265 research outputs found
Topological Designs
We give an exponential upper and a quadratic lower bound on the number of
pairwise non-isotopic simple closed curves can be placed on a closed surface of
genus g such that any two of the curves intersects at most once. Although the
gap is large, both bounds are the best known for large genus. In genus one and
two, we solve the problem exactly.
Our methods generalize to variants in which the allowed number of pairwise
intersections is odd, even, or bounded, and to surfaces with boundary
components.Comment: 14 p., 4 Figures. To appear in Geometriae Dedicat
On some applications of graph theory, I.
In a series of papers, of which the present one is Part I, it is shown that solutions to a variety of problems in distance geometry, potential theory and theory of metric spaces are provided by appropriate applications of graph theoretic results. © 1972
Close to Uniform Prime Number Generation With Fewer Random Bits
In this paper, we analyze several variants of a simple method for generating
prime numbers with fewer random bits. To generate a prime less than ,
the basic idea is to fix a constant , pick a
uniformly random coprime to , and choose of the form ,
where only is updated if the primality test fails. We prove that variants
of this approach provide prime generation algorithms requiring few random bits
and whose output distribution is close to uniform, under less and less
expensive assumptions: first a relatively strong conjecture by H.L. Montgomery,
made precise by Friedlander and Granville; then the Extended Riemann
Hypothesis; and finally fully unconditionally using the
Barban-Davenport-Halberstam theorem. We argue that this approach has a number
of desirable properties compared to previous algorithms.Comment: Full version of ICALP 2014 paper. Alternate version of IACR ePrint
Report 2011/48
The early evolution of the H-free process
The H-free process, for some fixed graph H, is the random graph process
defined by starting with an empty graph on n vertices and then adding edges one
at a time, chosen uniformly at random subject to the constraint that no H
subgraph is formed. Let G be the random maximal H-free graph obtained at the
end of the process. When H is strictly 2-balanced, we show that for some c>0,
with high probability as , the minimum degree in G is at least
. This gives new lower bounds for
the Tur\'an numbers of certain bipartite graphs, such as the complete bipartite
graphs with . When H is a complete graph with we show that for some C>0, with high probability the independence number of
G is at most . This gives new lower bounds
for Ramsey numbers R(s,t) for fixed and t large. We also obtain new
bounds for the independence number of G for other graphs H, including the case
when H is a cycle. Our proofs use the differential equations method for random
graph processes to analyse the evolution of the process, and give further
information about the structure of the graphs obtained, including asymptotic
formulae for a broad class of subgraph extension variables.Comment: 36 page
Short proofs of some extremal results III
We prove a selection of results from different areas of extremal combinatorics, including complete or partial solutions to a number of open problems. These results, coming mainly from extremal graph theory and Ramsey theory, have been collected together because in each case the relevant proofs are reasonably short
How Many Subpopulations is Too Many? Exponential Lower Bounds for Inferring Population Histories
Reconstruction of population histories is a central problem in population
genetics. Existing coalescent-based methods, like the seminal work of Li and
Durbin (Nature, 2011), attempt to solve this problem using sequence data but
have no rigorous guarantees. Determining the amount of data needed to correctly
reconstruct population histories is a major challenge. Using a variety of tools
from information theory, the theory of extremal polynomials, and approximation
theory, we prove new sharp information-theoretic lower bounds on the problem of
reconstructing population structure -- the history of multiple subpopulations
that merge, split and change sizes over time. Our lower bounds are exponential
in the number of subpopulations, even when reconstructing recent histories. We
demonstrate the sharpness of our lower bounds by providing algorithms for
distinguishing and learning population histories with matching dependence on
the number of subpopulations. Along the way and of independent interest, we
essentially determine the optimal number of samples needed to learn an
exponential mixture distribution information-theoretically, proving the upper
bound by analyzing natural (and efficient) algorithms for this problem.Comment: 38 pages, Appeared in RECOMB 201
The history of degenerate (bipartite) extremal graph problems
This paper is a survey on Extremal Graph Theory, primarily focusing on the
case when one of the excluded graphs is bipartite. On one hand we give an
introduction to this field and also describe many important results, methods,
problems, and constructions.Comment: 97 pages, 11 figures, many problems. This is the preliminary version
of our survey presented in Erdos 100. In this version 2 only a citation was
complete
Global aspects of the space of 6D N = 1 supergravities
We perform a global analysis of the space of consistent 6D quantum gravity
theories with N = 1 supersymmetry, including models with multiple tensor
multiplets. We prove that for theories with fewer than T = 9 tensor multiplets,
a finite number of distinct gauge groups and matter content are possible. We
find infinite families of field combinations satisfying anomaly cancellation
and admitting physical gauge kinetic terms for T > 8. We find an integral
lattice associated with each apparently-consistent supergravity theory; this
lattice is determined by the form of the anomaly polynomial. For models which
can be realized in F-theory, this anomaly lattice is related to the
intersection form on the base of the F-theory elliptic fibration. The condition
that a supergravity model have an F-theory realization imposes constraints
which can be expressed in terms of this lattice. The analysis of models which
satisfy known low-energy consistency conditions and yet violate F-theory
constraints suggests possible novel constraints on low-energy supergravity
theories.Comment: 41 pages, 1 figur
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