Self-organized criticality and the self-organizing map

Abstract

The self-organizing map (SOM), a biologically inspired, learning algorithm from the field of artificial neural networks, is presented as a self-organized critical (SOC) model of the extremal dynamics family. The SOM's ability to converge to an ordered configuration, independent of the initial state, is known and has been demonstrated, in the one-dimensional case. In this ordered configuration it is now indicated by analysis and shown by simulation that the dynamics of the SOM are critical. By viewing the SOM as a SOC system, alternative interpretations of learning, the organized configuration, and the formation of topographic maps can be made.

Constraining torsion with Gravity Probe B

“…As another illustrative example, we will develop in Section IX a family of tetrad theories in Riemann-Cartan space which linearly interpolate between GR and the Hayashi-Shirafuji theory. Although these particular Lagrangeans come with important caveats to which we return below (see also [126]), they show that one cannot dismiss out of hand the possibility that angular momentum sources non-local torsion (see also Table I). Note that the proof [106][107][108] of the oft-repeated assertion that a gyroscope without nuclear spin cannot feel torsion crucially relies on the assumption that orbital angular momentum cannot be the source of torsion.…”

“…After the first version of this paper was submitted, Flanagan and Rosenthal showed that the EinsteinHayashi-Shirafuji Lagrangian has serious defects [126], while leaving open the possibility that there may be other viable Lagrangians in the same class (where spinning objects generate and feel propagating torsion). The EHS Lagrangian should therefore not be viewed as a viable physical model, but as a pedagogical toy model giving concrete illustrations of the various effects and constraints that we discuss.…”

“…Both of these solutions can be embedded in RiemannCartan spacetime, and we will now present a more general two-parameter family of Lagrangians that interpolates between these two extremes, always allowing the Kerr metric and generally explaining the spacetime distortion with a combination of curvature and torsion. After the first version of this paper was submitted, Flanagan and Rosenthal showed that the Einstein-HayashiShirafuji Lagrangian has serious defects [126], while leaving open the possibility that there may be other viable Lagrangians in the same class (where spinning objects generate and feel propagating torsion). This Lagrangian should therefore not be viewed as a viable physical model, but as a pedagogical toy model admitting both curvature and torsion, giving concrete illustrations of the various effects and constraints that we discuss.…”

Some Thoughts on Energy Conditions and Wormholes

“…In fact, QI bounds have now been proven in a number of curved spacetimes as well [27,28,32,34,35,38]. In the short sampling time limit of these bounds, one does in fact recover the original flat spacetime bounds [27,28].…”

“…QI bounds have now been derived for free fields, including the electromagnetic field, the Dirac field, the massless and massive scalar fields, massive spin-one, and RaritaSchwinger fields using arbitrary (smooth) sampling functions [23,24,25,26,27,28,29,30,31,32,33,34,35,36,38,39,40]. These constraints have the form of an uncertainty principle-type bound.…”