542 research outputs found
Matroids with at least two regular elements
For a matroid , an element such that both and
are regular is called a regular element of . We determine completely the
structure of non-regular matroids with at least two regular elements. Besides
four small size matroids, all 3-connected matroids in the class can be pieced
together from or and a regular matroid using 3-sums. This result
takes a step toward solving a problem posed by Paul Seymour: Find all
3-connected non-regular matroids with at least one regular element [5, 14.8.8]
Splitters and Decomposers for Binary Matroids
Let denote the class of binary matroids with no minors
isomorphic to . In this paper we give a decomposition theorem
for , where is a certain 10-element rank-4
matroid. As corollaries we obtain decomposition theorems for the classes
obtained by excluding the Kuratowski graphs and . These decomposition
theorems imply results on internally -connected matroids by Zhou
[\ref{Zhou2004}], Qin and Zhou [\ref{Qin2004}], and Mayhew, Royle and Whitte
[\ref{Mayhewsubmitted}].Comment: arXiv admin note: text overlap with arXiv:1403.775
Unlabeled equivalence for matroids representable over finite fields
We present a new type of equivalence for representable matroids that uses the
automorphisms of the underlying matroid. Two matrices and
representing the same matroid over a field are {\it geometrically
equivalent representations} of if one can be obtained from the other by
elementary row operations, column scaling, and column permutations. Using
geometric equivalence, we give a method for exhaustively generating
non-isomorphic matroids representable over a finite field , where is
a power of a prime
A decomposition theorem for binary matroids with no prism minor
The prism graph is the dual of the complete graph on five vertices with an
edge deleted, . In this paper we determine the class of binary
matroids with no prism minor. The motivation for this problem is the 1963
result by Dirac where he identified the simple 3-connected graphs with no minor
isomorphic to the prism graph. We prove that besides Dirac's infinite families
of graphs and four infinite families of non-regular matroids determined by
Oxley, there are only three possibilities for a matroid in this class: it is
isomorphic to the dual of the generalized parallel connection of with
itself across a triangle with an element of the triangle deleted; it's rank is
bounded by 5; or it admits a non-minimal exact 3-separation induced by the
3-separation in . Since the prism graph has rank 5, the class has to
contain the binary projective geometries of rank 3 and 4, and ,
respectively. We show that there is just one rank 5 extremal matroid in the
class. It has 17 elements and is an extension of , the unique splitter
for regular matroids. As a corollary, we obtain Dillon, Mayhew, and Royle's
result identifying the binary internally 4-connected matroids with no prism
minor [5]
Trailing edge noise theory for rotating blades in uniform flow
This paper presents a new formulation for trailing edge noise radiation from
rotating blades based on an analytical solution of the convective wave
equation. It accounts for distributed loading and the effect of mean flow and
spanwise wavenumber. A commonly used theory due to Schlinker and Amiet (1981)
predicts trailing edge noise radiation from rotating blades. However, different
versions of the theory exist; it is not known which version is the correct one
and what the range of validity of the theory is. This paper addresses both
questions by deriving Schlinker and Amiet's theory in a simple way and by
comparing it to the new formulation, using model blade elements representative
of a wind turbine, a cooling fan and an aircraft propeller. The correct form of
Schlinker and Amiet's theory (1981) is identified. It is valid at high enough
frequency, i.e. for a Helmholtz number relative to chord greater than one and a
rotational frequency much smaller than the angular frequency of the noise
sources.Comment: 28 pages, 10 figures, submitted to Proceedings of the Royal Society
A: Mathematical, Physical and Engineering Sciences (final revision
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