577 research outputs found
Supersaturation for hereditary properties
Let be a collection of -uniform hypergraphs, and let . It is known that there exists such that the
probability of a random -graph in not containing an induced
subgraph from is . Let each graph in
have at least vertices. We show that in fact for every
, there exists
such that the probability of a random -graph in containing less
than induced subgraphs each lying in is at most
.
This statement is an analogue for hereditary properties of the
supersaturation theorem of Erd\H{o}s and Simonovits. In our applications we
answer a question of Bollob\'as and Nikiforov.Comment: 5 pages, submitted to European Journal of Combinatoric
Strictly monotonic multidimensional sequences and stable sets in pillage games
Let have size . We show that
there are distinct points such that for
each , the coordinate sequence is strictly
increasing, strictly decreasing, or constant, and that this bound on is
best possible. This is analogous to the \erdos-Szekeres theorem on monotonic
sequences in . We apply these results to bound the size of a stable set
in a pillage game. We also prove a theorem of independent combinatorial
interest. Suppose is a set of points in
such that the set of pairs of points not sharing a coordinate is
precisely . We show that , and
that this bound is best possible
Scripture and Self in Origen of Alexandria\u27s Exegetical Practice
In this dissertation I examine the nature of scripture and the self as presented by Origen of Alexandria. I argue that Christian scripture and the Christian self are constructed by exegetical practice; furthermore, in the case of Origen, I will demonstrate that Christian scripture and the Christian self are so closely related that it is best to speak of a scripture-self complex emerging out of his exegetical practice. I use a theory of structure as developed by William Sewell as a means to discuss both scripture and the self. As structures, scripture and the self are composed of resources and schemas that are paired together into meaningful wholes. That whole is a structure, which in turn structures other aspects of culture. However, resources and schemas are not automatically paired together. Rather, they are paired together by practices of historical agents who both shape structures and are shaped by them.
With this framework in mind, I discuss the ways in which exegetical practices pair resources and schemas together into meaningful wholes. There are two initial processes, the becoming scripture of biblical texts and the becoming the self of a human person, which I trace in Heracleon, Irenaeus, and Origen. I then argue that in the case of Origen, scripture and self mutually structure one another. I call these processes the anthropomorphizing of scripture and the scripturalizing of the self. These processes result in what I call a scripture-self complex, by this term I mean that scripture cannot be what scripture is without the self being what the self is and the self cannot be what the self is without scripture being what scripture is. Key texts for my study of Origen\u27s exegetical practices are his Commentary on the Gospel according to John, On First Principles, Homilies on Jeremiah, and finally, Commentary on the Song of Songs
Hypergraph containers
We develop a notion of containment for independent sets in hypergraphs. For
every -uniform hypergraph , we find a relatively small collection of
vertex subsets, such that every independent set of is contained within a
member of , and no member of is large; the collection, which is in
various respects optimal, reveals an underlying structure to the independent
sets. The containers offer a straightforward and unified approach to many
combinatorial questions concerned (usually implicitly) with independence.
With regard to colouring, it follows that simple -uniform hypergraphs of
average degree have list chromatic number at least . For this improves a bound due to Alon and is tight. For , previous bounds were weak but the present inequality is close to
optimal.
In the context of extremal graph theory, it follows that, for each
-uniform hypergraph of order , there is a collection of
-uniform hypergraphs of order each with copies of , such
that every -free -uniform hypergraph of order is a subgraph of a
hypergraph in , and where is
a standard parameter (there is a similar statement for induced subgraphs). This
yields simple proofs, for example, for the number of -free hypergraphs, and
for the sparsity theorems of Conlon-Gowers and Schacht. A slight variant yields
a counting version of the K{\L}R conjecture.
Likewise, for systems of linear equations the containers supply, for example,
bounds on the number of solution-free sets, and the existence of solutions in
sparse random subsets.
Balogh, Morris and Samotij have independently obtained related results.The first author was supported by a grant from the EPSRC.This is the author accepted manuscript. The final version is available from Springer at http://dx.doi.org/10.1007/s00222-014-0562-
Programmable Agents
We build deep RL agents that execute declarative programs expressed in formal
language. The agents learn to ground the terms in this language in their
environment, and can generalize their behavior at test time to execute new
programs that refer to objects that were not referenced during training. The
agents develop disentangled interpretable representations that allow them to
generalize to a wide variety of zero-shot semantic tasks
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