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How to manage remote teams
Tips (by DiSC style) for managers with remote teams
Tips for moving your DiSC® trainings online
Posted on March 27, 2020 in Everything Disc, Management
As more and more cities and states are asking businesses to switch to remote work to help slow the spread of COVID-19, countless managers are faced with not only recalibrating their own work reality, but guiding their employees through this great change as well.
"Trust and accountability is key," says Dana Montgomery, National Deputy Director for the Know the Signs program at Sandy Hook Promise. "You have to be someone people believe in and feel they can come to, and you have to model the behavior you want to see." Montgomery manages three region directors across the country who also manage teams of three to six people.
We talked to her about managing remote teams in general, and more specifically what she recommends to managers suddenly faced with this reality during the coronavirus pandemic. Here are some of her tips:
Acknowledge that this is challenging.
During this pandemic, many employees working from home are—at the same time—taking care of children whose schools have closed, supporting suddenly-unemployed partners, worrying about their aging relatives, and so much more. The shift to remote work is one of many huge changes for most people, and we need to acknowledge that.
Montgomery told us about conference calls they had last week and how her staff—99% of whom already work remotely—are facing many of the issues above. She really feels for folks trying to adjust to a new way of working on top of that. Montgomery says, "Employees knowing the organization empathizes and understands the complexity of their new reality is hugely important to instilling confidence and easing fears in the longer term."
Assume good intentions.
"This is so important," says Montgomery, "that it's one of our team expectations." She says so much gets lost when you aren't talking face-to face. "I'm not saying people shouldn't be held accountable for less than stellar behavior," she says, "but it's so easy to read into an email, or assign a tone that isn't really there. Dig deeper, ask people to explain what they meant by x, y, or z. This is especially important now when so many are feeling overwhelmed and out of sorts."
Limit the number of platforms you're using.
Montgomery says, "I would suggest picking one or two virtual communication tools to conduct day-to-day business, set expectations around how they should be utilized, and train people on how to use them effectively. Don't just share a new platform and expect everyone to start utilizing it."
When considering what tools to use, think about your team and what will work for them. For many, home Wi-Fi bandwidth is a challenge. If stipends are not available for employees to build up their in-house capacity and you want to do video conferencing, make sure you choose a platform that people can also join by phone.
Check in more often.
Things are changing rapidly in our world right now, and that's likely true for your workplace as well. At least once a week, have a staff check-in. Give yourself plenty of time to prepare for this meeting, so you can make sure you have collected all of the information your employees need to know and have figured out the best way to communicate it clearly and concisely. Be sure to leave a lot of time for staff to ask questions.
Create clear expectations.
If you lay down the team ground rules, Montgomery says, "people will become more confident and able to deliver in this new environment. It will also better allow the manager to understand who is on task and who is struggling."
Start by communicating your expectations around responding to email and keeping calendars updated. In a remote setting, these things are critical to a well-functioning team.
Montgomery recommends being specific about email response times. For example, "respond within 12 hours if it's urgent, or 24-48 hours if not." (Your response time expectations may be different, especially if you supervise part-time employees.) "I can't pop by my colleague's office to see if she received my email," Montgomery says, "and I can't ask her while we're both heating up lunch." If you don't hear back about an email you sent for several days, you don't know whether a response is coming, or if it slipped through the cracks.
Everyone should keep their calendars updated, including when they will be out of reach or on calls. Ask your team to decline or accept every meeting invitation they receive. Having to go through lots of back and forth to find a meeting time because people haven't clearly marked their availability on their calendars is a waste of everyone's time.
Encourage people to block off time for a lunch break. Also, some employees perform well when responding to emails as they come in, but others may prefer blocking off a time each day, say early mornings or late afternoons, where they are just focused on email and thus unavailable for meetings or other interruptions. "Allowing people space to claim their time is important," says Montgomery.
Model the behavior you want to see.
If you set email and calendar rules for your staff, be sure to follow those rules yourself. When you are communicating expectations to your team, "it's critical to make clear that this is not about being 'big brother,'" says Montgomery. It's about encouraging work-life balance and supporting your colleagues by not contacting them when they are off the clock. "I don't want to call one of my direct reports when they are with their kid at a doctor's appointment. I also don't want one of my team members to try to set up time with me because my calendar looks free, but I'm actually out of the office."
Speaking of work-life balance, managers should give some thought to the image they are projecting to their employees. If you tell your team they should be clocking out by 6 p.m. but they get emails from you sent at all hours of the night, they may wonder if you actually want them working around the clock as well. It's a reality of our work world that people often work more than 40 hours a week, and your job may require that of you and/or your team. Just be thoughtful about modeling sustainable work behavior. Even if you are writing emails until 11 p.m., could you save them as drafts and send them to your team first thing in the morning?
Do not deal with team conflict via email.
"Beware the pitfalls of relying too much on email in the absence of face-to-face," says Montgomery. She advises never to deal with conflict over email. This is also true of more complex discussions. "If you are on email exchange number two or three with someone and details are getting lost, or if something is just too complicated to explain in writing, pick up the phone or schedule a meeting," she says.
Think creatively about how to replace "water cooler" talk.
In most offices, there's quite a bit of casual conversation going on throughout the day as people pass through the space or congregate around the water cooler or coffee machine. In the absence of these informal interactions that help you know how your teammates are doing, you may have to build in some dedicated chat time. "Take the first five to ten minutes of a check-in to see how things are going," suggests Montgomery. You may not have needed to do that when you were all in an office together, but it may be beneficial now.
Montgomery also suggests setting up casual chats, like virtual coffee breaks or happy hours, and encouraging your team to do that with one another. The "water cooler" type chats among Montgomery's team often happen over text. "Some of this is about organizational culture," she says, "but I text with my team a ton. I rarely did this in my prior office job."
Don't expect business as usual.
For managers whose teams have moved to remote work with little time to prepare for the change, it's important to extend kindness and grace to both yourself and your team. This is a new situation for all of us, and you will likely be setting yourself up for failure if you are aiming for "business as usual."
This is an opportunity to build a culture of trust and compassion among your team. Remember: when managing remote teams, assume good intentions and acknowledge the challenges.
"Think creatively about how to keep team camaraderie," Montgomery advises. "Remember that a lot of people are working in ways they never would have chosen on their own. When all of this is over, you will want a team who felt supported by their employer."
Use Everything DiSC® Management Profile to discover your DiSC management style. You'll also learn how to adapt your style to manage, motivate, direct and develop your staff based on their own DiSC styles.
Building trust on virtual or remote teams, discprofiles.com blog
5 Behavioral Biases That Trip Up Remote Managers, Harvard Business Review
What It Takes to Run a Great Hybrid Meeting, Harvard Business Review
by Jessica Franken
Tags:management
Don't waste your one-on-one meetings
Employee retention starts with managers
8 common management issues
Management: What can you teach new managers?


Answering fundamental questions about musical preference, ability, and communication, the field of Musical Cognition and Development is critical to the understanding of how music is processed, grasped, and learned. This book covers the latest theoretical and practical techniques that explain meaning and understanding in music. Chapters offer cogent and concise insights giving up-to-date information and references. The book covers the most important topics in this field, including skill development in music performance, research on communicating music expressiveness, the neurobiology of music, the cognitive constraints in the listening process, and music and medicine as applied to neuroscience.


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DESIGN PHILOSOPHY & DESIGN STRATEGIES
INNOVATION / TECHNOLOGIES
PANELS & CERTIFICATIONS
Deniliquin Police Station
CCG Architects has recently completed the design and documentation for a new Regional Local Area Command Police Station at Deniliquin, NSW. Engaged as Principal Lead Consultant for the provision of Architectural, Structural Engineering and Services Engineering for the design, documentation and attendance during construction. Construction commenced in May 2016.
The site contains a number of buildings, including the former Police Barracks building, which was designed in the office of the Colonial Architect James Barnet and built in 1886. The heritage listed police barracks make a significant street scape contribution to the township due to its prominent corner location.
The Barracks will be retained and form an identity both culturally and aesthetically to the site. Setback from the Barracks the new police building will provide up to date facilities for the Deniliquin precinct. It will still engage the street and create a welcoming interface with the community both inside and out.
The proposed new building is a public building within a public sub-precinct with the hospital. It adopts a contemporary design utilising simple rectilinear forms, so that it does not compete with the former police barracks building. Wall surface materials have been selected in terra-cotta material that harmonises with the red brick of the heritage listed building. Landscaping proposed adjacent to the walls to the vehicle compound at the rear of the site to provide a softened appearance as the backdrop in the visual curtilage of the heritage listed police barracks.
« Central Station Dining Hall
PADSTOW MULTI STOREY COMMUTER CAR PARK »
Email: projects@ccgarchitects.com.au
Address: Level 2, 5 Wilson Street Newtown NSW 2042

redpajama_set_name: RedPajamaCommonCrawl


MONROE, WA – Canyon Creek Cabinet Co. recently launched new products in its custom framed Cornerstone and frameless Millennia cabinetry lines. Four new mitered doors that are variations of the classic shaker style join Canyon Creek's lineup of over 100 doors.
The company added Artisan 6, a package of finish enhancement distressing techniques that replicate the natural appearance of aged wood. A6 incorporates "wire brushing" to create the raised grain look of weathered wood. The addition of a glaze during the finishing process will further enhance the grain directional marks of this Artisan package, emphasizing the effect.
Canyon Creek also introduced a new decorative hardware program—in a variety of popular finishes and styles—to its four cabinet lines: Cornerstone, Millennia, Newcastle and Katana. More info on these new products can be found on the company website: CanyonCreek.com.
Canyon Creek Cabinet Co., located in Monroe, WA, is a leading manufacturer of affordable, custom frameless and framed style cabinetry for every room in the home. In 2009, Canyon Creek introduced its line of closet and storage products—Canyon Creek Closets Plus—creating one of the most extensive lines of custom cabinetry and closet storage systems available. In 2010, Canyon Creek launched Newcastle framed cabinetry, and in 2011, Katana frameless cabinetry. Both cabinet lines are designed to provide high-quality, entry-level product for today's price-competitive market.
Canyon Creek has been a recognized leader in the kitchen and bath industry since 1981 and has won many industry awards for its cabinetry and environmental practices. Maintaining its ISO 9001 certification since 2003, Canyon Creek was also chosen for the 2010 Association of Washington Business Green Manufacturing Award and the 2011 Manufacturer of the Year Award. In addition, Seattle Business magazine presented the company with their 2012 Washington Green 50 Manufacturing Award.


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rdom () { local IFS=\> ; read -d \< E C ;}
SEMANTIC_VERSION=$(while rdom; do
    if [[ $E = "SemanticVersion" ]]; then
        echo $C
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    fi
done < $SCRIPT_DIR/Module.xml)

# Get build date
BUILD_DATE=`date -u '+%Y-%m-%d %H:%M:%S'`
BUILD_DATE="$BUILD_DATE UTC"

# Get current commit
BUILD_COMMIT=`git rev-parse HEAD`
if [ "$GIT_BRANCH" != "" ]; then
  BUILD_COMMIT="$BUILD_COMMIT; $GIT_BRANCH"
fi

# Determine if there are pending changes (dirty working directory)
BUILD_IS_DIRTY="false"
if [ "$(git status --porcelain)" != "" ]; then
  BUILD_IS_DIRTY="true"
fi

cat >ProtobuildVersion.cs <<EOF
namespace Protobuild
{
    public static class ProtobuildVersion
    {
        public const string SemanticVersion = "$SEMANTIC_VERSION";

        public const string BuildDate = "$BUILD_DATE";

        public const string BuildCommit = "$BUILD_COMMIT";

        public const bool BuiltWithPendingChanges = $BUILD_IS_DIRTY;
    }
}
EOF

redpajama_set_name: RedPajamaGithub


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January 12, 2020 admindzuklu
SOCU charges…says Govt out to get JagdeoAhead of Guyana celebrating 52 years as an independent nation, former Head of State Donald Ramotar has noted with great concern that "the very heart" of the Guyanese democracy may be under threat following the recent arrests of several officials under the former People's Progressive Party/Civic (PPP/C) Government. He shared these sentiments in an interview with Guyana Times on Wednesday where he concurred with former Attorney General Anil Nandlall, that the charges pursed by the Special Organised Crime Unit (SOCU) is aimed to prosecute his predecessor, Bharrat Jagdeo who served as Finance Minister before his presidential reign."It is very clear that the real target is Jagdeo and that the regime is out to try to find some way to get him and that is why they're breaking all laws and precedents going back all of these years to try to see if they find any nonsense of what they may describe as misconduct in office. I think the Guyanese people are on to this military-type Government," Ramotar expressed.Commenting on the dangers such actions could place of the democratic process, he indicated that the country's economy and independence of various constitutional bodies and agencies could be infringed."It goes to the heart of our democracy and I think it will have some very serious consequences on our country as a whole because many banks and business people will be afraid to do business with the Government because they are going all these years to charge people. They are vicious and vindictive in their actions [which are] very discouraging to business people who would like to do business with the Government," he added, references the concern operatives within the Private Sector have raised.Following the high-profile arrests and $6 million bail each granted to former Finance Minister, Dr Ashni Singh and former National Industrial and Commercial Investments Limited (NICIL) Head Winston Brassington, former Finance Minister Sasenarine Kowlessar was recently arrested and questioned by SOCU. He served as Jagdeo's Finance Minister from 1999 to 2006. In Ramotar's view, the coalition Government is using Executive powers to "pressure" its opponents."[The Government] is undermining our democracy – the very fact that you are charging officials of the former Government. It's very clear that the APNU/AFC regime is undermining all our institutions and the Judiciary is not totally excluded from some the pressures that the Executive is putting in trying to undermine the constitutional bodies in our society including the Parliament and the Police. When one looks at some the decisions in the Judiciary, one wonders if some members are bowing and breaking to pressure to the Executive," he claimed.It was earlier this week that Nandlall expressed that the arrest of Kowlessar came as no surprise to members of the Opposition as they have always felt that the motive has been clear from day one. "It is clear beyond any doubt that these investigations are driven by a political and vindictive motive."Nandlall had said, ultimately, this strategy is designed to get to the period when Opposition Leader Bharrat Jagdeo was the Finance Minister. Jagdeo served as senior Finance Minister in May 1995 under Dr Cheddi Jagan. In December 1997, he was appointed as second Vice President.While noting that constitutional immunities prevent them from charging Jagdeo while he was President, Nandlall is of the opinion that the coalition Government will go as far as it take even if it meant returning to two decades ago to persecute him for transactions done while he was Finance Minister. Nandlall concluded that it is unfortunate that the Office of the Director of Public Prosecutions (DPP) does not see it as part of its constitutional mandate to protect the citizenry from these types of investigations, which are widely regarded as abusive and oppressive, but rather is part of the design that is promoting it. "If this is not the use of the criminal legal process to persecute political opponents, then I do not know what it is," he told this newspaper.
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People from around the country, weary with waiting for the Government to take the necessary action to address climate change, have decided to take matters into their own hands.
They have drafted a Declaration setting out what needs to be done, with people signing up to commit to making it happen. This will be done through community climate action plans where organisations investigate their own climate impact and make plans to reduce it. It is within these everyday groups that we are all contributing to climate change and it is within these groups where actions for a safe climate will make a difference.
"We know what has to be done. And it has to be done by people at all levels - citizens, businesses, local councils and government. We are not waiting - the problem is too serious and too urgent - and we are setting out to do it ourselves" said Dr Joanna Santa Barbara, national chair of Our Climate Declaration.
The project will be launched on Sunday 18 June, simultaneously in Auckland, Wellington, Christchurch and Dunedin. The launches will be video-linked, with each venue receiving some speakers from the others. It will also be live streamed to every device that wants to watch.
"We wanted to demonstrate that you don't have to fly or drive long distances to connect with others around the country" said Pat Baskett, organiser of the Auckland launch. This is the kind of world we are heading into to reduce our climate impact."
Help will be available to groups such as schools, churches, small business, sports clubs, medical practices, unions, farmers, and all places where people collectively use energy and can help protect the climate.
The Declaration is available at www.ourclimatedeclaration.org.nz and can be signed online. There is also advice on speaking to groups about climate change and how to develop local plans, which can be posted by community groups on the website for others to share. Details of the four launches are also available there, and the link for the live stream will be there from Saturday.
Endorsements have come in from Prof James Renwick, climate scientist; Phillip Mills, businessman; Ora Taiao, the NZ Climate and Health council, an organisation of health professionals advocating for healthy climate action; musicians such as Peter Scholes and Sean O'Connor; and celebrities such as Robyn Malcolm, who will all be featured at the launch.
"Eventually, when they see that citizens care enough to take their own action, governments will have to follow" said Santa Barbara.
More information can be found on the Our Climate Declaration website.


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Home » Celebrities » These Celebrities Shut Down Shipper Theories Completely
These Celebrities Shut Down Shipper Theories Completely
01/14/2021 Comments Off on These Celebrities Shut Down Shipper Theories Completely
Shipping might be some teenagers' favorite online pastime, but there are some A-listers who would rather not be a part of fans' one true pairing (OTP). The celebrities who shut down shipper theories shut them down hard, leaving little to fans' imaginations. Fan fiction is still, unfortunately for some, just fiction.
While shipping is a mostly-harmless common practice among fans of fictional TV shows, movies, and books, when it comes to IRL shipping, things get a little more complicated. Stans take body language, visual cues, and more into account when they ship real-life celebs. Some just make edits or GIFs to acknowledge their favorite actors' chemistry, but others go as far as writing stories and erotica about them. But when some shippers' fantasies begin turning into conspiracies that other fans believe and spread, they can negatively affect the celebs' mental health and real-life relationships, ruining the fun for everyone.
Shipping has become an undeniably large part of fan culture, and it's led some celebs to address their shippers directly. Unfortunately for the most dedicated shippers, celebs are usually denying their relationship rumors rather than confirming them. The revelation that one's ship might not ever come true is a sobering one, but sometimes a flat-out denial isn't enough to stop some fans from pursuing evidence they were right all along. Scroll through some of the most controversial ships below, and read why the celebs at the center of the fantasies shut down their fans' wildest dreams.
"Larry" – Louis Tomlinson & Harry Styles
Shippers of Louis Tomlinson and Harry Styles, or Larry shippers, are perhaps the most prominent shippers in 2021 despite the fact both stars have been in relationships with other people for most of their lives in the spotlight. One of the former One Direction bandmates shut down the shipper theories about them after the group's split in a pretty straightforward way.
In 2017, Tomlinson said he found the fantasies "disrespectful" to his girlfriend, Eleanor Calder, who he had gotten back with after dating originally from 2011 to 2015. He also said the rumors and fear that their every interaction would be analyzed under a microscope ultimately hurt his friendship with Styles. "It took away the vibe you get off anyone. It made everything, I think on both fences, a little bit more unapproachable," Tomlinson told The Sun.
"Camren" – Camila Cabello & Lauren Jauregui
Before Camila Cabello left Fifth Harmony in 2016, a loyal group of Harmonizers shipped Cabello with fellow group member Lauren Jauregui. While the stars were clearly friendly on camera and in front of fans onstage, they were definitely not in a relationship.
Jauregui, who openly identifies as bisexual, went off on Twitter in 2017 at Camren shippers, calling the theory "invasive, scary, delusional, [and] disrespectful." She also informed her timeline that the theory was "never real," and sent a message to other shippers about the impact the rumors can have on any given celeb. "[Y]ou never quite become ok with people sexualizing you and your friendships for their sick pleasure," she said.
In October 2020, Jauregui added more context to her 2017 tweets in an interview on Becky G's podcast En La Sala. "[The shipping] made me feel like a predator because the types of clips people would put together and the types of stories that people would write and the type of stuff, I was always the aggressor and I was always the one turning her," she said. "… To this day, [I] hyper-analyze every connection that I have with a girl because I don't want to make them feel like I'm looking at them that way. To this day, [fans are] convinced that this was real… And I can't do anything to change it."
"Trixya" – Trixie Mattel & Katya Zamolodchikova
Fans of Trixie Mattel and Katya Zamolodchikova have shipped the iconic duo for years, but some fans take their fantasies too far. Despite being super close and joking about sex onscreen, the former RuPaul's Drag Race stars never dated, and they don't appreciate erotic fan-fiction about them.
After they read some fan-fiction for a YouTube video in 2015, the two ended up on a panel at Drag Con and had an unrelenting fan read them their own erotica written about the drag queens.
Mattel recounted the incident while talking about doing live panels in 2018, sharing, "I said, 'No thank you!' but she proceeded to read it anyway. We don't have a whole lot of control!" Several fans also condemned the incident after it first happened in May 2017.
"Feltson" – Tom Felton & Emma Watson
Harry Potter fans love to ship fictional characters, but there are a few IRL fantasy couples Potterheads wish would get together, too. Emma Watson and Tom Felton, who played Hermione Granger and Draco Malfoy respectively, have long faced rumors of their potential romance.
Watson actually gave some credibility to the rumors in 2012 when she said she used to have a crush on Felton. "He was so cool. He totally knew [I had a crush]," she said on The Jonathan Ross Show. "And the thing is, he'd turn and go, 'Oh, I see her in a younger sisterly way.' And it just broke my heart, still does."
Potterheads rejoice, though, because while the duo may not be a couple in 2021, they have confirmed they are still friends to this day. At least there was a spark!
"Shamila" – Shawn Mendes & Camila Cabello
Some shipper theories really do come true! But, in the case of Shamila, not before the pop superstars came out the gate denying any and all romance rumors. Shamila shippers had their hopes and dreams shattered in October 2016 when Mendes told People he was only friends with his "I Know What You Did Last Summer" duet partner.
"We're just really close friends," Mendes said. "She is one of those people who is just very easy to connect with and still, to this day, is one of my greatest friends." At the time, they were seemingly telling the truth since Cabello went on to date Matthew Hussey in 2019 and Mendes went on at least one date with Hailey Baldwin in 2018.
In June 2019, however, Shamila entered the mainstream with their second duet, "Señorita." From then on, it was pretty clear the two were about to become an item. The pair confirmed they were dating in summer 2019, and Mendes even admitted he liked Cabello the whole time on The Late, Late Show with James Corden in 2020, just like fans suspected. Cue the PDA and Instagram makeout sessions.
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Tobiáš Adler přišel do Čech z Horního Rakouska. Se svým otcem Martinem a dědem Kryštofem působili nejprve jako sklářští tovaryši ve Fryšavě na Moravě. Tobiáš pak byl předním tovaryšem na sklárně u Doks v severních Čechách, odkud přešel do huti u Nespěřic. Kdy se tak přesně stalo, nevíme, ale první záznam se jménem Tobiáš Adler je ve zručské matrice k roku 1694. V huti u Nespěřic zaměstnával převážně bratry a později i syny. Huť vzkvétala a Tobiáš ji provozoval až do r. 1721, kdy odešel se synem Tobiášem do sklárny v Rožmitálu pod Třemšínem.

Tobiáš měl 5 synů - Františka Antonína, Tobiáše Matěje, Jana Bedřicha, Šimona Hanibala a Josefa.

František Antonín a Tobiáš Matěj provozovali hutě na Šumavě v prostoru Plání na Vimpersku a Železné Rudy. Obchodní blokády a krize sklářství během Napoleónských válek však způsobily, že se potomci Tobiáše a Františka na Šumavě přestali sklářstvím živit.

Tobiáš provozoval také huť v Rožmitálu pod Třemšínem a v letech 1749-1783 Skelnou Huť pod Brdcem.

František Antonín si na přelomu let 1711/1712 najal huť v Bošticích a provozoval ji asi do r.1714. V roce 1721 odkoupil od statku Těchobuz část lesa. Vybudoval zde huť Tallenberk, která pracovala až do r.1749. V roce 1721 se oženil v Ratajích s Annou Marii Gattermayerovou a jeho tchán mu snad jako věno zřídil sklárnu v Jelenově, kterou pak provozoval až do r.1735. Od roku 1722 až asi do r.1727 měl pak v nájmu i sklárnu ve Slavětíně. V roce 1724 koupil i statek Mezilesí a v roce 1735 statek Těchobuz. Zde zřídil další sklárnu, která pracovala až do r.1750, kdy byl provoz ukončen pro nedostatek dřeva. František Antonín zemřel v roce 1736.

Starou Huť u Nespeřic převzal po otci Jan Bedřich Adler, který se téhož roku oženil s Evou Terezii Gattermayerovou. Huť byla zrušena v r.1731 údajně pro nedostatek dřeva. Jan Bedřich založil blíže ke Kácovu pod obcí Kněž huť Marienthal. Zde ukončil provoz r.1744, kdy koupil statek Rušínov a Modletín a založil zde další sklárnu. Statek však sousedil s državou Auerspergu, se kterými se dostal do sporu. Jan Bedřich jim pak r. 1756 statek prodal. Jeho syn Ferdinand si před tím v r.1752 pronajal huť Brušoveckou u Herálce. Tato huť pak zanikla po r. 1760.

Šimon Hanibal se oženil také v roce 1721 a to s Alžbětou Lubiňskou z Uhl.Janovic. V roce 1726 založil v Loukově na panství lipnickém sklárnu Sv.Františka. V roce 1732 založil na vilémovickém statku u Ledče n. Sázavou Hanibalovskou sklárnu. Obě sklárny pak provozoval až do své smrti v roce 1742. Sklárna ve Vilémovicích pak zanikla. Sklárnu v Loukově provozovala dále Šimonova manželka Alžběta. Ta v roce 1743 získala statky Loukov a Budíkov. Na budíkovském statku u Kejžlic pak zřídila další huť, kterou vede až do své smrti. V roce 1759 František se sestrami tento statek prodávají. Pro syna Františka Karla již v roce 1749 koupila vdova Alžběta statek Častrov u Kamenice n.L. Zde zakládá František huť Drozdany. Huť v Loukově zůstává v držení sester Františka a vede jí nejprve Václav Čapek, manžel jedné ze sester a pak Guolfinger ze Steinsbergu, manžel druhé. V roce 1763 dědicové statek Loukov prodali, ale huť zde provozovali dále až do roku 1771, kdy jí získal Johann Ignác Eisner. R. 1779 přišel Častrovský statek Františka Karla do úpadku. František Karel je pak uváděn v roce 1788 jako skelmistr, který toho roku zemřel, na huti brumovské na moravsko-slovenských hranicích.

Poslední syn Josef skelmistrem nebyl. V roce 1727 se oženil v Brunstu u Železné Rudy s Ludmilou Eisnerovou a v roce 1743 umírá u svého otce na Rožmitálsku. Jeho syn Václav měl syna Michaela, Ten se oženil r.1797 na Zbraslavické huti s vlastní sestřenicí a téhož roku se stal skelmistrem v Dobré Vodě u Světlé nad Sázavou. V roce 1803 však koupil od skelmistra Pohla huť v Loukově, kde v roce 1821 zemřel. Před tím však v roce 1818 převzal do nájmu nově zřízenou huť sv.Josefa u Jilemu na Chotěbořském panství. Po Michaelově smrti se hutí ujal jeho syn Václav. Ten však zemřel již v r.1838. Další ze synů Michaela František měl v nájmu od roku 1829 huť v Ostředku. I když měl smlouvu uzavřenou na 8 let, již v roce 1830 od ní ustoupil a na huti pak pracoval jako skelmistr František Kavalier.

Odkazy

Reference

Literatura
 MAREŠ, František. České sklo: příspěvky k dějinám jeho až do konce XVIII. století. V Praze: Nákladem České akademie císaře Františka Josefa pro vědy, slovesnost a umění, 1893; 268 stran, 3 nečíslované listy příloh. Rozpravy České akademie císaře Františka Josefa pro vědy, slovesnost a umění v Praze; ročník II, třída I, číslo 1.
 VLASÁK, Antonín Norbert. Okres Uhlířsko-janovický v Čáslavsku: nástin historicko-archaeologický. V Táboře: A.N. Vlasák, 1885. 162 stran; Bibliotéka místních dějepisův pro školu a dům; svazek 12.
 ZAVADIL, Antonín Josef, ed. Kutnohorsko slovem i obrazem: Práce vešk. učitelstva okresu. Díl II, Místopis okresního hejtmanství. Část I-II. Kutná Hora: Šolc, 1912. 2 svazky.
 Jan Bárta: Staré skelmistrovské rody
 Jiří Adler: Posázavské skelmistrovské rody

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Tension mounts in Uttar Pradesh over conversions
LUCKNOW, JUL 13 (IANS) | Publish Date: 7/13/2019 11:23:43 AM IST
Tensions were building in the rural interiors of eastern Uttar Pradesh between fringe Hindu outfits and Christians over the issue of conversion.
While the outfits are accusing Christians of forcibly converting Hindus, attacks on churches and priests were also increasing. The police, in majority of the cases, were being accused of taking action only against Christians. Two priests in Rae Bareli, Azad and Kahadi Yadav, were recently arrested on charges of forcible conversions. On July 4, a local priest Jitendra Sahni was attacked and injured by 'Hindu' leaders.
Similar attacks have been reported from Jaunpur, Robertsganj, Varanasi and Gorakhpur in the eastern region. Jaunpur, in fact, witnessed violence and attacks on churches in September 2018 after which some churches had closed down.
Officials of the US Embassy then met the state's Minister of Minority Affairs Mohsin Raza and sought his intervention.
The minister said: "In December last year, a delegation came from the US Embassy and gave me a list of churches most of which were located in Jaunpur, Sultanpur, Azamgarh and some other districts and I spoke to the district officials. The churches were later reopened." Rajinder Chauhan, a priest in the Jaunpur church, says he was arrested after the church reopened. "I was sent to jail for 15 days because Hindu groups in connivance with the district administration does not want us to remain here," he said.
Police officials in these districts have claimed that they did not have the details of cases but said that they "often" received complaints of forced conversions.
According to a BBC report, the US-based 'Alliance Defending Freedom' (ADF) stated that more than 125 cases have been registered against Christians in this region and 110 priests were arrested on conversion charges. Chauhan said that more than 2,500 people come to attend his prayer meetings on Sundays.
"Surely I do not force 2,500 people to come and attend the prayer meeting. They come because they get peace of mind."
In Robertsganj, a priest Narendra Kumar was asked by the police whether he had the license to run a church. "When I said that no license was needed to run a church, they got it locked up," he added.
Defending the 'crusade' to check conversions, Harsh Agarwal, an RSS functionary in Robertsganj, said: "These people (Christians) are luring the poor with money and they abuse the Hindu religion. We have to stop this.
"We are not against the Christian religion but we are definitely going to oppose forced conversions."
Nirbhay Singh, the self-styled head of an organization called 'Hindu Yuva Shakti', has been accused of vandalizing churches.
"We are committed to 'ghar wapsi' (homecoming) of Hindus who have been misled. I am proud that I have brought 2,500 Hindus back into their parent religion."
-Tags:#Tension mounts in Uttar Pradesh


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Judge interdicts moe from procuring "le phirimela go tlhaba" novel
Court finds publisher breached contract
The applicant says the Publisher and Distributor failed to pay royalties
Ministry caught in between
Ministry suffers consequences of Publisher's breach
SESUPO RANTSIMAKO
FRANCISTOWN: The High Court has interdicted the Ministry of Education, Skills and Development from purchasing a Setswana novel titled "Le Phirimela go Tlhaba" from Pentagon Publishers or any of the agents.
The interdict was issued after the author Kedikilwe Oageng, represented by Uyapo Ndadi made an application to court alleging that Pentagon Publishers breached the terms of their publication contract. The contracted, dated May 27th, 2009 between Oageng and Pentagon Publishers, who were also the distributor of the book, provided that the company would pay royalties to Oageng for each book sold. Court documents reveal that Oageng claimed that Pentagon Publishers had previously failed to pay the agreed royalties at the rate of 15 percent per sold copy resulting in an initial law suit to recover money due to him in 2014. In spite of the successful litigation to recover the royalties, Oageng initiated fresh proceedings alleging that the publisher fell into arrears once again in March of the same year.
The Attorney General, representing the Ministry of Education, Skills and Development argued that Oageng could not initiate fresh proceedings arising from the same facts as those upon which he based his earlier litigation. Despite the provisions of State Proceedings Act, which prohibits the Court from issuing an Order interdicting government, Oageng wanted the High Court to interdict and restrain the Ministry of Education Skills and Development from procuring as well as purchasing the novel from Pentagon Publishers or any of its agents. The Court found that the author had sought an Order cancelling the contract between Pentagon Publishers and himself, arising from a breach of contract that arose after the initial judgement for royalties and dismissed the Attorney General's argument on the legal point. The Court, without making reference to the State Proceedings Act, in addition granted the interdict against the Ministry.
According to Oageng's Affidavit "on the 27th May 2009, I entered into a written agreement for the publishing and distributing of the Setswana novel with Pentagon Publishers, the terms of agreement was that the Publisher would pay royalty in 15 percent of every single copy sold. The Amounts are payable annually by the 15th March. Further the agreement stated that the Author is entitled to terminate the agreement in the event that the publisher fails to comply with any of the terms of the agreement giving the publisher three months' notice." By agreement between the legal representatives for Oageng and the Attorney General the facts were not disputed.
In the court papers the Author of the Setswana Novel submitted that the Ministry and Pentagon Publishers have a long-standing agreement from 2009, in terms of which the Publishers sells the novel to the Ministry at an agreed price and that the author would recover the royalties of all copies sold from it. "The Pentagon Publisher then breached the agreement we entered into and failed to pay the royalties as agreed. The Publisher fell into arrears and failed to pay the royalty for the accounting year of the 1st of March 2013 to the 28th of February 2014 and from March 2015 to date despite having sold copies of the novel to the Ministry and other customers. Consequently, I successfully sued Pentagon Publishers to recover the amount of in royalties. Unfortunately, I have not been able to successfully execute and recover the debt owed to me and the Publisher continues to be arrears of payment of royalties that are due post the judgement being issued by the court," the author claimed in his court papers.
As a result of the continuing breach of the contract Oageng issued a letter of demand to the Publisher on 12th December 2015 for breach of contract, demanding outstanding royalties and further cancelling the agreement. "The Publisher and the Ministry were informed of this cancellation of the agreement on the 28th January 2016 and as such asked to cease and desist from purchasing the book. Despite the cancellation of the agreement between Pentagon Publisher and myself, the latter continues to sell the novel to the Ministry and I do not get paid the royalties that is due to me," the author explained.
Delivering Judgment High Court Judge Tshepo Motswagole restrained and interdicted the Ministry from procuring and purchasing the novel from Pentagon Publishers or any of the agents. "I am satisfied that the Respondent's defences are untenable and lack merit, especially in the face of a formidable case put up by the Applicant. To permit such defenses to stand in the circumstances of this case will be tantamount to the legitimating of wrongful conduct, resulting in unjustly enriching the guilty party at the expense of the innocent. No evidence has been put before court to show that the Applicant renounced the right to benefit from the sweat of his labour as an Author, and that he has by conduct permitted the Respondent to take full benefit of the sale of the book. Therefore, the Applicant must accordingly succeed. It is hereby declared that the contract between the Applicant and Pentagon Publishers is cancelled. An order is hereby made interdicting and restraining the Ministry from purchasing and procuring a book titled "Le Phirimela go Tlhaba" from Pentagon Publishers or any of the agents," Motswagole ordered when delivering judgement.
At the time of press the Attorney General was not in a position to advise as to whether it would appeal the judgement and the interdict.
The fight for khama's mosu airstrip
A druglord's paradise!


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Q: Import CSV for products with new sub categories I'm importing a CSV file with new subcategories not already created.
Some times products imported correctly and others it gives me:
general system exception
Asymmetric transaction rollback

Anyone can help?

A: I had this issue recently and tried everything to resolve this.

*

*Full reset and manual reindex (completed successfully)

*Cleared Cache (cache:flush, cache:clean)

*cleared all static files

*Changed indexers to run on schedule rather than save

*Tried setting $this->_isRolledBack to false in Magento/framework/DB/Adapter/Pdo/Mysqlphp rollback()

*Reset Server, MySql and other services.

*Tried deleting all products, reindexing and importing

*full DB restoration

From my research the issue seems to point to a DB error, so I was confident that a full DB restoration would help. But after multiple restoration I still had the issue.
During the course of creating multiple clones, and trying to import products, I was successful on one instance. The difference was I removed all the image locations in my csv. (I had a lot of images with this import) But even if I reduced my product list to 100 it would still get the error. I was successfully able to import my products and later on imported the images in a separate spreadsheet.

redpajama_set_name: RedPajamaStackExchange


text:
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Bananacalypse – A Pointless Short Story About Panicking and Impending Doom>
Fiction, Short Stories
Dawn – Short Story
Kon July 6, 2018 /0 comments
Inspired by a prompt I saw a long time ago
The sun rose over the trees, announcing the start of a new day.
Monique glanced out of the window to see dawn approaching as she left her night shift at the hospital. As the doctor's assistant, she always had a busy shift. After all, her, the other assistant Melody, and Doctor Bennett were the only three citizens of their small town trained in medicine – and their small town kept them busy.
Monique had made it to her car when her phone buzzed. Report to ER ASAP. Frowning, Monique exited her car and made her way back through the small hospital.
The emergency room was small and often insufficient, but the three of them made it work. Monique had helped treat many emergencies in that room, from illnesses to injuries to the birth of their town's youngest citizen. None of that prepared her for what she saw.
Three teenagers – not much older than Monique's youngest brother – were sprawled out on cots. Each had burns decorating their skin, stopping at the edge of their clothes.
"What happened?" Monique asked.
"I don't know," Doctor Bennett replied. "Hand me the scanner."
Working quickly and efficiently, Doctor Bennett did his best to treat the three patients. Melody showed up about half an hour later, and she and Monique helped the best they can.
An hour later, the patients' vital signs were steady, but the burns hadn't began to heal.
Doctor Bennett slumped into his chair. "I have bad news."
"You found the cause of the burns?" Melody asked.
Doctor Bennett nodded. "Apparently, the patients contracted a disease that causes a reaction to the sun. I just hope it hasn't spread."
Monique looked around the room, noticing the large number of windows. Their little town had been built with natural light in mind, and with the exception of the shades from the trees, there was no place to hide from the sun.
"Is there a way you can make sure the disease isn't spreading?" Monique asked.
"I'll have to look into it," Doctor Bennett replied. "But the disease is viral. I believe that it will spread."
"So let's find some way to stop it," Melody said.
And they worked. The three of them put their heads together all morning to find the cause.
"I think I have it," Doctor Bennett announced around noon. "The three girls were bitten by the migrating beetles that have taken to our forests this spring."
"So, the beetles transmitted the disease?" Monique said, looking over the Doctor's shoulder at the research.
"See, the start of the disease is here around the bite," Doctor Bennett explained.
"But those beetles are infamous for biting everything," Melody pointed out.
"We can keep them away," Monique suggested. "It should be easy enough. The swarm isn't here until tonight – if we can figure out some way to repel them…"
When evening came, and the sun set, Doctor Bennett and his associates were sure they had found the secret to repelling the bugs.
"Thank you for your work, Doctor," the mayor said, after the repellant was deployed along the edge of the forest.
"They're coming!" Melody said.
Monique frowned and squinted at the forest. She didn't see anything…
But then she heard them, and they were loud. The buzzing sound grew in intensity, becoming almost painful, and a black cloud flew through the forest.
And right through their defenses.
"Run!" Somebody screamed.
Monique froze in her tracks as the bugs sped towards her. How had they failed?
"Monique!" Melody yelled, grabbing Monique's wrist. The two assistants sprinted away from the swarm as the bugs descended.
Monique's foot hit a rock, and she tumbled down, taking Melody with her. The bugs were on them in a second.
The bites didn't hurt, just tickled a little. Only a few bugs bit as the swarm flew by, and Monique hoped that maybe those bugs weren't infected.
"Ow," Melody complained, forcing herself off the ground. She helped Monique up, and the two of them walked towards the engulfed village.
When the swarm had passed, and only a few beetles remained, Doctor Bennett checked everyone for the disease. When he reached Monique, his face was ashen.
"It appears that all the bugs were infected," Doctor Bennett said. "And now we all are too. There might be a way to keep the burns from being fatal, but-"
Monique sighed and looked out at the villagers around her. In the distance, she could barely make out the forest. The sun rose over the trees, announcing the start of a new day.
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Going Binary: Captain Marvel Review


When Elynne Chaplik-Aleskow's youngest sister, Ivy (second from left), died in a plane crash at age 16, the remaining siblings decided to always honor the love they had shared. In doing so, Chaplik-Aleskow (far right) believes Ivy lives on in the hearts of her family and friends.
I believe in Ivy. She was my youngest sister. She died in a plane crash with my father when she was sixteen.
Ivy was born an old soul. There was something in her brown eyes that touched your core when she looked at you. When she smiled, you knew that she understood.
That was my nickname for her. "Love-love" described how I felt about her. It was a double love. We were thirteen years apart. She was my youngest sister and the daughter I would have wanted.
When Ivy was in high school, her English teacher wanted to promote her into Honors English. As was Ivy's style, she asked the family's advice, pondered it, and then came up with her own practical and clear decision.
On this issue she had decided to stay in regular-level English because she felt that her teacher was excellent. She said that when he spoke, he made her want to "hug a dictionary." Ivy was fourteen years old at the time of this particular insight.
She was the girl who befriended the underdog. If someone were made fun of by the group, Ivy would defend that person and protect his or her feelings. Her friends were a cross-section of many different types of people.
Ivy was quiet and gentle. Often she would observe others, not missing a thing. She was thoughtful, endearing, and loyal. She was her own person, her own young woman. Unknowingly she had no time to waste. She had only sixteen years to do what she was going to do.
Ivy observed that there were many people who were quietly giving of themselves but who were never noticed or acknowledged for their generosity of time and caring. After the plane crash, based on Ivy's philosophy, my family created the Ivy Lynn Chaplik Humanitarian Award at her high school.
Ivy's beauty lives on in her family's and friends' hearts. Each year in the hearts of the nominees for her award, her being is once again touched and her beliefs continue to inspire.
I have been blessed with three sisters: Linda, Susan, and Ivy. They are my treasures. They come from the same love from which I came. We are bound by DNA. Yet we are also connected by an invisible frame and foundation in which and upon which our lives have been shaped and implemented.
There will always be four. We are four. We exist as four. We loved and love as four. We are four sisters in life and in death.
Elynne Chaplik-Aleskow is founding general manager of WYCC-TV/PBS and Distinguished Professor Emeritus of Wright College in Chicago. Her nonfiction stories and essays are published in numerous anthologies and magazines. Her husband is her muse.


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Last Update 15:6
Doctors Without Borders says it plans to return to Somalia
AP , Friday 27 Jan 2017
Shabaab attacks Kenya army base in Somalia
Drought may add famine to Somalia's humanitarian woes
Three killed in Somalia tea shop blast, soldiers among casualties
Doctors Without Borders says it plans to return to Somalia, more than three years after closing its operations in the turbulent country.
The medical aid charity known by the French acronym MSF says its return to the Horn of Africa country will be a cautious one.
The charity pulled out of Somalia in August 2013 after 22 years of operations there. It cited "extreme attacks on its staff" in an environment where armed groups and civilian leaders increasingly allowed or condoned killings, assaults and abductions of humanitarian workers.
Somalia is trying to rebuild after decades of chaos that began in 1991 when warlords overthrew a longtime dictator. Extremist group al-Shabab continues to launch deadly attacks in the capital, Mogadishu, and elsewhere, threatening the country's first functioning central government in years.
Doctors Without Borders
10:14 English Premier League fixtures (19th matchday)


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Home › The Front › Recent
'Power' Season 6 Trailer Teases The "Final Betrayal"
Ghost can't trust anyone, not even his own son.
By Bernard "Beanz" Smalls , Staff Writer
@photosbybeanz
Meet Bernard Smalls, better known as Beanz. He is the Men's Lifestyle & Pop Culture writer for Cassius Life plus handles tech and video games for Hip-Hop Wired. Beanz has been with the company since 2019 and has been working as a writer for eight years. His favorite hobbies include video games, of course, and watching a lot of movies, mainly sci-fi. He also moonlights as a photographer in his spare time. @photosbybeanz
Follow @photosbybeanz
'Bel-Air' Season 2 Trailer Reveals OG Ashley, Tatyana Ali, Has Joined The Cast
EXCLUSIVE: Jacob Latimore, Tosin Cole, Rotimi & D.C. Young Fly Talk Remixing 'House Party'
August 25, we will be singing "they say it's a big rich town" for the final season. Starz's hit original series, Power returns for its sixth and final season. Yesterday (July 6) the network unveiled the trailer for the show, and it basically proves one thing, Ghost (Omari Hardwick) can't trust a single soul in his shaky circle, not even his own damn son.
All James St. Patrick wants to do is be a legitimate businessman, honor his deceased daughter by getting the Queens Child Project erected and be a loving father to his stubborn ass son Tariq. BUT, of course, that will prove to be difficult thanks in large part to his brother in crime, Tommy (Joseph Sikora) — now trying to cancel Christmas on him… AGAIN — to be the biggest dope dealer in NYC with the help of his boo Keisha (La La Anthony) who has a bone to with her best friend Tasha (Natura Naughton).
It also doesn't help that Riq is out here looking like a watered down version of his dad by getting into the drug game while at the same time partnering up with those who want to see the patriarch of the St. Patrick family six feet under. We can't forget that Dre (Rotimi) is still running the streets, and the Feds led by Cooper Saxe (Shane Johnson) are still hot on his tail and want to see the former drug kingpin in jail.
The 2:08 trailer confirms the return of Angela Valdez (Lela Loren) who was shot by Tommy in season 5's "cliffhanger" season finale. Despite viewers thinking it was the end of Ghost's legendary side chick, showrunner Courtney A.Kemp confirmed immediately that she would be returning for the final season so her presence in the trailer should come as no surprise to anyone.
So buckle up, it looks like we are in for a wild ride, season 6 will be bringing the twists and turns and somehow manage to spread it out through 15 episodes. Step into the trailer for Power's final season below and keep it locked here for full recaps.
La La Anthony Larenz Tate nl omari hardwick power starz


How To Rank Wish SMS/Quotes Collection Website?
Thread: How To Rank Wish SMS/Quotes Collection Website?
We know content is king. Google rank website depends on their unique content. But I'm curious about to know how to rank sms collection website? We search for various wish sms such as birthday sms, good morning sms etc. We find websites, most of them are post collection of sms/quotes. That means most of the contents are copy. So, how google treat them? Whats about their ranking factor. I want to start such kind of website. That is why I'm confused. I will post various sms collection as blog post which will be copied from different websites. Thanks in advance. Please share your thoughts.
From what your saying you'll be a copy cat. Copy cats won't rank unless you have a ton of good links, but that can kill you too.
Then what can I do? Suppose, I will post "50 Motivational Quotes For Your Life". Then I have to collect best 50 quotes from different site. am i right? Is that copy? How google will treat that?
You are copying, its as simple as that. why would google treat you better than the source. You need to add extra to the copied content so add value. There is of course a copyright issue. Being available on the web does not always mean free.
Rank with Quotes, Rank Without - Meaning?


text:
Riding Solo: Kaley Cuoco Literally Back In The Saddle One Day After Divorce Announcement
Source: FameFlynet
Sep. 27 2015, Updated 3:34 p.m. ET
One day after announcing her divorce from her husband of 21 months Ryan Sweeting, Kaley Cuoco found comfort in an old hobby -- horseback riding!
The actress got back in the saddle Saturday for a competition at Flintridge Riding Club in La Cañada Flintridge, California, People reports, and according to sources, she did not let troubles in her personal life prevent her from performing well at the event. "She was extremely focused during the event," an insider revealed. "You could tell that she was really enjoying being a part of the horse classic."
As Radar reported, Cuoco and Sweeting battled split rumors for months leading up to Friday when they announced their divorce. "Kaley Cuoco and Ryan Sweeting have mutually decided to end their marriage," Cuoco's rep said in a statement. "They ask for privacy at this time. No further statement will be issued regarding this matter."


Mini manicure- file, buff, polish on finger nails with nail art stickers- choose from over 20 great colours.
Peppermint foot soak and massage- soak feet in a warm peppermint bath followed by a heavenly massage using peppermint lotion and oils.
Facial mask with cucumbers- clean the face with warm water, apply our professional facial mask, then girls will sit back and relax with the cucumbers over their eyes.
Mini pedicure- soak feet, file, buff, polish on toe nails with nail art stickers- choose from over 20 great colours.
Hand and arm massage- A heavenly massage using a beautifully scented moisturiser.
An indulgent spa pamper party is the perfect solution for your teen's birthday party. Teenagers love being pampered just as much as mums do! Our party host will pamper each girl individually with her chosen treat. Whilst the other girls are waiting for their treat, our party host can get the girls involved in a beauty quiz and some games and activities. Whilst girls are waiting, they can munch on their party food, read mags, talk among themselves and of course gossip about boys. If you've already organised other activities or have a Wii/Play Station girls can participate in those activities too.
A scented candle and some relaxing spa music will set the mood for the girls to be treated. Every girl will be handed a spa bathrobe to wear during the party, this will get the girls into a pamper mood. Each girl will receive a mini manicure followed by a peppermint foot soak with a relaxing massage. The teenagers will love our funky nail polish selection with bright bold colours, glitter and shimmer finishes. Some nail art on finger nails will complete the look. Once the girls nails look 'super cool', they'll get a spritz of perfume to make them feel all grown up. Then they will sit back and relax with a facial mask and cucumbers placed on their eyes! This is usually the point when girls love to laugh and take photos of themselves. Our party host will pack up and let the girls have their fun. The mask will then be removed by the girls themselves after all the fun.
Space for party guests to lay back and be seated whilst receiving the foot soak treatment. The whole group soak their feet together. This is the same area required for girls to relax after the facial mask is applied and they put cucumber pieces over their eyes. e.g. lounge room, around dining table with the correct amount of chairs for each guest.
Goodie boxes filled with party favours, jewellery, nail polish and makeup- starting from $4.00.
Please keep in mind 8 children is manageable for one party host. If you have more kids and would like to have two entertainers please call 1300 544 403 or email us.


text:
\section{Introduction}\label{sec:bterm}
A word is a sequence of finite symbols or letters from a finite alphabet. Combinatorics on words is the study of mathematical and computational problems related to words. A word is said to be primitive if it is not a proper power of a shorter word \cite{lothaire1997combinatorics}. The notion of primitive words plays a central role in the field of combinatorics on words and algebraic coding theory.
Apart from this, primitive words have also received much attention in the area of formal langauges.
This paper presents a characterization of the set of a special type of primitive words which remain robust under an operation of insertion of a single letter from the underlying alphabet.

The relation between the language of primitive words and other formal languages has been
thoroughly explored \cite{lischke2011primitive,domosi1993formal,petersen1994ambiguity}.
In 1991, D\"{o}m\"{o}si et al. raised the problem whether the language of primitive words, $Q$, is context-free? This long standing open problem has been one of the main investigations about primitive words \cite{doi}. A linear time algorithm to test whether a given word is primitive is given in \cite{duval2004linear}.

The robustness of the language of primitive words is concerned with the preservation of primitivity of a primitive word with respect to the various point mutation
operations such as insertion, deletion or substitution of a symbol and homomorphism \cite{puaun2002robustness, srivastava2016robust}. A primitive word, $u$, is said to be ins-robust if a
word remains primitive on insertion of any letter in $u$ \cite{puaun2002robustness}. In this paper, we investigate the
language of ins-robust primitive words. In particular, our contributions are as
follows.
\begin{enumerate}[(a)]
\item We characterize ins-robust primitive words and identify several properties.
\item We show that the set of ins-robust primitive words is dense.
\item We prove that the language of ins-robust primitive words is not regular.
\item We show that the language of primitive words that are not ins-robust, is not context-free.
\item We give a lower bound on the number of ins-robust primitive words of a given
length.
\item We give a linear time algorithm to test if a word is ins-robust primitive.
\end{enumerate}

The paper is organized as follows. The next section reviews the
basic concepts on words and some existing results on robustness of primitive words.
In Section \ref{sec:ins} we characterize ins-robust primitive words and study
several properties of the language of ins-robust primitive words. In Section \ref{sec:den} we show that the set of ins-robust primitive words is dense. In Section \ref{sec:for} we give proof for the claim that language of ins-robust primitive words is not regular and the language of primitive words which is not ins-robust is not context-free.
In Section \ref{sec:cou} we give a lower bound on the number of ins-robust primitive words of
a given length.
In Section \ref{sec:alg} we give a linear time algorithm to recognize a
ins-robust primitive word.
Finally, conclusions and some open problems are presented in
Section \ref{sec:con}.

\section{Preliminaries}\label{sec:pre}
Let $V$ be a finite alphabet with at least two distinct elements. We call the elements of $V$ as letters or symbols. The set $V^*$ is the collection of all words (strings) generated by the letters from the alphabet $V$. The $concatenation$ of two words $u$ and $v$ is represented as $u.v$ or simply $uv$. The length of a word $w$ is denoted as $|w|$.
We use notation $|u|_a$  for the number of occurrences of symbol $a \in V$ in a word $u$.
The set $V^n$ is the set of all words having length $n$ over the alphabet $V$. We define $V^{*} = \bigcup_{n \in N} V^n$ and $V^{+} = V^{*}\setminus \{\lambda\}$, where $V^{0} = \{\lambda\}$ and $\lambda$ is the empty string, that is, $|\lambda| = 0$.
The set $V^*$ is the free monoid and $V^+ = V^* \setminus \{\lambda\}$ is the free semigroup under the concatenation operation.

Any set $L \subseteq V^{*}$ is a language over the alphabet $V$. A language $L$ over $V$ is
said to {\it reflective} if $uv \in L$ implies $vu \in L$ where $u,v \in V^*$. The word $u$ ($v$) is called {\it prefix} ({\it suffix}) of a word $w$ if $w$ can be written as $uv$. A prefix (suffix) with length $k$ of a word $u$ is denoted by $\tn{\it pref}(u,k)$
($\tn{suff}(u,k)$, respectively) where $k \in \{0, 1, \ldots, |u|\}$. For $k=0$, $\tn{\it pref}(u,0) = \tn{\it suff}(u, 0) = \lambda$. We use notation $\tn{\it pref}(u)$ to specify the set of all non-empty prefixes
of a word $u$. If $w$ can be written as $xyz$, where $y \in V^{+}$ and $x,z \in V^{*}$, then the word $y$ is called as a \textit{factor} of the word $w$. Let $w=a_1 \ldots a_n$ be a word, where $a_i \in V$ for $i \in
\{1, \ldots, n\}$.
 The reverse of the word $w$ is $rev(w) = a_n a_{n-1} \ldots a_2 a_1$.  The factor $a_i a_{i+1} \ldots a_j$ of $w$ is denoted by $w[i..j]$, where $i \leq j$. The cardinality of a set $X$ is denoted by $|X|$. The language of nonprimitive words is denoted as $Z$.
For  elementary notions and results in formal languages theory, we refer to \cite{hopcroft1979introduction}.

A well studied theme on words is properties about periodicity and primitivity \cite{lischke2011primitive,shyr1994non,petersen1994ambiguity,domosi1993formal,puaun2002robustness}
and associated counting and sampling problems \cite{lijun2001count}. A word $w = a_1 a_2 \ldots a_n$ is said to be periodic if $a_i = a_{i+p}$ for some $p > 0$ and $1 \le i \le n-p$. Such a number $p$ is called period of the word $w$. The ratio $ e = \frac{|w|}{ p} $ is called as the exponent of the word $w$. A word $w$ is said to be a $repetition$ if and only if $e \ge 2$. A $maximal \ repetition$ in a word is a factor which is a repetition such that its extension by one letter to the right or to the left yields a word with a larger period.  Equivalently, a factor $r = w[i..j]$ is a maximal repetition of a word $w$ if no factor of $w$ that contains $r$ as a proper factor has the same period as $r$ \cite{kolpakov1999finding,kolpakov1999maximal}. For example, the factor $ababa$ in the word $w=abaababaabaab$ is a maximal
repetition with period 2, while the factor $abab$ is not a maximal repetition.

 A word, $w$, is said to be primitive if $w$ cannot be expressed as a non-trivial power of another word. In other words, if $w = v^n$ for some word $v$ and $w$ is primitive then $n=1$ and $w=v$. As the definition is meaningful only when the underlying alphabet has at least two letters, we assume throughout that $V$ is a non-trivial alphabet with at least two distinct symbols. Primitive words have been extensively studied in the literature, see for example \cite{lischke2011primitive,puaun2002robustness,shyr1994non,petersen1994ambiguity,domosi1993formal}.

\section{Characterization of Ins-Robust Primitive Words}\label{sec:ins}
As mentioned in the previous section the set of all words over an alphabet can be partitioned in two sets, those which can not be written as power of any other smaller word, known as primitive words and the other remaining are non-primitive words. We can further partition the set of primitive words in different classes. In this section we study a class of primitive words that remains primitive on insertion of any arbitrary symbol from the alphabet. This special class is known as ins-robust primitive words. A formal definition is as follows.

\begin{definition}[Ins-Robust Primitive Word]
A primitive word $w$ of length $n$ is said to be ins-robust primitive word if the word
$$\tn{pref}(w, i)\ . \ a \ .\ \tn{suff}(w, n - i)$$ is a primitive word for all $i \in \{0, 1, \ldots, n \}$ where $a\in V$.
\end{definition}

There are infinitely many primitive words which are ins-robust. For example, the words $abba$ and $a^m b^n$ for $m,n \ge 2$ are ins-robust primitive words.
We denote the set of all ins-robust primitive words over an alphabet $V$
by $Q_I$. Clearly the language of ins-robust primitive words is a subset of the set
of primitive words, that is, $ Q_I \subset Q$.
The following results ensure that if a given word $w$ is not primitive then a word obtained after inserting a symbol from $V$ in $w$ will be primitive.
\begin{lemma}[\!\cite{puaun2002robustness}]\label{lem:ext}
If $u \in V^+$, $u \neq a^n$ for any $a \in V$, $n \geq 1$ then at least one
word among the words $u$ and $ua$ is primitive.
\end{lemma}
\begin{lemma}[\!\cite{puaun2002robustness}]\label{cor:prins}
If $u_1, u_2 \in V^+$, $u_1 u_2 \neq a^n$, for any $a \in V$, $n \geq 1$ then at least one
of the words among $u_1 u_2$, $u_1 a u_2$ is primitive.
\end{lemma}

The above results help us to completely characterize the primitive words which are not ins-robust.
Next we give a structural characterization of the words
that are in the set $Q$ but not in the set $Q_I$.

\begin{theorem} \label{thm:inschar}
A primitive word $w$ is not ins-robust if and only if $w$ can be expressed in the form of
$u^{r} u_1 u_2 u^{s}$ where $u = u_1 c u_2\in Q$, $u_1$, $u_2 \in V^*$, for some $c \in
V$, $r$, $s \ge 0$ and $r + s \ge 1$.
\end{theorem}

\begin{proof}
We prove the sufficient and necessary conditions below.
\begin{itemize}
\item[$(\Leftarrow)$] This part is straightforward. Let us consider a word $w = u^{k_1} u_1 u_2 u^{k_2}$ where $u_1 c u_2 = u$ for some $c \in V$.
The word $w$ is primitive by Lemma \ref{cor:prins}.
Now insertion of the letter $c$ in $w$ (between $u_1$ and $u_2$) gives the exact power of $u$ which become a non-primitive word. Hence, $w$ is not an ins-robust primitive word.\\
\item[$(\Rightarrow)$] Let $w$ be a primitive word but not ins-robust. Then there exists a
decomposition $w=w_1  w_2$ such that $w_1 c w_2$ is not a primitive word for some letter $c \in V$.
That is, $w_1 c w_2 = p^n $ for some $p \in Q$ and $n \ge 2$.
Therefore  $w_1  = p^r p_1 $ and  $ w_2 = p_2 p^s  $ for $r,s \ge 0 \ and \ r+s \ge 1$ such that $ p_1 c p_2 = p$.
Hence $w= p^r p_1  p_2 p^s$.
\end{itemize}
\end{proof}

\begin{definition}[Non-Ins-Robust Primitive Words]
A primitive word $w$ is said to be non-ins-robust if $w \in Q$ but $w \notin Q_I$. We denote the set of all non-ins-robust primitive words as $Q_{\overline{I}}$. So, $Q_{\overline{I}} = Q \setminus Q_I$, where `$\setminus$' is the set difference operator.
\end{definition}

The next theorem is about an equation in words and identifies a sufficient condition under which three words are power of a common word.
\begin{theorem}[\cite{lischke2011primitive}]\label{thm:equ}
If $u^m v^n = w^k \ne \lambda$ for words $u$, $v$, $w \in V^{*}$ and natural
numbers $m$, $n$, $k \ge 2$, then $u$, $v$ and $w$ are powers of a common word.
\end{theorem}

The following lemma is a consequence of the Theorem \ref{thm:equ} which states
that a word obtained by concatenating powers of two distinct primitive
words is also primitive.
\begin{lemma}[\cite{lischke2011primitive}] \label{lem:piqj}
If $p,q \in Q$ with $p \neq q$ then $p^i q^j \in Q$ for all $i$, $j \ge 2$.
\end{lemma}

\begin{proposition}
If $u ,v \in Q$, $u^m = u_1  u_2 $ and $v = u_1 c u_2$ for some $c \in V$ then $u^m v^n \in Q_{\overline{I}}$ for $m, n \ge 2$.
\end{proposition}
\begin{proof}
From Lemma \ref{cor:prins} we know that at least one of $u_1 u_2$ and $u_1 c u_2$ is primitive. Since $u^m = u_1  u_2$ for $m \ge 2$ and $v=u_1 c u_2$, therefore $v$ is primitive and so is $u^m v^n = u_1  u_2(u_1 c u_2)^n$. After insertion of the letter $c$ we will get $(u_1 c u_2)^{n+1}$
which is not a primitive word. However, by Lemma \ref{lem:piqj}, $u^m v^n$ is a primitive word for $m,n \ge 2$. Hence it is not a ins-robust word, that is, $u^m v^n \in Q_{\overline{I}}$.
\end{proof}

An existing result shows that if a word $w$ is primitive then $rev(w)$ is also primitive. We prove the similar notion for ins-robust primitive word.
\begin{lemma}
If $w \in Q_I$ then $rev(w) \in Q_I$.
\end{lemma}
\begin{proof}[By contradiction] Assume that for a word $w \in Q_I$, $rev(w)$ is not a ins-robust primitive word. i.e. $rev(w) = p^r p_1  p_2 p^s$ where $p = p_1 c p_2 \in Q$ for some $c \in V$.
Then the word $w = rev(rev(w)) =$ $rev(p^r p_1  p_2 p^s)$ =   $(rev(p))^s \ rev(p_2)$ $ \ rev(p_1) \ (rev(p))^r$ and $p= p_1 c p_2$, $rev(p) = rev(p_2) ~ c ~ rev(p_1)$. By Lemma \ref{cor:prins}, $w$ is not a ins-robust primitive word, which is a contradiction.
Therefore, if $w \in Q_I$ then $rev(w) \in Q_I$.
\end{proof}

We know that the language of primitive words $Q$ and the language of
non-primitive words $Z$ over an alphabet $V$ are reflective. Similarly, we have the property of reflectivity for the language of ins-robust primitive words $Q_I$.

\begin{lemma}[\! \cite{puaun2002robustness}] \label{lem:refl}
The languages $Q$ and $Z$ are reflective.
\end{lemma}

\begin{theorem} \label{thm:insref}
$Q_I$ is reflective.
\end{theorem}
\begin{proof}[By contradiction]
Let there be a word $w=xy \in Q_I$ such that $yx \notin Q_I$. Since $w \in Q_I$, hence $w \in Q$. By Lemma \ref{lem:refl}, we know that $Q$ is reflective. Therefore $yx \in Q$ and so $yx \in Q\setminus Q_I$, i.e. $yx \in Q_{\overline{I}}$. Using Theorem \ref{thm:inschar}, we have $yx = u^ru_1  u_2 u^s$ where $u =u_1 c u_2
\in V^*$ for some $c \in V$ and $r +s \ge 1$.
There can be three possibilities as follows.

\begin{description}
\item[\textbf{Case A}] If $y = u^{r_1} u'$, $x = u'' u^{r_2} u_1 u_2 u^s$ where $u = u' u''$ and $r_1 + r_2 + 1 = r$.

In this case $xy = u'' u^{r_2} u_1 u_2 u^s u^{r_1} u'$
= $(u'' u')^{r_2} u'' u_1 u_2 u' (u'' u')^{s + r_1}$.

Since $u = u_1 c u_2$, therefore $u'' u_1 c u_2 u' = u'' u u' = (u'' u')^2$.

Therefore $ (u'' u')^{r_2} u'' u_1 c u_2 u' (u'' u')^{s + r_1}$
= $(u'' u')^{s+r+1}$,
i.e. $xy \in Q_{\overline{I}}$, which is a contradiction.

\item[\textbf{Case B}] $y = u^{r} u'$, $x = u'' u^s$ where $u' u'' = u_1 u_2$
\begin{description}
\item[\textbf{Case B.1}] If $u' = u_1'$ and $u'' = u_1'' u_2 $ where $u_1' u_1'' = u_1$.

Since $u = u_1 c u_2 = u_1' u_1'' c u_2$.

In this case $xy =  u'' u^s u^{r} u' = u_1'' u_2 u^s u^r u_1'$.

Now $u_1'' c u_2 u^s u^r u_1' = (u_1'' c u_2 u_1')^{r+s+1}$.

Therefore $xy \in Q_{\overline{I}}$, a contradiction.

\item[\textbf{Case B.2}] If $u' = u_1 u_2'$ and $u'' = u_2'' $ where $u_2' u_2'' = u_2$. This is similar to Case B.1.
\end{description}
\item[\textbf{Case C}] If $y = u^{r} u_1 u_2 u^{s_1} u'$, $x = u'' u^{s_2}$ where $u = u' u''$. This case is similar to the Case A.
\end{description}
Hence $Q_I$ is reflective.
\end{proof}

\begin{corollary}
$Q_{\overline{I}}$ is reflective.
\end{corollary}
\begin{proof}
We prove it by contradiction. Let there be a word $w=xy \in Q_{\overline{I}}$ such that $yx \notin Q_{\overline{I}}$.
We have $xy \in Q$ and $Q$ is reflective, so $yx \in Q$.
 Therefore $yx \in Q \setminus  Q_{\overline{I}}$, i.e. $yx \in Q_I$. But $Q_I$ is reflective by Theorem \ref{thm:insref}, we have
$xy \in Q_I$, which is a contradiction. Hence $yx \in Q_{\overline{I}}$.
\end{proof}

\begin{theorem}\label{thm:cycl}
A word $w$ is in the set $Q_{\overline{I}}$ if and only if it is of the form $u^n u'$ or its cyclic permutation
for some $u \in Q$, $u = u' a$ , $a \in V$ and $n \ge 1$.
\end{theorem}
\begin{proof} We prove the sufficient and necessary conditions below.
\begin{description}
\item[($\Rightarrow$)] Let $w \in Q_{\overline{I}}$, then $w$ can be written as $w = u^r u_1  u_2 u^s$ for some $u \ (=u_1 a u_2) \in Q$ and $a \in V$.
Since $Q_{\overline{I}}$ is reflective, therefore $u_2 u^s u^r u_1  = (u_2
u_1 a)^{r+s} u_2 u_1$ is also in $Q_{\overline{I}}$.

\item[($\Leftarrow$)] If a word $w$ is a cyclic permutation of $u^n u'$ for $n
\ge 1$ and $u = u' a$ then after insertion of a symbol $a$, it gives a cyclic permutation of $u^{n+1}$ which
is non-primitive (since $Z$ is reflective). Therefore, $w \in
Q_{\overline{I}}$.
\end{description}
\end{proof}

We observe that a word $w$ is periodic with period $p$ ($\ge 2$) divides $|w| + 1$ and $p \le |w| $ then $w$ is non-ins-robust primitive word.
Since $Q_I$ is reflective, therefore any cyclic permutation of $w$ is also non-ins-robust primitive word.
We know that cyclic permutation of a primitive word is also primitive. Cyclic permutation of an ins-robust primitive word is primitive. In next result, we show that it remains ins-robust too.

\begin{corollary}
Cyclic permutation of a ins-robust primitive word is ins-robust.
\end{corollary}
\begin{proof}
Let $w \in Q_I$. Then cyclic permutation of $w$ will be $yx$ for some partition $w= xy$. Since $Q_I$ is reflective. Therefore $yx$ is also ins-robust primitive word.
This proves that any cyclic permutation of an ins-robust primitive word is ins-robust.
\end{proof}
Observe that not all infinite subsets of $Q$ are reflective. For example,
the subset $\{a^n b^n \mid n \geq 1\}$ of $Q$ over the alphabet $\{a, b\}$
is not reflective.

\section{Ins-robust Primitive Words and Density} \label{sec:den}
A language $L \subseteq V^*$ is called a \emph{dense} language if for every word $w \in V^*$, there exist words $x$, $y \in V^{*}$ such that $xwy \in L$.
A language $ L \subseteq V^*$ is called \emph{right $k$-dense} if for every $u \in V^*$ there exists a word $x \in V^*$ where $|x| \le k$ such that $ux \in L$. The language is said to be right dense if its right $k$-dense for every $k \ge 1$.

It is easy to see that $Q$ is right dense (see Lemma \ref{lem:ext}). The following result shows the relation between the language of primitive words $Q$ with the density.

\begin{theorem} \label{den1}
If $|w| = n$ and $wa^n \in Q_{\overline{I}}$ where $w \notin a^*$, then  $wa^n = u^2 u_1 u_2$, $u=u_1bu_2$ for $b \ne a$.
\end{theorem}

\begin{proof}
Let $wa^n \in Q_{\overline{I}}$. Then $wa^n = u^r u_1 u_2 u^s$, where $u = u_1 b u_2 $, for some $b \in V$.
We claim that $r=2 \ and \ s=0$.
If $s \ge 1$ Then $|u| \le n+1$.

\textbf{Case 1.} In this case we can have two cases depending on the length of $u$.

\textbf{Case 1a.} $|u| = n+1$ if $wa^n = u_1 u_2 u$.
Hence $u = ba^n$ for some $b \ne a$. Therefore $|u| = n+1$ and so $|u_1 u_2| = n $.
$|wa^n| = 2n+1$, which is a contradiction as $|wa^n|=2n$.

\textbf{Case 1b.} If $s \ge 1$ and $|u| \le n$, then $u = a^r$, where $r = |u|$, and therefore $u_1u_2 = a^{r-1}$.
$wa^n$ = $a^{2n}$. This case also leads to a contradiction.

Therefore $s=0$.
Hence $wa^n = u^r u_1 u_2$.

\textbf{Case 2.} $r=1$. This case is not possible.
Because in this case $|wa^n|= |u u_1u_2|=2n$, which implies $|u|= (2n+1)/2$ a non-integral value.

\textbf{Case 3.} If $r \ge 2$. Then $|wa^n| = |u^r u_1 u_2|  = ((r+1)|u|-1)$.
Since $|u_1u_2|$ = $|u|-1$, therefore $u_1u_2 = a^k$, where $k = |u| -1$.

If $r \ge 3$ then $|wa^n| = |u^r u_1 u_2|  = ((r+1)|u|-1)$.
In this case $|u u_1u_2|= {\frac{2n+1}{r+1}}*2 -1$ $\le n$. Therefore $u = a^{k+1}$.
Hence $r \ge 3$ is also not possible.

Thus the only possibility is $r=2$, $|wa^n| = u^2 u_1u_2$.
Since $u_1u_2 = a^k$ and $wa^n \in Q$ therefore $w, u \notin a^* $, and so
$u = a^{k_1}ba^{k_2}$ where $k_1 + k_2 =k$ and $b \ne a$.

\end{proof}

\begin{lemma} \label{denseins1}
Let $V$ be an alphabet, $w \in V^*$ , $|w|=n$ and $a \in V$. If $wa^n \in Q_{\overline{I}}$ then for $b \ne a$, $wb^n \in Q_I$.
\end{lemma}

\begin{proof} Let $wa^n \in Q_{\overline{I}}$. Then, by Theorem \ref{den1} we have, \\
$
\begin{array}{ll}
wa^n = u^2 u_1u_2 \tn{ and } u_1u_2 = a^k.\\
u=a^{k_1}ba^{k_2} \tn{ where } a \ne b.\\
\end{array}
$\\[5pt]
Let $wc^n$ be also in $ Q_{\overline{I}}$ for some $c \ne a$. Then, by Theorem \ref{den1} we have,\\
$
\begin{array}{ll}
wc^n = v^2 v_1v_2 \tn{ and } v_1v_2 = c^k.\\
v=c^{k_1'}dc^{k_2'} \tn{ where } c \ne d.
\end{array}
$\\
But since $|u| = |v|$ and $w = uu' = vv'$ where $u' = u u_1 u_2$ and $v' = v v_1 v_2$,
therefore $u=v$, that is,
$a^{k_1}ba^{k_2} = c^{k_1'}dc^{k_2'}$.\\
If $k_1 < k_1'$ then $a=b=c=d$, which is a contradiction. Alternatively, if $k_1 = k_1'$ then $a=c$ which is again a contradiction. Therefore $wc^n \in Q_I$.

\end{proof}

\begin{theorem}
The language $Q_I$ is dense over the alphabet $V$.
\end{theorem}
\begin{proof} Consider a word $w$. We only need to consider the case when $w \notin Q_I$, that is,  $w \in V^* \setminus Q_I$. By Lemma \ref{denseins1}, there exists $b \in V$ such that $wb^n \in Q_I$, where $n = |w|$.

\end{proof}

\section{Relation of $Q_I$ with the Other Formal Languages}\label{sec:for}
We now investigate the relation between the language of
ins-robust primitive words with the traditional languages in Chomsky hierarchy. We prove that the language of ins-robust primitive words over an alphabet is not regular and also show that the language of non-ins-robust primitive words is not context-free.
For completeness, we recall the pumping lemma for regular languages and pumpimg lemma for context-free languages which
will be used to show that $Q_{I}$ is not regular and
$Q_{\overline{I}}$ is not context-free respectively.

\begin{lemma}[Pumping Lemma for Regular Languages \cite{hopcroft1979introduction}]
 For a regular language $L$, there exists an integer $n > 0$ such that for every word $w \in L$ with $|w| \ge n$, there
exist a decomposition of $w$ as $w = xyz$ such that the following conditions
holds.
\begin{enumerate}[(i)]
\item $|y| > 0$,
\item $|xy| \le n$, and
\item $xy^i z \in L $ for all $i \ge 0$.
\end{enumerate}
\end{lemma}

Let us recall a result which will be used in proving that the language of ins-robust primitive words is not regular.
\begin{lemma}[\label{lem:sola}\cite{horvath2005language}]
For any fixed integer $k$, there exist a positive integer $m$ such that the
equation system $(k-j)x_j+j=m$, $j=0,1,2, \ldots, k-1$ has a nontrivial solution
with appropriate positive integers $x_1,x_2,\ldots,x_j > 1$.
\end{lemma}

\begin{theorem}
$Q_{I}$ is not regular.
\end{theorem}
\begin{proof}[By contradiction]
Suppose that the language of ins-primitive words $Q_{I}$ is regular. Then
there exist a natural number $n >0$ depending upon the number of states of finite automaton for
$Q_{I}$.

Consider the word $w=a^n b a^m b,m > n+1$ and $m \neq 2n$. Note that
$w$ is an ins-primitive word over $V$, where $|V| \ge 2$ and $a \ne b$. Since $w \in Q_{I}$
and $|w| \ge n$, then it must satisfy the other conditions of pumping Lemma
for regular languages. So there exist a decomposition of $w$ into
$x$, $y$ and $z$ such that $w= xyz,|y| >0$ and $x y^i z \in Q_{I}$ for all $i \ge 0$.

Let $x = a^k,\ y=a^{(n-j)},\ z=a^{j-k} b a^m b$. Now choose $i=x_j$ and
since we know by Lemma \ref{lem:sola} that for every $j
\in \{0,1, \ldots , n-1\}$, there exists a positive integer $x_j >1$ such that $x y^{x_j} z= a^k
a^{(n-j)x_j} a^{j-k} b a^m b = a^{(n-j)x_j+j} b a^m b =  a^m b a^m b = (a^m b)^2 \notin Q_{I}$ which is a contradiction. Hence the language of ins-primitive words $Q_{I}$ is not regular.

\end{proof}

\begin{lemma}[Pumping Lemma for Context-Free Languages \cite{hopcroft1979introduction}]
\label{lem:pcfl}
Let $L$ be a CFL. Then there exists an integer $n >0$ such that for every $u \in
L$ with $|u| \ge n$, $u$ can be decomposed into $vwxyz$ such that the following
conditions hold:
\begin{enumerate}[(a)]
 \item $|wxy| \le n$.
 \item $|wy| >0$.
 \item $v w^{i} x y^{i} z \in L$ for all $i \ge0$.
\end{enumerate}
\end{lemma}

\begin{theorem}\label{thm:cfl}
$Q_{\overline{I}}$ is not a context-free language for a binary alphabet.
\end{theorem}

\begin{proof}
Let $V= \{a, \ b \}$ be an alphabet. On contradiction, assume that $Q_{\overline{I}}$ is a context-free language. Let $p >0$ be
an integer which is the pumping length for the language $Q_{\overline{I}}$.
Consider the string $s = a^{p+1} b^{p+1} a^{p+1} b^p$, where $a , \ b \in V$ are
distinct. It is easy to see that $s\in Q_{\overline{I}}$ and $|s| \ge p$.

Hence, by the Pumping Lemma \ref{lem:pcfl}, $s$ can be written in the form $s = uvwxy$,
where $u, v, w, x,$ and $y$ are factors, such that $|vwx| \le p$, $|vx| \ge
1$, and $uv^iwx^iy$ is in $Q_{\overline{I}}$ for every integer $i \ge 0$. By the
choice of $s$ and the fact that $|vwx| \le p$, we have
one of the following possibilities for $vwx$:
\begin{enumerate}[(a)]
\item \label{e:a} $vwx = a^j$ for some $j \le p$.
\item \label{e:b} $vwx = a^jb^k$ for some $j$ and $k$ with $j+k \le p$.
\item \label{e:c} $vwx = b^j$ for some $j \le p$.
\item \label{e:d} $vwx = b^j a^k$ for some $j$ and $k$ with $j+k \le p$.
\end{enumerate}
In Case (\ref{e:a}), since $vwx = a^j$, therefore  $vx = a^ t$ for some $t \ge 1$ and hence $uv^iwx^iy = a^{p-t+1} b^{p+1} a^{p+1} b^p  \notin Q_{\overline{I}}$ for $i =0$.

Case (\ref{e:b}) can have several subcases.
\begin{enumerate}
\item  $v= a^{j_1}, \ w = a^{j_2}, \ x= a^{j_3}b^k$.
\item  $v= a^{j_1}, \ w = a^{j_2}b^{k_1}, \ x= b^{k_2}$.
\item  $v= a^{j}b^{k_1}, \ w = b^{k_2}, \ x= b^{k_3}$.
\end{enumerate}
In Case (1), Case (2) and Case (3) if we take $i = 4$, $uv^iwx^iy \ne Q_{\overline{I}}$.

Similarly, we can obtain contradiction in Case (\ref{e:c}) and Case (\ref{e:d}) by choosing a suitable $i$.
Therefore, our initial assumption that $Q_{\overline{I}}$  is context-free, must be false.
\end{proof}
Next we prove that the language of non-ins-robust primitive words is not context-free in general.
\begin{lemma}
The language $Q_{\overline{I}}$ is not context-free over an alphabet $V $ where $V$ has at least two distinct letters.
\end{lemma}
\begin{proof}
The proof of Theorem \ref{thm:cfl} can be generalized to arbitrary alphabet $V$ having
at least two letters. The set of all words over alphabet having greater than two distinct letters also contains the words with two letters. If $Q
_{\overline{I}}$ is assumed to be a CFL over $V $ where $|V| \ge 3$, then we can choose words of the form used in Theorem \ref{thm:cfl} and obtain a contradiction. Hence the language of non-ins-robust primitive words $Q_{ \overline{I}}$ is not context-free over $V$ where $|V| \ge 2$.

\end{proof}
\section{Counting Ins-Robust Primitive Words}\label{sec:cou}
In this section we give a lower bound on number of $n$-length ins-robust primitive words.
Let $V$ be an alphabet and $Z(k) = V^k \setminus Q$ be the set of $n$-length
non-primitive words.

We have the following result that gives the number of the primitive words of
length $m$.
\begin{proposition} [\cite{lijun2001count}] \label{count}
Let $m \in N$ and $m= {m_1}^{r_1} {m_2}^{r_2} \ldots {m_t}^{r_t}$ be the
factorization of $m$, where all $m_i , 1 \le i \le t$ , are prime and $m_i \ne m_j$ for $i\ne j$, then
the number of primitive words of length $m$ is equal to
$$
\begin{array}{rl}
|V|^m - & \sum\limits_{1\le i\le t}|V|^{\frac{m}{m_i}}
   + \sum\limits_{1\le i\le j \le t}|V|^{\frac{m}{m_i m_j}}\\\\
- & \sum\limits_{1\le i\le j \le k \le t}|V|^{\frac{m}{m_i m_j m_k}}
+
\cdots +(-1)^{t-1} |V|^{\frac{m}{m_1 m_2 \cdots m_t}}
\end{array}
$$
\end{proposition}

We observe that the deletion of a symbol from a $n$-length non-primitive word gives a maximum of $(n-1)$-different non-ins-robust primitive words  when the word is of type $a_1 a_2 \ldots a_{n}$ such that $a_i \ne a_{i+1}$ for $1 \le i \le n-1$ and minimum it can be $zero$ if the word is of type $a^r, \ r>2, \ a \in V$.
Given a word $w \in Z(n)$.
The number of words that can be obtained by deleting a symbol from $w$ is \\
$$0 \le |\{w_1 w_2 \mid w_1 a w_2 = w, ~w_1, w_2 \in V^*, a \in V \}| \le n.$$
We know from Lemma \ref{cor:prins} that a non-primitive word $w$ remains non-primitive after deleting a symbol $a$ if $w = a^n $ and $n \ge 3$. Otherwise a non-ins-robust primitive word.
$$Q_{\overline{I}}(n) = \{w_1 w_2 \mid w_1 a w_2 \in Z_{n+1}, a \in V, w_1, w_2 \in V^*\}$$

Therefore, the number of non-ins-robust primitive words of length $n$,
$Q_{\overline{I}}(n)$, is the difference between the number of all words obtained by deleting a symbol from
the words of set
$$Z(n+1) \setminus V^{n+1} \ where \ V^{n+1} =\{ a^{n+1} \mid a \in V \} \ for \ n \ge 2$$

We can find an upper bound on number of non-ins-robust primitive words of length
$n \ge 2$ as follows.
$$
\begin{array}{ll}
|Q_{\overline{I}}(n)|  = |\{w_1  w_2 \mid w =w_1 b w_2 \in Z(n+1)  \setminus V^{n+1} \\  ,w_1, w_2 \in V^*, b \in V \}|\\\\

|Q_{\overline{I}}(n)|  = |\{w_1  w_2 \mid w =w_1 a w_2 \in Z(n+1),\\  w_1, w_2 \in V^*, a \in V \}| - (n+1).|V|\\\\
 \le (n+1) . (|Z(n+1|) - |V|).
\end{array}
$$
From the Proposition \ref{count}, we know the number of primitive words of fixed
length. Thus the number of ins-robust-primitive words of length $n$, $Q_I(n)$, over an
alphabet $V$ is equal to $|Q_n| - |Q_{\overline{I}}(n)|$.

\section{Recognizing Ins-Robust Primitive Words} \label{sec:alg}
In this section, we give a linear time algorithm to determine if a given primitive word $w$ is ins-robust. We design the algorithm that exploits the property of the structure of ins-robust primitive words. We state some simple observations before presenting the algorithm.
The following theorem is based on the structure of ins-robust primitive word. We know that a word $w$ is ins-robust if it can not be written as $t^r t_1  t_2 t^s$ for $t \in Q$, $t = t_1 a t_2$ for some $a \in V$ , $r,s \ge 0$ and $r+s \ge 1$.

\begin{theorem} \label{insrp}
Let $u$ be a primitive word. Then $u$ will be \textbf{non-ins-robust} primitive word iff $uu$ contains at least one periodic word of length $|u|$ with period $p$ such that $p$ divides of length $|u| + 1$ and $p \le |u|$.
\end{theorem}

\begin{proof}
($\Rightarrow$) If $u$ is a non-ins-robust primitive word, then $u$ can be written as $t^r t_1  t_2 t^s$ for some primitive word $t$, $r+s \ge 1$ and $t = t_1 a t_2$ for some symbol $a \in V$ where $t_1 , t_2 \in V^*$.
$uu = t^r t_1 t_2 t^s t^r t_1 t_2 t^s$, This word contains a subword $ t_2 t^s t^r t_1$ of length $|u|$ that is $(t_2 t_1 a)^{r+s} t_2 t_1$ which is a periodic word with period $|t_2 t_1 a| = |t|$ which divides $|u|+1$.\\
($\Leftarrow$) Let $uu$ has a periodic substring of length $|u|$ with period p (divides $|u| +1$ and $p \le |u|$) where u is primitive word. Then $uu = t_1 x^r x' t_2$, where $t_1, t_2 \in V^*$, $|x^{r} x'|=|u|$, $x \in Q$, $r \ge 1$ and $x = x' a$ for some $a \in V$. $|t_1 t_2| = |u|$. Here we have two cases, either $x^r x'$ entirely contained in $u$ or some portion of $x^r x'$ contained in $u$.\\
\textbf{Case 1.} Let  $x^r x'$ entirely in $u$. Then $u$ is not ins-robust as $u = x^r x' $. \\
\textbf{Case 2.} Let some portion of $x^r x'$ contained in $u$. Since $uu = t_1 x^r x' t_2$, and $Z$ is reflective, therefore $t_2 t_1 x^r x' = u' u'$, where $u'$ is cyclic permutation of $u$. Hence, $u' = x^r x'$, is non-ins-robust. Since $Q_I$ is reflective, therefore $u$ is also non-ins-robust.

\end{proof}

\begin{corollary} \label{insr}
Let $u$ be a primitive word. Then $u$ will be \textbf{non-ins-robust} primitive word iff there exists a cyclic permutation of $u$, say $u'$, which is a periodic with period $p$ such that $p$ divides $|u| + 1$ and $p \le |u|$.
\end{corollary}

\begin{proof}
The proof follows from Theorem \ref{insrp}.

\end{proof}

Next we present a linear time algorithm to test ins-robustness of a primitive word by using the existing algorithm for finding maximal repetitions in linear time. For more details on the maximal repetition, see \cite{lothaire2005applied}.

\begin{algorithm}[!ht]
\caption{\textsc{IsInsRobust}}\label{inropr}
\begin{algorithmic}[1]
\Require A finite word $u$
\Ensure ``True'' if $u$ is an ins-robust primitive word, else ``False''
\Statex
\State Let $v \leftarrow uu$.\label{step}
\State $ S \leftarrow $ \Call{FindMaximalRepetitions}{$v$}\label{maxrep1}
\Comment $S$ is a set of pairs of period and length.
\ForAll{$(p_i, l_i) \in S$} \label{st1}
  \If {$|u| \mod p_i = 0$ \textbf{and} $p_i < |u|$} \label{prim} \label{alg:prim} \Comment Testing primitivity.
    \State \Return False \Comment{The word $u$ is not primitive.}
  \EndIf

  \If {$p_i \le |u|$ \textbf{and} $(|u| + 1) \mod p_i = 0$ \textbf{and} $l_i \ge |u|$}\label{alg:main}
    \State \Return False \Comment{The word $u$ is not ins-robust. (Corollary \ref{insr})}
  \EndIf
\EndFor\label{st2}
\State \Return True \Comment{The word $u$ is ins-robust.}
\end{algorithmic}
\end{algorithm}

\begin{theorem}
Let $w$ be a word given as input to Algorithm \ref{inropr}. The algorithm returns true if and only if the word $w$ is ins-robust.
\end{theorem}
\begin{proof}
In step (\ref{maxrep1}), the algorithm finds the maximal repetitions with their periods.
Since $Q_{\overline{I}}$ is closed under reflectivity, therefore  $uu$ has all the cyclic permutations of $u$. There is a periodic word $x^r x'$, a permutation of $u$ such that $x=x' a $ for some $a\in V$. Therefore $uu$ also has this periodic word which is proved in Theorem \ref{insrp}.
That is for a non-ins-robust primitive word $u$, $uu$ contains a periodic word of length at least $|u|$ with a period $p$ such that $p \ divides \ (|u|+1)$ and $p<|u|$.
This is explained in step (\ref{alg:main}) where $u$ is a primitive word step (\ref{alg:prim}).

\noindent  Otherwise $u$ is ins-robust primitive word.
\end{proof}

\begin{theorem}
The property of being ins-robust primitive is testable on a word of length $n$ in $O(n)$ time.
\end{theorem}

\begin{proof} The Step (\ref{step}) in Algorithm \ref{inropr} running time is an O(1) operation. In step (\ref{maxrep1}) maximal repetition algorithm given in \cite{lothaire2005applied} is used which has linear time complexity. Now from step (\ref{st1}) to step (\ref{st2}), the complexity depends on the cardinality of $S$, which is less than $n$. Hence it also has linear time complexity. Therefore by theorem (\ref{insrp}) testing ins-robustness for primitive word is linear.
\end{proof}

\section{Conclusions and Future Work} \label{sec:con}
We have discussed a special subclass of the language of primitive words which is known as
ins-robust primitive words. We have characterized ins-robust
primitive words and identified several properties and proved that the language
of ins-robust primitive words $Q_I$ is not regular. We also proved that the language of non-ins-robust primitive words $Q_{\overline{I}}$ is not context-free. We identified that $Q_I$ is dense over an alphabet $V$. We
have also presented a linear time algorithm to test if a given word is ins-robust primitive.
Finally, we have given a lower bound on the number of ins-robust primitive words of a given length.

There are several interesting questions that remain unanswered about
ins-robust primitive words. Some of them that we plan to explore in
immediate future are as follows. Is $Q_I ^{i}$ for $i \ge 2$ regular? It is known that $Q^i$ for $i \ge 2$ is regular \cite{doi}. We also
conjecture that the language of ins-robust primitive words $Q_I$ is not a
deterministic context-free language. We believe that the properties we have
identified for ins-robust primitive words will be helpful in answering
these questions.

redpajama_set_name: RedPajamaArXiv


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