1,102,213 research outputs found
Sharp bound on the number of maximal sum-free subsets of integers
Cameron and Erd\H{o}s asked whether the number of \emph{maximal} sum-free
sets in is much smaller than the number of sum-free sets. In
the same paper they gave a lower bound of for the
number of maximal sum-free sets. Here, we prove the following: For each , there is a constant such that, given any ,
contains maximal sum-free sets. Our
proof makes use of container and removal lemmas of Green, a structural result
of Deshouillers, Freiman, S\'os and Temkin and a recent bound on the number of
subsets of integers with small sumset by Green and Morris. We also discuss
related results and open problems on the number of maximal sum-free subsets of
abelian groups.Comment: 25 pages, to appear in the Journal of the European Mathematical
Societ
Cohomological aspects of Abelian gauge theory
We discuss some aspects of cohomological properties of a two-dimensional free
Abelian gauge theory in the framework of BRST formalism. We derive the
conserved and nilpotent BRST- and co-BRST charges and express the Hodge
decomposition theorem in terms of these charges and a conserved bosonic charge
corresponding to the Laplacian operator. It is because of the topological
nature of free U(1) gauge theory that the Laplacian operator goes to zero when
equations of motion are exploited. We derive two sets of topological invariants
which are related to each-other by a certain kind of duality transformation and
express the Lagrangian density of this theory as the sum of terms that are
BRST- and co-BRST invariants. Mathematically, this theory captures together
some of the key features of Witten- and Schwarz type of topological field
theories.Comment: 12 pages, LaTeX, no figures, Title and text have been slightly
changed, Journal reference is given and a reference has been adde
Sidon Sets in Groups and Induced Subgraphs of Cayley Graphs
Let S be a subset of a group G. We call S a Sidon subset of the first (second) kind, if for any x, y, z, w ∈ S of which at least 3 are different, xy ≠ zw (xy-1 ≠ zw-1, resp.). (For abelian groups, the two notions coincide.) If G has a Sidon subset of the second kind with n elements then every n-vertex graph is an induced subgraph of some Cayley graph of G. We prove that a sufficient condition for G to have a Sidon subset of order n (of either kind) is that (❘G❘ ⩾ cn3. For elementary Abelian groups of square order, ❘G❘ ⩾ n2 is sufficient. We prove that most graphs on n vertices are not induced subgraphs of any vertex transitive graph with <cn2/log2n vertices. We comment on embedding trees and, in particular, stars, as induced subgraphs of Cayley graphs, and on the related problem of product-free (sum-free) sets in groups. We summarize the known results on the cardinality of Sidon sets of infinite groups, and formulate a number of open problems.We warn the reader that the sets considered in this paper are different from the Sidon sets Fourier analysts investigate
On the sum of the Voronoi polytope of a lattice with a zonotope
A parallelotope is a polytope that admits a facet-to-facet tiling of
space by translation copies of along a lattice. The Voronoi cell
of a lattice is an example of a parallelotope. A parallelotope can be
uniquely decomposed as the Minkowski sum of a zone closed parallelotope and
a zonotope , where is the set of vectors used to generate the
zonotope. In this paper we consider the related question: When is the Minkowski
sum of a general parallelotope and a zonotope a parallelotope? We give
two necessary conditions and show that the vectors have to be free. Given a
set of free vectors, we give several methods for checking if is
a parallelotope. Using this we classify such zonotopes for some highly
symmetric lattices.
In the case of the root lattice , it is possible to give a more
geometric description of the admissible sets of vectors . We found that the
set of admissible vectors, called free vectors, is described by the well-known
configuration of lines in a cubic. Based on a detailed study of the
geometry of , we give a simple characterization of the
configurations of vectors such that is a
parallelotope. The enumeration yields maximal families of vectors, which
are presented by their description as regular matroids.Comment: 30 pages, 4 figures, 4 table
Universal terms for the entanglement entropy in 2+1 dimensions
We show that the entanglement entropy and alpha entropies corresponding to
spatial polygonal sets in dimensions contain a term which scales
logarithmically with the cutoff. Its coefficient is a universal quantity
consisting in a sum of contributions from the individual vertices. For a free
scalar field this contribution is given by the trace anomaly in a three
dimensional space with conical singularities located on the boundary of a plane
angular sector. We find its analytic expression as a function of the angle.
This is given in terms of the solution of a set of non linear ordinary
differential equations. For general free fields, we also find the small-angle
limit of the logarithmic coefficient, which is related to the two dimensional
entropic c-functions. The calculation involves a reduction to a two dimensional
problem, and as a byproduct, we obtain the trace of the Green function for a
massive scalar field in a sphere where boundary conditions are specified on a
segment of a great circle. This also gives the exact expression for the
entropies for a scalar field in a two dimensional de Sitter space.Comment: 15 pages, 3 figures, extended version with full calculations, added
reference
On the orbit closure containment problem and slice rank of tensors
We consider the orbit closure containment problem, which, for a given vector and a group orbit, asks if the vector is contained in the closure of the group orbit. Recently, many algorithmic problems related to orbit closures have proved to be quite useful in giving polynomial time algorithms for special cases of the polynomial identity testing problem and several non-convex optimization problems. Answering a question posed by Wigderson, we show that the algorithmic problem corresponding to the orbit closure containment problem is NP-hard. We show this by establishing a computational equivalence between the solvability of homogeneous quadratic equations and a homogeneous version of the matrix completion problem, while showing that the latter is an instance of the orbit closure containment problem. Secondly, we consider the notion of slice rank of tensors, which was recently introduced by Tao, and has subsequently been used for breakthroughs in several combinatorial problems like capsets, sunflower free sets, tri-colored sum-free sets, and progression-free sets. We show that the corresponding algorithmic problem, which can also be phrased as a problem about union of orbit closures, is also NP-hard, hence answering an open question by Bürgisser, Garg, Oliveira, Walter, and Wigderson. We show this by using a connection between the slice rank and the size of a minimum vertex cover of a hypergraph revealed by Tao and Sawin
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