24,450 research outputs found
Weighted interlace polynomials
The interlace polynomials introduced by Arratia, Bollobas and Sorkin extend
to invariants of graphs with vertex weights, and these weighted interlace
polynomials have several novel properties. One novel property is a version of
the fundamental three-term formula
q(G)=q(G-a)+q(G^{ab}-b)+((x-1)^{2}-1)q(G^{ab}-a-b) that lacks the last term. It
follows that interlace polynomial computations can be represented by binary
trees rather than mixed binary-ternary trees. Binary computation trees provide
a description of that is analogous to the activities description of the
Tutte polynomial. If is a tree or forest then these "algorithmic
activities" are associated with a certain kind of independent set in . Three
other novel properties are weighted pendant-twin reductions, which involve
removing certain kinds of vertices from a graph and adjusting the weights of
the remaining vertices in such a way that the interlace polynomials are
unchanged. These reductions allow for smaller computation trees as they
eliminate some branches. If a graph can be completely analyzed using
pendant-twin reductions then its interlace polynomial can be calculated in
polynomial time. An intuitively pleasing property is that graphs which can be
constructed through graph substitutions have vertex-weighted interlace
polynomials which can be obtained through algebraic substitutions.Comment: 11 pages (v1); 20 pages (v2); 27 pages (v3); 26 pages (v4). Further
changes may be made before publication in Combinatorics, Probability and
Computin
Layout of Graphs with Bounded Tree-Width
A \emph{queue layout} of a graph consists of a total order of the vertices,
and a partition of the edges into \emph{queues}, such that no two edges in the
same queue are nested. The minimum number of queues in a queue layout of a
graph is its \emph{queue-number}. A \emph{three-dimensional (straight-line
grid) drawing} of a graph represents the vertices by points in
and the edges by non-crossing line-segments. This paper contributes three main
results:
(1) It is proved that the minimum volume of a certain type of
three-dimensional drawing of a graph is closely related to the queue-number
of . In particular, if is an -vertex member of a proper minor-closed
family of graphs (such as a planar graph), then has a drawing if and only if has O(1) queue-number.
(2) It is proved that queue-number is bounded by tree-width, thus resolving
an open problem due to Ganley and Heath (2001), and disproving a conjecture of
Pemmaraju (1992). This result provides renewed hope for the positive resolution
of a number of open problems in the theory of queue layouts.
(3) It is proved that graphs of bounded tree-width have three-dimensional
drawings with O(n) volume. This is the most general family of graphs known to
admit three-dimensional drawings with O(n) volume.
The proofs depend upon our results regarding \emph{track layouts} and
\emph{tree-partitions} of graphs, which may be of independent interest.Comment: This is a revised version of a journal paper submitted in October
2002. This paper incorporates the following conference papers: (1) Dujmovic',
Morin & Wood. Path-width and three-dimensional straight-line grid drawings of
graphs (GD'02), LNCS 2528:42-53, Springer, 2002. (2) Wood. Queue layouts,
tree-width, and three-dimensional graph drawing (FSTTCS'02), LNCS
2556:348--359, Springer, 2002. (3) Dujmovic' & Wood. Tree-partitions of
-trees with applications in graph layout (WG '03), LNCS 2880:205-217, 200
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A resource guide to physical science/physics software for the Apple II, Atari, Pet, and TRS-80
The purpose of this resource guide is to present a list of the software available for high school physical science/physics for the four most popular microcomputers: Apple II, Pet, TRS-80, and Atari. The resource guide will consist of the following: 1. A list and description of the high school physical science/physics software available for AISD teachers in the various Resource Centers. 2. A list and description of the commercial software for high school physical science/physics. 3. Checklists for evaluating computer instructional programs. 4. An annotated bibliography of microcomputer publications.Science and Mathematics Educatio
Alternatives to Kronig-Kramers Transformation and Testing, and Estimation of Distributions
Two alternatives to Kronig-Kramers analysis of small-signal ac immittance data are discussed and illustrated using both synthetic and experimental data. The first, a derivative method of approximating imaginary-part response from real-part data, is found to be too approximate in regions where the imaginary-part varies appreciably with frequency. The second, a distribution of relaxation-times fitting method, is shown to be valuable for testing whether a data set satisfies the Kronig-Kramers relations and so is associated with a system whose properties are time-invariant. It also is valuable for estimating real- or imaginary-part response from the other part, usually with small error. Unlike Kronig-Kramers analysis, the second method usually requires no extrapolation outside the range of the measured data. Finally, this discrete-function method also allows one to estimate the distribution of relaxation times or activation energies associated with a given set of frequency-response data. This application is described and illustrated for both synthetic and experimental data and is shown to yield good but somewhat approximate results for the estimation of continuous distributions. It is particularly valuable for identifying response regions arising from a continuous distribution and distinguishing them from those associated with discrete time-constant response
A Tutorial on Clique Problems in Communications and Signal Processing
Since its first use by Euler on the problem of the seven bridges of
K\"onigsberg, graph theory has shown excellent abilities in solving and
unveiling the properties of multiple discrete optimization problems. The study
of the structure of some integer programs reveals equivalence with graph theory
problems making a large body of the literature readily available for solving
and characterizing the complexity of these problems. This tutorial presents a
framework for utilizing a particular graph theory problem, known as the clique
problem, for solving communications and signal processing problems. In
particular, the paper aims to illustrate the structural properties of integer
programs that can be formulated as clique problems through multiple examples in
communications and signal processing. To that end, the first part of the
tutorial provides various optimal and heuristic solutions for the maximum
clique, maximum weight clique, and -clique problems. The tutorial, further,
illustrates the use of the clique formulation through numerous contemporary
examples in communications and signal processing, mainly in maximum access for
non-orthogonal multiple access networks, throughput maximization using index
and instantly decodable network coding, collision-free radio frequency
identification networks, and resource allocation in cloud-radio access
networks. Finally, the tutorial sheds light on the recent advances of such
applications, and provides technical insights on ways of dealing with mixed
discrete-continuous optimization problems
Mobile vs. point guards
We study the problem of guarding orthogonal art galleries with horizontal
mobile guards (alternatively, vertical) and point guards, using "rectangular
vision". We prove a sharp bound on the minimum number of point guards required
to cover the gallery in terms of the minimum number of vertical mobile guards
and the minimum number of horizontal mobile guards required to cover the
gallery. Furthermore, we show that the latter two numbers can be calculated in
linear time.Comment: This version covers a previously missing case in both Phase 2 &
Towards Building Deep Networks with Bayesian Factor Graphs
We propose a Multi-Layer Network based on the Bayesian framework of the
Factor Graphs in Reduced Normal Form (FGrn) applied to a two-dimensional
lattice. The Latent Variable Model (LVM) is the basic building block of a
quadtree hierarchy built on top of a bottom layer of random variables that
represent pixels of an image, a feature map, or more generally a collection of
spatially distributed discrete variables. The multi-layer architecture
implements a hierarchical data representation that, via belief propagation, can
be used for learning and inference. Typical uses are pattern completion,
correction and classification. The FGrn paradigm provides great flexibility and
modularity and appears as a promising candidate for building deep networks: the
system can be easily extended by introducing new and different (in cardinality
and in type) variables. Prior knowledge, or supervised information, can be
introduced at different scales. The FGrn paradigm provides a handy way for
building all kinds of architectures by interconnecting only three types of
units: Single Input Single Output (SISO) blocks, Sources and Replicators. The
network is designed like a circuit diagram and the belief messages flow
bidirectionally in the whole system. The learning algorithms operate only
locally within each block. The framework is demonstrated in this paper in a
three-layer structure applied to images extracted from a standard data set.Comment: Submitted for journal publicatio
Experimental Study of the Sampled Labyrinth Chaos
In this paper, some new numerical as well as experimental results connected with the so-called labyrinth chaos are presented. This very unusual chaotic motion can be generated by mathematical model involving the scalar goniometrical functions which makes a three-dimensional autonomous dynamical system strongly nonlinear. Final circuitry implementation with analog core and digital parts can be used for modeling Brownian motion. From the viewpoint of generating chaotic motion by some electronic circuit, first step is to solve problems associated with the two-port nonlinear transfer functions synthesis. In the case of labyrinth chaos the finite dynamical range of the input variables introduced by the used active elements usually limits the performance greatly, similarly as it holds for the multi-grid spiral attractors. This paper shows an elegant way how to remove these obstacles by using uni-versal multiple-port with internal digital signal processing
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