78,074 research outputs found
Countable locally 2-arc-transitive bipartite graphs
We present an order-theoretic approach to the study of countably infinite
locally 2-arc-transitive bipartite graphs. Our approach is motivated by
techniques developed by Warren and others during the study of cycle-free
partial orders. We give several new families of previously unknown countably
infinite locally-2-arc-transitive graphs, each family containing continuum many
members. These examples are obtained by gluing together copies of incidence
graphs of semilinear spaces, satisfying a certain symmetry property, in a
tree-like way. In one case we show how the classification problem for that
family relates to the problem of determining a certain family of highly
arc-transitive digraphs. Numerous illustrative examples are given.Comment: 29 page
Maximal partial line spreads of non-singular quadrics
For n >= 9 , we construct maximal partial line spreads for non-singular quadrics of for every size between approximately and , for some small constants and . These results are similar to spectrum results on maximal partial line spreads in finite projective spaces by Heden, and by Gacs and SzAnyi. These results also extend spectrum results on maximal partial line spreads in the finite generalized quadrangles and by Pepe, Roing and Storme
On better-quasi-ordering classes of partial orders
We provide a method of constructing better-quasi-orders by generalising a
technique for constructing operator algebras that was developed by Pouzet. We
then generalise the notion of -scattered to partial orders, and use our
method to prove that the class of -scattered partial orders is
better-quasi-ordered under embeddability. This generalises theorems of Laver,
Corominas and Thomass\'{e} regarding -scattered linear orders and
trees, countable forests and N-free partial orders respectively. In particular,
a class of countable partial orders is better-quasi-ordered whenever the class
of indecomposable subsets of its members satisfies a natural strengthening of
better-quasi-order.Comment: v1: 45 pages, 8 figures; v2: 44 pages, 11 figures, minor corrections,
fixed typos, new figures and some notational changes to improve clarity; v3:
45 pages, 12 figures, changed the way the paper is structured to improve
clarity and provide examples earlier o
Bounded Representations of Interval and Proper Interval Graphs
Klavik et al. [arXiv:1207.6960] recently introduced a generalization of
recognition called the bounded representation problem which we study for the
classes of interval and proper interval graphs. The input gives a graph G and
in addition for each vertex v two intervals L_v and R_v called bounds. We ask
whether there exists a bounded representation in which each interval I_v has
its left endpoint in L_v and its right endpoint in R_v. We show that the
problem can be solved in linear time for interval graphs and in quadratic time
for proper interval graphs.
Robert's Theorem states that the classes of proper interval graphs and unit
interval graphs are equal. Surprisingly the bounded representation problem is
polynomially solvable for proper interval graphs and NP-complete for unit
interval graphs [Klav\'{\i}k et al., arxiv:1207.6960]. So unless P = NP, the
proper and unit interval representations behave very differently.
The bounded representation problem belongs to a wider class of restricted
representation problems. These problems are generalizations of the
well-understood recognition problem, and they ask whether there exists a
representation of G satisfying some additional constraints. The bounded
representation problems generalize many of these problems
Minimal Obstructions for Partial Representations of Interval Graphs
Interval graphs are intersection graphs of closed intervals. A generalization
of recognition called partial representation extension was introduced recently.
The input gives an interval graph with a partial representation specifying some
pre-drawn intervals. We ask whether the remaining intervals can be added to
create an extending representation. Two linear-time algorithms are known for
solving this problem.
In this paper, we characterize the minimal obstructions which make partial
representations non-extendible. This generalizes Lekkerkerker and Boland's
characterization of the minimal forbidden induced subgraphs of interval graphs.
Each minimal obstruction consists of a forbidden induced subgraph together with
at most four pre-drawn intervals. A Helly-type result follows: A partial
representation is extendible if and only if every quadruple of pre-drawn
intervals is extendible by itself. Our characterization leads to a linear-time
certifying algorithm for partial representation extension
The pointwise convergence of Fourier Series (I). On a conjecture of Konyagin
We provide a near-complete classification of the Lorentz spaces
for which the sequence of
partial Fourier sums is almost everywhere convergent along lacunary
subsequences. Moreover, under mild assumptions on the fundamental function
, we identify as
the \emph{largest} Lorentz space on which the lacunary Carleson operator is
bounded as a map to . In particular, we disprove a conjecture
stated by Konyagin in his 2006 ICM address. Our proof relies on a newly
introduced concept of a "Cantor Multi-tower Embedding," a special geometric
configuration of tiles that can arise within the time-frequency tile
decomposition of the Carleson operator. This geometric structure plays an
important role in the behavior of Fourier series near , being responsible
for the unboundedness of the weak- norm of a "grand maximal counting
function" associated with the mass levels.Comment: 82 pages, no figures. We have added the following items: 1) Section 5
presenting a suggestive example; 2) Section 6 explaining the fundamental role
of the so called grand maximal counting function; 3) Section 12 presenting a
careful analysis of the Lacey-Thiele discretized Carleson model and of the
Walsh-Carleson operator. Accepted for publication in J. Eur. Math. Soc.
(JEMS
On a Conjecture of EM Stein on the Hilbert Transform on Vector Fields
Let be a smooth vector field on the plane, that is a map from the plane
to the unit circle. We study sufficient conditions for the boundedness of the
Hilbert transform
\operatorname H_{v, \epsilon}f(x) := \text{p.v.}\int_{-\epsilon}^ \epsilon
f(x-yv(x)) \frac{dy}y where is a suitably chosen parameter,
determined by the smoothness properties of the vector field. It is a
conjecture, due to E.\thinspace M.\thinspace Stein, that if is Lipschitz,
there is a positive for which the transform above is bounded on . Our principal result gives a sufficient condition in terms of the
boundedness of a maximal function associated to . This sufficient condition
is that this new maximal function be bounded on some , for some . We show that the maximal function is bounded from to weak for all Lipschitz maximal function. The relationship between our results
and other known sufficient conditions is explored.Comment: 92 pages, 20+ figures. Final version of the paper. To appear in
Memoirs AM
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