text:
Researchers detect early signs of autism in at-risk infants
By Folio on January 7, 2021 No Comment
Some of the earliest signs apparent quite early in life – for many children even by the first birthday
Medical researchers reliably predicted autism spectrum disorder (ASD) in a large sample of at-risk children by identifying and tracking early behavioural signs at 12 months of age, according to study findings published Dec. 25 in Child Development.
Lonnie Zwaigenbaum
"This research really reinforces what parents have been telling us for so many years," said principal investigator Lonnie Zwaigenbaum, pediatrician and director of the Autism Research Centre within the University of Alberta's Faculty of Medicine & Dentistry.
"Some of the earliest signs may be subtle but they are apparent quite early in life – for many children even by the first birthday," said Zwaigenbaum, who holds the Stollery Children's Hospital Foundation Chair in Autism Research and is a Stollery Science Lab Distinguished Researcher.
Zwaigenbaum said specialist care can be difficult to access and a diagnosis of autism spectrum disorder is often not confirmed until age four or five, which is why being able to observe the signs before age three is key to earlier interventions and better outcomes.
"A timely diagnosis of ASD and referral for intensive behavioural and education interventions at the earliest possible age may lead to better long-term outcomes by capitalizing on the brain's neuroplasticity at younger ages," the researchers reported.
Early signs foreshadow later diagnoses
The team evaluated 681 infants using a Canadian-developed tool called the Autism Observation Scale for Infants (AOSI) to rate motor control and a broad range of behaviours such as eye contact, response to names, expression of positive emotion and interest in toys. The babies were followed at six, nine, 12, 15 and 18 months and then assessed at two and three years of age to determine which participants met criteria for diagnosis of autism spectrum disorder.
Five hundred of the children were younger siblings of children with ASD, considered nearly 20 times more likely to develop the syndrome than children in the general population.
The researchers found elevated AOSI scores at 12 months accurately identified 52 per cent of the children who were subsequently diagnosed with autism by age three. At six months, the only behaviour associated with a later autism diagnosis was reduced motor control.
The 'A' words – autism and advocacy – go hand in hand by Sharon McCarry
"By a year of age, we see a range of behaviours that really do map onto what we are looking for in older children, such as not responding when their name is called, not looking the parent in the eye, not smiling and engaging with parents and other people," Zwaigenbaum reported.
Zwaigenbaum pointed out that so far the results apply only to children who are at higher risk for autism because of the diagnosis of an older sibling, and the AOSI tool must be used in combination with others to confirm a diagnosis.
Next steps for his research include testing the tool and a related parent questionnaire in the general population and sharing diagnostic knowledge with community pediatricians so families don't face long waits for specialist care.
Encouraging 'joyful interactions'
Interventions for children showing signs of autism often focus on coaching parents on how to encourage their child to communicate and make more social connections.
"This approach recognizes the central role of the parent in helping their children learn as they develop," Zwaigenbaum said. "It's not just a brief session with a therapist, but instead thinking about how to incorporate learning opportunities throughout daily life through play, caregiving and new experiences.
"Children can benefit even after their parents receive a relatively brief 12 weeks of coaching," he added. "It's really about helping children develop functional skills so they can have joyful interactions with their family and the other people in their lives."
Autism affects roughly one in 66 Canadians and can lead to lifelong issues with social interactions, repetitive behaviours and unusual sensory sensitivities.
| By Gillian Rutherford
© Troy Media
Autism, Child care, Parenting, University of Alberta
Researchers detect early signs of autism in at-risk infants added by Folio on January 7, 2021
View all posts by Folio →


Defender 130 High Capacity Pick Up.
Vehicles affected by this recall may also be affected by 2017/16231 (https://www.productsafety.gov.au/recall/jaguar-land-rover-australia-2015...).
Certain fixings used in the front and rear hub assemblies may have overly brittle bolts installed. Should brittle bolts be present, there is a risk they could fracture.
If the defect occurs, other bolt fixings may be damaged. In the case of the front brakes, the fixings could puncture the front brake disc. Wheel lock up or braking instability could significantly increase the risk of a crash.
Owners will be notified and instructed to take their vehicles to a Land Rover authorised repairer who will replace all of the specific bolts within the wheel hub assembly. This will be completed on both front and rear wheels. If damage is observed the dealer will replace all associated components.


Deciduous perennial. Lovely, low-growing, groundcover for shady areas. Soft-green, scalloped foliage holds droplets of water giving them a silvery look. Delicate yellow flowers sprays extend above the foliage in summer. Moderate water needs once established. Adaptable to various soil conditions. Hardy to 0°F and below.


I'm a fierce and loyal friend!
Thank you so much, Mike! I bought a cheap bookcase-like piece w cubbies instead of shelves, but it's painted white & I want to distress it. Your Instructable came at exactly the right time! Don'tcha LUV it when that happens?!!
I luv the qtip idea! Thx 4 sharing!
So cool! I LUV books & the ones I used to recycle (or donate) can be used to make these beautiful flowers! Firstly, I can create for my own home (practice makes perfect, don'tcha know!). Secondly, to sell so that proceeds can be donated to literacy programs. Can't wait to try this. Thanks for sharing.


2) Light and Fluffy Spinach Quiche You can substitute any fresh or frozen vegetable you like for the spinach in this easy-to-make quiche. Watch the video to see how it all comes together.
10/06/2015�� Learn how to make a creamy and delicious mayonnaise without eggs. You only need 5 ingredients and a few minutes. You only need 5 ingredients and a few minutes. I don�t know about y�all, but for me, nothing says summer more than a potato salad, chicken salad, pasta salad or tuna salad.
Dairy Free Homemade Creamy Mayonnaise No Egg Recipe is very easy to make with only 4 basic ingredients and a blender. Using a high speed blender is the preferred choice to achieve the smooth creamy mayonnaise.
For anyone with an egg allergy, please note that while this recipe does not include any raw eggs, mayonnaise often contains eggs or egg products and Worcestershire Sauce and Fish Sauce often have anchovy as an ingredient. More on this after the recipe.


These lamentations may turn into legislation to shift the costs of work injuries onto workers, Medicaid, Medicare. Social Security and private health insurance as the Nebraska legislature convenes this week.
I am referring to the results of the 2018 Oregon Workers' Compensation Premium ranking which show that Nebraska and Iowa essentially changed spots from the 2016 study. According to the study, Iowa's workers' compensation premiums are slightly lower than Nebraska's. Iowa enacted anti-worker changes to their workers' compensation laws in 2017. Iowa Governor Kim Reynolds, who signed the anti-worker reforms, was criticized for accepting a plane ride to an Iowa State Cyclones bowl game from workers' compensation claims administrator Sedgwick.
Overall, Nebraska remains a friendlier legal climate for business than Iowa, according to the pro-business Institute for Legal Reform. Nebraska ranks 7th while Iowa ranks 13th according to the last study.
Business interests like to gripe that workers' compensation costs cause business to leave state's with high costs. But the same Nebraska Chamber of Commerce griping about workers' compensation costs in Nebraska is also complaining that there are more jobs than workers in Nebraska. Lack of jobs isn't a problem in Nebraska like it might be in other states.
Workers' compensation protects injured workers. In Nebraska the cost of this protection has declined for business and has not led the state to lose jobs. There are no good reasons to reduce workers' compensation benefits in Nebraska.
This entry was posted in Iowa, Nebraska, Sedgwick, Workers' Compensation and tagged Iowa, Nebraska, Sedgwick, workers' compensation.
Are Nebraska and Iowa taking different roads on workers' compensation?
Nebraska moved from the 32nd most expensive workers' compensation insurance premium rates to 26th, while Iowa moved from the 24th most expensive to 30th most expensive workers' compensation insurance premium rates, according to the 2018 Oregon Workers' Compensation Premium Ranking produced by the Oregon Department of Consumer and Business Services.
While Nebraska and Iowa both remained in the middle of the pack nationally, Iowa workers' compensation insurance rates decreased from 101 percent of the national median from the last study in 2016 to 96 percent of the national median in the current study. In contrast, Nebraska's workers compensation insurance rates increased from 91 percent of the median rate in 2016, to 100 percent of the median rate in 2018.
In short, workers' compensation insurance became comparatively cheaper in Iowa while it became comparatively more expensive in Nebraska when compared to other states.
Iowa made many changes to that state's workers' compensation laws in 2017, that could explain the relative decline in the cost of workers' compensation insurance in that state. If the relative price of workers' compensation insurance in Nebraska and Iowa continues to diverge, I would expect a strong push for workers' compensation reforms in Nebraska similar to those enacted in Iowa.
In other measures, Nebraska remains a friendly legal jurisdiction for business. According to the pro-tort reform Institute for Legal Reform, Nebraska has the seventh least costly legal system for business.
This entry was posted in Iowa, Nebraska, Workers' Compensation and tagged Iowa, Nebraska, workers' compensation.
This ProPublica/NPR series should be required reading for all who care about workers' compensation and how the process really works – or doesn't work – for those who are injured on the job. The series shows the very real cost to loved ones when the system doesn't hold up to the workers' compensation "grand bargain" that was entered into more than a century ago in many states, including Nebraska. Families are hurt economically, and necessary role changes occur when a spouse and/or children become caretakers, which is often the case.
Today's blog post is focused on an excellent and informative, but very sobering, interactive chart that looks at how workers' compensation "reforms" by state are occurring.
When benefits are reduced and medical care is denied, the burden of caring for injured workers shifts to taxpayers through social programs because the workers' compensation system has all too often come up short. This results in taxpayers subsidizing injured workers on the local and state levels through healthcare and the social safety net. At the same time, workers' compensation insurance premiums for businesses are at their lowest rate in 25 years, partially because the "reformed" workers' compensation systems can save businesses money while avoiding the costs of caring for these hurt workers.
I urge you to spend some time on the interactive graphic and see where your state stands in its support of injured workers. Although the firm's lawyers are licensed in Nebraska and Iowa, we work with many who are injured in other states as the need arises and have an extensive network of lawyers who we work with on workers' compensation, especially focusing on representing truck drivers.
This blog will feature continued commentary and analysis on the ProPublic/NPR report, as was first addressed last week. But if you have specific questions about an injured worker's situation and need help or are unsure what the next steps are, please contact an experienced workers' compensation attorney like those at Rehm, Bennett & Moore.
This entry was posted in Workers' Compensation Reform and tagged Iowa, legislation, National Public Radio, Nebraska, NPR, ProPublica, states.


Finally, Nancy Pelosi challenger Marcia Fudge (Leslie Jones) joined Ingraham. She said she was challenging Pelosi because "they can never find a way to make fun of me, a middle-aged black woman named Fudge." She also noted that Nancy Pelosi's baby sister was a redwood tree.
Ingraham's guests may be popular, but the star had to obviously scrape for "the few remaining businesses that are willing to be associated with me," including Fashion Catheters, Home Dog Baptism Kits, Teeny Tiny Turkey, Cash 4 Organs, and Volkswagen ("you know why"). Watch the video above.


Julián 'Alacrán' López: There's no need to be afraid to do something we're called to do. To fight our opponent. To win. It doesn't matter how many punches we take. It doesn't matter how many times we get knocked down, but how many times we get up and keep fighting. Nothing can defeat us but our own decisions. There's no need to be afraid of living.
How can you escape the circle of violence once you've entered it? With more violence? Or let's say "controlled" violence. Possible, but will our main character here be able to do so? You have to watch the movie to see that. It's a decent one, but not a great one. What seemed surprising at first, was the fact Javier Bardem was playing in it too ... might have to do with the fact another Bardem is on the cast list.
Whatever the case, the main actor is doing an OK job, portraying his dilemma. The tagged on love story (with added "color", no pun intended, but "story"/character serving) feels a bit off and put on, but it's overall believable (mostly). There is some nice advice to the boxing people, but I guess nothing someone already in the business wouldn't already know.


Boho Bedroom Decor Ideas : Indian Kantha Throw, Kantha Quilts & Kantha Blankets !! Boho Bedroom Decor Ideas : Indian Kantha Throw, Kantha Quilts & Kantha Blankets !!
Boho Bedroom Decor Ideas : Indian Kantha Throw, Kantha Quilts & Kantha Blankets !!
Hi Guys! Here I am presenting how we can give a completely bohemian and vintage look to your bedroom, living room and dining room using beautiful hand stitched Indian kantha throws, kantha quilts, baby kantha blankets and handmade kantha bedding set designed and developed only in "INDIA".
We at Jaipur handloom cherishes one of our handmade jewel we called them " Kantha quilts" or Kantha Throw .
If you don't have any color and pattern choices in your mind on how to give a unique look to your bedroom then you are at the right place. Beautifully hand stitched kantha quilts and kantha bedspread will make your bedroom magnificent. These kantha quilts come in multicolor and have various pattern that fits to all type of bedroom room interior. We can use them as kantha tablecloth, kantha table runner, kantha sofa throw, kantha beach blanket, kantha yoga mat, kantha bed cover, kantha picnic mat and many other ways.
What's Kantha Quilt or Kantha Throw ?
kantha throws and kantha quilts are a one-of-a-kind cotton textile. Infused with the history and pride of their rich heritage, these wonderfully artistic and brightly coloured quilts are made by stitching together layers of old saris. Kantha throws or quilts provide an amazing kaleidoscope of colour, completely transforming and brightening up any living space with their fantastic details and vibrant colours. In other words these vintage throws are essentially a patched, quilted and vividly embroidered textile made entirely out of used cloth, in most cases old saris.
I hope this information will help everyone who is looking for unique and hand crafted kantha bedding's , kantha quilts, kantha throws, kantha blankets, kantha table runner, kantha table cloth for their dreamy home.
Home / Bohemian Home Decor Ideas / Boho Bedroom Decor Ideas : Indian Kantha Throw, Kantha Quilts & Kantha Blankets !!


In our first teacher interview of 2017, Janice Paterson of Wormit Primary tells us more about how she's used picture books aimed at older pupils, and reveals how they can engage and challenge pupils of all reading abilities.
Picture books are for all ages, from little ones right up to grumpy adults!
I didn't know this before I became a teacher. I always assumed that picture books were only aimed at very young children, and were a stepping stone to chapter books. And then one day I was given The Rabbits by John Marsden and Shaun Tan, a jarring exploration of colonialism with stunning and disturbing visuals, and my perception changed forever.
Can you tell us some of the ways you've used picture books with older readers in the classroom, and about the impact these have had on pupils?
I have used picture books to illustrate and promote understanding of a topic - vivid images and short but powerful texts can really bring things to life. I often use such books to develop drama lessons too, which also helps develop understanding of a theme. I have a large coffee table style book of Norman Rockwell's paintings that is amazing for developing discussion of emotion, reactions and situations. The book tells the story of World War 2 with no words.
Sometimes children need to see as well as to read and it doesn't make them any less sophisticated.
Some picture books are wonderful for allowing older children to discuss issues such as loss and/or bereavement. Reading the story to older children as you would to young children gives a sense of security too. There are many picture books which have a big impact when an issue arises.
I also use picture books to develop writing: "Describe the picture but include the feelings." Reluctant readers feel really included when picture books are the source for a class discussion and they often are exceptionally perceptive and subsequently their status and self-esteem rises and it becomes win-win and they read more. It also allows me to promote comic books, graphic novels and the Manga genre. Sometimes children need to see as well as to read and it doesn't make them any less sophisticated.
It's easy to imagine that some older pupils will shun picture books, seeing them as babyish simply because of the format before they find out about the content of the book. Have you ever experienced this reaction from pupils? If so, have you been able to find a way round it?
I've never experienced this. I always talk about my passion for children's literature and my love of picture books with a new class and my pupils know I'll share material with them. When a picture book comes out it is a treat and older children love the 'childlike' experience of having a story with pictures read so they can see as well as listen.
What are some of your favourite picture books for older readers, and why do you like them?
Reluctant readers or just children who haven't been particularly engaged with literature can really see the value of reading when they experience the joy and subtlety of a good picture book.
My Father's Arms are a Boat by Stein Erik Lunde and Oyvind Torseter because it really allows children to open up about loss.
The Book of Mean People by Toni and Slade Morrison and Pascal LeMaitre because of the issue it discusses and because I can point out that the author of this picture book got the Nobel Prize for Literature.
The Red Tree by Shaun Tan because it looks at depression - something that older children are very aware of but it is still something that people are reluctant to discuss.
War Game by Michael Foreman because it helps children to understand World War 1 and it is a wonderful story. I have a movie of the story which we can then watch and discuss how a book becomes a film.
When the Wind Blows by Raymond Briggs as it allows discussion of scary topics in a safe (ish) way.
Fungus the Bogeyman by Raymond Briggs because it is gross and funny and children love it! Also, Raymond Briggs' illustrations are so amazing they merit discussion from an art perspective alone.
The Rabbit Problem by Emily Gravett because how else could you explain the Fibonacci Sequence to children?!
Norman Rockwell, 332 Magazine Covers by Christopher Finch because it tells a story from so many perspectives and because it makes children really look.
Do you think there's potential for picture books to inspire older pupils to read for pleasure?
Absolutely. As I mentioned above, when it is handled well and promoted as sophisticated and meaningful, older children often start to bring in picture books they have enjoyed and also buy them as well as traditional novels. We have times in school when our older readers share their favourite picture books and other books too, with younger children and each other. This all adds to the richness and value that is placed on all literature. Sometimes a picture book can be a springboard to other books, novels and comics. We hold monthly book groups and when children bring in graphic novels to discuss there is always much enthusiasm. Reluctant readers or just children who haven't been particularly engaged with literature can really see the value of reading when they experience the joy and subtlety of a good picture book.
Do you think picture books can be challenging to older pupils?
Janice previously featured on our Teachers and Librarians blog with her wonderful 'Book Reviews in a Box' project – read the blog post here.
Looking for some picture books for adult readers? Check out this fantastic list!
Janice Paterson is a Principal Teacher at Wormit Primary School in Fife.


My name is Ilnah, I live in Rio de Janeiro (Brazil).
I know Portuguese, Spanish, English (and I'm studying Japanese) so, if you know any of these languages you can write in them (but if you write in japanese, please use hiragana on top of the kanjis).
🐦🌿 I study Biology and I really like birds (specially parakeets) and plants!
✔ I LOVE Studio Ghibli movies, so if you have any postcards of their movies, send it to me!
🎨 I like Van Gogh and Monet!!!!


DoubleValueSum This class implements a value aggregator that sums up a sequence of double values.
LongValueMax This class implements a value aggregator that maintain the maximum of a sequence of long values.
LongValueMin This class implements a value aggregator that maintain the minimum of a sequence of long values.
LongValueSum This class implements a value aggregator that sums up a sequence of long values.
StringValueMax This class implements a value aggregator that maintain the biggest of a sequence of strings.
StringValueMin This class implements a value aggregator that maintain the smallest of a sequence of strings.
UniqValueCount This class implements a value aggregator that dedupes a sequence of objects.
UserDefinedValueAggregatorDescriptor This class implements a wrapper for a user defined value aggregator descriptor.
ValueAggregatorBaseDescriptor This class implements the common functionalities of the subclasses of ValueAggregatorDescriptor class.
ValueAggregatorCombiner<K1 extends WritableComparable,V1 extends Writable> This class implements the generic combiner of Aggregate.
ValueAggregatorJob This is the main class for creating a map/reduce job using Aggregate framework.
ValueAggregatorJobBase<K1 extends WritableComparable,V1 extends Writable> This abstract class implements some common functionalities of the the generic mapper, reducer and combiner classes of Aggregate.
ValueAggregatorMapper<K1 extends WritableComparable,V1 extends Writable> This class implements the generic mapper of Aggregate.
ValueAggregatorReducer<K1 extends WritableComparable,V1 extends Writable> This class implements the generic reducer of Aggregate.
$hadoop streaming -input /app/test.txt -output /app/test -mapper ./mapper -reducer aggregate -file mapper -jobconf mapred.reduce.tasks=1 -jobconf mapre.job.name="test"


"Loss of an erstwhile Kingdom to a Hamlet with an antagonistic name…!" The question "where is Manipur…?" looks to be funny and seems to be very easy to answer at sight. However, none of us could give the right answer as it is not so easy as it appears….
IT News Imphal, June 1 : Proscribed group PREPAK (PRO) today observed its 8th Progressive Day. A press statement of the outfit said that the main observances were held at the Central Head Quarter, General Head Quarters and other base areas . Leaders of the outfits and cadres attended the observance, the statement added.


The power of prosperity, increase and fertilitycan be found in this Moss Agate pendulum topped with a Green Man talisman.
The Green Man is a pagan nature spirit, and a symbol of connection to nature, a symbol of the underlying life-force, and of the renewed cycle of growth each spring. This talisman is excellent for those who wish for renewal, fertility or growth of any kind.


Complete instructions for collecting blood cultures by direct inoculation into the blood culture bottles and by collection into SPS vacutainer tubes can be found in specimen collection procedure. If the culture has been ordered with ARDS (often ordered by physicians when patient is receiving antibiotic therapy), please use the green- and purple-top blood culture bottles. Routine blood cultures are collected in blue- and purple-top blood culture bottles. Please refer to the specimen collection department of the Laboratory for detailed information regarding collection of pediatric specimens.
In order to emphasis the importance of a sterile site to prevent false positive results, the following information is listed here.
Clean skin over proposed site (area of 5cm) using 70% isopropyl alcohol for 30 seconds and allow to dry (10 sec).
Allow iodine to dry for 30 seconds (2% tincture) to 60 Seconds (povidone).
Clean top of collection bottle or tube for 10 seconds with 70% isopropyl alcohol immediately before collection. Allow to air dry for before contact with needle.
If patient is sensitive to iodine remove iodine following venipuncture using 70% isopropyl alcohol.
Clean access of catheter or line thoroughly for 30 seconds with 70% isopropyl alcohol. Allow to air dry for 30 seconds before contact with collection "needle".
Clean top of collection bottle or tube for 10 seconds with 70% isopropyl alcohol immediately before collection. Allow to air dry before contact with needle.
Collect a minimum of 2ml's of specimen, to remove contaminates and "sludge", in red top tube (without additives) before collection of blood culture specimen.


text:
.. _contributing:

:github_url: https://github.com/WIPACrepo/decotools

************
Contributing
************

If you are new to working with forks, check out `GitHub's working with forks article <https://help.github.com/articles/working-with-forks/>`_.

============================
Step 1: Creating a new issue
============================

- If you don't already have a `GitHub <http://www.github.com>`_ account, create one
- Go to the `decotools GitHub page <https://github.com/WIPACrepo/decotools>`_ and create a new issue by clicking on the "Issues" tab and then the "New issue" button

.. image:: _static/new-issue-button.png

==============================
Step 2: Forking the repository
==============================

(If you have an existing configured fork of decotools, you can skip to Step 4: Syncing an existing fork)

- From the decotools repository use the "Fork" button to fork the project into your GitHub account

.. image:: _static/fork-button.png

- This forked copy of decotools can now be cloned to your local machine using

.. code-block:: bash

    $ git clone https://github.com/<your_username>/decotools.git

=======================================
Step 3: Configuring a remote for a fork
=======================================

From your cloned copy of decotools from the previous step, list the existing remotes with

.. code-block:: bash

    $ git remote -v

You'll most likely see something like

.. code-block:: bash

    origin  https://github.com/<your_username>/decotools.git (fetch)
    origin  https://github.com/<your_username>/decotools.git (push)

To add the original decotools project repository as a remote (named "upstream") to your copy of decotools via

.. code-block:: bash

    $ git remote add upstream https://github.com/WIPACrepo/decotools.git

Now when you execute ``git remote -v``, the newly added upstream remote should be present

.. code-block:: bash

    origin  https://github.com/<your_username>/decotools.git (fetch)
    origin  https://github.com/<your_username>/decotools.git (push)
    upstream        https://github.com/WIPACrepo/decotools.git (fetch)
    upstream        https://github.com/WIPACrepo/decotools.git (push)

================================
Step 4: Syncing an existing fork
================================

To ensure that your existing fork is up-to-date with the original decotools repository, fetch the upstream commits via

.. code-block:: bash

    $ git fetch upstream

The output should look something like

.. code-block:: bash

    remote: Counting objects: xx, done.
    remote: Compressing objects: 100% (xx/xx), done.
    remote: Total xx (delta xx), reused xx (delta x)
    Unpacking objects: 100% (xx/xx), done.
    From https://github.com/WIPACrepo/decotools
     * [new branch]      master     -> upstream/master

Now the commits to the master branch of WIPACrepo/decotools are stored in your local upstream/master branch. At this point, you'll want to make sure (if you're not already) that you're on the master branch of your local repository

.. code-block:: bash

    $ git checkout master
    Switched to branch 'master'

Now you can merge the upstream/master branch into your master branch with

.. code-block:: bash

    $ git merge upstream/master

Now the master branch of your local copy of decotools should be up-to-date with the original decotools master branch!

===================================
Step 5: Create a new feature branch
===================================

Next, create a new branch for the feature you would like to develop with

.. code-block:: bash

    $ git checkout -b <new_feature_branch_name>

The output should be

.. code-block:: bash

    Switched to branch '<new_feature_branch_name>'

=========================
Step 6: Develop new code!
=========================

Now add your feature, bug fix, typo fix, etc.

=======================================
Step 7: Running tests with the new code
=======================================

Once your contribution has been added, you'll want to run the tests for this project to ensure that none of the code you added broke any tests. If you haven't already, make sure you have the necessary software installed for running the tests (pytest) via

.. code-block:: bash

    pip install -r requirements-dev.txt

Now the tests can be run by going to the root directory of your decotools repository and executing

.. code-block:: bash

    make tests

=====================
Step 8: Documentation
=====================

If necessary for your contribution, add the appropriate documentation to the files in the ``docs/`` directory

========================================
Step 9: Committing and uploading changes
========================================

Now the changes you've made are ready to be committed and uploaded to GitHub. Let git know which files you would like to include in your commit via

.. code-block:: bash

    $ git add <modifies_files_here>

and then commit your changes with

.. code-block:: bash

    $ git commit -m '<meaningful messages about the changes made>'

Now you can push this commit from your local repository to your copy on GitHub

.. code-block:: bash

    $ git push origin <new_feature_branch_name>

==================================
Step 10: Submitting a pull request
==================================

Finally, you can go to your copy of decotools on GitHub and submit a pull request by clicking the "Compare & pull request" button!

.. image:: _static/pull-request-button.png

redpajama_set_name: RedPajamaGithub


A Randall Road resident impaled her small right toe on a garden rake this morning.
Princeton Police, the Princeton First Aid and Rescue Squad, and EMS Technical Rescue went to the resident's home to help her.
The 55-year-old woman, who was conscious and alert, told emergency responders that she was gardening in her back yard when she accidentally stepped on the rake, which pierced her toe. EMS Rescue used a power tool to remove the tine from the rake. The resident was then transported to the University Medical Center at Plainsboro to have the tine surgically removed from her toe.
I just got a paper cut on my finger. Please call me for information for your story.


Zip Wrap Kwf 24 - Thermal Management Sleeving And Jackets - Thermal Management - Products - ZT | Zippertubing Co.
This Kevlar product is perfect for military or industrial applications!
Zip-Wrap® (KWF-24) is a medium weight, high-breakage strength, complete 360° thermal, and protection solution, for wires, cables, hoses or pipe management. Zip-Wrap® (Kevlar) can be customized to fit virtually any length and diameter, allowing for easy installation with a zipper closure system.
Zip-Wrap® (KWF-24) is comprised of a ballistic grade woven fabric Kevlar 24 style 745 outer jacketing material, resistant to most organic solvents, which provides strength, and abrasion protection reinforcement for wires or cables. The KWF-24 material is a lightweight, strong synthetic fiber that can be customized in application using multiple layers to create the needed degree of protection. Sewn in combination with a brass zipper closure system, using a high-temperature aramid Kevlar thread, resistant to bacteria and mildew, the total solution allows for the combined materials to create a durable 360° abrasion and thermal protection solution, that can be easily installed with no additional tools required.
There are so many uses for Kevlar, and that is why The Zippertubing® Company now offers this material as an addition to their ever-growing inventory of products and materials. This ballistic-grade, woven fabric Kevlar material is resistant to wear and has a high breakage strength, all while maintaining its lightweight properties. It can be configured for round, flat or rectangular wire or cable bundles, as well as other custom component configurations, and can utilize many of our different closure options.
This combination of features makes it the perfect solution for abrasion protection jackets that wrap around your critical components. Kevlar can also be applied in multiple layers for use in safety or personal protection. To determine the number of layers needed for your chosen application, testing should be completed under the given circumstances you require.
*Please note that this material will degrade in UV light, and may not retain its original properties when immersed in liquid.
Brass toothed zippers consist of individual brass metal interlocking teeth attached to a separating Aramid fabric tape which has a fixed start and stop point. The zipper is closed via a self-locking slider which make a zipper installation tool free. Brass toothed zippers are available as a medium or heavyweight, highly flexible, closure system for cables where a tool-free installation is desired and the open/closure cycle frequency is likely to be high. Toothed zippers perform best in round cable applications where the cable bundle diameters exceeds 1.0 inch and larger and on flat ribbon cables over 2.0" wide. Brass toothed zippers can be used in very large size applications by increasing the tooth size. Brass toothed zippers are stronger than molded toothed zippers and more immune to damage. However, unlike plastic zippers they do not have a "self-lubricating" quality and may require periodic lubrication to operate smoothly. Ideally all zippers should be used in applications that are free of dirt and debris which could cause jamming. Brass zippers can be cleaned with a small brush if contaminated and they are prone to corrosion unlike plastic zippers. Zippers cannot be cut to length on site at the time of installation so assembly length must be known at the time of order. Zippers items are typically commercial items and meet the manufactures performance specification; however they can be procured in compliance to A-A-55634. Available with optional black oxide tooth finish.
Color Brass with black tape (black oxide teeth optional) w/ 5mm tooth only.


text:
Haywood, Charles. (1967). Folk songs of the world, gathered from more than 100 countries, selected and edited, with commentary on their musical cultures and descriptive notes on each song : in the original languages with English translations, and with chord suggestions for instrumental acc. London : Barker
Haywood, Charles. Folk songs of the world, gathered from more than 100 countries, selected and edited, with commentary on their musical cultures and descriptive notes on each song : in the original languages with English translations, and with chord suggestions for instrumental acc. / Drawings by Carl Smith Barker London 1967
Haywood, Charles. 1967, Folk songs of the world, gathered from more than 100 countries, selected and edited, with commentary on their musical cultures and descriptive notes on each song : in the original languages with English translations, and with chord suggestions for instrumental acc. / Drawings by Carl Smith Barker London
| title=Folk songs of the world, gathered from more than 100 countries, selected and edited, with commentary on their musical cultures and descriptive notes on each song : in the original languages with English translations, and with chord suggestions for instrumental acc. / Drawings by Carl Smith
| author1=Haywood, Charles, 1901-
| publisher=Barker
Folk songs of the world, gathered from more than 100 countries, selected and edited, with commentary on their musical cultures and descriptive notes on each song : in the original languages with English translations, and with chord suggestions for instrumental acc. / Drawings by Carl Smith
Haywood, Charles, 1901-
London : Barker, [1967]
320 p. : illus. ; 30 cm.
Folk songs.
q 784.406 H427f Main Reading Room - Held offsite
Folk songs of many lands [music] / by Grace Castagnetta and Hendrik Willem Van Loon
More folk songs
David Lumsden, Trevor Lucas, Martin Wyndam-Read and Brian Mooney perform at the Australian Folk Night co...
David Lumsden performs at the Australian Folk Night concert in the Australian Folk Night oral history pr...
Martyn Wyndham-Read and David Lumsden perform at the Australian Folk Night concert in the Australian Fol...

redpajama_set_name: RedPajamaCommonCrawl


GDI Global Domains International Reviewed.
When you go to obtain website hosting, you shall need to choose a domain name for your website. This can be done through your host, which is likely to offer you a discount on the domain, such as a %age off of the price or even a free domain when you choose them for your website hosting needs. The most ideal domain would be your business name. However, for many reasons that might not be a possibility.
One reason might be that the domain is already taken, you might choices with different extensions such as .net, .us, .org, or other such extensions, but you have to remember that people usually remember .com, so you should strive for that. Another reason might be the length of your business name, such as Joe's Bar, Grill, and Dance club. Now, you can have a lengthy domain if you really want it like joesbargrilldanceclubdotcom, but look at that, would you remember it?


text:
Home Air BAE reveal 'directed energy laser and lens system'
The Laser Developed Atmospheric Lens (LDAL)
BAE reveal 'directed energy laser and lens system'
BAE Systems have revealed a concept that they believe battlefield commanders will deploy, a new type of 'directed energy laser and lens system', called a Laser Developed Atmospheric Lens.
The system is expected to be capable of enhancing a 'commanders ability to observe adversaries' activities over much greater distances than existing sensors.
At the same time, the lens could be used as 'a form of deflector shield' to protect friendly aircraft, ships, land vehicles and troops from incoming attacks by high power laser weapons.
According to a BAE press release:
"The Laser Developed Atmospheric Lens (LDAL) concept, developed by technologists at the Company's military aircraft facility in Warton, Lancashire, has been evaluated by the Science and Technology Facilities Council (STFC) Rutherford Appleton Laboratory and specialist optical sensors company LumOptica and is based on known science. It works by simulating naturally occurring phenomena and temporarily – and reversibly – changes the Earth's atmosphere into lens-like structures to magnify or change the path of electromagnetic waves such as light and radio signals.
LDAL is a complex and innovative concept that copies two existing effects in nature; the reflective properties of the ionosphere and desert mirages. The ionosphere occurs at a very high altitude and is a naturally occurring layer of the Earth's atmosphere which can be reflective to radio waves – for example it results in listeners being able to tune in to radio stations that are many thousands of miles away. The radio signals bounce off the ionosphere allowing them to travel very long distances through the air and over the Earth's surface. The desert mirage provides the illusion of a distant lake in the hot desert. This is because the light from the blue sky is 'bent' or refracted by the hot air near the surface and into the vision of the person looking into the distance.
LDAL simulates both of these effects by using a high pulsed power laser system and exploiting a physics phenomena called the 'Kerr Effect' to temporarily ionise or heat a small region of atmosphere in a structured way. Mirrors, glass lenses, and structures like Fresnel zone plates could all be replicated using the atmosphere, allowing the physics of refraction, reflection, and diffraction to be exploited."
Professor Nick Colosimo, BAE Systems said:
"Working with some of the best scientific minds in the UK, we're able to incorporate emerging and disruptive technologies and evolve the landscape of potential military technologies in ways that, five or ten years ago, many would never have dreamed possible."
Professor Bryan Edwards, Leader of STFC's Defence, Security and Resilience Futures Programme said: "
"For this evaluation project, STFC's Central Laser Facility team worked closely with colleagues at BAE Systems and by harnessing our collective expertise and capabilities we have been able to identify new ways in which cutting edge technology, and our understanding of fundamental physical processes and phenomena, has the potential to contribute to enhancing the safety and security of the UK."
Craig Stacey, CEO at LumOptica added:
"This is a tremendously exciting time in laser physics. Emerging technologies will allow us to enter new scientific territories and explore ever new applications. We are delighted to be working with BAE Systems on the application of such game-changing technologies, evaluating concepts which are approaching the limits of what is physically possible and what might be achieved in the future."
Merlin helicopter declares emergency after departing Culdrose
BAE Systems to support U.S. Harrier fleet to 2029
Ron5
Regurgitating week old news with no added value is a poor idea.
HMS Spey joins Royal Navy fleet
UK 'must learn lessons' of the Nagorno-Karabakh conflict
RAF deliver COVID19 vaccine to Gibraltar
UK to supply 100 military vehicles to Lebanon 'at no cost'
Britain purchases MK54 lightweight torpedo kits destined for P-8 Poseidon
RAF transport Zephyr 'High Altitude Pseudo-Satellite' to Australia for concept trials


text:
\section{Introduction}

In recent decades, the problem of dynamic consensus have received a lot of attention. The goal of this problem is to make a team of agents agree in a time-varying quantity through a distributed algorithm, by sharing information with their neighbors in a communication network. Some examples of applications of dynamic consensus include distributed estimation in sensor networks \cite{freeman2010}, distributed convex optimization \cite{Kia2015b}, distributed coordination in electrical grids \cite{Cherukuri2016} and distributed formation control \cite{porfiri2007}.

\RevAll{{In distributed average tracking (DAT) applications, dynamic consensus algorithms are used as virtual observers for the average of some time-varying reference signals, which can be tracked by a controller for a local physical system at each agent \cite{zhao2019,sen2020}. Depending on the order of the system it may be desirable for the dynamic consensus observer to obtain derivatives of the average signal \cite{sen2020}}. In this context, the works \cite{freeman2006,Solmaz2017,Olfati2007} propose linear dynamic consensus algorithms for scalar systems. However, these algorithms have the disadvantage of having a non-zero terminal error bound for some classes of reference signals. This issue has been tackled for scalar systems in  \cite{nostrati2012,freeman2019} and second order systems in \cite{Chen2012,ghan2019,zhao2019} by means of First Order Sliding Modes (FOSM), allowing exact convergence for more general classes of reference signals. However, these approaches suffer from the so-called chattering effect due to the discontinuous character of the FOSM \cite[Chapter 3]{fridman2002}. This makes the system sensitive to delays and noise. For the high-order case, some algorithms are able to obtain the average signal and its derivatives with exact convergence for vanishing reference differences in \cite{sen2020} and reference differences with a bounded high-order derivative in \cite{zhao2017}. Nonetheless, these approaches impose a higher communication burden since agents share all high-order errors instead of a single scalar. On the other hand, The Exact Dynamic Consensus of High Order (EDCHO) algorithm from \cite{aldana2021} manages to obtain the average signal and its high-order derivatives when the agents share a single scalar and the reference signals have differences with some bounded high-order derivative. Moreover, it employs employs High Order Sliding Modes (HOSM) which mitigates the chattering effect of FOSM algorithms. }

The main limitation of EDCHO is that the initial values of the agent states must be constrained within a specific surface. A similar condition is also required in \cite{chen2015,zhao2019,zhao2017,sen2020}.\RevTwo{ This issue dramatically limits its applicability since this initial requirement can be violated when agents connect or disconnect from the network, preventing the protocol from converging in an uncertain or time-varying environments as in the real network scenarios described in \cite{Zhang2013}.} The mismatch in the initial conditions has been tackled for linear algorithms in \cite[Page 57]{Solmaz2017} and for the robust FOSM algorithm in \cite{freeman2019}. However, the extension of these techniques to EDCHO is non-trivial due to its high-order non-linear system character.

Motivated by the previous discussion, we propose Robust EDCHO (REDCHO), which manages to achieve dynamic consensus with exact convergence towards the average signal and its high-order derivatives, regardless of isolated events of spontaneous  changes in the network. \RevTwo{This feature improves the applicability of REDCHO in uncertain or time-varying networks with respect to \cite{aldana2021}. In addition, the use of HOSM in REDCHO mitigates the effect of chattering when compared with \cite{freeman2019}. \RevFour{Moreover, REDCHO allows the agents to share only a scalar, reducing the communication burden compared to \cite{sen2020,zhao2017}.} }
\subsection{Notation}

Let $\mathds{1}_n = [1,\dots,1]^T\in\mathbb{R}^n$ and $\text{\normalfont I}_n\in\mathbb{R}^{n\times n}$ the identity matrix.  Let $\text{diag}(v), v\in\mathbb{R}^n$ represents a diagonal matrix whose diagonal is composed by $v$. Let $\otimes$ denote the Kronecker product Let $\text{sign}(x) = 1$ if $x> 0, \text{sign}(x)=-1$ if $x<0$ and $\text{sign}(0)=0$. Moreover, if $x\in\mathbb{R}$, let $\lceil x\rfloor^\alpha:= |x|^\alpha\text{sign}(x)$ for $\alpha>0$ and $\lceil x\rfloor^0:={\text{sign}}(x)$. When $x=[x_1,\dots,x_n]^T\in\mathbb{R}^n$, then $\sgn{x}{\alpha}:=\left[\sgn{x_1}{\alpha},\dots,\sgn{x_n}{\alpha}\right]^T$ for $\alpha\geq 0$. If $V:\mathbb{R}^n\to\mathbb{R}$ is continuously differentiable, denote $\frac{\partial V}{\partial x} = \left[\frac{\partial V}{\partial x_1},\dots,\frac{\partial V}{\partial x_n}\right]$. Furthermore, let $V:\mathbb{R}^n\to\mathbb{R}$ be continuous and $F:\mathbb{R}^n\to\mathbb{R}^n$, then $L_FV:\mathbb{R}^n\to\mathbb{R}$ denotes the Lie derivative. \RevTwo{Finally, let $x^{(\mu)}(t)$ denote the $\mu$-th derivative of a signal $x(t)$.}

\section{Problem statement}

Consider a multi-agent system distributed in a network $\mathcal{G}$. In the following $\mathcal{G}$ is an undirected connected graph of $n$ nodes characterized by its incidence matrix $D$ or its adjacency matrix $A$ \cite[Chapter 8]{Godsil}. We consider that each agent $i$ has access to a local time varying signal $u_i(t)\in\mathbb{R}$. Additionally, each agent is capable of communicating with their neighbors \RevFour{according to a communication topology defined by $\mathcal{G}$}.  Moreover, each agent $i$ \RevTwo{runs a local observer with} output $y_i(t)=[y_{i,0}(t),\dots,y_{i,m}(t)]^T\in\mathbb{R}^{m+1}$ \RevFour{where $m$ is a desired system order}. \RevFour{In order to reduce communication burden, we require all agents to share only $y_{i,0}$ instead of the whole $y_i(t)$}. The goal of the system is to achieve the following property.

\RevFour{
\begin{defn}\upshape{\bf Robust Exact Dynamic Consensus.} Let the average signal $
\bar{u}(t) := \frac{1}{n}\sum_{i=1}^nu_i(t)$. Then, the multi-agent system is said to achieve robust EDC, if the individual output signals for each agent comply
\begin{equation}
\label{eq:prob}
    \begin{aligned}
    &\lim_{t\to\infty}\left|y_{i,\mu}(t) - \bar{u}^{(\mu)}(t)\right| = 0,
    \end{aligned}
\end{equation}
for all $\mu\in\{0,\dots,m\}, i\in\{1,\dots,n\}$ regardless of  isolated events of spontaneous connection or disconnection of agents.
\end{defn}}
\vspace{-0.8cm}
\RevAll{\begin{rem}
Assume that all agents are provisioned with an observer complying the properties described before. Moreover, assume all agents have a local physical system with dynamics of relative degree $m$. Then, the local observer's output $y_i(t)$ can be used locally at each agent to solve a DAT problem. In this case, a local feedback control can be designed using the ideas from \cite{sen2020,ghan2019,zhao2019}.
\end{rem}
}
\section{REDCHO}
We propose a new algorithm, REDCHO, which manages to achieve robust EDC under mild assumptions on the reference signals. To present the algorithm, first let the auxiliary matrices
$$
\begin{aligned}
&\Gamma =
\begin{bmatrix}
-\gamma_0 & 1 & 0 &\cdots&0 \\
0 &-\gamma_1 & 1 & \ddots&\vdots \\\
\vdots&\ddots&\ddots&\ddots&0\\
\vdots& &\ddots&-\gamma_{m-1}&1 \\
0&\cdots&\cdots&0&-\gamma_{m}
\end{bmatrix},
G = \begin{bmatrix}
C\\
C\Gamma \\
\vdots \\
C\Gamma^{m}
\end{bmatrix}
\end{aligned}
$$
for design parameters $\gamma_0,\dots,\gamma_m>0$ and \RevFour{$C = [1,0,\dots,0]\in\mathbb{R}^{1\times(m+1)}$}. The structure of the REDCHO algorithm is proposed as
\RevFour{
\begin{equation}
\label{eq:redcho}
\begin{array}{ll}
    \dot{x}_{i,\mu}(t) &= k_\mu\theta^{\mu+1} \sum_{j=1}^na_{ij}\lceil y_{i,0}(t) - y_{j,0}(t) \rfloor^{\frac{m-\mu}{m+1}}\\&+\ x_{i,\mu+1}(t) - \gamma_\mu x_{i,\mu}(t),\ \ \ 0\leq \mu < m\\
    \dot{x}_{i,m}(t) &= k_m\theta^{m+1} \sum_{j=1}^na_{ij}\sgn{y_{i,0}(t) - y_{j,0}(t) }{0} \\&- \gamma_mx_{i,m}(t)\\
    y_{i,\mu}(t) &= u_i^{(\mu)}(t) - \sum_{\nu=0}^{m} G_{\mu+1,\nu+1} x_{i,\nu}(t).
\end{array}
\end{equation}
}where $G_{\mu+1,\nu+1}$ with $\mu,\nu\in\{0,\dots,m\}$ are the components of the matrix $G$ \RevFour{and $\theta\geq 1$ is a design parameter}. Furthermore, we consider the following assumption:
\begin{assum}
\label{as:u_signals}
Let
$w_i(t)=\left(\bar{u}^{(m+1)}(t) - {u}_i^{(m+1)}(t)\right) + \sum_{\mu=0}^m l_\mu \left(\bar{u}^{(\mu)}(t)-{u}_i^{(\mu)}(t)\right)$
where $l_0,\dots,l_m$ are the coefficients of the polynomial $(s+\gamma_0)\cdots(s+\gamma_m) = s^{(m+1)} + \sum_{\mu=0}^{m}l_\mu s^{\mu}$.
Thus, $|w_i(t)|\leq L, \forall t\geq t_0$ \RevFour{for fixed $\gamma_0,\dots,\gamma_m$ and known $L>0$}.
\end{assum}

{It is easy to show that the EDCHO algorithm from \cite{aldana2021} is a particular limiting case of REDCHO and can be recovered by choosing $\gamma_0=\cdots=\gamma_m=0$ \RevFour{and $\theta=1$}. However, EDCHO assumes that $\sum_{i=1}^nx_{i,\mu}(t_0)= 0$ is satisfied. This condition breaks easily, specially when agents connect or disconnect from the network. }The main result of this work, which is formally stated and shown in Section \ref{sec:main_result}, is that using the gains $k_0,\dots,k_m$ designed as in \cite[Theorem 7]{aldana2021} and under Assumption \ref{as:u_signals}, then REDCHO algorithm works even when $\sum_{i=1}^nx_{i,\mu}(t_0)\neq 0$ and thus achieves robust EDC. In order to formally show these facts, we provide some auxiliary results.

\section{Towards convergence of REDCHO}
The REDCHO algorithm \eqref{eq:redcho} can be written in partially vectorized form as:
\RevFour{
\begin{equation}
\begin{aligned}
\label{eq:main_algo_vec2}
    \dot{X}_{\mu}(t) &= X_{\mu+1}(t) + k_\mu\theta^{\mu+1} D\sgn{D^TY_{0}(t)}{\frac{m-\mu}{m+1}} - \gamma_\mu X_\mu(t)\\ &\text{for } 0\leq\mu < m, \\
    \dot{X}_{m}(t) &= k_m\theta^{m+1}D\sgn{D^TY_0(t)}{0} - \gamma_m X_m(t)\\
    Y_\mu(t) &= U^{(\mu)}(t)-\sum_{\nu=0}^{m} G_{\mu+1,\nu+1}X_{\nu}(t), \forall \mu\in\{0,\dots,m\}
\end{aligned}
\end{equation}
}
where we define $X_\mu(t):=[x_{1,\mu}(t),\dots,x_{n,\mu}(t)]^T, Y_\mu(t):=[y_{1,\mu}(t),\dots,y_{n,\mu}(t)]^T$, $U(t):=[u_1(t),\dots,u_n(t)]^T$ and \RevTwo{$D$ is the incidence matrix of $\mathcal{G}$}. Moreover, let $\mathbf{X}(t):=[X_0(t)^T,\dots,X_m(t)^T]^T, \mathbf{Y}(t):=[Y_0(t)^T,\dots,Y_m(t)^T]^T$, $\mathbf{U}(t):=\left[\left(U^{(0)}(t)\right)^T,\dots,\left(U^{(m)}(t)\right)^T\right]^T$ and
\RevFour{
$$
\mathbf{F}(Y_0(t);\theta) = \begin{bmatrix}
k_0\theta D\sgn{D^TY_{0}(t)}{\frac{m}{m+1}}\\
\vdots\\
k_m\theta^{m+1}D\sgn{D^TY_{0}(t)}{0}
\end{bmatrix}
$$
}
Using this notation we obtain the fully vectorized form of the algorithm:
\begin{equation}
\begin{aligned}
\label{eq:main_algo_vec3}
    \dot{\mathbf{X}}(t) &= (\Gamma\otimes \text{\normalfont I}_n)\mathbf{X}(t) + \mathbf{F}(Y_0(t);\theta)\\
    \mathbf{Y}(t) &= \mathbf{U}(t)- (G\otimes \text{\normalfont I}_n)\mathbf{X}(t)
\end{aligned}
\end{equation}
Both partial and fully vectorized versions of the algorithm \RevFour{will be used throughout this work}. Moreover, note that $G$ is the observability matrix of the pair $(\Gamma,C)$ which is invertible. Then, the dynamics of $\mathbf{Y}(t)$ result in
\begin{equation}
\label{eq:dyn_y}
    \begin{aligned}
    \dot{\mathbf{Y}} &= \dot{\mathbf{U}} - (G\otimes \text{\normalfont I}_n)(\Gamma\otimes \text{\normalfont I}_n)\mathbf{X} - (G\otimes \text{\normalfont I}_n)\mathbf{F}(Y_0;\theta) \\
    &=\dot{\mathbf{U}} + (G\otimes \text{\normalfont I}_n)(\Gamma\otimes \text{\normalfont I}_n)(G^{-1}\otimes \text{\normalfont I}_n)(\mathbf{Y}-\mathbf{U})\\& + (G\otimes \text{\normalfont I}_n)\mathbf{F}(Y_0;\theta) \\
    &=\dot{\mathbf{U}} + (G\Gamma G^{-1}\otimes \text{\normalfont I}_n)(\mathbf{Y}-\mathbf{U}) - (G\otimes \text{\normalfont I}_n)\mathbf{F}(Y_0;\theta)
    \end{aligned}
\end{equation}
since $\mathbf{X}(t) = -(G^{-1}\otimes \text{\normalfont I}_n)(\mathbf{Y}(t)-\mathbf{U}(t))$, and $(G\otimes \text{\normalfont I}_n)(\Gamma\otimes \text{\normalfont I}_n)(G^{-1}\otimes \text{\normalfont I}_n) = (G\Gamma G^{-1}\otimes \text{\normalfont I}_n)$.

We will show that $\mathbf{Y}(t)$ converges towards the average consensus vector $\left[\bar{u}(t)\mathds{1}_n^T,\dots,\bar{u}^{(m)}(t)\mathds{1}_n^T\right]^T$ asymptotically achieving EDC. This analysis is performed by decomposing $\mathbf{Y}(t)=(\text{\normalfont I}_{m+1}\otimes\mathds{1}_n)\bar{y}(t)+\mathbf{\tilde{Y}}(t)\in\mathbb{R}^{(m+1)n}$ in the consensus component $\bar{y}(t) = (\text{\normalfont I}_{m+1}\otimes \mathds{1}_n^T/n)\mathbf{Y}(t)\in\mathbb{R}^{m+1}$ and in the consensus error $\tilde{\mathbf{Y}}(t) = (\text{\normalfont I}_{m+1}\otimes P)\mathbf{Y}(t)\in\mathbb{R}^{(m+1)n}$ with $P=(\text{\normalfont I}_n-(1/n)\mathds{1}_n\mathds{1}_n^T)$. Therefore, convergence of $\mathbf{Y}(t)$ can be established by means of showing that $\bar{y}(t)$ converges exponentially to $[\bar{u}(t),\dots,\bar{u}^{(m)}(t)]^T$ and $\tilde{\mathbf{Y}}(t)$ converges in finite time to the origin as we do in the following sections. First, we provide some results regarding structural properties of the matrices $\Gamma, G$ and their relation to the signals $u_i(t)$ and $w_i(t)$ from Assumption \ref{as:u_signals}. These notions will be useful in subsequent proofs.

\begin{lem}
\label{le:w_dynamics}
Let the change of variables $\mathbf{W}(t)=[W_0^T,\dots,W_m^T]^T = (G^{-1}\otimes \text{\normalfont I}_n)\mathbf{U}(t)$ with $W_\mu(t)\in\mathbb{R}^n, \forall \mu\in\{0,\dots,m\}$. Moreover, define
\begin{equation}
\label{eq:Wmp1}
W_{m+1}(t) = U^{(m+1)}(t)+\sum_{\mu=0}^ml_\mu U^{(\mu)}(t)
\end{equation}
where $l_0,\dots,l_m$ are the coefficients of the polynomial $(s+\gamma_0)\cdots(s+\gamma_m) = s^{(m+1)} + \sum_{\mu=0}^{m}l_\mu s^{\mu}$. Then, we conclude that
\begin{equation}
\label{eq:dynU}
\dot{\mathbf{U}} = (\tilde{\Gamma}\otimes \text{\normalfont I}_n)\mathbf{U} + (B\otimes \text{\normalfont I}_n)W_{m+1}
\end{equation}
and
\begin{equation}
\label{eq:dynW}
\dot{\mathbf{W}} = (\Gamma\otimes \text{\normalfont I}_n)\mathbf{W} + (B\otimes \text{\normalfont I}_n)W_{m+1}
\end{equation}
with $B=[0,\dots,0,1]^T\in\mathbb{R}^{(m+1)\times 1}$ and
\begin{equation}
\label{eq:companion}
\tilde{\Gamma} = \begin{bmatrix}
0 & 1 & 0 &\cdots& 0 \\
\vdots & 0 & \ddots & \ddots &\vdots \\
\vdots&\vdots&\ddots& \ddots &0 \\
0 & 0&\cdots& 0& 1 \\
-l_0 & -l_1 & \cdots & \cdots&-l_m
\end{bmatrix}.
\end{equation}
\end{lem}
\vspace{1em}
\begin{pf}

First, let $U_\mu(t):=U^{(\mu)}(t)$ and note that $\dot{U}_\mu(t) = U_{\mu+1}(t)$ for $0\leq \mu<m$ and $\dot{U}_m(t) = W_{m+1}(t) - \sum_{\mu=0}^ml_\mu U_\mu(t)$ from the definition in \eqref{eq:Wmp1}. Writing this in complete vector form leads to \eqref{eq:dynU} directly. Now, rewrite \eqref{eq:Wmp1} as
\begin{equation}
\begin{aligned}
\label{eq:poly_conv}
W_{m+1}(t) &= \left(\frac{d^{m+1}}{dt^{m+1}} + \sum_{\mu=0}^ml_\mu\frac{d^\mu}{dt^{\mu}}\right)U(t) \\
&=\left(\frac{d}{dt}+\gamma_m\right)\cdots\left(\frac{d}{dt}+\gamma_0\right)U(t)
\end{aligned}
\end{equation}
by the relation between the coefficients $l_0,\dots,l_m$ and $\gamma_0,\dots,\gamma_m$. Thus, define $V_0(t) = U_0(t)$ and recursivelly $V_{\mu+1}(t) = (d/dt + \gamma_\mu)V_\mu(t)$ for $0\leq\mu< m$ from which $W_{m+1}(t) = (d/dt+\gamma_m)V_{m}(t)$ is obtained using \eqref{eq:poly_conv}. Equivalently, we have
$
\dot{V}_\mu(t) = V_{\mu+1}(t) - \gamma_\mu V_\mu(t),\  0\leq\mu<m
$
and $\dot{V}_m(t) = W_{m+1}(t)-\gamma_mV_m(t)$. Written in vector form, $\mathbf{V}(t) := [V_0(t)^T,\dots,V_m(t)^T]^T$ satisfies $\dot{\mathbf{V}}(t) = (\Gamma\otimes \text{\normalfont I}_n)\mathbf{V}(t) + (B\otimes \text{\normalfont I}_n)W_{m+1}(t)$. Now, we obtain the matrix which maps $\mathbf{V}(t)$ to $\mathbf{U}(t)$ by noting that with $V_0(t) \equiv U_0(t)= (C\otimes \text{\normalfont I}_n)\mathbf{V}(t)$,
$$
\begin{aligned}
U_1(t) = \dot{U}_0(t) &= (C\Gamma\otimes \text{\normalfont I}_n)\mathbf{V}(t) + (CB\otimes \text{\normalfont I}_n)W_{m+1}(t) \\&= (C\Gamma\otimes \text{\normalfont I}_n)\mathbf{V}(t)
\end{aligned}
$$
since $CB=0$ and continuing this procedure to obtain $U_\mu(t) = \dot{U}_{\mu-1}(t) = (C\Gamma^{\mu}\otimes \text{\normalfont I}_n)\mathbf{V}(t)$ since $C\Gamma^{\mu-1}B=0$ for $0\leq\mu<m$. This can be written as $\mathbf{U}(t) = (G\otimes \text{\normalfont I}_n)\mathbf{V}(t)$ or equivalently $\mathbf{V}(t) = (G^{-1}\otimes \text{\normalfont I}_n)\mathbf{U}(t) = \mathbf{W}(t)$. Therefore, $\mathbf{W}(t)$ satisfy \eqref{eq:dynW} concluding the proof.
\end{pf}
\begin{cor}
\label{cor:companion}
Let $\tilde{\Gamma},B$ be defined as in Lemma \ref{le:w_dynamics}. Then, $\tilde{\Gamma} = G\Gamma G^{-1}$ and $GB= B$.
\end{cor}
\vspace{-1em}
\begin{pf}
Let the change of variables from Lemma \ref{le:w_dynamics} as $\mathbf{U}(t) = (G\otimes \text{\normalfont I}_n)\mathbf{W}(t)$. Then,
$$
\begin{aligned}
\dot{\mathbf{U}}
&= (G\otimes \text{\normalfont I}_n)(\Gamma\otimes \text{\normalfont I}_n)\mathbf{W} + (G\otimes \text{\normalfont I}_n)(B\otimes \text{\normalfont I}_n)W_{m+1} \\
&=(G\Gamma G^{-1}\otimes \text{\normalfont I}_n)\mathbf{U} + (GB\otimes \text{\normalfont I}_n)W_{m+1}\\
\end{aligned}
$$
Comparing with \eqref{eq:dynU} completes the proof.
\end{pf}

\section{Convergence of the consensus components of REDCHO}
In this section we show the behaviour of $\bar{y}(t) = (\text{\normalfont I}_{m+1}\otimes \mathds{1}_n^T/n)\mathbf{Y}(t)$ as given in the following result.
\begin{lem}
\label{eq:conv_bar_y}
Let $\mathbf{Y}(t)\in\mathbb{R}^{n(m+1)}$. Then, with any $\gamma_0,\dots,\gamma_m>0$ and any initial conditions $\mathbf{Y}(t_0)$ for \eqref{eq:dyn_y}
it is satisfied that $\bar{y}(t)=(\text{\normalfont I}_{m+1}\otimes \mathds{1}_n^T/n)\mathbf{Y}(t)$ converge asymptotically towards $\mathbf{\bar{u}}(t)=(\text{\normalfont I}_{m+1}\otimes \mathds{1}_n^T/n)\mathbf{U}(t)=\left[\bar{u}(t)\mathds{1}_n^T,\dots,\bar{u}^{(m)}(t)\mathds{1}_n^T\right]^T$.
\end{lem}

\begin{pf}
First, recall that $G\Gamma G^{-1}=\tilde{\Gamma}$ from Corollary \ref{cor:companion} and obtain the dynamics of $\bar{y}(t)$ by multiplying \eqref{eq:dyn_y} by $(\text{\normalfont I}_{m+1}\otimes \mathds{1}_n^T/n)$ from the left:
$$
\begin{aligned}
\dot{\bar{y}} &= \mathbf{\dot{\bar{u}}} + (\text{\normalfont I}_{m+1}\otimes \mathds{1}_n^T/n)(\tilde{\Gamma}\otimes \text{\normalfont I}_n)(\mathbf{Y}-\mathbf{U})\\& + (\text{\normalfont I}_{m+1}\otimes \mathds{1}_n^T/n)(G\otimes \text{\normalfont I}_n)\mathbf{F}(Y_0;\theta)\\
&=\mathbf{\dot{\bar{u}}} + \tilde{\Gamma}(\bar{y}-\mathbf{\bar{u}}) +G(\text{\normalfont I}_{m+1}\otimes \mathds{1}_n^T/n)\mathbf{F}(Y_0;\theta)
\end{aligned}
$$
since $(\text{\normalfont I}_{m+1}\otimes \mathds{1}_n^T/n)(\tilde{\Gamma}\otimes \text{\normalfont I}_n)=(\tilde{\Gamma}\otimes\mathds{1}_n^T/n)=(\tilde{\Gamma}\otimes \text{\normalfont I}_1)(\text{\normalfont I}_{m+1}\otimes\mathds{1}_n^T/n) = \tilde{\Gamma}(\text{\normalfont I}_{m+1}\otimes \mathds{1}_n^T/n)$. Moreover, $G(\text{\normalfont I}_{m+1}\otimes \mathds{1}_n^T/n)\mathbf{F}(Y_0;\theta)=0$ since $\mathds{1}_n^TD=0$ \cite[Page 280]{Godsil}. Hence, we obtain $\dot{\bar{y}}(t) = \mathbf{\dot{\bar{u}}}(t) + \tilde{\Gamma}(\bar{y}(t) - \bar{\mathbf{u}})(t)$. Define the error $e(t) := \bar{y}(t) - \bar{\mathbf{u}}(t)$ to obtain $\dot{e}(t)=\tilde{\Gamma}e(t)$. Finally, note that $\tilde{\Gamma}$ is given in \eqref{eq:companion} and has characteristic polynomial $\prod_{\mu=0}^m(s+\gamma_\mu)=0$. Then, with $\gamma_0,\dots,\gamma_m>0$, $\tilde{\Gamma}$ has negative eigenvalues and $\bar{y}(t)-\mathbf{\bar{u}}(t)$ asymptotically converge to the origin.
\end{pf}

\section{Convergence of the consensus error}
In this section we show the behaviour of $\tilde{\mathbf{Y}}(t) = (\text{\normalfont I}_{m+1}\otimes P)\mathbf{Y}(t)$ where $P = (\text{\normalfont I}_n-(1/n)\mathds{1}_n\mathds{1}_n^T)$. First, we obtain how the dynamics of $\tilde{\mathbf{Y}}(t)$ relate to EDCHO.
\begin{lem}
Let  $\tilde{\mathbf{Y}}(t) = (\text{\normalfont I}_{m+1}\otimes P)\mathbf{Y}(t)$ where $P = (\text{\normalfont I}_n-(1/n)\mathds{1}_n\mathds{1}_n^T)$ and $\mathbf{Y}(t)$ satisfy \eqref{eq:dyn_y}. Moreover, let $\mathbf{W}(t) :=[W_0^T,\dots,W_m^T]^T= (G^{-1}\otimes \text{\normalfont I}_n)\mathbf{U}(t)$ with $W_\mu(t)\in\mathbb{R}^n, \forall\mu\in\{0,\dots,m\}$ \RevFour{ and $\Theta=\diag{1,\theta^{-1},\dots,\theta^{-m}}$. Then, $\mathbf{Z}(t) := [Z_0(t)^T,\dots,Z_m(t)^T]^T = (\Theta G^{-1}\otimes \text{\normalfont I}_n)\mathbf{\tilde{Y}}(t)$ with $Z_\mu(t)\in\mathbb{R}^n$, satisfy}
\RevFour{
\begin{equation}
\label{eq:dynZ}
    \begin{aligned}
        \dot{Z}_{\mu}(t) = \theta\Big(&Z_{\mu+1}(t) - k_\mu D\sgn{D^T Z_{0}(t)}{\frac{m-\mu}{m+1}}\\& - (\gamma_\mu/\theta) Z_\mu(t)\Big) \text{ for } 0\leq\mu \leq m-1, \\
    \dot{Z}_{m}(t) =\theta\Big(&PW_{m+1}(t)/\theta^{m+1}- k_mD\sgn{D^TZ_0(t)}{0}\\& - (\gamma_m/\theta)Z_m(t)\Big)
    \end{aligned}
\end{equation}
}
with $W_{m+1}(t)$ defined in \eqref{eq:Wmp1}.
\end{lem}

\begin{pf}
First, obtain the dynamics of $\tilde{\mathbf{Y}}(t)$ using \eqref{eq:dyn_y} and $\tilde{\Gamma}=G\Gamma G^{-1}$ from Corollary \ref{cor:companion}:
$$
\begin{aligned}
\frac{\text{d}}{\text{d}t}{\mathbf{\tilde{Y}}} &= (\text{\normalfont I}_{m+1}\otimes P)\dot{\mathbf{U}} + (\tilde{\Gamma}\otimes \text{\normalfont I}_n)\mathbf{\tilde{Y}} \\&- (\tilde{\Gamma}\otimes P)\mathbf{U} - (G\otimes P)\mathbf{F}(Y_0;\theta)
\end{aligned}
$$
However, note that $(\text{\normalfont I}_{m+1}\otimes P)\dot{\mathbf{U}}(t) = (\tilde{\Gamma}\otimes P)\mathbf{U}(t) + (B\otimes P)W_{m+1}(t)$ from \eqref{eq:dynU} in Lemma \ref{le:w_dynamics}. Then,
$$
    \frac{\text{d}}{\text{d}t}{\mathbf{\tilde{Y}}} = (\tilde{\Gamma}\otimes \text{\normalfont I}_n)\tilde{\mathbf{Y}} - (G\otimes P)\mathbf{F}(Y_0;\theta) + (B\otimes P)W_{m+1}
$$
Moreover, $(G\otimes P)\mathbf{F}(Y_0(t);\theta) = (G\otimes \text{\normalfont I}_n)\mathbf{F}(Y_0(t);\theta)$ since
\RevFour{
} $PD=(\text{\normalfont I}_n-(1/n)\mathds{1}_n\mathds{1}_n^T)D = D$. Furthermore, the dynamics of $\mathbf{Z}(t)$ are
\RevFour{$$
\begin{aligned}
\dot{\mathbf{Z}} &= (\Theta G^{-1}\otimes \text{\normalfont I}_n) \frac{\text{d}}{\text{d}t}\tilde{\mathbf{Y}} \\
&=(\Theta G^{-1}\tilde{\Gamma}G\Theta^{-1}\otimes \text{\normalfont I}_n)\mathbf{Z} \\
&- (\Theta\otimes \text{\normalfont I}_n)\mathbf{F}(Y_0;\theta) + (\Theta G^{-1}B\otimes P)W_{m+1} \\
&=(\Theta\Gamma\Theta^{-1}\otimes \text{\normalfont I}_n)\mathbf{Z}\\& - (\Theta\otimes \text{\normalfont I}_n)\mathbf{F}(Y_0;\theta) + \theta^{-m}(B\otimes \text{\normalfont I}_n)PW_{m+1}
\end{aligned}
$$
where Corollary \ref{cor:companion} was used and $(\Theta B\otimes P) = \theta^{-m}(B\otimes \text{\normalfont I}_n)(\text{\normalfont I}_1\otimes P) = \theta^{-m}(B\otimes \text{\normalfont I}_n)P$. In addition, note that}
\RevFour{
$$
\begin{aligned}
&(\Theta\otimes \text{\normalfont I}_{n})\mathbf{F}(Y_0;\theta)\\& = \begin{bmatrix}
\text{\normalfont I}_n & 0 & \cdots & 0 \\
0 & \text{\normalfont I}_n\theta^{-1} & \ddots & \vdots \\
\vdots& \ddots & \ddots & 0\\
0 &\cdots & 0& \text{\normalfont I}_n\theta^{-m}
\end{bmatrix}
\begin{bmatrix}
 k_0\theta D\sgn{D^TY_{0}}{\frac{m}{m+1}}\\
\vdots\\
k_m\theta^{m+1}D\sgn{D^TY_{0}}{0}
\end{bmatrix} \\&= \theta\mathbf{F}(Y_0;1)
\end{aligned}
$$
To simplify the $\Theta\Gamma\Theta^{-1}$ term, denote the nilpotent matrix $A_0:=\Gamma+\diag{\gamma}$ which does not depend on any of the parameters $\gamma=[\gamma_0,\dots,\gamma_{m}]^T$ and for which it can be verified $\Theta A_0 = \theta A_0 \Theta$. Hence,
$$
\begin{aligned}
&\Theta\Gamma\Theta^{-1}=(\Theta A_0-\Theta\diag{\gamma})\Theta^{-1} \\
&=(\theta A_0\Theta-\diag{\gamma}\Theta ) \Theta^{-1}= \theta (A_0-\diag{\gamma/\theta})=\theta\Gamma_\theta
\end{aligned}
$$
Where $\Gamma_\theta:=A_0-\diag{\gamma/\theta}$. Moreover, the first row of $G\Theta^{-1}$ is $C$ which leads to $Z_0(t)=Y_0(t)$. Hence, combining all these facts,}
\RevFour{
$$
\begin{aligned}
&\dot{\mathbf{Z}} =  (\theta\Gamma_\theta\otimes \text{\normalfont I}_n)\mathbf{Z} - \theta\mathbf{F}(Z_0;1) + \theta^{-m}(B\otimes \text{\normalfont I}_n)PW_{m+1}\\
&=\theta\left((\Gamma_\theta\otimes \text{\normalfont I}_n)\mathbf{Z} - \mathbf{F}(Z_0;1) + \theta^{-(m+1)}(B\otimes \text{\normalfont I}_n)PW_{m+1}\right)
\end{aligned}
$$}
Writing this equation in partially vertorized form we recover \eqref{eq:dynZ}, which completes the proof.
\end{pf}
Now, if we show that $\mathbf{Z}(t)$ converge to the origin in finite time, the same conclusion will apply to $\tilde{\mathbf{Y}}(t)$. \RevFour{Note that for given $\gamma_0,\dots,\gamma_m$,  $PW_{m+1}(t)/\theta^{m+1}\in[-L,L]^n/\theta^{m+1}\subseteq[-L,L]^n$ under Assumption \ref{as:u_signals} and $\theta\geq 1$}. Therefore, comparing \eqref{eq:dynZ} with the EDCHO error system \eqref{eq:main_algo_error2} in Appendix \ref{ap:edcho}, it would be the case that $\mathbf{Z}(t)$ reaches the origin if $\gamma_0=\cdots=\gamma_m=0$. In the following we will use homogeneity to show that even with those terms, stability of REDCHO will still be valid locally. To do so, we will decompose the right hand side of \eqref{eq:dynZ} in two parts, one similar to the right hand side of the EDCHO error system in \eqref{eq:main_algo_error2} and the other with the remaining linear terms. Let $\mathbf{H}:\mathbb{R}^{n(m+1)}\rightrightarrows\mathbb{R}^{n(m+1)}$ and $\mathbf{Q}:\mathbb{R}^{n(m+1)}\to\mathbb{R}^{n(m+1)}$ be defined as
\begin{equation}
\label{eq:HG}
    \mathbf{H}(\mathbf{Z}) = \begin{bmatrix}
Z_1 \\
\vdots\\
Z_m\\
[-L,L]^n
\end{bmatrix} - \mathbf{F}(Z_0;1), \ \ \RevFour{ \mathbf{Q}(\mathbf{Z}) = \begin{bmatrix}
-(\gamma_0/\theta)Z_0\\
\vdots\\
-(\gamma_m/\theta)Z_m
\end{bmatrix}}
\end{equation}
Then, \eqref{eq:dynZ} is equivalent to the differential inclusion \RevFour{$\dot{\mathbf{Z}}(t) \in \theta(\mathbf{H}(\mathbf{Z}(t)) + \mathbf{Q}(\mathbf{Z}(t)))$}. Moreover, let  $\mathbf{r} = [r_0\mathds{1}^T,\dots,r_m\mathds{1}^T]$ with $r_\mu := m+1-\mu, \forall \mu\in\{0,\dots,m\}$. Then, it can be verified that $\mathbf{H}(\bullet)$ and $\mathbf{Q}(\bullet)$ are $\mathbf{r}$-homogeneous of degrees $-1$ and $0$ respectively in the sense of   in the sense of Definition \ref{def:homo} in Appendix \ref{sec:homo}.

\begin{lem}
\label{le:z_convergence}
\RevFour{Let  $\mathcal{G}$ be a connected graph} and the pair $\mathbf{H}(\bullet)$, $\mathbf{Q}(\bullet)$ defined in \eqref{eq:HG}. Moreover, let some \RevFour{fixed $\gamma_0,\dots,\gamma_m>0$ so that there exists $L$ for which Assumption \ref{as:u_signals} is complied and $k_0,\dots,k_m$ chosen as in Theorem \ref{prop:edcho} in Appendix \ref{ap:edcho} for such $L$}. Then, there exists a neighborhood $\mathcal{R}_0\subset \mathbb{R}^{n(m+1)}$ of the origin such  that if $\mathbf{Z}(t_0)\in\mathcal{R}_0$, then the solution of \RevFour{$\dot{\mathbf{Z}}(t) \in \theta(\mathbf{H}(\mathbf{Z}(t))+ \mathbf{Q}(\mathbf{Z}(t)))$} converge to the origin in finite-time. Moreover, $\mathcal{R}_0$ can be made arbitrarily big by \RevFour{increasing $\theta$}.
\end{lem}
\begin{pf}
Consider $\gamma_0=\cdots=\gamma_m=0, \RevFour{\theta=1}$. Then, $\dot{\mathbf{Z}} \in \mathbf{H}(\mathbf{Z})$ is globally finite-time stable towards the origin by Theorem \ref{prop:edcho} from Appendix \ref{ap:edcho}. Moreover, recall that $\mathbf{H}(\bullet)$ is $\mathbf{r}$-homogeneous of degree $-1$. Hence, by Proposition \ref{prop:converse} in Appendix \ref{sec:homo} there exists scalar functions $V(\mathbf{Z}),W(\mathbf{Z})$ which are $\mathbf{r}$-homogeneous of degrees $k$ and $k-1$ respectively and comply $\dot{V}(\mathbf{Z}) = L_{\mathbf{H}}V(\mathbf{Z}) \leq -W(\mathbf{Z})\leq -\beta_mV(\mathbf{Z})^\frac{k-1}{k}$ with $0<\beta_m=\inf\{W(\mathbf{Z}):V(\mathbf{Z})=1\}$ using Proposition \ref{prop:homo_comparison}. Now consider $\gamma_0,\dots,\gamma_m>0$ with arbitrary $\theta\geq 1$ and the same Lyapunov function $V(\mathbf{Z})$ as before. In this case
\RevFour{
$$
\dot{V} = \theta\left(L_{\mathbf{H}}V + L_{\mathbf{Q}}V\right) \leq  \theta\left(-\beta_mV^{\frac{k-1}{k}} + L_{\mathbf{Q}}V\right)
$$}
Note that from Proposition  \ref{prop:lie_homo}  $L_{\mathbf{Q}}V(\mathbf{Z})=\frac{\partial V}{\partial \mathbf{Z}}\mathbf{Q}(\mathbf{Z})$ is $\mathbf{r}$-homogeneous of degree $k$ since $\mathbf{Q}(\bullet)$ is $\mathbf{r}$-homogeneous of degree $0$. Hence, $L_{\mathbf{Q}}V(\mathbf{Z})\leq  \beta_M V(\mathbf{Z})$ by Proposition \ref{prop:homo_comparison} and $\beta_M=\sup\{L_{\mathbf{Q}}V(\mathbf{Z}):V(\mathbf{Z})=1\}$. Note that $L_{\mathbf{Q}}V(\mathbf{Z})$ may be positive for some $\mathbf{Z}$. Therefore, $\beta_M$ may be positive too. Moreover,
\RevFour{
$$
\dot{V}\leq  \theta\left[-\beta_mV^{\frac{k-1}{k}} + \beta_MV\right]\leq -\theta\left[\beta_m - \beta_M V^{\frac{k}{k-1}} \right]V^{\frac{k-1}{k}}
$$
Denote with $\mathcal{R}_0\subseteq\mathbb{R}^{n(m+1)}$ any region in which
\begin{equation}
\label{eq:condition}
-\left[\beta_m - \beta_M V(\mathbf{Z})^{\frac{k}{k-1}}\right]\leq -c
\end{equation}}
\RevFour{is complied for some $c>0$ so that $\dot{V}(\mathbf{Z})\leq -\theta cV(\mathbf{Z})^{\frac{k-1}{k}}$ for any $\mathbf{Z}\in\mathcal{R}_0$. If $\beta_M\leq 0$, then \eqref{eq:condition} is complied for $c=\beta_m$ regardless of $\mathbf{Z}$ so that we can set $\mathcal{R}_0=\mathbb{R}^{n(m+1)}$. On the other hand, if $\beta_M>0$ then \eqref{eq:condition} is complied when
$
V(\mathbf{Z})^{\frac{k}{k-1}} \leq (\beta_m-c)/\beta_M
$ which is possible only for $c\in(0,\beta_m)$. Then, choose $\mathcal{R}_0$ with $c=\beta_m/2$ so that \eqref{eq:condition} is complied whenever $V(\mathbf{Z})\leq \beta_m/(2\beta_M)$. We can write explicitly $\mathcal{R}_0=\{\mathbf{Z}\in\mathbb{R}^{n(m+1)}:V(\mathbf{Z})\leq \beta_m/(2\beta_M)\}$ so that $\dot{V}(\mathbf{Z})\leq -(\theta\beta_m/2)V(\mathbf{Z})^{\frac{k-1}{k}}$ for any $\mathbf{Z}\in\mathcal{R}_0$ regardless of $\beta_M$.
}
\RevFour{
Note that due to Proposition \ref{prop:converse}, $V(\mathbf{Z})$ is continuously differentiable and we can write $\beta_M = \sup\left\{-\sum_{\mu=0}^m(\gamma_\mu/\theta)\frac{\partial V}{\partial Z_\mu}Z_\mu: V(\mathbf{Z})=1\right\} = \tilde{\beta}_M/\theta$ where $\tilde{\beta}_M:=\sup\left\{-\sum_{\mu=0}^m\gamma_\mu\frac{\partial V}{\partial Z_\mu}Z_\mu: V(\mathbf{Z})=1\right\}$ is a constant for fixed $\gamma_0,\dots,\gamma_m$. Thus, $\beta_m/(2\beta_M)=\theta (\beta_m/(2\tilde{\beta}_M))$ can be made arbitrarily big by increasing $\theta$, so that $\mathcal{R}_0$ can be made arbitrarily big as well. Finally, note that since $(k-1)/k\in(0,1)$, then $V(\mathbf{Z})$ will reach the origin in finite time \cite[Corolary 4.25]{bernuau2014} and so will $\mathbf{Z}(t)$ whenever $\mathbf{Z}(t_0)\in\mathcal{R}_0$, completing the proof.
}
\end{pf}

The previous result shows that trajectories of $\dot{\mathbf{Z}}(t)\in\theta(\mathbf{H}(\mathbf{Z}(t))+\mathbf{Q}(\mathbf{Z}(t)))$ reach the origin if $\mathbf{Z}(t_0)\in\mathcal{R}_0$ which motivates to study if diverging trajectories can be obtained for some $\mathbf{Z}(t_0)\notin\mathcal{R}_0$. In the following, we show that this is not possible and only a terminal bounded error is allowed.

\begin{lem}
\label{le:bounded_error}
Let the conditions of Lemma \ref{le:z_convergence} be satisfied. Thus, for any initial conditions $\mathbf{Z}(t_0)\in\mathbb{R}^{n(m+1)}$, there exists $T>0$ and a bounded neighborhood of the origin $\mathcal{R}_\infty$ such that solution of \RevFour{$\dot{\mathbf{Z}}(t) \in\theta( \mathbf{H}(\mathbf{Z}(t))+\mathbf{Q}(\mathbf{Z}(t)))$} comply $\mathbf{Z}(t)\in\mathcal{R}_\infty$ for $t\geq T+t_0$. Moreover, such neighborhood can be made arbitrarily big \RevFour{by increasing $\theta$}.
\end{lem}
\begin{pf}
We proceed very similarly to the proof of Lemma \ref{le:z_convergence}. Consider only $\dot{\mathbf{Z}}(t)=\theta\mathbf{Q}(\mathbf{Z}(t))$ using \eqref{eq:HG} and a Lyapunov function $V(\mathbf{Z}) = \sum_{\mu=0}^m({Z}_\mu^T{Z}_\mu)^{\frac{m+1}{2r_\mu}}$ \RevFour{ with $r_\mu := m+1-\mu, \forall \mu\in\{0,\dots,m\}$} obtaining
\RevTwo{
$$
\dot{V} = \sum_{\mu=0}^m\frac{m+1}{r_\mu}({Z}_\mu^T{Z}_\mu)^{\frac{m+1}{2r_\mu}-1}{Z}_\mu^T\left({-\gamma_\mu} {Z}_\mu\right)\leq {-\gamma_{\min}}V
$$
since $\gamma_{\min}:=\min\{\gamma_0,\dots,\gamma_m\}\leq \gamma_\mu$ and $1\leq\frac{m+1}{r_\mu}$, $\forall \mu\in\{0,\dots,m\}$.} Moreover, note that
$
V(\Delta_\mathbf{r}(\lambda)\mathbf{Z}) = \sum_{\mu=0}^m(\lambda^{2r_\mu}Z_\mu^TZ_\mu)^{\frac{m+1}{2r_\mu}} = \lambda^{m+1}V(\mathbf{Z})
$
and thus $V(\mathbf{Z})$ is $\mathbf{r}$-homogenenous of degree $m+1$ under the dilation $\Delta_{\mathbf{r}}(\lambda)\mathbf{Z} = [\lambda^{r_0}Z_0^T,\dots,\lambda^{r_m}Z_m^T]^T$. Let the same Lyapunov function for \RevFour{$\dot{\mathbf{Z}}(t) \in \theta( \mathbf{H}(\mathbf{Z}(t))+\mathbf{Q}(\mathbf{Z}(t)))$:
$$
\dot{V}= \theta\left(L_{\mathbf{Q}}V+L_{\mathbf{H}}V\right)\leq-{\gamma_{\min}}V + \theta L_{\mathbf{H}}V.
$$}
Note that from Proposition  \ref{prop:lie_homo}  $L_{\mathbf{H}}V(\mathbf{Z})$ is $\mathbf{r}$-homogeneous of degree $m$ since $\mathbf{H}(\bullet)$ is $\mathbf{r}$-homogeneous of degree $-1$. Hence, $L_{\mathbf{H}}V(\mathbf{Z})\leq  \beta_M' V(\mathbf{Z})^\frac{m}{m+1}$ by Proposition \ref{prop:homo_comparison} and $\beta_M'=\sup\{L_{\mathbf{H}}V(\mathbf{Z}):V(\mathbf{Z})=1\}$. Thus,
\RevFour{
$$
\dot{V}\leq-\left[{\gamma_{\min}} -\theta\beta_M' V^{-\frac{1}{m+1}}\right]V.
$$}
\RevFour{
Denote with $\mathcal{R}_\infty^c\subset\mathbb{R}^{n(m+1)}$ any region in which
\begin{equation}
    \label{eq:condition2}
    -\theta\left[\frac{\gamma_{\min}}{\theta} -\beta_M' V^{-\frac{1}{m+1}}\right] \leq -\theta c'
\end{equation}
for some $c'>0$ so that $\dot{V}(\mathbf{Z})\leq -\theta c'V(\mathbf{Z})$ for any $\mathbf{Z}\in\mathcal{R}_\infty^c$. If $\beta_M'\leq 0$, then we can set $\mathcal{R}_\infty^c=\mathbb{R}^{n(m+1)}$ with $c'=\gamma_{\min}/\theta$. On the other hand if $\beta_M'>0$, \eqref{eq:condition2} is equivalent to
$
V(\mathbf{Z})^{1/(m+1)}\geq \beta_M'/(\gamma_{\min}/\theta-c')
$. Then, choose $c'=\gamma_{\min}/(2\theta)$ so that we can write $\mathcal{R}_\infty^c=\{\mathbf{Z}\in\mathbb{R}^{n(m+1)} : V(\mathbf{Z})^{1/k}\geq 2\theta\beta_M'/\gamma_{\min}\}$ so that $\dot{V}(\mathbf{Z})\leq -(\gamma_{\min}/2)V(\mathbf{Z})$ for any $\mathbf{Z}\in\mathcal{R}_{\infty}^c$. Therefore, for any initial condition $\mathbf{Z}(t_0)\in\mathbb{R}^{n(m+1)}$ the trajectory of ${V(\mathbf{Z}(t))}$ will converge to $\mathcal{R}_\infty:=\mathbb{R}^{n(m+1)}\setminus\mathcal{R}_\infty^c$ after a finite time $T+t_0$ and comply $\mathbf{Z}(t)\in\mathcal{R}_\infty$ for all $t\geq T+t_0$. Finally, note that $\mathcal{R}_\infty$ can be made arbitrarily large by increasing $\theta$.}

\end{pf}
\section{Convergence of REDCHO}
\label{sec:main_result}
In this section we formally state the main result of this work.
\begin{thm}
\label{th:main}
\RevFour{
\RevFour{Let  $\mathcal{G}$ be a connected graph} and the pair $\mathbf{H}(\bullet)$, $\mathbf{Q}(\bullet)$ defined in \eqref{eq:HG}. Moreover, let some \RevFour{fixed $\gamma_0,\dots,\gamma_m>0$ so that there exists $L$ for which Assumption \ref{as:u_signals} is complied and $k_0,\dots,k_m$ chosen as in Theorem \ref{prop:edcho} in Appendix \ref{ap:edcho} for such $L$}. Then, there exists neighborhoods $\mathcal{R},\mathcal{R}'\subset \mathbb{R}^{n(m+1)}$ around consensus such that if the initial conditions comply $[y_{1,0}(t_0),\dots,y_{n,m}(t_0)]^T\in\mathcal{R}$, the REDCHO algorithm in \eqref{eq:redcho} achieves robust EDC. \RevEight{On the other hand, if $[y_{1,0}(t_0),\dots,y_{n,m}(t_0)]^T\notin\mathcal{R}$, \eqref{eq:redcho} will achieve at most a uniformly bounded terminal error $[y_{1,0}(t),\dots,y_{n,m}(t)]^T\in\mathcal{R}',\forall t\geq T$ around dynamic consensus after some finite time $T>0$. Moreover, the neighborhoods $\mathcal{R},\mathcal{R}'$ can be made arbitrarily big by increasing $\theta\geq 1$.}}
\end{thm}
\begin{pf}
First, decompose $\mathbf{Y}(t) = (I\otimes \mathds{1}_n)(\text{\normalfont I}_{m+1}\otimes\mathds{1}_n^T/n)\mathbf{Y}(t) + (\text{\normalfont I}_{m+1}\otimes P)\mathbf{Y}(t) = (\text{\normalfont I}_{m+1}\otimes \mathds{1})\bar{y}(t) + \tilde{\mathbf{Y}}(t)$. \RevTwo{Now, note that Lemma \ref{le:z_convergence} implies the existence of a neighborhood $\mathcal{R}_0$ such that if $\mathbf{Z}(t_0)\in\mathcal{R}_0$ with $\mathbf{Z}(t) := (\Theta G^{-1}\otimes \text{\normalfont I}_n)\tilde{\mathbf{Y}}(t)$ then convergence of $\mathbf{Z}(t)$ is achieved towards the origin. Hence, consider the biggest ball of radius $R(\theta)$, $\mathcal{B}_0(\theta)=\{\mathbf{Z}\in\mathbb{R}^{n(m+1)}:\mathbf{Z}^T\mathbf{Z}\leq R(\theta)^2\}$ such that $\mathcal{B}_0(\theta)\subseteq\mathcal{R}_0$ and $R(\theta)$ is an increasing function of $\theta$ due to the last part of Lemma \ref{le:z_convergence}. Now,
$$
\begin{aligned}
R(\theta)^2&\geq \mathbf{Z}^T\mathbf{Z}=\mathbf{Y}^T(\Theta (G^{-1})^TG^{-1}\Theta\otimes \text{\normalfont I}_n)\mathbf{Y}\\
&\geq \theta^{-2m}\mathbf{Y}^T((G^{-1})^TG^{-1}\otimes \text{\normalfont I}_n)\mathbf{Y}
\end{aligned}
$$
This implies that convergence of $\tilde{\mathbf{Y}}(t)$ towards the origin happens for any $\mathbf{\tilde{Y}}(t_0)\in\{\mathbf{Y}\in\mathbb{R}^{n(m+1)}:\mathbf{Y}^T((G^{-1})^TG^{-1}\otimes \text{\normalfont I}_n)\mathbf{Y}\leq (\theta^{m}R(\theta))^2 \}$ where the previous region can be made arbitrarily big by increasing $\theta$. This implies the existence of $\mathcal{R}\in\mathbb{R}^{n(m+1)}$ as required by the theorem. An identical argument can be made for $\mathcal{R}'\in\mathbb{R}^{n(m+1)}$ but using Lemma \ref{le:bounded_error} instead to conclude uniformly bounded trajectories for $\tilde{\mathbf{Y}}(t)$ implying uniformly bounded steady state error around dynamic consensus.} Furthermore, Lemma \ref{eq:conv_bar_y} imply that $\bar{y}(t)$ converge asymptotically towards $\mathbf{\bar{u}}(t)=(\text{\normalfont I}_{m+1}\otimes \mathds{1}_n^T/n)\mathbf{U}(t)=\left[\bar{u}(t)\mathds{1}_n^T,\dots,\bar{u}^{(m)}(t)\mathds{1}_n^T\right]^T$. Hence, when $\mathbf{Y}(t_0)\in\mathcal{R}$, then $\mathbf{Y}(t)$ converge asymptotically  towards $(\text{\normalfont I}_{m+1}\otimes \mathds{1}_n)\mathbf{\bar{u}}(t)$. Equivalently, $Y_\mu(t)\to \bar{u}^{(\mu)}(t)\mathds{1}_n$. Since no initialization condition is required, then \eqref{eq:redcho} achieves robust EDC.
\end{pf}
\vspace{-1cm}
\RevFour{
\begin{rem}
\label{rem:params}
Note that since the $\gamma_0,\dots,\gamma_m$ are fixed, the class of signals for which Assumption \ref{as:u_signals} is complied can be checked before-hand, so that the method remains fully distributed.
On the other hand, showing the same stability properties in the case when all agents have different parameters $\gamma_0^i,\dots,\gamma_m^i>0, i\in\{1,\dots,n\}$ require more complicated computations, but is straightforward using similar arguments as in this work. Thus, only the case with a single set of parameters for all agents is provided here for simplicity.
\end{rem}}

\section{Simulation examples}
\label{sec:simulations}
In the following we show some simulation scenarios designed to show the properties of the REDCHO protocol. The simulations were  implemented using explicit Euler method with time step $h=10^{-6}$ over \eqref{eq:redcho}.

\begin{example}
\label{ex:convergence}
In order to show the convergence properties as described in the analysis from the previous sections, we simulated \eqref{eq:redcho} for the network topology $\mathcal{G}$ shown in Figure \ref{fig:g1}. Moreover,the number of agents is $n=8$ and the signals $u_i(t)=a_i\cos(\omega_it)$ with amplitudes $a_i=0.95, 0.34, 0.58, 0.22, 0.75, 0.25, 0.50, 0.69$ and frequencies $\omega_i=0.70, 0.75, 0.27, 0.67, 0.65, 0.16, 0.11, 0.49$. Thus, we choose $m=2$, $\gamma_0=\gamma_1=\gamma_2=3$, $k_0=6, k_1=11, k_2=6, \theta=1.5$. Furthermore, initial conditions for \eqref{eq:redcho} were generated from a normal distribution with mean $1$ and variance $r=1$. Figure \ref{fig:convergence} shows the convergence of ${Y}_0(t),{Y}_1(t),{Y}_2(t)$ towards the signals $\bar{u}(t)\mathds{1},\dot{\bar{u}}(t)\mathds{1},\ddot{\bar{u}}(t)\mathds{1}$ in the first column. Moreover, by letting $\bar{y}(t)=[\bar{y}_0(t),\dots,\bar{y}_m(t)]^T$, we show the average consensus error $e_\mu(t):=\bar{y}_\mu(t)-\bar{u}^{(\mu)}(t), \mu=0,1,2$ in the second column of Figure \ref{fig:convergence}, \RevTwo{which converges asymptotically to the origin as expected from Lemma \ref{eq:conv_bar_y}.} The third column of Figure \ref{fig:convergence} \RevTwo{shows how the consensus errors $\tilde{Y}_0(t),\tilde{Y}_1(t),\tilde{Y}_2(t)$ converge to the origin in finite time as expected from Lemma \ref{le:z_convergence}.} This same experiment was repeated for different values of $r$ for which the norm of the average consensus errors $\|Y_\mu(t)-\bar{u}^{(\mu)}(t)\mathds{1}\|, \mu=0,1,2$ is shown in Figure \ref{fig:initial}. This experiment shows that even for big initial conditions, the algorithm manages to converge to EDC. \RevTwo{This is consistent with Lemma \ref{le:bounded_error} which implies that no diverging error trajectories are possible.}
\end{example}

\begin{example}
\label{ex:merge}
In order to show the robustness properties of the REDCHO protocol when the topology suffers from sudden changes, we simulated \eqref{eq:redcho} for the network topology shown in Figure \ref{fig:merge_graph} which changes from $\mathcal{G}_{t< 5}$ to $\mathcal{G}_{t\geq 5}$ at $t=5$. Consider the same configuration and signals as in the previous example. Figure \ref{fig:merge_trajectories} shows how the outputs of the REDCHO algorithm converge to EDC approximately at $t=0.5$. At $t=5$ the topology changes, but the REDCHO protocol manages to make all agents, the first four and the new ones, converge to EDC again even when the states didn't comply neither $\sum_{i=1}^4x_{i,\mu}(0)=0$ nor $\sum_{i=1}^8x_{i,\mu}(5)=0$  as required in \cite{aldana2021}. For comparison, consider the protocol in \cite{aldana2021} obtained by setting $\gamma_0=\gamma_1=\gamma_2=0$ in REDCHO. Moreover, initial conditions are changed so that $\sum_{i=1}^4x_{i,\mu}(0)=0$. Figure \ref{fig:merge_trajectories_edcho} shows the trajectories for the protocol in this case, where EDC is achieved before $t=5$. However, when the new agents merge to the network, the agents output converge to consensus towards a signal that diverges.
\end{example}

\begin{figure}
    \centering
    \includegraphics[width=0.2\textwidth]{ge.eps}
    \caption{The network topology $\mathcal{G}$ considered in Example \ref{ex:convergence}.}
    \label{fig:g1}
\end{figure}
\begin{figure}
    \centering
    \includegraphics[width=0.45\textwidth]{convergence_trajectories.eps}
    \caption{Convergence of the REDCHO algorithm for the scenario in Example \ref{ex:convergence}. The first column shows convergence of ${Y}_0(t),{Y}_1(t),{Y}_2(t)$ towards the signals $\bar{u}(t)\mathds{1},\dot{\bar{u}}(t)\mathds{1},\ddot{\bar{u}}(t)\mathds{1}$. The second column shows the average consensus error $e_\mu(t):=\bar{y}_\mu(t)-\bar{u}^{(\mu)}(t), \mu=0,1,2$. The third column shows the consensus errors $\tilde{Y}_0(t),\tilde{Y}_1(t),\tilde{Y}_2(t)$.}
    \label{fig:convergence}
\end{figure}
\begin{figure}
    \centering
    \includegraphics[width=0.45\textwidth]{initial_trajectories.eps}
    \caption{Convergence of the REDCHO algorithm for different magnitudes of initial conditions as described in Example \ref{ex:convergence}.}
    \label{fig:initial}
\end{figure}
\begin{figure}
    \centering
    \includegraphics[width=0.25\textwidth]{merge.eps}
    \caption{The network $\mathcal{G}$ considered in Example \ref{ex:merge}.}
    \label{fig:merge_graph}
\end{figure}
\begin{figure}
    \centering
    \includegraphics[width=0.45\textwidth]{merge_trajectories.eps}
    \caption{Trajectories for the REDCHO protocol in the scenario of Figure \ref{fig:merge_graph}, as presented in Example \ref{ex:merge}.}
    \label{fig:merge_trajectories}
\end{figure}
\begin{figure}
    \centering
    \includegraphics[width=0.45\textwidth]{merge_trajectories_EDCHO.eps}
    \caption{Trajectories for the protocol from \cite{aldana2021} scenario of Figure \ref{fig:merge_graph}, as presented in Example \ref{ex:merge}.}
    \label{fig:merge_trajectories_edcho}
\end{figure}

\RevFour{In the previous examples we showed the effectiveness of REDCHO against its non-robust version in \cite{aldana2021}. In the following we compare against other state of the art dynamic consensus methods, with particular focus on terminal precision for the EDC goal.}
\begin{example}
\label{ex:comparison}
\RevAll{
In this example, we compare REDCHO with the Boundary-layer (B-layer) approach  from \cite{zhao2017} and the High-Order Linear protocol (HOL) from \cite{sen2020}. Both previous methods are able to achieve consensus  towards the average signal and its derivatives, but are not robust and require to share a whole vector between agents. In addition, we compare with the First-Order Linear (FOL) protocol in \cite{Solmaz2017} and the First-Order Sliding Mode (FOSM) protocol in \cite{freeman2019}. \RevFour{Both protocols are robust, but cannot obtain the derivatives of the average signal by construction. Thus, a robust exact differentiator \cite{levant2003} is applied locally at each agent to obtain derivatives of the average signal.}}

\RevTwo{In this setting, consider $\mathcal{G}$ constructed as a ring topology of $n=20$ agents.} \RevAll{Similarly as before, consider signals $u_i(t)=a_i\cos(\omega_it)$ where the $a_i,\omega_i$ are not shown for brevity. Note that all approaches can handle these type of signals, with at least a bounded terminal consensus error regardless of the order of the algorithm. An order of $m=2$ was used for REDCHO, B-layer and HOL, and an exact differentiator of order $m$ is applied for FOL and FOSM. Hence, all algorithms are able to obtain up to the second derivative of the average signal. All algorithms were implemented with parameters of similar magnitude, chosen such to roughly match same settling time for the sake of fairness. Moreover, we show the resulting consensus errors for all algorithms with $h = 10^{-6}$ as shown in Figure \ref{fig:h6} and $h=10^{-3}$ in Figure \ref{fig:h3} to show how they degrade as the discretization becomes coarser. \RevTwo{In addition, we simulated that agent 1 fails at $t=25$ and resets its state}, allowing us to evaluate robustness of the algorithms.}

\RevAll{As it can be observed, HOL and FOL methods have similar low precision in all cases before $t=25$. The reason is that neither of these methods are able to achieve exact convergence for sinusoidal signals. However, their performance does not degrade significantly when the time step is increased. On the other hand, it can be noted that the FOSM approach have better performance than the linear approaches when $h=10^{-6}$ due to its theoretically exact convergence. However, it degrades significantly when $h$ is increased as shown in Figure \ref{fig:h3}. The reason is that this method suffers from the chattering effect which is amplified for the higher order derivatives due to the exact differentiator. Note that the B-layer and REDCHO approaches have similar performance before $t=25$ with both sampling step sizes and outperform the other methods with at least one order of magnitude of precision improvement when $h=10^{-3}$ as shown in Figure \ref{fig:h3}. However, after $t=25$ both B-layer and HOL converge to consensus only up to a constant error due to their lack of robustness as shown by the $\|Y_0(t)-\bar{u}(t)\mathds{1}\|$ curves in both Figures \ref{fig:h6} and \ref{fig:h3}. Although other methods manage to recover from the failure of agent 1, REDCHO is the one with the best performance in all cases.}

\begin{figure}
    \centering
    \includegraphics[width=0.45\textwidth]{h6_compt.eps}
\caption{Comparison of the magnitude of the average consensus errors for REDCHO, FOL, FOSM, HOL and B-layer in the case of a sampling step $h=10^{-6}$. }
    \label{fig:h6}
\end{figure}
\begin{figure}
    \centering
    \includegraphics[width=0.45\textwidth]{h3_compt.eps}
\caption{Comparison of the magnitude of the average consensus errors for REDCHO, FOL, FOSM, HOL and B-layer in the case of a sampling step $h=10^{-3}$. }
    \label{fig:h3}
\end{figure}

\end{example}

\section{Conclusions}
In this work we proposed the REDCHO protocol. This new protocol achieves exact consensus towards the average of time varying signals and its derivatives distributed through a network. Proofs of convergence of the algorithm are given even when agents connect or disconnect from the network.  Simulation scenarios were designed to confirm the advantages of the proposed protocol. \RevTwo{Still, the proposed methodology works only when the changes in the network are isolated events. An analysis for general uncertain, time-varying  networks with persistent fast changes will be explored in future work.}

redpajama_set_name: RedPajamaArXiv


In a report dated August 23, the US Department of Justice announced a slew of charges against Nebraska Railcar Cleaning Services and its two owners following an explosion in 2015 that killed two workers. The charges include conspiracy, violating worker safety standards, violating federal law that governs hazardous waste management and submitting false documents to a federal agency.
Nebraska Railcar Cleaning Services' president, Stephen Michael Braithwaite, and vice president, Adam Thomas Braithwaite, are both named in the 22-count indictment.
The company, its president and owner, Steven Michael Braithwaite, and its vice president and co-owner, Adam Thomas Braithwaite, were charged in a 22-count indictment with conspiracy, violating worker safety standards resulting in worker deaths, violating the Resource Conservation and Recovery Act (RCRA) which governs hazardous waste management, and submitting false documents to a federal agency.
Adam Braithwaite was also charged with perjury. The indictment was returned by a grand jury in Omaha, Nebraska and announced by Acting Assistant Attorney General for the Environment and Natural Resources Division Jeffrey H. Wood and United States Attorney Joseph P. Kelly.
"An important mission of the Office of Inspector General is to investigate allegations of fraud committed against the Department of Labor's regulatory agencies, including the Occupational Safety and Health Administration (OSHA). We will continue to work with our law enforcement partners to defend the missions of those regulatory agencies, which includes assuring safe and healthful working conditions for American workers," said Steven Grell, Special Agent-In-Charge, Dallas Region, US Department of Labor Office of Inspector General.
The indictment alleges that after a 2013 inspection of NRCS, Steve Braithwaite entered into a written agreement where he represented that NRCS had been testing for benzene since July 2014. After OSHA returned to NRCS in March 2015 to conduct a follow-up inspection and was turned away by Steve Braithwaite, Steve and Adam Braithwaite created documents that were submitted to OSHA to falsely show that NRCS had been purchasing equipment to test the contents of railcars for benzene and had taken other required safety precautions.
During inspections by the Nebraska Department of Environmental Quality and the US Environmental Protection Agency in 2013 and 2014 respectively, NRCS was informed that it was required to test its wastes to determine if they were hazardous in order to properly dispose of them, rather than send all untested waste to a landfill not permitted to receive hazardous waste. The indictment alleges that was not done before April 2015.
OSHA regulations under the Occupational Safety and Health Act require that the air in confined spaces such as rail tanker cars be tested for various gases including flammable and explosive ones before workers are allowed to enter, and that workers exposed to certain chemicals wear respirators for which they must be assessed and fit tested. EPA regulations under RCRA require assessments of wastes for whether they are hazardous and that hazardous wastes be treated and disposed of at appropriate facilities. Hazardous wastes include those that are ignitable and those that contain benzene.
According to the Centers for Disease Control and Prevention, benzene causes human cancer and has other health effects.


London Corporate's services covering accounting, payroll and tax compliance are designed for the smaller business where a dedicated finance function may not be present or is limited and requires supplementary support. We work closely with clients to provide a seamless bookkeeping service that can integrate with other third parties who may be preparing statutory accounts or audits. We also set up and manage registrations and returns for direct and indirect taxes such as VAT.
We can replicate UK services across a number of European jurisdictions in conjunction with the Warwick Legal Network and their local affiliations with a number of providers of accounting services.
For clients seeking tax planning advice regarding more complex issues including those affecting cross border transactions, London Corporate works with a specialist team for this purpose supplemented by overseas tax specialists qualified in their respective jurisdiction.