3,894 research outputs found
The modular geometry of Random Regge Triangulations
We show that the introduction of triangulations with variable connectivity
and fluctuating egde-lengths (Random Regge Triangulations) allows for a
relatively simple and direct analyisis of the modular properties of 2
dimensional simplicial quantum gravity. In particular, we discuss in detail an
explicit bijection between the space of possible random Regge triangulations
(of given genus g and with N vertices) and a suitable decorated version of the
(compactified) moduli space of genus g Riemann surfaces with N punctures. Such
an analysis allows us to associate a Weil-Petersson metric with the set of
random Regge triangulations and prove that the corresponding volume provides
the dynamical triangulation partition function for pure gravity.Comment: 36 pages corrected typos, enhanced introductio
Complexity of Inference in Graphical Models
Graphical models provide a convenient representation for a broad class of probability distributions.
Due to their powerful and sophisticated modeling capabilities, such models have
found numerous applications in machine learning and other areas. In this paper we consider the
complexity of commonly encountered tasks involving graphical models such as the computation
of the mode of a posterior probability distribution (i.e., MAP estimation), and the computation
of marginal probabilities or the partition function. It is well-known that such inference problems
are hard in the worst case, but are tractable for models with bounded treewidth. We ask
whether treewidth is the only structural criterion of the underlying graph that enables tractable
inference. In other words, is there some class of structures with unbounded treewidth in which
inference is tractable? Subject to a combinatorial hypothesis due to Robertson, Seymour, and
Thomas (1994), we show that low treewidth is indeed the only structural restriction that can
ensure tractability. More precisely we show that for every growing family of graphs indexed
by tree-width, there exists a choice of potential functions such that the corresponding inference
problem is intractable. Thus even for the "best case" graph structures of high treewidth, there is
no polynomial-time inference algorithm. Our analysis employs various concepts from complexity theory and graph theory, with graph minors playing a prominent role
IMPACT: The Journal of the Center for Interdisciplinary Teaching and Learning. Volume 1, Issue 1, Summer 2012
Impact: The Journal of the Center for Interdisciplinary Teaching & Learning is a peer-reviewed, biannual online journal that publishes scholarly and creative non-fiction essays about the theory, practice and assessment of interdisciplinary education. Impact is produced by the Center for Interdisciplinary Teaching & Learning at the College of General Studies, Boston University (www.bu.edu/cgs/citl)
IMPACT: The Journal of the Center for Interdisciplinary Teaching and Learning. Volume 1, Issue 1, Summer 2012
Impact: The Journal of the Center for Interdisciplinary Teaching & Learning is a peer-reviewed, biannual online journal that publishes scholarly and creative non-fiction essays about the theory, practice and assessment of interdisciplinary education. Impact is produced by the Center for Interdisciplinary Teaching & Learning at the College of General Studies, Boston University (www.bu.edu/cgs/citl)
A note on perfect partial elimination
In Gaussian elimination it is often desirable to preserve existing zeros (sparsity). This is closely related to perfect elimination schemes on graphs. Such schemes can be found in polynomial time. Gaussian elimination uses a pivot for each column, so opportunities for preserving sparsity can be missed. In this paper we consider a more flexible process that selects a pivot for each nonzero to be eliminated and show that recognizing matrices that allow such perfect partial elimination schemes is NP-hard
Locally Causal Dynamical Triangulations in Two Dimensions
We analyze the universal properties of a new two-dimensional quantum gravity
model defined in terms of Locally Causal Dynamical Triangulations (LCDT).
Measuring the Hausdorff and spectral dimensions of the dynamical geometrical
ensemble, we find numerical evidence that the continuum limit of the model lies
in a new universality class of two-dimensional quantum gravity theories,
inequivalent to both Euclidean and Causal Dynamical Triangulations
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