text:
## Sketches of syntax, a pipeline for Otto

Defining a good continuous integration and delivery pipeline syntax for Otto is one of the most important challenges in the entire project. It is one which I struggled with early in the project almost a year and a half ago. It is a challenge I continue to struggle with today, even as the puzzles pieces start to interlock for the multi-service system I originally imagined Otto to be. Now that I have started writing the parser, the pressure to make some design decisions and play them out to their logical ends is growing. The following snippet compiles to the current Otto intermediate representation and will execute on the current prototype agent implementation:

pipeline {
stages {
stage {
name = 'Build'
steps {
echo '>> Building project'
sh 'make all'
}
}
}
}

“It works!”

The reaction upon sharing this with friends and colleagues on Twitter was largely: “looks like Declarative [Pipeline].” The syntax above is just a simple shim to get the basics working, but the similarities are no accident. Jenkins Pipeline is the best publicly available syntax for describing a CI/CD pipeline (in my not-so-humble opinion). If I didn’t believe that I wouldn’t continue to advocate for its use at every possible turn.

For Otto however, the goal is not to create a Jenkins Pipeline knock-off. In this post I wanted to share some sketches of what I think Otto Pipelines should look like and why. For starters, the README has an incomplete list covering some high-level goals that I have for modeling continuous delivery. The examples/ directory also has a few sample .otto files which I will use for reference throughout this post.

It may also be worth reading my previous post describing Step Libraries if you haven’t already, since they play an integral part in making this syntax “go.”

With the pre-requisites out of the way, let’s walk through some syntax!

## Defining the available tools

use {
stdlib
'./common/slack'
}

Sprinkled at the top of Otto Pipelines is the use block. A common problem with Jenkins Pipeline is that the steps available in the Jenkinsfile is completely dependent on what plugins have been installed on the controller or what Shared Libraries have been configured. The use block effectively brings Otto Step Libraries into scope for the given pipeline. Because Step Libraries require no “controller-side” execution in Otto, each Otto Pipeline can use a completely different sets of steps for users to leverage in their workflow.

Open questions:

• Versioning for step libraries seems like it is worth doing, but what’s the right syntax for expressing it?
• Referring to step libraries by URL could be incredibly useful, but is it worth the complexity?

## Defining the execution environment

Next is declaring the execution environment for a stage/stages:

/* snip */
stage {
name = 'Build'

runtime {
docker {
image = 'ruby:2.6'
}
}

steps {
}
}

Resource allocation is one of the areas I am most excited to explore with Otto, but from a modeling standpoint and an execution standpoint. Jenkins was build before “cloud” was a thing, and arguably before “containers”, depending on whether or not any rabid Solaris users are within earshot. As such it has some pitfalls when mapping pipeline execution to these much more dynamic environments. On the flip side, newer CI/CD systems seem to have all gravitated towards container-all-the-things and typically don’t consider non-container workloads in any form, and will also usually require a Kubernetes clusters just to get started.

runtime {
arch = 'amd64'
linux {
pkgconfig = ['openssl', 'libxml-2.0']
}
python {
version = '~> 3.8.5'
virtualenv = true
}
}

For Otto I want to do better and started thinking about capabilities rather than fixed labels or names. In many cases, I don’t particularly care where my Rust project builds, just so long as it has cargo and an up-to-date stable rustc. Similarly for Python projects, I might need an execution environment with Python 3.x, virtualenv, and libxml2 installed. In most systems that precede Otto, administrators end up defining complex labels which users must know. Another way to paper over this complexity is to say “just bring your own container!” which pushes a lot of work back onto developers, typically leading to one-off Dockerfiles which just take an upstream image and add one or two dependencies.

With a capabilities-oriented model, the pipeline orchestration layer is no longer looking for machines labeled “linux-python” or and then hoping one is available. Instead the orchestrator can be smarter and find any available capacity to meet the capabilities request. I believe this approach can improve on overall system performance and scheduling. An idea which I have floating around as a draft RFC right now is basically to “auction” pipeline tasks to the lowest bidder. When I first started considering this idea, I found this paper titled Efficient Nash Equilibrium Resource Allocation Based on Game Theory Mechanism in Cloud Computing by Using Auction, which will likely guide the implementation of auctioneer quite a bit.

What remains to be seen is whether users are actually interested in expressing the capabilities that would be necessary to make a highly efficient resource auction practical.

Open questions:

• Do most developers think about what their pipeline needs in the same way I think about capabilities?
• How would an administrator define capabilities of a cloud-based VM template?

## Caching

From an operational standpoint, I think the most common problem of any CI system is overuse of remote resources by pipelines. This is not a niche problem, but rather something that affects practically everybody. Some will say “you should be caching and proxying all your remote resources!” which is simply not a practical solution for the vast majority of the users in the ecosystem. Many at users won’t be at organizations large enough to deploy such caching proxies.

stage {
name = 'Build'

cache {
// Create a cache named "gems", immutable for the rest of the Pipeline
gems = ['vendor/']
assets = ['dist/css/', 'dist/js/']
}

/* snip */
}
stage {
name = 'Test'
cache {
use gems
}
}

The cache block is intended to provide pipeline authors with a way to cache arbitrary sections of the workspace for later re-use in the pipeline across multiple agents. This is a pretty simple syntax addition, but something built into the Otto infrastructure from the beginning.

On the implementation side, this requires that archiving and retrieving these artifacts is relatively quick, which I don’t believe will be a major challenge.

Open questions:

• Is it sufficient to cache a file subtree and simply restore it into the same location in another agent’s workspace?
• Would this syntax accommodate the caching of Docker image layers?

## Composition and Re-use

Inevitably developers try to abstract common functionality and behaviors into re-usable components. Step Libraries can provide one flavor of this re-usability, but I don’t believe that it is sufficient. The ubiqitous adoption of YAML by newer CI/CD tools lead me to joke about the five stages of YAML wherein developers end up turning a declarative syntax into templates and then into just another turing-complete language.

In Jenkins we have seen numerous tools for templatizing jobs, pipelines, or other aspects of Jenkins configuration. Suffice it to say, there is a need to compose and re-use various aspects of pipelines.

For Otto, I have been playing around with a context-aware from keyword, such as below:

stage {
name = 'Test'
runtime {
from 'Build'
}
steps {
sh 'bundle exec rake spec'
}
}

In the above example, from instructs the pipeline to re-use the contents of the runtime block from the Build stage. My current thinking is that this simple use of from allows for pipeline-internal re-use of pieces without the need to set variables or turn this into a scripting language.

That said, re-usability within the pipeline isn’t where the main interest in “templates” lies.

I have been exploring the concept of a “blueprint” which can act as an re-usable unit of Otto Pipeline. I am imagining that these would be published and managed similarly to Step Libraries. In order to provide maximum flexibility, I think blueprints should be able to capture just about any snippet of the Otto Pipeline syntax for re-use, consider the following example to help make common Ruby gem build/test/publish pipelines cleaner:.

rubygem.blueprint

use {
stdlib
}

blueprint {
parameters {
rubyVersion {
description = 'Specify the Ruby container'
default = 'ruby'
type = string
}

deploy {
description = 'Push to rubygems.oorg'
default = true
type = boolean
}
}

plan {
stages {
stage('Build') {
runtime {
docker {
image = vars.rubyVersion
}
}

steps {
sh 'bundle install'
sh 'bundle exec rake build'
}
}

stage('Test') {
runtime { from 'Build' }
steps {
sh 'bundle exec rake test'
}
}

stage('Deploy') {
gates { enter { vars.deploy } }
runtime { from 'Build' }
steps {
sh 'bundle exec rake push'
}
}
}
}
}

This would then be later re-used within an Otto Pipeline by using the same from syntax as before:

pipeline {
from 'blueprints/rubygem'

/*
* which would be ordered after the blueprint's stages have completed
*/
}

Since from would be somewhat context-aware and would be able to pull all the right stages “into place” within the pipeline. I’m optimistic that this approach would allow the definition that includes just one stage for example, or other blocks which can be defined within the pipeline { }.

I am not yet sure what the right mechanism for passing parameters into the blueprint should be. Right now I am leaning towards keyword arguments on the from directive: from blueprint: 'blueprints/rubygem', rubyVersion: '2.6', deploy: false.I am not really sure what the implementation complexity of this approach will bring however.

Open Questions:

• Will treating from almost like a preprocessor directive allow the parser to successfully handle blueprints for arbitrary blocks of pipeline?
• Does this amount of composition alleviate the pressure that templates tend to solve for other systmes?

## Gates

The final bit of syntax I wish to discuss at the moment are “gates.” One of the least appreciated parts of just about every CD pipeline, gates define how the pipeline should behave differently under certain conditions, including pausing for user input or an external event.

From one of the modeling goals I had set:

External interactions must be model-able. Deferring control to an external system must be accounted for in a user-defined model. For example, submitting a deployment request, and then waiting for some external condition to be made to indicate that the deployment has completed and the service is now online. This should support both an evented model, wherein the external service “calls back” and a polling model, where the process waits until some external condition can be verified.

A contrived example of what this might look like for a pipeline which prepares a deployment whenever changes land in the main branch:

gates {
enter { branch == 'main' }

/*
* The exit block is where external stimuli back into the system
* should be modeled, providing some means of holding back the pipeline
* until the condition has been met
*/
exit {
input 'Does staging look good to you?'
}
}

Of anything discussed thus far, gates have the most runtime implementation requirements. In the primitive example above we have:

• A Git branch being referenced, which needs to be pulled into scope somehow/somewhere.
• An expression that needs to be evaluated in the service mesh before this stage of the pipeline is dispatched.
• An input step which should allow the agent which executed the stage to deallocate and pause further execution of the pipeline until some external event is provided.

The last item is the most challenging for me to think about from an implementation and modeling standpoint. Somewhere within Otto a state machine for each pipeline must be maintained, and once an input, webhook, or some other step is encountered, the state machine must pause for external actions. How those external actions should be wired in? Not sure! How those steps should be defined? Not sure!

There are so many open questions at this point.

Gates leave me with the most discomfort of any of my ideas for Otto. Done well, gates could provide a key component missing from many existing tools. The challenge is going to be finding the space between the pipeline modeling language and the execution engine which will accommodate them.

I still probably have more questions than answers at this point about how the pipeline modeling syntax should be defined and how it should execute. The one major lesson which I have learned from my time in the Jenkins project is that the pipeline syntax cannot be improved in isolation from the execution environment. There are many key design decisions which need to be made in both domains which will have major repercussions in the other.

I think back to the word used by a developer who read my thoughts on what I want to do with Otto:

Ambitious.”

As always, if you’re curious to learn more, you’re welcome to join #otto on the Freenode IRC network, or follow along on GitHub


text:
# CIPS-3D: A 3D-Aware Generator of GANs Based on Conditionally-Independent Pixel Synthesis

19 Oct 2021  ·  , , , ·

The style-based GAN (StyleGAN) architecture achieved state-of-the-art results for generating high-quality images, but it lacks explicit and precise control over camera poses. The recently proposed NeRF-based GANs made great progress towards 3D-aware generators, but they are unable to generate high-quality images yet... This paper presents CIPS-3D, a style-based, 3D-aware generator that is composed of a shallow NeRF network and a deep implicit neural representation (INR) network. The generator synthesizes each pixel value independently without any spatial convolution or upsampling operation. In addition, we diagnose the problem of mirror symmetry that implies a suboptimal solution and solve it by introducing an auxiliary discriminator. Trained on raw, single-view images, CIPS-3D sets new records for 3D-aware image synthesis with an impressive FID of 6.97 for images at the $256\times256$ resolution on FFHQ. We also demonstrate several interesting directions for CIPS-3D such as transfer learning and 3D-aware face stylization. The synthesis results are best viewed as videos, so we recommend the readers to check our github project at https://github.com/PeterouZh/CIPS-3D read more

PDF Abstract

## Results from the Paper Add Remove

Task Dataset Model Metric Name Metric Value Global Rank Benchmark
3D-Aware Image Synthesis FFHQ 256 x 256 pi-GAN FID 34.56 # 3
KID 26.58 # 3
3D-Aware Image Synthesis FFHQ 256 x 256 StyleNeRF FID 8.00 # 2
KID 3.70 # 2
3D-Aware Image Synthesis FFHQ 256 x 256 GIRAFFE FID 63.33 # 4
KID 50.94 # 4
3D-Aware Image Synthesis FFHQ 256 x 256 CIPS-3D FID 6.97 # 1
KID 2.87 # 1

format: plain


text:
# Contents

## Definition

A topological space $(X,\tau)$ is called metrisable if there exists the stucture of a metric space $(X,d)$ on the underlying set, such that $\tau$ is the corresponding metric topology.

## Metrisability theorem

Various theorems state sufficient conditions for a topological space to be metrisable:

(…)


text:
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# Changes in land use enhance the sensitivity of tropical ecosystems to fire-climate extremes

## Abstract

The Pantanal, the largest contiguous wetland in the world with a high diversity of ecosystems and habitat for several endangered species, was impacted by record-breaking wildfires in 2020. In this study, we integrate satellite and modeling data that enable exploration of natural and human contributing factors to the unprecedented 2020 fires. We demonstrate that the fires were fueled by an exceptional multi-year drought, but dry conditions solely could not explain the spatial patterns of burning. Our analysis reveals how human-caused fires exacerbated drought effects on natural ecosystem within the Pantanal, with large burned fractions primarily over natural (52%), and low cattle density areas (44%) in 2020. The post-fire ecosystem and hydrology changes also had strong ecological effects, with vegetation productivity less than − 1.5 σ over more than 30% of the natural and conservation areas. In contrast to more managed areas, there was a clear decrease in evaporation (by ~ 9%) and an increase in runoff (by ~ 5%) over the natural areas, with long-term impacts on ecosystem recovery and fire risk. This study provides the first tropical evidence outside rainforests of the synergy between climate, land management and fires, and the associated impacts on the ecosystem and hydrology over the largest contiguous wetlands in the world.

## Introduction

Vegetation disturbance from fires is an essential component of savanna and woodland ecosystems1. Fires clear older biomass, release nutrients, and open the canopy, stimulating new vegetation growth2,3,4. Irrespective of the ignition source, fires are often exacerbated by climate conditions such as warmer and drier conditions, drought, and heatwaves which increase vegetation flammability. In the contemporary tropics, most ignitions are from human origin5. Human influence in triggering and redistributing ignitions are well established over different areas6,7, with 84 to 97% of the wildfires in the United States being humans driven8,9, only 13% of the bushfires in Australia from natural causes6, and similar trends in other places in the world10. In contrast to global savanna11, the increase in managed land from deforestation, forest clearing for livestock grazing, and climate-change driven changes in extremes combined with increases in anthropogenic ignitions pose an increasing threat to the sustainability of tropical forests and woodlands12,13,14,15,16,17. Commensurate with the increase in global temperatures, droughts have been more severe18,19,20,21, with approximately 19% increase in the mean fire weather season length globally attributed to climate change variations22. These novel climate conditions have resulted in the emergence of a new fire type, synchronized fire events that affect significant portions of entire landscapes23,24 in different parts of the world. The magnitude of these recent events has fundamental consequences for trajectories of post-fire recovery, through alterations of hydrological fluxes and regional climate.

Coincident with the extreme droughts25,26,27,28,29, biomes in Brazil have been hotspots for forest fires over recent decades22,30. Similar to the fire occurrences in other parts of the world, human initiation is attributed to most fires in the humid tropical forest ecosystems of the Amazon16,31. The industrial agriculture and cattle ranching induced forest clearing with fires32,33 have increased the frequency of understory fires in recent years34 and fire might become the dominant driver of forest degradation under scenarios of future change35.

The Pantanal is a region in central South America consisting of the largest contiguous wetland in the world (Fig. 1), located mostly in the Brazilian state of Mato Grosso do Sul. The Pantanal hosts the highest concentration of wildlife in South America, which is sustained by the floodplains that are submerged in the rainy season and drained in the dry season36. Cattle ranching is a major part of the Pantanal economy, and 93% of the land on the Pantanal is private, which is where the industrialized farming typically occurs. The region also includes protected areas and indigenous lands, over which cattle ranching is minimal37. Despite the long history of the use of fires to clear the area for grazing, large-scale vegetation density in the region has largely remained unaffected by such activities38, particularly over the conservation areas. The Pantanal has been hit by an unprecedented drought since 201839,40, followed by extreme wildfires41,42. Here, we used burned area estimates from the Moderate Resolution Imaging Spectroradiometer (MODIS) sensor aboard the NASA Terra and Aqua satellites (MCD64A143), and found that approximately 3.7 Mha burned, representing 28.9% of the region burned from the 2020 fires (Fig. 1c), corroborating data from the ALARMES warning system44 from the Laboratory for Environmental Satellite Applications (LASA-UFRJ; https://lasa.ufrj.br/alarmes). The exceptional nature of these fires has raised widespread concerns about the conservation of the Pantanal landscape42,45,46, but much remains unknown about the precise drivers of the event and its long-term implications.

In this study, we use state-of-the-art land surface modeling and remote sensing data assimilation techniques to understand processes that led to the 2020 Pantanal fires and their impacts. First, using a land surface modeling integration47 that incorporates information from remote sensing inputs of vegetation changes, we assess the severity, duration, and extent of the recent drought. Second, we use remotely sensed estimates of burned areas to examine the changing influence of drought on fire instances over time and how the historical occurrences of fires increase over the natural areas and reduce over heavily managed regions. We combine model results with data about cattle density and land cover (Fig. 1a-b) to identify the outlier nature of the 2020 fires and interpret its drivers. Finally, we use the land surface model integration to assess the nature of the changes in the ecosystem and regional hydrology as a result of the fires by quantifying the unprecedented nature of the changes in vegetation, ecosystem productivity, and hydrology due to the fire-induced removal of vegetation. Combined, these analyses highlight how human activity in recent years has disproportionately made natural systems of the Pantanal more prone to fire occurrences during extreme droughts.

### The driving factors behind the 2020 Pantanal fires

Over 71% of the region, about 9.2 Mha, face standardized root zone soil moisture anomaly values below − 1 $$\sigma$$, indicating the unprecedented nature of the 2020 drought, both in terms of the magnitude and spatial extent (Fig. 1d). We examine the severity and extent of the recent drought over the Pantanal using deficits in antecedent precipitation and root zone soil moisture as analogs of drought conditions. Precipitation shortages represent the meteorological factors driving the anomalous dry conditions. Deficits of soil moisture resulting from the imbalance between moisture supply on the land surface and the losses from evaporation and runoff are typically used to characterize agricultural droughts48,49,50. Maps of root zone soil moisture anomalies for the dry season (Aug-Nov) across 2017 to 2020 (Fig. S1) indicate that, prior to 2018, the small root zone soil moisture anomalies are representative of the typical dry season. The drought onset is noticed in 2018, and by 2019 much of the region is in consistent negative soil moisture anomalies, exacerbating in 2020.

Historical trends (2003–2020) in MODIS-burned area43 stratified by cattle density (Fig. 2) and land cover type (Fig. S2) highlight the influence of the land use changes on the 2020 fires. Strikingly, we found that under similar drought conditions, natural landscapes were most sensitive to fire extremes. During 2020, 52% (362% increase compared to long-term mean) of protected areas and 44% (357% increase) of areas with very low cattle density burned, compared to 6% (16% increase) and 0.5% (8% decrease) of areas with medium and high cattle density, respectively (Fig. 2e). Similarly, 52% of all forests (692% increase compared to the long-term mean), 31% (814% increase) of wetlands have burned in 2020, compared to 27% (200% increase) of the grass and croplands and 21% (125% increase) of savannas, areas that are, to a large part, used for grazing (Fig. S2e). These patterns are also reflected in the historical time series, while there is a general correspondence between negative root zone soil moisture anomalies and the fire occurrences over the very low-density pasture areas, the strength of this relationship decreases with increasing cattle density (Fig. 2f).

The sensitivity of protected and extensively used land to the 2020 extreme fire occurrence is further highlighted by time series analysis. Figure 2f shows that the correlation between antecedent precipitation and burned area range from − 0.16 to 0.12 and the root zone soil moisture and burned area range from − 0.21 to − 0.03 during 2003–2019. Over the very low pasture areas, the association between precipitation and burned areas increases significantly (from − 0.16 to − 0.24) when 2020 data are considered, whereas no considerable change in the medium and high pasture density areas is seen. These results highlight the extreme and unequal impact of the 2020 drought, but also suggest that the sensitivity of natural ecosystems to fire has increased over time. Indeed, the observed trends in fraction of burned areas in the very low, low, and medium pasture areas were not significant during 2003–2019 (Table 1), but when 2020 data are included, there is a statistically significant increasing trend in the fraction of burned area over the protected and very low-density pasture areas. In contrast, no change is observed over the low and medium density areas where the relationship between burned area and drought remains weak. It is also notable that a statistically significant decreasing trend in the fire occurrences is seen over the high-density pasture areas, which is unaffected by the inclusion or exclusion of the 2020 data, consistent with increasing industrialization and cattle densities over time.

Similar results are obtained with comparisons of the fraction of burned areas and the associated root zone soil moisture anomalies over the four main land cover types (Fig. S2). The forests and wetlands are mostly natural areas whereas the crop, grass, and savanna areas are, to a large part, used for grazing. The natural vegetation areas show significantly higher percent of burns in 2020 compared to such occurrences in the observational record. On the other hand, the fraction of the burned areas over the crop, grass and savanna areas, is similar to the historical period prior to 2020. The anomalous nature of the fires is also seen over the wetlands, where fire occurrences are rare and range around 2%. In 2020, however, more than 10% of the wetlands burned. Our analysis quantifies that as much as 45% of the conserved areas have burn scar features in late 2020, significantly larger than the fire occurrences in the MCD64A1 data record. In the years prior to 2020, the percent of burns in the conserved areas never exceeded 18%.

Patterns similar to Figure 2f are observed when the level of association between antecedent moisture conditions and the fraction of burned area is stratified by the main land cover types and over the conservation and non-conservation areas (Fig. S3). Compared to the 2003–2019 period, there is a statistically significant change in the correlation between antecedent precipitation with the percent of burned area over the forests (a decrease of 36%) and wetlands (a decrease of 52%) when data from year 2020 is included (Fig. S3a). On the other hand, over the crop, grass and savanna areas, the changes in correlation between the antecedent precipitation anomalies and burned area with and without 2020 are not statistically significant. Similar patterns are observed between areas classified as reserves and non-reserves (Fig. S3b). Over the reserves, there is a statistically significant increase in the association between antecedent precipitation and the fraction of burned areas, when the 2020 record is considered, whereas over non-reserves no statistically significant change in the fire activity is observed for 2020.

We also examined the influence of large climate patterns (Fig. S3c) by examining the changes in correlation between the percentage of burned areas with the climate indices of the Atlantic Multidecadal Oscillation (AMO51), the Pacific Decadal Oscillation (PDO52) and the Multivariate El Nino Index (MEI53). The AMO represents the anomalous warming of the Atlantic Ocean, whereas PDO and MEI represent the anomalous warming of the equatorial and eastern tropical Pacific and tropical north Atlantic oceans, respectively. Consistent with prior studies over Amazon31,54 and Pantanal40, there is a general association between these climate indices and the drought and fire instances in the Pantanal. Though these correlation values show a marginal change when the year 2020 is included, compared to the period of 2002–2019, these changes are not statistically significant, suggesting that the increase in fire activity in 2020 is not solely explained by the anomalies in large scale climate variability indices. In contrast to the widely held assumption that remote areas are well protected from ecosystem degradation, our results highlight how the natural environments are more prone to fires under extreme drought conditions and human-induced ignitions, such as in 2020. Under such scenarios, fires spread over large areas without any suppression, increasing the difficulty of controlling fires42. Conservation strategies such as domesticated cattle exclosures will be less effective in such areas, under scenarios of future climate change with more frequent occurrences of dry extremes55.

To confirm the inferences from the correlation analysis, a Random Forest (RF) classifier model using the pasture density, landcover type, standardized anomalies of precipitation, root zone soil moisture, Leaf Area Index (LAI), and Gross Primary Production (GPP) as predictors and the burned area locations as the predictand is used (the larger scale climate indicators are not included given their lack of specific influence in the 2020 fires). The ordering of the importance of the predictors is generally consistent (Fig. S4) with the results from the one-variable at a time regression estimates described above, with the pasture density and the antecedent root zone soil moisture anomalies being the predictors with larger importance values. For example, the drop column importance values for the pasture density and antecedent root zone soil moisture are 0.28 each, higher than 0.21 for precipitation, 0.18 for landcover, 0.04 for LAI and 0.0 for GPP. The relative importance of the pasture density from this analysis also points to the role of human management in the fires over the Pantanal. This suggests human-caused fires often escape beyond their intended area in natural landscapes while controlled fires over areas with more grazing and pasture are not necessarily influenced by dry and drought conditions.

### Ecosystem and hydrology impacts from the 2020 Pantanal fires

The impact of fires on the ecosystem and hydrological conditions had strong ecological effects, though the exceptional drought is largely uniform across the whole region. Except for 2020, the correspondence between drought and negative vegetation anomalies is weak in this region (Fig. 3a; S5a-b), indicating that the Pantanal ecosystem is very resilient to moisture stress conditions. Moreover, while the entire region is in drought, the significant drop in LAI and GPP is primarily observed over areas such as the forests, wetlands, and conservation areas (Fig. 3b), with close correspondence to burned areas during 2020. Over the non-forested areas, a more moderate drop in LAI and GPP is observed. The results of this study also show that changes in precipitation are not the main driver of vegetation variability, consistent with prior studies that indicated that rainfall is not a good predictor of local vegetation structure in areas that flood seasonally56. The correlation between the standardized LAI and GPP anomalies and antecedent precipitation anomalies (Fig. S5a-b) is generally small (< 0.10), particularly in the 2002–2019 period. The significant negative anomalies in precipitation and vegetation states in 2020 increase these associations when the entire period is considered. This pattern is observed in the stratification with the dominant vegetation types and the pasture areas.

Our results show that antecedent vegetation conditions are not a major factor in the development of fires, rather, negative vegetation anomalies are observed following the fires (Fig. S5c-d). Compared to precipitation and root zone soil moisture, the antecedent vegetation anomalies are more weakly correlated with the burn areas. There is no statistically significant relationship between the antecedent LAI and burned areas prior to 2020, whereas the correlation between antecedent GPP and burned areas show little change with and without 2020, both over different land cover and pasture regimes. This is an indication that the potential fuel loads from the negative vegetation anomalies are not a major factor in the fire occurrences. On the other hand, the 2-month leading LAI and GPP anomalies show stronger correlations with the burned areas, indicating that the negative anomalies in vegetation variables follow the fire related burns or are consequences of the fires. The changes in the correlation between leading LAI/GPP and burn area in the 2002–2019 and 2002–2020 are statistically significant over the forests, confirming that the vegetation disturbances follow the fires over these areas. Over the crop, grassland, and savannas, the change in the correlations with and without 2020 is not statistically significant. During the 2003–2019 period, the average % of conservation areas with standardized LAI (GPP) values below − 1.5 $$\sigma$$ is only 3.3 (1.5), whereas in 2020, the % of areas with such large vegetation alterations jump to 35.6 (28.2) (Table 2). Similar large changes in vegetation anomalies are seen in areas with historically very low pasture density and forests, whereas over other areas, more moderate reductions in vegetation productivity are seen.

The annual average precipitation in 2020 was 40% below average fueling the drought and wildfires. A number of precipitation events in late October and early November, 2020 eventually helped in extinguishing the fires. During those rain events, the changes in ET and runoff had strong ecological effects with increasing runoff (with the interquartile range of 3–9%) and reduced ET (with the interquartile range of 0 to − 15%) over the burned areas of forests, conservation, and very low-density pasture (Fig. 3c), confirming that the removal of vegetation due to fire impacted the local hydrological response. Over other areas, the magnitude of such changes is smaller with a net impact of no clear shift in the hydrological regime (Fig. S6). Note that the increasing runoff and reducing ET impact from fires quantified here is likely underestimated, as factors such as water repellence of the soil from fires57, are not represented in our modeling system. As a consequence of the increased runoff, increased erosion and sediment yield can further contribute to the land degradation. The drastic changes in the local hydrology from the 2020 fires in the Pantanal underscore how the combination of droughts and human caused fires can have long lasting effects on the hydrological cycle and regional climate. These changes in turn form part of a positive feedback loop, with drier and warmer conditions, more open cover types, and enhanced fire risk.

### Implications

Understanding the complex feedback mechanisms involving fire, climate, vegetation, and human activity is crucial, but can only be achieved when there is comprehensive knowledge about these patterns. The sensitivity of natural landscapes to fire-driven degradation has been a concern across the southern Amazon for years, but here we demonstrate that the same mechanisms may be more universally applicable across the tropics. This study provides the first tropical evidence outside rainforests of the synergy between climate, land management and fires, and the associated impacts on the ecosystem and hydrology over the largest contiguous wetlands in the world. Though similar levels of drought anomalies have been observed in the past decades, the temporal extent of the 2020 drought exceeds what has been observed in prior years. Nevertheless, climate alone could not explain the unprecedented extent and location of wildfires in the Pantanal, in particularly during 2020, as previous observed over the Amazon rainforest16,31. Here, we demonstrate how cattle density, and associated shifts in land use drive regional fire patterns and its sensitivity to climate. Strikingly, natural areas, not human-dominated landscapes, were most sensitive to fire driven ecosystem degradation. While such conditions have been predicted to occur across parts of the southern Amazon34,35,58,59,60,61, here we provide large-scale evidence from the observational record. In addition, the biome-scale extent of the fires may further slow ecosystem recovery through its impacts on regional hydrology and climate. We observed a 30% decrease in the vegetation productivity, 9% decrease in ET and a 5% increase in runoff across the burned regions presenting an important feedback mechanism12,62,63. Our findings have global implications, as they demonstrate how tropical land management and fires impact the regional water cycle. Combined, these two factors can result in rapid degradation of natural areas previously assumed to be protected by their remoteness. Based on these findings, there is an urgent need to rethink tropical ecosystem conservation and adaptation strategies, using a framework that acknowledges the compounding, cascading and long-term effects of land management and climate. The quantification of such long-term impact assessments is left for a future work.

## Methods

### Modeling and data assimilation setup

A combination of physical modeling and machine learning approaches are employed in this study to examine the anomalous nature of the exceptional 2020 drought over the Pantanal and to identify their contribution to exacerbating the fires in this region. The Noah-MP land surface model (version 4.0.147,64) is forced with the surface meteorology from NASA’s Modern Era Retrospective-Analysis for Research and Applications, version 2 (MERRA265) and precipitation data from NASA’s Global Precipitation Measurement (GPM) Integrated Multi-satellitE Retrievals for GPM (IMERG66). MERRA2 data is available at hourly intervals and approximately about 50-km spatial resolution whereas IMERG provides precipitation data at every 30-minute intervals at, approximately, 10-km spatial resolution. The land surface model (LSM) simulations are conducted at 1-km spatial resolution over the domain shown in Fig. 1. The model configuration employs land cover data (at 1-km resolution) from the modified International Geosphere Biosphere Program (IGBP) Moderate Resolution Imaging Spectroradiometer (MODIS67) and soil parameters (at 1-km resolution) from the International Soil Reference and Information Centre (ISRIC68), and topography (at 90m resolution) from the Multi-Error Removed Improved-Terrain (MERIT69) digital elevation map. Statistical downscaling approaches are used to transform the coarse resolution MERRA2 and IMERG data to 1-km spatial resolution. The input meteorological fields of air temperature, humidity, surface pressure, wind, downward shortwave radiation, and downward longwave radiation are downscaled to 1-km by adjusting for terrain differences in elevation, slope, and aspect70. Precipitation fields at 1-km are generated by using the monthly high resolution WorldClim climatology71 to spatially disaggregate IMERG data to 1-km. The initial conditions for the LSM are generated through a long integration of the Noah-MP model starting in 2000. The model is cycled from year 2000 to 2020 and then reinitialized in 2000. All model integrations are conducted using the open source NASA Land Information System (LIS72) software.

Among other features, Noah-MP LSM includes a dynamic phenology model73 integrated with the Ball-Berry photosynthesis based stomatal resistance model74. In this study, we assimilate the 500m MCD15A2H collection 6 leaf area index (LAI) retrievals from the Moderate Resolution Imaging Spectroradiometer (MODIS) sensors aboard NASA’s Terra and Aqua satellites, using this feature in Noah-MP. Similar to the studies that describe the assimilation of MODIS LAI data75,76, a 1-dimensional Ensemble Kalman Filter (EnKF) algorithm is used to assimilate the LAI retrievals, with a 20 member model ensemble. The ensemble spread is created by applying small perturbations to the model states and meteorological inputs75. During assimilation, the prognostic leaf mass variable in the LSM is updated in response to the LAI inputs from the MODIS data. Before assimilation, the MCD15A2H data is aggregated to 1-km resolution and only data values flagged as ‘good quality’ considering factors such as cloud contamination, algorithm saturation, and detector signal quality are used in assimilation.

To evaluate the impact of the vegetation disturbances from the 2020 Pantanal fires on the local hydrology, an additional integration is conducted where the daily climatological vegetation conditions across the 2002–2019 time period are assimilated into the Noah-MP model (called the LAI-climo DA) during the year 2020. As the difference between the LAI-DA and LAI-climo-DA model runs is solely the vegetation changes from the 2020 fires, these two integrations are used to quantify the influence of the fire driven vegetation changes.

A multi-variable random forest (RF) classifier model77 focusing on the local scale variables, was developed to consider the joint or cross influence of the human management, fuel moisture and fuel load variables to fire occurrences over the Pantanal. RF models are non-parametric and are shown to work well with correlated and conditional variables and are robust in the presence of outliers and noise in the data78,79,80. RF also generates measures of feature importance, a score to indicate the usefulness of the input variable at predicting the target. Three different approaches are used to compute the feature importances generated by the RF model. The native approach with RF models computes the feature importances based on how much each feature decreases the variances of the optimality criteria used in the model. Similarly, the permutation feature importance method computes the feature importances by randomly reshuffling while preserving the distribution of each predictor to assess its influence on the model performance. The third approach, drop-column importance, excludes each feature one at a time and examines how it impacts the predictive accuracy.

The characterization of the extremes in hydrology and ecosystem variables is captured based on the outputs from the model integrations assimilating remotely sensed LAI. Deficits in precipitation and root zone soil moisture from the model integrations are assumed to represent the meteorological and agricultural droughts48,81. Specifically, the 12-month standard anomalies in precipitation and 6-month standardized anomalies in the top 1m root zone soil moisture across 2003–2020 are used as analogs of drought. This 12 month precipitation lag window is chosen based on the convention used at the NOAA National Centers for Environmental Information to characterize long-term and persistent droughts. A six-month and two-month window is chosen for capturing agricultural droughts based on the assumed temporal limits soil moisture and vegetation memory82,83,84, respectively.

### Ancillary datasets

A number of ancillary datasets are used in the analysis and evaluation of the results presented in the article. A map representing the variation in the potential density of cattle head per ranch (certified in the Cadastro Ambiental Rural in the Brazilian Pantanal)37 is used as an index of the level of pasture in this area. This map is estimated by the Mapbiomas project85 (https://mapbiomas.org/), which is a multi-institutional initiative to map the land use dynamics in Brazil and other tropical countries based on information from remote sensing and local information. There are large ranches in this region with 62.3% or more than 5000 ha37. As shown in Fig. 1, we define four categories of pasture with cattle heads per hectare ranging from 0 to 0.03, 0.03 to 0.1, 0.1 to 0.4, and > 0.4 for very low, low, medium, and high pasture density areas, respectively.

The impact of assimilating LAI data on various water and carbon fluxes and states is evaluated by comparing the model simulations with a number of remote sensing-based global reference data products (Extended Analysis 1). Soil moisture states from the model are evaluated by comparing against the Level 2 retrievals from NASA’s Soil Moisture Active Passive (SMAP86) mission. In addition to soil moisture, SMAP also provides estimates of vegetation optical depth (VOD), an analog of the above ground canopy biomass87, derived as part of the radiometric soil moisture retrieval. Here we use both VOD and soil moisture from the SMAP SPL2SMP_E product, which is available at 9-km spatial resolution through Backus-Gilbert interpolation applied to oversampled antenna measurements88. To evaluate the evapotranspiration (ET) estimates, MODIS thermal infrared (TIR) data based ET from the Atmosphere-Land Exchange Inverse (ALEXI89,90) model is used. The 5-km resolution gridded ALEXI ET estimates are used in this study. The subsurface moisture simulations are evaluated by comparing the monthly terrestrial water storage anomalies against those from NASA’s Gravity Recovery and Climate Experiment (GRACE) and GRACE-Follow On (GRACE-FO) satellites. Specifically, we use the Release 06 GRACE91 Level-2 Mass Concentration blocks (mascons) data products available from the Center for Space Research (CSR) at University of Texas. This product is available at 0.25 deg spatial resolution at monthly intervals. The impact of LAI DA on carbon fluxes is evaluated by comparing the simulated Gross Primary Production (GPP), which represents the total carbon fixation by plants through photosynthesis, against two remote sensing-based estimates: (1) remote sensing retrievals of Solar Induced Fluorescence (SIF) from the Global Ozone Monitoring Experiment-2 (GOME-2) aboard the MetOp-A satellite92,93 and (2) MODIS reflectance-based FLUXSAT product94. SIF represents the amount of solar radiation absorbed by chlorophyll and reemitted as fluorescence and is considered as a functional analog of GPP. FLUXSAT is developed through a data driven modeling approach by calibrating the MODIS reflectance information to ground measurements from the FLUXNET network. For uniformity of comparison, all reference data products are interpolated to the 1-km modeling grid in these evaluations.

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## Acknowledgements

The authors would like to thank W.M. Tomas for providing the pasture density map used in this study. Computing resources supporting this work were provided by the NASA High-End Computing (HEC) Program through the NASA Center for Climate Simulation (NCCS) at NASA Goddard Space Flight Center. RL was supported by Project Rede Pantanal (FINEP Grant 01.20.0201.00), CNPq (Grant Number 305159/2018-6) and FAPER (Grant Number E26/202.714/2019).

## Author information

Authors

### Contributions

S.K. and A.G. led the development of the manuscript and model simulations. R.L. assisted with the analysis and interpretations, particularly in the regional context. C.H. and S.M. helped with the use of various remote sensing datasets. N.A. provided insightful help in the structuring of the manuscript and the presentation of the analysis.

### Corresponding author

Correspondence to Sujay Kumar.

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### Competing interests

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Kumar, S., Getirana, A., Libonati, R. et al. Changes in land use enhance the sensitivity of tropical ecosystems to fire-climate extremes. Sci Rep 12, 964 (2022). https://doi.org/10.1038/s41598-022-05130-0

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# Numerical solution of a locally Lipschitz differential equation. (Gross-Pitaevskii)

A (very simplified) version of the GP-equation, can be written as

$$\frac{d\alpha}{dt}=-Ui\alpha^*\alpha^2=-Ui|\alpha|^2\alpha$$,

Where $$i$$ is the imaginary unit. In contrast to what I first thought, this problem seems not one-sided Lipschitz, so that an Implicit Euler method is generally not sufficient to tame numerical instability...; it is locally Lipschitz however.

Are there other implicit or explicit methods that are guaranteed to be numerically stable requiring only local Lipschitzness?


text:
### Beat the Streak: Day Six

In this blog post, I am going to show why the work I did on Day Three is so important, and how using the strategy I outlined in that post can improve your odds of beating the streak by a factor of 5-10!  On day three, I analyzed the situations under which you should select a player who you think will get a hit, as opposed to not selecting that player, and instead maintaining your current streak until the next day.  In summary, I found that your decision should be guided by your current streak, the number of games left in the season, your confidence in the player (how likely is he to get a hit?), and the distribution of likelihoods across all games in the season.  After solving for the optimal strategy, I was able to approximate the probability of winning under that strategy by making some simplifying assumptions about the probability distribution of the best player getting a hit on a given day.

I'm not going to get too deep into the math in this blog post, but I want to give you a little bit of intuition behind the math that went into it.  Let's say that you are very good at making BTS picks, and so 80% of your picks end up getting hits.  Then the probability of correctly picking 57 times in a row is $0.8^{57} \approx 0.000003$.  However, the probability of beating the streak is actually higher than this, because there are many 57-game windows that you could potentially beat the streak in.  The problem essentially reduces to: Consider a 183-length (# days in a season) bit string where each bit is 1 with probability 0.8 and 0 otherwise.  What is the probability that there are 57 1's in a row somewhere in the string?   This question can be answered by dynamic programming, and the probability is about $0.000078$.  Since in beat the streak you can make up to two picks per day, we can approximate that scenario by simply doubling the length of our bit string.  Note that this is an approximation, and to solve it exactly would require a more complex model.  For a bit string of length 350 (about double, but not quite), the probability of having a 57-bit substring of all 1's would be about $0.00018$ -- or 5620 to 1 odds.

If you are smart about when to take risks and when to be conservative by using the strategy I outlined on day three, you can improve your odds by a non-trivial amount.  In order to do that however, you need to have a way to assign a confident to the picks you make.  For example, if you have a method of selecting the best player every day and on average 80% of them end up getting a hit, you would have 5620 to 1 odds of winning using the simple strategy.  However, if you are able to assign a confidence to each of your picks, so that some of them have a better than 80% chance and other have a less than 80% chance of being successful, then you can wait for better opportunities when your streak is longer, but take more risks when your streak is shorter.  So even though your overall prediction accuracy is still 80%, by utilizing this strategy you can improve your odds of eventually obtaining a 57 games streak.

I wanted to get some concrete numbers to test this strategy, so I came up with an experiment where I calculated the probability of beating the streak under four situations:

1. The probability of getting a hit is $p$ every day.
2. The probability of getting a hit on a given day is sampled from a normal distribution with mean $p$ and standard deviation $0.01$ (68% of the time the probability will be within $[p-0.01,p+0.01]$).
3. The probability of getting a hit on a given day is sampled from a normal distribution with mean $p$ and standard deviation $0.02$ (68% of the time the probability will be within $[p-0.02,p+0.02]$).
4. The probability of getting a hit on a given day is sampled from a normal distribution with mean $p$ and standard deviation $0.03$ (68% of the time the probability will be within $[p-0.03,p+0.03]$).

I ran tests for $p$ ranging from $0.6$ to $0.9$.  My current models are around $0.73$ or $0.74$, but I am hoping to improve them to about $0.8$ by next season.  Below is a table summarizing the results.  The value in each cell is the odds of winning (e.g., for situation $1$ with $p=0.8$, the odds are 5620 to 1).

Odds Table
p
std=0.0
std=0.01
std=0.02
std=0.03
0.60
37500000000
14900000000
5460000000
1940000000
0.61
15000000000
6060000000
2270000000
824000000
0.62
6090000000
2510000000
960000000
355000000
0.63
2510000000
1050000000
411000000
155000000
0.64
1050000000
450000000
179000000
68700000
0.65
447000000
195000000
78800000
30900000
0.66
193000000
85400000
35200000
14000000
0.67
84200000
38000000
15900000
6480000
0.68
37300000
17100000
7310000
3020000
0.69
16800000
7820000
3400000
1430000
0.70
7620000
3620000
1600000
684000
0.71
3510000
1700000
762000
331000
0.72
1640000
804000
367000
162000
0.73
773000
386000
179000
80600
0.74
370000
188000
88600
40500
0.75
179000
92300
44300
20600
0.76
87500
45900
22400
10600
0.77
43300
23100
11500
5500
0.78
21700
11800
5930
2890
0.79
11000
6080
3110
1540
0.80
5620
3170
1650
831
0.81
2910
1670
886
454
0.82
1530
895
482
251
0.83
809
484
265
140
0.84
433
265
148
79.6
0.85
235
147
83.4
45.7
0.86
129
82.5
47.7
26.5
0.87
71.2
46.9
27.5
15.5
0.88
39.9
26.9
16.1
9.22
0.89
22.5
15.6
9.41
5.63
0.9
12.8
9.06
5.52
3.77

As you can probably see, this strategy significantly improves your odds of beating the streak, and the magnitude of the improvement increases with the standard deviation.  Without using the strategy, my odds of beating the streak would be about 370,000 to 1 ($p = 0.74$), but using this strategy I can improve them to 40,500 to 1 (assuming the standard deviation is $0.03$).  Similarly, if I ever improve my model to be 80% successful on average, I will be able to improve my odds from 5620 to 1 to 831 to 1 by using this strategy.

I showed in this post how you can non-trivially improve your odds of beating the streak by being smart about when to select 0, 1, or 2 players.  However, in order to use this strategy, you first need to be able to make well calibrated estimations about the likelihood that a particular player will get a hit in a given situation.  This is a very hard problem, and one that I have been struggling to answer for about a year now.  If you have any ideas on how to make well calibrated estimates, or if you would like to reproduce this work/ask any questions, feel free to reach out to me on Google+ or in the comments section below.


text:
Research Article

Moroccan Journal of Pure and Applied Analysis

, 2:8

First online:

Open Access

# Inclusion properties of Generalized Integral Transform using Duality Techniques

• Satwanti DeviAffiliated withDepartment of Applied Sciences, The NorthCap University Email author
• , A. SwaminathanAffiliated withDepartment of Mathematics, Indian Institute of Technology

## Abstract

Let $$W_\beta ^\delta (\alpha ,\gamma )$$ be the class of normalized analytic functions f defined in the region $$\left| z \right|<1$$ and satisfying

$$\text{Re}{{e}^{i\phi }}\left( (1-\alpha +2\gamma ){{(f/z)}^{\delta }}+\left( \alpha -3\gamma +\gamma \left[ (1-1/\delta (z{f}'/f)+1/\delta (1+z{f}''/{f}') \right] \right){{(f/z)}^{\delta }}(z{f}'/f)-\beta \right)>0,$$

with the conditions $$\alpha \ge 0,\beta {\rm{ }} \prec 1,\gamma \ge 0,\delta {\rm{ }} \succ 0$$ and ϕ ∈ ℝ. For a non-negative and realvalued integrable function λ(t) with $$\int_0^1 {\lambda (t)dt = {\rm{1}}}$$, the generalized non-linear integral transform is defined as

$$V_\lambda ^\delta (f)(z) = {\left( {\int_0^1 {\lambda (t){{(f(tz)/t)}^\delta }dt} } \right)^{1/\delta }}.$$

The main aim of the present work is to find conditions on the related parameters such that $$V_\lambda ^\delta (f)(z) \in W_{{\beta _1}}^{{\delta _1}}({\alpha _1},{\gamma _1}),$$ whenever $$f \in W_{{\beta _2}}^{{\delta _2}}({\alpha _2},{\gamma _2}).$$ Further, several interesting applications for specific choices of λ(t) are discussed.

### Key words and phrases

Integral Transforms Analytic functions Hypergeometric functions Convolution Duality techniques


text:
Quotient of a vector space by a linear finite group action

Let the cyclic group $\mathbb{Z}_n$ act on $\mathbb{C}^n$ (or on $\mathbb{R}^n$, I'm interested in both) by permuting coordinates. What does the topological quotient $Q$ by this group action look like? More explicitly, I'd like to identify points under the equivalence relation $x\sim y\Leftrightarrow x=g\cdot y$ for some $g\in\mathbb{Z}_n$. Is there some nice way to embed it as a subset of $\mathbb{C}^m$ for some $m$?

So far all I've thought of is that if $f: \mathbb{C}\to\mathbb{C}$ is any function, $e_f: \mathbb{C}^n\to\mathbb{C}^n$ is the application of $f$ elementwise, and $\mathcal{F}$ is the discrete Fourier transform, then the map $\mathcal{F}_f^n: \mathbb{C}^n\to\mathbb{C}^n$ defined by $x\mapsto \left(\mathcal{F}\left(e_f(x)\right)\right)^n$ is fixed by the group action and so defines a map out of $Q$. It's easy to see that this map is not injective, but perhaps by concatenating $\mathcal{F}_f^n$ for several different $f$ one can get an injective map and by choosing nicely-behaved $f$ one can get a nice embedding.

But I'd bet there are more illuminating embeddings. I'm happy to consider related questions where some bad points are removed from $\mathbb{C}^n$ (such as multiples of the all-ones vector), $n$ is assumed to be prime, etc. I'm also interested in other (abelian, so far) group actions, like $\mathbb{Z}_n\times\mathbb{Z}_n$ acting on $\mathbb{C}^{n\times n}$, so if there is a general theory of such quotient spaces, I'd be interested to learn about it. It seems like this sort of question must be well-studied, but I am not sure where exactly it fits, so feel free to re-tag.

• Do you want the quotient as a topological space? A scheme? A stack...? – Qiaochu Yuan Dec 1 '15 at 23:57
• I would like to understand it as a topological space or even just as a set. Or to understand why that is a bad idea. – Noah Stein Dec 2 '15 at 0:02
• My primary reason for this restriction is that I don't have a great feel for schemes and even less so for stacks. I know that they were created (at least in part) to enable you to talk about groups acting on schemes (or something like that), but I thought maybe when the space being acted on is just affine space, a more down-to-earth answer might be possible. – Noah Stein Dec 2 '15 at 0:08
• Maybe too complicated if you are interested in topological embeddings, but $\mathbb{C}^n$ is an affine variety, the quotient is also an affine variety, so there is an algebraic (hence also topological) embedding of $Q$ into some $\mathbb{C}^m$. Finding an algebraic embedding is equivalent to finding a set of algebra generators for the ring of polynomials in $n$ variables invariant under cyclic permutation of the variables, this question has some good info – Julian Rosen Dec 2 '15 at 2:10
• @MikeShulman: That won't work because (in the $n=2$ case, say) $(0,2)$ should be identified with $(2,0)$ but not with $(1,1)$. – Noah Stein Dec 2 '15 at 19:01

The action of $\mathbb{Z}_n$ on $\mathbb{C}^n$ can be diagonalized: it's conjugate to the action sending a vector $(z_0, z_1, \dots z_{n-1}) \in \mathbb{C}^{n-1}$ to
$$(z_0, \zeta_n z_1, \zeta_n^2 z_2, \dots \zeta_n^{n-1} z_{n-1}).$$
The quotient can be stratified according to which of the $z_i$ are nonzero (starting with $z_1$). On the open stratum where $z_1 \neq 0$, there's a unique representative of each orbit where $0 \le \text{arg}(z_1) < \frac{2\pi}{n}$. In general, on the stratum where $z_1 = \dots = z_{k-1} = 0$ and $z_k \neq 0$, there's a unique representative of each orbit where $0 \le \text{arg}(z_k) < \frac{2 \pi \gcd(k, n)}{n}$. Each stratum lies in the closure of the previous stratum.
There should be a similar and more complicated story for the action of $\mathbb{Z}_n$ on $\mathbb{R}^n$, coming from the decomposition of the latter into real irreducible representations.
• Your orbit representatives aren't quite unique because of the possibility $z_1=0$. – Julian Rosen Dec 2 '15 at 1:24
• I would like to easily define continuous maps out of $Q$, so having a continuous embedding of $Q$ into some vector space would make this trivial. This conjugacy was what brought the Fourier transform into the picture, but it wasn't immediately obvious to me how to choose representatives in a way that everything would be continuous. – Noah Stein Dec 2 '15 at 1:25


text:
#### Volume 19, issue 6 (2015)

 Recent Issues
 The Journal About the Journal Editorial Board Subscriptions Editorial Interests Editorial Procedure Submission Guidelines Submission Page Ethics Statement ISSN (electronic): 1364-0380 ISSN (print): 1465-3060 Author Index To Appear Other MSP Journals
Anti-trees and right-angled Artin subgroups of braid groups

### Sang-hyun Kim and Thomas Koberda

Geometry & Topology 19 (2015) 3289–3306
##### Abstract

We prove that an arbitrary right-angled Artin group $G$ admits a quasi-isometric group embedding into a right-angled Artin group defined by the opposite graph of a tree, and, consequently, into a pure braid group. It follows that $G$ is a quasi-isometrically embedded subgroup of the area-preserving diffeomorphism groups of the $2$–disk and of the $2$–sphere with ${L}^{p}$–metrics for suitable $p$. Another corollary is that there exists a closed hyperbolic manifold group of each dimension which admits a quasi-isometric group embedding into a pure braid group. Finally, we show that the isomorphism problem, conjugacy problem, and membership problem are unsolvable in the class of finitely presented subgroups of braid groups.

##### Keywords
right-angled Artin group, braid group, cancellation theory, hyperbolic manifold, quasi-isometry
##### Mathematical Subject Classification 2010
Primary: 20F36
Secondary: 53D05, 20F10, 20F67


text:
presentation of subgroup and quotient group.

If we have a presentation $\langle x_i: R_j\rangle$ of $G$,

(1) what does the presentation (in terms of the given generators and the given relations) of a subgroup of $G$ looks like? In particular, what does the presentation of a normal subgroup $N$ looks like?

(2) What does the presentation of the quotient group $G/N$ looks like?

It seems that a subgroup should have less generators and the same relations, a quotient group should have same generators but more relations. Is this true?

• Yes you can always find a presentation of a quotient group with the same generators and more relations. But a subgroup is more complicated, and in general it might need more generators, not less. There is a general method called the Reidemeister-Schreier algorithm for finding a presentation of a subgroup. – Derek Holt Oct 31 '16 at 19:27

Also, you can not create all subgroups in this way. A subgroup might in fact need more generators than the whole group. For example in the case without any relations (i.e. the free group), there is a nice (and possibly counter-intuitive) theorem that $\langle x_1,...,x_n:\emptyset\rangle$ is a subgroup of $\langle y_1, y_2 :\emptyset\rangle$ for any $k$.
(2) yes. Just add all relations of $N$ and you get the factor group $G/N$.
• In your answer to (1), you may need to add extra relations to get a presentation of the subgroup generated by a subset of the generators. It is even possible for the original $R$ to be finite, but for the subgroup not to be finitely presented. – Derek Holt Oct 31 '16 at 20:20


text:
# Is the sets of all maps from $\mathbb{N}$ to $\mathbb{N}$ countable? [duplicate]

Is the sets of all maps from $\mathbb{N}$ to $\mathbb{N}$ countable?

Attempts: I think it is uncountable. Consider $P(\mathbb{N})$ where $P(A)$ denote as the power sets of A. We know that $P(\mathbb{N})$ is uncountable. Moreover, by the axiom of choice, for any non empty set, we know that there exist a function $f: P(\mathbb{N})\to \mathbb{N}$ . Note that $P(\mathbb{N})$ have uncountable many elements and since each $f$ belongs to the set of all mapping from $\mathbb{N}$ to $\mathbb{N}$ so there exist infinite uncountably elements and hence this set are uncountable. Is my attempt work??

-

## marked as duplicate by Seirios, Martin Sleziak, Asaf Karagila, Stefan Hansen, Alexander Gruber♦Feb 26 '13 at 11:35

you're right, its uncountable. Don't need axiom of choice. Just use cantors diagonalization argument. The powerset of N is uncountable but it only contains finite sets and countable sets. –  user58512 Feb 26 '13 at 7:52
Do you know about cardinal exponentiation? The cardinality of the set of all maps $\mathbb{N}\to\mathbb{N}$ is $\aleph_0^{\aleph_0}\geq2^{\aleph_0}>\aleph_0$. –  Ben West Feb 26 '13 at 7:54
The axiom of choice really has nothing to do with the uncountablitiy of this set. –  Asaf Karagila Feb 26 '13 at 7:57
Once you know Cantor's theorem, take $\text{id} : \mathbb{N} \to \mathbb{N}$ and notice that for every non-empty $A \in \mathcal{P}(\mathbb{N})$ the restriction $\text{id} \mid_{A} : \mathbb{N} \to \mathbb{N}$ is of the required form. –  J.H. Feb 26 '13 at 8:04

Simply observe that $${\sf{card}} \left( \mathbb{N}^{\mathbb{N}} \right) \geq {\sf{card}} \left( 2^{\mathbb{N}} \right).$$ You can prove $2^{\mathbb{N}}$ to be uncountable without assuming the Axiom of Choice. Just apply Cantor’s Theorem, which states that there does not exist a surjection from a set $X$ onto its power set $\mathcal{P}(X)$ (Cantor’s Theorem can be established strictly within the framework of ${\sf{ZF}}$). In particular, there cannot exist a surjection from $\mathbb{N}$ onto $\mathcal{P}(\mathbb{N})$, which can be easily shown to be equipollent to $2^{\mathbb{N}}$.

-

Inside the set of functions $\def\N{\Bbb N}\N\to\N$ is the subset of functions $\N\to\{0,1\}$ which are precisely the characteristic functions of subsets of $\N$ (as indicated in this answer), and which subsets you already know to be uncountable un number. So certinaly $\N^\N$ is uncountable.

-


text:
# Electric forces and fields

1. Mar 28, 2006

### LezardValeth

a small 2 g ball is suspended by a 20 cm string in a uniform electric field of 1.0 x 10 to the fourth power N/C. If the ball is in equilibrium when the string makes a 15 degree angle with the vertical, what is the net charge on the ball?

I need someone to check my work for me

F=mg (mass times gravity)
F=(2)(9.8)
F = 19.6 N

E = F / q

19.6 / 1.0 x 10^4 = 1.96 x 10 ^-3

q= 1.96 x 10 ^-3

is that right at all ?

2. Mar 28, 2006

### nrqed

It's incorrect because you are not taking into account the tension in the string. And you are solving as if the forces are scalars. But they are vectors. You need to draw a free body diagram, decompose all the forces into x and y components and impose that the net force along the x axis is zero and the same for the net force along y.

Before doing that, you need to specify in what direction the electric field is pointing, though.

3. Mar 28, 2006

### LezardValeth

http://ourworld.cs.com/LezardV4leth/P1010062.JPG

theres the picture in the book along with the directions

the tension on the string you mean friction?

4. Mar 28, 2006

### nrqed

Ok I see now
No, I mean the tension in the string holding the charge. Have you done free body diagrams? Have you done problems in which a string is attached to an object? There is a tension force in the string.

First, you have to decide the sign of the charge.

Then, you will have to draw a FBD, decompose all the forces into x and y components and then impose that $\sum F_x = 0, \sum F_y=0$ and then solve for the charge.

Patrick

5. Mar 28, 2006

### LezardValeth

yes weve done FBD but I think weve only done that with collisions and kinetic energy. I dont recall doing anything with strings besides friction

6. Mar 28, 2006

### nrqed

I don't think we are using the same language at all here

Free body diagrams show all the forces acting on an object (which may be accelerating or not, maybe in motion or at rest). There is no kinetic energy involved, and it's rare that one draws a FBD during a collision (FBD involve forces and not energy or momentum, which is maybe what you are thinking about). Also, a string has nothing to do with friction!

This problem is too hard to explain through the Internet if you haven't worked with free body diagrams a bit before, or at least decomposed forces into x and y components.

7. Mar 28, 2006

### LezardValeth

lol =P I think my teacher just drew the picture and placed arrows or something. I thank you taking the time and helping me with this problem I really appreciate it

heres what I have so far http://ourworld.cs.com/LezardV4leth/13432.jpg

can you explain to me what tension is? or is there a technical term that I can look up in my text book for that?

8. Mar 29, 2006

### Hootenanny

Staff Emeritus
Tension is a force acting on a body in a direction as to produce strain (or a stretching effect).

-Hoot

9. Mar 29, 2006

### nrqed

Tension is just the force exterted by the string on the charge. (here it is acting along the red line in your diagram)

Consider a simpler problem: a mass hanging from a rope (no electric force). The mass is at rest. Clearly the net force on it is zero. There is obviously the gravitational force acting downward. There must be a force acting upward to cancel this. It's obviously produced by the string which is holding the mass. The force exterted by the string on the mass is what we call the tension force. In this case, it is obviously ${\vec T} = mg {\vec j}$ so that $\sum {\vec F} = {\vec T} + {\vec F_{grav}} = 0$. So the magnitude of the tension force, denoted T, is equal to mg in this case.

In your example, it won't be equal to mg because there is another force. You must call the magnitude of the tension force T and treat it as an unknown.

I have to run to classes. I will write more later today.

Patrick


text:
Cytoscape is a well-known bioinformatics tool for displaying and exploring biological networks. R is a powerful programming language and environment for statistical and exploratory data analysis. RCy3 uses CyREST to communicate between R and Cytoscape, allowing any graphs (e.g., igraph, graphNEL or dataframes) to be viewed, explored and manipulated with the Cytoscape point-and-click visual interface. Thus, via RCy3, these two quite different, quite useful bioinformatics software environments are connected, mutually enhancing each other, providing new possibilities for exploring biological data.

# 1 Installation

if(!"RCy3" %in% installed.packages()){
install.packages("BiocManager")
BiocManager::install("RCy3")
}
library(RCy3)

if(!"ndexr" %in% installed.packages()){
install.packages("BiocManager")
BiocManager::install("ndexr")
}
library(ndexr)

# 2 Prerequisites

In addition to these packages, you will need:

# 3 Finding networks

The Network Data Exchange (NDEx) is a platform for network storage, retrieval and exchange. Developed in close collaboration with Cytoscape, it is a natural partner for Cytoscape network queries and management.

ndexcon <- ndex_connect()

We will use ndexcon throughout the other ndexr calls. For example, a basic search.

networks <- ndex_find_networks(ndexcon, "Breast Cancer")
print(networks[,c("name","externalId","nodeCount","edgeCount")])

That print statement provides a nifty way to browse the search results. You’ll notice that we got results that hit each of the search terms individually, thus including any pathway with “cancer” in the name. That’s perhaps a bit too broad…

networks <- ndex_find_networks(ndexcon, "BRCA")
print(networks[,c("name","externalId","nodeCount","edgeCount")])

Ok. We can work with this list. Let’s use the first hit. Note: you are going to get different hits as this database changes over time, so proceed with any hit you like.

networkId = networks\$externalId[1]
network = ndex_get_network(ndexcon, networkId)
print(network)

# 4 Viewing networks

If we want to visualize and interact with this network, Cytoscape is our best choice. Both NDEx and Cytoscape work with the CX network data format, so the import can directly be managed by the CyNDEx component of Cytoscape, supported by RCy3…

brca.net.suid <- importNetworkFromNDEx(networkId)

Note: Do not use the “rcx” frunctions provided by ndexr to pass through igraph in order to import into Cytoscape. This unnecessary intermediate step will result in scientific notation values being translated to type “other” (denoted by “x”) and ultimately converted to Strings in Cytoscape. All style information is lost as well.

# 5 Sending networks to NDEx

Now say you have a network in Cytoscape and you want to send it to NDEx for storage, sharing or exchange. First, you need to have an NDEx account. Go to http://www.ndexbio.org to create one if you haven’t already. You will need your username and password coming up soon.

But first, let’s get a fresh network into Cytoscape. This could be any network, we’re just using the demo igraph network as an example.

ig2 <- makeSimpleIgraph()
net.suid <- createNetworkFromIgraph(ig2, 'Simple Network', 'Simple Collection')

And, now we send it off to NDEx.

user <- "your_NDEx_account_username"  #replace with your info
exportNetworkToNDEx(user, pass, isPublic=TRUE, network=net.suid)


text:
D. Rarity and New Dress
time limit per test
1 second
memory limit per test
256 megabytes
input
standard input
output
standard output

Carousel Boutique is busy again! Rarity has decided to visit the pony ball and she surely needs a new dress, because going out in the same dress several times is a sign of bad manners. First of all, she needs a dress pattern, which she is going to cut out from the rectangular piece of the multicolored fabric.

The piece of the multicolored fabric consists of $n \times m$ separate square scraps. Since Rarity likes dresses in style, a dress pattern must only include scraps sharing the same color. A dress pattern must be the square, and since Rarity is fond of rhombuses, the sides of a pattern must form a $45^{\circ}$ angle with sides of a piece of fabric (that way it will be resembling the traditional picture of a rhombus).

Examples of proper dress patterns: Examples of improper dress patterns: The first one consists of multi-colored scraps, the second one goes beyond the bounds of the piece of fabric, the third one is not a square with sides forming a $45^{\circ}$ angle with sides of the piece of fabric.

Rarity wonders how many ways to cut out a dress pattern that satisfies all the conditions that do exist. Please help her and satisfy her curiosity so she can continue working on her new masterpiece!

Input

The first line contains two integers $n$ and $m$ ($1 \le n, m \le 2000$). Each of the next $n$ lines contains $m$ characters: lowercase English letters, the $j$-th of which corresponds to scrap in the current line and in the $j$-th column. Scraps having the same letter share the same color, scraps having different letters have different colors.

Output

Print a single integer: the number of ways to cut out a dress pattern to satisfy all of Rarity's conditions.

Examples
Input
3 3
aaa
aaa
aaa

Output
10

Input
3 4
abab
baba
abab

Output
12

Input
5 5
zbacg
baaac
aaaaa
eaaad
weadd

Output
31

Note

In the first example, all the dress patterns of size $1$ and one of size $2$ are satisfactory.

In the second example, only the dress patterns of size $1$ are satisfactory.


text:
${{\boldsymbol \Lambda}}$ BARYONS($\boldsymbol S$ = $-1$, $\boldsymbol I$ = 0) ${{\mathit \Lambda}^{0}}$ = ${\mathit {\mathit u}}$ ${\mathit {\mathit d}}$ ${\mathit {\mathit s}}$ INSPIRE search

# ${{\boldsymbol \Lambda}{(2000)}}$ $I(J^P)$ = $0(1/2^{-})$

BARBARO-GALTIERI 1970 (in ${{\mathit \Sigma}}{{\mathit \pi}}$ ) and BRANDSTETTER 1972 (in ${{\mathit \Lambda}}{{\mathit \omega}}$ ) proposed a state at about this mass. Those analyses are considered to be obsolete, see NAKKASYAN 1975 and PDG 2018 .
 ${{\mathit \Lambda}{(2000)}}$ MASS $\approx2000$ MeV
 ${{\mathit \Lambda}{(2000)}}$ WIDTH


text:
# Math Help - integral question

1. ## integral question

h(x)= integral from 0 to e^x of ln(t) dt

solution

let u = e^x then du/dx =e^x

also dh/dx=dh/du du/dx

h'(x)=d/dx integral from 1 to e^x of ln(t) dt = d/du integral from 1 to u ln(t) dt times du/dx = ln u du/dx = (lne^x)*e^x = xe^x

for "d/du integral from 1 to u ln(t) dt times du/dx" above, why is the integral multiplied by the derivative of u WRT x

2. ## Re: integral question

Originally Posted by kingsolomonsgrave
[FONT=Times New Roman][SIZE=2][COLOR=#0e0e0e][FONT=Times New Roman][SIZE=2][COLOR=#0e0e0e]h(x)= integral from 0 to e^x of ln(t) dt

Do you really mean $\int_0^{e^x } {\ln (t)dt}$ or was that a typo and you want $\int_1^{e^x } {\ln (t)dt}~?$

3. ## Re: integral question

the second one, sorry

4. ## Re: integral question

$\frac{d}{dx} \int_a^u f(t) \, dt = f(u) \cdot \frac{du}{dx}$

$\frac{d}{dx} \int_1^{e^x} \ln(t) \, dt = \ln(e^x) \cdot e^x = x \cdot e^x$

5. ## Re: integral question

Originally Posted by kingsolomonsgrave
the second one, sorry

If $h(x)=\int_1^{e^x } {\ln (t)dt}$ then $h'(x)=xe^x~.$

Because $\ln(e^x)=x$ and the derivative of $e^x=e^x~.$


text:
# Quantization of Integrable Systems and Four Dimensional Gauge Theories

@article{Nekrasov2010QuantizationOI,
title={Quantization of Integrable Systems and Four Dimensional Gauge Theories},
author={Nikita A. Nekrasov and Samson L. Shatashvili},
journal={arXiv: High Energy Physics - Theory},
year={2010},
pages={265-289}
}
• Published 27 August 2009
• Physics, Mathematics
• arXiv: High Energy Physics - Theory
We study four dimensional N=2 supersymmetric gauge theory in the Omega-background with the two dimensional N=2 super-Poincare invariance. We explain how this gauge theory provides the quantization of the classical integrable system underlying the moduli space of vacua of the ordinary four dimensional N=2 theory. The epsilon-parameter of the Omega-background is identified with the Planck constant, the twisted chiral ring maps to quantum Hamiltonians, the supersymmetric vacua are identified with…
691 Citations

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