26,420 research outputs found
Young tableaux, canonical bases and the Gindikin-Karpelevich formula
A combinatorial description of the crystal B(infinity) for finite-dimensional
simple Lie algebras in terms of certain Young tableaux was developed by J. Hong
and H. Lee. We establish an explicit bijection between these Young tableaux and
canonical bases indexed by Lusztig's parametrization, and obtain a
combinatorial rule for expressing the Gindikin-Karpelevich formula as a sum
over the set of Young tableaux.Comment: 19 page
Ranking Functions for Size-Change Termination II
Size-Change Termination is an increasingly-popular technique for verifying
program termination. These termination proofs are deduced from an abstract
representation of the program in the form of "size-change graphs".
We present algorithms that, for certain classes of size-change graphs, deduce
a global ranking function: an expression that ranks program states, and
decreases on every transition. A ranking function serves as a witness for a
termination proof, and is therefore interesting for program certification. The
particular form of the ranking expressions that represent SCT termination
proofs sheds light on the scope of the proof method. The complexity of the
expressions is also interesting, both practicaly and theoretically.
While deducing ranking functions from size-change graphs has already been
shown possible, the constructions in this paper are simpler and more
transparent than previously known. They improve the upper bound on the size of
the ranking expression from triply exponential down to singly exponential (for
certain classes of instances). We claim that this result is, in some sense,
optimal. To this end, we introduce a framework for lower bounds on the
complexity of ranking expressions and prove exponential lower bounds.Comment: 29 pages
Mapping Twitter Topic Networks: From Polarized Crowds to Community Clusters
Conversations on Twitter create networks with identifiable contours as people reply to and mention one another in their tweets. These conversational structures differ, depending on the subject and the people driving the conversation. Six structures are regularly observed: divided, unified, fragmented, clustered, and inward and outward hub and spoke structures. These are created as individuals choose whom to reply to or mention in their Twitter messages and the structures tell a story about the nature of the conversatio
Public libraries play a central role in providing access to data and ensuring the freedom of digital knowledge.
Data connectivity is intrinsic to most of our daily lives. The place which exists in almost every community large or small, rural or urban, is the public library. Ben Lee argues that not only do libraries provide free access to data, but they do so in an environment which is trustworthy and neutral, geared to learning. Access to digital technology increasingly overlaps with access to opportunity and it is important to recognise the role public libraries already play (and have always played) in keeping the gate to knowledge open
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