2 research outputs found
On Selected Subclasses of Matroids
Matroids were introduced by Whitney to provide an abstract notion of independence.
In this work, after giving a brief survey of matroid theory, we describe structural results for various classes of matroids. A connected matroid is unbreakable if, for each of its flats , the matroid is connected%or, equivalently, if has no two skew circuits. . Pfeil showed that a simple graphic matroid is unbreakable exactly when is either a cycle or a complete graph. We extend this result to describe which graphs are the underlying graphs of unbreakable frame matroids. A laminar family is a collection \A of subsets of a set such that, for any two intersecting sets, one is contained in the other. For a capacity function on \A, let \I be %the set \{I:|I\cap A| \leq c(A)\text{ for all A\in\A}\}. Then \I is the collection of independent sets of a (laminar) matroid on . We characterize the class of laminar matroids by their excluded minors and present a way to construct all laminar matroids using basic operations. %Nested matroids were introduced by Crapo in 1965 and have appeared frequently in the literature since then. A flat of a matroid is Hamiltonian if it has a spanning circuit. A matroid is nested if its Hamiltonian flats form a chain under inclusion; is laminar if, for every -element independent set , the Hamiltonian flats of containing form a chain under inclusion. We generalize these notions to define the classes of -closure-laminar and -laminar matroids. The second class is always minor-closed, and the first is if and only if . We give excluded-minor characterizations of the classes of 2-laminar and 2-closure-laminar matroids
Coronal Heating as Determined by the Solar Flare Frequency Distribution Obtained by Aggregating Case Studies
Flare frequency distributions represent a key approach to addressing one of
the largest problems in solar and stellar physics: determining the mechanism
that counter-intuitively heats coronae to temperatures that are orders of
magnitude hotter than the corresponding photospheres. It is widely accepted
that the magnetic field is responsible for the heating, but there are two
competing mechanisms that could explain it: nanoflares or Alfv\'en waves. To
date, neither can be directly observed. Nanoflares are, by definition,
extremely small, but their aggregate energy release could represent a
substantial heating mechanism, presuming they are sufficiently abundant. One
way to test this presumption is via the flare frequency distribution, which
describes how often flares of various energies occur. If the slope of the power
law fitting the flare frequency distribution is above a critical threshold,
as established in prior literature, then there should be a
sufficient abundance of nanoflares to explain coronal heating. We performed
600 case studies of solar flares, made possible by an unprecedented number
of data analysts via three semesters of an undergraduate physics laboratory
course. This allowed us to include two crucial, but nontrivial, analysis
methods: pre-flare baseline subtraction and computation of the flare energy,
which requires determining flare start and stop times. We aggregated the
results of these analyses into a statistical study to determine that . This is below the critical threshold, suggesting that Alfv\'en
waves are an important driver of coronal heating.Comment: 1,002 authors, 14 pages, 4 figures, 3 tables, published by The
Astrophysical Journal on 2023-05-09, volume 948, page 7