheeger's
inequality, and the bound is optimal up to a constant factor for any k. Our
result shows that the spectral partitioning algorithm is a constant factor
approximation algorithm for finding a sparse cut if \lambda_k$ is a constant
for some constant k. This provides some theoretical justification to its
empirical performance in image segmentation and clustering problems. We extend
the analysis to other graph partitioning problems, including multi-way
partition, balanced separator, and maximum cut
Algebraic and Geometric intersection numbers for free groups
We show that the algebraic intersection number of Scott and Swarup for
splittings of free groups coincides with the geometric intersection number for
the sphere complex of the connected sum of copies of .Comment: 7 page
- …