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We give a sufficient condition for a Borel subset $E\subset X$ of a complete metric space with $\mathcal{H}^n(E)<\infty$ to be $n$-rectifiable. This condition involves a decomposition of $E$ into rectifiable curves known as an Alberti representation. Precisely, we show that if $\mathcal{H}^n|_E$ has $n$ independent Alberti representations, then $E$ is $n$-rectifiable. This is a sharp strengthening of prior results of Bate and Li. It has been known for some time that such a result answers many open questions concerning rectifiability in metric spaces, which we discuss. An important step of our proof is to establish the higher integrability of measures on Euclidean space satisfying a PDE constraint. These results provide a quantitative generalisation of recent work of De Philippis and Rindler and are of independent interest.

Protein evolution involves mutations occurring across a wide range of time scales. In analogy with other disordered systems, this dynamical heterogeneity suggests strong correlations between mutations happening at distinct sites and times. To quantify these correlations, we examine the role of various fluctuation sources in protein evolution, simulated using a data-driven epistatic landscape. By applying spatio-temporal correlation functions inspired by statistical physics, we disentangle fluctuations originating from the ancestral protein sequence from those driven by stochastic mutations along independent evolutionary paths. Our analysis shows that, in diverse protein families, fluctuations from the ancestral sequence predominate at shorter time scales. This allows us to identify a time scale over which ancestral sequence information persists, enabling its reconstruction. We link this persistence to the strength of epistatic interactions: ancestral sequences with stronger epistatic signatures impact evolutionary trajectories over extended periods. At longer time scales, however, ancestral influence fades as epistatically constrained sites evolve collectively. To confirm this idea, we apply a standard ancestral sequence reconstruction algorithm and verify that the time-dependent recovery error is influenced by the properties of the ancestor itself.

We investigate the question whether the $L^1(\mathbb R)$-norm of the second derivative of the uncentered Hardy-Littlewood maximal function can be bounded by a constant times the $L^1(\mathbb R)$-norm of the function itself. We give a positive answer for a class of functions that contains Sobolev functions on the real line which are decreasing away from the origin and even, and we provide a counterexample which is also decreasing away from the origin but not even., Comment: 26 pages

We introduce a data-driven epistatic model of protein evolution, capable of generating evolutionary trajectories spanning very different time scales reaching from individual mutations to diverged homologs. Our in silico evolution encompasses random nucleotide mutations, insertions and deletions, and models selection using a fitness landscape, which is inferred via a generative probabilistic model for protein families. We show that the proposed framework accurately reproduces the sequence statistics of both short-time (experimental) and long-time (natural) protein evolution, suggesting applicability also to relatively data-poor intermediate evolutionary time scales, which are currently inaccessible to evolution experiments. Our model uncovers a highly collective nature of epistasis, gradually changing the fitness effect of mutations in a diverging sequence context, rather than acting via strong interactions between individual mutations. This collective nature triggers the emergence of a long evolutionary time scale, separating fast mutational processes inside a given sequence context, from the slow evolution of the context itself. The model quantitatively reproduces epistatic phenomena such as contingency and entrenchment, as well as the loss of predictability in protein evolution observed in deep mutational scanning experiments of distant homologs. It thereby deepens our understanding of the interplay between mutation and selection in shaping protein diversity and novel functions, allows one to statistically forecast evolution, and challenges the prevailing independent-site models of protein evolution, which are unable to capture the fundamental importance of epistasis., Comment: 31 pages, 16 figures - final version

Solar radio bursts (SRBs), are emitted by electrons propagating through the corona and interplanetary space. Tracking such bursts is key to understanding the properties of accelerated electrons and radio wave propagation as well as the local plasma environment that they propagate through. Here, we present a novel multilateration algorithm called BayEsian LocaLisation Algorithm (BELLA). In addition, apparent SRB positions from BELLA are compared with comparable localisation methods and the predictions of solar wind models. BELLA uses Bayesian inference to create probabilistic distributions of source positions and their uncertainties. This facilitates the estimation of algorithmic, instrumental, and physical uncertainties in a quantitative manner. We validated BELLA using simulations and a Type III SRB observed by STEREO A/B and Wind. BELLA tracked the Type III source from $\sim$ 10--150 $R_{sun}$ (2-0.15 MHz) along a spiral trajectory. This allowed for an estimate of an apparent solar wind speed of $v_{sw} \sim$ 400 km s$^{-1}$ and a source longitude of $\phi_0 \sim$ 30deg. We compared these results with well-established methods of positioning: Goniopolarimetric (GP), analytical time-difference-of-arrival (TDOA), and Solar radio burst Electron Motion Tracker (SEMP). We found them to be in agreement with the results obtained by BELLA. Additionally, the results aligned with solar wind properties assimilated by the Heliospheric Upwind Extrapolation with time dependence (HUXt) model. We have validated BELLA and used it to identify apparent source positions as well as velocities and densities of the solar wind. Furthermore, we identified higher than expected electron densities, suggesting that the true emission sources were at lower altitudes than those identified by BELLA, an effect that may be due to appreciable scattering of electromagnetic waves by electrons in interplanetary space., Comment: 13 pages, 12 figures, in press, accepted by Astronomy and Astrophysics for publication

The Line Emission Mapper's (LEM's) exquisite spectral resolution and effective area will open new research domains in Astrophysics, Planetary Science and Heliophysics. LEM will provide step-change capabilities for the fluorescence, solar wind charge exchange (SWCX) and auroral precipitation processes that dominate X-ray emissions in our Solar System. The observatory will enable novel X-ray measurements of historically inaccessible line species, thermal broadening, characteristic line ratios and Doppler shifts - a universally valuable new astrophysics diagnostic toolkit. These measurements will identify the underlying compositions, conditions and physical processes from km-scale ultra-cold comets to the MK solar wind in the heliopause at 120 AU. Here, we focus on the paradigm-shifts LEM will provide for understanding the nature of the interaction between a star and its planets, especially the fundamental processes that govern the transfer of mass and energy within our Solar System, and the distribution of elements throughout the heliosphere. In this White Paper we show how LEM will enable a treasure trove of new scientific contributions that directly address key questions from the National Academies' 2023-2032 Planetary Science and 2013-2022 Heliophysics Decadal Strategies. The topics we highlight include: 1. The richest global trace element maps of the Lunar Surface ever produced; insights that address Solar System and planetary formation, and provide invaluable context ahead of Artemis and the Lunar Gateway. 2. Global maps of our Heliosphere through Solar Wind Charge Exchange (SWCX) that trace the interstellar neutral distributions in interplanetary space and measure system-wide solar wind ion abundances and velocities; a key new understanding of our local astrosphere and a synergistic complement to NASA IMAP observations of heliospheric interactions..., Comment: White Paper for the Line Emission Mapper Astrophysics APEX X-ray Probe

Generative probabilistic models emerge as a new paradigm in data-driven, evolution-informed design of biomolecular sequences. This paper introduces a novel approach, called Edge Activation Direct Coupling Analysis (eaDCA), tailored to the characteristics of RNA sequences, with a strong emphasis on simplicity, efficiency, and interpretability. eaDCA explicitly constructs sparse coevolutionary models for RNA families, achieving performance levels comparable to more complex methods while utilizing a significantly lower number of parameters. Our approach demonstrates efficiency in generating artificial RNA sequences that closely resemble their natural counterparts in both statistical analyses and SHAPE-MaP experiments, and in predicting the effect of mutations. Notably, eaDCA provides a unique feature: estimating the number of potential functional sequences within a given RNA family. For example, in the case of cyclic di-AMP riboswitches (RF00379), our analysis suggests the existence of approximately $\mathbf{10^{39}}$ functional nucleotide sequences. While huge compared to the known $< \mathbf{4,000}$ natural sequences, this number represents only a tiny fraction of the vast pool of nearly $\mathbf{10^{82}}$ possible nucleotide sequences of the same length (136 nucleotides). These results underscore the promise of sparse and interpretable generative models, such as eaDCA, in enhancing our understanding of the expansive RNA sequence space., Comment: 33 pages (including SI)

One of the greatest challenge facing current space weather monitoring operations is forecasting the arrival of coronal mass ejections (CMEs) and Solar Energetic Particles (SEPs) within their Earth-Sun propagation timescales. Current campaigns mainly rely on extreme ultra-violet and white light observations to create forecasts, missing out many potential events that may be hazardous to Earth's infrastructure undetectable at these wavelengths. Here we introduce the SURROUND mission, a constellation of CubeSats each with identical radio spectrometers, and the results of the initial Phase-0 study for the concept. The main goal of SURROUND is to monitor and track solar radio bursts (SRBs), widely utilised as a useful diagnostic for space weather activity, and revolutionise current forecasting capabilities. The Phase-0 study concludes that SURROUND can achieve its mission objectives using 3 - 5 spacecraft using current technologies with feasible SEP and CME forecasting potential: a first for heliospheric monitors., Comment: 12 pages, 3 figures, 2 tables, submitted to Planetary, Solar and Heliospheric Radio Emissions IX, held 26-28 September, 2022 [ACCEPTED]; Edited by: C. K. Louis, C. M. Jackman, G. Fischer, A. H. Sulaiman, P. Zucca, Publishers: Dublin Institute for Advanced Studies and Trinity College Dublin. (these authors contributed equally to this work: D. M. Weigt, L. A. Ca\~nizares)

Our main result is a weighted fractional Poincar\'e-Sobolev inequality improving the celebrated estimate by Bourgain-Brezis-Mironescu. This also yields an improvement of the classical Meyers-Ziemer theorem in several ways. The proof is based on a fractional isoperimetric inequality and is new even in the non-weighted setting. We also extend the celebrated Poincar\'e-Sobolev estimate with $A_p$ weights of Fabes-Kenig-Serapioni by means of a fractional type result in the spirit of Bourgain-Brezis-Mironescu. Examples are given to show that the corresponding $L^p$-versions of weighted Poincar\'e inequalities do not hold for $p>1$., Comment: 26 pages, 2 figure. Added Figure 2 and Remark 3.4

Jupiter's magnetosphere is considered to be the most powerful particle accelerator in the Solar System, accelerating electrons from eV to 70 MeV and ions to GeV energies. How electromagnetic processes drive energy and particle flows, producing and removing energetic particles, is at the heart of Heliophysics. Particularly, the 2013 Decadal Strategy for Solar and Space Physics was to "Discover and characterize fundamental processes that occur both within the heliosphere and throughout the universe". The Jovian system offers an ideal natural laboratory to investigate all of the universal processes highlighted in the previous Decadal. The X-ray waveband has been widely used to remotely study plasma across astrophysical systems. The majority of astrophysical emissions can be grouped into 5 X-ray processes: fluorescence, thermal/coronal, scattering, charge exchange and particle acceleration. The Jovian system offers perhaps the only system that presents a rich catalog of all of these X-ray emission processes and can also be visited in-situ, affording the special possibility to directly link fundamental plasma processes with their resulting X-ray signatures. This offers invaluable ground-truths for astrophysical objects beyond the reach of in-situ exploration (e.g. brown dwarfs, magnetars or galaxy clusters that map the cosmos). Here, we show how coupling in-situ measurements with in-orbit X-ray observations of Jupiter's radiation belts, Galilean satellites, Io Torus, and atmosphere addresses fundamental heliophysics questions with wide-reaching impact across helio- and astrophysics. New developments like miniaturized X-ray optics and radiation-tolerant detectors, provide compact, lightweight, wide-field X-ray instruments perfectly suited to the Jupiter system, enabling this exciting new possibility., Comment: A White Paper for the 2024-2033 Solar and Space Physics (Heliophysics) Decadal Survey

During its 53-day polar orbit around Jupiter, Juno often crosses the boundaries of the Jovian magnetosphere (namely the magnetopause and bow shock). From the boundary locations, the upstream solar wind dynamic pressure can be inferred, which in turn illustrates the state of compression or relaxation of the system. The aim of this study is to examine Jovian radio emissions during magnetospheric compressions, in order to determine the relationship between the solar wind and Jovian radio emissions. In this paper, we give a complete list of bow shock and magnetopause crossings (from June 2016 to August 2022), along with some extra informations (e.g. solar wind dynamic pressure and position of the standoff distances inferred from Joy et al. (2002)). We then select two compression events that occur in succession (inferred from magnetopause crossings) and we present a case study of the response of the Jovian radio emissions. We demonstrate that magnetospheric compressions lead to the activation of new radio sources. Newly activated broadband kilometric emissions are observed almost simultaneously to compression of the magnetosphere, with sources covering a large range of longitudes. Decametric emission sources are seen to be activated more than one rotation later only at specific longitudes and dusk local times. Finally, the activation of narrowband kilometric radiation is not observed during the compression phase, but when the magnetosphere is in its expansion phase.

The Line Emission Mapper (LEM) is an X-ray Probe for the 2030s that will answer the outstanding questions of the Universe's structure formation. It will also provide transformative new observing capabilities for every area of astrophysics, and to heliophysics and planetary physics as well. LEM's main goal is a comprehensive look at the physics of galaxy formation, including stellar and black-hole feedback and flows of baryonic matter into and out of galaxies. These processes are best studied in X-rays, and emission-line mapping is the pressing need in this area. LEM will use a large microcalorimeter array/IFU, covering a 30x30' field with 10" angular resolution, to map the soft X-ray line emission from objects that constitute galactic ecosystems. These include supernova remnants, star-forming regions, superbubbles, galactic outflows (such as the Fermi/eROSITA bubbles in the Milky Way and their analogs in other galaxies), the Circumgalactic Medium in the Milky Way and other galaxies, and the Intergalactic Medium at the outskirts and beyond the confines of galaxies and clusters. LEM's 1-2 eV spectral resolution in the 0.2-2 keV band will make it possible to disentangle the faintest emission lines in those objects from the bright Milky Way foreground, providing groundbreaking measurements of the physics of these plasmas, from temperatures, densities, chemical composition to gas dynamics. While LEM's main focus is on galaxy formation, it will provide transformative capability for all classes of astrophysical objects, from the Earth's magnetosphere, planets and comets to the interstellar medium and X-ray binaries in nearby galaxies, AGN, and cooling gas in galaxy clusters. In addition to pointed observations, LEM will perform a shallow all-sky survey that will dramatically expand the discovery space., Comment: 18 pages. White paper for a mission concept to be submitted for the 2023 NASA Astrophysics Probes opportunity. v2: All-sky survey figure expanded, references fixed. v3: Added energy resolution measurements for prototype detector array. v4: Author list and reference fixes

Several strategies have been recently proposed in order to improve Monte Carlo sampling efficiency using machine learning tools. Here, we challenge these methods by considering a class of problems that are known to be exponentially hard to sample using conventional local Monte Carlo at low enough temperatures. In particular, we study the antiferromagnetic Potts model on a random graph, which reduces to the coloring of random graphs at zero temperature. We test several machine-learning-assisted Monte Carlo approaches, and we find that they all fail. Our work thus provides good benchmarks for future proposals for smart sampling algorithms.

Predicting protein-protein interactions from sequences is an important goal of computational biology. Various sources of information can be used to this end. Starting from the sequences of two interacting protein families, one can use phylogeny or residue coevolution to infer which paralogs are specific interaction partners within each species. We show that these two signals can be combined to improve the performance of the inference of interaction partners among paralogs. For this, we first align the sequence-similarity graphs of the two families through simulated annealing, yielding a robust partial pairing. We next use this partial pairing to seed a coevolution-based iterative pairing algorithm. This combined method improves performance over either separate method. The improvement obtained is striking in the difficult cases where the average number of paralogs per species is large or where the total number of sequences is modest., Comment: 19 pages

This contribution focuses on the fascinating RNA molecule, its sequence-dependent folding driven by base-pairing interactions, the interplay between these interactions and natural evolution, and its multiple regulatory roles. The four of us have dug into these topics using the tools and the spirit of the statistical physics of disordered systems, and in particular the concept of a disordered (energy/fitness) landscape. After an introduction to RNA molecules and the perspectives they open not only in evolutionary and synthetic biology but also in medicine, we will introduce the important notions of energy and fitness landscapes for these molecules. In Section III we will review some models and algorithms for RNA sequence-to-secondary-structure mapping. Section IV discusses how the secondary-structure energy landscape can be derived from unzipping data. Section V deals with the inference of RNA structure from evolutionary sequence data sampled in different organisms. This will shift the focus from the `sequence-to-structure' mapping described in Section III to a `sequence-to-function' landscape that can be inferred from laboratory evolutionary data on DNA aptamers. Finally, in Section VI, we shall discuss the rich theoretical picture linking networks of interacting RNA molecules to the organization of robust, systemic regulatory programs. Along this path, we will therefore explore phenomena across multiple scales in space, number of molecules and time, showing how the biological complexity of the RNA world can be captured by the unifying concepts of statistical physics., Comment: 19 pages, 6 figures, to appear in "Spin Glass Theory and Far Beyond - Replica Symmetry Breaking after 40 years" (edited by P Charbonneau, E Marinari, G Parisi, F Ricci Tersenghi, G Sicuro and F Zamponi)

The emergence of new variants of SARS-CoV-2 is a major concern given their potential impact on the transmissibility and pathogenicity of the virus as well as the efficacy of therapeutic interventions. Here, we predict the mutability of all positions in SARS-CoV-2 protein domains to forecast the appearance of unseen variants. Using sequence data from other coronaviruses, pre-existing to SARS-CoV-2, we build statistical models that do not only capture amino-acid conservation but more complex patterns resulting from epistasis. We show that these models are notably superior to conservation profiles in estimating the already observable SARS-CoV-2 variability. In the receptor binding domain of the spike protein, we observe that the predicted mutability correlates well with experimental measures of protein stability and that both are reliable mutability predictors (ROC AUC ~0.8). Most interestingly, we observe an increasing agreement between our model and the observed variability as more data become available over time, proving the anticipatory capacity of our model. When combined with data concerning the immune response, our approach identifies positions where current variants of concern are highly overrepresented. These results could assist studies on viral evolution, future viral outbreaks and, in particular, guide the exploration and anticipation of potentially harmful future SARS-CoV-2 variants., Comment: 21 pages + supplementary information

We consider the maximal operator with respect to uncentered cubes on Euclidean space with arbitrary dimension. We prove that for any function with bounded variation, the variation of its maximal function is bounded by the variation of the function times a dimensional constant. We also prove the corresponding result for maximal operators with respect to collections of more general sets than cubes. The sets are required to satisfy a certain inner cone star condition and in addition the collection must enjoy a tiling property which for example the collection of all cubes does enjoy and the collection of all Euclidean balls does not., Comment: The results are generalized from cubes to \(L\)-complete \(\Lambda\)-stars. Formulations and minor corrections of proofs

Boltzmann machines are energy-based models that have been shown to provide an accurate statistical description of domains of evolutionary-related protein and RNA families. They are parametrized in terms of local biases accounting for residue conservation, and pairwise terms to model epistatic coevolution between residues. From the model parameters, it is possible to extract an accurate prediction of the three-dimensional contact map of the target domain. More recently, the accuracy of these models has been also assessed in terms of their ability in predicting mutational effects and generating in silico functional sequences. Our adaptive implementation of Boltzmann machine learning, adabmDCA, can be generally applied to both protein and RNA families and accomplishes several learning set-ups, depending on the complexity of the input data and on the user requirements. The code is fully available at https://github.com/anna-pa-m/adabmDCA. As an example, we have performed the learning of three Boltzmann machines modeling the Kunitz and Beta-lactamase2 protein domains and TPP-riboswitch RNA domain. The models learned by adabmDCA are comparable to those obtained by state-of-the-art techniques for this task, in terms of the quality of the inferred contact map as well as of the synthetically generated sequences. In addition, the code implements both equilibrium and out-of-equilibrium learning, which allows for an accurate and lossless training when the equilibrium one is prohibitive in terms of computational time, and allows for pruning irrelevant parameters using an information-based criterion.

We prove sharp local and global variation bounds for the centred Hardy--Littlewood maximal functions of indicator functions in one dimension. We characterise maximisers, treat both the continuous and discrete settings and extend our results to a larger class of functions., Comment: 18 pages, 3 figures

Every 19 years, Saturn passes through Jupiter's 'flapping' magnetotail. Here, we report Chandra X-ray observations of Saturn planned to coincide with this rare planetary alignment and to analyse Saturn's magnetospheric response when transitioning to this unique parameter space. We analyse three Director's Discretionary Time (DDT) observations from the High Resolution Camera (HRC-I) on-board Chandra, taken on November 19, 21 and 23 2020 with the aim to find auroral and/or disk emissions. We infer the conditions in the kronian system by looking at coincident soft X-ray solar flux data from the Geostationary Operational Environmental Satellite (GOES) and Hubble Space Telescope (HST) observations of Saturn's ultraviolet (UV) auroral emissions. The large Saturn-Sun-Earth angle during this time would mean that most flares from the Earth-facing side of the Sun would not have impacted Saturn. We find no significant detection of Saturn's disk or auroral emissions in any of our observations. We calculate the 3$\sigma$ upper band energy flux of Saturn during this time to be 0.9 - 3.04 $\times$ 10$^{14}$ erg cm$^{-2}$ s$^{-1}$ which agrees with fluxes found from previous modelled spectra of the disk emissions. We conclude by discussing the implications of this non-detection and how it is imperative that the next fleet of X-ray telescope (such as Athena and the Lynx mission concept) continue to observe Saturn with their improved spatial and spectral resolution and very enhanced sensitivity to help us finally solve the mysteries behind Saturn's apparently elusive X-ray aurora., Comment: 8 pages, 3 figures, accepted for publication in MNRAS

During their evolution, proteins explore sequence space via an interplay between random mutations and phenotypic selection. Here we build upon recent progress in reconstructing data-driven fitness landscapes for families of homologous proteins, to propose stochastic models of experimental protein evolution. These models predict quantitatively important features of experimentally evolved sequence libraries, like fitness distributions and position-specific mutational spectra. They also allow us to efficiently simulate sequence libraries for a vast array of combinations of experimental parameters like sequence divergence, selection strength and library size. We showcase the potential of the approach in re-analyzing two recent experiments to determine protein structure from signals of epistasis emerging in experimental sequence libraries. To be detectable, these signals require sufficiently large and sufficiently diverged libraries. Our modeling framework offers a quantitative explanation for the variable success of recently published experiments. Furthermore, we can forecast the outcome of time- and resource-intensive evolution experiments, opening thereby a way to computationally optimize experimental protocols., Comment: 16 pages, 14 figures

Generative models emerge as promising candidates for novel sequence-data driven approaches to protein design, and for the extraction of structural and functional information about proteins deeply hidden in rapidly growing sequence databases. Here we propose simple autoregressive models as highly accurate but computationally efficient generative sequence models. We show that they perform similarly to existing approaches based on Boltzmann machines or deep generative models, but at a substantially lower computational cost (by a factor between $10^2$ and $10^3$). Furthermore, the simple structure of our models has distinctive mathematical advantages, which translate into an improved applicability in sequence generation and evaluation. Within these models, we can easily estimate both the probability of a given sequence, and, using the model's entropy, the size of the functional sequence space related to a specific protein family. In the example of response regulators, we find a huge number of ca. $10^{68}$ possible sequences, which nevertheless constitute only the astronomically small fraction $10^{-80}$ of all amino-acid sequences of the same length. These findings illustrate the potential and the difficulty in exploring sequence space via generative sequence models., Comment: 12 pages, 4 Figures + Supplementary Material

We establish that the map $f\mapsto |\nabla \mathcal{M}_{\alpha}f|$ is continuous from $W^{1,1}(\mathbb{R}^d)$ to $L^{q}(\mathbb{R}^d)$, where $\alpha\in (0,d)$, $q=\frac{d}{d-\alpha}$ and $\mathcal{M}_{\alpha}$ denotes either the centered or non-centered fractional Hardy--Littlewood maximal operator. In particular, we cover the cases $d >1$ and $\alpha \in (0,1)$ in full generality, for which results were only known for radial functions., Comment: 12 pages, 1 figure

Inverse statistical physics aims at inferring models compatible with a set of empirical averages estimated from a high-dimensional dataset of independently distributed equilibrium configurations of a given system. However, in several applications such as biology, data result from stochastic evolutionary processes, and configurations are related through a hierarchical structure, typically represented by a tree, and therefore not independent. In turn, empirical averages of observables superpose intrinsic signal related to the equilibrium distribution of the studied system and spurious historical (or phylogenetic) signal resulting from the structure underlying the data-generating process. The naive application of inverse statistical physics techniques therefore leads to systematic biases and an effective reduction of the sample size. To advance on the currently open task of extracting intrinsic signals from correlated data, we study a system described by a multivariate Ornstein-Uhlenbeck process defined on a finite tree. Using a Bayesian framework, we can disentangle covariances in the data corresponding to their multivariate Gaussian equilibrium distribution from those resulting from the historical correlations. Our approach leads to a clear gain in accuracy in the inferred equilibrium distribution, which corresponds to an effective two- to fourfold increase in sample size., Comment: 34 pages 10 figures