17,633 research outputs found
Renormalization of noncommutative phi 4-theory by multi-scale analysis
In this paper we give a much more efficient proof that the real Euclidean phi
4-model on the four-dimensional Moyal plane is renormalizable to all orders. We
prove rigorous bounds on the propagator which complete the previous
renormalization proof based on renormalization group equations for non-local
matrix models. On the other hand, our bounds permit a powerful multi-scale
analysis of the resulting ribbon graphs. Here, the dual graphs play a
particular r\^ole because the angular momentum conservation is conveniently
represented in the dual picture. Choosing a spanning tree in the dual graph
according to the scale attribution, we prove that the summation over the loop
angular momenta can be performed at no cost so that the power-counting is
reduced to the balance of the number of propagators versus the number of
completely inner vertices in subgraphs of the dual graph.Comment: 34 page
Polyhedral characteristics of balanced and unbalanced bipartite subgraph problems
We study the polyhedral properties of three problems of constructing an
optimal complete bipartite subgraph (a biclique) in a bipartite graph. In the
first problem we consider a balanced biclique with the same number of vertices
in both parts and arbitrary edge weights. In the other two problems we are
dealing with unbalanced subgraphs of maximum and minimum weight with
nonnegative edges. All three problems are established to be NP-hard. We study
the polytopes and the cone decompositions of these problems and their
1-skeletons. We describe the adjacency criterion in 1-skeleton of the polytope
of the balanced complete bipartite subgraph problem. The clique number of
1-skeleton is estimated from below by a superpolynomial function. For both
unbalanced biclique problems we establish the superpolynomial lower bounds on
the clique numbers of the graphs of nonnegative cone decompositions. These
values characterize the time complexity in a broad class of algorithms based on
linear comparisons
Generalized Paley graphs and their complete subgraphs of orders three and four
Let be an integer. Let be a prime power such that if is even, or, if is odd. The
generalized Paley graph of order , , is the graph with vertex set
where is an edge if and only if is a -th power
residue. We provide a formula, in terms of finite field hypergeometric
functions, for the number of complete subgraphs of order four contained in
, , which holds for all . This generalizes
the results of Evans, Pulham and Sheehan on the original (=2) Paley graph.
We also provide a formula, in terms of Jacobi sums, for the number of complete
subgraphs of order three contained in , . In
both cases we give explicit determinations of these formulae for small . We
show that zero values of (resp.
) yield lower bounds for the multicolor diagonal Ramsey
numbers (resp. ). We state explicitly these
lower bounds for small and compare to known bounds. We also examine the
relationship between both and ,
when is prime, and Fourier coefficients of modular forms
A necessary and sufficient condition for lower bounds on crossing numbers of generalized periodic graphs in an arbitrary surface
Let , and be a graph, a tree and a cycle of order ,
respectively. Let be the complete join of and an empty graph on
vertices. Then the Cartesian product of and can be
obtained by applying zip product on and the graph produced by zip
product repeatedly. Let denote the crossing number of
in an arbitrary surface . If satisfies certain connectivity
condition, then is not less than the sum of the
crossing numbers of its ``subgraphs". In this paper, we introduced a new
concept of generalized periodic graphs, which contains . For a
generalized periodic graph and a function , where is the number
of subgraphs in a decomposition of , we gave a necessary and sufficient
condition for . As an application, we
confirmed a conjecture of Lin et al. on the crossing number of the generalized
Petersen graph in the plane. Based on the condition, algorithms
are constructed to compute lower bounds on the crossing number of generalized
periodic graphs in . In special cases, it is possible to determine
lower bounds on an infinite family of generalized periodic graphs, by
determining a lower bound on the crossing number of a finite generalized
periodic graph.Comment: 26 pages, 20 figure
The split-and-drift random graph, a null model for speciation
We introduce a new random graph model motivated by biological questions
relating to speciation. This random graph is defined as the stationary
distribution of a Markov chain on the space of graphs on .
The dynamics of this Markov chain is governed by two types of events: vertex
duplication, where at constant rate a pair of vertices is sampled uniformly and
one of these vertices loses its incident edges and is rewired to the other
vertex and its neighbors; and edge removal, where each edge disappears at
constant rate. Besides the number of vertices , the model has a single
parameter .
Using a coalescent approach, we obtain explicit formulas for the first
moments of several graph invariants such as the number of edges or the number
of complete subgraphs of order . These are then used to identify five
non-trivial regimes depending on the asymptotics of the parameter . We
derive an explicit expression for the degree distribution, and show that under
appropriate rescaling it converges to classical distributions when the number
of vertices goes to infinity. Finally, we give asymptotic bounds for the number
of connected components, and show that in the sparse regime the number of edges
is Poissonian.Comment: added Proposition 2.4 and formal proofs of Proposition 2.3 and 2.
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