The Modular Homology of Inclusion Maps and Group Actions

Communictated by the Managing Editors Let 0 be a finite set of n elements, R a ring of characteristic p>0 and denote by Mk the R-module with k-element subsets of 0 as basis. The set inclusion map: Mk Mk&1 is the homomorphism which associates to a k-element subset 2 the sum (2)=11+12+}}}+1kof all its (k&1)-element subsets 1i. In this paper we study the chain 0 M 0 M 1 M 2}}}M k M k+1 M k+2}}} (*) arising from. We introduce the notion of p-exactness for a sequence and show that any interval of (*) not including Mn 2 or Mn+1 2 respectively, is p-exact for any prime p>0. This result can be extended to various submodules and quotient modules, and we give general constructions for permutation groups on 0 of order not divisible by p. If an interval of (*) , or an equivalent sequence arising from a permutation group on 0, does include the middle term then proper homologies can occur. In these cases we have determined all corresponding Betti numbers. A further application are p-rank formulae for orbit inclusion matrices. 1996 Academic Press, Inc. 1

Adaptive Gaussian inverse regression with partially unknown operator

This work deals with the ill-posed inverse problem of reconstructing a function $f$ given implicitly as the solution of $g = Af$, where $A$ is a compact linear operator with unknown singular values and known eigenfunctions. We observe the function $g$ and the singular values of the operator subject to Gaussian white noise with respective noise levels $\varepsilon$ and $\sigma$. We develop a minimax theory in terms of both noise levels and propose an orthogonal series estimator attaining the minimax rates. This estimator requires the optimal choice of a dimension parameter depending on certain characteristics of $f$ and $A$. This work addresses the fully data-driven choice of the dimension parameter combining model selection with Lepski's method. We show that the fully data-driven estimator preserves minimax optimality over a wide range of classes for $f$ and $A$ and noise levels $\varepsilon$ and $\sigma$. The results are illustrated considering Sobolev spaces and mildly and severely ill-posed inverse problems

Anelastic dynamo models with variable electrical conductivity: an application to gas giants

The observed surface dynamics of Jupiter and Saturn is dominated by a banded system of zonal winds. Their depth remains unclear but they are thought to be confined to the very outer envelopes where hydrogen remains molecular and the electrical conductivity is small. The dynamo maintaining the dipole-dominated magnetic fields of both gas giants likely operates in the deeper interior where hydrogen assumes a metallic state. Here, we present numerical simulations that attempt to model both the zonal winds and the interior dynamo action in an integrated approach. Using the anelastic version of the MHD code MagIC, we explore the effects of density stratification and radial electrical conductivity variation. The electrical conductivity is mostly assumed to remain constant in the thicker inner metallic region and it decays exponentially towards the outer boundary throughout the molecular envelope. Our results show that the combination of stronger density stratification and weaker conducting outer layer is essential for reconciling dipole dominated dynamo action and a fierce equatorial zonal jet. Previous simulations with homogeneous electrical conductivity show that both are merely exclusive, with solutions either having strong zonal winds and multipolar magnetic fields or weak zonal winds and dipole-dominated magnetic fields. All jets tend to be geostrophic and therefore reach right through the convective shell in our simulations. The particular setup explored here allows a strong equatorial jet to remain confined to the weaker conducting outer region where it does not interfere with the deeper seated dynamo action. The flanking mid to high latitude jets, on the other hand, have to remain faint to yield a strongly dipolar magnetic field. The fiercer jets on Jupiter and Saturn only seem compatible with the observed dipolar fields when they remain confined to a weaker conducting outer layer.Comment: 16 pages, 11 figures, 2 tables, submitted to PEP

Correlated continuous-time random walks: combining scale-invariance with long-range memory for spatial and temporal dynamics

Standard continuous time random walk (CTRW) models are renewal processes in the sense that at each jump a new, independent pair of jump length and waiting time are chosen. Globally, anomalous diffusion emerges through action of the generalized central limit theorem leading to scale-free forms of the jump length or waiting time distributions. Here we present a modified version of recently proposed correlated CTRW processes, where we incorporate a power-law correlated noise on the level of both jump length and waiting time dynamics. We obtain a very general stochastic model, that encompasses key features of several paradigmatic models of anomalous diffusion: discontinuous, scale-free displacements as in Levy flights, scale-free waiting times as in subdiffusive CTRWs, and the long-range temporal correlations of fractional Brownian motion (FBM). We derive the exact solutions for the single-time probability density functions and extract the scaling behaviours. Interestingly, we find that different combinations of the model parameters lead to indistinguishable shapes of the emerging probability density functions and identical scaling laws. Our model will be useful to describe recent experimental single particle tracking data, that feature a combination of CTRW and FBM properties.Comment: 25 pages, IOP style, 5 figure