On an origin of numerical diffusion: Violation of invariance under space-time inversion

The invariant properties of the convection equation du/dt + adu/dx = 0 (where d is the partial differential operator) with respect to spatial reflection, time reversal, and space-time inversion are studied. Generally, a finite-difference analog of this equation may possess some or none of these properties. It is shown that, under certain conditions, the von Neumann amplification factor of an analog satisfies a special relation for each invariant property this analog possesses. Particularly, an analog is neutrally stable and thus free of numerical diffusion if it possesses the invariant property related to space-time inversion. It is also explained why generally (1) an upwind scheme possesses neither the invariant property related to spatial reflection nor that related to space-time inversion, and (2) an explicit scheme possesses neither the invariant property related to time reversal nor that related to space-time inversion. Extension to the viscous case and a remarkable connection between the current work and a new numerical framework for solving conservation laws are also discussed

Tetrahedron and 3D reflection equations from quantized algebra of functions

Soibelman's theory of quantized function algebra A_q(SL_n) provides a representation theoretical scheme to construct a solution of the Zamolodchikov tetrahedron equation. We extend this idea originally due to Kapranov and Voevodsky to A_q(Sp_{2n}) and obtain the intertwiner K corresponding to the quartic Coxeter relation. Together with the previously known 3-dimensional (3D) R matrix, the K yields the first ever solution to the 3D analogue of the reflection equation proposed by Isaev and Kulish. It is shown that matrix elements of R and K are polynomials in q and that there are combinatorial and birational counterparts for R and K. The combinatorial ones arise either at q=0 or by tropicalization of the birational ones. A conjectural description for the type B and F_4 cases is also given.Comment: 26 pages. Minor correction

Power as Control and the Therapeutic Effects of Hegel’s Logic

Rather than approaching the question of the constructive or therapeutic character of Hegel’s Logic through a global consideration of its argument and its relation to the rest of Hegel’s system, I want to come at the question by considering a specific thread that runs through the argument of the Logic, namely the question of the proper understanding of power or control. What I want to try to show is that there is a close connection between therapeutic and constructive elements in Hegel’s treatment of power. To do so I will make use of two deep criticisms of Hegel’s treatment from Michael Theunissen. First comes Theunissen’s claim that in Hegel’s logical scheme, reality is necessarily dominated by the concept rather than truly reciprocally related to it. Then I will consider Theunissen’s structurally analogous claim that for Hegel, the power of the concept is the management of the suppression of the other. Both of these claims are essentially claims about the way in which elements of the logic of reflection are modified and yet continue to play a role in the logic of the concept

The Local Field Potential Reflects Surplus Spike Synchrony

The oscillatory nature of the cortical local field potential (LFP) is commonly interpreted as a reflection of synchronized network activity, but its relationship to observed transient coincident firing of neurons on the millisecond time-scale remains unclear. Here we present experimental evidence to reconcile the notions of synchrony at the level of neuronal spiking and at the mesoscopic scale. We demonstrate that only in time intervals of excess spike synchrony, coincident spikes are better entrained to the LFP than predicted by the locking of the individual spikes. This effect is enhanced in periods of large LFP amplitudes. A quantitative model explains the LFP dynamics by the orchestrated spiking activity in neuronal groups that contribute the observed surplus synchrony. From the correlation analysis, we infer that neurons participate in different constellations but contribute only a fraction of their spikes to temporally precise spike configurations, suggesting a dual coding scheme of rate and synchrony. This finding provides direct evidence for the hypothesized relation that precise spike synchrony constitutes a major temporally and spatially organized component of the LFP. Revealing that transient spike synchronization correlates not only with behavior, but with a mesoscopic brain signal corroborates its relevance in cortical processing.Comment: 45 pages, 8 figures, 3 supplemental figure

A semi-Lagrangian scheme for Hamilton–Jacobi–Bellman equations with oblique derivatives boundary conditions

We investigate in this work a fully-discrete semi-Lagrangian approximation of second order possibly degenerate Hamilton–Jacobi–Bellman (HJB) equations on a bounded domain O⊂ RN (N= 1 , 2 , 3) with oblique derivatives boundary conditions. These equations appear naturally in the study of optimal control of diffusion processes with oblique reflection at the boundary of the domain. The proposed scheme is shown to satisfy a consistency type property, it is monotone and stable. Our main result is the convergence of the numerical solution towards the unique viscosity solution of the HJB equation. The convergence result holds under the same asymptotic relation between the time and space discretization steps as in the classical setting for semi-Lagrangian schemes on O= RN. We present some numerical results, in dimensions N=1,2, on unstructured meshes, that confirm the numerical convergence of the scheme

Dispersion relations for stationary light in one-dimensional atomic ensembles

We investigate the dispersion relations for light coupled to one-dimensional ensembles of atoms with different level schemes. The unifying feature of all the considered setups is that the forward and backward propagating quantum fields are coupled by the applied classical drives such that the group velocity can vanish in an effect known as "stationary light". We derive the dispersion relations for all the considered schemes, highlighting the important differences between them. Furthermore, we show that additional control of stationary light can be obtained by treating atoms as discrete scatterers and placing them at well defined positions. For the latter purpose, a multi-mode transfer matrix theory for light is developed