240 research outputs found
A note on leapfrogging vortex rings
In this paper we provide examples, by numerical simulation using the Navier-Stokes equations for axisymmetric laminar flow, of the 'leapfrogging' motion of two, initially identical, vortex rings which share a common axis of symmetry. We show that the number of clear passes that each ring makes through the other increases with Reynolds number, and that as long as the configuration remains stable the two rings ultimately merge to form a single vortex ring
Historical roots of gauge invariance
Gauge invariance is the basis of the modern theory of electroweak and strong
interactions (the so called Standard Model). The roots of gauge invariance go
back to the year 1820 when electromagnetism was discovered and the first
electrodynamic theory was proposed. Subsequent developments led to the
discovery that different forms of the vector potential result in the same
observable forces. The partial arbitrariness of the vector potential A brought
forth various restrictions on it. div A = 0 was proposed by J. C. Maxwell;
4-div A = 0 was proposed L. V. Lorenz in the middle of 1860's . In most of the
modern texts the latter condition is attributed to H. A. Lorentz, who half a
century later was one of the key figures in the final formulation of classical
electrodynamics. In 1926 a relativistic quantum-mechanical equation for charged
spinless particles was formulated by E. Schrodinger, O. Klein, and V. Fock. The
latter discovered that this equation is invariant with respect to
multiplication of the wave function by a phase factor exp(ieX/hc) with the
accompanying additions to the scalar potential of -dX/cdt and to the vector
potential of grad X. In 1929 H. Weyl proclaimed this invariance as a general
principle and called it Eichinvarianz in German and gauge invariance in
English. The present era of non-abelian gauge theories started in 1954 with the
paper by C. N. Yang and R. L. Mills.Comment: final-final, 34 pages, 1 figure, 106 references (one added with
footnote since v.2); to appear in July 2001 Rev. Mod. Phy
Computational Eulerian Hydrodynamics and Galilean Invariance
Eulerian hydrodynamical simulations are a powerful and popular tool for
modeling fluids in astrophysical systems. In this work, we critically examine
recent claims that these methods violate Galilean invariance of the Euler
equations. We demonstrate that Eulerian hydrodynamics methods do converge to a
Galilean-invariant solution, provided a well-defined convergent solution
exists. Specifically, we show that numerical diffusion, resulting from
diffusion-like terms in the discretized hydrodynamical equations solved by
Eulerian methods, accounts for the effects previously identified as evidence
for the Galilean non-invariance of these methods. These velocity-dependent
diffusive terms lead to different results for different bulk velocities when
the spatial resolution of the simulation is kept fixed, but their effect
becomes negligible as the resolution of the simulation is increased to obtain a
converged solution. In particular, we find that Kelvin-Helmholtz instabilities
develop properly in realistic Eulerian calculations regardless of the bulk
velocity provided the problem is simulated with sufficient resolution (a factor
of 2-4 increase compared to the case without bulk flows for realistic
velocities). Our results reiterate that high-resolution Eulerian methods can
perform well and obtain a convergent solution, even in the presence of highly
supersonic bulk flows.Comment: Version accepted by MNRAS Oct 2, 2009. Figures degraded. For
high-resolution color figures and movies of the numerical simulations, please
visit
http://www.astro.caltech.edu/~brant/Site/Computational_Eulerian_Hydrodynamics_and_Galilean_Invariance.htm
On equal temperament
In this article, I use Stengers’ (2010) concepts of ‘factish’, ‘requirements’ and ‘obligations’, as well as Latour’s (1993) critique of modernity, to interrogate the rise of Equal Temperament as the dominant system of tuning for western music. I argue that Equal Temperament is founded on an unacknowledged compromise which undermines its claims to rationality and universality. This compromise rests on the standardization which is the hallmark of the tuning system of Equal Temperament, and, in this way, it is emblematic of Latour’s definition of modernity. I further argue that the problem of the tuning of musical instruments is one which epitomizes the modern distinction between the natural and the social. In turn, this bears witness to what Whitehead calls the ‘bifurcation of nature’. Throughout this article, using the work of Stengers and Latour, I seek to use tuning as a case study which allows social research to talk both of the natural and of the social aspects of music and tuning, without recourse to essentialism or simple social construction. In this way, my argument seeks to avoid bifurcating nature
Collective Animal Behavior from Bayesian Estimation and Probability Matching
Animals living in groups make movement decisions that depend, among other factors, on social interactions with other group members. Our present understanding of social rules in animal collectives is based on empirical fits to observations and we lack first-principles approaches that allow their derivation. Here we show that patterns of collective decisions can be derived from the basic ability of animals to make probabilistic estimations in the presence of uncertainty. We build a decision-making model with two stages: Bayesian estimation and probabilistic matching.
In the first stage, each animal makes a Bayesian estimation of which behavior is best to perform taking into account personal information about the environment and social information collected by observing the behaviors of other animals. In the probability matching stage, each animal chooses a behavior with a probability given by the Bayesian estimation that this behavior is the most appropriate one. This model derives very simple rules of interaction in animal collectives that depend only on two types of reliability parameters, one that each animal assigns to the other animals and another given by the quality of the non-social information. We test our model by obtaining theoretically a rich set of observed collective patterns of decisions in three-spined sticklebacks, Gasterosteus aculeatus, a shoaling fish species. The quantitative link shown between probabilistic estimation and collective rules of behavior allows a better contact with other fields such as foraging, mate selection, neurobiology and psychology, and gives predictions for experiments directly testing the relationship between estimation and collective behavior
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