354,381 research outputs found
Fractal fractal dimensions of deterministic transport coefficients
If a point particle moves chaotically through a periodic array of scatterers
the associated transport coefficients are typically irregular functions under
variation of control parameters. For a piecewise linear two-parameter map we
analyze the structure of the associated irregular diffusion coefficient and
current by numerically computing dimensions from box-counting and from the
autocorrelation function of these graphs. We find that both dimensions are
fractal for large parameter intervals and that both quantities are themselves
fractal functions if computed locally on a uniform grid of small but finite
subintervals. We furthermore show that there is a simple functional
relationship between the structure of fractal fractal dimensions and the
difference quotient defined on these subintervals.Comment: 16 pages (revtex) with 6 figures (postscript
Modified iterative versus Laplacian Landau gauge in compact U(1) theory
Compact U(1) theory in 4 dimensions is used to compare the modified iterative
and the Laplacian fixing to lattice Landau gauge in a controlled setting, since
in the Coulomb phase the lattice theory must reproduce the perturbative
prediction. It turns out that on either side of the phase transition clear
differences show up and in the Coulomb phase the ability to remove double Dirac
sheets proves vital on a small lattice.Comment: 14 pages, 8 figures containing 23 graphs, v2: 2 figures removed, 2
references adde
Hierarchical models of rigidity percolation
We introduce models of generic rigidity percolation in two dimensions on
hierarchical networks, and solve them exactly by means of a renormalization
transformation. We then study how the possibility for the network to self
organize in order to avoid stressed bonds may change the phase diagram. In
contrast to what happens on random graphs and in some recent numerical studies
at zero temperature, we do not find a true intermediate phase separating the
usual rigid and floppy ones.Comment: 20 pages, 8 figures. Figures improved, references added, small
modifications. Accepted in Phys. Rev.
The Minimum Shared Edges Problem on Grid-like Graphs
We study the NP-hard Minimum Shared Edges (MSE) problem on graphs: decide
whether it is possible to route paths from a start vertex to a target
vertex in a given graph while using at most edges more than once. We show
that MSE can be decided on bounded (i.e. finite) grids in linear time when both
dimensions are either small or large compared to the number of paths. On
the contrary, we show that MSE remains NP-hard on subgraphs of bounded grids.
Finally, we study MSE from a parametrised complexity point of view. It is known
that MSE is fixed-parameter tractable with respect to the number of paths.
We show that, under standard complexity-theoretical assumptions, the problem
parametrised by the combined parameter , , maximum degree, diameter, and
treewidth does not admit a polynomial-size problem kernel, even when restricted
to planar graphs
Coupling of hard dimers to dynamical lattices via random tensors
We study hard dimers on dynamical lattices in arbitrary dimensions using a
random tensor model. The set of lattices corresponds to triangulations of the
d-sphere and is selected by the large N limit. For small enough dimer
activities, the critical behavior of the continuum limit is the one of pure
random lattices. We find a negative critical activity where the universality
class is changed as dimers become critical, in a very similar way hard dimers
exhibit a Yang-Lee singularity on planar dynamical graphs. Critical exponents
are calculated exactly. An alternative description as a system of
`color-sensitive hard-core dimers' on random branched polymers is provided.Comment: 12 page
Algorithms, Reductions and Equivalences for Small Weight Variants of All-Pairs Shortest Paths
APSP with small integer weights in undirected graphs [Seidel'95, Galil and
Margalit'97] has an time algorithm, where
is the matrix multiplication exponent. APSP in directed graphs with small
weights however, has a much slower running time that would be
even if [Zwick'02]. To understand this bottleneck, we
build a web of reductions around directed unweighted APSP. We show that it is
fine-grained equivalent to computing a rectangular Min-Plus product for
matrices with integer entries; the dimensions and entry size of the matrices
depend on the value of . As a consequence, we establish an equivalence
between APSP in directed unweighted graphs, APSP in directed graphs with small
integer weights, All-Pairs Longest Paths in DAGs with small
weights, approximate APSP with additive error in directed graphs with small
weights, for and several other graph problems. We also
provide fine-grained reductions from directed unweighted APSP to All-Pairs
Shortest Lightest Paths (APSLP) in undirected graphs with weights and
APSP in directed unweighted graphs (computing counts mod
).
We complement our hardness results with new algorithms. We improve the known
algorithms for APSLP in directed graphs with small integer weights and for
approximate APSP with sublinear additive error in directed unweighted graphs.
Our algorithm for approximate APSP with sublinear additive error is optimal,
when viewed as a reduction to Min-Plus product. We also give new algorithms for
variants of #APSP in unweighted graphs, as well as a near-optimal
-time algorithm for the original #APSP problem in unweighted
graphs. Our techniques also lead to a simpler alternative for the original APSP
problem in undirected graphs with small integer weights.Comment: abstract shortened to fit arXiv requirement
Small covers and the equivariant bordism classification of 2-torus manifolds
Associated with the Davis-Januszkiewicz theory of small covers, this paper
deals with the theory of 2-torus manifolds from the viewpoint of equivariant
bordism. We define a differential operator on the "dual" algebra of the
unoriented -representation algebra introduced by Conner and Floyd, where
. With the help of -colored graphs (or mod 2 GKM graphs), we
may use this differential operator to give a very simple description of tom
Dieck-Kosniowski-Stong localization theorem in the setting of 2-torus
manifolds. We then apply this to study the -equivariant unoriented bordism
classification of -dimensional 2-torus manifolds. We show that the
-equivariant unoriented bordism class of each -dimensional 2-torus
manifold contains an -dimensional small cover as its representative, solving
the conjecture posed in [19]. In addition, we also obtain that the graded
noncommutative ring formed by the equivariant unoriented bordism classes of
2-torus manifolds of all possible dimensions is generated by the classes of all
generalized real Bott manifolds (as special small covers over the products of
simplices). This gives a strong connection between the computation of
-equivariant bordism groups or ring and the Davis-Januszkiewicz theory of
small covers. As a computational application, with the help of computer, we
completely determine the structure of the group formed by equivariant bordism
classes of all 4-dimensional 2-torus manifolds. Finally, we give some essential
relationships among 2-torus manifolds, coloring polynomials, colored simple
convex polytopes, colored graphs.Comment: 32 pages, updated version with the title of paper changed and a large
expansio
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