text:
Salads, products with onions sold at Walmart, Kroger, H-E-B part of recall and USDA public health alert
Kelly Tyko, USA TODAY
Various products including salads sold at some Walmart, Kroger and H-E-B stores are included in a new public health alert because they contain recalled onions.
The Department of Agriculture's Food Safety and Inspection Service issued the alert Wednesday night for ready-to-eat meat and poultry products that contain "onions that have been recalled by Thomson International Inc. due to concerns that the products may be contaminated with Salmonella Newport."
On Thursday, Taylor Farms Texas, announced in a recall notice, it was "voluntarily recalling products containing onions as a result of the expanded onion recall initiated by Thomson International, Inc., which resulted in a recall by Taylor Farms' onion supplier." The products were sold at some Walmart and Kroger stores.
Also on Thursday, the USDA's Food Safety and Inspection Service added to its public health alert a product sold by Amana Meat Shop and Smokehouse in Amana, Iowa.
Thomson International of Bakersfield, California, on Saturday recalled all red, white, yellow and sweet yellow onions shipped from May 1 through the present out of concern they might be affected, according to the Food and Drug Administration.
The onion recall came after health officials announced they were investigating a salmonella outbreak tied to red onions, which has caused nearly 400 cases reported in 34 states this month.
However, while the red onions potentially contaminated with salmonella were shipped to supermarkets and restaurants in all 50 states and Canada, the products in the new health alert "were shipped to retail locations in Arkansas, Louisiana, Mississippi, Oklahoma and Texas."
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"FSIS is issuing this public health alert out of the utmost of caution to ensure that consumers are aware that these products, which bear the USDA mark of inspection, should not be consumed," the USDA said in a statement. "As more information becomes available, FSIS will update this public health alert."
The products have the establishment number "P-34733" or "34733" inside the USDA mark of inspection or printed on the container. See product labels on the USDA page.
These ready-to-eat meat and poultry items were produced by Taylor Farms on July 30 and 31. The affected products include:
7.25-ounce plastic sealed container labeled as "Sausage Breakfast Scramble Bowl" with lot code TFD212AU8 and TFD213AU8 and with a best if used by 08/06/2020 or 08/07/2020.
6.2-ounce plastic sealed container labeled as "Taylor Farms Cheddar Cheese & Chicken Salad Snack Tray" with use by date 08/06/20 or 08/07/20 and lot code TFD212AU7 and TFD213AU7.
41.35-ounce plastic bags containing "Chicken Salad" with use by date 08/04/20 or 08/05/20 and lot codes TFD212AU8 and TFD213AU8.
10-ounce plastic sealed container labeled as "Chicken Salad Deli Snack" with lot codes TFD212AU3 and TFD213AU3 with best by dates 08/06/2020 or 08/07/2020.
7.75-ounce plastic sealed container labeled as "H.E.B. Shake Rattle Bowl SOUTHWEST SALAD with CHICKEN" and a best if used by date of "Aug 10/2020 and lot code TFD213AU20."
17.25-ounce plastic sealed container labeled as "Marketside SOUTHWEST STYLE SALAD WITH CHICKEN" with a best if used by date of "08/11/20 or 08/12/2020 and lot codes TFD212AU26 or TFD213AU26."
According to the health alert, there have been "no confirmed reports of illness due to consumption of the FSIS-regulated products produced containing these onions."
With the Centers for Disease Control and Prevention, the FDA is investigating a multi-state outbreak of salmonella newport infections possibly linked to the onions. But so far, the FDA has found no specific source of contamination has been identified is also investigating other potential sources of contamination, the agency says.
Those who become ill from salmonella often develop diarrhea, fever, and abdominal cramps from six hours to six days after exposure, the CDC says. People with more severe cases may have a high fever, aches, headaches, lethargy, rashes, and blood in the urine or stool. The illness can be fatal; salmonella leads to about 420 deaths annually in the U.S., according to the CDC.
"FSIS is concerned that some product may be in consumers' refrigerators," the health alert said. "Consumers who have purchased these products are urged not to consume them. These products should be thrown away or returned to the place of purchase."
For information, consumers can call Thomson International at 661-845-1111.
Taylor Farms recall
The following products are part of a new recall announced Thursday:
Taylor Farms Macaroni Salad, 9.9-ounce container: UPC 030223015642, lot codes TFD212 and TFD213. Best if used by Aug. 5 and 6. Distributed in Louisiana and Texas Kroger stores.
Taylor Farms Rotini Pasta Salad, 10-ounce container: 030223002789, lot codes TFD212 and TFD213. Best if used by Aug. 5 and 6. Distributed in Louisiana and Texas Kroger stores.
Taylor Farms Chicken Salad Croissant Sandwich, 6.25-ounce: UPC 03223028611, lot codes TFD212 and TFD213. Best if used by Aug. 3 and 4. Distributed in Louisiana and Texas Kroger stores.
Marketside Diced Yellow Onion 3/8", 8-ounce tray: UPC 681131328739, lot code TFD212. Best if used by Aug. 11. Distributed in Walmart stores in Colorado, Iowa, Kansas, Missouri, Nebraska, North Dakota, Oklahoma and South Dakota.
Marketside Diced Mirepoix, 10-ounce tray: UPC 681131276511, lot code TFD212. Best if used by Aug. 10. Distributed in Walmart stores in Colorado, Iowa, Kansas, Missouri, Nebraska, North Dakota, Oklahoma and South Dakota.
Marketside Fajita Stir Fry, 8-ounce tray: UPC 681131093026, lot code TFD212. Best if used by Aug. 10. Distributed in Walmart stores in Colorado, Iowa, Kansas, Missouri, Nebraska, North Dakota, Oklahoma and South Dakota.
Taylor Farms Rotini Pasta Salad, 10-ounce container: UPC 030223110095, lot codes TFD212 and TFD213. Best if used by Aug. 4 and 5. Distributed in Walmart Texas stores.
According to the recall notice, customers who have purchased these products are "urged not to consume the products and should dispose of the recalled products immediately." For more information, call 855-455-0098 between 9 a.m. and 5 p.m. CST Monday through Friday.
Amana Meat Shop and Smokehouse recall
These following products were produced by Amana Meat Shop and Smokehouse on May 14, 2020, June 2, 18, and 29, 2020 and July 2, 9, 21, 2020:
1.25-lb quart jar containing "Amana Meat Shop and Smokehouse Ham Water added in Vinegar Pickle: and lot codes E2620, F0320, F3720, F5420, G0620, G1720 and G3920. The products, which bear the establishment number "EST. 2357" inside the USDA mark of inspection, were shipped to retail locations in California, Illinois, Iowa, and South Carolina, and were also available for mail orders.
Contributing: Mike Snider, USA TODAY
Follow USA TODAY reporter Kelly Tyko on Twitter: @KellyTyko
This article originally appeared on USA TODAY: Onion recall: Taylor Farms products, Walmart Marketside part of recall
Nova Scotia looking for ways to attract offshore wind industry
It may be many years before turbines jut out of the Atlantic Ocean around Nova Scotia to spin wind into energy for the electric grid, but the growth of the offshore wind industry in other parts of the world has sparked the province's interest. "In Nova Scotia we are blessed with a long coastline. We're blessed with a tremendous wind resource," said Alisdair McLean, executive director of the Offshore Energy Research Association, which is using $50,000 from the provincial government to commission a report on offshore wind potential. The association wants to find out how Nova Scotia might use government policy to attract the offshore wind industry to build wind farms off the province's coasts, where winds are typically stronger than over land. "The purpose of our work is to try to understand whether it could be viable in Nova Scotia or not," McLean said in a recent interview. "Although it is viable in other jurisdictions, clearly, every jurisdiction has its own peculiarities. And so our role here is to evaluate whether or not it could be useful and productive and affordable in Nova Scotia." Nova Scotia already has hundreds of turbines at dozens of wind farms across the province. They generate about 20 per cent of the electricity on Nova Scotia's grid, and make up the single largest renewable energy source, but they're all onshore. Offshore wind energy farms were established in parts of Europe in the 1990s and the industry is still growing. The European Union released an offshore renewable energy strategy last fall that includes a goal of increasing offshore wind capacity five-fold by 2030, and five-fold again by 2050. The offshore wind industry started developing in the U.S. more recently, particularly along the Eastern Seaboard. According to the American Office of Energy Efficiency and Renewable Energy, the country's first offshore wind farm started operating off Rhode Island in 2016. Targets difficult to meet All the while, Nova Scotia has renewable energy targets that are proving difficult to meet. In 2020, renewable energy sources made up about 30 per cent of electricity on the grid — 10 per cent shy of what the province had hoped to achieve by last year. After the already-troubled Muskrat Falls hydroelectric project faced further delays in 2020, Nova Scotia's 40 per cent renewable energy target has been pushed back to 2022. McLean said "there's no way" any offshore wind projects would develop quickly enough to help achieve that goal, but offshore could fit in with the province's longer-term goals, like reducing greenhouse gas emissions to net-zero by 2050. Dan Roscoe said he expects to see more onshore wind farms built in Nova Scotia before any offshore development begins. Roscoe, an engineer and lead of renewable energy at Cape Breton University's Verschuren Centre for Sustainability in Energy and the Environment, said jurisdictions where offshore wind has taken off typically have space constraints on land, which is not currently a problem in Nova Scotia. "I think what it comes down to is what are we going to use that offshore wind energy for? And how can we get it to a spot where it's going to be cost-competitive?" Roscoe said. According to Nova Scotia Power's latest resource plan, released last fall, the cost of generating offshore wind in Nova Scotia in 2021 would be about double the cost of onshore wind — $113 per megawatt hour versus $56 per megawatt hour. The utility has forecast the cost of generating energy from both sources will drop, and the gap between them will tighten. By 2045, NSP's plan predicts onshore wind will cost $44 per megawatt hour and offshore will cost $59 per megawatt hour. Roscoe said wind energy costs are coming down in part because of the economies of scale. Wind turbines are getting bigger, with some recent offshore projects boasting capacities as high as one gigawatt — "which would be more than half the demand and roughly half of the needs of Nova Scotia, all in one project." In order to make projects of that size viable in Nova Scotia, Roscoe said some of the power generated would likely have to be sold for export. Researchers pinpoint best locations for wind energy Two researchers at Dalhousie University's Renewable Energy Storage Lab just published some findings that they think might help with the cost of wind energy. Mechanical engineering professor Lukas Swan and research engineer Nathaniel Pearre mapped out the best places around the Maritime provinces for wind farms, with an eye to mitigating one of the resource's major drawbacks: variability. Because wind speed and strength are constantly changing, the amount of energy generated through wind turbines is also constantly changing. At peak, there can be congestion on the grid. At base, wind power needs to be backed up by other energy sources. Swan and Pearre's mapping study shows potential wind farm locations around the Maritimes that could naturally balance what's already on the grid. "To put it into a bullet point … if you build out more resources towards the periphery of the area, the better," Pearre said in an interview. In Nova Scotia, the offshore area around Sable Island scored well, according to the researchers, as did parts of the Bay of Fundy, on the New Brunswick side. Onshore, they found potential in parts of Cape Breton and northwestern New Brunswick. Swan said he thinks offshore wind's potential deserves more exploration in Nova Scotia, but he isn't counting on the industry exploding any time soon. "Everything is harder than it looks and probably takes a little bit longer than you'd think." MORE TOP STORIES


Ever dream about producing your own clothing line? Then you need to know about the Tech Pack, a vital tool used in today's apparel manufacturing industry to provide the critical blueprint necessary to mass produce a garment according to modern standards and technology. Learn the fundamental elements necessary to complete a tech pack for an apparel manufacturing company from a digital platform. This course is ideal for anyone interested in a job with a clothing company in the areas of fashion design or product development, for designers who need to update their knowledge and skills to re-enter the fashion industry, and for small-scale designers starting a clothing line that requires mass production. Skills in sewing and patternmaking are helpful, but not required. Bring to class a 4GB or larger USB flash drive.


Get your heart racing at Queensland's number 1 wagering destination and pay homage to one of Australia's greatest horses - Apache Cat.
Apache Cat Bar epitomises the true-blue spirit of its sunburnt homeland with a wide selection of beverages showcasing Australian brews in a laid-back atmosphere.
Apache Cat Bar features a range of local and imported alcoholic and non alcoholic beverages. Please ask our friendly staff for a current menu of offerings.
*Last race will be the closest race time to the advertised closing time on the day. Please check with TAB Host for last race details.


The three earned a seal of approval to join Silla's cool Hwarang last week, and ended the day on a high note. What would be in store for them this week?
Don't you worry about a thing, since Kriesha's tours defy ordinary! Today's Gyeongju itinerary is a fantastic collaboration between activities and history!
The first stop is an amusement park, a world of fantasy filled with hopes and dreams. The three run into a happy problem of what to pick from all the rides they want to try! What rides would they be on?
There is a special way to look around historic relics these days?!
Simply looking around it is a thing of the past!
Taking selfies at the relics is the thing these days.
What selfies do these kids of the day take at the relic?
Curious of proof shots by insiders?
Join us on Travel Agency at 10:30 am on Thursday, December 28th!


I've recently moved to smaller premise and i'm instantly noticing the plasma fumes, much more noticable that mig fumes, even though I have a water table on my cnc table.
I'm wondering where is the best place to extract from, the width of hood and fan size. I can either add a hood at the cutter or make a larger hood to cover the whole 4x 4 table and extract from a single apex point. Currently have a lot of 6" 150mm hose should that be a good diameter.
Any ideas or links welcome.
That would give a fan and ducting size of 630mm diameter.
The 0.75 m/s comes from the HSE recommendation for venting of plasma tables.
I went for something smaller as I did not want to cut that size hole in the wall, so I went for 500mm diameter and uprated the fan to move more pressure. It's louder but I can vary the speed as required.
I vented from under the table, to prevent the finish face from discolouring etc.
How does your table work?
Industrial tables usually have water baths which significantly reduce emissions to the workspace.
I currently use a water table, which is a 100% necessity with plasma. TBH I use the plasma infrequently but when its on it runs on and off all day. i'll probably have to use a standard 300mm dia hose because thats the max exhaust size I can use to the outside. Will extend the hood and fan power accordingly.


text:
14-year-old basketball player impaled by broken floor
By: Ashton Edwards , Tribune Media Wire
Posted at 8:44 AM, Feb 16, 2016
MIDDLETON, Wis. — A 14-year-old basketball player is recovering after a freak accident during a game. The girl fell and was partially impaled by a splintered floorboard.
"I mean, there's no rhyme or reason to this one," tournament organizer Perry Hibner said.
The gym was filled with confusion after the girl went flying into the floor and couldn't get up.
"And the official said, 'What do you mean? Are you hurt?' And she's like, 'There's a piece of wood stuck to me," Hibner said.
The board had split in the middle and stuck into the girl from Wisconsin Heights, Wis. Her mom, the coach, was quickly by her side.
"It was 3 to 4 inches long and in some parts, probably a quarter to a half inch deep," Hibner said.
Minutes later, emergency crews showed up and everyone was asked to leave the gym.
"We thought they were going to have to remove an entire board and take the board to the hospital," Hibner said.
That didn't have to happen, but the eighth-grader was loaded on a stretcher and spent the night at the hospital. None of her internal organs were hit by the wood.
"I don't know if we'll ever hear something like this ever happening again anywhere," Hibner said.
Hibner is especially surprised it happened in the gym, where the floor is just 15 years old. A replacement sliver of wood was glued in and garbage cans were put in place until a permanent fix is made.
"I've got a tournament here in two weeks, and I have every belief that we'll be playing on this court that as well," Hibner said.
Hibner is confident the rest of the court will hold up and said he doesn't expect any legal action against the school district.
"Kind of like a strike of lightning where you just don't think those things are going to happen. It was a complete fluke," Hibner said. "I just think sometimes strange things happen and there's no explanation for the phenomena."
Even though it was a scary situation for players and parents, Hibner said the girl who was injured is expected to be ready to play in the near future."

redpajama_set_name: RedPajamaCommonCrawl


MPP Laurie Scott claims that when the Ford government does get around to doing something about climate change "it's not going to be footed by the taxpayers." Of course, this was exactly the case for the cap-and-trade program this government just cancelled.
Under cap and trade, large polluters had a choice of either buying emission permits or reducing emissions. Many chose to do both.
Funds from permit sales were directed to greenhouse gas reduction projects, such as the Haliburton bioheat project. Ms. Scott goes on to suggest that the government "is looking at all projects" before cancellation, but again what we actually see is the government simply taking an axe to anything funded by cap and trade, including money-saving energy efficiency projects for schools and hospitals.
Clearly they are desperate to recover every penny they can in order to return it to the large polluters who are now out of pocket and not likely to agree to the government's magic-wand plan to make its financial obligations disappear. As your front-page article on Durham's climate efforts makes clear, the costs of inaction on climate change are going to dwarf the "savings" generated by cancelling cap and trade. Crying in our $1 beer is not going to help.


Raleigh International is a Sustainable Development charity working in Youth Development; access to Water, Sanitation and Hygiene (WASH); Livelihood Improvements and Natural Resources Management sectors. We work across the globe providing young people with unique opportunities for personal and professional development. Raleigh International in Nepal implements the project Inspiring Youth Through Active Citizenship since 2016.
On our programmes you will be working side by side with local communities and international volunteers in various villages of Gorkha or Makwanpur districts for 12 weeks . Please note: The role is voluntary and unpaid – but you are provided with accommodation, transport, food and medical cover throughout the programme.
Responsible to implement the WASH projects. Initiate in the technical study and resource development such as water scheme development and training manuals to support volunteer learning. Provide conceptual guidance and technical inputs in project development and capacity building of partners. Maintain the quality coordination with the private sector, local service providers and local institutions for effective service delivery in WASH.
Bachelor's degree in social sciences (economics, sociology, business management, water resources management or relevant field) or civil engineering.
Minimum 3 years of experience working in the WASH sector, Experience in planning, implementation and monitoring of WASH components in communities and schools. Skills in training, coaching and capacity building of partners and CBOs.
Applications will be accepted and reviewed on a rolling basis until the position is filled. only shortlisted candidates will be contacted.


The Sia lantern is the very essence of elegance - we absolutely love it.
Tall and slender (you can go taller with our large model!), it has a slightly aged black-metal finish, which will look great anywhere - in a hallway, on a dining table, lighting up late drinks in the garden…we suspect it's going to be on the move quite a lot!


text:
between the acts
(c) Julie Raby
Waiting For Godot (Theatre Royal Newcastle, 23rd April 2009)
Many of my recent posts have commented on how much the performances have referenced theatre. I couldn't discuss this production of Waiting for Godot without commenting on the metatheatre.
The set is a derelict theatre. We watch the action through two frames, one is the proscenium arch of the Theatre Royal and the other is the run down, tumbling brickwork of the a theatre no longer in use. It could have been bombed during the second world war or it had fallen into disrepair and out of fashion. There is a wonderful moment when Estragon (Mckellan) leaves his boots on stage at the end of the first half and they are present in front of the safety curtain for the whole interval. The tree on stage is barren and resembles the gallows, or the cross, and in the second half it sprouts leaves. Does this signal there is hope after all?
The production was a very funny production, though Beckett is funny of course. Stewart and McKellan work with the lines and there is clearly a rapport between the two characters. There is also a sadness as well. Estragon forgets things and he is starting to rely on Vladimir to remind him.
Waiting for Godot is about the relationship between two men, whose life is about waiting around for something to happen. There's slapstick and elements of music hall. There's also that sense of people getting old. The constant repetition of the word 'nothing' and the sight of two old men helping each other in the wilderness, really highlighted the echoes of Shakespeare's King Lear in Beckett's play. At the same time it is like Laurel and Hardy or Morecombe and Wise. There's humour is sadness and we don't know whether to laugh or cry.
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Julie RabyLeave a comment
When We Are Married (WYP 21st April 2009)
I think I was surprised. I had read other reviews and I didn't think that I would enjoy this production, but I had a really good time. it just shows though how diverse the reviews can be. The Telegraph gives the production 5 stars, but Charles Hutchinson in the Yorkshire Evening Press found flaws and The Stage reviewer wasn't that impressed either. I tended to side with Charles Spencer on this production. For me, the strength was that the production was clearly set in a period of time. When I went to see the play, I had in mind Boeing Boeing, which just doesn't hold true for 2009. WYP's When We Were Married didn't pretend it was relevant to 2009 and so it became a very entertaining evening. The female characters seemed to be much more developed than those in Boeing Boeing and had comments and views on the situation they found themselves in. Even though the status quo was maintained at the end of the play, it felt that the characters has been through a transforming experience. So even though there was a traditional ending, I went away feeling things might be slightly different for the couples, even if it was only slightly different.
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The Tempest – again (Sheffield Lyceum, 23rd April 2009)
Having seen this Baxter Theatre/RSC production in Stratford (Courtyard Theatre), I was really keen to see it in a different playing space. The proscenium arch theatre in Sheffield did present a very different viewing experience, but none of exceitement I felt when I saw the production in February was lost.
In a proscenium arch theatre, the audience are separated from the performance. It is like watching a framed picture and as you are aware of rows of the backs of other members of the audience in front of you there isn't that sense of being really close to the action, but feeling at a distance from the actors, dancers and musicians. In Sheffield the house lights are down, whereas in Stratford they were up so there was much more sense of being aware that the actors could see you and were responding to you.
At Sheffield, I was much more aware of the changing colours. The strong green and blue backgrounds contrasted with the very dark backgrounds when Caliban was on stage. I hadn't really seen this in Stratford. In Stratford, I watched the performance from several different angles. In Sheffield all the action was in front of me. The action is much more contained on a proscenium arch stage. In Stratford the sea serpent and the mariners use the whole stage, which means they are surrounded by the audience. The Sycorax puppet enters from different parts of the theatre making it very magical.
I felt the production worked well in the different space, but it does highlight how the Courtyard stage draws the audience in and how close you are to the actors and action.
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Doctor Who, Planet of the Dead (BBC 1, 11th April 2009)
I am a great fan of Doctor Who and look forward to each episode. Unfortunately, the special 'Planet of the Dead' was not, in my view, one of the best episodes of the TV programme.
With any Doctor Who episode, viewers are confronted with the layers of narratives surrounding the production. It is as if the viewer feels that they are being let into the 'secrets', but it is more like a publicity campaign constantly keeping Doctor Who in the public eye. For this episode we were told all the details of the red bus. For example, we were supplied with images of Michelle Ryan and David Tennant on the bus and we were privy to the news that the bus had been damaged in transit to Dubai. David Tennant produced a blog and we saw more of the red bus in his blog.
All this was really interesting and certainly kept my attention, but the none of the pre programme stuff really gave any in depth information on the programme itself. They were all, of course – teasers.
I thought the opening of the programme was really strong. It set a tone and aesthetic which I felt if the programme had sustained would have made an excellent episode. The kind of James Bond/Jewell thief cliche could have been developed, but apart from Lady Christina de Souza carting round a bag full of tools and being able to get the diamond and clamps, this aspect wasn't really developed.
Ok, so the episode moved on and there was the London bus on a rainy evening and maybe things would develop well from this. The idea that the Doctor catches a bus is wonderful, there are so many ways this could be developed and then the bus goes through a wormhole and everything is very disjointed from then on.
The title suggested for me something like Sixth Sense and having a character who had a kind of sixth sense gave the impression that we would really feel surounded by ghosts and not, as it turned out, on a planet which had been invaded by a swarm of insects, or stingrays – I wasn't sure what.
The bus in the middle of the desert at times looked like a toy bus on the beach. As I knew this was Dubai it felt too much like a London bus in Dubai.
Captain Erisa Magambo is a great character, but UNIT and no Martha or mention of Martha baffled me slightly and I didn't get the Professor at all.
This is Doctor Who and there were good moments, such as the Doctor's teeth , but not the bus flying off at the end – why?.
This episode did not explore a concept in the detail and really take an idea forward as other episodes such as 'Gridlock' (traffic jams), 'The Sontaron Strategem' (SatNav) or 'Rise of Cyberman' do. As a viewer I could see the possibilities as if the production team had set some really exciting possiblities up, but had then tried to create an episode of a patchwork of unrelated ideas. Normally this would be great, but on this occasion it didn't work for me.
I am now looking forward to 'The Waters of Mars', but maybe staying clear of the pre show publicity and hoping for a classic episode.
Julie Raby
I am a lecturer in HE. My research interest is Contemporary Shakespeare Performance.
More about my work on julieraby.com
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Recent Posts: Julie Raby
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Case Study: Using Screencast-O-Matic in teaching


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      $scope.experiencePairs = createExperiencePairs();
    }
  };

  $scope.nextExperiencePair = function(experiencePair, winner) {
    experiencePair.winner = winner;

    if(winner == 0) {
      experiencePair.experience1.timesWon++;
      experiencePair.experience2.timesLost++;
    } else {
      experiencePair.experience2.timesWon++;
      experiencePair.experience1.timesLost++;
    }

    var experience1 = createPlayerFromExperience(experiencePair.experience1);
    var experience2 = createPlayerFromExperience(experiencePair.experience2);

    $scope.batch.push([experience1, experience2, winner]);

    $scope.experiencePairCounter++;
    if($scope.experiencePairCounter == $scope.experiencePairs.length) {
      glickoService.updateRatings($scope.batch);

      updateExperiences();
    }
  };

  function createExperiencePairs() {
    var experiencePairs = [];

    for(var i = 0; i < $scope.eligibleExperiences.length; i++) {
      for(var j = i + 1; j < $scope.eligibleExperiences.length; j++) {
        experiencePairs.push(new ExperiencePair($scope.eligibleExperiences[i], $scope.eligibleExperiences[j]));
      }
    }

    return experiencePairs;
  };

  function createPlayerFromExperience(experience) {
    if(!$scope.experiencePlayerSet[experience.$id]) {
      $scope.experiencePlayerSet[experience.$id] = glickoService.createRatingExperience(experience);
    }

    return $scope.experiencePlayerSet[experience.$id];
  };

  function updateExperiences() {
    angular.forEach($scope.experiencePlayerSet, function(value, key) {
      var experience = $scope.experiences.$getRecord(key);
      experience.rating = value.getRating();
      experience.rd = value.getRd();
      experience.vol = value.getVol();
      $scope.experiences.$save(experience);
    });
  };
});

redpajama_set_name: RedPajamaGithub


, we selects the very best selections with best quality just for you all, and now this pictures is usually one among images collections in our greatest images gallery in relation to New Fall Wedding Invitations and Rsvp Cards. I'm hoping you might enjoy it.
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text:
Q: Android how to video record, upload, transcode, download, play I'm researching the development of an Android (2.2) app/service that will enable users to record short (I do emphasize short, < 30seconds) video on their phones and then upload that video (HTTP) to a server that will then transcode the video to other formats.  That same user can download videos from other Android users and play them.
Now, I get a bit lost with everyones recommended approaches to all the issues in doing something like this because I haven't seen any ask this in a cohesive context.  Ideally I would like a non commercial solution to this (as in no vendor/service being needed for the the video hosting/transcoding), but, feel free to include those as a recommendation (I've marked this as a wiki) as I know many like to use youtube and vimeo for the middle layer in all this.
The questions are

*What server technologies do you
   recommend for hosting and
   transcoding?

*What technology do you
   recommend for streaming the video (it
   would be nice to offer a high and
   low quality encoding depending on
   the users network connection)

*What video format and software do you recommend for converting the uploaded video on the server to be viewable later by other Android owners.

*Im assuming it's bad to do any transcoding on the phone prior to upload (battery/proc issues), but, if I'm wrong with that assumption what do you recommend?
Some things that may help you...

*The video will only need to render on an Android device, and in the future in a webkit html5 browser.

*Bandwidth isnt cheap (even with numerous 30 second videos), so a good mix of video quality and video file size is important (streaming if needed to ensure quality vs. download).

*This is for android 2.2 devices with a video camera of course and medium to high density screen of 800x400 min.

*Open source solutions (server to receive the uploads, code to do the transcoding, server to do the streaming) are preferred, but not required.

*CDN's are an option, but I don't think that really figures in to the picture right now.

A: Check out this page to see all the video formats that Android supports for encoding and decoding.
http://developer.android.com/guide/appendix/media-formats.html
For encoding use FFmpeg or a service like encoding.com

redpajama_set_name: RedPajamaStackExchange


You'll see in the title I haven't defined what better is exactly, only that this will be a journey to better. Who knows where the game of golf will take me and I'll be using this blog to reflect every couple days or so.
So today I had another lesson and we worked a ton on alignment, and making sure that my upper body doesn't lean too far away from the target. This causes my centered turn to end up being around my hips and spine, but keeps my chest behind the ball which causes chunks, no matter how far over I slide my hips. Instead, now my hips slide and as a result my chest hangs back but is slightly ahead of the ball giving me clean contact. Not to mention, controlling my starting line is getting much easier.


text:
Q: Access to XMLHttpRequestfrom origin has been blocked by CORS policy: No 'Access-Control-Allow-Origin' header is present on the requested resource I am calling an API (asp.net web api) from Angular 9 but sometimes I am getting Cross origin error. The debugger is hitting the API method and it returns the values as expected but in the browser I am getting 500 status with Access-Control-Allow-Origin header issue. Also this error does displays always only sometimes in particular for some particular responses. so can anyone help me here?
Here are some sample codes.
[Route("api/product/Selection")]
[HttpPut()]
public IHttpActionResult Selection(VOW.Api.ProjectManager.Models.IndustrialSilencer product, string currency, string markup, string userTier2Markup, string projectId)
        {
           //some code
            ISM.SelectOutput(ref output, product, currency, Convert.ToDouble(markup), Convert.ToDouble(userTier2Markup), projectInfo.ExchangeRate);
            return Ok(output);
        }

      const S = this.http.put<Silencer>(this.baseurl + "/Selection", product, opts).subscribe(r => {

        S.unsubscribe();

        observer.next(r);
        observer.complete();

      },
        e => {
          S.unsubscribe();
          observer.error(e); // because its not any http call
          observer.complete();
        },
        () => {
          S.unsubscribe();
        }
      );

public class RouteConfig
    {
        public static void RegisterRoutes(RouteCollection routes)
        {
            routes.IgnoreRoute("{resource}.axd/{*pathInfo}");

            routes.MapRoute(
                name: "Default",
                url: "{controller}/{action}/{id}",
                defaults: new { controller = "Home", action = "Index", id = UrlParameter.Optional }
            );
        }
    }

A: CORS stands for Cross-Origin Resource Sharing. The problem is not in your (front-end) side but the "problem" is in the back-end side.
They are blocking the adress from which you want to access the API. That's it. When they "unblock" or give you some key which you usually add at the end of an URL, the problem will be gone.
Hope this helps.

A: I used to have this intermitent error and mine resolved by adding on the proxy.conf.json (on the root folder) the following data:
{
  "/api/*": {
    "target": "http://localhost:53648",
    "secure": false,
    "logLevel": "debug",
    "changeOrigin": true
  }
}

I added on target the url for the api and changeOrigin to true, also when I serve or build the app I have to do it like this.
ng serve webApp --proxy-config proxy.conf.json

But like on the comment before this one. Most of the CORS issues are backend related. this helped me because the backend was already working correctly.


This vintage style print has been created with much love.
The Blueberry Wreath design is hand drawn by the artist, Jenny Richardson, and features calligraphy style wording.
This vintage style print has been created with much love. It echoes the words we whisper to our beautiful girls every day.
The Blueberry Wreath design is hand drawn by the artist, Jenny Richardson, and features calligraphy style wording. This piece is printed in a monochrome colour palette, and is a beautiful choice of fine art for your little one's bedroom or nursery.
Jenny is a mother of three beautiful little ones and an artist based in Esperance, Western Australia. She creates fine art illustrations and hand drawn custom name prints for your darlings bedrooms and nurseries. She has a background in Agribusiness/Aquaculture, but has always loved Art and drawing. Her art and creations are based on her children's and even her own upbringing; my children are big berry eaters, our love of the ocean and all things marine. Jenny became quite obsessed with creating something special for children's bedrooms and through having her own babies. She wanted unique fine art for their walls. We spend so much time in their rooms that they understand the importance to make it a special beautiful world for them, and let's face it, for us parents also.


text:
Driver rescued from floodwaters in Maury County
By MIKE CHRISTEN mchristen@c-dh.net
Jan 12, 2020 at 11:19 AM Jan 12, 2020 at 6:00 PM
As storms continued to barrage Middle Tennessee on Saturday, Maury County's emergency responders saved a man trapped atop a pickup truck that was swept away in floodwaters.
The Maury County Fire Department water rescue team was dispatched to Rally Hill Road in rural northeastern Maury County at 4:40 p.m. on Saturday afternoon.
"Units placed multiple teams of rescuers down stream on both sides of the river and used chainsaws to access the river's edge," District Chief Richard Schatz said in a prepared statement "An inflatable raft with five swift water technicians was launched from the roadway and floated down to the truck where the subject was placed in a life jacket and placed in the raft. The raft was then pulled to shore by SHFD members. The occupant was checked by EMS."
Other than being cold from exposure to the chilly floodwaters, the subject was uninjured, Schatz said.
The volunteer department was assisted by the Maury County Office of Emergency Management, Maury Regional EMS, Maury County Sherriff's Department, Spring Hill Police Department and a representative from the Franklin Fire Department also responded.
"Special thanks to all of our partner agencies who made this an absolute success and brought this to a safe conclusion," Schatz said.
Before the rescue, the driver attempted to go through a low water crossing that is typically covered with a small amount of water. But after two days of rainfall, the small creek that runs through the rural road had transformed into a swift river.
The pickup truck was swept approximately 150 yards down stream and the subject was trapped on top of the vehicle that was partially submerged in the swift waters before he was brought to safety.
"We would like to remind the public that even a few inches of water can sweep a vehicle away," Schatz said. "Turn around, don't drown."
Although the rains have stopped, some flooding may continue and local emergency responders remind drivers to take precautions.
More than 6 inches of standing water or 4 inches of moving water can be damaging to most vehicles, according to The Tennessee Emergency Management Agency (TEMA), along with the First Tennessee Area Agency on Aging and Disability (FTAAAD).
"If you are in a flooded situation, if you don't know, don't go," Jeremy Finley, Columbia Fire and Rescue assistant chief previously told The Daily Herald.
"You don't know how deep that water is, you don't know what is in that water," Finley continued. "The biggest thing is just stay away from moving water when you don't know what is under it. Anytime you are going to be around water, have on a life vest and follow codes set by the Tennessee Wildlife Resources Agency."
Columbia Fire and Rescue recommends the following safety tips:
◆ Never drive around barricades shutting down a roadway. Even when it appears safe, flooding may have caused an unseen safety problem with the roadway.
◆ Do not drive through flooded areas, even if it looks shallow enough to cross. The large majority of deaths due to flash flooding occur with people driving through flooded areas. Water only a foot deep can displace a 1500-pound vehicle. Two feet of water can easily carry most automobiles.
◆ Roads concealed by water may not be intact.
◆ Do not cross flowing streams of water on foot where water is above your ankles.
◆ Do not allow children to play around streams, drainage ditches or viaducts, storm drains, or other flooded areas.
◆ Be especially cautious at night. It's harder to recognize water danger in low light situations.
◆ Monitor U.S. National Weather Service (NWS) or other weather reporting sources for updated information on flooding.


Losing weight can be a challenge for anyone. The greater the amount of weight we have to lose, it seems, the worse our eating habits are, and the bigger our obstacles to weight loss become. One of the biggest obstacles in weight loss can be unhealthy food choices.
Choosing the wrong foods, on a daily basis, can be what is keeping your weight exactly where it is, or even worse making the scale move up. Unfortunately, there are many unhealthy food choices available today that are dressed up as healthy foods. Here we are going to discuss some of these "healthy" unhealthy food choices and how they could be limiting your weight loss.
What is the SnackWell Effect?
To put it simply, when people buy low fat, low calorie "healthy" alternatives to their snack food and junk food they eat more of it. When someone buys a 100 calorie per serving ice cream, they are more likely to eat 2 or 3 servings than if they had just purchased the regular 150-calorie ice cream.
If you must keep snack food and junk food in the house then keep the real thing. Real chips, cookies, and ice cream will leave you more satisfied (because you feel like you indulged) and you are less likely to overeat them because you know they aren't healthy.
The most important things when eating these foods is moderation and portion control.
When we think of a healthy snack, we think of fruit. Fruit is an amazing snack that is full of fiber, natural sugars, water, and a ton of vitamins and minerals. The problem with fruit is that it doesn't stay in the car very long, and it can spoil easily. So many people simply switch to dried fruit as an alternative to the fresh thing, thinking that this will still give them all the benefits of the healthy fruit.
A half a cup of grapes contains 32 calories and about 7.5 g of sugar.
A half cup of raisins contains 246 calories and 51 g of sugar.
These same nutritional differences can be seen when looking at all dried fruits. They instantly become sugary substitutes, and you might as well have a Snickers bar (250 calories and only 27 g sugar).
The whole time you're snacking on dried fruit, it's easy to fool yourself into thinking that you are making a healthy food choice. However, when trying to reduce our body mass dried fruit should be one of the last things we reach for – grab the fresh thing instead.
Meet with a Dietician or Nutritionist and they are likely to ask you to do a dietary recall journal. The problem with these (your dietician knows this) is that we underreport what we eat.
The key to healthy eating habits comes from knowing what we are putting into our bodies. It is easy to say I had a half a cup of mashed potatoes when I have no earthly idea what a half cup of potatoes looks like.
One strategy to combat the habit of underreporting food intake is to keep a diet journal that uses weights and measurements. Using a small food scale and measuring cups you can accurately report how much food you are eating.
Whether you are dressing up your junk food, or over indulging on something, you thought would be healthy, our food choices are what can make or break our weight loss. Knowing what we eat and how much of it, is the key to successful weight loss. Start today by making smart, healthy choices and by being conscious of what you put into your body.


text:
Q: Getting rid of border-bottom in submenus The active link in my navigation has a border-bottom: 1px solid #424242 but I don't want to have border-bottom in the dropdown. As you can see when you mouseover the dropdown the links jump. I've been really trying but I can't find a solution for this. May be I have to refactor the html. I'm confused! Any help? This is my fiddle. Thanks.

A: from the looks of it to have the sub-menu items with no border, it would appear as if this would work. In your fiddle you cant see the subs but the code is there for them so:
.top-bar-section .has-dropdown li > a {
    border-bottom: none;
}

A: You need to remove .top-bar-section ul li a:hover from:
.top-bar-section ul li a:hover, .top-bar-section li.active:not(.has-form) a:not(.button), .top-bar-section li.active:not(.has-form) a:not(.button):hover {
    color: #424242;
    background-color: #f3f4f6;
    background: #f3f4f6;
    border-bottom: 2px solid #424242;
}

DEMO
EDIT:
A better solution is to remove the border only in sublists.
.top-bar-section ul ul a:hover {
    border-bottom: none;
}

And this is a possible solution to deal with jumping:
.top-bar-section > ul > li > a {
    border-bottom: 2px solid transparent;
}

DEMO2


text:
\section{Introduction}

The notion of asynchronous control system \cite{Hassibi:Boyd:How1999} often denotes models for asynchronously occurring discrete events which trigger a continuous time system, e.g., control systems in which signals are transmitted over an asynchronous network. In this paper we coin the notion of asynchronous discrete dynamical system to denote models with two inherently different discrete time scales. Little is known about about linear discrete-time systems consisting of coupled components each of which has its own potentially different time scale.  This is despite the fact that asynchronous discrete-time phenomena are observed in the real world, both in the human society and in the animal kingdom. A number of disciplines within the social and natural sciences have attempted to formally explore these phenomena.

For example, in economics, asynchronous time scales arise naturally since \cite{T} \emph{``decisions by economic agents are reconsidered daily or hourly, while others are reviewed at intervals of a year or longer''}. These intervals are driven by various costs and benefits that individuals, companies and governments face if they want to reconsider their decisions \cite{K}. Therefore there exist doubts about the prevalence of a synchronous approach and it has been argued \cite{LM} that \emph{``\ldots the synchronized move is not an unreasonable model of repetition in certain settings, but it is not clear why it should necessarily be the benchmark setting.''}

Similarly, in biology, \cite{Murray2} examined, e.g., wolf activities occurring over yearly, seasonal and daily time scales, which significantly affects the modeling of wolf-deer interactions. Interestingly, periodically varying insect populations with distinct life cycle periods coexist in many regions around the world. Their periods range from the most commonly observed one year, through many species with periods of two or three years to periods of 13 and 17 years of periodic cicadas of genus \emph{Magicada}, see the survey paper \cite{HVS}. For other examples of coupled systems and models with different time scales in social and natural sciences, see, e.g., \cite{LM, LS1, LS2, SBCS}.

From the mathematical point of view, there has been a considerable interest in the role of timing structures both from the numerical as well as analytical point of view. A well-developed mathematical theory on dynamic equations on a single time-scale can be found, e.g., in \cite{BP, H}. Note that even the elementary notion of stability depends strongly on the underlying time scale \cite{PSW}. This effect is apparent when we explore asynchronous linear systems below, focusing on two periodic time scales with different periods in particular.

There exist sporadic continuous-time approaches in which distinct periodic physical processes (e.g., mechanical, thermal, diffusion, chemical) are considered. The temporal homogenization technique \cite{AP,YF} considers such continuous phenomena in the special case when there co-exist fast and slow oscillatory processes with significantly different periods. The large ratio of the two periods allows to split the analysis in local and global problems. Our approach not only considers discrete time instead but is more general in the sense that arbitrary (not necessarily significantly different) periods are taken into account.

The paper is organized as follows. In $\S$\ref{sec:pf} we formulate two-dimensional asynchronous dynamical systems and introduce the necessary notation. In $\S$\ref{sec:linear:system} we associate an extended four-dimensional system to the asynchronous linear one and study the solution operator of the system. In $\S$\ref{sec:timeone} we analyze an interpolated dynamical system on a finer time scale. Consequently, we provide results and examples for various special cases in $\S$\ref{sec:synchronous}-$\S$\ref{sec:commensurable}. We conclude by formulating open questions and identifying directions for further research in $\S$\ref{sec:final:remarks}.

\section{Problem formulation -- 2 equations}\label{sec:pf}
For $\rho>0$, we define the periodic (or regular) one-sided discrete time scale
\begin{equation*}
   \mathbb{T}_\rho
   \coloneqq
   \left\lbrace 0, \rho, 2\rho, \ldots \right \rbrace,
\end{equation*}
and the (forward) difference operator $\Delta_\rho \colon \mathbb{R}^{\mathbb{T}_\rho} \to \mathbb{R}^{\mathbb{T}_\rho}$ by
\begin{equation*}
   \Delta_\rho x(t)
   =
   \frac{x(t+\rho)-x(t)}{\rho}
   \qquad
   (t \in \mathbb{T}_\rho),
\end{equation*}
for $x \colon \mathbb{T}_\rho \to \mathbb{R}$.
The lag operator  $\cdot^{*\rho} \colon \mathbb{R}_{\geq 0} \to \mathbb{T}_\rho$ on $\mathbb{R}_{\geq 0} \coloneqq \{x \in \mathbb{R} \,|\, x \geq 0\}$ is defined by
\begin{equation*}
   t \mapsto t^{*\rho}
   \coloneqq
   \rho\left\lfloor \frac{t}{\rho} \right\rfloor
   =
   \max \left\lbrace s\in\mathbb{T}_\rho \colon s\leq t \right\rbrace
   \qquad
   (t \in \mathbb{R}_{\geq 0}),
\end{equation*}
and satisfies $t-t^{*\rho} \in [0,\rho)$ for $t \in \mathbb{R}_{\geq 0}$.

Let $\nu > 0$. Using the lag operator $\cdot^{*\nu}$, a function $z_{\mathbb{T}_\nu} \colon \mathbb{T}_\nu \to \mathbb{R}$ on $\mathbb{T}_\nu$ can naturally be extended to a function $z_{\mathbb{R}_{\geq 0}} \colon \mathbb{R}_{\geq 0} \to \mathbb{R}$ on $\mathbb{R}_{\geq 0}$ by defining
\begin{equation*}
   z_{\mathbb{R}_{\geq 0}}(t)
   \coloneqq
   z_{\mathbb{T}_\nu}(t^{*\nu})
   \qquad
   (t \in \mathbb{R}_{\geq 0}).
\end{equation*}
The extended function $z_{\mathbb{R}_{\geq 0}}$ realizes the principle of \emph{sample and hold} in systems theory \cite[Section 1.4]{Ogata1995} and can be used in the analysis of discrete-time equations \cite{Slavik2012}. For every $\tau \in \mathbb{T}_\nu$ the value $z_{\mathbb{T}_\nu}(\tau)$ is sampled and held constant for one period $\nu$, i.e.,
\begin{equation*}
   z_{\mathbb{R}_{\geq 0}}(t)
   =
   z_{\mathbb{T}_\nu}(\tau)
   \qquad
   (t \in [\tau, \tau+ \nu)).
\end{equation*}
Let $\mu > 0$. The restriction $z_{\mathbb{T}_\mu} \coloneqq z_{\mathbb{R}_{\geq 0}}|_{\mathbb{T}_\mu}$ of $z_{\mathbb{R}_{\geq 0}}$ to $\mathbb{T}_\mu$, satisfies
\begin{equation*}
   z_{\mathbb{T}_\mu}(t)
   =
   z_{\mathbb{T}_\nu}(t^{*\nu})
   \qquad
   (t \in \mathbb{T}_\mu),
\end{equation*}
and reads the value of $z_{\mathbb{T}_\nu}$ which was sampled at $t^{*\nu}$ and held until $t \in \mathbb{T}_\mu$.

We define an asynchronous discrete dynamical system in the following way. Consider periods $\mu, \nu > 0$ and the matrix
\begin{equation*}
  P
  =
  \begin{pmatrix}
    \alpha & \beta \\
    \gamma & \delta
  \end{pmatrix}
  \in \mathbb{R}^{2\times 2}.
\end{equation*}
Then the two coupled equations on the time scales $\TS_\mu$ and $\TS_\nu$
\begin{equation}\label{e:problem:2}
   \begin{cases}
      \Delta_\mu x(t) = \alpha x(t) + \beta y(t^{*_\nu}), & t\in\TS_\mu, \\
      \Delta_\nu y(t) = \gamma x(t^{*_\mu}) + \delta y(t), & t\in\TS_\nu,
   \end{cases}
\end{equation}
are called $(\mu,\nu)$-asynchronous difference equation (or $(\mu,\nu)$-asynchronous discrete time dynamical system) with parameters $P$. See Figure \ref{f:53} for an illustration of $\TS_\mu$ and $\TS_\nu$.
\begin{figure}
\begin{center}
\includegraphics[width=12cm]{pic_asynchronous_ADDS35.pdf}
\vspace*{-2ex}
\end{center}
\caption{Time scales $\mathbb{T}_3$ and $\mathbb{T}_5$ of a $(3,5)$-asynchronous discrete dynamical system \eqref{e:problem:2}.}\label{f:53}
\end{figure}

A tuple $(x,y)$ of sequences $x \colon \TS_\mu \rightarrow \mathbb{R}$, $y \colon \TS_\nu \rightarrow \mathbb{R}$, is called \emph{solution of} \eqref{e:problem:2}, if it satisfies \eqref{e:problem:2}.

Throughout the paper we use the notation $\sigma(t)$, $\sigma:\mathbb{T}\rightarrow\mathbb{T}$, to denote the successor of an element $t$ of a given time scale $\mathbb{T}$. This operator is commonly known as the forward-jump operator in the theory of time scales \cite{BP}. Naturally, its value depends strongly on the considered time scale, e.g., $\sigma(0)=\rho$ for $\mathbb{T}=\mathbb{T}_\rho$.

\begin{example}\label{x:VAR}
In economics, dynamics of variables naturally includes data of different frequencies.  A typical problem arises from different natural time scales. Let us consider a simplistic model of fiscal spending involving gross domestic product (GDP) and government spending. GDP data are usually observed quarterly and would naturally imply a time scale $\mathbb{T}_{3}$ with $\mu=3$ months. On the other hand, the government spending is determined by a time series with yearly frequency as government budget is approved once a year. In this case the natural time scale $\mathbb{T}_{12}$ has a frequency of $\nu=12$ months.

Denoting GDP by $x$ and government spending by $y$, a synchronized simplistic fiscal model has the form:
\begin{equation}\label{eq:synchronous:fiscal}
\begin{cases}
      \Delta x(t) = \alpha x(t) + \beta y(t), & t\in\mathbb{T}_{12}, \\
      \Delta y(t) = \gamma x(t) + \delta y(t), & t\in\mathbb{T}_{12},
   \end{cases}
\end{equation}

Alternatively, we could consider the asynchronous model \eqref{e:problem:2} and allow GDP $x$ and government spending $y$ to follow their own time scales - quarterly $\mathbb{T}_{3}$ for $x$ and annual $\mathbb{T}_{12}$ for $y$.

Apparently, the asynchronous model is closer to how economic agents observe macroeconomic data. In this paper we are mainly interested in qualitative differences between the synchronous model \eqref{eq:synchronous:fiscal} and the asynchronous model \eqref{e:problem:2}. Naturally, from the point of view of macroeconomic applications there are numerous other questions, which we do not discuss here. Are the real data time series better explained by the asynchronous model \eqref{e:problem:2}? Can the asynchronous formulation \eqref{e:problem:2} provide better forecasts of the future?

\end{example}

\section{Linear system representation}\label{sec:linear:system}

It is well-known that for difference equations a delay can be eliminated by increasing the dimension of the system (see, e.g., \cite{EZ}). We associate a 4-dimensional linear system to \eqref{e:problem:2} by storing appropriate delayed values of $x(t), y(t)$ in auxiliary variables $\underline{x}(t), \underline{y}(t)$. For this to work, we need to incorporate all times on which dynamics happens in \eqref{e:problem:2}, i.e., we consider the union
\[
  \mathbb{T} \coloneqq \TS_\mu \cup \TS_\nu
\]
of the two time scales of \eqref{e:problem:2}.

\begin{theorem}[Linear system representation]\label{thm:representation}
Let $\mu,\nu>0$ and $\mathbb{T} = \TS_\mu \cup \TS_\nu$. Then there exists a unique $A:\mathbb{T}\rightarrow \mathbb{R}^{4\times 4}$ and the corresponding four dimensional dynamic equation
\begin{equation}\label{e:linear:system}
\begin{pmatrix}
u \\
\underline{u} \\
v \\
\underline{v}
\end{pmatrix} (\sigma(t)) =
 A(t) \begin{pmatrix}
u \\
\underline{u} \\
v \\
\underline{v}
\end{pmatrix} (t), \qquad t\in\mathbb{T}, \begin{pmatrix}
u \\
\underline{u} \\
v \\
\underline{v}
\end{pmatrix} \in\mathbb{R}^4,
\end{equation}
such that for arbitrary sequences $x:\TS_\mu\rightarrow\mathbb{R}$ and $y:\TS_\nu\rightarrow\mathbb{R}$ the following statements are equivalent:
\begin{enumerate}
  \item[(i)] $(x,y)$ is a solution of \eqref{e:problem:2},

  \item[(ii)] the solution $(u,\underline{u},v,\underline{v}) \colon \mathbb{T} \rightarrow \mathbb{R}^4$  of \eqref{e:linear:system} with $u(0)=x(0)$, $\underline{u}(0)=0$, $v(0)=y(0)$, $\underline{v}(0)=0$ satisfies $u|_{\TS_\mu}= x$ and $v|_{\TS_\nu}= y$.
\end{enumerate}
\end{theorem}

\begin{proof}
Consider the following difference equation on $\mathbb{T}$:
\begin{numcases}{\label{(a)}\hspace*{-1.6cm}u(\sigma(t))=}
   u(t) \quad \text{if } \sigma(t) \notin \TS_\mu, & \label{(a1)}
   \\[1ex]
   (1 + \alpha \mu)u(t) + \beta \mu v(t) \quad \text{if } \sigma(t) \in \TS_\mu \wedge t \in \TS_\mu\cap\TS_\nu, & \label{(a2)}
   \\[1ex]
   (1 + \alpha \mu)u(t) + \beta \mu \underline{v}(t) \quad \text{if } \sigma(t) \in \TS_\mu \wedge t \notin \TS_\mu\cap\TS_\nu,& \label{(a3)}
\end{numcases}
\addtocounter{equation}{-1}
\begin{numcases}{\label{(b)}\underline{u}(\sigma(t))=}
   \setcounter{equation}{4}
   u(t) \quad \text{if } t\in\TS_\mu\cap\TS_\nu \vee (t\notin \TS_\nu \wedge \sigma(t)\in \TS_\nu),\phantom{aaaaaaaaaaaaaa} & \label{(b1)}
   \\[1ex]
   \underline{u}(t) \quad \text{otherwise,} & \label{(b2)}
\end{numcases}
\addtocounter{equation}{-1}
\begin{numcases}{\label{(c)}\hspace*{-1.6cm}v(\sigma(t))=}
   \setcounter{equation}{6}
   v(t) \quad \text{if } \sigma(t) \notin \TS_\nu, & \label{(c1)}
   \\[1ex]
   (1 + \delta \nu)v(t) + \gamma \nu u(t)\quad \text{if } \sigma(t) \in \TS_\nu \wedge t \in \TS_\mu\cap\TS_\nu, & \label{(c2)}

   \\[1ex]
   (1 + \delta \nu)v(t) + \gamma \nu \underline{u}(t) \quad \text{if } \sigma(t) \in \TS_\nu \wedge t \notin \TS_\mu \label{(c3)}
\end{numcases}
\addtocounter{equation}{-1}
\begin{numcases}{\label{(d)}\underline{v}(\sigma(t))=}
   \setcounter{equation}{9}
   v(t)  \quad \text{if } t\in\TS_\mu\cap\TS_\nu \vee (t\notin \TS_\mu \wedge \sigma(t)\in \TS_\mu),\phantom{aaaaaaaaaaaaaa} & \label{(d1)}
    \\[1ex]
   \underline{v}(t) \quad \text{otherwise.} \label{(d2)}
\end{numcases}
It is of the form \eqref{e:linear:system}. We first prove the following statement:
\begin{equation}\label{e:statement}
   (u,\underline{u},v,\underline{v}) \text{ solves }
   \eqref{(a)}
   \quad \Rightarrow \quad
   (u|_{\TS_\mu}, v|_{\TS_\nu}) \text{ solves }  \eqref{e:problem:2}.
\end{equation}
To this end, let $(u,\underline{u},v,\underline{v}) \colon \mathbb{T} \rightarrow \mathbb{R}^4$  be a solution of \eqref{(a)}. In order to prove \eqref{e:statement}, we show that
\begin{alignat}{2} \label{e:proof:4:u}
   u(t+\mu) &= (1+\alpha \mu) u(t) + \beta \mu v(t^{*_\nu}) &&\qquad (t \in \TS_\mu),
\\ \label{e:proof:4:v}
   v(t+\nu) &= \gamma \nu u(t^{*_\mu}) + (1+\delta \nu) v(t)  &&\qquad  (t \in \TS_\nu).
\end{alignat}
We only show \eqref{e:proof:4:u}, the latter relation \eqref{e:proof:4:v} is proved analogously. To show \eqref{e:proof:4:u}, assume that $t \in \mathbb{T}_\mu$, then $t + \mu \in \mathbb{T}_\mu$, too. Define
\begin{equation*}
   r \coloneqq \max \{s \in \mathbb{T} \colon s < t + \mu\}.
\end{equation*}
Obviously, $t \leq r < t + \mu$. We distinguish between the following cases.

Case 1: If $r \in \mathbb{T}_\mu \cap \mathbb{T}_\nu$, then clearly $r = t = t^{*\nu}$ and \eqref{e:proof:4:u} follows from (\ref{(a)}b).

Case 2: If $r \not\in \mathbb{T}_\mu \cap \mathbb{T}_\nu$, then we can either have  $r \in \mathbb{T}_\mu \setminus \mathbb{T}_\nu$ or  $r \in \mathbb{T}_\nu \setminus \mathbb{T}_\mu$:

Case 2.1: If $r \in \mathbb{T}_\mu \setminus \mathbb{T}_\nu$, then $r=t$ and (\ref{(d)}i)-(\ref{(d)}j) imply that
\begin{equation}\label{e:case2.1}
   \underline{v}(r) = \underline{v}(t) = v(t^{*\nu}).
\end{equation}

Case 2.2: If $r \in \mathbb{T}_\nu \setminus \mathbb{T}_\mu$, then (\ref{(d)}j) yields
\begin{equation*}
   \underline{v}(\sigma(t)) = \underline{v}(s) = \underline{v}(r),
\end{equation*}
for all $s\in\mathbb{T}_\nu$ with $\sigma(t)<s<r$. Consequently, (\ref{(d)}i) implies that
\begin{equation}\label{e:case2.2}
   \underline{v}(r) = \underline{v}(\sigma(t)) = v(t^{*\nu}).
\end{equation}
By assumptions of Case 2 and (\ref{(a)}a), $u(r) = u(t)$. Relations \eqref{e:case2.1} and \eqref{e:case2.2} yield that $\underline{v}(r) = v(t^{*\nu})$. Hence (\ref{(a)}c) implies that
\begin{align*}
   u(t + \mu)
   &=
   (1 + \alpha \mu) u(r) + \beta \mu \underline{v}(r)
\\
   &=
   (1 + \alpha \mu) u(t) + \beta \mu v(t^{*\nu}),
\end{align*}
i.e., \eqref{e:proof:4:u} holds in this case as well.

$(i)\Rightarrow (ii)$.
Let $(x,y)$ be a solution of \eqref{e:problem:2} and let $(u,\underline{u},v,\underline{v}) \colon \mathbb{T} \rightarrow \mathbb{R}^4$ be the solution of \eqref{(a)} with $u(0)=x(0)$, $\underline{u}(0)=0$, $v(0)=y(0)$, $\underline{v}(0)=0$. Then by \eqref{e:statement}, $(u|_{\TS_\mu}, v|_{\TS_\nu})$ solves \eqref{e:problem:2}. Since $u(0)=x(0)$, $v(0)=y(0)$, it follows that $u|_{\TS_\mu}= x$ and $v|_{\TS_\nu}= y$.

$(ii)\Rightarrow (i)$. Let $x:\TS_\mu\rightarrow\mathbb{R}$ and $y:\TS_\nu\rightarrow\mathbb{R}$ be given and let $(u,\underline{u},v,\underline{v}) \colon \mathbb{T} \rightarrow \mathbb{R}^4$ be the solution of \eqref{(a)} with $u(0)=x(0)$, $\underline{u}(0)=0$, $v(0)=y(0)$, $\underline{v}(0)=0$. Then by (ii), $u|_{\TS_\mu}= x$ and $v|_{\TS_\nu}= y$, and \eqref{e:statement} implies that $(x,y)$ solves \eqref{e:problem:2}.
\end{proof}

Using the indicator function $\indicator_M \colon \mathbb{R} \to \{0,1\}$ for a set $M \subseteq \mathbb{R}$,
\begin{equation*}
   \indicator_{M}(t)
   \coloneqq
   \begin{cases}
      1 & \text{if } t \in M,
   \\
      0 & \text{if } t \not\in M,
   \end{cases}
\end{equation*}
we get the following explicit representation of the coefficient matrix $A(t)$ of the linear system representation \eqref{e:linear:system}.

\begin{corollary}[Explicit form of linear system representation]\label{c:matrices}
Under the assumptions of Theorem \ref{thm:representation} the coefficients $A \colon \mathbb{T} \rightarrow \mathbb{R}^{4 \times 4}$ are given by
\begin{equation*}
   A(t)
   \coloneqq
   A_{i j k \ell}
\end{equation*}
with $(i,j,k,\ell) = \big(\indicator_{\mathbb{T}_{\mu}}(t), \indicator_{\mathbb{T}_{\mu}}(\sigma(t)), \indicator_{\mathbb{T}_{\nu}}(t), \indicator_{\mathbb{T}_{\nu}}(\sigma(t))\big)$ and $A_{i j k \ell}$ as defined by one of the $9$ cases listed in Table \ref{t:9cases}.
\end{corollary}

\begin{proof}
For each $t \in \mathbb{T} = \mathbb{T}_\mu \cup \mathbb{T}_\nu$ there are $3$ possible disjoint cases, either
\begin{equation*}
   t \in \mathbb{T}_\mu \cap \mathbb{T}_\nu
   \qquad \text{or} \qquad
   t \in \mathbb{T}_\mu \setminus \mathbb{T}_\nu
   \qquad \text{or} \qquad
   t \in \mathbb{T}_\nu \setminus \mathbb{T}_\mu.
\end{equation*}
Similarly, there are $3$ cases for $\sigma(t) \in \mathbb{T}$. Consequently, there are $9$ possible combinations for the values of the indicator functions in the quadruple
\begin{equation*}
   (i,j,k,\ell)
   \coloneqq
   \big(\indicator_{\mathbb{T}_{\mu}}(t),
   \indicator_{\mathbb{T}_{\mu}}(\sigma(t)),
   \indicator_{\mathbb{T}_{\nu}}(t),
   \indicator_{\mathbb{T}_{\nu}}(\sigma(t))\big)
\end{equation*}
which are listed and illustrated in Table \ref{t:9cases}. For each of those cases, we can use the linear system representation \eqref{(a)} to compute $A_{i j k \ell}$ as listed in Table \ref{t:9cases}.
\end{proof}

Consequently, we are able to introduce a solution operator of \eqref{e:problem:2}.

\begin{corollary}[Solution operator for asynchronous discrete time dynamical system]\label{c:representation:matrix}
Let $\mathbb{T}=\mathbb{T}_\mu \cup \mathbb{T}_\nu$, $(\mathbb{T})^2_{\geq} \coloneqq \{(t,t_0) \in \mathbb{T} \times \mathbb{T} \colon t \geq t_0\}$ and let $\Phi \colon (\mathbb{T})^2_{\geq} \rightarrow \mathbb{R}^{4 \times 4}$, $(t,t_0) \mapsto \Phi(t,t_0)$, denote the evolution operator (2-parameter process) of \eqref{(a)}, i.e., for $t_0 \in \mathbb{T}$
\[
  (t,t_0) \mapsto \Phi(t,t_0) (x_0, \underline{x}_0, y_0, \underline{y}_0)^{\top}
\]
is the solution of the initial value problem \eqref{(a)}, $(x(t_0), \underline{x}(t_0), y(t_0), \underline{y}(t_0)) = (x_0, \underline{x}_0, y_0, \underline{y}_0)$. Define the \emph{solution operator for \eqref{e:problem:2}} as $\Psi \colon (\mathbb{T})^2_{\geq} \rightarrow \mathbb{R}^{2 \times 2}$, $(t,t_0) \mapsto \Psi(t,t_0)$,
\begin{equation*}
  \Psi(t,t_0)
  \coloneqq
  \begin{pmatrix}
     \Phi_{11}(t,t_0) & \Phi_{13}(t,t_0)
  \\
     \Phi_{31}(t,t_0) & \Phi_{33}(t,t_0)
  \end{pmatrix}.
\end{equation*}
Then for $(x_0,y_0) \in \mathbb{R}^2$ and $x, y \colon \mathbb{T} \rightarrow \mathbb{R}$ with
\begin{equation*}
   \begin{pmatrix}
     x(t)
   \\
     y(t)
  \end{pmatrix}
  \coloneqq
   \Psi(t,0)
   \begin{pmatrix}
     x_0
   \\
     y_0
   \end{pmatrix},
\end{equation*}
the tuple $(x_{| \mathbb{T}_\mu}, y_{| \mathbb{T}_\nu})$ of restrictions is a solution of \eqref{e:problem:2}.
\end{corollary}

\begin{proof}
Let $t\in\mathbb{T}$. The set $\mathbb{T} \cap [0,t]$ contains finitely many elements $0 = t_0 < t_1 < \dots < t_k = t$. Theorem~\ref{thm:representation} implies that with the matrices $A(t_i)$ from Corollary \ref{c:representation:matrix} and Table \ref{t:9cases},
\begin{align*}
	(x(t),\underline{x}(t),y(t),\underline{y}(t))^\top
	&\coloneqq \Phi(t,0) (x_0,0,y_0,0)^\top \\
	&= \Phi(t_k,t_{k-1}) \cdots \Phi(t_2,t_1) \Phi(t_1,t_0) (x_0,0,y_0,0)^\top \\
	 &= A(t_{k-1}) \cdots A(t_1)A(t_0) (x_0,0,y_0,0)^\top
\end{align*}
is a solution of \eqref{(a)}. Writing $\Phi = (\Phi_{i j})_{i,j=1, \dots, 4}$, we get for $t \in \mathbb{T}$
\[
\Phi(t,0) =
	\begin{pmatrix}
	\Phi_{11} & \Phi_{12} & \Phi_{13} & \Phi_{14} \\
	\Phi_{21} & \Phi_{22} & \Phi_{23} & \Phi_{24} \\
	\Phi_{31} & \Phi_{32} & \Phi_{33} & \Phi_{34} \\
	\Phi_{41} & \Phi_{42} & \Phi_{43} & \Phi_{44} \\
	\end{pmatrix}(t,0)
\]
and hence
\begin{align*}
\begin{pmatrix}
x(t)\\
y(t)
\end{pmatrix}
&=
\begin{pmatrix}
\pi_1 \circ \Phi(t,0)\cdot (x_0,0,y_0,0)^\top \\
\pi_3 \circ \Phi(t,0)\cdot (x_0,0,y_0,0)^\top
\end{pmatrix}
\\
&=
\begin{pmatrix}
\Phi_{11} & \Phi_{13} \\
\Phi_{31} & \Phi_{33}
\end{pmatrix} (t,0)\cdot (x_0,y_0)^\top \\
&= \Psi(t,0) \cdot (x_0,y_0)^\top.
\end{align*}
By Theorem \ref{thm:representation}, the tuple $(x_{| \mathbb{T}_\mu}, y_{| \mathbb{T}_\nu})$ solves \eqref{e:problem:2}.
\end{proof}

\begin{remark}
Note that $\Psi(t,t_0)$ is not a $2$-parameter process on $\mathbb{T}$, since for arbitrary $t, \tau, s \in \mathbb{T}$,
\begin{equation}\label{e:semigroup}
   \Psi(t, \tau) \Psi(\tau, s) = \Psi(t, s)
\end{equation}
does not necessarily hold. However, \eqref{e:semigroup} holds for all $t, \tau, s \in \mathbb{T}_\mu \cap \mathbb{T}_\nu$. Moreover, if $\mu$ and $\nu$ are commensurable, i.e., $\mathbb{T}_\mu \cap \mathbb{T}_\nu = \mathbb{T}_T$ with $T = \operatorname{lcm}(\mu, \nu)$, then for $k \in \mathbb{Z}$
\begin{equation*}
   \Psi(T, 0) = \Psi((k + 1) T, k T).
\end{equation*}
Consequently, $ \Psi(kT,0) = \prod_{i=1}^k \Psi(T,0)$ for $k \in \mathbb{N}$ in this case.
\end{remark}

We conclude this section with a simple illustration of Corollary \ref{c:representation:matrix}.

\begin{example}\label{x:12}
Let us consider asynchronous dynamics with $\mu=1$ and $\nu=2$, i.e.,
\begin{align*}
\TS_\mu &= \{0,1,2, \ldots \}, \\
\TS_\nu &= \{0,2,4, \ldots \}, \\
\mathbb{T} &= \TS_\mu\cup\TS_\nu = \{0,1,2, \ldots \}.
\end{align*}
Since $0,2\in\TS_\mu\cap\TS_\nu$ and $1\in\TS_\mu\setminus\TS_\nu$ we get that $\Phi(2,0)=A_{1101}\cdot A_{1110}$ where (see Table \ref{t:9cases}):
\[
A_{1110} = \begin{pmatrix}
 1+\alpha & 0 & \beta  & 0 \\
 1 & 0 & 0 & 0 \\
 0 & 0 & 1 & 0 \\
 0 & 0 & 1 & 0 \\
\end{pmatrix},
\quad
A_{1101} = \begin{pmatrix}
 1+\alpha  & 0 & 0 & \beta  \\
 1 & 0 & 0 & 0 \\
 0 & 2 \gamma  & 1+2 \delta & 0 \\
 0 & 0 & 0 & 1 \\
\end{pmatrix}.
\quad
\]
Consequently,
\[
\Phi(2,0)=
\begin{pmatrix}
 (1+\alpha)^2 & 0 & (1+\alpha) \beta +\beta  & 0 \\
 1+\alpha & 0 & \beta  & 0 \\
 2 \gamma  & 0 & 1+2 \delta & 0 \\
 0 & 0 & 1 & 0
\end{pmatrix},
\quad
\Psi(2,0)=\begin{pmatrix}
 (1+\alpha)^2 & (1+\alpha) \beta +\beta   \\
 2 \gamma  &  1+2 \delta
\end{pmatrix}.
\]
\end{example}

\section{Interpolated dynamics}\label{sec:timeone}
Recall that for $p \in \mathbb{N}$ a nonsingular matrix $M \in \mathbb{R}^{n \times n}$ has a complex $p$-th root $\sqrt[p]{M} \in \mathbb{C}^{n \times n}$ \cite[Section 7.1, pp.\ 173-174]{Hi}. If $\mu,\nu>0$ are commensurable, i.e., $\mathbb{T}_\mu \cap \mathbb{T}_\nu = \mathbb{T}_T$ with $T > 0$, there exist $k, \ell \in \mathbb{N}$ with $k \mu = \ell \nu = T$. We define the time scale $\mathbb{T}_\tau$ with
\begin{equation*}
   \tau
   \coloneqq
   \tfrac{T}{k \ell}
   =
   \tfrac{\mu}{\ell}
   =
   \tfrac{\nu}{k}.
\end{equation*}
$\mathbb{T}_\tau$ refines or interpolates $\mathbb{T}_\mu$ as well as $\mathbb{T}_\nu$, since $\mathbb{T}_\mu \subseteq \mathbb{T}_\tau$ and $\mathbb{T}_\nu \subseteq \mathbb{T}_\tau$. We can construct an associated complex difference equation
\begin{equation}\label{e:problem:uv}
   \begin{pmatrix}
      u(t+\tau) \\ v(t+\tau)
   \end{pmatrix}
   =
   B
   \begin{pmatrix}
      u(t) \\ v(t)
   \end{pmatrix}
   \qquad
   (t \in \mathbb{T}_\tau),
\end{equation}
which interpolates \eqref{e:problem:2}, by defining
\[
   B
   \coloneqq \sqrt[k \ell]{\Psi(T,0)} \in \mathbb{C}^{2 \times 2},
\]
and
\begin{equation}\label{e:timeone}
   \Psi(t,s)
   \coloneqq
   B^{\frac{t-s}{\tau}}
   \qquad
   \text{for } t, s, \in \mathbb{T}_\tau, t \geq s.
\end{equation}
By \eqref{e:timeone}, $B = \Psi(\tau,0)$ and $\Psi(T,0) = B^{k \ell}$.

For every solution $(x,y)$ of \eqref{e:problem:2} and solution $(u,v)$ of \eqref{e:problem:uv} with $x(0) = u(0)$, $y(0) = v(0)$, we have
\[
   \begin{pmatrix}
      x(t) \\ y(t)
   \end{pmatrix}
   =
   \begin{pmatrix}
      u(t) \\ v(t)
   \end{pmatrix}
   \qquad \text{for all } t \in \TS_T.
\]
Note that in general $x(t) = u(t)$ does not necessarily hold for $t \in \mathbb{T}_\mu \setminus \TS_T$, also $y(t) = v(t)$ is not necessarily true for $t \in \mathbb{T}_\nu \setminus \TS_T$.

We illustrate these notions by going back to Example \ref{x:12}.
\begin{example}\label{x:12:time1map}
Let us consider the asynchronous discrete time dynamical system \eqref{e:problem:2} with $\mu=1$, $\nu=2$ and parameters $\alpha=2$, $\beta=1$, $\gamma=-1$ and $\delta=1$. Following Example \ref{x:12}, we observe that
\[
\Psi(2,0)= \begin{pmatrix}
 9 & 4   \\
 -2  &  3
 \end{pmatrix}.
\]
Since $T=\operatorname{lcm}(1, 2)=2$, we compute the matrix square root of $\Psi(2,0)$ and determine that
\[
   B \coloneqq \Psi(1,0) \coloneqq \sqrt{\Psi(2,0)} =
\begin{pmatrix}
 -\sqrt{5}+2 \sqrt{7} & -2 \sqrt{5}+2 \sqrt{7} \\
 \sqrt{5}-\sqrt{7} & 2 \sqrt{5}-\sqrt{7}
 \end{pmatrix}
 \approx
 \begin{pmatrix}
 3.05543 & 0.819367 \\
 -0.409683 & 1.82638
 \end{pmatrix}.
\]
\end{example}

\section{Synchronous case $\mu=\nu$}\label{sec:synchronous}
Let us consider the standard synchronous setting $\mu = \nu$ as a benchmark case first. We have $\mathbb{T}=\TS_\mu=\TS_\nu$ and the system \eqref{e:problem:2} can be rewritten as
\begin{equation}\label{e:problem:2:equal}
\begin{cases}
\Delta_\mu x (t) = \alpha x(t) + \beta y(t),\\
\Delta_\mu y (t) = \gamma x(t) + \delta y(t),
\end{cases}
\qquad (t\in\mathbb{T}).
\end{equation}
In order to be able to compare various asynchronous dynamics later, we use special notation for the solution operator $\Psi$ which indicates the periodicities via upper indices
\[
\Psi^{\mu,\mu}(\mu,0) :=\left(\begin{array}{cc}
1+\mu\alpha & \mu\beta \\
\mu\gamma & 1+\mu\delta
\end{array}
 \right).
\]
We can rewrite system \eqref{e:problem:2:equal} as
\[
\binom{x(t+\mu)}{y(t+\mu)}  =  \left(\begin{array}{cc}
1+\mu\alpha & \mu\beta \\
\mu\gamma & 1+\mu\delta
\end{array}
 \right)\binom{x(t)}{y(t)} = \Psi^{\mu,\mu}(\mu,0) \binom{x(t)}{y(t)}.
\]
Denoting by $\lambda(M)$ the set of all eigenvalues of a matrix $M$, we can claim the following result, an alternative of a well-known result from the theory of discrete-dynamical systems, see \cite{KP}.
\begin{theorem}\label{thm:synchronous}
Let $\lambda(\Psi^{\mu,\mu}(\mu,0))\subset B(0,1)$ or, equivalently, $\lambda(P)\subset B\big(-\frac{1}{\mu},\frac{1}{\mu} \big)$. Then $o \coloneqq (0,0) \in \mathbb{R}^2$ is a globally asymptotically stable solution of \eqref{e:problem:2:equal}, i.e., every solution $(x,y)$ of \eqref{e:problem:2:equal} satisfies
\[
\lim\limits_{t\rightarrow\infty} x(t) = 0 = \lim\limits_{t\rightarrow\infty} y(t).
\]
\end{theorem}
\section{Special case $\mu\in\mathbb{N}$ and $\nu=1$}\label{s:subset}\label{sec:multiple}
Next, we consider the case when either $\mu=k\nu$ or $\nu=k\mu$ for some $k,\mu,\nu\in\mathbb{N}$, i.e., either $\mathbb{T}_\mu\subset \mathbb{T}_\nu$ or $\mathbb{T}_\nu\subset \mathbb{T}_\mu$. Without loss of generality we only focus on the case $\mu\in\mathbb{N}$ and $\nu=1$, i.e.,
\[
\mathbb{T}_\mu = \left\lbrace 0, \mu, 2\mu, \ldots \right \rbrace = \mu \mathbb{T}_\nu ,
\]
and problem \eqref{e:problem:2} could be rewritten as (similarly as in Examples \ref{x:12} or \ref{x:dynamical:equivalence})
\begin{equation}\label{e:problem:2:mu:1}
\binom{x(t+\mu)}{y(t+\mu)}  =  \left(\begin{array}{cc}
1+\mu\alpha & \mu\beta \\
\sum_{i=0}^{\mu-1} \gamma (1+\delta)^i & (1+\delta)^\mu
\end{array}
 \right)\binom{x(t)}{y(t)} =  \Psi^{\mu,1}(\mu,0) \binom{x(t)}{y(t)},
\end{equation}

with the solution operator
\[
 \Psi^{\mu,1}(\mu,0) \coloneqq \left(\begin{array}{cc}
1+\mu\alpha & \mu\beta \\
\sum_{i=0}^{\mu-1} \gamma (1+\delta)^i & (1+\delta)^\mu
\end{array}
 \right).
\]

We have the following result regarding the stability of the origin in this case:

\begin{theorem}\label{thm:mu:1}
Let $\lambda( \Psi^{\mu,1}(\mu,0))\subseteq B(0,1)$. Then $o$ is a globally asymptotically stable solution of the $(\mu,1)$-discrete dynamical system \eqref{e:problem:2}, i.e., every solution $(x,y)$ of \eqref{e:problem:2:mu:1} satisfies
\[
\lim\limits_{t\rightarrow\infty} x(t) = 0 = \lim\limits_{t\rightarrow\infty} y(t).
\]
\end{theorem}

\begin{proof}
We divide the proof into two parts. First, we show that
\begin{equation}\label{e:Amo:Tx}
\lim\limits_{\substack{t\rightarrow\infty \\ t\in \mathbb{T}_\mu}} \binom{x(t)}{y(t)} = \binom{0}{0}.
\end{equation}
and next we show that this is also true for $t\in \mathbb{T}_\nu$.

The assumption $\lambda(\Psi^{\mu,1}(\mu,0))\subseteq B(0,1)$ implies that the spectral radius of $\Psi^{\mu,1}(\mu,0)$ satisfies
\[
\rho(\Psi^{\mu,1}(\mu,0)) = \max\{|\lambda_1|,|\lambda_2|\} < 1.
\]
Consequently, a standard argument (e.g., \cite[Theorem 4.4]{KP}) implies that for $\kappa$  with $\rho(\Psi^{\mu,1}(\mu,0))< \kappa <1$ there exists $K>0$ so that for each $t\in \{0,\mu, 2\mu, 3\mu, \ldots\}$ we have
\begin{equation}\label{e:Amo:Tx:estimate}
\left\lVert \binom{x(t)}{y(t)} \right\rVert = \left\lVert \binom{x(n\mu)}{y(n\mu)} \right\rVert \leq K \kappa^n \left\lVert \binom{x(0)}{y(0)} \right\rVert, \qquad (n\in\mathbb{N}).
\end{equation}
This implies that \eqref{e:Amo:Tx} holds.

Assume now that $t\notin \mathbb{T}_\mu$, i.e., $t=n\mu + m$, with $n\in\mathbb{N}$ and $m\in \{1,2,\ldots, \mu-1 \}$. Apparently, $x(n\mu)=x(n\mu+1)=x(n\mu+2)=\cdots=x(n\mu+\mu-1)$ and we have
\begin{align*}
y(n\mu+1) &= \gamma x(n\mu) + (1+\sigma) y(n\mu) \\
y(n\mu+2) &= \gamma x(n\mu+1) + (1+\sigma) y(n\mu+1) \\
	&= (\gamma + \gamma(1+\sigma)) x(n\mu) + (1+\sigma)^2 y(n\mu) \\
& \hspace*{1.2ex} \vdots \\
y(n\mu+m) &= \left(\sum_{i=0}^{m-1} \gamma (1+\delta)^i \right) x(n\mu) + (1+\sigma)^m y(n\mu).
\end{align*}
Consequently,  we can write
\[
\binom{x(t)}{y(t)} = \binom{x(n\mu+m)}{y(n\mu+m)}=
\left(\begin{array}{cc}
1 & 0 \\
\sum_{i=0}^{m-1} \gamma (1+\delta)^i & (1+\delta)^m
\end{array}
 \right)\binom{x(n\mu)}{y(n\mu)}.
\]

Then we have\footnote{For $M\in\mathbb{R}^{N\times N}$ we use the spectral matrix norm $\lVert M \rVert $
\[
\lVert M \rVert = \lVert M \rVert_{\mathop{spec}} := \sup \{\lVert Mx \rVert_2: \|x\|_2=1 \},
\]
which is equal to the largest singular value of the matrix $M$, see L\"{u}tkepohl \cite[Chapter 8]{L}.
}
\begin{align*}
\left\lVert \binom{x(t)}{y(t)} \right\rVert = \left\lVert \binom{x(n\mu+m)}{y(n\mu+m)} \right\rVert &\leq  \left\lVert \left(\begin{array}{cc}
1 & 0 \\
\sum_{i=0}^{m-1} \gamma (1+\delta)^i & (1+\delta)^m
\end{array}
 \right) \right\rVert \cdot
 \left\lVert \binom{x(n\mu)}{y(n\mu)} \right\rVert \\
 & \leq L \left\lVert \binom{x(n\mu)}{y(n\mu)} \right\rVert,
\end{align*}
where $L$ is a constant defined by
\[
L = \max\limits_{m=1,2,\ldots, \mu-1} \left\lVert \left(\begin{array}{cc}
1 & 0 \\
\sum_{i=0}^{m-1} \gamma (1+\delta)^i & (1+\delta)^m
\end{array}
 \right) \right\rVert.
\]

Consequently, the estimate \eqref{e:Amo:Tx:estimate} implies that
\[
\left\lVert \binom{x(t)}{y(t)} \right\rVert = \left\lVert \binom{x(n\mu+m)}{y(n\mu+m)} \right\rVert \leq
L K \kappa^n \left\lVert \binom{x(0)}{y(0)} \right\rVert,
\]
which finishes the proof.
\end{proof}

The following counterexamples show that neither the asymptotic stability of $(\mu,\mu)$-dynamics \eqref{e:problem:2:equal} implies the asymptotic stability of $(\mu,1)$-dynamics \eqref{e:problem:2:mu:1} with the same parameters $\alpha, \beta, \gamma, \delta$, nor vice versa.

\begin{example}
If
\[
P= \left(
\begin{array}{cc}
 -\frac{1}{16} & \frac{1}{8} \\
 -\frac{1}{8} & -\frac{1}{16} \\
\end{array}
\right)
\]
and $\mu=7$, then the origin in the $(7,7)$-dynamics \eqref{e:problem:2:equal} is not asymptotically stable, since
\[
\Psi^{7,7}(7,0)= \left(
\begin{array}{cc}
 \frac{9}{16} & \frac{7}{8} \\
 -\frac{7}{8} & \frac{9}{16} \\
\end{array}
\right),
\]
and both eigenvalues satisfy $| \lambda_{12}| \approx 1.04 >  1$. However, the origin in the $(7,1)$-dynamics \eqref{e:problem:2:mu:1} is asymptotically stable, since the eigenvalues of
\[
\Psi^{7,1}(7,0) = \left(
\begin{array}{cc}
 \frac{9}{16} & \frac{7}{8} \\
 -\frac{97\, 576\, 081}{134\, 217\, 728} & \frac{170\,859\,375}{268\,435\,456} \\
\end{array}
\right)
\]
satisfy $| \lambda_{12}| \approx 0.997 < 1$.
\end{example}

\begin{example}
If
\[
P= \left(
\begin{array}{cc}
 -\frac{1}{11} & \frac{1}{10} \\
 -\frac{2}{15} & \frac{1}{15} \\
\end{array}
\right)
\]
and $\mu=3$, then the origin in the $(3,3)$-dynamics \eqref{e:problem:2:equal} is asymptotically stable, since
\[
\Psi^{3,3}(3,0)= \left(
\begin{array}{cc}
 \frac{8}{11} & \frac{3}{10} \\
 -\frac{2}{5} & \frac{6}{5} \\
\end{array}
\right),
\]
and both eigenvalues satisfy $| \lambda_{12}| \approx 0.996 <1$. However, the origin in the $(3,1)$-dynamics \eqref{e:problem:2:mu:1} is not asymptotically stable, since the eigenvalues of
\[
\Psi^{3,1}(3,0)= \left(
\begin{array}{cc}
 \frac{8}{11} & \frac{3}{10} \\
 -\frac{1\, 442}{3\, 375} & \frac{4\, 096}{3\,375} \\
\end{array}
\right),
\]
satisfy $| \lambda_{12}| \approx 1.006 > 1$.
\end{example}

In the same spirit, neither the asymptotic stability of $(1,1)$-dynamics \eqref{e:problem:2:equal} implies the asymptotic stability of $(\mu,1)$-dynamics \eqref{e:problem:2:mu:1} with the same parameters $\alpha, \beta, \gamma, \delta$, nor vice versa.
\begin{example}
If
\[
P= \left(\begin{array}{cc}
 -8 & -1 \\
 \frac{1089}{40} & 4 \\
\end{array}
\right)
\]
 then the origin in the $(2,1)$-dynamics \eqref{e:problem:2:mu:1} is asymptotically stable, since the eigenvalues of
\[
\Psi^{2,1}(2,0)=\left(\begin{array}{cc}
 -17 & -2 \\
 \frac{1089}{8} & 16 \\
\end{array}
\right)
\]
are both equal to $-\frac{1}{2}$. At the same time, the origin is unstable in  the $(1,1)$-dynamics \eqref{e:problem:2:mu:1}, since one of the eigenvalues of
\[
\Psi^{1,1}(1,0)= P =\left(\begin{array}{cc}
 -8 & -1 \\
 \frac{1089}{40} & 4 \\
\end{array}
\right)
\]
is $\frac{1}{20} \left(-40-3 \sqrt{390}\right) \approx -4.96$.
\end{example}

\begin{example}
Finally, we can trivially observe that for
\[
P=\left(
\begin{array}{cc}
 -\frac{7}{4} & 0 \\
 0 & -\frac{7}{4} \\
\end{array}
\right)
\]
the origin of the $(\mu,1)$-dynamics, $\mu\in\mathbb{N}$, is stable if and only if $\mu=1$, since
\[
\Psi^{\mu,1}(\mu,0) = \left(
\begin{array}{cc}
 1-\frac{7 \mu}{4} & 0 \\
 0 & \left(-\frac{3}{4}\right)^\mu \\
\end{array}
\right)
\]
and $1-\frac{7 \mu}{4} < -1$ for $\mu=2,3,\ldots$
\end{example}

\section{Commensurability case $\mu,\nu\in\mathbb{N}$}\label{sec:commensurable}
In this section we consider a general situation in which $\mu,\nu\in\mathbb{N}$ are not multiples of each other, i.e., $T=\mathrm{lcm}\{\mu,\nu\}>\max\{\mu,\nu\}$. We consider the following time scales
\begin{align*}
\mathbb{T}_\mu &= \left\lbrace 0, \mu, 2\mu, \ldots \right \rbrace, \\
\mathbb{T}_\nu &= \left\lbrace 0, \nu, 2\nu, \ldots \right \rbrace, \\
\mathbb{T} &= \mathbb{T}_\mu \cup \mathbb{T}_\nu, \\
\mathbb{T}_T &= \left\lbrace 0, T, 2T, \ldots \right \rbrace = \mathbb{T}_\mu \cap \mathbb{T}_\nu.
\end{align*}

For given parameters $P=\left(\begin{array}{cc}
\alpha & \beta \\
\gamma & \delta
\end{array}
 \right)$ we can use the matrices $A_{ijkl}$ from Table \ref{t:9cases} to construct the evolution operator (see Corollaries \ref{c:matrices} and \ref{c:representation:matrix}):
\[
\Phi^{\mu,\nu} (T,0) = \prod_{m\in\mathbb{T}\cap[0,T)} A_{i_mj_mk_ml_m},
\]
and the solution operator (matrix) $\Psi^{\mu,\nu} (T,0)$ defined by
\[
\Psi^{\mu,\nu} (T,0)\coloneqq
  \begin{pmatrix}
     \Phi^{\mu,\nu}_{11}(T,0) & \Phi^{\mu,\nu}_{13}(T,0)
  \\
     \Phi^{\mu,\nu}_{31}(T,0) & \Phi^{\mu,\nu}_{33}(T,0)
  \end{pmatrix}.
\]

We have the following result.

\begin{theorem}\label{thm:mu:nu}
Let $\mu, \nu\in \mathbb{N}$ and $\lambda( \Psi^{\mu,\nu} (T,0))\subseteq B(0,1)$. Then $o$ is a globally asymptotically stable solution of \eqref{e:problem:2}, i.e., every solution of \eqref{e:problem:2} satisfies
\[
\lim\limits_{t\rightarrow\infty} x(t) = 0 = \lim\limits_{t\rightarrow\infty} y(t).
\]
\end{theorem}
\begin{proof}
The proof follows the ideas of the proof of Theorem \ref{thm:mu:1}.

First, we show that
\begin{equation}\label{e:Amn:TT}
\lim\limits_{\substack{t\rightarrow\infty \\ t\in \mathbb{T}_T}} \binom{x(t)}{y(t)} = \binom{0}{0}.
\end{equation}
First, we choose $\kappa$ such that $\rho(\Psi^{\mu,\nu}(T,0))< \kappa <1$. Then, there exists $K>0$ so that for each $t\in \mathbb{T}_T\cap\mathbb{N}$ we have
\begin{equation}\label{e:Amn:Tx:estimate}
\left\lVert \binom{x(t)}{y(t)} \right\rVert = \left\lVert \binom{x(nT)}{y(nT)} \right\rVert \leq K \kappa^n \left\lVert \binom{x(0)}{y(0)} \right\rVert, \quad n\in\mathbb{N},
\end{equation}
which implies \eqref{e:Amn:TT}.

Next, we focus on the values of $x(t)$ and $y(t)$ on the intervals $(mT,(m+1)T)\cap\mathbb{T}$ for some $m\in\mathbb{N}$. Since the evolution operator $\Phi^{\mu,\nu}$ is defined as a product of matrices $A_{ijkl}$ from Table \ref{t:9cases} and the solution operator $\Phi^{\mu,\nu}$ as the submatrix of $\Psi^{\mu,\nu}$, we observe that with the constant
\[
L = \max\limits_{t\in\mathbb{T}\cap(0,T)} \| \Psi^{\mu,\nu}(t,0) \|,
\]
we have for all $t\in\mathbb{T}\cap(0,T)$ and $n\in\mathbb{N}$,
\[
\left\lVert \binom{x(nT+t)}{y(nT+t)} \right\rVert \leq L \left\lVert \binom{x(nT)}{y(nT)} \right\rVert.
\]
Employing the estimate \eqref{e:Amn:Tx:estimate} we get
\[
\left\lVert \binom{x(nT+t)}{y(nT+t)} \right\rVert \leq
L K \kappa^n \left\lVert \binom{x(0)}{y(0)} \right\rVert,
\]
and the proof is complete.
\end{proof}

Before we illustrate Theorem \ref{thm:mu:nu} we introduce the notion of dynamically equivalent asynchronous dynamical systems.

\begin{definition}
Let $\mu,\nu,\hat{\mu},\hat{\nu}$ be commensurable, i.e., there exists $T=\mathrm{lcm}\lbrace \mu,\nu,\hat{\mu},\hat{\nu} \rbrace $. We say that a $(\mu,\nu)$-asynchronous discrete dynamical system with parameters $P$ and a $(\hat{\mu},\hat{\nu})$-discrete dynamical system with parameters $\hat{P}$ are \emph{dynamically equivalent} if the solution operators satisfy
\[
\Psi^{\mu,\nu}(T,0) = \hat{\Psi}^{\hat{\mu},\hat{\nu}}(T,0).
\]
\end{definition}

\begin{example}\label{x:dynamical:equivalence}
The following two asynchronous discrete dynamical systems are dynamically equivalent because in both cases they lead to the dynamics with solution operator
\[
\Psi^{\mu,\nu}(6,0)=\left(
\begin{array}{cc}
 -\frac{7}{10} & \frac{1}{10} \\
 -\frac{9}{16} & \frac{29}{80}
\end{array}
\right).
\]
Since $\lambda(\Psi^{\mu,\nu}(6,0))=\left\{\frac{1}{160} \left(-27-\sqrt{5785}\right),\frac{1}{160} \left(\sqrt{5785}-27\right)\right\}\approx\{-0.644,0.306\}$, the origin is asymptotically stable in both cases.
\begin{itemize}
\item[(a)] \textbf{$(2,3)$-asynchronous discrete dynamics}. If we consider parameters
\[
P=\begin{pmatrix}
    \alpha & \beta \\
    \gamma & \delta
  \end{pmatrix} =  \left(
\begin{array}{cc}
 -1 & \frac{1}{5} \\
 \frac{1}{4} & -\frac{1}{4} \\
\end{array}
\right),
\]
\begin{figure}
\begin{center}
\includegraphics[width=10cm]{pic_asynchronous_ADDS23.pdf}
\includegraphics[width=10cm]{pic_asynchronous_ADDS61.pdf}
\vspace*{-2ex}
\end{center}
\caption{Time scales related to dynamically equivalent (2,3)- and (6,1)-asynchronous discrete dynamical systems from Example \ref{x:dynamical:equivalence}.}\label{f:dynamical:equivalence:ts}
\end{figure}

we have in this case (see Figure \ref{f:dynamical:equivalence:ts} and Table \ref{t:9cases}) the evolution operator
\begin{align*}
\Phi^{2,3}(6,0)&=A_{1101}A_{0110}A_{1001}A_{1110}\\ &=\left(
\begin{array}{cccc}
 (2 \alpha +1)^3+6 \beta  \gamma  & 0 & (2 \alpha +1) (2 (2 \alpha +1) \beta +2 \beta )+2 \beta  (3 \delta +1) & 0 \\
 (2 \alpha +1)^2 & 0 & 2 (2 \alpha +1) \beta +2 \beta  & 0 \\
 3 (2 \alpha +1) \gamma +3 (3 \delta +1) \gamma  & 0 & (3 \delta +1)^2+6 \beta  \gamma  & 0 \\
 3 \gamma  & 0 & 3 \delta +1 & 0 \\
\end{array}
\right).
\end{align*}
Consequently, the solution operator is
\begin{align*}
\Psi^{2,3}(6,0)&=\left(
\begin{array}{cc}
 (2 \alpha +1)^3+6 \beta  \gamma  & (2 \alpha +1) (2 (2 \alpha +1) \beta +2 \beta )+2 \beta  (3 \delta +1) \\
 3 (2 \alpha +1) \gamma +3 (3 \delta +1) \gamma  & (3 \delta +1)^2+6 \beta  \gamma  \\
\end{array} \right) \\
&=\left(
\begin{array}{cc}
 -\frac{7}{10} & \frac{1}{10} \\
 -\frac{9}{16} & \frac{29}{80} \\
\end{array}
\right).
\end{align*}
\item[(b)] \textbf{$(6,1)$-asynchronous discrete dynamics}. If we consider parameters
\[
\displaystyle \hat{P}=\begin{pmatrix}
    \hat{\alpha} & \hat{\beta} \\
    \hat{\gamma} & \hat{\delta }
  \end{pmatrix} =  \begin{pmatrix}
 -\frac{17}{60} & \frac{1}{60} \\[1ex]
 -\frac{\frac{-9}{16}}{\sum_{i=0}^5  (\sqrt[6]{\frac{29}{60}})^i} & -1+\sqrt[6]{\frac{29}{60}} \\
\end{pmatrix}\approx \begin{pmatrix}
 -.283 & .017 \\
 -.137 & -.156 \\
\end{pmatrix},
\]
then we have (see Figure \ref{f:dynamical:equivalence:ts} and Table \ref{t:9cases})
\begin{align*}
\hat{\Phi}^{6,1}(6,0)&=\hat{A}_{0111}(\hat{A}_{0011})^4 \hat{A}_{1011}
=\begin{pmatrix}
 1+6 \hat{\alpha} & 0 & 6 \hat{\beta}  & 0 \\
 1 & 0 & 0 & 0 \\
 \hat{\gamma}\sum_{i=0}^5  (1+\hat{\delta})^i & 0 & (1+\hat{\delta} )^6 & 0 \\
  \hat{\gamma}\sum_{i=0}^4  (1+\hat{\delta})^i & 0 & (1+\hat{\delta} )^5 & 0 \\
\end{pmatrix}.
\end{align*}
Consequently, the solution operator is
\begin{align*}
\hat{\Psi}^{6,1}(6,0)&=
\begin{pmatrix}
 1+6 \hat{\alpha} &  6 \hat{\beta} \\
 \hat{\gamma}\sum_{i=0}^5  (1+\hat{\delta})^i & (1+\hat{\delta} )^6  \\
\end{pmatrix} =
\begin{pmatrix}
 -\frac{7}{10} & \frac{1}{10} \\
 -\frac{9}{16} & \frac{29}{80} \\
\end{pmatrix}.
\end{align*}
\end{itemize}
Observant readers may have noted that the parameters $\hat{P}$ can be derived backwards so that the solution matrices $\Psi^{2,3}(6,0)$ and $\hat{\Psi}^{6,1}(6,0)$ have the same form. Similarly, one could find parameters sets $\bar{P}$ for, e.g., dynamically equivalent $(1,1)$-, $(2,1)$-, $(3,2)$-, $(1,3)$-asynchronous discrete dynamical systems.
\end{example}

\begin{remark}
Note that the notion of dynamical equivalence of asynchronous discrete dynamical systems could have been alternatively introduced via the induced time-$1$ dynamics. $(\mu,\nu)$-asynchronous discrete dynamical system with parameters $P$ and a $(\hat{\mu},\hat{\nu})$-discrete dynamical system with parameters $\hat{P}$ are \emph{dynamically equivalent} if the induced time-1 operators defined by \eqref{e:timeone} are equal, i.e.,
\[
\Psi^{\mu,\nu}(1,0)=\hat{\Psi}^{\hat{\mu},\hat{\nu}}(1,0).
\]

Note that in Example \ref{x:dynamical:equivalence} both asynchronous discrete dynamical systems are associated with the complex time-1 solution operator
\[
\Psi(1,0)=\sqrt[6]{\Psi^{\mu,\nu}(6,0)}\approx \left(
\begin{array}{cc}
 0.804\, +0.492 i & 0.002\, -0.049 i \\
 -0.01+0.275 i & 0.822\, -0.027 i \\
\end{array}
\right).
\]
\end{remark}

\section{Final remarks}\label{sec:final:remarks}
Our ideas can in principle be extended to dynamical systems with more equations, e.g., 3 asynchronous discrete equations. Naturally, such a process could be computationally demanding.

However, there are two essential questions which remain open even in the case of two asynchronous equations \eqref{e:problem:2}.

First, note that the most general case we have studied was the case of commensurable $\mu,\nu$, i.e., the situation in which there exists $T=\mathrm{lcm}\{\mu,\nu\}.$ However, the cornerstone of our approach, the construction of a solution operator $\Psi(T,0)$ on the intersection time scale $\mathbb{T}_T=\mathbb{T}_\mu\cap \mathbb{T}_\nu$ cannot be used in the situation when $\mu, \nu\in\mathbb{R}^+$ are incommensurable (for example $\mu=1$ and $\nu=\pi$, etc.). In this case, there is no least common multiple $T$. The open question is how to study such $(\mu,\nu)$-asynchronous discrete dynamical systems. Under which condition is the origin of a $(\mu,\nu)$-asynchronous discrete dynamical system with incommensurable $\mu,\nu$ asymptotically stable?

The second question is motivated by counterexamples in Section \ref{s:subset} where we showed that for a given set of parameters $P$ the asymptotic stability of origin in $(\mu,1)$-asynchronous discrete dynamical systems, $\mu\in\mathbb{N}$ neither implies nor is implied by the asymptotic stability of the origin of $(\mu,\mu)$- or $(1,1)$-synchronous discrete dynamical system. Under which assumptions does the asymptotic stability of the origin in $(\mu,\nu)$-dynamics imply the asymptotic stability of the origin in $(\hat{\mu},\hat{\nu})$-dynamics?

From the point of view of applications, there are also natural questions. We can illustrate one of the key ones by our little Example \ref{x:VAR}. In the case of macroeconomic time series, can we show that in some specific instances, a variant of our asynchronous model \eqref{e:problem:2} explains the real asynchronous time series better than standard synchronous fiscal models \eqref{eq:synchronous:fiscal}? Naturally, asynchronous systems would create a realm of questions in econometrics related to the estimation of parameters, etc.

\subsubsection*{Acknowledgements}
The authors are grateful to Michal Franta, Jan Libich and Eduard Rohan for their insights from econometrics, economics and computational mechanics. PS acknowledges the support of the project LO1506 of the Czech Ministry of Education, Youth and Sports under the program NPU I.

{

redpajama_set_name: RedPajamaArXiv