6,001 research outputs found
Space-Efficient DFS and Applications: Simpler, Leaner, Faster
The problem of space-efficient depth-first search (DFS) is reconsidered. A
particularly simple and fast algorithm is presented that, on a directed or
undirected input graph with vertices and edges, carries out a
DFS in time with bits of working memory, where is the
(total) degree of , for each , and . A slightly more complicated variant of the algorithm works in the same
time with at most bits. It is also shown that a DFS can
be carried out in a graph with vertices and edges in
time with bits or in time with either
bits or, for arbitrary integer , bits. These
results among them subsume or improve most earlier results on space-efficient
DFS. Some of the new time and space bounds are shown to extend to applications
of DFS such as the computation of cut vertices, bridges, biconnected components
and 2-edge-connected components in undirected graphs
The risk of misclassifying subjects within principal component based asset index.
The asset index is often used as a measure of socioeconomic status in empirical research as an explanatory variable or to control confounding. Principal component analysis (PCA) is frequently used to create the asset index. We conducted a simulation study to explore how accurately the principal component based asset index reflects the study subjects' actual poverty level, when the actual poverty level is generated by a simple factor analytic model. In the simulation study using the PC-based asset index, only 1% to 4% of subjects preserved their real position in a quintile scale of assets; between 44% to 82% of subjects were misclassified into the wrong asset quintile. If the PC-based asset index explained less than 30% of the total variance in the component variables, then we consistently observed more than 50% misclassification across quintiles of the index. The frequency of misclassification suggests that the PC-based asset index may not provide a valid measure of poverty level and should be used cautiously as a measure of socioeconomic status
Enhancing non-melonic triangulations: A tensor model mixing melonic and planar maps
Ordinary tensor models of rank are dominated at large by
tree-like graphs, known as melonic triangulations. We here show that
non-melonic contributions can be enhanced consistently, leading to different
types of large limits. We first study the most generic quartic model at
, with maximally enhanced non-melonic interactions. The existence of the
expansion is proved and we further characterize the dominant
triangulations. This combinatorial analysis is then used to define a
non-quartic, non-melonic class of models for which the large free energy
and the relevant expectations can be calculated explicitly. They are matched
with random matrix models which contain multi-trace invariants in their
potentials: they possess a branched polymer phase and a 2D quantum gravity
phase, and a transition between them whose entropy exponent is positive.
Finally, a non-perturbative analysis of the generic quartic model is performed,
which proves analyticity in the coupling constants in cardioid domains
The Tensor Track, III
We provide an informal up-to-date review of the tensor track approach to
quantum gravity. In a long introduction we describe in simple terms the
motivations for this approach. Then the many recent advances are summarized,
with emphasis on some points (Gromov-Hausdorff limit, Loop vertex expansion,
Osterwalder-Schrader positivity...) which, while important for the tensor track
program, are not detailed in the usual quantum gravity literature. We list open
questions in the conclusion and provide a rather extended bibliography.Comment: 53 pages, 6 figure
Double Scaling in Tensor Models with a Quartic Interaction
In this paper we identify and analyze in detail the subleading contributions
in the 1/N expansion of random tensors, in the simple case of a quartically
interacting model. The leading order for this 1/N expansion is made of graphs,
called melons, which are dual to particular triangulations of the D-dimensional
sphere, closely related to the "stacked" triangulations. For D<6 the subleading
behavior is governed by a larger family of graphs, hereafter called cherry
trees, which are also dual to the D-dimensional sphere. They can be resummed
explicitly through a double scaling limit. In sharp contrast with random matrix
models, this double scaling limit is stable. Apart from its unexpected upper
critical dimension 6, it displays a singularity at fixed distance from the
origin and is clearly the first step in a richer set of yet to be discovered
multi-scaling limits
On singular Lagrangians affine in velocities
The properties of Lagrangians affine in velocities are analyzed in a
geometric way. These systems are necessarily singular and exhibit, in general,
gauge invariance. The analysis of constraint functions and gauge symmetry leads
us to a complete classification of such Lagrangians.Comment: AMSTeX, 22 page
- …