Scale dependence of cosmological backreaction

Due to the non-commutation of spatial averaging and temporal evolution, inhomogeneities and anisotropies (cosmic structures) influence the evolution of the averaged Universe via the cosmological backreaction mechanism. We study the backreaction effect as a function of averaging scale in a perturbative approach up to higher orders. We calculate the hierarchy of the critical scales, at which 10% effects show up from averaging at different orders. The dominant contribution comes from the averaged spatial curvature, observable up to scales of 200 Mpc. The cosmic variance of the local Hubble rate is 10% (5%) for spherical regions of radius 40 (60) Mpc. We compare our result to the one from Newtonian cosmology and Hubble Space Telescope Key Project data.Comment: 6 pages, 2 figures; v3: substantial modifications, new figure

Rethinking the Micro-Foundation of Opinion Dynamics: Rich Consequences of an Inconspicuous Change

Public opinion formation faces unprecedented challenges such as radicalization, echo chambers, and opinion manipulations. Mathematical modeling plays a fundamental role in understanding how social influence shapes individuals' opinions. Although most opinion dynamics models assume that individuals update their opinions by averaging others' opinions, we point out that this taken-for-granted mechanism features a non-negligible unrealistic implication. By resolving the shortcomings of weighted averaging in the framework of cognitive dissonance theory and network games, we derive a new micro-foundation of opinion dynamics, which happens to be the weighted-median mechanism. Empirical validation indicates that the weighted-median mechanism significantly outperforms the weighted-averaging mechanism in predicting individual opinion shifts. Compared with some widely-studied averaging-based models, the weighted-median model, despite its simplicity in form, replicates more realistic features of opinion dynamics, and exhibits richer phase-transition behavior depending on more delicate and robust network structures. Our new model provides an untouched perspective on the study of opinion formation processes and broadens the applicability of opinion dynamics models

Continuous data assimilation with blurred-in-time measurements of the surface quasi-geostrophic equation

An intrinsic property of almost any physical measuring device is that it makes observations which are slightly blurred in time. We consider a nudging-based approach for data assimilation that constructs an approximate solution based on a feedback control mechanism that is designed to account for observations that have been blurred by a moving time average. Analysis of this nudging model in the context of the subcritical surface quasi-geostrophic equation shows, provided the time-averaging window is sufficiently small and the resolution of the observations sufficiently fine, that the approximating solution converges exponentially fast to the observed solution over time. In particular, we demonstrate that observational data with a small blur in time possess no significant obstructions to data assimilation provided that the nudging properly takes the time averaging into account. Two key ingredients in our analysis are additional boundedness properties for the relevant interpolant observation operators and a non-local Gronwall inequality.Comment: 44 page

Beyond Hebb: Exclusive-OR and Biological Learning

A learning algorithm for multilayer neural networks based on biologically plausible mechanisms is studied. Motivated by findings in experimental neurobiology, we consider synaptic averaging in the induction of plasticity changes, which happen on a slower time scale than firing dynamics. This mechanism is shown to enable learning of the exclusive-OR (XOR) problem without the aid of error back-propagation, as well as to increase robustness of learning in the presence of noise.Comment: 4 pages RevTeX, 2 figures PostScript, revised versio