16,551 research outputs found
Vertex identifying codes for the n-dimensional lattice
An -identifying code on a graph is a set such that for
every vertex in , the intersection of the radius- closed neighborhood
with is nonempty and different. Here, we provide an overview on codes for
the -dimensional lattice, discussing the case of 1-identifying codes,
constructing a sparse code for the 4-dimensional lattice as well as showing
that for fixed , the minimum density of an -identifying code is
.Comment: 10p
An improved lower bound for (1,<=2)-identifying codes in the king grid
We call a subset of vertices of a graph a -identifying
code if for all subsets of vertices with size at most , the sets
are distinct. The concept of
identifying codes was introduced in 1998 by Karpovsky, Chakrabarty and Levitin.
Identifying codes have been studied in various grids. In particular, it has
been shown that there exists a -identifying code in the king grid
with density 3/7 and that there are no such identifying codes with density
smaller than 5/12. Using a suitable frame and a discharging procedure, we
improve the lower bound by showing that any -identifying code of
the king grid has density at least 47/111
Automated Discharging Arguments for Density Problems in Grids
Discharging arguments demonstrate a connection between local structure and
global averages. This makes it an effective tool for proving lower bounds on
the density of special sets in infinite grids. However, the minimum density of
an identifying code in the hexagonal grid remains open, with an upper bound of
and a lower bound of . We present a new, experimental framework for producing discharging
arguments using an algorithm. This algorithm replaces the lengthy case analysis
of human-written discharging arguments with a linear program that produces the
best possible lower bound using the specified set of discharging rules. We use
this framework to present a lower bound of on
the density of an identifying code in the hexagonal grid, and also find several
sharp lower bounds for variations on identifying codes in the hexagonal,
square, and triangular grids.Comment: This is an extended abstract, with 10 pages, 2 appendices, 5 tables,
and 2 figure
Improved Bounds for -Identifying Codes of the Hex Grid
For any positive integer , an -identifying code on a graph is a set
such that for every vertex in , the intersection of the
radius- closed neighborhood with is nonempty and pairwise distinct. For
a finite graph, the density of a code is , which naturally extends
to a definition of density in certain infinite graphs which are locally finite.
We find a code of density less than , which is sparser than the prior
best construction which has density approximately .Comment: 12p
HP-sequence design for lattice proteins - an exact enumeration study on diamond as well as square lattice
We present an exact enumeration algorithm for identifying the {\it native}
configuration - a maximally compact self avoiding walk configuration that is
also the minimum energy configuration for a given set of contact-energy
schemes; the process is implicitly sequence-dependent. In particular, we show
that the 25-step native configuration on a diamond lattice consists of two
sheet-like structures and is the same for all the contact-energy schemes,
; on a square lattice also, the
24-step native configuration is independent of the energy schemes considered.
However, the designing sequence for the diamond lattice walk depends on the
energy schemes used whereas that for the square lattice walk does not. We have
calculated the temperature-dependent specific heat for these designed sequences
and the four energy schemes using the exact density of states. These data show
that the energy scheme is preferable to the other three for both
diamond and square lattice because the associated sequences give rise to a
sharp low-temperature peak. We have also presented data for shorter (23-, 21-
and 17-step) walks on a diamond lattice to show that this algorithm helps
identify a unique minimum energy configuration by suitably taking care of the
ground-state degeneracy. Interestingly, all these shorter target configurations
also show sheet-like secondary structures.Comment: 19 pages, 7 figures (eps), 11 tables (latex files
Detecting Topological Order with Ribbon Operators
We introduce a numerical method for identifying topological order in
two-dimensional models based on one-dimensional bulk operators. The idea is to
identify approximate symmetries supported on thin strips through the bulk that
behave as string operators associated to an anyon model. We can express these
ribbon operators in matrix product form and define a cost function that allows
us to efficiently optimize over this ansatz class. We test this method on spin
models with abelian topological order by finding ribbon operators for
quantum double models with local fields and Ising-like terms. In
addition, we identify ribbons in the abelian phase of Kitaev's honeycomb model
which serve as the logical operators of the encoded qubit for the quantum
error-correcting code. We further identify the topologically encoded qubit in
the quantum compass model, and show that despite this qubit, the model does not
support topological order. Finally, we discuss how the method supports
generalizations for detecting nonabelian topological order.Comment: 15 pages, 8 figures, comments welcom
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