36,831 research outputs found
Parking functions, labeled trees and DCJ sorting scenarios
In genome rearrangement theory, one of the elusive questions raised in recent
years is the enumeration of rearrangement scenarios between two genomes. This
problem is related to the uniform generation of rearrangement scenarios, and
the derivation of tests of statistical significance of the properties of these
scenarios. Here we give an exact formula for the number of double-cut-and-join
(DCJ) rearrangement scenarios of co-tailed genomes. We also construct effective
bijections between the set of scenarios that sort a cycle and well studied
combinatorial objects such as parking functions and labeled trees.Comment: 12 pages, 3 figure
Nonlinear nonlocal multicontinua upscaling framework and its applications
In this paper, we discuss multiscale methods for nonlinear problems. The main
idea of these approaches is to use local constraints and solve problems in
oversampled regions for constructing macroscopic equations. These techniques
are intended for problems without scale separation and high contrast, which
often occur in applications. For linear problems, the local solutions with
constraints are used as basis functions. This technique is called Constraint
Energy Minimizing Generalized Multiscale Finite Element Method (CEM-GMsFEM).
GMsFEM identifies macroscopic quantities based on rigorous analysis. In
corresponding upscaling methods, the multiscale basis functions are selected
such that the degrees of freedom have physical meanings, such as averages of
the solution on each continuum.
This paper extends the linear concepts to nonlinear problems, where the local
problems are nonlinear. The main concept consists of: (1) identifying
macroscopic quantities; (2) constructing appropriate oversampled local problems
with coarse-grid constraints; (3) formulating macroscopic equations. We
consider two types of approaches. In the first approach, the solutions of local
problems are used as basis functions (in a linear fashion) to solve nonlinear
problems. This approach is simple to implement; however, it lacks the nonlinear
interpolation, which we present in our second approach. In this approach, the
local solutions are used as a nonlinear forward map from local averages
(constraints) of the solution in oversampling region. This local fine-grid
solution is further used to formulate the coarse-grid problem. Both approaches
are discussed on several examples and applied to single-phase and two-phase
flow problems, which are challenging because of convection-dominated nature of
the concentration equation
Exploring Fijian high school studentsâ conceptions of averages
This paper focuses on part of a much larger study that explored form five (14 to 16 year-old) studentsâ ideas in statistics. A range of ideas was explored, including the studentsâ ideas about measures of centre and graphical representations. Studentsâ ideas about measures of centre were analysed and categories of responses identified. While students could compute mean and median, they were less competent with tasks that involved constructing meanings for averages. This could be due to an emphasis in the classroom on developing procedural knowledge or to linguistic and contextual problems. Some students used strategies based on prior school and everyday experiences. The paper concludes by suggesting some implications for mathematics education
Fast, adaptive, high order accurate discretization of the Lippmann-Schwinger equation in two dimension
We present a fast direct solver for two dimensional scattering problems,
where an incident wave impinges on a penetrable medium with compact support. We
represent the scattered field using a volume potential whose kernel is the
outgoing Green's function for the exterior domain. Inserting this
representation into the governing partial differential equation, we obtain an
integral equation of the Lippmann-Schwinger type. The principal contribution
here is the development of an automatically adaptive, high-order accurate
discretization based on a quad tree data structure which provides rapid access
to arbitrary elements of the discretized system matrix. This permits the
straightforward application of state-of-the-art algorithms for constructing
compressed versions of the solution operator. These solvers typically require
work, where denotes the number of degrees of freedom. We
demonstrate the performance of the method for a variety of problems in both the
low and high frequency regimes.Comment: 18 page
A numerical method for junctions in networks of shallow-water channels
There is growing interest in developing mathematical models and appropriate
numerical methods for problems involving networks formed by, essentially,
one-dimensional (1D) domains joined by junctions. Examples include hyperbolic
equations in networks of gas tubes, water channels and vessel networks for
blood and lymph in the human circulatory system. A key point in designing
numerical methods for such applications is the treatment of junctions, i.e.
points at which two or more 1D domains converge and where the flow exhibits
multidimensional behaviour. This paper focuses on the design of methods for
networks of water channels. Our methods adopt the finite volume approach to
make full use of the two-dimensional shallow water equations on the true
physical domain, locally at junctions, while solving the usual one-dimensional
shallow water equations away from the junctions. In addition to mass
conservation, our methods enforce conservation of momentum at junctions; the
latter seems to be the missing element in methods currently available. Apart
from simplicity and robustness, the salient feature of the proposed methods is
their ability to successfully deal with transcritical and supercritical flows
at junctions, a property not enjoyed by existing published methodologies.
Systematic assessment of the proposed methods for a variety of flow
configurations is carried out. The methods are directly applicable to other
systems, provided the multidimensional versions of the 1D equations are
available
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