686 research outputs found
Random Distances Associated with Trapezoids
The distributions of the random distances associated with hexagons, rhombuses
and triangles have been derived and verified in the existing work. All of these
geometric shapes are related to each other and have various applications in
wireless communications, transportation, etc. Hexagons are widely used to model
the cells in cellular networks, while trapezoids can be utilized to model the
edge users in a cellular network with a hexagonal tessellation. In this report,
the distributions of the random distances associated with unit trapezoids are
derived, when two random points are within a unit trapezoid or in two neighbor
unit trapezoids. The mathematical expressions are verified through simulation.
Further, we present the polynomial fit for the PDFs, which can be used to
simplify the computation.Comment: 12 pages, 4 figure
Kirchhoff index, multiplicative degree-Kirchhoff index and spanning trees of the linear crossed polyomino chains
Let be a linear crossed polyomino chain with four-order complete
graphs. In this paper, explicit formulas for the Kirchhoff index, the
multiplicative degree-Kirchhoff index and the number of spanning trees of
are determined, respectively. It is interesting to find that the Kirchhoff
(resp. multiplicative degree-Kirchhoff) index of is approximately one
quarter of its Wiener (resp. Gutman) index. More generally, let
be the set of subgraphs obtained by deleting vertical
edges of , where . For any graph , its Kirchhoff index and number of spanning trees are
completely determined, respectively. Finally, we show that the Kirchhoff index
of is approximately one quarter of its Wiener index
Signless Laplacian spectral radius and fractional matchings in graphs
A fractional matching of a graph is a function giving each edge a
number in such that for each vertex
, where is the set of edges incident to . The
fractional matching number of , written , is the
maximum value of over all fractional matchings. In this
paper, we investigate the relations between the fractional matching number and
the signless Laplacian spectral radius of a graph. Moreover, we give some
sufficient spectral conditions for the existence of a fractional perfect
matching
Random Distances Associated with Arbitrary Polygons: An Algorithmic Approach between Two Random Points
This report presents a new, algorithmic approach to the distributions of the
distance between two points distributed uniformly at random in various
polygons, based on the extended Kinematic Measure (KM) from integral geometry.
We first obtain such random Point Distance Distributions (PDDs) associated with
arbitrary triangles (i.e., triangle-PDDs), including the PDD within a triangle,
and that between two triangles sharing either a common side or a common vertex.
For each case, we provide an algorithmic procedure showing the mathematical
derivation process, based on which either the closed-form expressions or the
algorithmic results can be obtained. The obtained triangle-PDDs can be utilized
for modeling and analyzing the wireless communication networks associated with
triangle geometries, such as sensor networks with triangle-shaped clusters and
triangle-shaped cellular systems with highly directional antennas. Furthermore,
based on the obtained triangle-PDDs, we then show how to obtain the PDDs
associated with arbitrary polygons through the decomposition and recursion
approach, since any polygons can be triangulated, and any geometry shapes can
be approximated by polygons with a needed precision. Finally, we give the PDDs
associated with ring geometries. The results shown in this report can enrich
and expand the theory and application of the probabilistic distance models for
the analysis of wireless communication networks.Comment: 16 pages, 14 figure
Random Distances Associated with Arbitrary Triangles: A Systematic Approach between Two Random Points
It has been known that the distribution of the random distances between two
uniformly distributed points within a convex polygon can be obtained based on
its chord length distribution (CLD). In this report, we first verify the
existing known CLD for arbitrary triangles, and then derive and verify the
distance distribution between two uniformly distributed points within an
arbitrary triangle by simulation. Furthermore, a decomposition and recursion
approach is applied to obtain the random point distance distribution between
two arbitrary triangles sharing a side. As a case study, the explicit
distribution functions are derived when two congruent isosceles triangles with
the acute angle equal to form a rhombus or a concave 4-gon.Comment: 22 pages, 15 figure
Distance Distribution Between Two Random Nodes in Arbitrary Polygons
Distance distributions are a key building block in stochastic geometry
modelling of wireless networks and in many other fields in mathematics and
science. In this paper, we propose a novel framework for analytically computing
the closed form probability density function (PDF) of the distance between two
random nodes each uniformly randomly distributed in respective arbitrary
(convex or concave) polygon regions (which may be disjoint or overlap or
coincide). The proposed framework is based on measure theory and uses polar
decomposition for simplifying and calculating the integrals to obtain closed
form results. We validate our proposed framework by comparison with simulations
and published closed form results in the literature for simple cases. We
illustrate the versatility and advantage of the proposed framework by deriving
closed form results for a case not yet reported in the literature. Finally, we
also develop a Mathematica implementation of the proposed framework which
allows a user to define any two arbitrary polygons and conveniently determine
the distance distribution numerically.Comment: submitted for possible publicatio
Recursion-based Analysis for Information Propagation in Vehicular Ad Hoc Networks
Effective inter-vehicle communication is fundamental to a decentralized
traffic information system based on Vehicular Ad Hoc Networks (VANETs). To
reflect the uncertainty of the information propagation, most of the existing
work was conducted by assuming the inter-vehicle distance follows some specific
probability models, e.g., the lognormal or exponential distribution, while
reducing the analysis complexity. Aimed at providing more generic results, a
recursive modeling framework is proposed for VANETs in this paper when the
vehicle spacing can be captured by a general i.i.d. distribution. With the
framework, the analytical expressions for a series of commonly discussed
metrics are derived respectively, including the mean, variance, probability
distribution of the propagation distance, and expectation for the number of
vehicles included in a propagation process, when the transmission failures are
mainly caused by MAC contentions. Moreover, a discussion is also made for
demonstrating the efficiency of the recursive analysis method when the impact
of channel fading is also considered. All the analytical results are verified
by extensive simulations. We believe that this work is able to potentially
reveal a more insightful understanding of information propagation in VANETs by
allowing to evaluate the effect of any vehicle headway distributions.Comment: 6 page
Resistance distance-based graph invariants and spanning trees of graphs derived from the strong product of and
Let be a graph obtained by the strong product of and , where
. In this paper, explicit expressions for the Kirchhoff index,
multiplicative degree-Kirchhoff index and number of spanning trees of are
determined, respectively. It is surprising to find that the Kirchhoff (resp.
multiplicative degree-Kirchhoff) index of is almost one-sixth of its
Wiener (resp. Gutman) index. Moreover, let be the set of
subgraphs obtained from by deleting any vertical edges of ,
where . Explicit formulas for the Kirchhoff index and
the number of spanning trees for any graph are
completely established, respectively. Finally, it is interesting to see that
the Kirchhoff index of is almost one-sixth of its Wiener index
Beyond Powers of Two: Hexagonal Modulation and Non-Binary Coding for Wireless Communication Systems
Adaptive modulation and coding (AMC) is widely employed in modern wireless
communication systems to improve the transmission efficiency by adjusting the
transmission rate according to the channel conditions. Thus, AMC can provide
very efficient use of channel resources especially over fading channels.
Quadrature Amplitude Modulation (QAM) is an ef- ficient and widely employed
digital modulation technique. It typically employs a rectangular signal
constellation. Therefore the decision regions of the constellation are square
partitions of the two-dimensional signal space. However, it is well known that
hexagons rather than squares provide the most compact regular tiling in two
dimensions. A compact tiling means a dense packing of the constellation points
and thus more energy efficient data transmission. Hexagonal modulation can be
difficult to implement because it does not fit well with the usual power-
of-two symbol sizes employed with binary data. To overcome this problem,
non-binary coding is combined with hexagonal modulation in this paper to
provide a system which is compatible with binary data. The feasibility and
efficiency are evaluated using a software-defined radio (SDR) based prototype.
Extensive simulation results are presented which show that this approach can
provide improved energy efficiency and spectrum utilization in wireless
communication systems.Comment: 9 page
Random Distances Associated with Rhombuses
Parallelograms are one of the basic building blocks in two-dimensional
tiling. They have important applications in a wide variety of science and
engineering fields, such as wireless communication networks, urban
transportation, operations research, etc. Different from rectangles and
squares, the coordinates of a random point in parallelograms are no longer
independent. As a case study of parallelograms, the explicit probability
density functions of the random Euclidean distances associated with rhombuses
are given in this report, when both endpoints are randomly distributed in 1)
the same rhombus, 2) two parallel rhombuses sharing a side, and 3) two
rhombuses having a common diagonal, respectively. The accuracy of the distance
distribution functions is verified by simulation, and the correctness is
validated by a recursion and a probabilistic sum. The first two statistical
moments of the random distances, and the polynomial fit of the density
functions are also given in this report for practical uses
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