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Dein erstes Forum » the second Detroit Grand Prix race of the
Online-Spiele Tips & Tricks » Deine erste Kategorie » Dein erstes Forum » the second Detroit Grand Prix race of the
#1 | the second Detroit Grand Prix race of the 24.04.2019 02:53
DETROIT -- Helio Castroneves celebrated in his signature style, climbing a fence, in the same place where he did it for the first time many years ago. Castroneves crew joined him above the track in front of roaring fans and it was fitting because his behind-the-scenes teammates helped him have his way with the competition. "I did not expect that," he said after easily winning the second Detroit Grand Prix race of the weekend Sunday, finishing 1.6836 seconds ahead of Penske Racing teammate Will Power. "They deserved it. "That was great to see them there." It was here in Detroit where Castroneves raced to the first of his 29 victories in 2000 and scaled the safety fence. And, the 39-year-old Brazilian is still winning and climbing with more composure. "I was able to hold my emotions better," he said. "I guess Im getting older." Team owner and unofficial race promoter Roger Penske was not a part of the fence-climbing celebration, but he had to be one of the happiest guys on Belle Isle because Castroneves and Power finished first and second. "You dream about these weekends," Penske said. "To be as strong as we were and see both guys in the winners circle." Castroneves lead was so large that when he made his final pit stop on Lap 49 he still was ahead when he got back on the track. The competition got closer after cautions led to restarts with seven and three laps left, but Castroneves could not be caught in part because Power did not want to risk ruining his teammates path to victory by possibly hitting him. "Because of Roger, I definitely wasnt going to race him hard," Power said. Castroneves has 29 IndyCar victories, tying Rick Mears for 11th on the career. "Oh, really?" Castroneves asked. "Wow. What an honour." Castroneves finished 0.060 seconds behind Ryan Hunter-Reay last week in the Indianapolis 500 in his bid to join Mears as a four-time winner in open-wheel racings signature event. Power won Saturday and played a big part in a sloppy start Sunday. He was penalized for avoidable contact on the opening lap, failing to turn right and causing Josef Newgarden to hit him from behind to trigger a three-car crash. That led to the first of two cautions within the first ten laps after a false start briefly delayed the race beginning. Despite a drive-through penalty, Power was able to pull into contention later in the race with aggressive moves. No one, though, was faster than Castroneves. Hunter-Reay, meanwhile, had a poor ending to a rough weekend after the biggest win of his career. He started 21st in the 22-car field on Saturday and Sunday because of crashes. Hunter-Reay ended the first race by crashing on the final lap and the second one did not last as long due to an electrical problem knocking him out after 61 laps. "You name it, we had the problem this weekend," he said. "Im just glad to be getting out of here." Hunter-Reay left Detroit -- heading to New York for an appearance Monday night with David Letterman -- without the IndyCar points lead. He entered the weekend with a 40-point lead on Power and exits it in third, 27 points behind Power and eight more behind Castroneves. "Major hit," Hunter-Reay acknowledged. Honda had dominated the Detroit Grand Prix the previous two years -- in the shadow of Chevys world headquarters -- and had the fastest car last week at the Indy 500. The engine manufacture did not fare better than fifth on Sunday with James Hinchcliffe following Charlie Kimball and Scott Dixon. Pole-sitter Takuma Sato was spun twice, dropping him to 18th. Juwan Howard Jersey . Blatter also told reporters Saturday after meeting with Qatars emir that the decision to award the tournament to the desert nation is "not reversible." There have been calls to move the tournament because of Qatars intense heat. Bradley Beal Jersey .5 million. University officials released the term sheet signed by Harsin Wednesday, the day the former Broncos walk-on player and Boise native was named the successor to Chris Petersen, who left for Washington last week. http://www.cheapwizardsjerseyssale.com/k...wizards-jersey/. Solomon Elimimian did not make the trip with the team after suffering what appeared to be a right leg injury in the teams regular season finale against the Calgary Stampeders. Jeff Green Jersey . -- Nick Bjugstad snapped out of his scoring slump and spoiled Drew MacIntyres first NHL start. Washington Wizards Jerseys . "Uuufff," was all shortstop Adeiny Hechavarria could come up with after Stantons latest mammoth shot.OTTAWA -- Bryan Murray wants to do right by Jason Spezza, but not at the expense of the Ottawa Senators. Ottawas general manager said hes had discussions about trading his captain but will not make a move unless he can get an acceptable return. "I dont think when you trade a player of Jasons calibre that you ever win," Murray said on Thursday. "I think we try to get the best return we can and then if its not satisfactory we cant do it and he has a year left on his contract." Spezza had 23 goals and 66 points in 75 games last season. He has 251 goals and 687 points in 686 career NHL games. Spezza has one year remaining on his contract with a cap hit of US$7 million. He has a limited no-trade clause, which allowed him to submit a list of ten teams where he would not want to play. Its believed he would prefer to move out of Canada and the glare of the spotlight that comes from playing north of the border. Ideally, its believed Murray would like to get a player who can come in and play on the teams top two lines, a highly-ranked prospect and perhaps a first-round pick at the upcoming NHL Entry Draft, seeing as the Senators dont have one this year. Murray said he has heard from teams, and speculation has Anaheim and St. Louis as the front-runners. "All we do in our business is investigate when theres a request by a player," said Murray. "You investigate possibilities and you talk about it and negotiate and when you negotiate it sometimes takes a week. Not very often its one phone call, sometimes many and in this case it appears that there will be many." The 31-year-old Spezza has spent 11 seasons with the Senators, the last as captain, but at the conclusion of this season it was clear he was unhappy and rumours of a trade request soon surfaced. From the moment Spezza arrived in Ottawa as a 19-year-old, he has been a lightning rod for criticism by both fans and media and it appears hes grown tired of it all. "The only explanation I have is that he would like to try something different," said Murray. "He thinks that the fingers are pointed at him quite often in Ottawa. Winning or losing he gets credit or he gets bblamed and I think he feels its time to do something else.dddddddddddd" When asked whether or not he felt that was a legitimate cause for a trade request Murray said, "thats not for me to judge." Its not the first time a star player has requested a move under Murrays watch. The Senators traded Dany Heatley to San Jose in 2009 after Heatley requested a move. "Im always disappointed that we havent been able to satisfy players, whether it be Jason or any other player that has ever asked for a trade," said Murray. "But its probably a little more common place now." With the departure of longtime captain Daniel Alfredsson last season, Spezza was the reins of the club. Expectations were that the added responsibility might translate to better play away from the puck. Over the course of the season it appeared head coach Paul MacLean grew frustrated with his captains play and that the two didnt always see eye-to-eye. "Jason told me this at the end, he understood what Paul was trying to do and thats not the reason (he wants a trade)," said Murray. "Thats basically what he told me." Despite the NHL entry draft being just over a week away Murray says he feels no sense of urgency to complete the trade. At this point all it entails is a little more work on his part. "Its just another number of phone calls you have to make on a daily basis almost to see if you can do something to satisfy the player and the team," said Murray. "Let me assure you were looking to satisfy our hockey team. Were looking to win here. Were trying to get return that will give us that chance. It just makes it a little bit busier thats all." Notes: The Senators have signed C Mike Hoffman to a one-year, two-way deal. C Stephane DaCosta turned down an offer by the Senators and appears poised to play in the KHL this coming season. Assistant general manager Pierre Dorion admitted the team has held discussions with G Robin Lehner regarding a contract extension for the restricted free agent. The Senators have made an offer to Milan Michalek and are confident a deal can be made with the veteran winger. Blackhawks Jerseys Store Cheap Wild Jerseys Cheap Red Wings Jerseys Cheap Maple Leafs Jerseys Penguins Jerseys Store Capitals Jerseys For Sale Blues Jerseys Store Cheap Kings Jerseys Adidas Lightning Jerseys Stars Jerseys For Sale Cheap Predators Jerseys Ducks Store Sharks Jerseys For Sale Cheap Sabres Jerseys Rangers Jerseys For Sale ' ' '
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EDITORIAL COMMENTARY
The enormous economic burden of slow learners: A wake up call
M Parakh
Department of Pediatrics, Dr. S N Medical College, Jodhpur, Rajasthan, India
Date of Web Publication 14-Oct-2019
Department of Pediatrics, Dr. S N Medical College, Jodhpur, Rajasthan
DOI: 10.4103/jpgm.JPGM_394_19
Parakh M. The enormous economic burden of slow learners: A wake up call. J Postgrad Med 2019;65:199-200
Parakh M. The enormous economic burden of slow learners: A wake up call. J Postgrad Med [serial online] 2019 [cited 2022 Jan 18];65:199-200. Available from: https://www.jpgmonline.com/text.asp?2019/65/4/199/269171
Children with borderline intellectual functioning ("slow learners") form a significant section of the student population and face a lot of academic difficulties. The exact prevalence of slow learners in a large country like India is not yet clearly determined. Nor has the economic burden of this underrecognized yet prevalent clinical condition been addressed adequately. In this journal issue, Karande et al[1] have evaluated the economic burden of slow learners by estimating its direct, indirect and intangible costs. Their study reveals that economic burden of slow learners is enormous (intangible > indirect > direct costs).[1] Tuitions are the most costly component of direct costs. Parental loss of earnings is the most costly component of indirect costs.[1] The results of this study should serve as an eye-opener to the authorities dealing with ensuring inclusive education in our country.
Karande et al[1] have addressed a very important and timely topic and highlighted its impact on the family and community. In the past, a few studies have tried to look into the social and academic issues of slow learners. However, no previous study has looked into the economic burden of slow learners. An extensive review of the literature revealed that this study is an unique attempt to address this important but neglected condition.
The methodology of this study conducted by Karande et al[2] is robust: an adequate sampling size and collecting all the necessary data to estimate the direct (both medical and non-medical), indirect and intangible costs. However the study is a single center study and the sampling method followed was by the nonprobability method, which could introduce a bias in the results and its interpretation. This study has documented that the economic burden of slow learners in the city of Mumbai is huge not only for the afflicted families, but also for the healthcare provider, and ultimately for society at large. "Indirect costs" far outweigh "direct costs" of slow learners (62.9% vs 37.1%) and the hypothetical "intangible costs" of slow learners is much higher than the "total costs" of slow learners. Furthermore, Karande et al[1] have reported that "longer duration of poor school performance" is an independent predictor of increased "direct costs" and "total costs."
The authors have also highlighted that most of the slow learners continue to be diagnosed late (mean age 14.5 years), which delays or makes them miss out on their educational rehabilitation. This study [1] also highlights that slow learners need to be identified at a much younger age to ensure that they receive remedial eduction and individual attention from their teachers to learn the school curriculum. Early diagnosis and developing facilities for remedial education will definitely support parents to reduce expenditure on private tuitions which comprises the major chunk of direct costs.
Other researchers have also recently highlighted the enormous economic burden of children with chronic conditions.[2],[3] Hoefgen et al[2] from US have estimated the economic burden of children having a single minor chronic disease (asthma, attention-deficit/ hyperactivity disorder, diabetes mellitus, and depression). Miller et al[3] from US have also highlighted the enormous burden of five chronic conditions (asthma, epilepsy, hypertension, food allergies, and diabetes) in children.
Karande et al[1] have also highlighted that almost 25% of slow learners have co-morbid attention-deficit/ hyperactivity disorder and/or a chronic medical condition (namely: refractive error, epilepsy, hemiparesis, cogenital heart disease, hypothyroidism) which need treatment. One of the main challenges for India is that health spending is mostly out-of-pocket and both the urban and rural poor cannot afford private care.[4] It is very likely that parents of slow learners will be finding it very difficult to manage the medical and educational costs of their children; as both costs are chronic and substantial.
Karande et al[1] have made a very important and valid appeal to school managements to set up resource rooms to ensure that these children receive regular and affordable remedial education. Such a step would aid the concept of inclusive education and also reduce the economic burden on the afflicted parents.
Based on the results of this study, it is prudent to state that health care managers in this country need to formulate a policy for developing a focussed infrastructure having integrated educational, health care and vocational facilities. This will reduce the cost of education and health care of slow learners and will help in appropriate employment/placement of these children. This model is being tried in many western countries but requires development of a clear policy targeted for the Indian setting.[5] It can therefore be envisaged that in future rehabilitation professionals in India will work at the interface of clinical and educational settings to support slow learners/children with disabilities at school which can eventually reduce the financial burden on parents, community and healthcare system.
Development of an ideal economic model which has the best academic and health outcome for Indian children will be very complex because of high variation in demographic variables across the length and breadth of this vast country. In a study conducted in rural Sri Lanka, it was suggested that students with learning difficulties actively engage in learning when an integrated approach utilising thematic units reflecting the students' world is used along with a behavioural approach to teaching-learning.[6] In future, the economics of this approach or other similar models can be studied in the Indian setting especially for rural non-English speaking children.
It is imperative that a comprehensive national registry be developed for children with academic difficulties, which would help to set up a focused state funded financial and logistic support for these children and their families.[7],[8] This will help in obviating the limitations pointed out in the current study. The current study sets up the stage to design future studies looking further into the direct, indirect and intangible costs to tease apart every part of economics involved in the care of slow learners. The present study can help school authorities and the government plan and allocate resources which are specific in their objectives to reduce the overall financial burden and also optimize the outcome of slow learners.
Karande S, Ramadoss D, Gogtay N. Economic burden of slow learners: A prevalence-based cost of illness study of its direct, indirect, and intangible costs. J Postgrad Med 2019;65:219-26.
Hoefgen ER, Andrews AL, Richardson T, Hall M, Neff JM, Macy ML, et al. Health care expenditures and utilization for children with noncomplex chronic disease. Pediatrics 2017;140:e20170492.
Miller GF, Coffield E, Leroy Z, Wallin R. Prevalence and costs of five chronic conditions in children. J Sch Nurs 2016;32:357-64.
Nair SR. Relevance of health economics to the Indian healthcare system: A perspective. Perspect Clin Res 2015;6:225-6.
Ng SL, Lingard L, Hibbert K, Regan S, Phelan S, Stooke R, et al. Supporting children with disabilities at school: Implications for the advocate role in professional practice and education. Disabil Rehabil 2015;37:2282-90.
Wickremesooriya S. Teaching children with learning difficulties via community-based rehabilitation projects in rural Sri Lanka. Disabil CBR Inclusive Dev 2016;26:53-81.
Miller B, Vaughn S, Freund L. Learning disabilities research studies: Findings from NICHD funded projects. J Res Educ Eff 2014;7:225-31.
Padhy SK, Goel S, Das SS, Sarkar S, Sharma V, Panigrahi M. Prevalence and patterns of learning disabilities in school children. Indian J Pediatr 2016;83:300-6.

redpajama_set_name: RedPajamaCommonCrawl


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/*
 * To change this template, choose Tools | Templates
 * and open the template in the editor.
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package com.github.manitourobotics.robot.commands;

import edu.wpi.first.wpilibj.smartdashboard.SmartDashboard;

/**
 *
 * @author robotics
 */
public class MoveBarrelServoUp extends CommandBase {

    // The purpose of the timeout is to allow this command to be added to a command group and
    // Not execute the next command right away
    private double timeout;

    // timeoutEnabled allows one to call the command without a timeout by changing the
    // isFinished conditions
    private boolean timeoutEnabled = false;

    public MoveBarrelServoUp() {
        // Use requires() here to declare subsystem dependencies
        // eg. requires(chassis);
        requires(barrelStopper);
    }

    public MoveBarrelServoUp(double timeout) {
        this();
        timeoutEnabled = true;
        setTimeout(timeout);
    }

    // Called just before this Command runs the first time
    protected void initialize() {
        barrelStopper.setServoUp();
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    // Called repeatedly when this Command is scheduled to run
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}

redpajama_set_name: RedPajamaGithub


In Roker v. Tower Hill Preferred Insurance Co., 40 FLW D764b (Fla. 2nd DCA March 27, 2015), the Second District Court of Appeal was asked to review a Summary Judgment granted to the homeowners insurer in a sinkhole case.
The insurance company and the homeowner agreed that sinkhole activity caused damage to the insured home. The insurance company's engineer recommended a subsurface remediation play which only called for compaction and chemical grouting. Tower Hill instructed its insured that it would only pay to remediate the sinkhole if the insured must enter into a contract for subsurface repairs in accordance with its engineering report.
The insured obtained a second opinion on the adequacy of the insurance company's proposed subsurface remediation plan. The second opinion found the insurance company's plan deficient, and proposed an alternate remediation plan which included compaction, chemical grouting and underpinning.
(a) Subject to paragraph (b), if a sinkhole loss is verified, the insurer shall pay to stabilize the land and building and repair the foundation in accordance with the recommendations of the professional engineer as provided under s. 627.7073, and in consultation with the policyholder, subject to the coverage and terms of the policy. The insurer shall pay for other repairs to the structure and contents in accordance with the terms of the policy.
(b) The insurer may limit its payment to the actual cash value of the sinkhole loss, not including underpinning or grouting or any other repair technique performed below the existing foundation of the building, until the policyholder enters into a contract for the performance of building stabilization or foundation repairs.
Tower Hill's argument that the policy language of the sinkhole endorsement required Roker to enter into a contract based on Tower Hill's engineer's protocol as a condition precedent to payment is similarly without merit. The policy merely provides that Tower Hill will cover the necessary repairs "[i]n accordance with the recommendations of the professional engineer who verifies the presence of a [s]inkhole [l]oss in compliance with Florida sinkhole testing standards." The policy does not specify that the insurance company's engineer unilaterally dictates the method of subsurface repairs. The parties do not dispute the fact that all three engineers who evaluated Roker's home complied with Florida sinkhole testing standards.
To the extent that Tower Hill is arguing that section 627.7073(1)(c) creates a presumption which justifies mandating that the insured accept the recommendation of an insurer's expert, the idea that an insurance company is entitled to rely on that presumption in the litigation context was rejected by the Florida Supreme Court in Warfel, 82 So. 3d at 57. In Warfel, the Florida Supreme Court explained that "[t]he application of a presumption as alleged and argued by [the insurer] at trial, that an insured could not overcome this presumption, would render any portion of section 627.7073 unconstitutional and inconsistent with all other provisions of the sinkhole statutes." Id. at 58. Ultimately, the supreme court held that the presumption disappeared once evidence rebutting it was introduced. Id. at 59. Because Roker submitted evidence rebutting Tower Hill's report, Tower Hill may not rely on section 627.7073(1)(c) to disprove the existence of a material fact.
The question of which recommended method of subsurface repair is sufficient to repair Roker's home is a question for the jury. Roker cooperated with Tower Hill throughout the claims process and exercised her statutory right to reject the neutral evaluator's recommendation. Tower Hill refused to pay anything for subsurface repair after being provided with a contract for repairs based on the recommendations of a qualified engineer. There remains a bona fide dispute as to the proper method of subsurface repair and, thus, Roker's loss.


Professor Colin Raston, from Flinders University, has the singular honour of winning an Ig Nobel prize, and international acclaim in less than a year, and with the same device.
Professor Raston and his team of researchers developed the thermos-sized Vortex Fluidic Device (VFD) with which they successfully unboiled an egg, winning them the Iggie.
Now they have successfully synthesised lidocaine with the VFD, a process which Professor Raston says "signals a paradigm shift in pharmaceutical manufacture", allowing production of lidocaine in high need areas, such as war zones and developing countries.
"The VFD uniquely controls how drug molecules can be made, and this is under continuous flow, such that research in making drug molecules can be readily translated into industry, avoiding conventional scale up problems and large reaction vessels — just leave the VFD running to make as much as you need," he said.


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Ukraine Tour Guides interview. Places the Locals Like – Kiev Private Tours
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Ukraine Tour Guides interview. Places the Locals Like
We asked Ukrainian Tour Guides who work in Kiev which places they like most and recommend tourists to visit. The list includes both popular among tourists and less known places of Kyiv.
My name is Olga, I'm 33 years old, I've been working as a guide for more than 5 years.
In 2013, I was actively taking part in EuroMaidan, Revolution of Dignity, so as the eyewitness I consider it my duty to tell people what happened, what pain we have endured.
When on Maidan, the tourist without much knowledge of what was going on here four years ago, probably will not notice the Alley dedicated to the Heroes of the Heavenly Hundred and will never know how many people died on February 20, 2014. This date will always be in our memory.
And to see why people were ready to die, why they wanted to change the executives in the country, change the system and destroy corruption, go to Mezhyhirya – a territory of 140 hectares, which earlier was a residence of former Ukrainian President V. Yanukovych. He spent 2 billion dollars he has earlier stolen to build a palace with all the furniture covered with gold; the area with artificial lakes and black swans, golf course, private zoo and helipad. This place is undeniably beautiful and at the same time it resent the insatiability of Ukrainian officials. Just imagine: the average pension is 70 dollars, and the president lives in a posh mansion worth 2 billion USD.
Tatiana, 30 years, works as a guide in Kiev for 2 years. I adore Ukrainian culture and traditions, Ukrainian vichyvanki and countryside. To feel it, I always go to Pirogovo, and I advise you to go there too.
In the village of Pirogovo you have a unique opportunity to get acquainted with the peasant life, architecture and creativity of Ukraine in its natural environment. The museum features more than three hundred interesting buildings and outbuildings, illustrating the life of ordinary Ukrainian village. Right at the entrance to the museum, you will feel beautiful Ukrainian spirit and a wonderful atmosphere. I remember once I told my mother it is a pity there is no asphalt, but only sand on the main road in Pirogovo. But now I understand that if there was asphalt around, there would be no such feeling of atmosphere in Pirogovo. On the territory there is a Church. People go there for weddings and baptisms of toddlers, as well as ordinary ceremonies of worship. The church is in excellent condition; there is a bell beside, and flowers are carefully planted around it. The houses are a little aged, although they look great. It's so interesting to go inside and study the household items our ancestors used every day.
Visitors of the open air museum relax on the field with the mills: some are taking photos, and some are riding horses, while others simply enjoy the nature and singing of birds. On weekends, craftsmen gather in Pirogovo to conduct master classes. Just imagine what a beauty it is: divine nature around, white huts, folk music, flowers, trees … peace and harmony! And the main thing is that with every step your mood is getting better! Here you can also buy pastries, vareniki, sausages, shish kebab, and various drinks. Pirogovo is charming at any time of the year. And most importantly, it's a big Ukrainian soul you can come back again and again.
Elena, 25 years old, works as guide in Kiev for 3 years.
I love Kiev Street Art. It's hard to count how many murals are in Kiev now. During 2014-2016, artists from Ukraine, Spain, Argentina, Australia, Germany, Great Britain and Portugal decorated dozens of houses with murals in Kiev, and this trend continues.
Murals have already become a true decoration of Kiev and have every chance to become its new landmark.
And in November 2017 Kyiv mural on the building of the regional department of the Main Directorate of the Ministry of Internal Affairs of Ukraine was recognized as one of the best in the world.
Every month Dutch magazine Street Art Today selects the seven most beautiful murals in the world.
The Kiev graffiti "Rise Up in the Dirt" is seventh in this list; the author is BKfoxx.
To finish the mural, the author worked even during the snowfalls and rain. This is the first police building in Ukraine, which was decorated with a bright mural.
According to users of social networks, more than 700 paint cans were used to create this mural.
The mural shows scattered parts and debris with the chamomile growing in the middle. The man in jeans and army boots sat down to protect the flower with his hands.
Contact Guide me UA to book Tours in Kiev:
+380 66 067 90 89 WhatsApp, Viber, SMS, Calls
Skype: Guide me UA
www.guideme.com.ua
Also, follow us on Instagram, Twitter and Facebook.
Cultural, Historical, Holidays, Restaurants, Travel
10 Kiev Bars to Relax – Nightlife in Ukraine
Ukraine Best Cities to Visit on Vacation
Thank you for your blog post. Really thank you! Awesome.


text:
MDW21 CALENDAR OF EVENTS
SEE WHAT'S NEXT!
Brookdale Park (Rose Garden)
Brookdale Park (Rose Garden), Bloomfield, NJ 07003, USA
Celebrate the best of public art this side of Passaic River Our 15-mile circuit will include a few stops to rest and take in stunning examples of mural art that spans the ages!
Become a Member and Save
Hidden Montclair LIVE!
Glenridge Ave x Bloomfield Ave
Oct 10, 2021, 3:00 PM – 4:00 PM EDT
Glenridge Ave x Bloomfield Ave, 441 Bloomfield Ave, Montclair, NJ 07042, USA
We're surrounded by homes, businesses, and infrastructure packed so tightly together that it's often hard to notice and appreciate the tiny details in the built environment.
MADFlash Film Fest
MDW21 HUB
Oct 09, 2021, 7:30 PM – 10:00 PM EDT
MDW21 HUB, 641 Bloomfield Ave, Montclair, NJ 07042, USA
Welcome to the second annual MADFlash Film Fest – 100-second Dash! Join us for the screening of 29 official selections from around the world!
Sonic Forests Concert
Undisclosed, Montclair, NJ, USA
Imagine the sway of your body, suspended among the trees and surrendering to the sounds of the forest. At the moment when day begins to turn into night, we'll gather under the fall skies for a very special evening of listening.
RE-imagine Montclair as S.A.F.E.
MDW 21 HUB
MDW 21 HUB, 641 Bloomfield Ave, Montclair, NJ 07042, USA
Through large visual displays, verbal presentation and small group discussion we will breakdown the Streets Are For Everyone Complete Streets Implementation Plan so you can understand it.
Toney's Brook Expedition – South
Rand Park
Oct 09, 2021, 12:00 PM – 2:00 PM
Rand Park, 100 Chestnut St, Montclair, NJ 07042, USA
Grab your walking boots and follow Toney's Brook, Montclair's fabled waterway that courses north to south through diverse neighborhoods.
Toney's Brook Expedition – North
Oct 09, 2021, 9:30 AM – 12:00 PM
Montclair, 288 Bellevue Ave, Montclair, NJ 07043, USA
Pop Up N' Dance!
Fri, Oct 08
Oct 08, 2021, 8:00 PM – 11:00 PM
The DJ trio of Butta, KayTee, and REBEXX will be taking over the HUB and playing the dance tunes to keep you on the floor all night long!
Open House Montclair
Galleries across Montclair
Galleries across Montclair, Montclair, NJ, USA
Artists in the (Open) House, all across Montclair! For two hours, some of Montclair's finest, boldest galleries and art spaces invite your curiosities in to discover the many ways we activate our imaginations, our bodies and our environments.
Developing for a Post-Suburban Future
This panel brings together leaders in the design, policy, and planning fields to examine New Jersey's economic and environmental futures.
"Tell Your Montclair Story" Installation
Oct 08, 2021, 12:00 PM EDT – Oct 10, 2021, 4:00 PM EDT
If Montclair were a radio station, what would it sound like? What unique stories would it tell? What compelling questions would it ask and what complex problems would it address? And just WHO would be the voices driving it all? We think it's YOU! Visit Radio Free Montclair's beta recording booth!
Jersey 4: The Future Exhibition
Oct 03, 2021, 4:00 PM EDT – Oct 10, 2021, 2:00 PM EDT
Teams of students from NJIT, Kean University and Montclair State University exhibit their project in response to this year's design challenge – a time capsule sent from the year 2121 back to 2021!
"If Not Now, When" Exhibition
Oct 01, 2021, 6:00 PM – Oct 10, 2021, 7:00 PM
If we've learned anything from the past year, it's that life can change in an instant, and the future we want won't design itself. How do we take part in the process, and what will connect us with our uncertain future? This year's theme brings five artists and designers together, asking us to listen


Houseplants make our indoor environments better in so many ways! They clean air, provide oxygen, and have proven psychological benefits. However, we hear way too often that people love houseplants but can't keep them alive. We will walk you through which houseplants are best for different situations, the top performers for indoor air purification, and best practices to make anyone a successful houseplant parent!
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text:
\section{Introduction}
\IEEEPARstart{S}{ynchronization} of coupled dynamical systems have been widely studied over
the past decades \cite{Wu}-\cite{Zhang2}, which can be
characterized by that all oscillators approach to a uniform dynamical
behavior and generally assured by the couplings among nodes and/or
external distributed and cooperative control.

In most existing works on linearly coupled dynamical systems, each node
needs to gather its own state and neighbors states and update them
spontaneously or in a fixed sampling rate, which may cost much. In order to
reduce the sampling rate of the coupling between nodes, specific
discretization is necessary. As pointed out by \cite{Astrom}, event-based
sampling was proved to possess better performance than sampling
periodically in time. Hence, \cite{Mazo}-\cite{Wang11} suggested that the
event-based control algorithms to reduce communication and computation load
in networked coupled systems. And \cite{Wang11}-\cite{HZ} showed that the
event-based control maintains the control performance. Event-based control
seems to be suitable for coupled dynamical systems with limited resources
and many works addressed the event-triggered algorithms. \cite{Dimarogonas}
considered centralized formulation, distributed formulation event-driven
strategies for multi-agent systems and proposed a self-triggered setup,
by which continuous measuring of the neighbor states can be avoided.
\cite{Johannesson}-\cite{Rabi} studied the stochastic event-driven
strategies. \cite{Seyboth} introduced event-based control strategies for both networks of
single-integrators with time-delay in communication links and networks of
double-integrators. By using scattering
transformation, \cite{YuHan} investigated the output synchronization
problem of multi-agent systems with event-driven communication in the
presence of constant communication delays. In some cases, event-driven strategies
for multi-agent systems can be regarded as linearization and discretization
process. For example, as mentioned in \cite{LC2004,LC2007}, the following
algorithm
\begin{align}
x^{i}(t+1)=f(x^{i}(t))+c_{i}\sum_{j=1}^{m}a_{ij}(f(x^{j}(t)))\label{cml}
\end{align}
can be a variant of the event triggering (distributed, self triggered) model for
consensus problem. In centralized control, the bound for
$(t_{k+1}^{i}-t_{k}^{i})=(t_{k+1}-t_{k})$ to reach synchronization was
given in the paper \cite{LC2004} when the coupling graph is indirected and
in \cite{LC2007} for the directed coupling graph.

Motivated by these works, we apply the idea of event-triggered sampling
scheme to the coupling configurations to realize synchronization of linearly coupled
dynamical systems. Here, for each node, the coupling term is set
up from the information of its local neighborhood at the last event time
and the event is triggered by some criteria derived from the information of
its local neighborhood. That is, once the triggering rule of node is
satisfied, the coupling term of this node is updated. Thus, the coupling terms are piece-wise constant
between two neighboring event times. We consider two scenarios:
continuous monitoring and discrete monitoring. Continuous monitoring means
that each node can observe its neighborhood's instantaneous information,
but discrete monitoring means that each node can only obtain the its
neighborhood's information at this event triggered time. As a payoff for
small cost of discrete monitoring, the triggering events happen more
frequently than continuous-time monitoring. For each scenario, it is shown
that the proposed event-triggered strategies guarantee the performances of
the nominal systems.

This paper is organized as follows. In sec. II, we propose event-trigger
coupling strategies to guarantee synchronization by employing continuous
monitoring. In sec. III, we consider discrete monitoring. Simulations are
given in sec. IV to verify the theoretical results. We conclude this paper
in sec. V.

\section{Continuous-time Monitoring}\label{2.1}

We consider the following network of coupled dynamical systems with piece-wise constant linear couplings:
\begin{eqnarray}
\frac{dx^{i}(t)}{dt}=f(x^{i}(t))-c\sum_{j=1}^{m}L_{ij}\Gamma(x^{j}(t_{k}^{i})-x^{i}(t_{k}^{i})),\nonumber\\
~~t_{k}^{i}\le t<t_{k+1}^{i},~i=1,\cdots,m.\label{cds}
\end{eqnarray}
Here, $x^{i}(t)\in\mathbb R^{n}$ denotes the state vector of node $i$, the
continuous map $f(\cdot):\mathbb R^{m}\to\mathbb R^{m}$ denotes the identical node
dynamics if there is no coupling. $L=[L_{ij}]_{i,j=1}^{m}\in\mathbb R^{m,m}$ is
the Laplacian matrix of the underlying bi-graph $\mathcal G=\{V,E\}$, with
the node set $V$ and link set $E$: for each pair of nodes $i\ne j$,
$L_{ij}=-1$ if $i$ is linked to $j$ otherwise $L_{ij}=0$, and
$L_{ii}=-\sum_{j=1}^{m}L_{ij}$; the graph that we consider in this paper is undirected and connected,
so $L$ is irreducible and symmetric. $c$ is the uniform coupling strength
at all nodes, and $\Gamma\in\mathbb R^{n,n}$ denotes the inner configuration matrix.
Let $0=\lambda_{1}(L)<\lambda_{2}(L)\le\cdots\le\lambda_{m}(L)$ be the
eigenvalues of $L$ with counting the multiplicities.

The increasing triggering event time sequence
$\{t_{k}^{i}\}_{k=1}^{\infty}$ (to be defined) are node-wise for
$i=1,\cdots,m$. At time $t$, each node $i$ collects its neighbor's state with
respect to an identical time point $t_{k_{i}(t)}^{i}$ with
$k_{i}(t)=arg\max_{k'}\{t^{i}_{k'}\le t\}$.

For the node dynamics map $f$, we suppose it belong to some map class
$Quad(P,\alpha\Gamma,\beta)$ for some positive definite matrix
$P\in\mathbb R^{n,n}$, constant $\alpha\in\mathbb R$ and positive constant $\beta>0$,
i.e.,
\begin{align}
\nonumber&(u-v)^{\top}P\bigg[f(u)-f(v)-\alpha\Gamma(u-v)\bigg]\\
&\le-\beta(u-v)^{\top}P(u-v)\label{quad}
\end{align}
holds for all $u,v\in\mathbb R^{n}$. In fact, we do not need this $Quad$ condition
(\ref{quad}) holds for all $u,v\in\mathbb R^{n}$ but for a region
$\Omega\subset\mathbb R^{n}$, which is assumed to contain a global attractors of
the coupling systems (\ref{cds}).

We highlight the basic idea behind the setup of the coupling term above. Instead of using the spontaneous state from the neighborhood to
realize synchronize, an economic alternative for the node $i$ is to use the
neighbor's constant states at the nearest time point $t^{i}_{k}$ until some
pre-defined event is triggered at time $t^{i}_{k+1}$; then the incoming
neighbor's information is updated by the states at $t^{i}_{k+1}$ until the
next event is triggered, and so on. The event is defined based on the
neighbor's and its own states with some prescribed rule. This process goes
on through all nodes in a parallel fashion.

To depict the event that triggers the next coupling time point, we
introduce the following candidate Lyapunov function:
\begin{align}
\nonumber V(t)& =-\frac{1}{2}\sum_{i=1}^{m}
\sum_{j=1}^{m}L_{ij}(x^{i}(t)-x^{j}(t))^{\top}P(x^{i}(t)-x^{j}(t))\\
&=x^{\top}(t)(L\otimes P)x(t)
\label{V}
\end{align}
with $x=[{x^{1}}^{\top}(t),\cdots,{x^{m}}^{\top}(t)]^{\top}\in\mathbb R^{nm}$ and
$\otimes$ represents the Kronecker product. For a compact expression, we
denote $F(x)=[f(x^{1})^{\top},\cdots,f(x^{m})^{\top}]^{\top}$. Then, the
derivative of $V(t)$ along (\ref{cds}) is
\begin{align}
\nonumber\frac{d}{dt}V(t)\left|_{(\ref{cds})}\right.=&2x^{\top}(t)(L\otimes P)\bigg\{F(x(t))-\alpha (I_m\otimes\Gamma)x(t)\\
&-[(c L-\alpha I_m)\otimes\Gamma] x(t)+ce(t)\bigg\}
\label{dV1}
\end{align}
where $e(t)=[e_1^{\top}(t),\cdots,e^{\top}_m(t)]^{\top}$ and
$$e_i(t)=\sum_{j}L_{ij}\Gamma\left(x^j(t)-x^j(t_{k_i(t)}^i)-x^i(t)+x^i(t_{k_i(t)}^i)\right).$$
By assuming $f\in Quad(P,\alpha\Gamma,\beta)$, we have
\begin{align}\label{a}
&\frac{d}{dt}V(t)\left|_{(\ref{cds})}\right.\nonumber\\
\le&-2\beta x^{\top}(L\otimes P)x-2x^{\top}[L(c L-\alpha I_{m})\otimes(P\Gamma)]x\nonumber\\
&+2cx^{\top}(L\otimes P)e\nonumber\\
\le&-2\beta' x^{\top}(L\otimes P)x-2(\beta-\beta')x^{\top}(L\otimes P)x\nonumber\\
&-2(c\lambda_{2}(L)-\alpha)x^{\top}[L\otimes(P\Gamma)]x+2cx^{\top}(L\otimes P)e
\end{align}
with any $0<\beta'<\beta$. Pick a constant $\upsilon>0$, then
\begin{align}\label{b}
\nonumber2x^{\top}(L\otimes P)e&\leq {\upsilon}x^{\top}(L^2\otimes P^2)x+\frac{1}{\upsilon} e^{\top}e\\
&\leq {\upsilon\lambda_m(L)\lambda_m(P)}x^{\top}(L\otimes P)x+\frac{1}{\upsilon}e^{\top}e.
\end{align}
Substitute inequality (\ref{b}) into (\ref{a}), we have
\begin{align}\label{c}
\nonumber \frac{d}{dt}V(t)\left|_{(\ref{cds})}\right.&\le -2(\beta-\beta')V(t)
-2(c\lambda_{2}(L)-\alpha)V(t)\\
&+[-2\beta'+
{c\upsilon\lambda_m(L)\lambda_m(P)}]V(t)+\frac{c}{\upsilon}e^{\top}e.
\end{align}
Denote $$z_i(t)=\sqrt{\sum_{j\neq
i}(-L_{ij})(x^j(t)-x^i(t))^{\top}P(x^j(t)-x^i(t))}.$$ Then, we have
$V(t)=\sum_{i=1}^m z_i^2(t)$.

Denote $\|\cdot\|$ the Euclidean norm, i.e., for any vector $\xi\in\mathbb
R^n$, $\|\xi\|=\sqrt{\xi_1^2+\cdots+\xi_n^2}$. For a matrix $A\in\mathbb
R^{n\times n}$, the spectral norm of $A$ is induced from Euclidean norm,
i.e., $\|A\|=\sqrt{\lambda_{max}(A^{\top}A)}$. Hence, $\|A\xi\|\leq
\|A\|\|\xi\|$ always holds, which will be used later as default. Moreover,
$\|x\|_{P}=\sqrt{x^{\top}Px}$ for some positive definite matrix
$P\in\mathbb R^{n,n}$. Thus, we have the following theorem
\begin{theorem}\label{thm1}
Suppose that $f\in Quad(P,\alpha\Gamma,\beta)$ with
positive matrix $P$ and $\beta>0$, $c\lambda_{2}(L)>\alpha$ and
$P\Gamma$ is semi-positive definite. Pick $\beta'<\beta$. Then either one of
the following two updating rules can guarantee that system (\ref{cds})
synchronizes:
\begin{enumerate}
\item Set $t^{i}_{k+1}$ as the time point by the rule
\begin{align}
t_{k+1}^{i}=&\max_{\tau}\left\{\tau\ge t_{k}^{i}:\right.\nonumber\\
&\left.~\|e_i(\tau)\|
\le\frac{\beta'}{c\sqrt{\lambda_m(L)\lambda_m(P)}}z_i(\tau)\right\};
\label{event1}
\end{align}

\item Set $t^{i}_{k+1}$ as the time point by the rule
\begin{eqnarray}
t_{k+1}^{i}=\max\left\{\tau\ge t_{k}^{i}:~\|e_i(\tau)\|\le a\exp{(-b\tau)}\right\}.
\label{event2}
\end{eqnarray}
\end{enumerate}
\end{theorem}
\begin{proof}
Noting that $c\lambda_{2}(L)>\alpha$ holds and $P\Gamma$ is semi-positive definite. By (\ref{c}), we have
\begin{align}\label{Vcase1}
\nonumber\frac{d}{dt}V(t)\left|_{(\ref{cds})}\right.\leq& -2(\beta-\beta')V(t)+[c\upsilon\lambda_m(L)\lambda_m(P)\\
\nonumber&-2\beta']V(t)+\frac{c}{\upsilon}\|e(t)\|^2\\
\nonumber=& -2(\beta-\beta')V(t)+\sum_{i=1}^m [c\upsilon\lambda_m(L)\lambda_m(P)\\
&-2\beta']z_i^2(t)+\sum_{i=1}^m\frac{c}{\upsilon}\|e_i(t)\|^2
\end{align}
holds for any $\upsilon>0$.

(1).
In case of
\begin{equation}
\label{est1}
\|e_i(t)\|^2\leq \frac{\upsilon}{c}[2\beta'-c\upsilon\lambda_m(L)\lambda_m(P)]z_i^2(t)
\end{equation}
for some constant $\upsilon>0$, we have
\begin{equation}
\label{dV}
\frac{d}{dt}V(t)\left|_{(\ref{cds})}\right.\leq -2(\beta-\beta')V.
\end{equation}
This implies that $V(t)$ converges to $0$ exponentially. Note
$$\max_{\upsilon>0}\frac{\upsilon}{c}[2\beta'-c\upsilon\lambda_m(L)\lambda_m(P)]
=\frac{\beta'^{2}}{c^{2}\lambda_{m}(L)\lambda_m(P)}.$$ Then, we take
$\upsilon=\frac{\beta'}{c\lambda_{m}(L)\lambda_m(P)}$, which guarantees that (\ref{event1})
holds.

(2). In case of
\begin{equation}
\label{est2}
\|e_i(t)\|\leq a\exp{(-bt)},
\end{equation}
we have
\begin{align*}
\frac{d}{dt}V(t)\left|_{(\ref{cds})}\right.\leq& -2(\beta-\beta')V+[c\upsilon\lambda_m(L)\lambda_m(P)\\
&-2\beta']V
+\frac{a^2cm}{\upsilon} \exp{(-2bt)}.
\end{align*}
Pick $\upsilon= \frac{2\beta'}{c\lambda_m(L)\lambda_m(P)}$. Then, we have
\begin{align}\label{Vcase2}
\nonumber\frac{d}{dt}V(t)\left|_{(\ref{cds})}\right.&\leq -2(\beta-\beta')V\\
&+\frac{a^2c^2m\lambda_m(L)\lambda_m(P)}{2\upsilon\beta'} \exp{(-2bt)},
\end{align}
which implies that $V(t)$ converges to $0$ exponentially.

\end{proof}
In fact, in (\ref{event1}), if $\tau=t_k^i$ but the system does not synchronize, then the left-hand term
$\|e_i(t_k^i)\|=0$ and at least one agent has positive right-hand term,
which means the next inter-event interval of one agent must be positive.
While in (\ref{event2}), if $\tau=t_k^i$ but the system does not synchronize, the left-hand term equals $0$ and
the right-hand term is positive for all nodes. Hence, the inter-event intervals of all
agents are positive. But in case the derivative of $e_i(t)$ is sufficiently
large, the inter-event interval might tends to $0$. Since the
dynamics of $e_i(t)$ is highly related to the property of $f(\cdot)$, towards a lower-bound of the inter-event intervals, we suppose that $f(\cdot)$ is
Lipschitz in the following theorem.

It should be highlighted that if $f$ is Lipschitz and $P\Gamma$ is semi-positive
definite, then $f\in Quad(P,\alpha\Gamma,\beta)$ with
$\beta=\frac{\alpha\lambda_1(P\Gamma)}{\lambda_m(P)}-\frac{\lambda_m(P)}{\lambda_1(P)}$.
Thus, we have

\begin{theorem}\label{thm2}
Suppose $f\in Quad(P,\alpha\Gamma,\beta)$ with positive matrix $P$ and
$\beta>0$, satisfies Lipschitz condition with Lipschitz constant $L_f$,
and there exists some $\sigma$ (possibly negative) such that
\begin{eqnarray}
(u-v)^{\top}P(f(u)-f(v))\ge\sigma (u-v)^{\top}P(u-v)\label{vi}
\end{eqnarray}
for all $u,v\in\mathbb R^{n}$.  $c\lambda_{2}(L)>\alpha$ and $P\Gamma$ is semi-positive
definite. For any $\beta'<\beta$, any initial condition and any time $t\leq
0$, we have
\begin{enumerate}
\item
With the updating rule (\ref{event1}), at least one agent has next
inter-event interval, which is lower-bounded by a common constant
$\tau_{O}>0$.
in addition, if there exists $\varsigma>0$ such that $z_{i}^{2}(t)\ge
\varsigma V(t)$ for all $i=1,\cdots,m$ and $t\ge 0$, then the next inter-event
interval of every agent is strictly positive and is lower-bounded by a
common constant.
\item With the updating rule (\ref{event2}), the next inter-event interval of every
agent is strictly positive and is lower-bounded by a common constant.
\end{enumerate}
\end{theorem}
\begin{proof}
(1).  Note
\begin{align*}
\dot{x}^j(t)-\dot{x}^i(t)=&f(x^j)-f(x^i)-c[(L\otimes\Gamma)x(t_{k_j(t)}^j)]_j\\
&+c[(L\otimes\Gamma)x(t_{k}^i)]_i,~~~~~t\in[t_k^i,t_{k+1}^i).
\end{align*}
Combining with the facts that $f$ is Lipschitz and $V(t)$ is decreasing, we have
\begin{equation}\label{deri_dis}
\|\dot{x}^j(t)-\dot{x}^i(t)\|\leq L_f\|x^j(t)-x^i(t)\|+c\|\Gamma\|V(t_k^i).
\end{equation}
According to
\begin{equation}\label{dyn_e}
e_i(t)=\sum_{j}L_{ij}\Gamma\int_{t_k^i}^t[\dot{x}^j(s)-\dot{x}^i(s)]ds
\end{equation}
and inequality (\ref{deri_dis}), we have
\begin{align}\label{e}
\|e_i(t)\|\leq
&\frac{1}{\sqrt{\lambda_1(P)}}\left(L_f\|\Gamma\|+mc\|\Gamma\|^2\right)\sqrt{2V(t_k^i)}(t-t_k^i).
\end{align}
And, noting $V(t)=\sum_{i=1}^{m}z_{i}^{2}(t)$, there exists $i_{*}$ such that
\begin{eqnarray}\label{z}
z_{i_{*}}(t)\ge\sqrt{\frac{1}{m}V(t)}.
\end{eqnarray}
From the condition (\ref{vi}), (\ref{dV1}) gives
\begin{eqnarray}
\dot{V}&\ge&2\sigma x^{\top}(t)(L\otimes P)x(t)-2cx^{\top}(t)[L^{2}\otimes(P\Gamma)^{s}]x(t)
\nonumber\\
&&+c2x^{\top}(t)(L\otimes P)e(t).\label{xx1}
\end{eqnarray}
Noting
\begin{eqnarray*}
u^{\top}[L^{2}\otimes(P\Gamma)^{s}]u\le \lambda_{m}(L)\frac{\|(P\Gamma)^{s}\|}{\lambda_{1}P)}u^{\top}[L\otimes(P)]u,
\end{eqnarray*}
for all $u\in\mathbb R^{mn}$,
\begin{eqnarray*}
2x^{\top}(L\otimes P)e&\geq& -{\upsilon}x^{\top}(L^2\otimes P^2)x-\frac{1}{\upsilon} e^{\top}e\\
&\geq& -{\upsilon\lambda_m(L)\lambda_m(P)}x^{\top}(L\otimes P)x-\frac{1}{\upsilon}e^{\top}e,
\end{eqnarray*}
and event (\ref{event1}), (\ref{xx1}) gives
\begin{eqnarray*}
\dot{V}(t)\ge \varpi V(t)
\end{eqnarray*}
for all $t$ before the next triggering time, with
\begin{eqnarray*}
\varpi&=&2\sigma-2\lambda_{m}(L)\frac{\|(P\Gamma)^{s}\|}{\lambda_{1}(P)}-\upsilon\lambda_{m}(L)
\lambda_{m}(P)\\
&&-\frac{1}{\upsilon}\frac{\beta'}{c\lambda_{m}(L)\lambda_{m}(P)}.
\end{eqnarray*}
Thus, we have $V(t)\ge V(t_{k}^{i_{*}})\exp(\varpi (t-t^{i_{*}}_{k}))$. Combined with (\ref{e}) and (\ref{z}), this implies that for each $t\le t^{i_{*}}_{k}+\tau_{O}$, where $\tau_{O}$ satisfies
\begin{eqnarray*}
\frac{\left(L_f\|\Gamma\|+mc\|\Gamma\|^2\right)\sqrt{2}}{\sqrt{\lambda_1(P)}}\tau_{O}
=\frac{\beta'\sqrt{\frac{1}{m}\exp(\varpi\tau_{O})}}{c\sqrt{\lambda_m(L)\lambda_m(P)}}
\end{eqnarray*}
we have the inequality in (\ref{event1}) holds. Therefore, we have the next triggering time should be larger than $t^{i_{*}}_{k}+\tau_{O}$.

In addition, if $z_{i}^{2}(t)\ge \varsigma V(t)$ for all $i=1,\cdots,m$ and $t\ge 0$, replace
(\ref{z}) by
\begin{eqnarray}
z_{i}(t)\ge \sqrt{\varsigma V(t)}
\end{eqnarray}
for all $i=1,\cdots,m$. Then following the same arguments after (\ref{z}), we can conclude that we have the inequality in (\ref{event1}) holds for all $t\ge t_{k}^{i}+\tau_{O'}$ with some positive $\tau_{O'}$ satisfying:
\begin{eqnarray*}
\frac{\left(L_f\|\Gamma\|+mc\|\Gamma\|^2\right)\sqrt{2}}{\sqrt{\lambda_1(P)}}\tau_{O'}
=\frac{\beta'\varsigma\sqrt{\exp(\varpi\tau_{O'})}}{c\sqrt{\lambda_m(L)\lambda_m(P)}}.
\end{eqnarray*}

(2). Under updating rule (\ref{event2}).
By inequality (\ref{Vcase2}), we get
$$
V(t)\leq \rho \exp{[-2\min(b,\beta-\beta')t]}
$$
with $\rho=V(0)+\frac{a^2c^2m\lambda_m(L)\lambda_m(P)}{4v\beta'(\beta-\beta')}+1$.  Hence, combined with (\ref{e}), this gives
\begin{align*}
\|e_i(t)\|\leq &\sqrt{\frac{2\rho}{\lambda_1(P)}} \left(L_f\|\Gamma\|+mc\|\Gamma\|^2\right)\\
&\times \exp{[-\min(b,\beta-\beta')t_k^i]}(t-t_k^i).
\end{align*}
Therefore, (\ref{event2}) is guaranteed by the following inequality
\begin{align*}
&\sqrt{\frac{2\rho}{\lambda_1(P)}} \left(L_f\|\Gamma\|+mc\|\Gamma\|^2\right)
\exp{[-\min(b,\beta-\beta')t_k^i]}(t-t_k^i)\\
& \leq a\exp{(-bt)}.
\end{align*}
Since at time $t=t_k^i$, $e_i(t)=0$ holds. Based on rule (\ref{event2}),
the next event will not trigger until $e_i(t)=a\exp{(-bt)}$. Thus, the
inter-event intervals $\tau=t_{k+1}^i-t_k^i$ is lower bounded by the
solution $\tau_D$ of the following equation
\begin{align*}
&\sqrt{\frac{2\rho}{\lambda_1(P)}} \left(L_f\|\Gamma\|+mc\|\Gamma\|^2\right)
\exp{[-\min(b,\beta-\beta')t_k^i]}\tau_D\\
&= a\exp{[-b(\tau_D+t_k^i)]}.
\end{align*}
It can be seen that this equation has a positive solution. This completes the proof.
\end{proof}
\begin{remark}
The updating rules (\ref{event1}) and (\ref{event2}) are different but
closely related to each other in some respects. It can be seen from
inequalities (\ref{Vcase1}) and (\ref{Vcase2}) used in the derivation that
the convergence behavior for (\ref{event1}) might be better than
(\ref{event2}). However, it makes rule (\ref{event1}) more complicated than
(\ref{event2}), since each agent should receive the message of the states
of its neighborhood but rule (\ref{event2}) does not need. Therefore, rule
(\ref{event1}) costs more updating times than (\ref{event2}). Moreover, as
shown by Theorem \ref{thm2}, rule (\ref{event2}) can guarantee the
positivity of the intervals to next updating time for all agents but rule
(\ref{event1}) can only guarantee it for at least one agent at each time or
for all nodes under some specific additional conditions.
\end{remark}
\section{Discrete-time Monitoring}\label{2.2}
By the discrete-time monitoring strategy, each node $i$ only needs its
local neighborhood's state at time-points $t^{i}_{k}$, $k=1,2,\cdots$. By
this way, the design of the next $t^{i}_{k+1}$ depends only on the local
states at time $t^{i}_{k}$, other than the triggering event (\ref{event1}),
(\ref{event2}), which requires the continuous time states. For early works, see \cite{LC2004,LC2007} for reference.

Consider system (\ref{cds}) and  the candidate Lyapunov
function $V(x)$ with its derivative (\ref{dV1}). To propose a triggering criterion,
which depends only on $t_{k}^{i}$ by the criterion (\ref{event1}) in
Theorem \ref{thm1}, we need to estimate the bounds of
$(x^{q}(t)-x^{q}(t_{k}^{l}))-(x^{i}(t)-x^{i}(t_{k}^{l}))$ for any $q,i$
with $L_{iq}\neq 0$ and $x^{i}(t)-x^{j}(t)$ for any $i\ne j$.

First, we estimate the lower-bound of $x_i(t)-x_j(t)$, which satisfies
\begin{eqnarray*}
\frac{d[x^{i}(t)-x^{j}(t)]}{dt}=[f(x^{i}(t))-f(x^{j}(t))]+\theta_{i}(t_{k_i(t)}^i)-\theta_{j}(t_{k_j(t)}^j)
\end{eqnarray*}
by provided the initial values at $t^{i}_{k}$: $x^{i}(t^{i}_{k})$ and $x^{j}(t^{i}_{k})$.
This can be generalized as
\begin{eqnarray}
\begin{cases}\frac{du}{dt}=f(u(t))+\theta&u(0)=u_{0}\\
\frac{dv}{dt}=f(v(t))+\vartheta&v(0)=v_{0}.
\end{cases}\label{r2}
\end{eqnarray}

Suppose that the solutions  satisfy the following inequality:
\begin{eqnarray}
\bigg[(u(t)-v(t))^{\top}{P}(u(t)-v(t))\bigg]^{1/{2}}\geq\varrho(t,\theta,\vartheta,u_{0},v_{0})
\end{eqnarray}
Here $\varrho$ can be regarded as the lower-bound estimation of the distance (in $P$-norm) between two trajectories:
\begin{align*}
&\|u(t)-v(t)\|_{P}\\
=&\bigg\|\int_{0}^{t}[f(u(s))-f(v(s))]ds
+(\theta-\vartheta)t
-(u_0-v_0)\bigg\|_{P}.
\end{align*}

To specify $\varrho$, the celebrated Gronwall-Bellman inequality is used,
which can be verified straightforwardly and described as follows:
\begin{lemma} \cite{Gron,Bell}
For a nonnegative differentiable function $x(t)$ and two continuous functions: $\alpha(t)$ and $\beta(t)$, defined on interval $[a,b]$, if
\begin{eqnarray*}
\dot{x}\ge\alpha(t)x(t)+\beta(t)
\end{eqnarray*}
for all $t\in[a,b]$, then we have
\begin{eqnarray}
&&x(t)\ge x(a)\exp\left(\int_{a}^{t}\alpha(s)ds\right)\nonumber\\
&&+\int_{a}^{t}\beta(s)
\exp\left(\int_{s}^{t}\alpha(u)du\right);\label{gw1}
\end{eqnarray}
for two continuous functions $x(t)$ and $\beta(t)$, and an integrable
function $\alpha(t)$, defined on interval $[a,b]$, if $\beta(\cdot)$ is
nonnegative and
\begin{eqnarray*}
x(t)\le \alpha(t)+\int_{a}^{t}\beta(s)x(s)ds
\end{eqnarray*}
for all $t\in[a,b]$, then we have
\begin{eqnarray}
x(t)\le \alpha(t)+\int_{a}^{t}\alpha(s)\beta(s)\exp\left(\int_{s}^{t}\beta(u)du\right).\label{gw2}
\end{eqnarray}
\end{lemma}
By this lemma, $\varrho$ is a nonnegative-valued continuous map and satisfies (i). $\varrho(\cdot,\theta,\theta,u_{0},u_{0})\equiv 0$; (ii). $\varrho(0,\cdot,\cdot,u_{0},u_{0})\equiv 0$.
For example, assuming that condition (\ref{vi}) holds,
we have
\begin{align*}
&\frac{d}{dt}[(u(t)-v(t))^{\top}P(u(t)-v(t))]\left|_{(\ref{r2})}\right.\\
=&2(u-v)^{\top}P[f(u)-f(v)+\theta-\vartheta]\\
\ge&{2\sigma}(u-v)^{\top}P(u-v)-\mu (u-v)^{\top}P(u-v)\\
&-\frac{1}{\mu}(\theta-\vartheta)^{\top}P(\theta-\vartheta)
\end{align*}
for any $\mu>0$. By the Gronwall-Bell inequality (\ref{gw1}), we have
\begin{align*}
&(u(t)-v(t))^{\top}P(u(t)-v(t))\\
&\ge\exp{[({2\sigma}-\mu)t]}(u_{0}-v_{0})^{\top}P(u_{0}-v_{0})\\
&-\frac{(\theta-\vartheta)^{\top}P(\theta-\vartheta)/\mu}{2\sigma-\mu}\bigg\{\exp[({2\sigma}-\mu)t]-1\bigg\}
\end{align*}
which is positive for a small interval of $t$, starting from $0$, and $u_{0}\ne v_{0}$.

It can be seen that (\ref{vi}) holds for a large class of functions
$f(\cdot)$. For example,  if there exists some $\sigma\in\mathbb R$ such that
\begin{eqnarray}
\{P\frac{\partial f}{\partial x}(x)\}^{s}\ge\sigma P\label{Pf}
\end{eqnarray}
for all $x\in\mathbb R^{n}$ and some $\sigma\in\mathbb R$, then
\begin{eqnarray*}
&&(u-v)^{\top}P[f(u)-f(v)]=\int_{0}^{1}(u-v)^{\top}\\
&&\{P\frac{\partial f}{\partial x}(\lambda(v-u)+v)\}^{s}(u-v)d\lambda\ge\sigma (u-v)^{\top}P(u-v),
\end{eqnarray*}
which implies (\ref{vi}). In particular, if the Jacobin of $f(\cdot)$ is
bounded, namely, $ \|{\partial f}/{\partial x}\|\ge J_{f}$ for some $J_f>0$,
then we have
\begin{eqnarray*}
\{P\frac{\partial f}{\partial x}(y)\}^{s}\ge-J_{f}\|P\|,
\end{eqnarray*}
which implies $\sigma=-J_{f}\frac{\|P\|_{2}}{\lambda_{\min}(P)}$ in
(\ref{Pf}).

Second, we consider the differential equations (\ref{r2}) and suppose that the
solutions of (\ref{r2}) satisfy the following inequality:
\begin{eqnarray}
\|(u(t)-u_{0})-(v(t)-v_{0})\|\le\rho(t,\theta,\vartheta,u_{0},v_{0}),\label{r1}
\end{eqnarray}
where $\rho$ is nonnegative-valued continuous map that depends on the node
dynamics map $f(\cdot)$, the initial value $u_{0},v_{0}$ and inputs
$\theta,\vartheta$, and satisfies  $\rho(0,\cdot,\cdot,\cdot,\cdot)\equiv
0$. Geometrically, $\rho$ is an upper-bound estimation of the difference
between the displacements of two trajectories of (\ref{r2}) with respect to
their initial locations,
\begin{align*}
&\|(u(t)-u_{0})-(v(t)-v_{0})\|\\
=&\left\|\int_{0}^{t}[f(u(s))-f(v(s))]d s+(\theta-\vartheta) t\right\|.
\end{align*}

For example, if $f(\cdot)$ is Lipschitz (on the two trajectories):
$\|f(u(s))-f(v(s))\|\le L_{f}\|u(s)-v(s)\|$ for all $s\ge 0$, then we have
\begin{align*}
&\|(u(t)-u_{0})-(v(t)-v_{0})\|\\
\le& L_{f}\int_{0}^{t}\|(u(s)-u_{0})-(v(s)-v_{0})\|ds\\
&+(\|\theta-\vartheta\|+L_f\|u_{0}-v_{0}\|)~t.
\end{align*}
By the Gronwall inequality (\ref{gw2}), we have
\begin{align}
\label{Lips}
\nonumber&\|(u(t)-u_{0})-(v(t)-v_{0})\|\\
\le&\frac{(\|\theta-\vartheta\|+L_f\|u_{0}-v_{0}\|)}{L_{f}}[\exp(L_{f}t)-1].
\end{align}
It can be seen that the upper-bound equals to zero if $t=0$.

It can be seen that the estimation of $\rho$ and $\varrho$ substantially
depends on the form of $f(\cdot)$. There might not be a unified approach to
give precise estimation for general $f(\cdot)$ but might be done case by
case. Therefore, an efficient but cost way is to use integrators that
simulate the node dynamics of $\dot{u}=f(u)+\theta$ as the generators of
$\rho$ and $\varrho$. These generators are independent of the states of the
nodes and so parallel to the networked systems. Figs. \ref{rho}  and
\ref{varrho} show the configurations of the generators of $\rho$ and
$\varrho$ respectively.

\begin{figure
\centering
\includegraphics[width=0.5\textwidth]{rho.eps}
\caption{$\rho$ generator.} \label{rho}
\end{figure}

\begin{figure
\centering
\includegraphics[width=0.5\textwidth]{varrho.eps}
\caption{$\varrho$ generator.} \label{varrho}
\end{figure}

Let
\begin{align*}
&\theta^{i}_{k}=-c\sum_{j=1}^{m}L_{ij}(x^{j}(t^{i}_{k})-x^{i}(t^{i}_{k})),\\
&\theta_{k_{j}(t)}^{j}=-c\sum_{l=1}^{m}L_{jl}[x^{l}(t^{j}_{k_{j}(t)})-x^{j}(t^{j}_{k_{j}(t)})]
\end{align*}
Then, for $\tau\ge t^{i}_{k}$, we have
\begin{align*}
&\|e_{i}(\tau)\|\le\sum_{q\ne i}(-L_{iq})\|\Gamma\|\rho(\tau-t_{k}^{i},\theta^{i}_{k},\theta_{k_{q}(\tau)}^{q},x^{i}(t_{k}^{i}),x^{q}(t^{i}_{k})),\\
&z_{i}(\tau)\ge\sqrt{\sum_{j\ne i}(-L_{ij})\varrho^{2}(\tau-t_{k}^{i},\theta^{i}_{k},\theta_{k_{j}(\tau)}^{j},x^{i}(t_{k}^{i}),x^{j}(t^{i}_{k}))}.
\end{align*}

With these assumptions and Theorem \ref{thm1}, we have the following result.
\begin{theorem}\label{thm3}
Suppose that $f\in Quad(P,\alpha\Gamma,\beta)$ with positive matrix $P$ and $\beta>0$ such that $P\Gamma$ is semi-positive definite. $c\lambda_{2}(L)>\alpha$. For any positive $\beta'<\beta$, set $\xi^{i}_{k}$ by:
\begin{align}
\xi^{i}_{k}=\max\bigg\{&\xi:~\sum_{q\ne i}(-L_{iq})\|\Gamma\|\rho(\xi,\theta^{i}_{k},\theta_{k_{q}(\xi+t_k^i)}^{q},x^{i}(t_{k}^{i}),x^{q}(t^{i}_{k}))
\nonumber\\
\nonumber&\le\frac{\beta'}{c\sqrt{\lambda_{m}(L)\lambda_m(P)}}\bigg[\sum_{q\ne i}(-L_{iq})\varrho^{2}(\xi,\theta^{i}_{k},\theta_{k_{j}(\xi+t_k^i)}^{j},\\
&~~~~~~~~~~~~~~~~~~
~~~~~~~~~x^{i}(t_{k}^{i}),x^{j}(t^{i}_{k}))\bigg]\bigg\}.\label{event3}
\end{align}
The event timing $\{t^{i}_{k}\}$ are set by the following scheme:
\begin{enumerate}
\item Initialization: $t_{0}^{i}=0$ for all $i=1,\cdots,m$;
\item For node $i$, set $\xi^{i}_{k}$ via its neighbor's and its own current states and diffusion by (\ref{event3});
\item If one of its neighbors, for example, $j$, triggers at $t=t^{j}_{k'+1}$ ($k'$ is the latest event at node $j$ before $t$), then  replace $\theta^{j}_{k'}$ by $\theta^{j}_{k'+1}$ in (\ref{event3}) and go to Step 2;
\item Let $t^{i}_{k+1}=t^{i}_{k}+\xi^{i}_{k}$, the event triggers at node $i$ by changing $t^{i}_{k}$ in (\ref{cds}) to $t^{i}_{k+1}$.
\end{enumerate}
Then, system (\ref{cds}) synchronizes.
\end{theorem}
The proof of this theorem can be derived by using (\ref{event1}) in Theorem
\ref{thm1} directly.

\begin{remark}
Firstly,  if node $i$ has a neighbor $j$ satisfying $x^{j}(t^{i}_{k})\ne
x^{i}(t^{i}_{k})$, then $\xi_{k}^{i}$ in (\ref{event3}) is well defined. In
fact, in (\ref{event3}), if $\xi=0$, the left-hand side of the inequality in
(\ref{event3}) equals zero while the right-hand side is nonzero. Therefore, by
the continuous dependence of the parameters in the system (\ref{cds}),
(\ref{event3}) has indeed a positive maximum $\xi$.

Secondly, each node $i$ needs to know the states of itself and
its neighbors. In details, when one node $j$ is triggered, it sends off its new
coupling terms, $\theta_{j}(t^{j}_{k_{j}(t)})$, to all its neighbors for
their updating the estimation of $\xi^{i}_{k}$ for their next updating
times.
\end{remark}

\begin{remark}
In case that $x^{j}(t^{i}_{k})=x^{i}(t^{i}_{k})$ for all neighbors $j$ of node $i$, both left and right sides equal zero, which might lead to a Zeno behavior. To avoid the Zeno behavior, we provide a triggering event, which depends only on $t_{k}^{i}$ by the rule (\ref{event2}) in Theorem \ref{thm1}. Here, we only need to estimate the bounds of $(x^{q}(t)-x^{q}(t_{k}^{l}))-(x^{i}(t)-x^{i}(t_{k}^{l}))$ for any $q,i$ with $L_{iq}\neq 0$. Note
\begin{align*}
&(x^{q}(t)-x^{q}(t_{k}^{l}))-(x^{i}(t)-x^{i}(t_{k}^{l}))\\
=&\int_{t_{k}^l}^t\left[f(x^q(s))-f(x^i(s))+\theta_{k_{q}(s)}^{q}-\theta^{i}_{k}\right]ds.
\end{align*}
Combing with  $\|\theta^{i}_{k}-\theta_{k_{q}(s)}^{q}\| \leq M$, where $M>0$ is some constant, we suppose that
the solutions of (\ref{r2}) satisfy the following inequality:
\begin{eqnarray}
\|(u(t)-u_{0})-(v(t)-v_{0})\|\le\rho_1(t,u_{0},v_{0}).\label{rho2}
\end{eqnarray}
Then, for $\tau\ge t^{i}_{k}$, we have
\begin{align*}
\|e_{i}(\tau)\|\le\sum_{j\ne i}(-L_{ij})\|\Gamma\|\rho_1(\tau-t_{k}^{i},x^{i}(t_{k}^{i}),x^{j}(t^{i}_{k})).
\end{align*}
\end{remark}

\begin{theorem}\label{thm4}
Suppose that $f\in Quad(P,\alpha\Gamma,\beta)$ with positive matrix $P$ and $\beta>0$ such that $P\Gamma$ is semi-positive definite. $c\lambda_{2}(L)>\alpha$. For any positive $\beta'<\beta$, set inter-event interval $\xi^{i}_{k}$ by:
\begin{align}\label{event4}
\nonumber\xi^{i}_{k}=\max\bigg\{&\xi:~\sum_{j\ne i}(-L_{ij})\|\Gamma\|\rho_1(\xi,x^{i}(t_{k}^{i}),x^{j}(t^{i}_{k}))\\
&\le a\exp{[-b(\xi+t_k^i)]}\bigg\}.
\end{align}
The event times $\{t^{i}_{k}\}$ are set by the following scheme:
\begin{enumerate}
\item Initialization: $t_{0}^{i}=0$ for all $i=1,\cdots,m$;
\item For node $i$, search $\xi^{i}_{k}$ via its neighbor's and its own current states by (\ref{event3});
\item Triggers node $i$ by changing $t^{i}_{k}$ in (\ref{cds}) to $t^{i}_{k+1}=t^{i}_{k}+\xi^{i}_{k}$,.
\end{enumerate}
Then, system (\ref{cds}) synchronizes.
\end{theorem}

This theorem can be derived by using (\ref{event2}) in Theorem \ref{thm1}.

It should be highlighted that under rule (\ref{event4}), every node does not need to know the coupling terms of neighbors anymore and the inter-event intervals have a lower-bound.

In fact, in (\ref{event4}), if $\xi=0$, the left-hand part equals zero while the right-hand is nonzero. Therefore, according to the continuous dependence of the parameters in the system (\ref{cds}), (\ref{event4}) has indeed a positive maximum argument $\xi_k^i$.

Similar to Theorem \ref{thm2}, we have
\begin{theorem}\label{thm5}
Suppose that $f\in Quad(P,\alpha\Gamma,\beta)$ with positive matrix $P$ and $\beta>0$, satisfies Lipschitz condition with Lipschitz constant $L_f$, and there exists some $\sigma$ (possibly negative) such that (\ref{vi}) holds for all $u,v\in\mathbb R^{n}$ . $c\lambda_{2}(L)>\alpha$ and $P\Gamma$ is semi-positive definite. For any $\beta'<\beta$, any initial condition and any time $t\leq 0$, we have
\begin{enumerate}
\item[(1)]
under the updating rule (\ref{event3}), there exists $\tau_{O}>0$ such that there exists at least one agent
$k\in\{1,\cdots,m\}$ such that the next inter-event interval is strictly
positive and has the lower-bound $\tau_{O}$; in addition, if there exists $\varsigma>0$ such that $z_{i}^{2}(t)\ge \varsigma V(t)$ for all $i=1,\cdots,m$ and $t\ge 0$, then the next inter-event interval of every
agent is strictly positive and has a common positive lower-bound.
\item[(2)] suppose $f(\cdot)$ is Lipschitz with constant $L_f$. Then,
under the updating rule (\ref{event4}), the next inter-event interval of every agent is strictly positive and has a common lower-bound $\tau_D'$.
\end{enumerate}

\end{theorem}

\begin{remark}
In comparison with the continuous-time monitoring, the discrete-time
monitoring works well particularly when the states of nodes cannot be
monitored spontaneously. Generally speaking, the main difference between
these two monitoring strategies is that continuous-time monitoring
determines the next updating time in an on-line way, based on the
spontaneous information of states of nodes. Instead, the discrete-time
monitoring predicts the next updating time. Therefore, the discrete-time
monitoring costs less for collecting state information than continuous-time
monitoring. However, as a trade-off, it needs more calculations in
predicting the next updating time, as mentioned in (\ref{event3}) or
(\ref{event4}).
\end{remark}

\section{Examples}
In this section, we present two examples to illustrate the theoretical results. The system is an array of $10$ linearly coupled Chua circuits with the node dynamics
\begin{eqnarray}
f(z)=\left[\begin{array}{c}p*(-z_{1}+z_{2}-g(z_{1}))\\
z_{1}-z_{2}+z_{3}\\
-q*z_{2}\end{array}\right],
\end{eqnarray}
where $g(z_{1})=m_{1}*z_{1}+1/2*(m_{0}-m_{1})*(|z_{1}+1|-|z_{1}-1|)$, with the parameters $p=9.78$, $q=14.97$, $m_{0}=-1.31$ and $m_{1}=-0.75$, which implies that the intrinsic node dynamics (without diffusion) have a double-scrolling chaotic attractor \cite{chua}. The coupling graph topology is shown in Fig. \ref{topology}. $L$ is picked as the Laplacian of the graph where each link has uniform weight $1$. Then, the largest and smallest nonzero eigenvalues equal to $\lambda_{2}=0.8363$ and $\lambda_{m}=7.3484$ respectively. Let $P=\Gamma=I_{3}$. To estimate the parameter $\beta$ in the $Quad$ condition, noting the Jacobin matrices of $f$ is one of the following
\begin{align*}
&A_{1}=\left[\begin{array}{lll}-2.445&9.78&0\\
1&-1&1\\0&-14.97&0\end{array}\right]\\
&A_{2}=\left[\begin{array}{lll}3.0318&9.78&0\\
1&-1&1\\0&-14.97&0\end{array}\right]
\end{align*}
then we can estimate $\beta'=\alpha-\lambda_{\max}((A_{2})^{s})=\alpha-\chi$, where $
\chi=9.1207$ is the upper-bound of the largest eigenvalues of the symmetry parts of all possible Jacobin matrices of $f$.

\begin{figure
\centering
\includegraphics[width=0.3\textwidth]{Network_topology.eps}
\caption{Topology of the graph of the coupled system and the pinned node.} \label{topology}
\end{figure}

The ordinary differential equation (\ref{cds}) is numerically solved by the
Euler method with a time step $0.001$ (sec) and the time duration of the
numerical simulations is $[0,2]$(sec).

\subsection{Continuous-time monitoring}

First, we consider the updating rule (\ref{event1}). According to $c\lambda_{2}>\alpha$, where $\lambda_{2}(L)$ is the smallest eigenvalue of $L$, except the unique zero eigenvalue, the supremum of the term $\beta'/(c\sqrt{\lambda_m})$ is estimated as follows:
\begin{eqnarray*}
\sup\frac{\beta'}{c\sqrt{\lambda_m(L)}}=\frac{c\lambda_2(L)-\lambda_{\max}(A_{2}^{s})}{c\sqrt{\lambda_m(L)}}\to\frac{\lambda_{2}(L)}{\sqrt{\lambda_m(L)}}
\end{eqnarray*}
as $~c\to\infty$, by picking $\alpha=c\lambda_{2}(L)$.
Fig. \ref{ets} shows the variation of $\beta'/(c\sqrt{\lambda_m(L)})$ with respect to $c$. In this example, we pick $c=20.3281$, which implies $\beta'/(c\sqrt{\lambda_m(L)})=0.1450$. We employ the updating rule (\ref{event1}) in Theorem \ref{thm1}. Fig. \ref{variation_x_con1} presents the dynamics of each component of the $10$ nodes and show that the coupled system (\ref{cds}) reaches synchronization. Fig. \ref{variation_v_all} shows that $V(t)$ decreases with respect to time and converges toward zero as time goes to infinity.

\begin{figure
\centering
\includegraphics[width=0.5\textwidth]{variation_to_c.eps}
\caption{Variation of $\beta'/(c\sqrt{\lambda_m})$ under continuous-time monitoring for synchronization with respect to the coupling strength $c$.} \label{ets}
\end{figure}

\begin{figure
\centering
\includegraphics[width=0.5\textwidth]{variation_x_con1.eps}
\caption{Dynamics of components of the coupled system (\ref{cds}) under the event triggering rule (\ref{event1}). }
\label{variation_x_con1}
\end{figure}

Second, we employ the updating rule (\ref{event2}). We take the same value
of $c$ as above and $a=0.5$, $b=0.5$. Fig.
\ref{variation_x_con2} presents the dynamics of each
component of the $10$ nodes and show that the coupled system (\ref{cds})
reaches synchronization. Fig. \ref{variation_v_all} shows that $V(t)$  decreases
with time and converges to zero as time goes to infinity.

\begin{figure
\centering
\includegraphics[width=0.5\textwidth]{variation_x_con2.eps}
\caption{Dynamics of components of the coupled system (\ref{cds}) under the event triggering rule (\ref{event2}). }
\label{variation_x_con2}
\end{figure}

\subsection{Discrete-time monitoring}

First, we employ the updating rule (\ref{event3}). The term $\beta'/(c\sqrt{\lambda_{m}})$ can be directly derived from the arguments above. We pick the same $c$ and then $\beta'/(c\sqrt{\lambda_{m}})$ is the same as above. We employ the event trigger algorithm (\ref{event3}) in Theorem \ref{thm3}. Fig. \ref{variation_x_dis1} presents the dynamics of each component of the $10$ nodes and shows that the coupled system (\ref{cds}) reaches synchronisation. Fig. \ref{variation_v_all} shows that $V(t)$ decreases with time and converges to zero.

\begin{figure
\centering
\includegraphics[width=0.5\textwidth]{variation_x_dis1.eps}
\caption{Dynamics of components of the coupled system (\ref{cds}) under the event triggering rule (\ref{event3}). }
\label{variation_x_dis1}
\end{figure}

Second, we consider the updating rule (\ref{event4}).  We pick the same $c$ as above and $a=0.5$, $b=0.5$. Fig. \ref{variation_x_dis2} presents the dynamics of each component of the $10$ nodes and show that the coupled system (\ref{cds}) reach synchronization.  Fig. \ref{variation_v_all} shows that $V(t)$ decreases with respect to time and converges toward zero as time goes to infinity.

\subsection{Performance comparison}
In comparison, we consider the original linear coupled system as follows:
\begin{eqnarray}
\frac{dx^{i}(t)}{dt}=f(x^{i}(t))-c\sum_{i=1}^{m}L_{ij}\Gamma(x^{j}(t)-x^{i}(t)),\label{org}
\end{eqnarray}
for $i=1,\cdots,m$. By the same setups of model and numerical approach as
above, its performance in terms of converge rates of $V(t)$ is shown
similar with those of event-triggered rules (\ref{event1}) and
(\ref{event3}), as comparatively shown by Fig. \ref{variation_v_all}. As for the performance of rules
(\ref{event2}) and (\ref{event4}), since the exponential convergence rates
are pre-designed, as shown by (\ref{event2}) and (\ref{event4}), it is not
surprising that their convergence rates are not as good as (\ref{org}).
However, their updating times of these rules are much less than those of
rules  (\ref{event1}) and (\ref{event3}), as comparatively shown in Figs. \ref{triggering_times_1}-\ref{triggering_times_2}.

\begin{figure
\centering
\includegraphics[width=0.5\textwidth]{variation_x_dis2.eps}
\caption{Dynamics of components of the coupled system (\ref{cds}) under the event triggering rule (\ref{event4}). }
\label{variation_x_dis2}
\end{figure}

\begin{figure
\centering
\includegraphics[width=0.5\textwidth]{variation_v_all.eps}
\caption{Dynamics of $V(t)$ for systems with continuous updating, continuous monitoring under rules (\ref{event1}),(\ref{event2}), discrete monitoring under rules (\ref{event3}), (\ref{event4}).}
 \label{variation_v_all}
\end{figure}

\begin{figure
\centering
\subfigure[under updating rules (\ref{event1}) and (\ref{event3})]{
\includegraphics[width=.5\textwidth]{triggering_times_1.eps}
\label{triggering_times_1}
}
\subfigure[under updating rules (\ref{event2}) and (\ref{event4})]{
\includegraphics[width=.5\textwidth]{triggering_times_2.eps}
\label{triggering_times_2}
}
\caption{Histogram of triggering times of each node in $[1,2]$(sec).}
\end{figure}

\section{conclusion}
In this paper, we employed event-triggered coupling configurations to
realize synchronization for linearly coupled dynamical systems. We studied
both continuous monitoring and discrete monitoring schemes: continuous
monitoring scheme means that each node collects its neighborhood's
instantaneous state, and discrete monitoring scheme means that each node
obtains its neighborhood's states at the event triggered time. The
event-triggered rules were proved to perform well and can exclude Zeno
behaviors, as proved for some cases and illustrated by simulations. We showed that there are
trade-offs between better performance in terms of fast convergence and less
updating time slots, and between more cost in observation of states and
more computation load of predicting next updating times. One step further,
there are a few issues, including how to estimate the number of updating
time slots and its dependence on the parameters in the rule and the
structure of network structure, which merits the future research.

\section*{Acknowledgement}
The authors are very grateful to reviewers for their useful comments
and suggestions.

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Here are a few recent articles that are worth reading given recent discussions about VBACs, cesarean sections (elective or otherwise), and the ever-climbing cesarean rate.
Two weeks before Kristi Ashley gave birth to a son in 2007, an ultrasound exam estimated the baby at a hefty 12 pounds, 10 ounces — too big, her doctor believed, for a safe vaginal delivery. After the child weighed in at 9 pounds, 4 ounces in the delivery room, Ashley came to believe that the planned cesarean section she had, with its attendant pain, long recovery and what she called "emotional damage," may have been a rush to judgment.
"It's very hard to go up against your physician, especially at the 12th hour," said Ashley, 38, of Hopewell Junction. "I think doctors are very quick these days to get scared. They would rather opt for the surgical solution."
Determined to avoid another surgical birth and aided by a supportive doctor, hospital and birthing coach, Ashley last month did something that has become increasingly rare for post-cesarean women today: She gave birth vaginally, to another son.
In an era of soaring malpractice premiums, technology that sometimes sets off false alarms, physicians pressed for time and mothers-to-be conflicted by fear, cesarean-section birth is soaring to its highest levels ever. Read the rest of the article here.
The next article by the same author, Modern medicine increasingly intervenes in the birth process, discusses many of the same issues, in addition to the increasing medicalization of childbirth.
In the decade through 2002, something momentous happened to babies in the wombs of American women, especially white women. The average time fetuses spent there decreased from 40 weeks to 39.
The decline, reported in a 2006 study in the medical journal Seminars in Perinatology, appears to have little to do with nature.
Instead, earlier births may be the outcome of "increased use of induction (of labor) and other obstetric interventions such as cesarean delivery," said a January report by the U.S. Centers for Disease Control. Prematurity rose 20 percent since 1990, the report said, and the rate of low birth-weight babies hit a 40-year high.
Researchers, midwives, birth coaches and mothers point to such data as symptoms of a flawed system of birthing in America, one they say over-manages, over-medicates and over-monitors labor and delivery, often leading to unnecessary cesarean-section births. Read the rest of the article here.
The last article is a glimmer of hope amidst the gloom of our contemporary obstetrical culture. In C-section births fall, one hospital has lowered its cesarean rate (18% last year, usually around 16%). Some of the key practices the hospital has adopted are taking a midwifery approach to childbearing with a focus on facilitating spontaneous, natural births, minimizing the routine use of technology and interventions, and offering and encouraging VBACs.
While more and more women choose to undergo Cesarean section births despite a national push by the federal government to decrease the number, the local rate has declined and is well below the state average.
North Adams Regional Hospital performs significantly fewer c-sections than other hospitals around the state — an average of 18 percent of all births at the hospital compared to the state average of 34 percent, according to reports released by the state Department of Public Health.
The hospital also has a better prenatal care record, according to the reports: 94 percent of women giving birth have had nine or more prenatal care visits versus the state average of 87 percent having that many visits.
"I think what is being reflected in our numbers is that we are taking a more 'midwifery' approach with our practice then before," Robin Rivinus, a certified nurse midwife with Northern Berkshire Obstetrics & Gynecology at the hospital, said last week. "It means that we do fewer unnecessary interventions — inductions, Cesarean sections, episiotomies. We treat childbirth as the normal, natural thing that it is. We only step in when it's medically necessary, which is much better for both the mother and the baby." Read the rest of the article here.
Thank you for posting all three of these. It's nice to hear some noise being made about this outside our little blogosphere...and nice to see that at least one hospital knows what to do about this calamity!
Hilarious! These were all of the articles I was going to blog tonight!! Now I can just send a convenient link right to this post.
I found that part about "shortening the gestational age" to be a bit disturbing. I wonder what the long-term effects are going to be of all this.
The more I think about this subject, the more I think that this problem has two roots.
1)C/sections are thought of as normal. We need to see a shift of thought on what normal childbirth is, much like we've been seeing in the breastfeeding/formula problem. My oldest child is 5, and I've really been seeing improvement in this area. I'm 23 weeks, and instead of getting formula samples, I've been getting breastfeeding books and DVDs (same doctor's office).
2)Something needs to take place in our legal drive by lawsuit world. A lot of docs need to know that if they do everything they are trained to do, they don't have to worry about being sued because something "happened" that was beyond their control.


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Pubs crisis spreads to Mitchells & Butlers as 1,300 workers are axed
Friday November 27 2020, 9.00am, The Times
M&B owns 1,650 outlets under brands including All Bar One and Harvester. The job losses represent 3 per cent of its workforce
MAUREEN MCLEAN/REX
More than a thousand pub sector job cuts were announced yesterday as Mitchells & Butlers became the latest operator to respond to the coronavirus crisis.
M&B, which owns brands including All Bar One and Harvester, said it had made 1,300 workers redundant and closed a small number of sites owing to reduced trade.
It said that the cuts had come "despite our best efforts to protect as many jobs as we can" and represented only 3 per cent of its total workforce of 43,000.
News of the cuts came as Fuller, Smith & Turner confirmed its previously announced decision to cut about 350 jobs, although it said it had managed to redeploy about 250 workers whose jobs were at risk of redundancy. Most of the


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Can Biden or Any Democrat Avoid a Brokered Convention?
by Doug Johnson Hatlem
My #10at10 2020 Democratic Primary Model is now live. It includes delegate projections down to the state and Congressional District level (State Senate District in Texas) for every state voting from the Iowa Caucus on February 3rd through to Super Tuesday, when thirteen states vote a month later. The Model uses a variety of factors in an attempt to route around the difficult data-based issues that menace the potential success of such a projection.
There were 177 unique state polls in the three week periods before relevant contests in the 2016 Democratic Primary. Eighty of them (45%) missed outside their margin of error (MoE) in projecting Vermont Senator Bernie Sanders' share of the vote. Fifty-six (32%) underestimated his share of the vote outside their MoE. Of the twenty-four polls that overestimated his share, twenty-two were in nine southern states including Texas. The scientific standard for accuracy in polling is 95% – for example "accurate to within +3.2% 19 times out of 20." Across all 177 polls, Sanders' share of the vote was underestimated by an average of 3.4%.
Three-quarters, or more [pdf], of likely Democratic voters for the 2020 primary have not settled into support for a single candidate. The contest itself will take place over a nineteen week period in which several of the candidates will drop out if they have not already in the next six months. All of this should give some idea of why models projecting delegate totals by candidate for the Primary are exceedingly rare and worthy of suspicion. At this stage, what follows is not a prediction of where things will likely wind up, but of how things would go if this model is reasonably accurate and if the Iowa caucuses were imminent. Nevertheless, the model does include defensible innovations to account for obvious data difficulties.
A big question looming over the Primary is whether any candidate can prevent a brokered convention. Even though Biden has lost ten or more points on a national lead, the former Vice-President maintains a large popular vote advantage and a delegate lead in thirteen of seventeen states that will vote through Super Tuesday. Taken together, Biden could manage, if the model is correct, to spin 44% (642 of 1470) of pledged delegates available through March 3rd out of just 28% of the national vote on average. Still, Gaffe Master Flash would have to recover serious momentum to avoid at least a second ballot at the Democratic Convention scheduled for July 2020 in Milwaukee. After his debacle last month, the debate on Wednesday may be the biggest moment in his decades as a career politician.
In Iowa, a tight race between the top-4 candidates suggests Biden, if the caucuses were tomorrow, would likely capture less than 25% of the vote but punch out a four delegate win (15-11) over Massachusetts Senator Elizabeth Warren, with Sanders (10) not far behind and California Senator Kamala Harris (5) trailing but viable statewide with 14.6% of the popular vote. In New Hampshire, Biden enjoys a sixpoint lead in the popular vote projection, but shares the delegate lead, nine a piece,with Sanders on account of the vagaries of Democratic delegate math.
Sanders is the only candidate besides Biden that the model sees winning delegates in all seventeen states and competing seriously in every one of the four major regions nationally. Sanders' strength in every region flows out of his racially diverse, working class coalition that continues to hold strong appeal for voters under forty-five years of age. The model has Sanders at 15% (the cut off by delegate contest) or greater with voters in each of the four major groups by race or ethnicity, with White, non-Hispanic voters as his weakest point (14.9%) and Hispanic voters (20.2%) as his strongest.
Elizabeth Warren would be projected currently to win no delegates in Alabama and South Carolina and just fourteen delegates total in four other Southern states with large African American populations. Warren is strong in many other parts of the country, however, including now pushing Sanders (30 delegates) and Biden (34) in her home state of Massachusetts where the model would project her to pick up 27 delegates. Warren also does well on the left coast in California (98 delegates, good enough currently for 2nd there) but also in places like Colorado (17 of 75 delegates, 3rd) and even Texas (47 of 228, 3rd).
Could Warren and Sanders combine their delegates at a contested convention to put one of them forward as the nominee? Perhaps. But one or both would have to substantially improve their current standing. With just enough delegates for the two of them together (657) to push past Biden alone (643), they would, if the contest started today, have to catch fire in the final three months of the primary or broker a much larger deal to prevent Harris, candidates with much smaller delegate totals, or the superdelegates, which return this time for the second ballot, from easily pushing Biden over the top.
Harris is projected to do reasonably well in California and also, potentially, in the South. As the Model was initially being built out, Harris was enjoying the crest of her post-first-debate surge, hitting an average of 15% nationally. She has since lost 3.2% on average nationally while Biden has recovered three points. Dipping below the magic 15% threshold has seen Harris go from a projection of 275 delegates or more through March 3, 2020 to her current state in the initial full launch at just 122 delegates, almost all of them (97) from California. Even at her best, however, Harris polled fairly poorly with Hispanic, Asian, Indigenous, and "other" survey respondents. Outside the South and California, Harris' best state in the model is Iowa, showing that a well organized, legacy media supported campaign could see her capitalize on a traditional Democratic Primary trajectory for previously not well-known candidates.
South Bend Mayor Pete Buttigieg polls fifth in terms of popular vote in the Model at 5.8% in strict average, 4.9% as weighted together with the demographic projection portion and numbers from my preliminary model, which will continue to play a small part (2-8%) going forward. Buttigieg's failure to gain any traction with people of color and his bad luck in being a politician from Indiana have him stuck on zero in the #10at10 2020 Democratic Primary Model's delegate count through Super Tuesday. Former Texas Congressman Beto O'Rourke is at 3% and 11 delegates, all from Texas. Minnesota Senator Amy Klobuchar is at less than 1.5% but 27 delegates, all from Minnesota. Former U.S. Housing and Urban Development Secretary Julián Castro appeal to Hispanic voters sees him at 10 delegates split between California and Texas.
There is a long way to go, with many assumptions to be challenged and at least moderately surprising, cart tipping events to unfold. So long as it can reasonably account for those dips and doodles, I will fully update the #10at10 2020 Model bi-weekly, with some adjustments around more sporadic polling and events ongoing between full updates.
Will the Democratic Convention be brokered or contested – the "Holy Grail" of political reportage? If the #10at10 2020 Democratic Primary Model is accurate and the contest started tomorrow, the answer would be technically yes but practically no. It is extremely unlikely absent a major change in one candidate's fortunes that any of then will reach 1885 delegates or 50% on the first ballot. But if Biden can reach 44% by delegate total through Super Tuesday and maintain or increase that share as candidates drop out after March 3rd, he would almost certainly ask Harris to be his Vice-President. Superdelegates would rubber stamp the arrangement, and the whole deal would be widely understood months ahead ofMilwaukee.
Brokered ahead of the convention and requiring a second ballot, but not actually contested.
That, however, would not have been the conclusion ten days ago with Harris at 15% nationally and Biden at 25%. A candidate with less than 30% support nationally breezing to a second-ballot win would sit horribly with large segments of the Democratic electorate. The #10at10 2020 Model is built around the likelihood that three or more candidates will regularly poll over 15% nationally. In that case, Biden is unlikely to maintain a 10%+ lead over the long haul.
Doug Johnson Hatlem writes on polling, elections data, and politics. For questions, comments, or to inquire about syndicating this weekly column for the 2020 cycle in your outlet, he can be contacted on Twitter @djjohnso (DMs open) or at djjohnso@yahoo.com (subject line #10at10 Election Column).


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Suffolk County Meets Inner Mongolia at UFC Boston
Fightland Blog
By Sascha Matuszak
Photo by Mitch Viquez/Zuffa LLC via Getty Images
It only matters when you have skin in the game, said the gambler at the bar, throwing back another shot of Patron and reaching for a chicken wing. Partisan fans all around him, green and yellow, silver and blue, but the gambler wore black and white. He went deep into the numbers with me, the hedge and side bets, the money he stood to win, and the data and hunches behind it all. He tried to explain the web of information and bets that ensured he would either win big or small, but not lose.
For me and most of the known world, the game on TV was an insignificant showdown, something nobody cared about, except for the hometown fans, and the man with skin the game. But after listening to the gambler talk, I started watching the game differently. Suddenly, just by way of a transfer of information, I had skin in the game.
I thought of that gambler as I spoke to Chris Wade on the telephone about the wrestling scene out in Long Island. Chris was born and raised in Islip in Suffolk County: blue collar and rough, brimming with hard nosed talent just now making a run at the UFC. Chris won titles as a wrestler in high school, moved on to kickboxing, and now he's fighting Zhang Lipeng on Sunday in Boston.
Photo by Christopher Cherry
Did you know that Chris Weidman is also from a tiny town on Long Island, called Baldwin, and that there is a lot more where he came from? I didn't. Ryan Laflare, Al Iaquinta, Eddie Gordon, Gian Villante. Chris Wade. All Long Island guys, from Suffolk and Nassau counties, living near each other, training with each other fighting and grinding.
We talked about his only pro MMA loss, to a Dagestani named Ozzy Dugulubgov. Ozzy tried to finish him with every swing, and for Chris it was eye-opening, not just because he let himself get dragged into the other guy's world, but that there was a world to get dragged into. A world filled with hulking, bearded mountain men with wide chins of stone, great wrestling, and, seemingly, aggression in their DNA. It reminded him of home.
"There is this something about the people here on the island, the way of life, the grind," he told me. "You can't just have an easy life; if you don't get ahead, don't out grind and outwork people, you're going to have to pack up your bags and move somewhere else."
The world of hard-nosed wrestlers coming out of Suffolk and Nassau. If you don't know, now you're wondering. A little information was transferred there, as I listened to Chris talk about fighting on the island, and now suddenly I had skin in the game.
I know Chris Wade's opponent well enough. Zhang Lipeng is another Mongolian representing China's tiny MMA scene in the UFC. He's the TUF: China lightweight champ and I watched him beat up Brendan O'Reilly last year in Macao. When I saw him last, it was in Beijing. He was wearing round spectacles and talking softly about his buddies the Leech and Ning Guangyou, this upcoming fight, his first outside of China, and what Chinese fighters need to be doing right now.
"We, as Chinese fighters in the UFC, need to be reaching for bigger, grander goals," he said. "We are on an international stage now and should be thinking about being the best in the world, not just within China. I'm setting my sights on the top contenders in my weight class—great fighters like Anthony Pettis. Aiming high is the only way to get there."
Zhang feels like he represents a nation every time he fights now. For the UFC, and for many of his friends and compadres, every win for Zhang Lipeng increases the strength and scope of the China MMA scene. If the scene is to flourish, if he is to have a community to fall back on, he must win. For the past month, Zhang has been staying with Scott Sheeley and his family in Ohio. Sheeley is Cung Le's longtime coach and confidan and he and Zhang Lipeng have spent the last few weeks training together in Ohio, shooting guns, and putting together a game plan for Chris Wade. Scott will be in Zhang's corner come Sunday.
I've had skin in that game since 2011, when I saw my first fight in Hohhot, Inner Mongolia, met Zhang's friend and former coach Vaughn Anderson, and immersed myself in the world of Chinese Sanda fighters moving to MMA.
The Zhang - Wade fight is on the preliminary card. A lot of fans won't even see the struggle taking place between a wrestler from Suffolk an another wrestler from Inner Mongolian, between the communities gathered around them, both nascent and ambitious, world's apart. Those who do see it might focus on beer and wings instead, or on the aftermath of the football game that just went down.
I'm the gambler in this scenario. It's my job to transfer just enough information to get you wondering, to drag you a few steps into their worlds, to force you to have skin in the game.
Check out these related stories:
Let a Thousand Flowers Bloom: China Opens up the Fight Game
The Silk Road: Traveling through Asia's Fight Circuit
A Watershed Moment for China's Regional MMA Circuit
Written by: Sascha Matuszak
Jan 15 2015 Like · Comment · Share Tags: Zhang Lipeng, china, UFC Boston, MMA
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