3,959 research outputs found
The paradox of the clumps mathematically explained
The lumpy distribution of species along a continuous one-dimensional niche axis recently found by Scheffer and van Nes (Scheffer and van Ness 2006) is explained mathematically. We show that it emerges simply from the eigenvalue and eigenvectors of the community matrix. Both the transient patterns—lumps and gaps between them—as well as the asymptotic equilibrium are explained. If the species are evenly distributed along the niche axis, the emergence of these patterns can be demonstrated analytically. The more general case, of randomly distributed species, shows only slight deviations and is illustrated by numerical simulation. This is a robust result whenever the finiteness of the niche is taken into account: it can be extended to different analytic dependence of the interaction coefficients with the distance on the niche axis (i.e., different kernel interactions), different boundary conditions, etc. We also found that there is a critical value both for the width of the species distribution s and the number of species n below which the clusterization disappear
Interior regularity criteria for suitable weak solutions of the Navier-Stokes equations
We present new interior regularity criteria for suitable weak solutions of
the 3-D Navier-Stokes equations: a suitable weak solution is regular near an
interior point if either the scaled -norm of the velocity
with , , or the -norm of the
vorticity with , , or the
-norm of the gradient of the vorticity with , , , is sufficiently small near
A geometric condition implying energy equality for solutions of 3D Navier-Stokes equation
We prove that every weak solution to the 3D Navier-Stokes equation that
belongs to the class and \n u belongs to localy
away from a 1/2-H\"{o}lder continuous curve in time satisfies the generalized
energy equality. In particular every such solution is suitable.Comment: 10 page
Partial Regularity of solutions to the Four-dimensional Navier-Stokes equations at the first blow-up time
The solutions of incompressible Navier-Stokes equations in four spatial
dimensions are considered. We prove that the two-dimensional Hausdorff measure
of the set of singular points at the first blow-up time is equal to zero.Comment: 19 pages, a comment regarding five or higher dimensional case is
added in Remark 1.3. accepted by Comm. Math. Phy
Spatial correlation as leading indicator of catastrophic shifts
Generic early-warning signals such as increased autocorrelation and variance have been demonstrated in time-series of systems with alternative stable states approaching a critical transition. However, lag times for the detection of such leading indicators are typically long. Here, we show that increased spatial correlation may serve as a more powerful early-warning signal in systems consisting of many coupled units. We first show why from the universal phenomenon of critical slowing down, spatial correlation should be expected to increase in the vicinity of bifurcations. Subsequently, we explore the applicability of this idea in spatially explicit ecosystem models that can have alternative attractors. The analysis reveals that as a control parameter slowly pushes the system towards the threshold, spatial correlation between neighboring cells tends to increase well before the transition. We show that such increase in spatial correlation represents a better early-warning signal than indicators derived from time-series provided that there is sufficient spatial heterogeneity and connectivity in the syste
Biaxial Testing of Elastomers: Experimental Setup, Measurement and Experimental Optimisation of Specimen’s Shape
The present article deals with the setup and the control of a biaxial tension test device for characterising the material properties of elastomers. After a short introduction into the experimental setup a brief explanation of the benefits of a biaxial tension test is given. Furthermore the analysis of this test will be discussed. Therefore, the used optical field measurement by digital image correlation for analysing the strains is shortly introduced to the reader. Additionally, the basic concepts of the calculation of an inverse boundary problem for identifying the material’s parameters are imposed. However the main focus is laid on the experimental optimisation of the specimen’s geometry, whereupon a nearly hyperelastic, incompressible silicone is used to get the experimental results. The resulting geometry will be specially fitted to the requirements of elastomers. The tested geometries and the evaluation of the experiments will be explained as well as the resulting quality factor for the suitability of a specimen’s shape. After all, a short validation of the foregoing considerations will be presented
Climbing Escher's stairs: a way to approximate stability landscapes in multidimensional systems
Stability landscapes are useful for understanding the properties of dynamical
systems. These landscapes can be calculated from the system's dynamical
equations using the physical concept of scalar potential. Unfortunately, for
most biological systems with two or more state variables such potentials do not
exist. Here we use an analogy with art to provide an accessible explanation of
why this happens. Additionally, we introduce a numerical method for decomposing
differential equations into two terms: the gradient term that has an associated
potential, and the non-gradient term that lacks it. In regions of the state
space where the magnitude of the non-gradient term is small compared to the
gradient part, we use the gradient term to approximate the potential as
quasi-potential. The non-gradient to gradient ratio can be used to estimate the
local error introduced by our approximation. Both the algorithm and a
ready-to-use implementation in the form of an R package are provided
Independent analysis of the orbits of Pioneer 10 and 11
Independently developed orbit determination software is used to analyze the
orbits of Pioneer 10 and 11 using Doppler data. The analysis takes into account
the gravitational fields of the Sun and planets using the latest JPL
ephemerides, accurate station locations, signal propagation delays (e.g., the
Shapiro delay, atmospheric effects), the spacecrafts' spin, and maneuvers. New
to this analysis is the ability to utilize telemetry data for spin, maneuvers,
and other on-board systematic effects. Using data that was analyzed in prior
JPL studies, the anomalous acceleration of the two spacecraft is confirmed. We
are also able to put limits on any secondary acceleration (i.e., jerk) terms.
The tools that were developed will be used in the upcoming analysis of recently
recovered Pioneer 10 and 11 Doppler data files.Comment: 22 pages, 5 figures; accepted for publication in IJMP
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