19,150 research outputs found
Lebesgue â–¼-Measure and Riemann â–¼-Integration
Throughout the study of dynamic equations on time-scales, one traditionally finds many theorems involving the â–² case (i.e., â–²-derivatives, â–²-measure, â–²-integrability, etc.) with only a brief mention of the â–¼case. This thesis is designed to give a greater understanding of â–¼-measurability and Riemann â–¼-integration, as well as to give a comparison between â–² and â–¼-measurability. Furthermore, a Mathematica program for finding the Riemann â–¼-integral of a function on various time-scales is provided
Formal vs self-organised knowledge systems: a network approach
In this work we consider the topological analysis of symbolic formal systems
in the framework of network theory. In particular we analyse the network
extracted by Principia Mathematica of B. Russell and A.N. Whitehead, where the
vertices are the statements and two statements are connected with a directed
link if one statement is used to demonstrate the other one. We compare the
obtained network with other directed acyclic graphs, such as a scientific
citation network and a stochastic model. We also introduce a novel topological
ordering for directed acyclic graphs and we discuss its properties in respect
to the classical one. The main result is the observation that formal systems of
knowledge topologically behave similarly to self-organised systems.Comment: research pape
Stretched Exponential Relaxation Arising from a Continuous Sum of Exponential Decays
Stretched exponential relaxation of a quantity n versus time t according to n
= n_0 exp[-(lambda* t)^beta] is ubiquitous in many research fields, where
lambda* is a characteristic relaxation rate and the stretching exponent beta is
in the range 0 < beta < 1. Here we consider systems in which the stretched
exponential relaxation arises from the global relaxation of a system containing
independently exponentially relaxing species with a probability distribution
P(lambda/lambda*,beta) of relaxation rates lambda. We study the properties of
P(lambda/lambda*,beta) and their dependence on beta. Physical interpretations
of lambda* and beta, derived from consideration of P(lambda/lambda*,beta) and
its moments, are discussed.Comment: 8 pages, 10 figures; version to be published in Phys. Rev.
3D Printing A Pendant with A Logo
The purpose of this short paper is to describe a project to manufacture a
3D-print of a pendant that includes a logo. The methods described in this paper
involve processing the image of the logo through a Mathematica script. These
methods can be applied to many logos and other images. With the Mathematica
script, a STereoLithography (.stl) file is created that can be used by a 3D
printer. Finally, the object is created on a 3D printer. We assume that the
reader is familiar with the basics of 3D printing.Comment: 8 pages, 7 figure
Airy Distribution Function: From the Area Under a Brownian Excursion to the Maximal Height of Fluctuating Interfaces
The Airy distribution function describes the probability distribution of the
area under a Brownian excursion over a unit interval. Surprisingly, this
function has appeared in a number of seemingly unrelated problems, mostly in
computer science and graph theory. In this paper, we show that this
distribution also appears in a rather well studied physical system, namely the
fluctuating interfaces. We present an exact solution for the distribution
P(h_m,L) of the maximal height h_m (measured with respect to the average
spatial height) in the steady state of a fluctuating interface in a one
dimensional system of size L with both periodic and free boundary conditions.
For the periodic case, we show that P(h_m,L)=L^{-1/2}f(h_m L^{-1/2}) for all L
where the function f(x) is the Airy distribution function. This result is valid
for both the Edwards-Wilkinson and the Kardar-Parisi-Zhang interfaces. For the
free boundary case, the same scaling holds P(h_m,L)=L^{-1/2}F(h_m L^{-1/2}),
but the scaling function F(x) is different from that of the periodic case. We
compute this scaling function explicitly for the Edwards-Wilkinson interface
and call it the F-Airy distribution function. Numerical simulations are in
excellent agreement with our analytical results. Our results provide a rather
rare exactly solvable case for the distribution of extremum of a set of
strongly correlated random variables. Some of these results were announced in a
recent Letter [ S.N. Majumdar and A. Comtet, Phys. Rev. Lett., 92, 225501
(2004)].Comment: 27 pages, 10 .eps figures included. Two figures improved, new
discussion and references adde
Accurate macroscale modelling of spatial dynamics in multiple dimensions
Developments in dynamical systems theory provides new support for the
macroscale modelling of pdes and other microscale systems such as Lattice
Boltzmann, Monte Carlo or Molecular Dynamics simulators. By systematically
resolving subgrid microscale dynamics the dynamical systems approach constructs
accurate closures of macroscale discretisations of the microscale system. Here
we specifically explore reaction-diffusion problems in two spatial dimensions
as a prototype of generic systems in multiple dimensions. Our approach unifies
into one the modelling of systems by a type of finite elements, and the
`equation free' macroscale modelling of microscale simulators efficiently
executing only on small patches of the spatial domain. Centre manifold theory
ensures that a closed model exist on the macroscale grid, is emergent, and is
systematically approximated. Dividing space either into overlapping finite
elements or into spatially separated small patches, the specially crafted
inter-element/patch coupling also ensures that the constructed discretisations
are consistent with the microscale system/PDE to as high an order as desired.
Computer algebra handles the considerable algebraic details as seen in the
specific application to the Ginzburg--Landau PDE. However, higher order models
in multiple dimensions require a mixed numerical and algebraic approach that is
also developed. The modelling here may be straightforwardly adapted to a wide
class of reaction-diffusion PDEs and lattice equations in multiple space
dimensions. When applied to patches of microscopic simulations our coupling
conditions promise efficient macroscale simulation.Comment: some figures with 3D interaction when viewed in Acrobat Reader. arXiv
admin note: substantial text overlap with arXiv:0904.085
Building Advocacy Capacity: Where Grantees Started
Describes the baseline levels of core advocacy capacities of groups participating in Consumer Voices for Coverage, a twelve-state initiative to build consumer organizations' network and advocacy capacity. Discusses lessons learned and recommendations
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