Torsion, Chern-Simons Term and Diffeomorphism Invariance

In the $torsion \otimes curvature$ approach of gravity Chern-Simons modification has been considered here. It has been found that Chern-Simons contribution to the bianchi identity has become cancelled from that of the scalar field part. But "homogeneity and isotropy" consideration of present day cosmology is a consequence of the "strong equivalence principle" and vice-versa.Comment: 8 page

Absolute Objects and Counterexamples: Jones-Geroch Dust, Torretti Constant Curvature, Tetrad-Spinor, and Scalar Density

James L. Anderson analyzed the novelty of Einstein's theory of gravity as its lack of "absolute objects." Michael Friedman's related work has been criticized by Roger Jones and Robert Geroch for implausibly admitting as absolute the timelike 4-velocity field of dust in cosmological models in Einstein's theory. Using the Rosen-Sorkin Lagrange multiplier trick, I complete Anna Maidens's argument that the problem is not solved by prohibiting variation of absolute objects in an action principle. Recalling Anderson's proscription of "irrelevant" variables, I generalize that proscription to locally irrelevant variables that do no work in some places in some models. This move vindicates Friedman's intuitions and removes the Jones-Geroch counterexample: some regions of some models of gravity with dust are dust-free and so naturally lack a timelike 4-velocity, so diffeomorphic equivalence to (1,0,0,0) is spoiled. Torretti's example involving constant curvature spaces is shown to have an absolute object on Anderson's analysis, viz., the conformal spatial metric density. The previously neglected threat of an absolute object from an orthonormal tetrad used for coupling spinors to gravity appears resolvable by eliminating irrelevant fields. However, given Anderson's definition, GTR itself has an absolute object (as Robert Geroch has observed recently): a change of variables to a conformal metric density and a scalar density shows that the latter is absolute.Comment: Minor editing, small content additions, added references. Forthcoming in_Studies in History and Philosophy of Modern Physics_, June 200

Integrable discretizations of some cases of the rigid body dynamics

A heavy top with a fixed point and a rigid body in an ideal fluid are important examples of Hamiltonian systems on a dual to the semidirect product Lie algebra $e(n)=so(n)\ltimes\mathbb R^n$. We give a Lagrangian derivation of the corresponding equations of motion, and introduce discrete time analogs of two integrable cases of these systems: the Lagrange top and the Clebsch case, respectively. The construction of discretizations is based on the discrete time Lagrangian mechanics on Lie groups, accompanied by the discrete time Lagrangian reduction. The resulting explicit maps on $e^*(n)$ are Poisson with respect to the Lie--Poisson bracket, and are also completely integrable. Lax representations of these maps are also found.Comment: arXiv version is already officia

Incidence of qq-statistics in rank distributions

We show that size-rank distributions with power-law decay (often only over a limited extent) observed in a vast number of instances in a widespread family of systems obey Tsallis statistics. The theoretical framework for these distributions is analogous to that of a nonlinear iterated map near a tangent bifurcation for which the Lyapunov exponent is negligible or vanishes. The relevant statistical-mechanical expressions associated with these distributions are derived from a maximum entropy principle with the use of two different constraints, and the resulting duality of entropy indexes is seen to portray physically relevant information. While the value of the index $\alpha$ fixes the distribution's power-law exponent, that for the dual index $2-\alpha$ ensures the extensivity of the deformed entropy.Comment: Santa Fe Institute working paper: http://www.santafe.edu/media/workingpapers/14-07-024.pdf. see: http://www.pnas.org/content/early/2014/09/03/1412093111.full.pdf+htm

Statistical-thermodynamical foundations of anomalous diffusion

It is shown that Tsallis' generalized statistics provides a natural frame for the statistical-thermodynamical description of anomalous diffusion. Within this generalized theory, a maximum-entropy formalism makes it possible to derive a mathematical formulation for the mechanisms that underly Levy-like superdiffusion, and for solving the nonlinear Fokker-Planck equation.Comment: 13 pages, 8 figures; to appear in special issue of Braz. J. Phys. as invited revie

‘One must eliminate the effects of … diffuse circulation [and] their unstable and dangerous coagulation’: Foucault and beyond the stopping of mobilities

Foucault spent time investigating the stopping of mobilities, notably when studying carceral spaces such as asylums and prisons which effectively immobilise their inmates at a societal scale. In Discipline and Punish, he speculates on how such spaces are designed to put a stop to casual ‘nomadisms’. The purpose here is to inspect this aspect of Foucault’s thinking, particularly to recover what he also said about the regulation and cultivation of mobilities within the depths of immobility. Attention is also drawn to an engagement with mobility-immobility appearing in Foucault’s little-discussed Psychiatric Power lectures, prompted by the ideas and practices of Edouard Seguin, an educator of ‘idiot’ children, whose own words provide additional ‘empirical’ weight to an emerging argument. Reading the unabridged English translation of Madness and Civilization, a final claim is that Foucault’s phenomenology of ‘madness’ depends upon unruly mobilities within the asylum, the very stuff of ‘unstable and dangerous coagulation’. The overall ambition is to furnish an alternative account of Foucault and mobilities, concentrating on those Foucauldian texts initially seeming the least promising for scholars of mobilities

19th century real analysis, forward and backward

19th century real analysis received a major impetus from Cauchy's work. Cauchy mentions variable quantities, limits, and infinitesimals, but the meaning he attached to these terms is not identical to their modern meaning. Some Cauchy historians work in a conceptual scheme dominated by an assumption of a teleological nature of the evolution of real analysis toward a preordained outcome. Thus, Gilain and Siegmund-Schultze assume that references to limite in Cauchy's work necessarily imply that Cauchy was working with an Archi-medean continuum, whereas infinitesimals were merely a convenient figure of speech, for which Cauchy had in mind a complete justification in terms of Archimedean limits. However, there is another formalisation of Cauchy's procedures exploiting his limite, more consistent with Cauchy's ubiquitous use of infinitesimals, in terms of the standard part principle of modern infinitesimal analysis. We challenge a misconception according to which Cauchy was allegedly forced to teach infinitesimals at the Ecole Polytechnique. We show that the debate there concerned mainly the issue of rigor, a separate one from infinitesimals. A critique of Cauchy's approach by his contemporary de Prony sheds light on the meaning of rigor to Cauchy and his contemporaries. An attentive reading of Cauchy's work challenges received views on Cauchy's role in the history of analysis, and indicates that he was a pioneer of infinitesimal techniques as much as a harbinger of the Epsilontik.Comment: 28 pages, to appear in Antiquitates Mathematica