130,462 research outputs found
Transition from connected to fragmented vegetation across an environmental gradient: scaling laws in ecotone geometry
A change in the environmental conditions across space—for example, altitude or latitude—can cause significant changes in the density of a vegetation type and, consequently, in spatial connectivity. We use spatially explicit simulations to study the transition from connected to fragmented vegetation. A static (gradient percolation) model is compared to dynamic (gradient contact process) models. Connectivity is characterized from the perspective of various species that use this vegetation type for habitat and differ in dispersal or migration range, that is, “step length” across the landscape. The boundary of connected vegetation delineated by a particular step length is termed the “ hull edge.” We found that for every step length and for every gradient, the hull edge is a fractal with dimension 7/4. The result is the same for different spatial models, suggesting that there are universal laws in ecotone geometry. To demonstrate that the model is applicable to real data, a hull edge of fractal dimension 7/4 is shown on a satellite image of a piñon‐juniper woodland on a hillside. We propose to use the hull edge to define the boundary of a vegetation type unambiguously. This offers a new tool for detecting a shift of the boundary due to a climate change
On Hull-Variation Problem of Equivalent Linear Codes
The intersection () of a linear code and its Euclidean dual (Hermitian dual ) is called the Euclidean
(Hermitian) hull of this code. The construction of an entanglement-assisted
quantum code from a linear code over or depends
essentially on the Euclidean hull or the Hermitian hull of this code. Therefore
it is natural to consider the hull-variation problem when a linear code is transformed to an equivalent code . In this paper
we introduce the maximal hull dimension as an invariant of a linear code with
respect to the equivalent transformations. Then some basic properties of the
maximal hull dimension are studied. A general method to construct
hull-decreasing or hull-increasing equivalent linear codes is proposed. We
prove that for a nonnegative integer satisfying , a
linear self-dual code is equivalent to a linear -dimension hull
code. On the opposite direction we prove that a linear LCD code over satisfying and is equivalent to a linear
one-dimension hull code under a weak condition. Several new families of
negacyclic LCD codes and BCH LCD codes over are also constructed.
Our method can be applied to the generalized Reed-Solomon codes and the
generalized twisted Reed-Solomon codes to construct arbitrary dimension hull
MDS codes. Some new EAQEC codes including MDS and almost MDS
entanglement-assisted quantum codes are constructed. Many EAQEC codes over
small fields are constructed from optimal Hermitian self-dual codes.Comment: 33 pages, minor error correcte
Remarks on the geometry of coordinate projections in R^n
We study geometric properties of coordinate projections. Among other results,
we show that if a body K in R^n has an "almost extremal" volume ratio, then it
has a projection of proportional dimension which is close to the cube. We
compare type 2 and infratype 2 constant of a Banach space. This follows from a
comparison lemma for Rademacher and Gaussian averages. We also establish a
sharp estimate on the shattering dimension of the convex hull of a class of
functions in terms of the shattering dimension of the class itself.Comment: Israel Journal of Mathematics, to appea
Convexity in partial cubes: the hull number
We prove that the combinatorial optimization problem of determining the hull
number of a partial cube is NP-complete. This makes partial cubes the minimal
graph class for which NP-completeness of this problem is known and improves
some earlier results in the literature.
On the other hand we provide a polynomial-time algorithm to determine the
hull number of planar partial cube quadrangulations.
Instances of the hull number problem for partial cubes described include
poset dimension and hitting sets for interiors of curves in the plane.
To obtain the above results, we investigate convexity in partial cubes and
characterize these graphs in terms of their lattice of convex subgraphs,
improving a theorem of Handa. Furthermore we provide a topological
representation theorem for planar partial cubes, generalizing a result of
Fukuda and Handa about rank three oriented matroids.Comment: 19 pages, 4 figure
Reverse Chv\'atal-Gomory rank
We introduce the reverse Chv\'atal-Gomory rank r*(P) of an integral
polyhedron P, defined as the supremum of the Chv\'atal-Gomory ranks of all
rational polyhedra whose integer hull is P. A well-known example in dimension
two shows that there exist integral polytopes P with r*(P) equal to infinity.
We provide a geometric characterization of polyhedra with this property in
general dimension, and investigate upper bounds on r*(P) when this value is
finite.Comment: 21 pages, 4 figure
Faster Geometric Algorithms via Dynamic Determinant Computation
The computation of determinants or their signs is the core procedure in many
important geometric algorithms, such as convex hull, volume and point location.
As the dimension of the computation space grows, a higher percentage of the
total computation time is consumed by these computations. In this paper we
study the sequences of determinants that appear in geometric algorithms. The
computation of a single determinant is accelerated by using the information
from the previous computations in that sequence.
We propose two dynamic determinant algorithms with quadratic arithmetic
complexity when employed in convex hull and volume computations, and with
linear arithmetic complexity when used in point location problems. We implement
the proposed algorithms and perform an extensive experimental analysis. On one
hand, our analysis serves as a performance study of state-of-the-art
determinant algorithms and implementations. On the other hand, we demonstrate
the supremacy of our methods over state-of-the-art implementations of
determinant and geometric algorithms. Our experimental results include a 20 and
78 times speed-up in volume and point location computations in dimension 6 and
11 respectively.Comment: 29 pages, 8 figures, 3 table
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