419,667 research outputs found
Grid classes and partial well order
We prove necessary and sufficient conditions on a family of (generalised)
gridding matrices to determine when the corresponding permutation classes are
partially well-ordered. One direction requires an application of Higman's
Theorem and relies on there being only finitely many simple permutations in the
only non-monotone cell of each component of the matrix. The other direction is
proved by a more general result that allows the construction of infinite
antichains in any grid class of a matrix whose graph has a component containing
two or more non-monotone-griddable cells. The construction uses a
generalisation of pin sequences to grid classes, together with a number of
symmetry operations on the rows and columns of a gridding.Comment: 22 pages, 7 figures. To appear in J. Comb. Theory Series
C*-algebras associated to coverings of k-graphs
A covering of k-graphs (in the sense of Pask-Quigg-Raeburn) induces an
embedding of universal C*-algebras. We show how to build a (k+1)-graph whose
universal algebra encodes this embedding. More generally we show how to realise
a direct limit of k-graph algebras under embeddings induced from coverings as
the universal algebra of a (k+1)-graph. Our main focus is on computing the
K-theory of the (k+1)-graph algebra from that of the component k-graph
algebras.
Examples of our construction include a realisation of the Kirchberg algebra
\mathcal{P}_n whose K-theory is opposite to that of \mathcal{O}_n, and a class
of AT-algebras that can naturally be regarded as higher-rank Bunce-Deddens
algebras.Comment: 44 pages, 2 figures, some diagrams drawn using picTeX. v2. A number
of typos corrected, some references updated. The statements of Theorem 6.7(2)
and Corollary 6.8 slightly reworded for clarity. v3. Some references updated;
in particular, theorem numbering of references to Evans updated to match
published versio
Critical random graphs : limiting constructions and distributional properties
We consider the Erdos-Renyi random graph G(n, p) inside the critical window, where p = 1/n + lambda n(-4/3) for some lambda is an element of R. We proved in Addario-Berry et al. [2009+] that considering the connected components of G(n, p) as a sequence of metric spaces with the graph distance rescaled by n(-1/3) and letting n -> infinity yields a non-trivial sequence of limit metric spaces C = (C-1, C-2,...). These limit metric spaces can be constructed from certain random real trees with vertex-identifications. For a single such metric space, we give here two equivalent constructions, both of which are in terms of more standard probabilistic objects. The first is a global construction using Dirichlet random variables and Aldous' Brownian continuum random tree. The second is a recursive construction from an inhomogeneous Poisson point process on R+. These constructions allow us to characterize the distributions of the masses and lengths in the constituent parts of a limit component when it is decomposed according to its cycle structure. In particular, this strengthens results of Luczak et al. [1994] by providing precise distributional convergence for the lengths of paths between kernel vertices and the length of a shortest cycle, within any fixed limit component
On rational homology disk smoothings of valency 4 surface singularities
Thanks to the recent work of Bhupal, Stipsicz, Szabo, and the author, one has
a complete list of resolution graphs of weighted homogeneous complex surface
singularities admitting a rational homology disk ("QHD") smoothing, i.e., one
with Milnor number 0. They fall into several classes, the most interesting of
which are the three classes whose resolution dual graph has central vertex with
valency 4. We give a uniform "quotient construction" of the QHD smoothings for
these classes; it is an explicit Q-Gorenstein smoothing, yielding a precise
description of the Milnor fibre and its non-abelian fundamental group. This had
already been done for two of these classes in a previous paper; what is new
here is the construction of the third class, which is far more difficult. In
addition, we explain the existence of two different QHD smoothings for the
first class.
We also prove a general formula for the dimension of a QHD smoothing
component for a rational surface singularity. A corollary is that for the
valency 4 cases, such a component has dimension 1 and is smooth. Another
corollary is that "most" H-shaped resolution graphs cannot be the graph of a
singularity with a QHD smoothing. This result, plus recent work of
Bhupal-Stipsicz, is evidence for a general
Conjecture: The only complex surface singularities with a QHD smoothing are
the (known) weighted homogeneous examples.Comment: 28 pages: title changed, typos fixed, references and small
clarifications adde
Large components in random induced subgraphs of n-cubes
In this paper we study random induced subgraphs of the binary -cube,
. This random graph is obtained by selecting each -vertex with
independent probability . Using a novel construction of
subcomponents we study the largest component for
, where , . We prove that there exists a.s. a unique largest
component . We furthermore show that , and for , holds.
This improves the result of \cite{Bollobas:91} where constant is
considered. In particular, in case of , our
analysis implies that a.s. a unique giant component exists.Comment: 18 Page
Asymmetric evolving random networks
We generalize the poissonian evolving random graph model of Bauer and Bernard
to deal with arbitrary degree distributions. The motivation comes from
biological networks, which are well-known to exhibit non poissonian degree
distribution. A node is added at each time step and is connected to the rest of
the graph by oriented edges emerging from older nodes. This leads to a
statistical asymmetry between incoming and outgoing edges. The law for the
number of new edges at each time step is fixed but arbitrary. Thermodynamical
behavior is expected when this law has a large time limit. Although (by
construction) the incoming degree distributions depend on this law, this is not
the case for most qualitative features concerning the size distribution of
connected components, as long as the law has a finite variance. As the variance
grows above 1/4, the average being <1/2, a giant component emerges, which
connects a finite fraction of the vertices. Below this threshold, the
distribution of component sizes decreases algebraically with a continuously
varying exponent. The transition is of infinite order, in sharp contrast with
the case of static graphs. The local-in-time profiles for the components of
finite size allow to give a refined description of the system.Comment: 30 pages, 3 figure
Nonlinear hyperbolic systems: Non-degenerate flux, inner speed variation, and graph solutions
We study the Cauchy problem for general, nonlinear, strictly hyperbolic
systems of partial differential equations in one space variable. First, we
re-visit the construction of the solution to the Riemann problem and introduce
the notion of a nondegenerate (ND) system. This is the optimal condition
guaranteeing, as we show it, that the Riemann problem can be solved with
finitely many waves, only; we establish that the ND condition is generic in the
sense of Baire (for the Whitney topology), so that any system can be approached
by a ND system. Second, we introduce the concept of inner speed variation and
we derive new interaction estimates on wave speeds. Third, we design a wave
front tracking scheme and establish its strong convergence to the entropy
solution of the Cauchy problem; this provides a new existence proof as well as
an approximation algorithm. As an application, we investigate the
time-regularity of the graph solutions introduced by the second author,
and propose a geometric version of our scheme; in turn, the spatial component
of a graph solution can be chosen to be continuous in both time and space,
while its component is continuous in space and has bounded variation in
time.Comment: 74 page
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