10,281 research outputs found
A Theme-Rewriting Approach for Generating Algebra Word Problems
Texts present coherent stories that have a particular theme or overall
setting, for example science fiction or western. In this paper, we present a
text generation method called {\it rewriting} that edits existing
human-authored narratives to change their theme without changing the underlying
story. We apply the approach to math word problems, where it might help
students stay more engaged by quickly transforming all of their homework
assignments to the theme of their favorite movie without changing the math
concepts that are being taught. Our rewriting method uses a two-stage decoding
process, which proposes new words from the target theme and scores the
resulting stories according to a number of factors defining aspects of
syntactic, semantic, and thematic coherence. Experiments demonstrate that the
final stories typically represent the new theme well while still testing the
original math concepts, outperforming a number of baselines. We also release a
new dataset of human-authored rewrites of math word problems in several themes.Comment: To appear EMNLP 201
Donaldson-Thomas invariants, torus knots, and lattice paths
In this paper we find and explore the correspondence between quivers, torus
knots, and combinatorics of counting paths. Our first result pertains to quiver
representation theory -- we find explicit formulae for classical generating
functions and Donaldson-Thomas invariants of an arbitrary symmetric quiver. We
then focus on quivers corresponding to torus knots and show that their
classical generating functions, in the extremal limit and framing , are
generating functions of lattice paths under the line of the slope .
Generating functions of such paths satisfy extremal A-polynomial equations,
which immediately follows after representing them in terms of the Duchon
grammar. Moreover, these extremal A-polynomial equations encode
Donaldson-Thomas invariants, which provides an interesting example of
algebraicity of generating functions of these invariants. We also find a
quantum generalization of these statements, i.e. a relation between motivic
quiver generating functions, quantum extremal knot invariants, and -weighted
path counting. Finally, in the case of the unknot, we generalize this
correspondence to the full HOMFLY-PT invariants and counting of Schr\"oder
paths.Comment: 45 pages. Corrected typos in new versio
On the finite presentation of subdirect products and the nature of residually free groups
We establish {\em{virtual surjection to pairs}} (VSP) as a general criterion
for the finite presentability of subdirect products of groups: if
are finitely presented and
projects to a subgroup of finite index in
each , then is finitely presentable, indeed there
is an algorithm that will construct a finite presentation for .
We use the VSP criterion to characterise the finitely presented residually
free groups. We prove that the class of such groups is recursively enumerable.
We describe an algorithm that, given a finite presentation of a residually free
group, constructs a canonical embedding into a direct product of finitely many
limit groups. We solve the (multiple) conjugacy problem and membership problem
for finitely presentable subgroups of residually free groups. We also prove
that there is an algorithm that, given a finite generating set for such a
subgroup, will construct a finite presentation.
New families of subdirect products of free groups are constructed, including
the first examples of finitely presented subgroups that are neither
nor of Stallings-Bieri typeComment: 44 pages. To appear in American Journal of Mathematics. This is a
substantial rewrite of our previous Arxiv article 0809.3704, taking into
account subsequent developments, advice of colleagues and referee's comment
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