849 research outputs found
On Vertex Identifying Codes For Infinite Lattices
PhD Thesis--A compilation of the papers: "Lower Bounds for Identifying Codes
in Some Infinite Grids", "Improved Bounds for r-identifying Codes of the Hex
Grid", and "Vertex Identifying Codes for the n-dimensional Lattics" along with
some other resultsComment: 91p
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
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
Three Puzzles on Mathematics, Computation, and Games
In this lecture I will talk about three mathematical puzzles involving
mathematics and computation that have preoccupied me over the years. The first
puzzle is to understand the amazing success of the simplex algorithm for linear
programming. The second puzzle is about errors made when votes are counted
during elections. The third puzzle is: are quantum computers possible?Comment: ICM 2018 plenary lecture, Rio de Janeiro, 36 pages, 7 Figure
Diddy: a Python toolbox for infinite discrete dynamical systems
We introduce Diddy, a collection of Python scripts for analyzing infinite
discrete dynamical systems. The main focus is on generalized multidimensional
shifts of finite type (SFTs). We show how Diddy can be used to easily define
SFTs and cellular automata, and analyze their basic properties. We also
showcase how to verify or rediscover some results from coding theory and
cellular automata theory.Comment: 12 page
Lessons Learned in the Selection and Development of Test Cases for the Aeroelastic Prediction Workshop: Rectangular Supercritical Wing
The Aeroelastic Prediction Workshop brought together an international community of computational fluid dynamicists as a step in defining the state of the art in computational aeroelasticity. The Rectangular Supercritical Wing (RSW) was chosen as the first configuration to study due to its geometric simplicity, perceived simple flow field at transonic conditions and availability of an experimental data set containing forced oscillation response data. Six teams performed analyses of the RSW; they used Reynolds-Averaged Navier-Stokes flow solvers exercised assuming that the wing had a rigid structure. Both steady-state and forced oscillation computations were performed by each team. The results of these calculations were compared with each other and with the experimental data. The steady-state results from the computations capture many of the flow features of a classical supercritical airfoil pressure distribution. The most dominant feature of the oscillatory results is the upper surface shock dynamics. Substantial variations were observed among the computational solutions as well as differences relative to the experimental data. Contributing issues to these differences include substantial wind tunnel wall effects and diverse choices in the analysis parameters
Hydrogen Epoch of Reionization Array (HERA)
The Hydrogen Epoch of Reionization Array (HERA) is a staged experiment to
measure 21 cm emission from the primordial intergalactic medium (IGM)
throughout cosmic reionization (), and to explore earlier epochs of our
Cosmic Dawn (). During these epochs, early stars and black holes
heated and ionized the IGM, introducing fluctuations in 21 cm emission. HERA is
designed to characterize the evolution of the 21 cm power spectrum to constrain
the timing and morphology of reionization, the properties of the first
galaxies, the evolution of large-scale structure, and the early sources of
heating. The full HERA instrument will be a 350-element interferometer in South
Africa consisting of 14-m parabolic dishes observing from 50 to 250 MHz.
Currently, 19 dishes have been deployed on site and the next 18 are under
construction. HERA has been designated as an SKA Precursor instrument.
In this paper, we summarize HERA's scientific context and provide forecasts
for its key science results. After reviewing the current state of the art in
foreground mitigation, we use the delay-spectrum technique to motivate
high-level performance requirements for the HERA instrument. Next, we present
the HERA instrument design, along with the subsystem specifications that ensure
that HERA meets its performance requirements. Finally, we summarize the
schedule and status of the project. We conclude by suggesting that, given the
realities of foreground contamination, current-generation 21 cm instruments are
approaching their sensitivity limits. HERA is designed to bring both the
sensitivity and the precision to deliver its primary science on the basis of
proven foreground filtering techniques, while developing new subtraction
techniques to unlock new capabilities. The result will be a major step toward
realizing the widely recognized scientific potential of 21 cm cosmology.Comment: 26 pages, 24 figures, 2 table
Synthesis of Frame Field-Aligned Multi-Laminar Structures
In the field of topology optimization, the homogenization approach has been
revived as an important alternative to the established, density-based methods
because it can represent the microstructural design at a much finer
length-scale than the computational grid. The optimal microstructure for a
single load case is an orthogonal rank-3 laminate. A rank-3 laminate can be
described in terms of frame fields, which are also an important tool for mesh
generation in both 2D and 3D.
We propose a method for generating multi-laminar structures from frame
fields. Rather than relying on integrative approaches that find a
parametrization based on the frame field, we find stream surfaces, represented
as point clouds aligned with frame vectors, and we solve an optimization
problem to find well-spaced collections of such stream surfaces. The stream
surface tracing is unaffected by the presence of singularities outside the
region of interest. Neither stream surface tracing nor selecting well-spaced
surface rely on combed frame fields.
In addition to stream surface tracing and selection, we provide two methods
for generating structures from stream surface collections. One of these methods
produces volumetric solids by summing basis functions associated with each
point of the stream surface collection. The other method reinterprets point
sampled stream surfaces as a spatial twist continuum and produces a
hexahedralization by dualizing a graph representing the structure.
We demonstrate our methods on several frame fields produced using the
homogenization approach for topology optimization, boundary-aligned, algebraic
frame fields, and frame fields computed from closed-form expressions.Comment: 19 pages, 18 figure
- …