On optimal truncation of divergent series solutions of nonlinear differential systems; Berry smoothing

We prove that for divergent series solutions of nonlinear (or linear) differential systems near a generic irregular singularity, the common prescription of summation to the least term is, if properly interpreted, meaningful and correct, and we extend this method to transseries solutions. In every direction in the complex plane at the singularity (Stokes directions {\em not} excepted) there exists a nonempty set of solutions whose difference from the ``optimally'' (i.e., near the least term) truncated asymptotic series is of the same (exponentially small) order of magnitude as the least term of the series. There is a family of generalized Borel summation formulas $\mathcal{B}$ which commute with the usual algebraic and analytic operations (addition, multiplication, differentiation, etc). We show that there is exactly one of them, $\mathcal{B}_0$, such that for any formal series solution $\tilde{f}$, $\mathcal{B}_0(\tilde{f})$ differs from the optimal truncation of $\tilde{f}$ by at most the order of the least term of $\tilde{f}$. We show in addition that the Berry (1989) smoothing phenomenon is universal within this class of differential systems. Whenever the terms ``beyond all orders'' {\em change} in crossing a Stokes line, these terms vary smoothly on the Berry scale $\arg(x)\sim |x|^{-1/2}$ and the transition is always given by the error function; under the same conditions we show that Dingle's rule of signs for Stokes transitions holds

Retrogressive thaw slumps and active layer detachment slides in the Brooks Range and foothills of northern Alaska: terrain and timing

Thesis (Ph.D.) University of Alaska Fairbanks, 2015Permafrost degradation is widespread throughout the circumpolar north, occurring by multiple modes and mechanisms on many types of landscapes. The pan-Arctic rate of permafrost degradation is reportedly increasing, and permafrost carbon and nitrogen release are likely to be major contributors to global atmospheric greenhouse gas concentrations in coming decades. Locally, liberation of previously frozen substrates, organic materials, and nutrients alters the ecology of receiving streams, causes ecological and hydrobiogeochemical impacts in lake ecosystems, and impacts vegetation through disturbance, nutrient release, and succession on altered surfaces. Understanding the diverse modes of permafrost landscape response to climate, within time and space, is critical to questions of future impacts and feedbacks to climate change. Active layer detachment sliding and retrogressive thaw slumping are important modes of upland permafrost degradation and disturbance throughout the low arctic, and have been linked with climate warming trends, ecosystem impacts, and permafrost carbon release. In the Brooks Range and foothills study region of northwest Alaska, active layer detachment slides and retrogressive thaw slumps are widespread and prominant modes of permafrost degradation. Their distribution has been partially correlated with landscape properties, especially upper permafrost characteristics. However, drivers of active layer detachment slide and retrogressive thaw slump distribution and initiation triggering mechanisms, are poorly understood in this region, and detailed spatial distribution of permafrost characteristics is particularly lacking for the entire area. To better understand retrogressive thaw slump initiation triggers, this research used archived ERS-1 synthetic aperture RADAR data (1997-2010) to determine the year of first detection for 21 active retrogressive thaw slumps in the Noatak Basin, and examined weather records from remote and regional weather stations (1992-2011), along with satellite image-derived seasonal snowpack distribution (2000-2012) for correlations among weather, snowpack duration, and the timing of retrogressive thaw slump initiation. Most slumps first appeared within a 13 month span beginning June of 2004. Early summer 2004 was distinct in the weather records for anomalously warm early thaw-season temperatures, intense rainfall events in May, and unusually early dissipation of the annual snowpack. Results suggest that, regionally, retrogressive thaw slump initiation may be clustered in time, in response to seasonal shifts or anomalous weather events, and that future landscape response to climate change may depend on the nature and timing of climate change as much as overall magnitude. SS_para>The project examined inter-related and co-varying terrain properties at specific sites to identify relationships among terrain properties and permafrost characteristics. Consistent relationships among vegetation, surficial geology and permafrost characteristics were found using multiple factor analysis ordination of empirical data from diverse field sites throughout the region. Ordination results suggest relevant relationships among these factors to support regional-scale spatial analysis of terrain and permafrost properties. Field sites were also found to form consistent groupings from hierarchical clustering of ordination results, suggesting that relationships among these factors remain relevant across diverse gradients of landscape conditions in the region. Several thousand observed feature locations of active layer detachment slides and retrogressive thaw slumps were then used to examine region-wide terrain suitability based on terrain properties including: surficial geology, topography, geomorphology, vegetation and hydrology. Structural equation modeling and integrated terrain unit analyses confirmed and identified the nature and relative strength of relationships among terrain factors explaining observed feature distribution. These results may partially correspond with permafrost ground ice conditions as well, which is further supported by our ordination results. Analysis results drove mapped estimates of terrain suitability for active layer detachment slides and retrogressive thaw slumps across the region, enabling better estimates of permafrost carbon vulnerable to release, and ecosystems potentially impacted by these modes of permafrost degradation. Up to 57% of the study region may contain 'suitable' terrain for one or both of these features. Results support a 'state factor' approach as a useful organizing framework for assessing and describing terrain suitability, and for examining drivers of permafrost characteristics

The Electron Temperature Gradient in the Galactic Disk

We derive the electron temperature gradient in the Galactic disk using a sample of HII regions that spans Galactocentric distances 0--17 kpc. The electron temperature was calculated using high precision radio recombination line and continuum observations for more than 100 HII regions. Nebular Galactocentric distances were calculated in a consistent manner using the radial velocities measured by our radio recombination line survey. The large number of nebulae widely distributed over the Galactic disk together with the uniformity of our data provide a secure estimate of the present electron temperature gradient in the Milky Way. Because metals are the main coolants in the photoionized gas, the electron temperature along the Galactic disk should be directly related to the distribution of heavy elements in the Milky Way. Our best estimate of the electron temperature gradient is derived from a sample of 76 sources for which we have the highest quality data. The present gradient in electron temperature has a minimum at the Galactic Center and rises at a rate of 287 +/- 46 K/kpc. There are no significant variations in the value of the gradient as a function of Galactocentric radius or azimuth. The scatter we find in the HII region electron temperatures at a given Galactocentric radius is not due to observational error, but rather to intrinsic fluctuations in these temperatures which are almost certainly due to fluctuations in the nebular heavy element abundances. Comparing the HII region gradient with the much steeper gradient found for planetary nebulae suggests that the electron temperature gradient evolves with time, becoming flatter as a consequence of the chemical evolution of the Milky Way's disk.Comment: 43 pages, 9 figures (accepted for publication in the ApJ

Deformations with a resonant irregular singularity

I review topics of my talk in Alcal\ue1, inspired by the paper [1]. An isomonodromic system with irregular singularity at z= 1e (and Fuchsian at z=0) is considered, such that z= 1e becomes resonant for some values of the deformation parameters. Namely, the eigenvalues of the leading matrix at z= 1e coalesce along a locus in the space of deformation parameters. I give a complete extension of the isomonodromy deformation theory in this case

High-Mass Star Formation in the Outer Scutum-Centaurus Arm

The Outer Scutum-Centaurus (OSC) spiral arm is the most distant molecular spiral arm in the Milky Way, but until recently little was known about this structure. Discovered by Dame and Thaddeus (2011), the OSC lies $\sim$15 kpc from the Galactic Center. Due to the Galactic warp, it rises to nearly 4$^{\circ}$ above the Galactic Plane in the first Galactic quadrant, leaving it unsampled by most Galactic plane surveys. Here we observe HII region candidates spatially coincident with the OSC using the Very Large Array to image radio continuum emission from 65 targets and the Green Bank Telescope to search for ammonia and water maser emission from 75 targets. This sample, drawn from the WISE Catalog of Galactic HII Regions, represents every HII region candidate near the longitude-latitude (l,v) locus of the OSC. Coupled with their characteristic mid-infrared morphologies, detection of radio continuum emission strongly suggests that a target is a bona fide HII region. Detections of associated ammonia or water maser emission allow us to derive a kinematic distance and determine if the velocity of the region is consistent with that of the OSC. Nearly 60% of the observed sources were detected in radio continuum, and over 20% have ammonia or water maser detections. The velocities of these sources mainly place them beyond the Solar orbit. These very distant high-mass stars have stellar spectral types as early as O4. We associate high-mass star formation at 2 new locations with the OSC, increasing the total number of detected HII regions in the OSC to 12.Comment: 14 pages text and tables + 10 pages supplemental figure

Extension of formal conjugations between diffeomorphisms

We study the formal conjugacy properties of germs of complex analytic diffeomorphisms defined in the neighborhood of the origin of ${\mathbb C}^{n}$. More precisely, we are interested on the nature of formal conjugations along the fixed points set. We prove that there are formally conjugated local diffeomorphisms $\phi, \eta$ such that every formal conjugation $\hat{\sigma}$ (i.e. $\eta \circ \hat{\sigma} = \hat{\sigma} \circ \phi$) does not extend to the fixed points set $Fix (\phi)$ of $\phi$, meaning that it is not transversally formal (or semi-convergent) along $Fix (\phi)$. We focus on unfoldings of 1-dimensional tangent to the identity diffeomorphisms. We identify the geometrical configurations preventing formal conjugations to extend to the fixed points set: roughly speaking, either the unperturbed fiber is singular or generic fibers contain multiple fixed points.Comment: 34 page